Solid State Physics Lecture Notes
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Lecture Notes of Solid State Physics Carlo E. Bottani a.y. 2017/18 - February 2018
1
Contents 1 Solid bodies 1.1 Order and symmetry . . . . . . . . . . . . . . . . . . . 1.1.1 Simple Crystals: lattices and translational order 1.1.2 Complex crystals: lattice and basis . . . . . . . 1.2 Crystal binding . . . . . . . . . . . . . . . . . . . . . . 1.3 Tensorial observables . . . . . . . . . . . . . . . . . . .
. . . . .
5 6 7 13 13 15
2 Scattering theory 2.1 Elementary theory of elastic scattering . . . . . . . . . . . . . 2.2 Quantum particle scattering amplitude: Born approximation 2.3 Elastic scattering - Bragg law . . . . . . . . . . . . . . . . . . 2.4 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . .
16 17 19 22 24
3 Electronic States 3.1 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Reduction to the first Brillouin zone . . . . . . . . . . . . . . 3.3 Born-von Karman boundary conditions . . . . . . . . . . . . 3.4 Free electron gas . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Plane waves - Fermi energy - Density of states . . . . 3.4.2 Chemical potential and electronic specific heat . . . . 3.4.3 Quasi-particles and Fermi liquid theory (a qualitative introduction) . . . . . . . . . . . . . . . . . . . . . . 3.5 Band structure: nearly free electron approximation . . . . . 3.5.1 Free electron diffraction . . . . . . . . . . . . . . . . 3.5.2 Fermi surface and density of states . . . . . . . . . . 3.6 Band structure of tightly bound electrons . . . . . . . . . . . 3.6.1 LCAO Method . . . . . . . . . . . . . . . . . . . . . 3.7 Electron dynamics in external fields . . . . . . . . . . . . . . 3.7.1 Equivalent Hamiltonian. Effective mass theorem . . . 3.7.2 Impurity levels . . . . . . . . . . . . . . . . . . . . . 3.7.3 Semiclassical dynamics in a crystal . . . . . . . . . . 3.7.4 Limitations in the effective mass description . . . . . 3.7.5 Electric current - Electrons and holes . . . . . . . . . 3.7.6 Excitons . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
29 31 35 37 39 39 41
. . . . . . . . . . . . .
43 44 44 46 47 50 51 52 55 55 58 59 64
4 Adiabatic theorem and vibrational motions
. . . . .
. . . . .
. . . . .
64
2 5 Lattice dynamics 5.1 Lattice specific heat: Einstein model . . . . . . . . . . . . . 5.2 Lattice dynamics: Born - Von Karman model . . . . . . . . 5.3 Linear chain with two atoms per cell: acoustic modes and optical modes . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simple 1D crystal: acoustic phonons and elastic waves . . . 5.5 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Specific heat: Debye model . . . . . . . . . . . . . . . . . . . 5.7 Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
76 78 80 82 88 93
6 Optical properties 6.1 Kramers e Kronig relations . . . . . . . . . . . 6.2 Probability per unit time of optical transitions 6.3 Interband optical transitions . . . . . . . . . . 6.3.1 Direct transitions . . . . . . . . . . . . 6.3.2 Indirect transitions . . . . . . . . . . . 6.4 Intraband transitions and plasmons . . . . . .
. . . . . .
. . . . . .
98 100 103 105 105 108 111
. . . . . .
112 . 112 . 116 . 118 . 119 . 120 . 122
7 Transport phenomena 7.1 Phenomenology . . . . . . . . . . . . . . 7.2 Kinetic theory . . . . . . . . . . . . . . . 7.2.1 Appendix: Liouville theorem . . 7.3 Electrical conductivity . . . . . . . . . . 7.4 Electronic thermal conductivity . . . . . 7.5 Lattice thermal conductivity (insulators)
. . . . . .
. . . . . .
. . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
8 Appendix: Quantum approximate methods 8.1 Time-dependent perturbations . . . . . . . . . . . . 8.1.1 Harmonic perturbation . . . . . . . . . . . . 8.1.2 Photon-electron interactions: electric dipole rules . . . . . . . . . . . . . . . . . . . . . . 8.2 Static perturbation theory . . . . . . . . . . . . . . 8.2.1 Adiabatic switching on . . . . . . . . . . . . 8.2.2 Degenerate energy levels . . . . . . . . . . . 8.3 Dirac delta function . . . . . . . . . . . . . . . . . . 8.4 Green functions . . . . . . . . . . . . . . . . . . . . 8.5 Selfconsistent mean field . . . . . . . . . . . . . . . 8.6 Semiclassical dynamics in a conservative force field
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
67 . 67 . 69
124 . . . . . . 124 . . . . . . 128 selection . . . . . . 129 . . . . . . 130 . . . . . . 130 . . . . . . 132 . . . . . . 133 . . . . . . 135 . . . . . . 136 . . . . . . 139
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9 Appendix: Elasticity and Elastic Waves 142 9.1 Stresses, strains and elastic constants . . . . . . . . . . . . . . 142 9.2 The acoustic waves and their phonons . . . . . . . . . . . . . . 146
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Foreword To make a proper use of these notes, students should carefully read this foreword. The present text is the English translation of an original Italian version used till the second semester of a.y. 2011/12. The translation was quickly accomplished to be ready for the 2013/14 course of Solid State Physics and was far from being completely satisfactory. A better edition is now available for the 2017/18 course. Concerning the content, although I have taken the occasion to introduce some minor changes, the nature of the notes remains the same: a short and synthetic outline of the theoretical part of the lectures on Solid State Physics I gave at Politecnico di Milano in the last eight years. The contents of more practical lectures (very specific examples, exercises and description of experiments), which also belong to the course program and are possible objects of the final exam, are almost not included. Anyhow this notes, initially born as a personal notebook of the teacher and not intended for student studying, should be read only as an additional didactic aid and should never be used in place of a good textbook of Solid State Physics, which remains an indispensable tool for any student. For my lectures I have taken inspiration from many famous textbooks like: J.M. Ziman, Theory of solids (2nd edition), Cambridge University Press (1995); C. Kittel, Introduction to Solid State Physics (8th edition), John Wiley &Sons (2004); N.W. Ashcroft and N.D. Mermin, Solid State Physics, HRW International Editions (1981); W.A. Harrison, Solid State Theory, Dover Publications (1979); F. Bassani e U.M. Grassano, Fisica dello Stato Solido, Bollati Boringhieri (2000). Students should choose one of them and study it carefully. A further reason not to use these notes as a unique reference text is their intrinsic degree of difficulty: many comments, clarifications and examples I always make during lectures are not reported here. Without them understanding very synthetic written considerations may be rather hard. Yet I hope these notes will be of some help for those students who mainly like the very few fundamental principles upon which every physical theory is based: a dry skeleton of fundamental concepts. A last comment: understanding these notes requires a good knowledge of basic quantum mechanics even though the appendices introduce some more advanced topics, e.g. approximate methods like perturbation theory.
4
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Figure 1: Crystals
1
Solid bodies
From a phenomenological and macroscopic point of view a solid body is characterized by both a proper shape and a proper volume (while a liquid owns only a proper volume and both gases and vapours neither). All solids mechanically resist actions tending to varying their shape and/or their volume. The mechanical response, described by a stress-strain law1 , is reversible (often linear) as long as the applied stresses do not exceed critical values: the so called elastic regime. In this last condition shape and volume are thermodynamic state variables like temperature. Beyond such critical stress values solid bodies undergo irreversible processes, e.g. plastic deformation and fracture. Crystalline solids become liquid at a characteristic temperature (melting point) with a first order phase transition. Disordered solids instead exhibit different irreversible processes leading anyhow to a fluid state beyond specific temperature values. Some solid bodies (for instance tungsten oxide 3 ) directly vaporize without passing through the liquid state: this phase transition is called sublimation. From a microscopic (atomistic) point of view solid bodies can be roughly classified as crystalline (monolithic ordered atomic structures, see below), polycrystalline (bodies assembled by many individual crystalline portions called grains, joined together through disordered transition regions or grain boundaries) or amorphous (disordered atomic structures). In the last twenty years, with the advent of nanotech1
More precisely one should speak of stress tensor and strain tensor. See Appendix on elasticity and elastic waves
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nology in the realm of materials science, also the terms nanocrystalline and cluster assembled have been introduced, making the classification less sharp and more complex. In principle the main goal of solid state physics should be the forecast of the atomic structure of solid bodies (and thus of all their properties) solely on the base of quantum mechanics and statistical thermodynamics. This objective is impossible to be reached. The reason is that the underlying quantum many body problem is far too difficult to be solved. In the case of crystals the atomic structure is usually assumed as given (most often based on experimental results like X-Ray diffraction) and then quantum mechanics is used to compute the properties of that particular structure. E.g. we are not able to understand from first principles why solid copper is a face centered cubic crystal with a cubic cell of 3.61 Å edge but, assuming this structure as true, we understand why copper is a noble metal, we can compute its specific heat, electrical and thermal conductivities, optical reflectivity, thermal expansion and many other properties in good agreement with experimental data. This is possible because, given the reference structure, many properties can be reconducted to elementary quantum excitations that, to a first approximation, beahave as ensambles of independent particles which are born, move and die within the body. There exist two types of elementary excitations: quasi-particles (quasi-electrons, holes, polarons) and collective excitations (phonons, plasmons, polaritons, magnons). All these excitations are deeply connected with the real particles like electrons and nuclei which, in the normal energy range of interest for solid state physics, are stable particles whose number is conserved. In the following we will deal mainly with quasi-electrons (which will be simply called electrons), holes and phonons.
1.1
Order and symmetry
Among solid bodies crystals are particularly important. Such bodies are ideally characterized by long range order of atomic positions, which gives crystals a unique property: translational symmetry. The term atomic position used in the crystallographic language should be understood, from a more general point of view, as mean atomic position in conditions of thermodynamic equilibrium and not as instantaneous atomic position. In this respect symmetry itself is a thermodynamic property of crystals. In real crystals atoms vibrate around their stable mean positions (stable provided the temperature is far from the melting point) due to thermal agitation. Furthermore lattice defects (vacancies, interstitials, dislocations ecc), breaking the translational symmetry, are unavoidably present in real crystals. Lattice dynamics studies atomic vibrational motions. For didactic reasons we start
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describing the properties of ideal crystals, free of both lattice defects and thermal motions. First let us consider the so called simple crystals in which atomic positions coincide with the points of a Bravais lattice. 1.1.1
Simple Crystals: lattices and translational order
A Bravais lattice is defined as the discrete infinite set of points (vectors) given by the following formula: n = 1 a1 + 2 a2 + 3 a3 1 2 3 being a triplet of integer numbers and a1 a2 a3 being the primitive vectors (not all in the same plane) of the primitive cell. The primitive cell, which contains only one lattice point, is the volume of space that, when translated through all the vectors in a Bravais lattice, just fills all of space without either overlapping itself or leaving voids. The vectors a are said to generate or span the lattice. Given a Bravais lattice, the primitive vectors are univocally identified (primitive unit cell ) provided the volume of the cell is minimal and the three vectors connect the origin of the cell to nearest neighbors (that is, to lattice points of minimal distance). Alternatively the Wigner and Seitz method can be used. The lattice point taken as origin is connected to all equivalent nearest neighbors generating a set of segments. Then every segment is bisected by a normal plane. The set of all such intersecting planes defines a polyhedron: the Wigner-Seitz cell, containing only one Bravais lattice point, whose volume is identical to that of the primitive unit cell. The advantage of using the Wigner-Seitz cell as the primitive one is that all the symmetry properties of the Bravais lattice (also the non-translational ones, see below) can be directly seen by inspection of it. The figures also illustrate different kinds of cells: one instance of elementary cell (minimal volume, but the connected points are not all nearest neighbors) and one instance of conventional cell (non minimal volume; moreover the one to one correspondence between lattice point n and cell is broken).
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A lattice translation Tnn0 is the vector difference between two vectors n 0 and n and can thus be expressed by the same type of formula representing a 0 0 lattice point. If one assumes n = 0, Tn0 = Tn = n. Any two points r and r within the crystal volume (in general neither coinciding with any lattice point) are physically equivalent if they are connected by a lattice translation 0
r = r + Tn = r+1 a1 + 2 a2 + 3 a3 A physical observable of the crystal, e.g. the electric charge density, turns out to be invariant with respect to any Tn : 0
(r ) = (r + Tn ) = (r)
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0
is then a multiply periodic function of position. In one dimension = + and (+) = (). Thus () is expressible as a Fourier series +∞ X
() =
2
(1)
=−∞
whose Fourier coefficients are computed as 1 =
Z
2
()−
(2)
0
2
The reciprocal lattice vectors = can be written again as
+∞ X
() =
are then introduced and the series
(3)
=−∞
The last is surely a periodic function because 2
= = = 2() = 1
(4)
In three dimensions, for a lattice generated by periodic translation of a primitive cell in the shape of a right-angled parallelepiped, the reciprocal lattice vectors can be written in the simple form g =
2 2 2 u + u + u 1 2 3
and the density can be expressed as X (r) = g ·r
(5)
(6)
=
1
Of course it turns out that
Z
(r)−g ·r r
(7)
g ·T = 1
(8)
In the general case the following more complex definition of the primitive vectors of the reciprocal lattice is needed: a2 × a3 |a1 · a2 × a3 | a3 × a1 = |a1 · a2 × a3 | a1 × a2 = |a1 · a2 × a3 |
b1 = b2 b3
(9) (10) (11)
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then one writes g = 2(b1 + b2 + b3 )
(12)
being the volume of the primitive cell given by ( ) = |a1 · a2 × a3 |
(13)
In this way it is always true that a · b = and the (8) is authomatically satisfied. Fundamental theorems 1. Every reciprocal lattice vector g is perpendicular to a family of (direct) lattice planes with Miller indices2 . 2. If the components of g have no common factors, then in general the first neighbour planes distance is related to g by =
2 |g |
(14)
and, for cubic crystals (s.c., b.c.c., f.c.c.), with a conventional cubic cell of edge , by (15) = √ 2 + 2 + 2 3. The volume of the primitive cell of the reciprocal lattice can be computed as ( ) =
8 3 8 3 = ( ) |a1 · a2 × a3 |
(16)
4. The direct lattice is the reciprocal of its reciprocal lattice. For the primitive cell of the reciprocal lattice the most common definition is the Wigner-Seitz one: the corresponding cell is called first Brillouin zone. The zone center is called Γ point. Besides from its translational symmetry each Bravais lattice is characterized by its point symmetry. The point group is the set of symmetry operations which leave a lattice point (taken as origin) fixed. A point group contains rotations of an angle 2 (with = 2 3 4 6) about crystal 2
(hkl) denotes a plane that intercepts the three points 1 , 2 , and 3 , or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane, in the basis of the lattice vectors. If one of the indices is zero, it means that the planes do not intersect that axis (the intercept is "at infinity").
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Figure 2: axes passing through the fixed point, the inversion (r → −r) and the rotoinversions (equivalent to reflections across crystal planes passing through the fixed point) consisting of a rotation followed by the inversion (in the group algebra the corrisponding operation is the product of the two distinct operations). All Bravais lattices own the inversion symmetry with respect to any lattice point. The space group of the lattice contains both the symmetry translations and the symmetry operations belonging to the point group. Groups theory shows that (in three dimensions) do exist only fourteen (14) different Bravais lattices (result obtained by the French physicist Auguste Bravais in the year 1845) each charatterized by a specific primitive cell (a specific triplet of vectors a1 a2 a3 ). The 14 Bravais lattices distribute themselves among the seven (7) crystal systems (syngonies), illustrated in the figure, for there are only 7 point groups compatible with translational symmetry.
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Figure 3:
12
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Complex crystals: lattice and basis
Figure 4: In the most general case a crystal can be conceptually built decorating every primitive cell of a Bravais lattice with a specific set of atoms called the basis of the crystal. The basis is then the elementary building block which, periodically translated along the space directions defined by vectors a1 a2 a3 , reproduces the whole (in principle infinite) crystal structure. The possible point symmetry (compatible with the translational one) in the presence of a basis is defined by the thirtytwo (32) crystallographic point groups. The overall set of symmetry translations and of non-translational symmetry operations3 which leave the crystal invariant can be further classified among two hundred and thirty (230) space groups. In the following we often shall refer only to simple crystals, unless the discussed physics does require the presence of a basis.
1.2
Crystal binding
A further different classification of crystals hinges on the type of (chemical/physical) binding responsible for their cohesion. Thinking a solid body as a complex many body system made of "valence" electrons and "lattice" ions (the ions in turn being made of both nuclei and "core" electrons), we give the following definition of cohesion energy: the energy needed to disassemble the crystal into a set of neutral atoms positioned each other an 3
Combining the symmetry of a Bravais lattice with that of the basis, new symmetry operations can arise such glide planes and screw axes.
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infinite distance apart (without loosing any original particle). Now thinking to the reverse process (at zero temperature), starting from the disassembled state, the formation of the stable final bound state of the crystal, occupying a finite volume, must correspond to an energy gain (lowering of the total energy) equal to (defining) the cohesion energy. The main acting forces are of electrostatic origin (repulsive between charges of same sign: electron-electron, nucleus-nucleus; attractive between charges of opposite sign: electron-nucleus). Magnetic forces, associated with elementary magnetic moments, connected with electronic and nuclear spins, are not equally important, with some exceptions. Gravitational forces are negligible (except for oriented crystal growth). Yet, only in the case of ionic solids (see below) the role of electrostatic interactions is fully understandable in terms of classical physics. To understand both crystal binding and properties in general a quantum approach is needed. Within this quantum description key items are the overlapping of atomic orbitals (to create extended states), the exchange interactions (related to exchange symmetry of the electrons wave function and thus, in a single particle mean field approximation, to Pauli exclusion principle) and, in some cases, the zero point energy (a consequence of Heisenberg uncertainty principle). An essential role is played by the charge spatial distribution depending on individual presence probability densities of valence electrons and of lattice ions. In this respect metal crystals and ionic crystals constitute two different extreme cases. In the former case the wave functions of valence electrons fill all crystal volume with a weak periodic inhomogeneity of the probability density (quasi-plane wave states) within simple lattices with the highest packing index, mainly face centered cubic (f.c.c.), e.g. , , , and hexagonal closed packed (h.c.p.), e.g. , , , but also body centered cubic (b.c.c.), e.g. , , . Here lattice ions are treated, to a first approximation, as hard spheres embodied in an almost uniform valence electrons cloud. In the latter case (perfect ionic binding) the crystal is a lattice of positive and negative ions (e.g. the salt + · − ) and extended (Bloch) states for valence electrons do not exist, being all electrons in localized atomic states. In the case of single element insulating crystals, e.g. diamond or semiconducting, e.g. Si and , binding is covalent. Extended states of valence electrons exhibit a strong interionic localization (bonds) from which strong bonding anisotropy follows in rather open structures with atomic tetrahedral coordination based on 3 atomic orbitals hybridization. III-V semiconductors, e.g. , or II-VI, e.g. , show a situation intermediate between covalent and ionic binding (there are extended states with strong bonding directionality and partial charge accumulation- of opposite sign- on pairs of nearest lattice ions). At very low temperature noble gases crystallize in a lattice of spherical neutral atoms bonded together because of
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the weak interaction between instantaneous atomic dipoles, fluctuating with zero average (van der Waals forces). Even in this case there are no extended stationary electronic states. Lastly we find layered crystals composed by stacking individual atomic planes (quasi 2D crystals), with strong covalent in-plane bonds and weak interplanar van der Waals interactions. graphite belongs to this class. Graphite has a hexagonal Bravais lattice with a basis of four carbon atoms. In plane bonding is due to overlapping of 22 hybrid atomic orbitals (with a trigonal coordination) while charge fluctuations in the cloud of 2 electrons, originating from atomic orbitals perpendicular to covalent graphenic planes, provides weak van der Waals interactions between pairs of such planes. Thus graphite is a solid lubricant due to the easy glide between adjacent planes. 2 orbitals give rise to extended interplanar quasifree electronic states responsible for electrical conductivity and most optical properties of this crystal. Graphite can be conceived as a sort of molecular crystal, being the graphenic planes the composing molecules. Recently single graphenic planes have been isolated as true 2D crystals (graphene) with exotic optoelectronic properties deriving from electronic bands with linear dispersion associated with the so called massless Dirac fermions.
1.3
Tensorial observables
The properties of a given crystal depend from both its structure (including defects) and chemical composition. The physical, chemical... observables can be classified as scalar (rank 0 tensors), vectorial (rank 1 tensors) and tensorial (rank ≥ 2 tensors). Without pretending any generality, we examine by means of an example the role of symmetry in determining the crystal properties. Here we consider only spatially homogeneous observables (the local quantities are averaged over a primitive cell). In particolar we examine dielectric susceptibility χ representing the response of an insulating crystal to an external electrostatic field E. Introducing the polarization vector P (mean electric dipole moment per unit volume) it turns out that (for not too intense fields and excluding ferroelectric crystals4 ) = ◦
(17)
In this formula, as in all tensorial equations, the sumPfrom 1 to 3 over all pairs of repeated indexes is understood (in our case =13 ) and ◦ is the vacuum dielectric constant. χ = is a rank 2 tensor. In isotropic solids the e (as a function of temperaferroelectric crystals can own a permanent polarization P ture) indipendently from the application of an external field. In this case the law must be modified as = ◦ + e 4
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tensor is diagonal: = . In this case the nine components of χ collapse into a unique scalar and the vector P is always parallel to the electric field E. In general there is a variable degree of crystalline anisotropy. Let us now consider the effect of a coordinates transformation, for instance a rotation. After this geometric operation the new coordinates (e.g. of a lattice point) can be expressed in terms of the old ones as: 0
= (18) ³ ´ 0 where the matrix = cos d is made by the cosines of the angles between each new coordinate axis and each old one. Then linear algebra gives us the tensor transformation formula: 0
(19)
=
If the considered rotation is a symmetry operation of the crystal (that is, 0 it belongs to the crystal space group) then χ = χ and the constraints 5 − = 0 lower the number of independent components of tensor χ. The number of independent tensor components can be reduced further by other physical laws not directly related with the space symmetry of the crystal. For example, the stress tensor is symmetric = (it has only 6 independent compenents) because of the requirements of static equilibrium with respect to rigid rotations. The strain tensor, related to the displacement vector u by definition as = (12)( + ), is symmetric too by its definition itself. It follows that the elastic constants tensor (a 4 rank tensor), expressing the elastic linear response of a crystal in the generalized Hooke’s law = , consists of no more than 36 independent components (in front of a total number of components 34 = 81). Different considerations of energetic character lower that number further to 21. Every residual reduction is caused by symmetry, down to cubic lattices (s.c., b.c.c. and f.c.c.) described by only 3 independent elastic constants and to the limiting case of isotropic solids (e.g. polycrystalline bodies) described by only 2 independent elastic constants. A complete description of the role of symmetry on tensor observables requires the use of group theory.
2
Scattering theory
To ascertain the static (average) structure of solids elastic scattering of probe particles is utilized. Using undulatory terms one can say that the structural 5
0
These conditions can be written in compact form as χ = aχa−1 = χ, introducing the matrix a−1 , the inverse of matrix a.
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information is extractable from diffraction of the wave associated with the particle used as probe. For X-rays the probe particle is a photon and the wave is of electromagnetic nature. When electrons or neutrons are employed as probes, the wave is their quantum wave function. In the case of crystals, from the measurement of the intensity of the scattered wave as a function of scattering angle it is possible to reconstruct both the Bravais lattice and the atomic positions of the basis. In the following mainly the quantum scattering of non-zero rest mass particles, like electrons and neutrons, will be treated. Because of different interactions (electrons interact with internal charges through electromagnetic forces and have a small penetration power; neutrons interact only with nuclei through strong forces at very short distances and also sense nuclear spins) electrons are suitable to explore surface crystallography or that of thin films, while neutrons give us bulk information and are very sensitive probes of light elements solids, in particular if solids contain hydrogen and its isotopes. Even X-Rays can provide bulk information. For X-Rays the interaction takes place mainly with valence and core electrons. In this case, beyond elastic scattering, also Compton scattering is relevant.
2.1
Elementary theory of elastic scattering
Some general theoretical quantities involved in the explanation of elastic scattering/diffraction experiments off crystals can be introduced in the following semi-phenomenological way. Let us consider a crystal hit by a monochromatic plane wave. In the case of an electromagnetic wave we could write its incident electric field as E◦ exp(k · r). Let us ussume that the pointlike atoms of the crystal, positioned at the points n of its Bravais lattice, react to the incident field each emitting in turn a spherical wave E◦ exp(k · n)n exp( |r − n| ) |r − n| whose amplitude is proportional to the incident field in n and to an atomic scattering amplitude n . Here = |k | = 2, being the wavelength of the incident wave. We define the scattered wave vector as k = r |r| = r in such a way that |k | = |k | (elastic scattering). The angle between k and k is the scattering angle . We set our detector of scattered waves atq the point r very far from the (finite) ¡ ¢2 p crystal in such a way that |r − n| = 1 − 2 r·n + ' 1 − 2 r·n ' ¡ ¢ √ r·n 1 − ' − k · n, where we have used 1 − ' 1 − 2 valid if ¿ 1 (far field approximation). Then the total waveX amplitude at E◦ exp(k · the detector can be written as: E(r) ≈ E◦ exp(k · r) + n
n)n exp() exp(−k · n)Now, if the incident beam is collimated and if
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the angle is not too small, in the simplest Xcase of identical atoms (n = ) exp [− (k − k ) · n], or, introwe can write: E(r) ≈ (E◦ exp()) n X exp (−Q · n). ducing the transferred wave vector Q = k −k , E(r) ≈ (E◦ exp()) n
The measured intensity of the scattered wave is thus proportional to ¯ ¯2 ¯X ¯ ¯ ¯ 2 exp (−Q · n)¯ (Q) ∝ |E(r)| ∝ ¯ ¯ n ¯
which is a function of the scattering angle as |Q| = 2 sin(2)In the case X of a cubic lattice n = u + u + u , the sum exp (−Q · n) is
the product of three unidimensional sums of the type
n −1 X
exp (− ) =
=0
1−exp(− ) 1−exp(− )
(a partial sum of the geometric series
∞ X
exp (− ) =
=0 1 ). 1−exp(− )
In the crystal there are 3 primitive cells. We can trans-
form this last result, without changing its squared modulus, as this way (Q) sin2 ( 2) sin2 ( 2)
2 ( 2) sin2 ( 2) sin2 ( 2) ∝ sin . sin2 ( 2) sin2 ( 2) sin2 ( 2)
sin( 2) sin( 2)
In
The study of the function
shows that,
I(x) 100
75
50
25
0 -15
-10
-5
0
5
10
15 x
= = 10 for big enough N, it is almost zero everywhere except where = (2) ,
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being any integer (positive or negative, including zero). There the function (an extremely sharp maximum) is equal to 2 Remembering the expression for the reciprocal lattice vectors of a cubic lattice, we conclude that the condition to have maxima in the scattering intensity (constructive interference) is Q = g for any triplet also corresponding to a family of crystal planes of Miller indeces . In the next subsection this selection rule is further examined in a more rigorous way for particle scattering and shown to be fully general.
2.2
Quantum particle scattering amplitude: Born approximation
Let us consider a steady flow of fast particles of mass , initially free and all with the same kinetic energy ~2 |k |2 2, which interact with a crystal ideally without thermal motions. This may be a beam of electrons or neutrons. The stationary state of a single particle that interacts with the crystal is determined by the mean potential energy field (r) which has the same symmetry of the direct lattice, and the wave function of a scattered particle is governed by the time-independent Schroedinger equation. ~2 2 − ∇ (r)+ (r)(r) =(r) 2 If |k |2 = |k|2 =
2 = 2 ~2
(20)
(21)
the equation can be written as: ¡ 2 2¢ 2(r) (r) ∇ + (r) = ~2
(22)
The initial state of incoming particles is described by the plane wave (r) =k ·r
(23)
p = ~k = ~k
(24)
where is the linear momentum of the particles and k the incident wave vector. In general an incident probability current density vector is collectively associated to states like this j =
~ ~k (∗ (r)∇ (r) − (r)∇∗ (r)) = 2
(25)
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The plane wave satisfies the homogeneous equation (empty lattice): ¡ 2 2¢ (26) ∇ + (r) =0 ¡ ¢ 0 The Green function (r r ) of operator ∇2 +2 is, by definition, the solution of the non homogeneous equation ¡ 2 2¢ 0 0 (27) ∇ + (r r ) = (r − r ) 0
where (r) is the Dirac delta function. Using (r r ) the more complex equation ¡ 2 2¢ ∇ + (r) = (r) (28) can be solved by means of a superposition integral: Z 0 0 0 (r) = (r)+ (r r )(r )r
(29)
The explicit form of the Green function turns out to be (see e.g. Davydov, Quantum Mechanics, Pergamon Press): 0
(r r ) = −
0 r−r
0 4 |r − r |
(30)
The initial Schroedinger equation transforms then into the integral equation: (r) = (r)−
2~2
Z
0 r−r
0 0 0 0 (r ) (r )r |r − r |
(31)
¯ 0¯ If ¯r ¯ 1 (far field approximation) the spherical wave emitted from 0 point r within the scattering volume can be approximated by the product of a spherical wave emitted from the origin (always taken within the scattering volume) and of a plane wave emitted in the direction of wave vector k = r 0 r−r
−k ·r0 ≈ 0 |r − r |
(32)
being k = r the scattered wave vector 6 . The angle between k and k is called scattering angle. The direction of k can be given by the angles r q ¯ ¯ ³ 0 ´2 ³ 0 0 0¯ ¯ ¯r − r ¯ = 1 − 2 r·r + ' 1 − 2 r·r ' 1 − √ where we have used 1 − ' 1 − 2 valid if ¿ 1 6
0
r·r
´
0
' − k · r
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and of spherical coordinates and the infinitesimal solid angle about this direction is Ω = sin . The radial probability current density vector µ ¶ ~ (r) ∗ (r) ∗ (r) − (r) (33) j (r) = 2 is associated to the scattered particle beam. The integral equation can be written again as k ·r + (r) = (34) Seen from afar, the set of scattering centers (contained within the scattering volume ), when hit by a monochromatic plane wave, generates a modulated spherical wave characterized by the scattering amplitude Z 0 0 0 0 −k ·r = − (r ) (r )r (35) 2 2~
Within Born approximation potential energy is considered as a perturbation with respect to kinetic energy. Consequently under the integration symbol 0 0 one writes (r ) ≈k ·r . In this way multiple scattering is neglected and the first order scattering amplitude is obtained as: Z 0 0 0 = − (r )−Q·r r = (36) 2 2~
k + Q|(r)|k = − 2~2 where Q = k − k
(37)
is the transferred wave vector. Within this approximation the scattering amplitude is proportional to the Fourier transform of index Q of interaction potential energy. Perturbations produced by the corresponding force field generate transitions between initial states |k and final states |k + Q = |k both described by plane waves (far from the scattering volume). The quantities k |(r)|k are called matrix elements of perturbation operator (r). The differential cross section (Ω) = (number of scattered particles per unit time within the solid angle Ω)/(flux of incident particles) = ( 2 Ω ) turns out to be: (38) (Ω) = | |2 Ω
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Solid State Physics Lecture Notes
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Elastic scattering - Bragg law
In case of elastic scattering initial kinetic energy is conserved and thus |k| = |k| Furthermore, in crystals the potential energy owns the lattice periodicity and can be represented as the Fourier series X g ·r (39) (r) =
g being the reciprocal lattice vectors, with Fourier coefficients Z 1 = (r)−g ·r r Then the scattering amplitude can be written as: Z X 0 0 0 g ·r −Q·r = − r = 2 2~ Z 0 X 0 (g −Q)·r r = − 2 2~
(40)
(41)
If the scattering volume is much bigger than |Q|−3 and also, as it is generally true in most scattering experiments, than , and if the far field approximation is still valid, in practice the integration can be extended over all space ( → ∞), so obtaining = −
4 2 X (Q − g ) ~2
(42)
From this last result it is seen that the scattering amplitude is different from zero (in the real case, it is maximum) if and only if Q = g . Formula (42) has a simple physical meaning thanks to the properties of reciprocal and direct lattices. A particular family of direct lattice planes owns as normal vectors the reciprocal lattice vectors labelled by the triplet of least integers () (identical to crystallographic Miller indices of that family of planes) and by all tripletys {} that are obtained from () multiplying every index by the same integer . E.g. in a simple cubic lattice: (110),(220),(330),..... Vectors g220 ,g330 ,..... share the same direction with vector g110 and for the moduli we have: |g220 | = 2 |g110 |, |g330 | = 3 |g110 |,.....√ The first neighbour planes distance of this family is 110 = 2 |g110 | = 2 and represents the space period of the Bravais lattice along direction 110 perpendicular
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to all considered planes. Let us consider the necessary condition to have maximum scattering amplitude: Q = k − k = g
(43)
|k − g |2 = |k |2
(44)
|k |2 − 2k · g + |g |2 = |k |2
(45)
and write it as that is For elastic scattering and introducing the Bragg angle as the complement to 900 of the angle between k and g , one gets |g | − 2 |k | sin = 0
(46)
This law tells us that maximum scattering is obtained when the projection of particle wave vector onto the vector g equals half modulus of g (see Umklapp below). For = 2 |g | and = 2 |k | is the de Broglie wavelength of incident particles, one eventually obtains 2 sin =
(47)
the well known Bragg law (which gives the angular positions of Bragg peaks, being the Bragg angle one half of scattering angle: consider the vector diagram of equation k − k = g ). To arrive at (42) we used the result: Z 0 0 (48) (g −Q)·r r = (2)3 (g − Q) easily obtainable in 1D as Z
+∞
= 2()
(49)
−∞
considering that (see Appendix) Z +∞ 1 = 2 −∞
Z + 1 lim = →∞ 2 − sin() = () = lim →∞
(50)
is anyhow useful to describe the interaction with small crysThe form sin() tals and is the basis for the experimental determination of crystalline domains size analyzing the shape of diffraction peaks. If a family of lattice
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planes {} satisfies Bragg law g − Q = 0, the peak intensity (area) is proportional to the squared modulus of and dipends on the matter distribution (distribution of scattering centers) within a single primitive cell, that is on the basis containing atoms. In the case of neutron scattering, due to the extremely small interaction range,Pin (40) one can write (being the integral limited to only one cell) (r) = =1 (r − r ). The r are the positions of the basis nuclei of primitive cell and all coordinates refer to the local origin of the cell itself. This form of interaction energy between probe particles and crystal atoms is called Fermi pseudo-potential. The are the scattering amplitudes of nuclei. Then it turns out: Z X 1 = (r − r )−g ·r r = =1 1 X −g ·r (51) = =1 The quantity =
X
−g ·r
(52)
=1
is called the geometric structure factor. The intensity of a single Bragg peak () is then proportional to the squared modulus of . Sometimes, due to a particular basis structure generating distructive interference, may be null and the corresponding Bragg peak disappears (see below). In the case of both X-Rays and electrons is given by the same formula but with electronic quantities (Q = g ) substituting the nuclear constants . The (Q) is an individual atomic property mainly related to core electrons. Quantities (Q) are called atomic form factors (see below).
2.4
X-Ray Diffraction
For X-Ray elastic scattering (XRD: X-Ray Diffraction ) most considerations already introduced for particle scattering remain valid. In particular similar formulae for both scattering amplitude and differential cross section are obtained, provided the role of (r) is played by the electronic charge density (r). In fact the wave equation, in the monochromatic case, for the electric field propagating in a medium of index of refraction (r) can be
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Solid State Physics Lecture Notes
reconducted to the vector Helmholtz equation7 µ ¶ 2 2 2 ∇+ 2 E(r) = 0
25
(53)
Taking into account that (above plasma frequency, see eq. 428), for X-Rays we can write (r) 2 = 1 − (54) 2 0 one gets the equation ¶ µ 2 (r) 2 E(r) (55) ∇ + 2 E(r) = 2 0 with 2 2 = 2 . This equation is quite similar to the scalar equation 22. In our analogy the role of is then played by : Z 1 = (r)−g ·r r (56) if (r) =
X
=1
(r − r )
(57)
(in the assumption, not applicable to covalent solids, that the electron charge density is the sum of the electron densities of each atom ):
∝ = = =
1 1 1 1
Z
X
=1
Z
X
=1
X
=1
X
(r − r )−g ·r r = (r − r )−g ·(r−r ) −g ·r r =
−g ·r
Z
(r)−g ·r r =
−g ·r (g )
(58)
=1
where 7
The propagation of an electromagnetic wave in a mean with refractive index is de2 2 scribed by equation ∇2 E = 2 E 2 , where E is the electric field. When E = E(r) exp(−ω), the Helmholtz equation is obtained.
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Solid State Physics Lecture Notes
(Q) =
Z
(r)−Q·r r
26
(59)
once that the diffraction condition is set Q = g . In case of X ray diffraction we define the structure factor as: X = −g ·r (g )
(60)
=1
where depends on the nature of the pth atom.
Monoatomic lattice with a basis: X −g ·r = (g ) = (g )
(61)
=1
P We can calculate = =1 −g ·r in case of a bcc lattice considered as a simple cubic lattice with a basis: = 1 2 r1 = 0 r2 = (2)(x + y + z) g = (2)(x + y + z) = 1 + exp[−g · ((2)(x + y + z))]½= ++
= 1+exp(−(++)) = 1+(−1)
=
2 when + + is even 0 when + + is odd
¾
in reciprocal space this converts the simple cubic lattice with side 2 (the reciprocal of a simple cube with side ) in a fcc lattice with a conventional cell with side 4, i.e. exactly the reciprocal of a bcc with side , for which is 1 (see picture 6.11 Ashcroft-Mermin). P Then we calculate = =1 −g ·r in case of a fcc considered as a simple cubic lattice with a basis: = 1 2 r1 = 0 r2 = (2)(x + y) r3 = (2)(y + z) r4 = (2)(x + z) g = (2)(x + y + z)
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= 1 + exp[−g · ((2)(x + y))] + exp[−g · ((2)(x + z))] + exp[−g · ((2)(y + z))] = =½ 1 + exp(−( + )) + exp(−( + )) + exp(−( + )) = ¾ 0 if in : 2 are even and 1 is odd or 2 odd and 1 even = 4 if in : 3 are even or 3 are odd in reciprocal space this converts the simple cubic lattice with side 2 (the reciprocal of a simple cube with side ) in a bcc with a conventional cell with side 4, i.e. exactly the reciprocal of a fcc with side , for which is 1 (see picture 6.11 Ashcroft-Mermin). P Finally we calculate = =1 −g ·r in case of diamond lattice considered as a fcc with a basis: = 1 2 r1 = 0 r2 = (4)(x + y + z) g = (b1 + b2 + b3 ) b1 = (2)(y + z − x) b2 = (2)(z + x − y) b3 = (2)(x + y − z) ⎧= 1 + exp(− 2 ( + + )) = ⎫ ⎨ 2 if + + is two times an even number ⎬ 1 ± , if + + is odd = ⎩ ⎭ 0 if + + is two times an odd number
Bragg formulation. As has already been discussed in the particle scattering, Q = g is equal to 2 sin =
(62)
where is the spacing between to planes in the family of crystal planes with indices (), is the angle between k (and k ) and the plane, is the wave length of radiation X, in an integer number. Q = g is equal Q = k − k = 2 2 sin = 2 = g . Von Laue formulation. Since |k | = |k | = , then Q = k − k = g k = k − g by squaring both sides of this (and since = |k − g |) we obtain:
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Solid State Physics Lecture Notes
k · Powder diffraction:
g 1 = |g | |g | 2
28
(63)
= 2 where is the angle of the diffraction (angle between k and k ). 2 sin
(64)
OBSERVATIONS: The intensity of diffraction is proportional to the Fourier transform of the static autocorrelation function: (Q) is the scattering amplitude, (Q) is the intensity experimentally measured with scattering wave vector Q = k −k defined by experimental conditions.
(Q) ∼
Z
(r)−Q·r r Z
−Q·r0
Z
(Q) ∼ |(Q)| ∼ (Q) (Q) ∼ (r ) r ∗ (r)Q·r r = Z Z Z ∗ −Q·R = (r + R) (r) rR = (R)−Q·R R (65) 2
∗
0
0
(R) is called static autocorrelation function or Patterson function.
(R) = x-rays: neutrons:
Z
(r + R) ∗ (r)r Z (R) = (r + R)∗ (r)r 2
(R) =
Z X
= 2
X
(r + R − r )
(R − (r − r ))
X
(r − r )r = (66)
The formula for neutrons refers to a monoatomic material, with equal for any atom.
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Liquids (neutron scattering):
2
(Q) ∼
= 2
Z X
X
(R − (r − r ))−Q·R R =
−Q·(r −r ) = 2
2
à X
1+
2
= +
Z X
X X
6=
−Q·(r −r )
!
=
(R)−Q·R R =
µ ¶ Z 2 −Q·R = 1 + (R) R
(67)
where (R) is the pair-correlation function, i.e. (R)R gives the mean numbers of atoms in a volume R around distance R from one atom in the liquid ( (R) = (R): static non local homogeneity hypothesis). In the hypothesis of statistical isotropy ((R) = ()): µ ¶ Z −Q·R (Q) ∼ 1 + () R = µ ¶ Z 2 − cos 2 2 sin = = 1 + () ¶ µ Z sin 2 2 = 1 + 4 () 2
(68)
where () is the radial distribution function, i.e. 42 () is the mean number of atoms located in a spherical shell with thickness centered on an atom in the liquid. () exhibits characteristic oscillations with maxima in values of corresponding to a short range order (dynamic statistical order: first neighbours, second neighbours, etc.). A monoatomic gas doesn’t exhibit characteristic oscillations in ().
3
Electronic States
Physical and chemical properties in solids strictly depend on electronic motion, both stationary and not. Considering valence electrons motion as independent from motion of ions in the lattice (or from motion of nuclei, since
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core electrons can be considered strictly bounded to them) provides just a good approximation that will be examined in detail at the beginning of chapter 4. See chapter 4 also for a better comprehension of this complex issue. However in this chapter we will assume this independency and, moreover, we will assume nuclei as stationary in their equilibrium positions. If these equilibrium positions form a Bravais lattice (in case with a basis) the reduced many-electron problem can be simplified again and considered as a one-electron problem in a mean periodic potential (see also paragraph immediately after equation (220)). We start trying to solve the equation for stationary states for an isolated multielectron system:
where and
b (r) = () (r) H
b=H b0 = b + (r) + (r) H b =
X |pb |2
2
=−
~2 X 2 ∇r 2
(69) (70) (71)
is the total kinetic energy operator for valence electrons, (r) =
2 1X 2 6= 40 |r − r |
(72)
is the Coulomb potential energy associated with repulsive interactions between electrons, XX 2 (73) (r) = − 4 |r − l| 0 l
is the Coulomb potential energy associated with attractive interactions electronion. For simplicity we considered a simple monoatomic crystal: l indicating the positions of nuclei/ions and their bare charge. r = (r1 s1 r2 s2 r s r s )
represents the set of dynamic coordinates (position and spin) of all valence electrons. We neglect relativistic effects (including spin-orbit interactions and direct magnetic spin-spin interactions). With a procedure similar to the Hartree or Hartree-Fock self-consistent mean-field approximation used for many-electron
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atoms (see appendix), we simplify the previous many-body problem to an effective one-electron problem b (r ) = (r ) H
(74)
|pb |2 b + h (r )i + (r ) (75) H = 2 hi representing the average value of electron-electron interactions. Here we do a further approximation: we assume that the mean periodic potential h (r )i + (r ) = (r ) is the same for every electron (a good approximation for macroscopic crystals): for this reason by now on we drop the subscript . From a physical point of view the sum of h (r)i and of the bare periodic potential (r) is equivalent to the electrostatic screening of the latter. In other words: the generic valence electron moves in an attractive periodic potential (r) resulting from the interaction with lattice ions, whose positive charge is partially screened from the mean negative charge distribution of all other valence electrons. This mean field approach is static in nature and neglects the dynamic correlation effects present within the screening electrons cloud surrounding ions (and the screening holes cloud surrounding every generic electron, see below the introduction to the quasiparticle concept). In a Fourier representation of periodic potential this Hartree-Fock approximation can be described by a dielectric function depending on wave vector (see Bloch’s theorem). This dielectric function substitutes 0 in a formula in which the Fourier transform of (r) appears.
3.1
Bloch’s Theorem
b n is defined as follows: The translation operator T
b n (r) = (r + Tn ) = (r+1 a1 + 2 a2 + 3 a3 ) T
(76)
b n () = (+Tn ) = (+) T
(77)
In a one dimensional space:
We consider now the problem of determining stationary electronic states in the lattice within an independent electron approximation in a mean periodic potential: (78) (+Tn ) = (+) = ()
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We have to solve the eigenvalue equation: b =
where
2 b = b + () 2 b is invariant under translations such as T b n: Since
b n] = 0 b T [
we can actually write
(79)
(80)
(81)
b n ()() b n () b b + )( + ) = () b T T = (
From a general principle of quantum mechanics, if the commutator between two operators vanishes, then the two operators share the same set of eigenb n commute, it is possible to write: b and T functions. In this case, since b n () = (n)() T
b n is an additive operator, which means: T
b n2 () = (+1 + 2 ) = b n1 T T
(82)
(83)
b n1 +n2 () = (+(1 + 2 )) = T
Using now the eigenvalue equation we obtain:
(n1 )(n2 )() = (n1 + n2 )()
(84)
(n1 )(n2 ) = (n1 + n2 )
(85)
This equation is satisfied by (n) =
(86)
for any choice of the complex number , provided that it has dimensions of reciprocal length. Exploiting periodicity we apply the normalization condition within a single primitive cell of volume : Z ¯ ¯2 ¯b ¯ |()| = 1 = ¯Tn ()¯ = Z 2 2 = |(n)| |()| Z
2
(87)
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i.e. |(n)|2 = 1 = | |2
(88)
=
(89)
b n () = (+Tn ) = (+) = () T
(90)
() = ()
(91)
(+Tn ) = (+) = ()
(92)
then, necessarily with real. Therefore:
This property is satisfied by any () (Bloch wavefunction):
in which is a periodic function in . From a physical (intuitive) point of view it is important to consider that Bloch functions are plane waves modulated by a periodic factor. b ( () is a stationary state of an electron in To any eigenfunction of a periodic potential) b () = () (93)
a wave vector is associated such that equation (90) holds, and therefore also equations (91) and (92) are true. Stationary energy levels associated to the electron depend on wave vector too. In 3D we generalize the above result using the vector notation: (r + T ) = k·T k (r)
(94)
k (r) = k (r)k·r
(95)
k (r + T ) = k (r)
(96)
where and
b = −~∇ to a Bloch wave and to a Applying the momentum operator p plane wave leads to two different result. For a Bloch wave: bk (r) 6= ~kk (r) p
(97)
This means that a Bloch wave is not an eigenfunction of momentum operator as instead a simple plane wave is. Thus the term ~k is called
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quasimomentum or crystal momentum. The mean momentum (the quantum expectation value of momentum in a Bloch state), instead, is equal to (for a complete derivation see Appendix E, Ashcroft and Mermin): p =
k =v ~ k
(98)
k = Normally Bloch stationary states are propagating waves unless k 0 (In which case we have a standing wave. This can happen at specific points: see below other typical bands structures k ). The existence of stationary electronic propagating waves in a periodic accelerating potential is a purely quantum effect. Since in real crystals the electron motion usually encounters some resistance, there must be something that alters the perfect periodic structure of the crystal, a periodicity defect. Insertion of the Bloch wave function in the form 91 into 3D version of eq. 93 yields the following eigenvalue problem:
− id est:
¢ ~2 ¡ 2 ∇ k + 2k · ∇k − 2 k + (r)k = k k 2 "
# (b p + ~k)2 + (r) k = k k 2
b = −~∇ is the momentum operator. where p
(99)
(100)
For any fixed k, the eigenvalue problem (100) defines a spectrum of discrete eigenvalues k = k . Every energy level k is labelled with a branch index which stands for the set of quantum numbers needed to label the eigenvalue spectrum at fixed k. Usually these k are degenerate energy levels: all states belonging to the same branch can be written as different k·r Bloch functions , where now represents the subset of k () = k () quantum numbers required to span all states. depend on the nontranslational part of the space group of the crystal. By fixing and letting k vary we obtain a hypersurface (a line in 1D, a surface in 2D): a specific electronic branch of the band structure of the crystal. Letting both and k vary we obtain the entire band structure. In this context a specific band is the whole energy sub-interval spanned by all k values. Sometimes the word band is used to indicate a branch, generating some confusion. In particular, adopting our nomenclature, we can say that, in some crystal structures, different energy bands can be (partially) superimposed. As far as the physical meaning of the single electron energy levels k is concerned, if the periodic potential is the self consistent one obtained by the Hartee-Fock method
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(see appendix on self consistent mean field) assuming a Slater determinant crystal wavefunction, describing all itinerant electrons, built by all occupied spin-orbitals, with orbital parts constituted by the one-electron Bloch states k () = k ()k·r , we can state two relevant results. First (Koopman’s theorem): −k equals the ionization energy of a single electron initially in state k () being the global initial state the minimum energy one (ground state). Second: k − k ≈ ∆ approximately equals the transition energy between the initial state k (), belonging to the global ground state, and the final state k (), belonging to an excited global state. The two results are tenable provided all other spin-orbital are unchanged in the involved process (no relaxation). The fact that both the Coulomb and the exchange integrals present in Hartee-Fock theory scale as 1 where is the number of primitive cells make the above statements particularly good for itinerant states such as Bloch waves (see e.g. the tight binding method). For a proof (including the definitions of Coulomb and exchange integrals) see the advanced textbook G. Grosso & G. Pastori Parravicini Solid State Physics (2-nd ed.) Academic Press (2014), pag. 146. If instead the present version of the Density Functional Approximation is used, the k have generally no direct physical meaning (ibidem pag. 163).
3.2
Reduction to the first Brillouin zone
In one dimension. Let us first consider = ( being any reciprocal lattice vector). In this case (), associated to the eigenvalue , is a periodic function with the same space periodicity of the lattice. ( + ) = ( + )(+) =
= ()
(101)
= ()
We can generalize that to a 3D situation as: g (r + T ) = g (r)
(102)
We consider now, for the sake of simplicity, a generic Bloch wavefunction = in a one dimensional situation; let , and , be fixed. () b and related to the eigenvalue/energetic The state , eigenfunction of level , is also eigenfunction of bn and thus related to the eigenvalue :
bn =
Since there is a one-to-one correspondence between the eigenvalue and the energy level , and since is a periodic function of with
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periodicity = 2, also will be a periodic function of with the same periodicity. The representation of the states as functions of , with −∞ +∞ (repeated-zone scheme), is so highly redundant, since the same branch segment is shown in all infinitely many equivalent wave vector periods. Fixing univocally the corresponding to a specific energy level/state is now necessary. This is achieved using the reduced-zone scheme where we consider only the wave vectors belonging to the first Brillouin zone, in such a way to associate to any state of branch a single wave vector . In one dimension this corresponds to limiting the vector within the interval: − ≤ (103) A third representation is the extended-zone scheme in which in any Brillouin zone there is just one electronic branch. Starting in the first zone and increasing, in any direction, the modulus of the wave vector, we will assign to any zone only one specific branch , in such a way to progessively increase the index (the first branch in the first zone, the second branch in the second zone and so on).
Umklapp of free electron parabola . As a limiting case, we consider now an empty (periodic!) lattice. This physically corresponds to a vanishingly small periodic potential. The Bloch states reduce to plane waves −12 k·r (they describe the motion of a free electron) and the energy levels associated to them are given by the following formula, which can be considered also as a valid representation in the extended-zone scheme with an arbitrary space period. ~2 |k|2 k = 2
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The picture shows the result of this fictitious periodicity of an empty lattice. In order to obtain the reduced-zone scheme we must back translate (umklapp) of reciprocal lattice vectors, the different pieces of the parabola from any higher order Brillouin zone into the first Brillouin zone. A real periodic potential would introduce a distortion of the parabola and, generally, the opening of a prohibited energy gap between two consecutive bands. Writing the Bloch functions in their characteristic expressions can definitely persuade us that the states described with wave vector k and with wave vector k0 = k − g are exactly the same state. 0
k ·r −g·r k·r k·r (r) = = = = k (r) k (r) k0 (r) k0 (r) k0 −g·r , since −g·r and (r) In the formula we defined k (r) = k0 (r) k ¯0 ¯ ¯ ¯2 in the direct lattice have the same periodicity. Notice that ¯k (r)¯ = ¯ ¯2 ¯ ¯ ¯k−g (r)¯ .
3.3
Born-von Karman boundary conditions
Since real crystals are finite, they are not perfectly periodic. Yet, by neglecting surface states and introducing appropriate boundary conditions compatible with translational invariance, Bloch’s theorem still holds and provides a suitable approximation for the real case. In one dimension, considering N (N1) primitive cells in a crystal of length = we can write the periodic boundary conditions as: ( + ) = () = ()
(104)
which requires = 1This last equation implies that the possible (allowed) wave vectors belong to the discrete set of values: =
2
(105)
In the (− ≤ ) we have exactly N independent corresponding to integer varying between − 2 and 2 : −
≤ 2 2
(106)
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Since ∆ = +1 − = 2 2, wave vector nearly form a continuous set of values. The situation in 3D is very similar to this one. We consider = 1 2 3 cells, in the first Brillouin zone the allowed wave vectors are given by the following formula: ¶ µ 1 2 3 k(1 2 3 ) = 2 (107) b1 + b2 + b3 1 2 3 where −
≤ 2 2 = 1 2 3
(108)
Taking into account spin degeneracy and Pauli principle, it follows that a single branch k of the band structure (in the first Brillouin zone) can guest up to 2 electronic states (Bloch waves). As a result, knowing the band structure of a crystal, including the existence and width of forbidden energy gaps, it is possible to foresee whether the material is a conductor (metal) or a semiconductor/insulator. As an example we consider a monovalent element which crystallizes as a simple crystal, such as (a body centered cubic structure). In this case, in the ground state, the set of valence electrons occupies just the first half of k1 , leaving the second half empty. If accelerated by an external electrical field (see below "Single electron Dynamics") there will be a net current generated by the highest energy electrons making transitions to close empty energy states belonging to the same band: such a crystal is a conductor. Following similar lines of reasoning, to get a more general rule of thumb, for the sake of simplicity we assume that contiguous allowed bands, spanned by the different branches k , are separated by forbidden energy intervals called band gaps, i.e. ranges of energy that an electron in the solid may not have (see e.g. "Nearly-free electron bands"). Let be the atoms valence and let be the number of atoms in the basis; if is odd, the crystal should be a conductor, otherwise if is even, the crystal should be an insulator (or a semiconductor, if the width of the first prohibited band is less than or equal to ). However some real situations are more complicated and there are a lot of important counterexamples of our rough rule. For example all of the elements in the second column of the periodic table ( = 2) form metallic crystals, even though they are simple crystals (one atom per primitive cell). In these cases we should consider the complex topology of the real band structure in three dimensional space. Different branches with energy k can exhibit a partial superposition of the relative energy bands, along different directions in space. This is true for the above case (e.g for ): as a consequence the first band is not full
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and the second is also partly occupied giving rise to a metallic behavior. We will see that the empty states of the first band seem to behave as positive charge carriers (holes). In one dimension superposition of bands is not allowed because it would imply branch crossing: in a 1D world would be an insulator (or a semiconductor) according to our rule of thumb.
3.4 3.4.1
Free electron gas Plane waves - Fermi energy - Density of states
If we neglect completely the periodic potential, the one electron Hamiltonian coincides with the kinetic energy operator: 2
b = − ~ ∇2 2 and the equation for stationary states becomes
(109)
~2 2 − ∇ k (r) = k k (r) (110) 2 We already know that eigenvalues (degenerate energy levels) in this case are given by the following expression ~2 (2 + 2 + 2 ) ~2 |k|2 = (111) 2 2 and that the corresponding eigenfunctions are the plane waves 1 (112) k (r) = √ k·r where is the crystal volume. Vectors k are the discrete ones given by the periodical boundary conditions. If we consider the set of allowed wave vectors (corresponding to a lattice of points in space) we can notice that there is a volume (2)3 corresponding to any vector/point/orbital state (plane wave). The dispersion relation (111) shows that the √ locus of points of constant energy is a spherical surface of radius () = 2~. The number of states with energy between 0 and is then k =
4 ()3 () = 2 3 3 (2)
(113)
the factor 2 is introduced to take into account the spin degeneracy. The density of states thus assumes expression µ ¶3 2 2 √ (114) () = = 2 2 ~2
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The free electron gas ground state (at absolute zero) is determined by ordering the states of the system (112) by increasing energy, and by consecutively filling up the unoccupied quantum states with the lowest energy, according to the Pauli exclusion principle. When all the electrons have been put in, the Fermi energy is the energy of the highest occupied state. The corresponding spherical surface has a radius of = ( ). The Fermi wave vector is obtained from (113) by equalizing ( ) to the number of electrons contained in the volume µ ¶ 13 2 = 3 ¡ ¢ 23 ~2 3 2 (115) = 2 Both Fermi wave vector and Fermi energy depend on valence electron density (that is the number of valence electrons per unit volume). The probability that, at a temperature , a state with energy will be occupied is given by the Fermi-Dirac distribution (| ) (| ) =
1 exp( − )
(116)
+1
where is the chemical potential of an electron. Since depends weakly on the temperature (see "Electronic specific heat"), in most of the cases the approximation ( ) ≈ (0) = can be used. Wecan now introduce now the density of occupied states D(| ) = () (| ) = 2 2
µ
2 ~2
¶ 32
√ exp( − )+1
Obviously for an isolated conductor we have Z Z ∞ D(| ) = () = 0
(117)
(118)
0
When the Fermi wavelength = 2 has a value comparable with the interatomic distances (in 1 when = 2) we expect a Bragg reflection (electron diffraction, see "Scattering theory") to occur off a family of lattice planes for electrons on the Fermi surface. In this case the free electron model fails.
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Chemical potential and electronic specific heat
We start with a property (pointed out by Sommerfeld) of Fermi-Dirac distribution related to the fact that by differentiating the (116) with respect to energy we obtain a function similar (but not identical) to the Dirac delta function (red curve showed in the figure, with a negative sign). The width at half maximum of this function is ∼ ¿ ≈ . Thus the exact delta function behaviour is exhibited only at → 0 f
3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2
2.5
3 E
In this picture chemical potential is 2 a.u.
To make the above assumption stronger, we observe that: Z ∞ (| ) = (0| ) − (∞| ) = 1 − 0
The property is the following Z Z ∞ ()(| ) ≈ 0
0
2 () + 6
µ
¶
( )2
(119)
(120)
=
For a proof see, e.g., Ziman, Principles of the theory of solids 2-nd ed., Cambridge University Press. There, to derive the previous formula, the following identity (obtained by integration by parts) is used: µ ¶ Z ∞ Z ∞ (| ) () (| ) = () − (121) 0 0 R 0 0 where () = 0 ( ) . Then (120) can be written in another form (useful in applications) as: ¶ ¶ µ µ Z ∞ 2 2 (| ) ≈ () + () − ( )2 (122) 2 6 0 =
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When the temperature dependent term can be neglected it is possible to write −(| ) ≈ ( − ). We can now use eq. (120) to determine the dependence of the chemical potential on temperature. We write (118) as µ ¶ Z Z 2 () ≈ () + ( )2 (123) = 6 0 0 Since
Z
Z
() =
0
0
() +
Z
()
(124)
It is possible to obtain µ ¶ Z 2 () ≈ ( )2 ≈ ( ) ( − ) 6
(125)
Resolving the previous formula with respect to the chemical potential we get µ ¶ 2 ≈ − ln () ( )2 (126) 6 The apparently rough approximation ≈ is usually acceptable even beyond room temperature. The electronic contribution to specific heat could also be found by differentiating the internal energy with respect to temperature, being Z ∞ = ()(| ) (127) 0
Here we use a more direct approach. Because of the exclusion principle, at temperature , only electrons =
Z
+
−
2
2
() ≈ ( )
(128)
are thermally excited. Each of them contributes to the part of internal energy depending on the temperature = with an individual contribution Then ≈
¤ £ = ( ) ( )2 = 22 ( )
(129)
The exact calculation would give us the constant 2 3 instead of the prefactor 2 of the previous formula. In conductors this contribution to specific heat is important at low temperature and gives us a way to measure ( ).
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Quasi-particles and Fermi liquid theory (a qualitative introduction)
The dependence of density of occupied states on temperature shows us that, due to the exclusion principle, just a tiny amount of electrons with energy close to Fermi energy (in an interval ± 2 ) can be thermally excited. In the scheme of single particle states in a mean field, we can say that the excited state is defined both by the wave functions of the (few) electrons with an energy bigger than the Fermi one (more precisely, bigger than the chemical potential) and by the wave functions of the (many) remaining electrons with an energy smaller than the Fermi one. In the ground state all states up to are occupied and all states above it are empty. is of the order of some (for example in the value is = 31 ), while at room temperature we have ≈ 25×10−3 . The above "electron representation" of the excited state is very redundant. In order to get closer to the (quasi)electrons and holes representation we will use later, we introduce here, in a qualitative way, some important concepts due to Lev Landau. We linearize the dispersion relation about the energy point ≈ ¶ µ ≈+ ( − ) = ( − ) where the Fermi velocity is = }−1 () and where = } is the momentum of the quasi particles. If we consider as the new zero of the energy scale, the dispersion relation becomes = ( − ). This is valid only for excited electrons with . In the limit of validity of the linearization we have unoccupied states if . Afterwards we will see that the excited state can be described by few particles with negative charge (electrons) for which the previous dispersion relation is valid with = 0 and = and by equally few particles with positive charge (holes) for which is valid the dispersion relation obtained by the previous formula multiplied by −1: = − = ( − ) with . For the holes this transformation is equivalent to a time reversal transformation, as originally shown by Feynman. Electrons attract holes and holes attract electrons. Electrons with a cloud of holes and holes with a cloud of electrons are called quasi-particles and acquire an effective mass. Furthermore, since the dynamic cloud is not perfectly stationary, the quasi-particles have a finite life time () beyond which they are annihilated by collisions. Even neglecting this electron-electron effects (the cloud), just introducing the periodic potential, the velocity and the mass of electrons and holes can be changed.The above considerations will help us to explain why the mass of an electron in a crystal lattice is different from that of the electron in vacuum. Moreover
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two quasi-electrons repel each other less than two electrons separated by the same distance in vacuum. This can be represented by introducing a dielectic constant different from the vacuum one. All the same, mutatis mutandis, is valid for interactions between two holes and for electron-hole interactions. The number of quasi-particles is not constant (as is the number of electrons) but depends on temperature.The complex dynamic system described in this paragraph is called Fermi liquid. The theoretical framework needed to take into account many body effects in condensed matter is a particular aspect of Quantum Field Theory. We will not enter this complex field. The interested reader is referred to R.D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2-nd ed. Dover 1992.
3.5 3.5.1
Band structure: nearly free electron approximation Free electron diffraction
If we allow alkaline atomic elements to crystallize, due to their weak atomic first ionization potential, we can consider a correspondingly weak periodic potential () and treat it as a static perturbation (in comparison to kinetic energy). Then standard 2-nd order perturbation theory (see Appendix on "Approximate Methods") gives us the following expression for the perturbed energy levels (): ¯ ¯2 0 ¯ ¯ 2 k| (r)|k 2 ¯ ¯ X ~ |k| ³ ´ (k) ≈ + k|(r)|k + (130) 2 0 2 ~2 2 |k| − |k | k 6=k 2 Here we have adopted the Dirac notation. The unperturbed states | are the plane waves in the vacuum lattice (see above). We have to examine the matrix elements Z 0 0 1 1 √ −k·r (r) √ k ·r r = k| (r)|k ≡ (131) Z 0 1 −(k−k )·r (r)r =
The periodic potential energy () can be represented as a Fourier series X (r) = g g·r (132) g
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where the are the reciprocal lattice vectors and where the Fourier coefficients g can be written as: Z 1 (r)−g·r r (133) g = Thus: Z X 0 1 k| (r)|k = g g·r −(k−k )·r r = g Z 0 1 X = g (g−k+k )·r r =g k0 k−g g 0
(134)
In order to obtain (134) we used the periodic boundary conditions which give us a discretized set of wave vectors (105). A similar treatment can be found above in "Scattering theory" where a continuous set of wave vectors 0 is used. So, unless = − , the elements of the matrix are zero and the R 1 diagonal element |()| = () represents the mean value of potential energy. If we set this value to zero we can write the expression for perturbed energy levels as: ~2 |k|2 X + (k) ≈ 2 g6=0
|g |2 ¡ 2 ¢ ~2 |k| − |k − g|2 2
(135)
This representation makes sense for any with the exception for the ones that satisfy the degeneracy condition (525) (0) (k) = (0) (k − g)
(136)
|k| = |k − g|
(137)
|k|2 = |k|2 −2k · g + |g|2
(138)
i.e., using the (111): This condition is equivalent to
and consequently to k·
|g| g = |g| 2
(139)
For a better understanding of this equation in terms of Bragg reflection read the subsection Elastic Scattering - Bragg law. This condition in one dimension is satisfied by = ± = ± (). In this case the perturbation theory
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for degenerate energy levels (see "Approximate methods") must be applied first, using only two basis wave functions as 1 1 ± = √ (140) ∓ √ − 2 2 ¡ ¢ ~2 2 so splits into two The unperturbed degenerate level (0) (± ) = 2 non-degenerate energy levels
¯ ~2 ³ ´2 ¯¯ ¯ = ± ¯ 2 ¯ 2 ±
between which a forbidden energy gap is generated ¯ ¯ ¯ ¯ = 2 ¯ 2 ¯
¯ ¯2 If we examine the probability densities ¯± ¯ we can find that ¶ µ ¯ + ¯2 2 ¯ ¯ = sin2 ( ) µ ¶ ¯ − ¯2 2 ¯ ¯ = cos2 ( )
(141)
(142)
(143) (144)
¯ − ¯2 ¯ ¯ has its maximum values where = (we can find the minima of the potential in correspondence with the ions: lattice points), while ¯ + ¯periodic ¯ ¯2 has its maximum values at interstitial positions between ions, i.e. at ¡ ¢ + 12 (maxima of the periodic potential). This explains the difference of energy between the two different perturbed states. Since the degeneration has been removed it is now possible to use the general formula for non-degenerate energy levels for all wave vectors in the first Brillouin zone, including the ones at the zone boundaries, to further improve the accuracy of the method. 3.5.2
Fermi surface and density of states
The Fermi surface is the set of surfaces characterized by equations = (k) for each partially occupied zone . The bigger is the periodic potential, the stronger is the deviation of the shape of this surface from that of the free electron sphere. The density of states as a function of energy for a given branch is derived by integrating the density of states 2 (2)3 in space k over the infinitesimal volume bounded by the two surfaces with equation = (k) and + = (k) and dividing the result by :
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Solid State Physics Lecture Notes
Z 2 () = k = () = (2)3 Z Z 2 = ⊥ = 3 3 4 |∇k | (2)
47
(145)
where is the infinitesimal element of the first surface while ⊥ represents the local distance between the two surfaces. We used the following formula: = |∇k | ⊥ . Thus the total density of states is: Z X X (146) () = 3 () = 4 |∇k |
If we plot the density of states as a function of energy, where the mean velocity of a Bloch state k = (∇k ~) is zero (this is a critical point in the first Brillouin zone) we find discontinuities in () called Van Hove singularities (the virtual discontinuities in () are smoothed out by integration). Since (−k) =(k), as stated by the Kramers theorem (symmetry − ), Γ [(k = 0)] is always a critical point. In lattices such as f.c.c., because of symmetry reasons, also [k =(2)(100)] and [k =(2)(12 12 12)] are critical points. Segment Γ is called Λ while segment Γ is called ∆. For a given branch , close to a critical point it is possible to write = (k) as a quadratic form; using the directions of the local principal axes we get: =0 + 2 + 2 + 2
(147)
The generic critical point is indicated with where represents the number of negative coefficients in the above expression. As an example we consider the density of states near 0 in an isotropic situation: = = = 0. Using polar coordinates |k| we obtain: Z p |k|2 sin = 2 32 −0 (148) () = 3 4 |k|= − 2 |k| 4 0 Using the effective mass ∗ approximation (see below) in this case we would get = ~2 2∗ . Similar considerations may be applied to the study of phonon dispersion relations (see below) in order to obtain their density of states in terms of frequency.
3.6
Band structure of tightly bound electrons
While in the interstitial space between two lattice ions, where the periodic potential is nearly constant, the Bloch wave functions are very similar to
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plane waves, close to the positions where lattice ions are located Bloch waves resemble localized atomic orbitals. In other words we assume that in the vicinity of each lattice point in position n the periodic potential can be approximated by the atomic potential of a single ion located in the same position n (r − n). In order to describe electronic bands in crystals with strongly bounded electrons we try to develop a Bloch state as an appropriate sum, over all positions in the crystal, of atomic orbitals (r − n) centered on ions themselves ( is the set of quantum numbers defining one of the possible stationary state for an electron in the mean field of central forces of a single atom). Thus satisfies the stationary Schroedinger equation (we put = 0): ∙ ¸ ~2 2 − ∇ +0 (r) (r)= (r) (149) 2 Then a trial Bloch function of the whole crystal can be written as: 1 X k·n (r − n) (150) k (r) = √ n
Computing k (r + l), being l too a lattice translation like n, and multiplying the right side of the equation by k·l −k·l we get : 1 X k·(n−l) (r − (n − l)) = k (r + l) = k·l √ n 1 X k·h (r − h) = k·l k (r) = k·l √ h Since both l and n − l = h are lattice translations, we have just verified that (150) satisfies Bloch’s theorem. Neglecting superposition integrals, function (150) satisfies the normalization condition, i.e.: Z
∗ k (r) k (r)r
Z 1 X X k·(n0 −n) 0 = 1+ ∗ (r − n) (r − n )r ≈1 n 0 n 6=n
The expectation value of energy in state k will be then: ∙ ¸ Z ~2 2 ∗ (k) = k (r) − ∇ +(r) k (r)r 2
(151)
(152)
where (r) is the exact periodic potential. Introducing ∆(r) = (r)−0 (r), a function with maximum at r = 0 (we set lattice point n as origin of coordinates), and again neglecting some superposition integrals, we get the final result:
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Solid State Physics Lecture Notes
(k) =
X
Eh k·h
49
(153)
h
with:
E0 ≈ Z Eh6=0 ≈ ∗ (r + h)∆ (r) (r)r
(154)
Equation (153) is the exact Fourier representation of a periodic function of k (see below Wannier theorem); here we use for it the approximate expression (154) of Fourier coefficients valid within the tight binding approximation. Because of the nature of both ∆ (r) and (r), we expect Eh to be very small, with the exception of first neighbors. In one dimension ( = ), considering that E± is negative due to the attractive nature of the periodic potential with respect to the vacuum level, it is possible to write: () ≈ − 2 |E± | cos()
(155)
So we have a band with a minimum min = − 2 |E± | at = 0 and a maximum max = + 2 |E± | at the zone boundary edges where = ±: therefore the bandwidth is 4 |E± |. In case of a strong bond the hopping probability from one site to the next (proportional to Eh ) is small and the band is quite flat (localized atomic level). E(k)
ak
"tight binding" band The black curve represents (), the red curve the atomic level, the green curve the parabolic approximation. We have set min = 0. For small values of we can approximate the band as: ~2 2 ()≈ ∗ 2
(156)
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Comparing with (155) we obtain for the effective mass: ∗ =
~2 22 |E± |
(157)
So, electrons in bands of strongly localized states have a big effective mass. Remembering the discrete allowed wavevectors of the first Brillouin zone, in a crystal (a polystable system with ions - i.e. potential wells) the original atomic level splits in a band of levels () as a result of the above hopping events (transitions) between different wells. Jumps are promoted by ∆ , acting as a perturbation operator in (154) which resemble a matrix element of a transition in perturbation theory (see Appendices). 3.6.1
LCAO Method
A "tight binding" energy band built with a given atomic state can exhibit a partial energy superposition with another one built with a different state (corresponding to a different atomic energy level ). Along a given direction this generally happens when |k| is greater than a threshold value in the first zone. In this case there is a loss of significance in the concept of distinct band and band (take the example of band 4 and band 3 in transition elements as shown, e.g., in Ziman, Theory of Solids (2-nd ed.) pag. 113). This inconvenient can be eliminated using a linear combination of all (or almost all) different atomic orbitals (and not only one) corrisponding approximately to the same atomic energy ( = within the same atomic schell, for example the valence shell 22 of Carbon, containing four valence electrons and eight possible atomic orbitals, some filled and some empty) and so obtaining a generalization of the above trial tight binding wave functions, that is the LCAO Bloch functions: 1 X X k·n (r) = √ (r − n) k n
The variational coefficients can be determined minimizing ∙ ¸ Z ~2 2 ∗ (k) = k (r) − (r)r ∇ +(r) k 2
(158)
(159)
In interstitial regions atomic states (r − n) and Bloch states are very different one another; this happens because of energy barriers between pairs of ions which appear in the periodic potential and which alter the potential wells produced by isolated ions. Besides bound atomic states are an incomplete set for the description of a Bloch state with energy greater than periodic
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potential maxima (non tunneling states), where the energy spectrum is basically continuous (see unbound states with positive energy in isolated atoms). In the figure below the function ∆ is shown together with two different energy levels greater and smaller than periodic potential maxima. Thus it was necessary to look for better approximate methods based on basis functions closer to exact Bloch functions, but the description of such methods is beyond the scope of these lecture notes.
∆() and periodic potential
3.7
Electron dynamics in external fields
Being a consequence of translational invariance, equation (153) has a general validity and gives us the possibility to prove a theorem which is very crucial in the study of electron dynamics, when the crystal is plunged in an external field. The statement of Wannier theorem is contained in the following identity (−∇)k (r) = (k)k (r)
(160)
b = (−∇) is the operator obtained from function (k) in which where the following substitution is made: k → − ∇
(161)
We prove the theorem dropping subscript and considering only one band in a one dimensional situation where the lattice positions are given by = . b = (−∇), built starting from (k) expressed as We apply operator the Fourier series (153), to Bloch wave function k () and perform just a few steps:
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(−
Solid State Physics Lecture Notes
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X )k () = E k () = (162) X 2 1 E (1 + + ()2 2 + )k () = = 2 X X E k (+) = E k () = ()k () =
c.v.d. In the last step we used Bloch’s theorem. 3.7.1
Equivalent Hamiltonian. Effective mass theorem
We consider now the motion of an electron in an external field (for example an electric field); let us assume that the electron initially occupies a Bloch state in a partially occupied band. Let (r) be the extra potential energy (to be added to the periodic potential (r)) gained by the electron plunged in the external field. The wave function of the electron obeys Schroedinger equation: b0 + (r) ~ = (163) where 2 b0 = − ~ ∇2 + (r) (164) 2 and the isolated crystal problem b0 k (r) = (k)k (r)
(165)
has already been solved. We look for a solution of (163) as a wave packet of Bloch states with time dependent coefficients: X k ()k (r) (166) (r ) = k
where the sum is over an appropriately limited (with respect to zone boundary) interval of values around a specific mean k. Since the set of allowed k is almost continuous in a macroscopic crystal, it is even possible to write: Z (r ) = k ()k (r)k (167)
This dynamic description of an accelerated wave packet is substantially a particle description (within the entire crystal volume). In fact in this way the
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uncertainty in the instantaneous position of the electron is greater than the lattice parameter but less than the minimum linear size of the crystal volume (yet, see below Limitations in the effective mass description). Introducing (167) in (163) we get: Z b 0 k (r)k + (r) (168) ~ = k ()
b0 and the Wannier theorem, it is possible Using the eigenvalue equation of to write: Z ~ = k ()(−∇)k (r)k + (r) = (169) Z = (−∇) k ()k (r)k + (r) = = (−∇) + (r)
i.e.:
= [(−∇) + (r)] We can now define the equivalent Hamiltonian operator as: ~
b = (−∇) + (r)
(170)
(171)
It is equivalent to the former Hamiltonian operator: 2
b = − ~ ∇2 + (r) + (r) 2
(172)
Formally (−∇) can be viewed as the kinetic energy operator associated with a particle immersed in an external field (r). Operator (−∇) contains both the inertial properties of the particle and the effect of the periodic potential. The result in (170), together with eqs. (174) and (176), is called effective mass theorem for a Bloch-waves packet. If we assume that k correspond to the minimum of a partially occupied band (it could be the bottom (0) of the conduction band in a semiconductor), in 1 it is possible to write: ~2 2 ()≈ (0) + (173) 2∗ where ~2 ´ (174) ∗ = ³ 2 () 2
0
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So the corresponding equivalent Hamiltonian is: 2 b = − ~ ∇2 + (0) + (r) 2∗
(175)
If we count energy from (0), we get:
2 b = − ~ ∇2 + (r) 2∗
(176)
This Hamiltonian leads the motion of an electron which in the absence of an external force F = −∇ (r) (177) would be free but with a mass different from the one in vacuum: it is just the effective mass ∗ , a concept we have already introduced in qualitative terms. In this way the periodic potential has been included in the equivalent 2 ~2 kinetic energy operator − 2 ∗∇ . If we, e.g., carefully observe the trend of LCAO bands around the top of the valence band and around the bottom of the conduction band in the first Brillouin zone and if, in particular, we apply the previous procedure to the top of the valence band, we will notice that in this case electrons have a negative effective mass, whose absolute value is generally different from that of electrons at the bottom of conduction band. E 8
6
4
2 0 -2.5
-1.25
0
1.25
2.5 k
The implied negative kinetic energy is difficult to conceive, unless we remove this oddness reversing the energy axis, counting energy downwards from the top of the valence band. This seems not to make any sense for valence band electrons, but will be useful, and physically sound, for holes.
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Impurity levels
If a tetravalent semiconductor such as is doped with substitutional atoms with five valence electrons such as , the exceeding electron can be bound near the donating atom in a localized state belonging to an energy level within the forbidden energy gap just below the conduction band. Due to thermal agitation this electron may gain enough energy to occupy a new (propagating) Bloch state in the conduction band and potentially become a charge carrier under the effect of an external field; in this case the initial localized state has been thermally ionized. In a situation like this one the semiconductor is said to be doped and the arsenic atom is called a donor. The energy levels of this impurity can be described as follows. The exceeding electron coming from atom , in a crystal, is plunged in an effective electric field generated by the positive ion acquiring an additional Coulomb-like potential energy: 2 (r) = − (178) 4 where is the distance between the electron and the ion and is the electric permittivity of . The wave function of the exceeding electron obeys the effective Schroedinger equation: µ ¶ ~2 2 2 − ∗∇ − = (179) 2 4 The energy eigenvalues are the hydrogen-like levels: = = −
∗ 4 82 2 2
(180)
These levels lie just below the conduction band. In fact in ∗ = 10 and = 10 0 and the ionization energy will be 1 ≈ 10−3 rydberg (1 rydberg 4 = 8 2 2 ≈ 136 eV) which is an amount of energy lower than at room 0 temperature (25 × 10−3 eV). This means that doping is an effective way to increase the electron density in the conduction band of . Since the Bohr 2 ∗ radius in ∗ = ∗ 2 is a factor ( 0 ) = 100 greater than the radius of hydrogen atom, the effective mass theorem can be used in this situation (the radius is greater than the lattice constant). 3.7.3
Semiclassical dynamics in a crystal
If (r) depends weekly on position, it is possible that the mean de Broglie wavelength = 2 of electrons, conceived as Bloch waves packets, is short enough not to produce strong electron-wave diffraction. In this case we cope
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with a semiclassical situation and it isn’t necessary to solve equation (170), but we can instead start with an equivalent classical Hamiltonian. Here we follow the most straightforward path to obtain the result. Let us begin with the original classical Hamiltonian: |p|2 = + (r) + (r) 2
(181)
using the Jordan quantization rules (first quantization) r→ r p→ −
~∇
(182) (183)
we obtain the Hamiltonian operator 2
b = − ~ ∇2 + (r) + (r) 2
(184)
Applying the Wannier theorem to the band structure = (k)
(185)
k → − ∇
(186)
and using the substitution we get the equivalent Hamiltonian operator: b = (−∇) + (r)
(187)
In a semiclassical situation it is possible to build the equivalent classical Hamiltonian by reversing the quantization rules (quantization dismounting) r→ r
(188)
p −∇ → ~
(189)
and so obtaining = Now, using Hamilton’s equations
³p´ ~
+ (r)
r = p p = − r
(190)
(191)
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we get ¡ ¢ p~ r = p (r) p = − r
(192)
i.e.: r 1 (k) = ~ k (r) k = − ~ r
(193) (194)
Differentiating the first equation with respect to time and substituting in it k from the second equation we obtain the Newton’s law: µ ¶ (r) 1 2 (k) 2 r − (195) = 2 2 ~ kk r This defines the tensor: µ
1 ∗ (k)
¶
=
1 2 (k) ~2
(196)
In 1 the effective mass is thus defined as: ∗ () =
~2
(197)
2 () 2
and it is a function of v,m
2
1
0 -2.5
-1.25
0
1.25
2.5 k
-1
-2
Speed and effective mass vs.
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Analyzing the figure we can understand some interesting applications. Imagine that an electron in a LCAO conduction band at a time = 0 has = 0. The electron is then accelerated by a constant electric field E = −u () ( 0). The electron feels the force F = − u = u . Integrating the (194) we get: 1 = (198) } As we can see the wave vector grows linearly with . In the reduced zone scheme, when = , a Bragg reflection happens; instantaneously the 0 wave vector becomes = − = − 2 = − and then it starts growing linearly again till the new reflection at the edge of the Brillouin zone occurs. In the figure are shown in black the trend of the mean velocity of the electron and his effective mass as functions of (red curves refer to the valence band). At first ( = 0 and = 0) the mass is constant and positive and the velocity grows linearly. At = (2) the mass becomes infinite, initially positive then negative, and in a while it stabilizes on a negative value; the velocity, instead, has already gained its maximum so it decreases until its value becomes zero at the edge of the Brillouin zone. These anomalies in the Bloch waves packet show that this description of the electron motion isn’t completely classical. After the Bragg reflection everything starts all over again: thus a constant force would produce a periodic motion! In the real situation (see Transport properties) due to scattering off phonons and lattice imperfections the electron wave vector has always limited to small values close to = 0 being cutt off by scattering processes and the periodic motion can be hardly observed experimentally. 3.7.4
Limitations in the effective mass description
Under too either intense or fast-varying fields, interband transitions become possible (tunneling under the energy barrier of hight ) and the concept of effective mass cannot be applied any more, at least in clear rigorous terms (see Optical properties). It is possible to define a upper circular frequency limit as: ~ and a upper limit on the applied electric field as: ≤
(199)
(200) where is the lattice parameter. Actually these conditions are neither restrictive enough nor unique. In a device the particle description (wave packet) |E| ≤
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needs the charge carrier localization ∆ to be appropriate with respect to the overall linear dimension: ∆ should be should be small enough so that the charge carriers could be considered as particles with respect to the size of the device itself, but, at the same time, big enough to have a quick reaction ∆ to external stimuli. The uncertainty ∆ of wave vector of a Bloch waves packet corresponds to an energy uncertainty of thermal origin: ~2 (∆)2 = 2∗ Using the Heisenberg’s uncertainty principle ∆ =
~∆∆ ≥ ∆∆ ≥
(201)
(202) (203)
we get ∆ ≥
∆ ≥ √ ∗ 2
(204) (205)
At 300 with = 1 (199) gives us a bandwidth of 2.4 x 1014 Hz while (204) gives 6.3 x 1012 Hz. At 300 (205) gives r ∆ ≥ 76 (206) ∗ In a device with an effective mass of 0.067 the smallest dimension should be about 29.4 . 3.7.5
Electric current - Electrons and holes
We want now to deepen former considerations considering a semiconductor with a completely occupied valence band and a completely empty conduction band.
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If at = 0 we apply force F, the response of each electron in the valence band can be treated as in the previous chapter. The trend of dynamic variables is shown in the first column of the following picture.
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Solid State Physics Lecture Notes
At any instant in time the sum of the wave vectors is X = 0
61
(207)
and the total electric current is proportional to (which we are going to call current for the sake of simplicity): =
X X 1 k =0 (−) h i = (−) ~ k
(208)
We have so obtained the crucial result that a completely occupied band cannot conduct current. Let us assume now that, initially, a single electron occupies the conduction band, thus leaving an empty state in the valence band.
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Solid State Physics Lecture Notes
At instant 1 we have, in the conduction band, X = 5 0
62
(209)
and the current is µ ¶ X X 1 k 1 k = = (−) h5 i = (−) (−) h i = (−) 6= 0 ~ k ~ k 5 (210) While, in the valence band, X = −5 = 3 0 (211)
with a current equal to µ ¶ X X 1 k 1 k = (−) h3 i = (−) (−) h i = (−) = = ~ k ~ k 3 (212)
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Therefore the total current is = + . Considering a subsequent instant = 2 it is possible to notice that the sum of wave vectors associated to all occupied states in the valence band decreases (it increases as the electric field increases) and that the total energy increases. The energy of the empty state decreases while the wave vector and the energy associated to the electron in the conduction band increase. The set of electrons in the valence band behaves as a single positive particle (with a wave vector opposite to the one of the empty state) whose energy increases, if we reverse the energy axes. This particle has a positive effective mass, at least in correspondence with small wave vectors, while electrons would have a negative effective mass. The picture below represents the two possible representations: the electron representation is shown in the left column while in the right one the electron-hole representation is depicted.
Basically this is the concept of hole in a dynamic band representation. Since it is a positive particle, the hole could also be considered as spatially localized describing it as an appropriate Bloch waves packet in the electron-hole representation. The electron-hole representation is a quite general concept that can be applied to all systems of Fermions in the framework of second quantization (see e.g. R.D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2-nd ed. Dover 1992). In doped with trivalent impurities (acceptors), such as , in order
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to complete tetrahedral bonding the substitutional atom captures a valence electron from , thus becoming a negative ion. Thereby in valence band remains an empty state that could be described by a hole. This hole is attracted by the impurity. As long as it is in a localized state, the hole could be described as stated in section Impurity levels and its energy levels are situated in the gap just above the valence band. If then the hole is thermally ionized and submitted to an external electric field, it can carry charge and contribute to the total current, as already shown for electrons. In this case we have a p-type semiconductor. 3.7.6
Excitons
In an intrinsic semiconductor each electron-hole pairs could also be generated by the absorption of a photon with energy greater than (see Optical properties). The couple is in a bound metastable state whose energy levels can be calculated using the same method presented in section Impurity levels introducing the reduced mass of the system, a sort of hydrogen atom where the nucleus is the hole (Wannier exciton). In optical absorption experiments peaks related to transitions between exciton levels can be measured in the vicinity of the absorption edge.
4
Adiabatic theorem and vibrational motions
Modelling a crystal as a set of independent valence electrons moving in a mean perfectly periodic potential generated by the interaction with a rigid lattice of ions does not explain many important physical facts. Among these: the temperature dependence of specific heat, thermal expansion, melting (thermodynamic properties), the temperature dependence of electrical resistivity of conductors and of thermal conductivity of insulators, low temperature superconductivity (transport properties), optic absorption of ionic crystals in the infrared, Brillouin and Raman scattering of light (optic properties). In this section we discuss an indispensable approximation for a feasible theory of all the above phenomena: the Born-Oppenheimer approximation or adiabatic theorem (principle). Before starting the treatment, we warn the reader that this approximation does not explain other important physical facts, like superconductivity, which require an "ad hoc" approach. In the previous sections we assumed that each independent valence electron in a solid moved in a mean periodic field caused by the lattice of ions and the other electrons (the symmetry of this field is the same as that of the space group of the considered crystal). Now we want to consider both the thermal
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motion of ions about the official atomic points (at a given temperature) and the motion of valence electrons in the crystal volume; the instantaneous position of a generic ion with charge is the sum of a vector of the Bravais lattice l and a displacement ul () (for the sake of simplicity, for the moment we consider a simple crystal, a crystal with a basis will be considered later). The dynamic variables of this quantum many-body problem are the displacements ul and the instantaneous positions r of all valence electrons. Thus the crystal Hamiltonian is: b = b + b + (r) + (r u) + (u) H
(213)
where r is the set of all electronic coordinates and u is the set of all ionic coordinates. ~2 X 2 ∇r (214) b = − 2 is the total kinetic energy operator of electrons; ~2 X 2 b = − ∇ul 2 l
(215)
is the total kinetic energy operator of ions;
2 1X (r) = 2 6= 40 |r − r |
(216)
is the repulsive Coulomb potential energy of electrons; (r u) = −
XX
l
2 40 |r − l − ul |
(217)
is the attractive Coulomb potential energy of ions and electrons; (u) =
1X 2 2 2 lh6=l 40 |l + ul − h − uh |
(218)
is the repulsive Coulomb potential energy of ions. First step. Since the electron mass is much smaller than the mass of an ion (if we consider a single proton, we have: = 1836 ) as a first approximation we can assume that the electrons wavefunction (r|u) has only a parametric dependence on displacements u. In particular, considering stationary states () (r|u) exp(− }), the (r|u) can be thought as the eigenfunctions of the reduced Hamiltonian (in which the term b has been neglected) :
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Solid State Physics Lecture Notes
b0 = b + (r) + (r u) H
66
(219)
b0 (r|u) = () (u) (r|u) H
(220)
b=H b0 + b + (u) H
(221)
()
Electron energy levels (u) have a parametric dependence on the instantaneous positions of ions l + u. The wavefunctions (r|u) are many-electron functions (states). However it is possible to solve problem (220) using a mean field approximation (for example Hartree’s approximation) and obtain a set of single electron states (independent particles or quasi-particles, see above). Setting u = 0 in the single-electron mean-field Schrödinger equation, the total mean potential energy h (r)i + (r 0) exhibit the translational invariance of the Bravais lattive and it is possible to use Bloch’s theorem to calculate monoelectronic states (band structure). As a second step, we look b now for eigenfunctions of the full Hamiltonian H b HΨ(r u) = Ψ(r u)
where we use the superposition principles of states as: X Ψ(r u) = () (u) (r|u)
(222)
(223)
Multiplying the eigenvalue equation (222) by ∗ (r|u) and integrating over the electronic coordinates r ( (r|u) form a set of orthonormal functions), we get: h i () () b + () (u) + (u) () (224) (u) = (u)
where we have neglected several non-adiabatic terms which are listed below. In this approximation for each electronic stationary state (r|u) we find a spectrum of vibrational states () (u) corresponding to vibrational energy () () levels . In equation (224) the approximate total electronic energy (u) has the role of that part of potential energy which depends on ions motion. In order to obtain (224) we neglected the following terms: ¶ ´ µZ ~2 X ³ () ∗ − 2∇u (u) · (r|u)∇u (r|u)r + (225) 2 µZ ¶ ~2 X () ∗ 2 (u) − (r|u)∇u (r|u)r 2
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Solid State Physics Lecture Notes
The matrix indexed by and Z ∗ (r|u)∇u (r|u)r exhibits diagonal elements equal to zero, since Z Z ∗ (r|u)∇u (r|u)r = ∇u ∗ (r|u) (r|u)r
67
(226)
(227)
is the gradient of the (constant) total number of electrons with respect to ion displacements. Diagonal elements of the other matrix: Z −~2 (228) ∗ (r|u)∇2u (r|u)r 2 are multiples of the total kinetic energy of electrons by a factor and, thus, negligible. In fact in the worst case, i.e. in case of very strong electronion interaction, it is possible to write (r|u) = (r − u) and diagonal elements become: Z
5 5.1
−~2 2 ∇u (r − u)r = − u) 2
Z
−~2 2 ∇ (r − u)r 2 r (229) Generally, instead, all non-diagonal elements are different from zero thus possibly leading to transitions between electronic states while ions are moving (electron-phonon interaction). ∗ (r
∗ (r − u)
Lattice dynamics Lattice specific heat: Einstein model
In solid insulators and at high temperatures the Dulong and Petit’s law is valid. This law states that the molar specific heat is constant and equal to 3 where = ' 831 −1 −1 is the ideal gas constant, is the Avogadro constant and the Boltzmann constant. In contrast with what classical statistical mechanics states (equipartition theorem), experimentally tends to decrease towards zero at low temperatures. Einstein gave a first explanation of this behaviour developing a quantum oversimplified model of vibrational motion of nuclei in crystals. In this model Einstein assumes that every nucleus performs harmonic oscillations about its equilibrium position; all atoms oscillate with the same frequency . In light
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of modern quantum lattice dynamics, this assumption provides satisfactory results only at high temperatures, when the motions of nuclei can be considered as independent. However Einstein model explains in a qualitative way the dependence of on temperature when → 0. Each nucleus is considered a quantum harmonic oscillator with oscillation frequency equal to ; the spectrum of energy levels is then: µ ¶ 1 = + (230) ~ 2
Neglecting the zero-point energy, with a simple derivation we can obtain the mean energy associated to each oscillator: ~
=
~
= ~
(231)
−1 it is the energy associated to a single quantum of vibrational energy ~ multiplied by the oscillator occupation number given by the Bose-Einstein distribution.
=
1
(232)
~
−1 Now it is possible to conclude that the internal energy of the crystal (we are considering one mole) is: = 3 (233)
Thus is equal to: =
µ
¶
~
3 (~ )2 = µ ~ ¶2 2 −1
(234)
showing a temperature trend qualitatively similar to the experimentally observed one: lim ( ) = 0
(235)
lim ( ) = 3
(236)
→0
→∞
Around the Einstein temperature = ~ the curve exhibits a very sudden exponential change in the slope. Examining more carefully the experimental results we can notice that, when , tends to 0 more slowly than as it is stated by the exponential decay in the Einstein model. The low
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temperature empirical trend is ∝ 3 . In addition, at high temperatures the experimental is not exactly constant but it slowly rises as the temperature increases. This last effect would require the inclusion of anharmonic corrections (see below) and, being almost negligible, will not be considered further. In order to improve the agreement between theory and experiment at low temperatures, it is necessary to leave behind the Einstein model and to develop a more realistic model, able to better describe the true lattice dynamics. In particular at low temperatures motions of two near nuclei are generally correlated and this lowers the intensity of internal forces within the couple.
5.2
Lattice dynamics: Born - Von Karman model
On the basis of the adiabatic principle a full quantum treatment of lattice dynamics could be built. An alternative approach, phenomenological lattice dynamics, can be started with an initially classical description. The reason for the "quasi-classical" vibrational behaviour of nuclei in many circumstances is related to their very short thermal De Broglie wavelength. Let us consider a particle of mass accomplishing an oscillatory motion. The quantum mechanical version of virial theorem states that: 2 h i = hi where is the kinetic energy and a potential energy of the type ∝ . Thus for a harmonic oscillator h i = h i like in classical mechanics and h2 i h + i = 2 h i. Then h i can be estimated as 2 = 32 and an average De Broglie wavelenth of thermal origin as = √ 2 = √3 . Inserting h i
˚ which is the mass of the proton at room temperature one finds ≈ 145 shorter than the typical lattice spacing. This is even more true for heavier nuclei. According to adiabatic approximation we assume the existence of a vibrational potential energy which is a function of displacements of nuclei with respect to their official positions in the crystal when considered with no thermal motions (from now on we will use the Born and Huang notation) µ ¶ = Rn + r = r (237) n = 1 a1 + 2 a2 + 3 a3 + r µ ¶ is the position, with respect to the origin 0, of nucleus with r n mass situated in the primitive cell (which contains atoms) identified by lattice point Rn . r is the position of nucleus with respect to vector Rn (internal origin of primitive cell n). In the crystal, ideally with no
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thermal motions, each nucleus is located exactly in its official position and the potential energy attains its minimum value. We introduce now the displacements of atomic nuclei: µ ¶ µ ¶ µ ¶ 0 (238) =r −r u n n n 0
r indicates the instantaneous position of that nucleus ( n ), which fluctuates due to vibrational motions. If thermal motions slightly perturb the periodic structure of the crystal (in thermodynamic equilibrium this means that crystal temperature is much less than melting temperature) a series expansion of potential energy is feasible and can be arrested to the second order (the minimum value is put equal to zero): 1
13
1 XXX = Φ 0 2 0 0 0 nn
µ
0
0 n n
¶
µ
n
¶
0
µ
where we introduced the interatomic force constants tensor ∙ µ µ 0 ¶ 0 ¶¸ = Φ 0 Φ 0 0 n n n n defined as: Φ 0 0
µ
0
0 n n
¶
=
µ
2 ¶ µ 0 ¶ 0 0 n n
0
0 n
¶
(239)
(240)
(241)
The double sum nn is to be extended to all crystal cells for both n and 0 () n . Physically is the energy (u) + (u) = (u) which appears in eq. (224), including here the general situation of crystal with a basis, and () it is related to the electronic state (u) (for example the ground state) being a perturbation of it. This approximation is called harmonic approximation. Higher order terms in the series expansion of potential energy, the anharmonic terms, should be introduced in order to explain melting, thermal expansion and thermal conductivity. Harmonic approximation explains essential aspects of lattice vibrations-photon scattering (Brillouin and Raman scattering), the dependence of specific heat on temperature, infrared optical properties of ionic crystals, insulators and semiconductors and electrical resistivity in conductors (electron-lattice vibrations scattering). Remembering that a quantum treatment will be finally necessary, we start developing first a classical dynamics description. The vibrational kinetic energy of the
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crystal can be then written as: ¯ µ ¶¯2 1 ¯ 1 XX ¯¯ = ¯¯u˙ n ¯ 2 n
(242)
Introducing the Lagrangian = − , the Lagrange equations
µ ¶− µ ¶ =0 u˙ u n n
(243)
give birth to a system of classical (Newtonian) dynamic equations, a system µ ¶ of 3 equations in the 3 variables n u ¨
µ
n
¶
=−
1 XX 0
n
Φ
0
µ
0
0 n n
¶
:u
µ
0
0 n
¶
(244)
where : is a rows by columns product of the square matrix corresponding to 0 tensor Φ = [Φ 0 ] multiplied by vector u =[ 0 ] ( = 1 2 3). F
µ
n
¶
=−
1 XX 0
n
0
Φ
µ
0
0 n n
¶
:u
µ
0
0 n
¶
(245)
it is the total elastic force, due to displacements of all the other nuclei, applied to nucleus situated in cell n. Consequently it is clear the operational physical meaning of µ µ 0 ¶ 0 ¶ ˜ 0 ≡ −Φ 0 0 0 n n n n which is the component of the force applied on nucleus in cell caused by 0 0 0 a unitary displacement along direction of nucleus in cell .The number of independent components in tensor Φ = [Φ 0 ] depends on the point symmetry of the crystal. If tensor Π =[Π 0 ] represents a symmetry operation of the point group of the crystal (for example a rotation of order around an axis which passes through a fixed point) then (due to the transformation law of 0 tensors after a change in coordinates given by Π : r = Π : r) it must be 0
Φ = ΠΦΠ−1 = Φ
(246)
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where Π−1 is the inverse of matrix Π (Π is a unitary matrix: det Π = 1, and Π = Π−1 ). Applying a relation like that to any operation in the group we obtain some constraints determining the independent components of Φ. Now, due to the translational symmetry of the crystal it is possible to write 0 n = n + h, where h ≡ Th is a lattice translation (in the same way in this paragraph we will write Rn as n). So we can write: µ ¶ µ 0 ¶ µ 0 ¶ 0 Φ =Φ (247) =Φ 0 n n+h h n n 0
Notice that Φ depends only on the difference n − n = h. We can eventually write the dynamic equations as u ¨
µ
n
¶
=−
1 XX h
0
Φ
µ
0
h
¶
:u
µ
0
n+h
¶
(248)
Where the sum over h involves all the lattice translations which join cell n with the other ones (including h = 0). Applying periodic boundary conditions, as in the case of electrons, and thus imposing that µ ¶ µ ¶ (249) u =u n + ( − 1)a n ( = 1 2 3; = 1 2 3 total number of cells) the dynamic equations (248) maintain a translational invariance, although they refer to a finite number of nuclei. The fundamental solutions (lattice waves) should then satisfy Bloch theorem in its discrete formulation, id est: µ ¶ µ ¶ u =u q·n (250) n 0 To every lattice wave at least one wave vector q is associated such that the previous equation is satisfied. As in the case of electrons (k vectors), the set of q vectors is that composed by the vectors contained in the first Brillouin zone. Bloch theorem helps us to reduce the set of 3 dynamic equations to a set of 3 equations which describes the vibrations of the nuclei contained in cell 0. Dropping subscript 0 and considering explicitly the dependence on q, the new reduced set of equations becomes à ! 1 ³ 0 ´ X µ 0 ¶ X q·h : u q u ¨ ( q) = − Φ (251) h 0
h
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We introduce now the dynamic matrix 0 ¶ µ 0 ¶ XΦ h = q·h D √ q 0 h
µ
(252)
as the discrete Fourier Transform of interatomic force constant tensor. It is then convenient to write system (251) as: µ 0 ¶ 1 ³ 0 ´ √ Xp u : u q ¨ ( q) = − 0 D (253) q 0
We now look for elementary harmonic solutions as: 1 (q)e ( q) −(q) u ( q) = p
(254)
where we introduced the polarization unit vector e ( q) and the scalar complex amplitude (q). Substituting this type of solution into the dynamic equations we obtain the homogeneous system of linear algebraic equations (eigenvalue problem): µ 0 ¶ ¾ ³ 13 1 ½ ´ X X 0 2 0 − (q) 0 0 0 q = 0 (255) q 0 0
from which it is clear that, for any q, the squares of the natural frequencies are the eigenvalues of the dynamic matrix while the polarization unit vectors are its eigenvectors.The eigenvalues are computed solving the equation (solution condition of the above system): ½ µ 0 ¶ ¾ 2 − (q) 0 0 = 0 det 0 q For any value of vector q (there are independent q vectors in the first Brillouin zone, as just remembered) the equation has 3 countable solutions 2 (q) = 2 (q) characterized by a branch index . The dynamic matrix is self-adjoint: thus we have real eigenvalues 2 (q) and a full set of orthonormal eigenvectors: 1 13 X X
X
³ ´ 0 = 0 ∗ ( q ) q
³ 0 ´ ∗ ( q ) 0 q = 0 0
(256) (257)
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Solution functions = (q) (only positive roots have physical significance as vibrational frequencies) are called dispersion relations. Considering vibrational motions of nuclei in a primitive cell we cope with 3 degrees of freedom. Among them 3 refer to the center of mass of the basis (acoustic branches), while the other 3 −3 refer to internal motion of the basis (optical branches). The term acoustic comes from the fact that, when q → 0, acoustic motions (if coherently, and not thermally, excited) coincide with the macroscopic elastic waves characterized by (q) = (q|q|)q, where (q|q|), which depends only on the direction of the propagation q|q|, is the speed of sound and corresponds to a quasi-longitudinal wave (1 = ) or to two different quasi-transverse waves (2 = 1 , 3 = 2 ). These three speeds of sound are generally different one another. The term quasi indicates that the polarization of these waves is mainly longitudinal or mainly transverse. Along particular symmetry directions and in specific crystals (the cubic ones, as an example) these waves are completely longitudinal or completely transverse. The term optical derives from the fact that the corresponding modes can have a fluctuating electric dipole determining the optical infrared absorption in the crystal or can have associated a fluctuation of electrical polarizability determining, in the crystal, the vibrational Raman scattering of photons (in the infrared, visible and ultraviolet spectrum). If the crystal is a Bravais lattice with a basis limited to a single atom per primitive cell, only acoustic modes can exist. In silicon (face centered cubic lattice with 2 atoms per primitive cell: diamond structure) in addition to the 3 acoustic branches there are 3 × 2 − 3 = 3 optical branches: one of them is longitudinal, the other two are transverse. Using again Bloch theorem it is now possible to express the most general solution of the dynamic equations as the following linear combination of all lattice waves:
u
µ
n
¶
13 X X 1 =p (q)e ( q ) q·n − (q) q∈BZ
(258)
Introducing the 3 normal coordinates (q) = (q)− (q) ((q) are the amplitudes of normal coordinates) it is possible to write for the real displacements: ! Ã µ ¶ 13 X X 1 1 p = u (q )e ( q ) q·n + n 2 q∈BZ
Substituting these solutions in the vibrational Hamiltonian
c °2017-2018 Carlo E. Bottani
=
1 XX n
Solid State Physics Lecture Notes
µ ¶ u˙ u˙ n
µ
n
¶
75
−
and, finally, using the properties of polarization unit vectors and the fact that ions displacements (although they have been written so far as complex quantities) components are real quantities, we get the most important issue in the harmonic approximation: 13 ª 1 X X© | (q )|2 + 2 (q) |(q )|2 = 2 q∈BZ
where (q ) =
˙ ) (q
(259)
(260)
are the kinetic momenta conjugated to normal coordinates (q ). We have thus proved that the vibrational behaviour of the crystal is equivalent to that of a set of 3 independent harmonic oscillators with frequencies 2 (); there is a one to one correspondence between oscillators and lattice waves (normal modes) (q ). Collective motions (q ) are harmonic motions, while the motion of a single nucleus is not. For this reason the (q ) are called collective coordinates and their energy quanta } () behave as Bose particles, or quasi-particles (see below); as a matter of fact they are not associated to individual particles (which would be individually localized in the limit of uncertainty principle). The introduction of normal coordinates decouples the system of harmonic dynamic equation describing the motion of nuclei and produces 3 independent elementary equations: ¨(q ) + 2 (q)(q ) = 0
(261)
which can be obtained from using the Hamilton equations: ˙ ) = (q (q) − = ˙ (q ) (q)
(262) (263)
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5.3
Solid State Physics Lecture Notes
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Linear chain with two atoms per cell: acoustic modes and optical modes
Let us consider a 1 crystal with lattice parameter and a basis with two ions of mass 1 and 2 (1 2 ) located in the official (frozen) positions µ ¶ 1 = µ µ ¶ ¶ 1 2 = + 2 and take only into account the interactions between them and with their first neighbours. It is not necessary for the second ion to be located in the middle of the cell. However we made this choice in order to use a unique interatomic force constant . Considering the interaction between ions inside the cell = 0 and with ions in the adjacent cells = ±1, we find nonzero force constants which determine the dynamics of ions in the cell = 0 (we drop subscripts 11 of Φ) ¶ ¶ µ ¶ µ µ 21 12 11 = −; = −; Φ = 2; Φ Φ 0 0 0 ¶ ¶ µ ¶ µ µ 12 21 22 = − = −; Φ = 2; Φ Φ −1 1 0 by means of which we can build the dynamic matrix : ! µ 0 ¶ à 2 − √ − (1 + ) 1 1 2 = 2 − √1 2 (1 + + ) 2 The eigenvalues of this matrix are the solutions of the following equation (solubility conditions of system(255)): µ ¶µ ¶ ³ ´ 2 2 42 2 2 − − − cos2 =0 1 2 1 2 2 i.e.: 212 ()
= ∓
where =
r ³ ´ ³´ sin2 1−4 2
1 2 and = 1 + 2
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are, respectively, the reduced mass and the total mass (mass of the center of mass) of the basis. When → 0, we asymptotically have r || 1 ≈ 2 r µ ¶ 2 2 2 2 ≈ 1− (264) 8 while near the edges of the first Brillouin zone (where = ±) we get r 2 1 = r 1 2 2 = (265) 2 The first branch of eigenvalues has an acoustic nature (see next section) while the second one an optical (see Optical properties below) nature. Moreover between 1 () and 2 () pthere is a gap of prohibited frequencies. Frequencies greater than 2 (0) = 2 are not possible. 1.5 1.25 y 1 0.75 0.5 0.25 0 -2.5
-1.25
0
1.25
2.5
x
Dispersion relations in the linear biatomic chain (y=
p
)
In figure the dispersion relations in the first Brillouin zone ( = ≤ ) with 1 = 22 are shown; the minimum forbidden frequency interval in correspondence with the edges of the zone is clearly visible.
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5.4
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Simple 1D crystal: acoustic phonons and elastic waves
This case is even simpler than the previous one: the linear atomic chain is a 1D simple crystal with only one lattice ion (nucleus) of mass per primitive cell of size (the 1D Bravais lattice positions are = ). We still consider only first neighbors interactions of strength . The problem is so simple that we can solve it directly using Newton’s law for the motion of a generic nucleus interacting with nuclei − 1 and + 1. In this way we get the system of ( = 0 1 2 − 1) coupled equations: ¨ = −2 + −1 + +1
(266)
Since we cope with a periodic lattice (1D crystal), Bloch theorem (in its discrete form) can be used: (267) = 0 in this way we get a single equation which describes the motion of the nucleus located in the first cell ( = 0): µ ¶ µ ¶ ³´ ³´ 2 2 − ¨0 = − 0 + 0 0 = − (1 − cos())0 + (268) We now look for a harmonic solution such as − 0 () = √ and we get the solubility condition: ³ ´ ³´ 2 2 sin =4 2
i.e.
r ¯ ³ ´¯ ¯ ¯ =2 ¯sin ¯ 2
(269)
(270)
(271)
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1
y 0.75
0.5
0.25
0 -2.5
-1.25
0
1.25
2.5
p x Dispersion relation y= 2
In figure the dispersion relation in the first Brillouin zone (− = ≤ ) is shown. In the limit → 0 r () ≈ || (272) while
r (± ) = 2
(273)
is the crystal cutoff frequency for acoustic modes. When → 0, = (2) and the lattice wave feels the crystal as a continuous elastic medium. This directly follows from the dynamic equation rewritten as ¶ µ ³´ −2 + −1 + +1 (274) ¨ = 3 2 which, for long wavelengths, can be well approximated by the wave equation
2 2 = 2 2
(275)
where we have introduced = 3 (mass density), = (Young’s modulus) and ( ) (continuous field of elastic displacements). The solutions of this last equation are elastic propagating waves ( ) = (−) with speed of sound = =
s
(276)
(277)
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and linear dispersion relation () = ||
(278)
In the discrete (atomic) case, comparing the limits of the dispersion relations when → 0, we notice that r (279) =
In other words in a crystal it is possible to identify the acoustic modes with long wavelength with the elastic waves in the same crystal considered as a continuous medium (see below Debye model).
5.5
Phonons
Referring to the situation of the previous case, we now introduce the most general solutions of dynamic equations as a linear combination of all lattice waves: X 1 () −() () = √ ∈BZ
(280)
The N discrete allowed wavevectors are those imposed by the periodic boundary conditions −1 = 0 (281) for we can write: − +1 2 2
() =
X
√ − 2( )
(282)
We define: = −
(283)
as the normal coordinates and r ¯ µ ¶¯ ¯¯ ¯¯ sin = 2 ¯ ¯
(284)
as the corresponding proper frequencies. We can now write the vibrational Lagrangian as: 0 −1 0−1 1 X 1 X 2 = ˙ − (+1 − )2 2 2
(285)
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and the Hamiltonian as =
0 −1 0−1 1 X 1 X 2 + (+1 − )2 2 2 2
where =
= ˙ ˙
(286)
(287)
are the conjugated momenta. Substituting (282) in the Lagrangian and in the Hamiltonian, after some lenghty computation we omit here (imposing () to be real), we can finally get: ¾ X ½¯¯ ¯¯2 2 2 ˙ ¯ ¯ − | |
− +1 2 2
1 = 2
1 = 2 where
(288)
− +1 2 2
X © 2 ª | | + 2 | |2
(289)
=
˙
(290)
are the canonical conjugated momenta. As in the general 3D case, the periodic monoatomic linear chain can be considered as a set of N independent harmonic oscillators, each oscillator being represented by a normal coordinate. Introducing now the Hamiltonian operator ˆ = 1 2 where
¯ ¯2 ¾ X ½¯¯ ¯¯2 ¯ ¯ 2 ¯ˆ ¯ + ¯ˆ ¯
− +1 2 2
(291)
ˆ = −~
(292)
it is immediately possible to switch to a quantum description of lattice vibrations: the crystal is, from a vibrational point of view, equivalent to a system of quantum independent harmonic oscillators. In the 3D general case each oscillator with normal coordinate (q ) owns a frequency (q ) and a polarization unit vector e(q ) and can assume the discrete energy levels ¶ µ 1 (q ) = (q ) + ~(q ) (293) 2
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where (q ) = 0 1 2 3 The total vibrational energy is thus equal to: ¶ 13 µ X X 1 (q ) + ~(q ) = 2 q∈BZ
(294)
(q ) represents the quantum number -(phonon population number) associated to the mode (normal coordinate) (q ). The set of 3 numbers (q ) completely determines the vibrational quantum state of the crystal at zero temperature. In thermal equilibrium at temperature , the vibrational contribution to the internal energy of the crystal is: ¶ 13 µ X X 1 h(q )i + ~(q ) = 2 q∈BZ where h(q )i =
1 ~ (q)
(295)
(296)
−1 is the Bose-Einstein distribution function. Thus, within the harmonic approximation, the vibrational properties of the crystal is described by an ideal Bose-Einstein gas of phonons. Sometimes the term phonon is used to indicate the collective coordinates (and the normal modes) (q ) and not only their quantum vibrational energy ~(q ). Also in interactions with electrons and photons (emission, absorption and scattering events), the quantized lattice vibrations behave in the same way as particles associated to collective coordinates characterized by a crystal momentum ~q (in the reduced zone scheme or in the repeated zone scheme) and by an energy ~(q ) (which is a periodic function of q in the repeated zone scheme); lattice waves can be distinguished in acoustic and optical phonons. When, in an interaction process, only phonons with wave vectors situated in the first Brillouin zone are involved, the process is called N process (normal), otherwise, if some wavevectors exceed the the first Brillouin zone, we have a U process (umklapp).
5.6
Specific heat: Debye model
As in the case of electronic states, in a macroscopic crystal the phonons wave vectors (where coincides with the total number of primitive cells) quasi-continuously occupy the first Brillouin zone in reciprocal space. This suggests to approximate this quasi-continuous filling with a truly continuous one and to introduce the concept of density of vibrational normal modes (if
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instead we considered a nanometric crystal, this continuous approximation could not be used). In general in a crystal of volume containing primitive cells, the elementary volume of reciprocal space associated to a single wave vector is equal to: 83 8 3 8 3 = = |a1 · a2 × a3 |
(297)
as turns out from periodic boundary conditions. Furthermore if we consider a monoatomic simple crystal, to any wave vector there correspond three normal acoustic modes (one longitudinal and two transverse). For any polarization (phonon branch) the above quantity may be defined density of normal modes: it is constant in reciprocal space but, in general, it is frequency dependent in frequency space. To get this frequency dependent density, we start with the quantity () which measures how many normal modes with a given polarization have frequency within 0 and . Fixing the dispersion relation = (q) defines a surface in reciprocal space and this surface encloses a volume () such that: () (298) () = ¡ 83 ¢
Now the density of modes as a function of can be computed as the derivative of (): () (299) () = In 1 the constant frequency surface degenerates into two discrete points and the volume () into a linear segment () bounded by these points. For example, in the monoatomic 1 chain treated above, for a given the equation r ¯ ³ ´¯ ¯ ¯ =2 (300) ¯sin ¯ 2
has the two roots
⎛ ⎞ µ ¶ 1 2 arcsin ⎝ q ⎠ ± () = ± 2 1
(301)
⎞ ⎛ µ ¶ 1 4 arcsin ⎝ q ⎠ () = 2+ () = 2 1
(302)
which defines a segment of length:
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thus we have:
⎞
⎛
1 () 2 arcsin ⎝ q ⎠ () = ¡ 2 ¢ = 2 1
84
(303)
to which there corresponds the 1D density: ⎞⎤ ⎡ ⎛ 1 ⎣ 2 1 arcsin ⎝ q ⎠⎦ = q q () = 2 1 1 1−
(304) 1 2 4
In one dimension the density of acoustic p modes diverges in correspondence of the maximum possible frequency 2 at zone boundary, where the phonon group velocity is zero, while it is nearly constant for small wave vectors, where the group velocity is constant and equal to the speed of sound. In view of the difficulties of generalization of this procedure when applied to a complex 3 crystal whose lattice dynamics is governed by realistic atomic force constants, Peter Debye introduced an approximate model for the derivation of () considering the crystal as an isotropic elastic continuous body. In this way only non dispersive acoustic modes (elastic waves) do exist. The discrete nature of the real finite crystal is lumped into a maximum allowed frequency, called the Debye frequency . In a continuous and isotropic elastic solid body (see Appendix Elasticity and Elastic Waves) a dynamic deformation process can generally be described as the superposition of a simple dilatation (compression) field and by a simple shear strain field. This corresponds to a total displacement field u u = u + u
(305)
∇ × u = 0 ∇ · u = 0
(306) (307)
where
Displacement fields u and u obey these two decoupled wave equations: 2 u = 2 ∇2 u 2 2 u = 2 ∇2 u 2
(308) (309)
where =
s
+ 43
(310)
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is the longitudinal speed of sound and =
r
(311)
is the transverse speed of sound. In previous equations is the mass density, is the bulk modulus and is the shear modulus. These two elastic constants can be expressed by a function of the couple of technical constants (Young’s modulus) and (Poisson’s ratio) using the following formulae: 3(1 − 2) = 2(1 + )
(312)
=
(313)
For thermodynamic stability reasons it is always . In case of a simple uniaxial deformation in the direction of axis : = and = = − . is the axial stress while , , are the three diagonal components of the strain tensor. Moreover it is possible to write: + + =
+ + =
(314)
We call = − 13 ( + + ) the hydrostatic pressure and we notice that = −
(315)
If we consider, for example, the longitudinal acoustic branch, the dispersion relation is: q = |q| = 2 + 2 + 2 (316)
and, fixing , the (316) represents in space q the equation of a spherical surface with radius . Then it is: 4 () = 3
() =
4 3
() =
³ ´3
¡ 83 ¢
µ
µ ¶3
3 6 2 3
= ¶
(317)
3 6 2 3
=
2 22 3
(318) (319)
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i.e. we have a parabolic density of modes. Repeating all this for the two transverse branches (degenerate in the continuous and isotropic elastic mean) we get the total density of modes: 2 () = 2 3 2
(320)
1 2 1 = 3+ 3 3
(321)
where
The Debye frequency is then obtained from condition: Z () = 3
(322)
0
as:
µ
¶ = 18 With the previous formula it is possible to write: 3
2 3
() =
92 3
(323)
(324)
Now we can generalize the specific heat expression in the Einstein model (see above). We consider the Einstein formula still valid for the single mode with frequency but now we sum up over all the modes using () as a weighting function: =
µ
¶
=
Z
0
2
~
(~) µ ~ ¶2 () 2 −1
(325)
This expression for the specific heat fits rather well most experimental data. Every solid is characterized by its Debye temperature: Θ =
~
(326)
Setting: ~ (327) we can rewrite the expression of the specific heat as (assuming = ): µ ¶3 Z Θ 4 = 3 3 (328) Θ ( − 1)2 0 =
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At low temperatures (Θ → ∞) the integral tends to a constant and the specific heat tends to zero as 3 . At high temperatures Θ tends to zero, thus the integral can be approximated as: µ ¶3 Z Θ Z Θ 4 Θ 2 3 ≈ 3 = (329) ( − 1)2 0 0 and the specific heat obeys the classical Dulong and Petit law: ≈ 3 = 3
(330)
Thus a crystal behaves in a classical way when Θ and in a quantum way if Θ . For example diamond (one of the two allotropes of carbon in the solid state) with Θ = 1860 and silicon with Θ = 625 both exhibit a quantum thermodynamic behaviour at room temperature (300 ).
List of Debye temperatures
The figure below shows the universal trend of Debye specific heat (328) as a function of the reduced temperature = Θ :
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1
Cv/3R
0.75
0.5
0.25
0 0
0.5
1
1.5
2 x
Debye specific heat The success of the Debye model finds a simple physical justification in the behaviour of Bose-Einstein factor as a function of frequency at low tempera−
~ (q)
tures. As a matter of fact when → 0 we have that (q) ≈ : normal modes with a lower frequency has a higher occupational factor. These modes are just the acoustic modes about the center of the Brillouin zone. For the contribution of these very modes the Debye model provides a very good approximation. Simple crystals have only acoustic modes, while more complicated complex crystals need a better theoretical description including the optical modes.
5.7
Polaritons
The denomination optical modes originates from the fact that these modes determine the linear optical properties of polar (ionic) crystals in the infrared. These crystals are non monoelemental crystals in which the chemical bond is, at least partially, of ionic character. The extreme case is constituted by alkali metals halides (for example + − ) in which the interatomic forces have a purely electrostatic nature and the electron states are localized on individual ions. However also the cases of III-V compounds, such as and II-VI compounds, such as (zincblende), should be considered. In these crystals, where the bonds are mixed covalent/ionic, the electrons eigenstates are Bloch waves and the usual band structure picture holds. All these structures share the absence of a center of symmetry and, as a consequence, in every primitive cell they exhibit an electric dipole moment (we have a total electric dipole of ions forming the crystal basis). A characteristic crystallografic structure is zincblende: its Bravais lattice is f.c.c. (as in the case of − or Si); the basis is formed by two atoms located in (0,0,0) and in (1/4,1/4,1/4), all along the principal diagonal of the conventional cubic
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cell, in units of (cubic lattice parameter). We now imagine that in (0,0,0) the positive ion is situated, while in (1/4,1/4,1/4) the negative one. Considering the propagation of electromagnetic waves along direction (111), we can approximate the lattice dynamic as the one of a linear chain with two atoms per cell (see above), as long as the branch 2 () of the dispersion relations is concerned. We study the forced response of the chain under the action of the (local) electric field of an electromagnetic wave E = E0 (−)
(331)
which propagates in the same direction (111). The corresponding 3 problem is very difficult to be analyzed since we should consider both the longitudinal and transverse polarizations, for both the field of the wave in the crystal (within matter also longitudinal electromagnetic waves may exist) and for the field of atomic displacements. Using a simplified model, regardless of the polarization we can write directly the equations of the forced motion of ions as ( is the absolute value of an ion charge) ¶ µ µ ¶ µ ¶ µ ¶ 2 2 1 1 + + + 1 ¨ = −2 −1
2 ¨
µ
2
¶
1
1
2
2
((
+
)−)
+0 µ ¶ µ ¶ µ ¶ 2 1 1 = −2 + + + +1 ((
−0
+
)−)
(332)
(333)
Assuming that the wavelength = 2 is much greater than , we apply Bloch theorem to previous equations (as already done for the chain with only one atom per cell) in the hypothesis that = (this could be justified by quantum theory). In this way it is possible to considerably simplify the equations of motion: 2 (1 − 2 ) + 1 2 = + (1 − 2 ) − 2
¨1 = − ¨2
0 − 1 0 − 2
(334)
Subtracting the second equation from the first, and introducing the reduced mass , we get: 2 0 − 2 (1 − 2 ) + (1 − 2 ) = 2
(335)
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Introducing now the normal coordinate = 1 − 2 and the characteristic frequency r 2 = = 2 ( = 0) (336)
we obtain:
0 − 2 + 2 = 2
The forced solution is: =
0 − − 2
2
(337)
(338)
Thus the associated electric dipole moment is: = =
2 0 − 2 − 2
(339)
and the polarizability of the single cell (of the basis) = (0 ) may be written as: 1 2 = (340) 0 2 − 2
Neglecting local-field effects, if there are cells per unit volume, the dielectric susceptibility of the crystal can be written as: = =
1 2 0 2 − 2
Consequently the dielectric constant becomes: µ ¶ 2 = 0 (1 + ) = 0 1 + 2 − 2 where we introduced the plasma frequency of lattice ions: s 2 = 0
(341)
(342)
(343)
q Note that when the frequency is equal to = 2 + 2 we have () = 0. Moreover it is (Lyddane-Sachs-Teller relation): (0) 2 = 2 (∞)
(344)
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¶ µ 2 (0) = (∞) 1 + 2
91
(345)
epsilon rel. 3.75
2.5
1.25
0 0
0.5
1
1.5
2
2.5
omega/omegaT -1.25
Relative dielectric constant of GaAs As shown in the picture (black curve), in the frequency interval ∆ between and the dielectric constant is negative. Since the speed of propagation of the electromagnetic wave is: r 0 = ≈ (346) where is the refractive index, we have that in the interval ∆ the electromagnetic wave cannot propagate in the crystal. An e.m. wave like that would be totally reflected from the surface of the crystal (Restrahlen effect). Thus, independently from periodicity effects, in a ionic crystal there is an energy gap prohibited to photons ∆ = ~∆. If in the model we include dissipative interactions, the trend of the dielectric function (the real part) is the one without divergences shown in the picture (red curve: absorption and dispersion in infrared). The second figure shows both the real and the imaginary part of the refractive index, including dissipation (cfr eq. 347).
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2.5
2
1.5
1
0.5
0 0
0.5
1
1.5
2
omega/omrgaT
Complex index of refraction (GaAs) In order to describe the absorption (attenuation of the electromagnetic wave) we introduce a complex wave vector. Thus the dispersion relation of electromagnetic waves in the crystal can by described as: r 0 = = = r³ (347) ´ 2 1 + 2 −2
Solving with respect to we get the complex wave vector: sµ ¶ 2 1+ 2 = − 2 ck
(348)
5
3.75
2.5
1.25
0 0
0.5
1
1.5
2
omega/omegaT
Polariton dispersion relations In the last figure (relative to not including dissipation) the real part of the product of the wave vector and the speed of light in vacuum as a function of the frequency is shown in black. The imaginary part, corresponding to the attenuation of the electromagnetic wave, is shown in red. The group
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velocity can never exceed . The asymptotic phononic and photonic trends are evident. The mixed photon-optical phonons excitations introduced in this section are quantized (their energies are ~()) and they are called polaritons. In a more realistic 3 model, in which transverse optical phonons are distinguished from longitudinal optical phonons (and transverse e.m. wave are distinguished from longitudinal ones) we have = (0) and = (0). Moreover the e.m. transverse waves couple exclusively with transverse optical phonons and they can propagate outside the interval ∆. When = mixed longitudinal optical phonon/longitudinal e.m. wave, stationary modes exist. In the previous picture the relative dispersion relation would be represented as a vertical line. Generally and lie in the infrared (in = 51 × 1013 s−1 and = 55 × 1013 s−1 ). In a ionic crystal a moving electron carries with itself a cloud of optical phonons which changes its mass (the mass increases): this quasi-particle is called polaron. Also in non-polar crystals, such as silicon, optical modes interact with photons, but this interaction causes nonlinear optical properties (Raman scattering) and the effect is evident also in the visible and in the ultraviolet. From a quantum point of view the description of scattering processes needs a second order correction in dynamic perturbation theory (see Appendix: Quantum approximate methods).
5.8
Inelastic scattering
Let us assume that the interaction potential in a crystal can be described as a sum of atomic potentials (for the sake of simplicity consider a simple crystal in which the atoms occupy the nodes of the Bravais lattice): X (r) = (r − rn ) n
As we already learned in the theory of scattering, the scattering amplitude is given by the following expression: Z Z X −Q·r (Q) ∼ (r) r ∼ −Q·r (r − rn )r = = =
Z X
X
n
−Q·(r−rn ) −Q·rn (r − rn )r =
−Q·rn
n
∼
n
(Q)
X n
Z
−Q·(r−rn ) (r − rn )r ∼
−Q·rn
(349)
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P Where n −Q·rn is the structure factor of the whole crystal, and the atomic shape factor is: Z −Q·R (R)r
(Q) =
When atoms occupy equilibrium positions (there are no lattice vibrations) rn = n =1 a1 + 2 a2 + 3 a3 it is
X n
−Q·n = (Q − g)
where g is a vector of the reciprocal lattice, and Bragg law for elastic diffraction is recovered. In fact for any vector m in the lattice: X X X −Q·n = −Q·(n+m) = −Q·m −Q·n (350) n
n
n
P
i.e. is either n −Q·n = 0 or, if Q = g (a vector of the reciprocal lattice), P it−Q·n = . n Otherwise, if atoms vibrate about their equilibrium positions, we can write their displacements as a superposition of all normal modes summing over index q. For the sake of simplicity we drop branch index ; moreover there is not any index since we are considering just one atom per cell: X rn = n+ (un (q) + u∗n (q)) (351) q
where we added the complex conjugate in order to have real displacements. The sum is confined in the first Brillouin zone and the sum over is implied. Using Bloch theorem we again write un (q) = u0 (q)q·n and u0 (q) = (q)e(q)−(q) where e(q) is the polarization vector of the normal mode q, and (q) √ is the amplitude of the corresponding normal coordinate, equal to (q) in the notation used in the description of lattice vibrations. Thus considering equations (351) and (349) we have: X X ∗ −Q·rn = −Q·(n+ q (un (q)+un (q))) = (Q) ∼ n
=
X n
−Q·n
Y q
n
−Q·(un (q)+u∗n (q))
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Considering only small lattice displacements ∗
−Q·(un (q)+un (q)) ≈ 1 − Q · (un (q) + u∗n (q)) − |Q · u0 (q)|2 + and just the linear terms, i.e.: X Y ∗ −Q·(un (q)+un (q)) ≈ 1 − Q · (un (q) + u∗n (q)) q
(352)
(353)
q
we get:
(Q) ∼ =
n
Ã
n
−Q·n −
X
X
−Q·n
1−
X
= (Q − g) − −
X q
X q
−Q·n
n
X q
Q · (un (q) + X q
!
u∗n (q))
Q · (u0 (q)q·n +u∗0 (q)−q·n ) =
(Q · e(q)) (q)−(q) (q)
(Q · e(q)) (q)
=
X
X −Q·(n−q) + n
−Q·(n+q)
n
where normal coordinates have real amplitudes. We consider now the Bravais lattice inversion symmetry, in this case the lattice sum over q is equal to the one over −q, and we have: X (Q) ∼ (Q − g) − (Q · e(q)) (q)−(q) (Q − q − g)+ −
X q
q
(Q · e(q)) (q)(q) (Q − q − g)
(354)
We now apply the Fermi’s golden rule (where is the transition probability per unit time), id est (Q) ∼ = 2 |h1|0 |2i|2 (2 − 1 ± ~), ~ in the form valid for interaction with a time harmonic potential 0 ± (in this case we have a term ±~ in a Dirac delta function which represents the conservation of energy). We consider as initial state |1i ∼ k ·r , and as final state h2| ∼ k ·r (initially free particles with a defined wavevector interacting with a crystal). InR this case the matrix element h1|0 |2i is equal to the scattering amplitude 0 (r)−Q·r r , where Q = k − k . Thus the first term in equation (354) gives us the elastic contribution in the expression of the intensity and is proportional to (Q − g)(2 − 1 ) | (Q)|2 . In this way we find the Bragg law, with the condition that scattered particles and incident particles share the same energy. The second and the third term give us an inelastic contribution in the expression of intensity, corresponding to a first
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P order inelastic scattering, and proportional to | (Q)|2 q |Q · e(q)|2 2 (q) (Q − q − g)(2 − 1 ± ~(q)). This means that there is an active energy exchange mechanism between the particle (a neutron, for example) or between a photon of the electromagnetic field (light or X rays, for example) and the quanta of vibrational energy. A vibrational quantum with energy ~(q) can be created or destructed. Factor (Q − q − g) imposes the relation: Q = k − k = q + g (355) If the equality is satisfied when g = 0, id est Q is in the first Brillouin zone of the reciprocal lattice, we have a Normal or N process, otherwise, if Q is outside the first Brillouin zone, id est equation (355) is satisfied when g 6= 0, we have a Umklapp or U process. In the first situation equation (355) can be seen as a principle of conservation of quasi-momentum (exchange between the crystalline momentum ~q of a phonon and the momentum of the particle): ~Q = ~k − ~k = ~q In the second situation the additional term ~g corresponds to a term of variation of the momentum associated to the motion of the center of mass of the entire crystal. The Umklapp process can be considered as the creation (or destruction) of a phonon associated to a Bragg reflection. In a quantum description of lattice vibrations, the mean value (it is a thermal mean) of h 2 (q)i depends on the population of the vibrational mode with wave vector q, described by the Bose-Einstein distribution. Actually, if we considered the quadratic terms in the expansion (352), both the elastic term and the inelastic one in the expression of intensity should be multiplied by a factor of: Y 2 (1 − |Q · u0 (q)|2 ) = − q |Q·u0 (q)| = −2 q
where −2 is the Debye-Waller factor, where = =
1X 1X |Q · u0 (q)|2 = |Q · e(q)|2 2 (q) = 2 q 2 q 2 (q) 1X |Q · e(q)|2 2 q
√ In the last expression we wrote the normal coordinate (q) with the notation used in the description of lattice vibrations ((q) shouldn’t be confused with the absolute value of Q = k − k , which is a scattering wave vector). When lattice vibrations are considered, the intensity of Bragg peaks
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(and the one of inelastic peaks) is thus decreased due to a disorder which introduces a "noise" in the crystal translational symmetry. In a D quantum E 2 (q) description of lattice vibration, the mean value (thermal mean) of ¡ ¢ ~ is equal to (q) + 12 , where (q) is the occupation number for bosons in the quantum harmonic oscillator corresponding to the mode q: (q) =
1 exp( ~(q) )
−1
The Debye-Waller factor depends on the temperature, and in particular is proportional to at high temperatures (above the Debye temperature), while it assumes a constant value, still different from zero (due to the vibrational zero-point energy), at low temperatures. If we decide to consider also the nonlinear terms, which contain products of more than one term such as Q · (un (q) + u∗n (q)), in the expression of intensity we would have also terms corresponding to second order scattering (with a smaller intensity with respect to the first order), id est corresponding to an exchange of energy and momentum with two or more phonons (multiple phonons processes). Space-time autocorrelation function: In a "static" crystal we have that: Z (Q) ∼ (Q) ∼ (R)−Q·R R which is the static structure factor, is equal to the spatial Fourier transform of the static autocorrelation function: Z (R)= (r + R) ∗ (r)r In a crystal or in a mean with vibrational motions we have that: Z (Q ) ∼ (Q ω) ∼ (R)−Q·R R which is the dynamic structure factor of the crystal, it is equal to the spacetime Fourier transform of the dynamic autocorrelation function (where the time integration is done on time intervals larger than characteristic times in vibrational motions, the integral corresponds to a thermal average operation):
(R) ∼
Z Z
∗
(r + R + ) (r )r ∼
¿Z
À (r + R) (r 0)r ∗
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Optical properties
Generally the linear optical properties in solids (dispersion, absorption, reflection, refraction) depend on macroscopic interaction processes between photons, electrons and phonons. In the figure below the qualitative trend of the absorption coefficient (358) in a doped polar semiconductor is shown where it is possible to see all possible microscopic mechanisms.
Absorption coefficient In this section we will analyze explicitly just the principal band and the continuous background (in the previous section we introduced the polaritonic absorption), however the principles already treated would give us the possibility of understanding from a quantum point of view all the peaks in the figure. To start we build a general connection between the macroscopic optical parameters (refractive index and extinction coefficient) and the transition probabilities between different quantum states in a solid excited by an electromagnetic radiation. The electric field of a monochromatic plane wave, with polarization unit vector e, which propagates in a solid along the axis is described, in the typical complex form, by expression:
E = 0 e(−) (356) n o ( −) Thus the effective electric field will be e Re 0 . Macroscopically the dispersion and absorption (extinction) processes can be described introducing a complex wave vector e = + and we can write: E = 0 e− (−)
(357)
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Id est, we have a wave attenuation. The intensity of the wave (averaged ³ n o´2 = 12 |E |2 , exhibits an over a period), proportional to Re 0 (−) exponential decay in space: = 0 −2 = 0 −Σ
(358)
Introducing the dispersion relation e e
= with complex refractive index: 0
(359)
00
e = +
we have
Σ=2
³ ´
(360)
”
(361) 0
”
We now the complex relative dielectric constant e = +√ = ¡ introduce 0¢ 1 + + ” ( is the complex dielectric susceptibility). Since e ≈ e , it is: ³ 0 ´2 ³ 00 ´2 0 0 = 1 + = − (362) ´ ³ ´ ³ ” 0 00 = ” = 2 Using the second equation in (362) we can thus write: Σ=
³ ´ ”
0
But, since the imaginary part in the refractive index is always less than the real part, using the first equation in(362) we get: Σ=
³ ´
”
p ' 1 + 0
³ ´
”
the approximation being valid especially about the absorption frequencies, 0 where ' 0. Now we want to underline the connection between the extinction coefficient and the microscopic transition probabilities (per unit time) induced by the coupling between the solid and the electromagnetic field of the wave. We consider now the electromagnetic energy density associated to the wave (averaged on a period):
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100
(363)
(364)
The conservation equation for the electromagnetic energy density in the stationary state is: =− (365) where is the power density absorbed by the solid; the same power is lost by the wave. So it is: = 0 (366) Since the intensity of the wave is proportional to , decreases in space as stated by the law (358) thus it is: ³ ´ ” 1 ³ 0 ´2 = − 0 Σ = − 0 2 0 |E |2 2
(367)
1 = 0 ” () |E |2 2
(368)
Using, again, the second equation in (362), we finally get:
Now, introducing the probability per unit volume and time of a single microscopic event which leads to the extinction of a photon to happen, we get: X 1 0 ” () |E |2 = ~ (369) 2 The quantity can be estimated using the Fermi’s golden rule (see Appendix: Quantum approximate methods). Solving with respect to ” () we get: µ ¶ 2~ X ” () = (370) 0 |E |2
6.1
Kramers e Kronig relations ”
Once the ” () = () has been calculated with (369), it is possible to 0 0 obtain () = () − 1 using an integral relation (and vice versa). It is
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a general equation which comes both from the linear relation between the polarization vector P() and the electric field () and from the principle of causality (linear response theory). In presence of delay effects (due to dissipation and dispersion), the most general linear relation between P() and () can be written as: Z ∞ P() = 0 ( )( − ) (371) 0
If (− ) = (− ) it is P() = 0 (); which gives () its own meaning. In particular () is equal to zero when 0 (principle of causality: effects cannot precede causes): so we integrate between 0 and ∞. Equation (371) implies also the time invariance (the hereditary integral is a convolution: id est ( ) = ( − )). If the polarizing field is harmonic: () = exp(−) we have µZ ∞ ¶ ( ) − (372) P() = 0 0
So P() = P exp(−) and we can write the important relation:
being
e() P = 0 e() =
Z
∞
( )
(373)
(374)
0
from which it results immediately that: e(−) = e∗ ()
(375)
More generally e() is a tensor e () which depends on the symmetry of the crystal. In cubic crystals it reduces to a scalar quantity. Considering the principle of causality, in equation (374) we could integrate between −∞ and +∞ thus concluding that e() is the Fourier transform of (). We can make now an analytic continuation of e() in the upper semiplane, introducing 0 0 ” the complex frequency = + ” . Since lim” −→∞ − = 0, e() is analytic in the upper semiplane. As a consequence (using Cauchy theorems): I e() = 0 (376) −
for any circuit in the upper semiplane which does not contain . Let us assume ≥ 0 on the real axis. We chose as a circuit = 1 ∪ 2 ∪ 3 ∪ 4 0 where 1 (− ≤ ≤ −; ” = 0), 2 ( = + = − 0), 3 (+ ≤
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0
≤ ; ” = 0) and 4 ( = = 0 − ). Solving the four line integrals in the limit −→ 0 and −→ ∞, we get from (376): +∞ Z −∞
0
e( ) 0 − e () = 0 0 −
(377)
where we should consider the integral as : Cauchy principal value: lim→0 We can rewrite the previous equation as: 1 e() =
+∞ Z
hR −
0
e( ) 0 0 −
−∞
(378)
Introducing again the real and the imaginary part of the susceptibility, we obtain the Kramers e Kronig relations: 1 () = 0
+∞ Z
0
” ( ) 0 0 −
−∞ +∞ Z
1 () = − ”
−∞
0
(379)
0
( ) 0 0 −
(380)
0
() is the Hilbert transform of ” () and vice versa. Since from (375) we 0 get that () is an even function, while ” () is an odd function, multiplying 0 the numerator and denominator by + , we can rewrite equations (379) as: 2 () = 0
+∞ Z 0
2 ” () = −
0
0
” ( ) 0 0 2 − 2 +∞ Z 0
0
(381)
0
( ) 0 02 2 −
(382)
From the first equation of (381) we obtain the static relative dielectric constant as: +∞ Z ” 0 ( ) 0 2 0 0 (383) (0) = 1 + (0) = 1 + 0 0
−∞
+
R +∞ i . +
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The (383) is called sum rule. As a direct application of (381), we assume that, in (369), ∝ ( − 0 ) (384) So from (370) it is ” () = ( − 0 ), where is a constant. Using the (383) we notice that = 0 [ (0) − 1] 2. Now, using the first equation in (381), we finally get: ¤ £0 20 (0) − 1 0 0 (385) () = 1 + () = 1 + 20 − 2 Tthis equation can be compared with the one obtained in the polariton theory.
6.2
Probability per unit time of optical transitions
When a crystal is perturbed by an electromagnetic wave whose electric field in vacuum is 1 E = e0 (k·r−) + (386) 2 in (370) the transition probability per unit time (and volume) may be evaluated by means of the Fermi’s golden rule (see Appendix: Quantum approximate Methods). If we consider only the first order absorption processes in the perturbation theory (the initial state is the ground state with single electron () energies ) the rule is: ´ 2 ¯¯ c ¯¯2 ³ () () = ¯h | |i¯ − − ~ ~
(387)
where the matrix elements of the time independent part of the perturbation operator can be written as: c|i = h |
0 h|k·r e·b p|i 2
(388)
b = −~∇ is the momentum operator and is the electron mass in where p vacuum. In order to explain the (388) we should remember that, using the vacuum gauge, the electromagnetic field can be derived using just the vector potential A (with divergence equal to zero: ∇ · A = 0) as: A B = ∇×A E = −
(389)
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In this situation the Hamilton function of a particle with charge and inertial momentum p =v, moving in an external electromagnetic field, can be written as: |p − A|2 (390) = 2 If the charge is also under the effect of an additional electrostatic field (due to other fixed charges) with potential (r), and thus it has a potential energy of (r) = (r), the total Hamilton function becomes: |p − A|2 + (r) (391) 2 Developing the square of the absolute value, and substituting to p the momentum operator, we obtain the quantum Hamiltonian operator as: =
b = b0 + b
where
(392)
|b p|2 b + (r) (393) 0 = 2 |A|2 b = − A·b p+ ≈ − A·b p (394) 2 In the last approximation we neglected the quadratic term, according to first order perturbation theory. This is consistent with the linear response theory we are developing, see (371). In this way we cannot take into account nonlinear optical phenomena. This treatment of e.m. radiation-matter interaction in which particles obey the Schroedinger nonrelativistic quantum mechanics and e.m. fields obey Maxwell equations is called semiclassical. This semiclassical treatment does not explain spontaneous emission and high energy relativistic processes. If the wave electric field has an expression such as (386) then, using (389), the expression of the vector potential becomes: 0 (k·r−) + e 2 So, if the charged particle is an electron with = −, it is: A=
(395)
c− + b = 0 k·r e·b p− + = (396) 2 We neglect the c.c., accounting for stimulated emission in which the initial state is an excited state whose energy is higher than that of the final state, and get the (387) and the (388) in the form: ³ ´ ¯2 2 2 |0 |2 ¯¯ () () k·r ¯ = h| e·b p|i (1 − ) − − ~ (397) ~ 4 2 2
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Using the (370) we get: ” () =
³ ´ ¯2 2 X ¯¯ k·r ¯ (1 − ) () − () − ~ (398) e·b p |i h | 0 2 2
where we introduced the probability that the initial state is occupied and the probability 1 − that the final state is unoccupied. Using the first of 0 0 the Kramers and Kronig equations we get () = 1 + (). Id est the unity plus a sum of other terms such as the one in the right side of equation (385): one term for each possible transition between the initial and the final state of each electron in the solid (assuming unitary volume).
6.3 6.3.1
Interband optical transitions Direct transitions
As an application of (398) we consider an intrinsic semiconductor at low temperature: initially the valence band is completely occupied and the conduction band completely unoccupied, then an e.m. radiation represented by the plane wave (386) hits the crystal. Transitions between occupied Bloch states in the valence band |i = k (r)k ·r
(399)
with energy (k ) and unoccupied Bloch states in conduction band | i = k (r)k ·r
(400)
with energy (k ) occur, generating electron-hole pairs. Considering the matrix elements in the (388) we notice that they are proportional to: Z ¡ ¢ k·r −k ·r k·r −~h| ∇|i = −~ ∗ ∇ k (r)k ·r r (401) k (r)
where the integral is over the whole crystal volume. Introducing the relative electronic coordinates r0 = r − n (where n is a generic node of the Bravais lattice) and integrating over the volume of a single primitive cell, we can write: Z ³ ´ 0 X 0 0 0 0 k·r −(k −k−k )·n −k ·r k·r k ·r0 ∗ (r ) ∇ (r ) r h | ∇|i = k k n
0
Using (350), we notice that the lattice sum
(402) −(k −k−k )·n is zero, unless: n
P
k − k = k + g
(403)
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where g is a vector of the reciprocal lattice. This conservation principle of the total wave vector in the crystal is a consequence of the translational symmetry. Considering just the first BZ (g = 0) and observing that the photon wave vector k is negligible with respect to the dimensions of the Brillouin zone (electric dipole approximation), the previous relation becomes k = k . So in the band representation (reduced zone scheme) the allowed transitions are only the vertical ones. Since each initial state is orthogonal to each final state we finally get: Z 0 0 0 k·r −~h | ∇|i ≈ ∗ (404) k (r ) (−~∇) k (r )r
0
Using the following relation: h|b p|i =
´ ³ (0) (0) − h|r|i ~
(405)
which comes from the time derivative of operator r: b r p 1 h b i = r0 = ~
we get: k·r
−~h|
∇|i ≈ (k )
where:
Z
(406)
0
0
0
0
0
∗ k (r )r k (r )r
(407)
[ (k ) − (k )] (408) ~ Here we can notice how the transition probability depends on the crystal point symmetry. In case of a crystal with a center of symmetry, a transition can only occur when the parity of the initial state is different from that of the final state. Now we can finally get a more compact form of the imaginary part of the dielectric susceptibility as a sum over all k0 in the first Brillouin zone: 2 X ” () = | (k)|2 [ (k) − (k) − ~] (409) 2 2 0 k (k ) =
where:
(k) = (k )
Z
0
0 ∗ k (r )
³ ´ 0 0 0 e · r k (r )r
(410)
Generally (k) can be considered as almost constant, so: 2 | |2 X () ≈ [ (k) − ~] 0 2 2 k ”
(411)
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where (k) = (k) − (k) is the vertical difference in energy between the conduction and the valence band. The latter equation can be rewritten as: 2 | |2 ~2 ” () ≈ (~) (412) 0 2 (~)2 where: (~) =
X k
[ (k) − ~]
(413)
is by definition the joined density of states. Considering that the density of states in the BZ is 2 (2)3 and considering a unitary volume, the joined density of states can also be written as: Z 2 (~) ≈ [ (k) − ~] k (414) (2)3 and, using (146):
Z 1 k (415) (~) ≈ 3 4 |∇k (k)| where k is the constant energy surface of equation (k) = ~. Equation (414) is equal to (415) due to the following property of the Dirac delta function: P Z ( ) [()] () = 0 (416) | ( )| where are the roots of equation () = 0. The assumption we made that both the band and are doubly degenerate because of the spin, is always valid for centrosymmetric crystals but it is not applicable to, e.g., zincblende structures. We have a critical point whenever ∇k (k) vanishes; the joined density of states exhibist then a Van Hove singularity (147). Assuming parabolic (and isotropic) bands around Γ we can write: ~2 |k|2 (k) = (k) − (k) ≈ + 2∗
(417)
where:
∗ (0)∗ (0) (418) ∗ (0)+∗ (0) is the reduced mass of the electrons and holes effective masses and = (0) − (0). Now, applying the methods learned in subsection Density of states, we get, in the vicinity of the fundamental absorption edge, the approximate form: ∗ =
52
(~) =
2
(∗ )32 p ~− 3
(419)
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. Band structure in Germanium 6.3.2
” of Germanium
Indirect transitions
In the previous subsection the absorption properties of a semiconductor with a direct gap such as , and all the II-VI compound are explained. But, when the maximum of the valence band and the minimum of the conduction band do not share the same k the related direct transition is not allowed not being vertical (see the figure below). In this case to find out the absorption threshold one must consider more complex transition paths involving phonons.
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Indirect interband transitions This is the case, for example, of , , . In silicon, e.g., the transition (Γ) → () is prohibited by the electric dipole selection rules. In presence of phonons an indirect transition can happen in two subsequent steps. These indirect processes are also important in semiconductors with a direct gap when vertical transitions are forbidden by the point symmetry (eq. 407), even though only in Γ, as it happens in 2 . Referring to the adiabatic theorem (eq. 220) the interaction (r u) can be written as the following lattice sum: X X (r u) ≈ (r − n) − un ·∇ (r − n) (420) n
n
where we introduced the screened pair potentials (r − n − un ). Developing un in normal coordinates (258) and considering, for the sake of simplicity, only one active normal mode, the electron-phonon perturbation operator can be written as: Ã ! X (q) b = − √ e (q) ·∇ (r − n)q·n − (q) + (421) 2 n
The term − (q) is responsible for the absorption (destruction) of a phonon, while the c.c. is responsible for the stimulated emission (creation) of a phonon. A deeper analysis of the matrix elements of operator (421) between initial and final states such as |() (u) (r|0) (see adiabatic theorem in order to understand the meaning of symbols) shows that the allowed transitions conserve the total wave vector (see also equation (355) and inelastic scattering theory): k − k = ±q + g (422)
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Normal events correspond to g = 0 and Umklapp events to g 6= 0. In the presence of the perturbation (421) and of the photon-electron perturbation b (396) the indirect transition can occur in two different ways. In the first one there is at first a direct vertical transition (Γ) → (Γ) and then, due to a creation or a destruction of a phonon with wave vector q(), the quasi horizontal transition (Γ) → () occurs. In the second one at first, due to a creation or destruction of a phonon with wave vector q(), the transition (Γ) → () takes place and then there is the vertical transition () → () caused by the absorption of a photon. The probability per unit time of these events can be calculated using the Fermi’s golden rule up to second order: ¯ à !¯2 X ¯¯ 2 X h | b |ih| b |i h | b |i ¯¯ b |ih| = + ¯ ¯ (423) ¯~ − − ~ − ± ~ (q) ¯
k k
× [ (k ) − (k ) − ~ ± ~ (q)]
Since the previous formula contains (q), the amplitude of the phonon normal coordinate, the probability of indirect transitions depends on temperature. (q) is a fluctuating random variable whose quadratic mean value can be calculated by means of the Bose-Einstein distribution. We can actually use the fact that the mean total energy of the oscillator is twice its mean potential energy: ® 2 (q) 2 (q) ∝
~ (q)
~ (q)
(424)
−1
In (423) the factor multiplying the delta function weakly depends on the wave vector. Moreover, since the second denominator is always greater than the first one, the second type of transition can be neglected. In this way, repeating considerations similar to those in the previous subsection, the imaginary part of for an indirect transition reduces to: Z Z 2 ®2 ” [ (k ) − (k ) − ~ ± ~(q)] k k (425) ∝ (q )
Assuming parabolic bands around stationary points we get, near the indirect band gap = () − (Γ), ” ∝ (~ ∓ ~(q) − )2
(426)
when ~ ≥ ± (q) and 0 otherwise. Neglecting the dispersion of phonon bands (as in the Einstein model), we set as mean frequency (q) = (q),
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the frequency corresponding to wave vector q = q . Because the (425) strongly depends on temperature it is possible to experimentally ascertain the existence of an indirect gap measuring the temperature dependence of absorption. The electron-hole pairs with different wave vectors created during indirect processes generally have a longer lifetime than that of the pairs with the same wave vector generated in direct processes.
6.4
Intraband transitions and plasmons
In metals and in extrinsic semiconductors, besides the interband transitions at high frequencies already treated in the previous subsection, there are quasifree charge carriers which populate partially occupied bands and cause the continuous background shown in the introductive figure on optical absorption. A quantum treatment of intraband indirect transitions (vertical intraband transitions do not exist) could proceed exactly as in the previous subsection just considering occupied and unoccupied states in the same band. The states associated to quasi-free electrons are perturbed by both electronphonon and photon-electron interactions. The model adopted to describe this situation is similar to that used in the theory of stationary transport phenomena (see below) in which just the electron-phonon interaction is considered (for the dynamic part) and the states are Bloch wave packets obeying semiclassical equations. Here we limit ourselves to some simple considerations in the free electrons approximation. We proceed in analogy with the coupling between photons and oscillating ions plasma described in subsection polaritons. We can apply the same theory to the jellium: a plasma constituted by free electrons and by a uniform background of positive ions. The main difference lies in the fact that here the resonance frequency corresponding to binding energy is zero: thus wepwrite = 0. Besides the plasma frequency is that of free electrons = 2 . The dielectric constant in the plasma then becomes: µ ¶ 2 = 0 (1 + ) = 0 1 − 2 (427) and the refractive index:
r
2 (428) 2 Consequently the dispersion relation of transverse e.m. waves becomes: sµ ¶ 2 = 1− 2 (429) =
1−
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2.5
2
1.5
1
0.5
0 0
0.5
1
1.5
2 omega
Jellium dispersion relation
A thorough analysis of Maxwell equations in jellium would show that transverse e.m. waves can propagate only if their frequency is above the plasma frequency (real , black line in the figure). Below the plasma frequency waves are damped (imaginary , red in the figure). Exactly at the plasma frequency ( ) = 0 stationary longitudinal waves, associated to a longitudinal oscillating electric field, are established in jellium (mixed collective non propagating modes). These stationary waves are quantized and their energy quanta ~ are called plasmons. In metals the plasma frequency occurs in the ultraviolet. Plasmons are the evidence of collective motions whose understanding requires going beyond the approximation of independent electrons used in normal band theory. The oversimplified model we have used here has classical bases (Drude-Lorentz theory). A quantum description of the free electron gas would lead to a wave vector depending dielectric function (spatial dispersion) as it is introduced in the theory of potential screening (217).
7 7.1
Transport phenomena Phenomenology
In a crystal containing mobile charges (conductor or semiconductor) the presence of an electric field E of external origin causes an irreversible charge transport phenomenon (electric current) removing the body from thermodynamic equilibrium. On the other hand, in all crystals a temperature gradient ∇ causes an irreversible heat flux called conduction. From a macroscopic
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point of view (adopted by the thermodynamics of irreversible processes) when the cause of the flux is small, we are within the linear response regime and two famous phenomenological laws, Ohm’s local law and Fourier’s law, well describe both transport processes: j = E
(430)
q = −k ∇
(431)
where j is the charge current density vector. Its flux through a surface represents the amount of charge per unit time, current , flowing through the surface: Z = j · n (432)
The principle of electric charge conservation can be written as the continuity equation for the charge density (charge per unit volume). + ∇ · j =0
(433)
In the Fourier’s law q is the heat current density vector. Its flux through a surface represents the amount of heat flowing through the surface per unit time: Z † = q · n (434)
q is strictly connected to the entropy current density vector q whose divergence appears in the balance equation for entropy density (entropy per unit volume) ³q ´ + ∇· = [] (435) [] is the entropy production (inherently positive). The latter equation shows, in a local form, the second principle of thermodynamics. For electrical conductivity we use the symbol while we use the symbol k for thermal conductivity. More precisely in crystals both conductivities are second order tensors and the two phenomenological laws should be generalized as: =
13 X
(436)
= −
13 X
(437)
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These new laws show that generally the current density vectors have not the same direction of the electric field and of the temperature gradient. As a matter of fact even more general kinetic coefficients exist: a temperature gradient can generate an electric current and an electric field can create a heat flux. The kinetic microscopic theory (see below) explains these correlated effects called thermoelectric effects (Seebeck and Peltier effects). Furthermore, the Onsager theorem, based on microscopic reversibility of dynamic laws, states that the two conductivities are symmetric tensors: = , = . Since every symmetric tensor, with an appropriate rotation of the coordinate system, can be reduced to the diagonal form: ⎞ ⎛ 1 0 0 σ = ⎝ 0 2 0 ⎠ (438) 0 0 3 it turns out that the number of independent components of both σ and k can be, at most, three. Due to crystal symmetry this number can be further decreased. Actually, let R be a symmetry operation and R the unitary matrix which transforms the atom coordinates (position vectors) according to the symmetry operation, then it should be: 0
σ = RσR −1 = σ
(439)
Repeating this consideration for each symmetry operation, it is shown that triclinic, monoclinic and orthorhombic crystals have three independent components: 1 6= 2 6= 3 ; tetragonal, trigonal and hexagonal crystals have two independent components: 1 = 2 6= 3 ; and cubic crystals only one: 1 = 2 = 3 (isotropy). The situation is analogous for thermal conductivity. As far as the temperature dependence is concerned, the electrical resistivity ( = 1) of metals shows a monotonically increasing trend. At low temperatures it increases as 0 + 5 . 0 is called residual resistivity depending on static lattice defects (vacancies/interstitials, dislocations, packing defects). At high temperatures the resistivity increases linearly: ∝ . Generally, as stated by Matthiessen law, the total resistivity is the sum of both an athermal (0 ) and a thermal contribution.
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Electric conductivity vs T The case of semiconductors is more complex since their resistivity depends also on the possible doping and its treatment needs special considerations. In metals the thermal conductivity at low temperatures linearly increases with , reaches a maximum value and then rapidly decreases until it reaches a constant value at high temperatures. In insulators, instead, at low temperatures the thermal conductivity increases as 3 , reaches a maximum value and then decreases as −1 at high temperatures. In metals, at very low temperatures, the ratio between electrical and thermal conductivity is substantially proportional to temperature and does not depend on the particular metal (Wiedemann-Franz law).
Metals
Insulators
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Kinetic theory
Since transport coefficients refer to irreversible processes in non-equilibrium conditions, for a theoretical treatment we cannot exploit the traditional bridge connecting the microscopic (quantum) dynamic description to equilibrium macroscopic properties (functions of state): the equilibrium statistical mechanics. As long as electrons are concerned, for equilibrium situations major information is given by the Fermi-Dirac distribution (116). In the present non-equilibrium case we should use, instead, the Boltzmann kinetic theory adapted to electrons in the semiclassical dynamic scheme. At first we start with a completely classical description of a gas with non interacting electrons. In the phase space (r p) of one electron we can introduce the single particle distribution function: (r p)rp = representing the number of electrons whose instantaneous state is within a volume Γ = rp. = = (r p)rp represents the probability of occupation of the instantaneous state (r p). If, moreover, the electrons are under the action of an external field of forces F(r) = −∇U (r), the motion is led by the single particle Hamilton function = |p|2 2 + U (r). For a given energy , the dynamic trajectory of an electron in phase space obeys Hamilton equations (191) from which one can demonstrate the Liouville theorem: (r p) (and, consequently, (r p)) is constant along all dynamic trajectories: (r p) =0
(440)
The proof is shown in the appendix below. If now we want to introduce in the simplest way the effect of interactions (collisions) with other particles (electrons and/or phonons) we can correct ad hoc the Liouville theorem as follows: µ ¶ (r p) (r p) (r p) − 0 (r p) = (441) =− (p) The previous equation is based on the heuristic hypothesis that, in the absence of external forces and temperature gradients, thermal equilibrium is restored by scattering between particles in a mean relaxation time (p). In the classical case 0 (r p) is the Maxwell-Boltzmann distribution 0 (r p) = −1 exp(−0 (r p) ) where = is the partition function.
Z
exp(−0 (r p) )rp
(442)
(443)
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Computing the total derivative in equation 441 and using the Hamilton equations we get the Boltzmann equation: ³ p ´ (r p) − 0 (r p) + F(r) · + · =− p r (p)
(444)
Using the effective mass theorem (equivalent Hamiltonian) we now modify the above equation to make it applicable to a quantum electrons system. Electrons will be described by semiclassical Bloch wave packets moving in a partially occupied band = (k) neglecting interband transitions. This is the case of a conductor under the effect of a weak electric field E: F(r) = −E. Employing the results of subsection Semiclassical dynamics, we can now write the Boltzmann equation in the stationary case = 0 as: F(r) (r k) (r k) (r k) − 0 (k) · + v(k) · =− ~ k r (k) where: v(k) =
1 (k) ~ k
(445)
(446)
In the fermion quantum situation 0 (k) ≈
exp
h
1 (k)− (r)
i
(447) +1
is the Fermi-Dirac distribution. The density of occupied states in k space depends also on r and its expression per unit volume is: D(r k| (r)) =
2 (r k) (2)3
(448)
where we have considered the spin degeneracy and assumed a local temperature field (r) with a weak gradient. We can thus apply a linear response approximation. To do that we rewrite equation (445) as: (r k) = 0 (k) − (k)
F(r) (r k) (r k) · − (k)v(k) · ~ k r
(449)
We solve now this equation approximately up to the first step of an iterative procedure: (r k) ≈ 0 (k) − (k)
0 (k) F(r) 0 (k) · − (k)v(k) · ~ k r
(450)
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Computing the partial derivatives 0 (k) 0 = ~ v(k) k µ ¶ 0 (k) 0 0 (k) − = = − r r (r) r we finally get: ∙ ¸ 0 ( − ) v(k) · F(r) − = (r k) ≈ 0 (k) − (k) (r) r = 0 (k) + (r k) 7.2.1
118
(451)
(452)
Appendix: Liouville theorem
The motion of a set of N particles in phase space is equivalent to the motion of a fluid with density = (r p); thus the conservation principle of mass corresponds to the conservation of the number of particles. This principle can be written in the form of a continuity equation: + ∇· (v) = 0 In the phase space of a particle (independent particles gas in an external field): µ µ ¶ X ¶ 3 3 (r p) X + + =0 =1 =1 Using the Hamilton equations we get: (r p) + { } = 0 where: 3 3 X X − { } = =1 =1
is the Poisson bracket. The total derivative of , as already discussed in the derivation of (444), is: (r p) = + { } and, consequently, it is immediately obtained that = 0 (c.v.d.). Note that the last equation is very similar to that for the time derivative of a quantum operator b b b h b bi = + . (453) ~
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Electrical conductivity
Once we have solved the Boltzmann equation (445) in the linear response regime (eq. 452), the current density vector in the local Ohm law (430) can be computed (as a function of the electric field) as the integral over the whole k space : Z Z −2 j = − D(r k| (r))v(k)k = (r k)v(k)k (454) (2)3 To a first approximation, neglecting the thermoelectric effects due to the thermal gradient, we get Z 22 j= (k)( − ) (v(k) · E) v(k)k (2)3 Where, based on Kramers theorem ((−k) =(k) → v(−k) = −v(k)) and 0 (−k) = 0 (k), we have taken into account that Z 2 (455) 0 (k)v(k)k = 0 (2)3 and made the further approximation 0 ≈ −( − ). Thus (see subsection density of states): Z 22 j= (k)( − ) (v(k) · E) v(k) 3 (2) |∇k (k)| and finally 2
j =
Z
(k) (v(k) · E) v(k) (k)
(456)
where the integral is over the Fermi surface (k) = ; we now define the density of states on the Fermi surface as: (k) =
1 2 3 (2) |∇k (k)|
(457)
The electric conductivity tensor can be eventually written in the form: Z 2 σ= (k) (k)v(k)v(k) (458)
Notice that only electrons on the Fermi energy can contribute to the charge transport process. In metals the dependence of conductivity on temperature is lumped in (k) = (k), on the Fermi surface, which is mainly due to
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electron-phonon scattering. The probability of a scattering event per unit time is ( (k))−1 and can be computed by means of the Fermi’s golden rule up to first order using the matrix elements of an appropriate electron-phonon interaction operator. At high temperatures this implies a dependence of ( (k))−1 on h2 (q)i which, in turn, is ∝ and explains the linear trend of resistivity with respect to temperature. At low temperatures, instead, elastic scattering with lattice imperfections prevails, and this explains the athermal residual resistivity. At intermediate temperatures a simple theory combining the different processes in a sufficiently accurate way does not exist. Actually it is very complicated to account for (Umklapp) processes. In simple metals the Fermi surface is spherical (free electrons model), to a first approximation, and all the quantities that have to be integrated are constant. So in this case we get, after a few simple steps: 2 (2∗ )32 p = 3 2 2 ~3 2
(459)
Using the expressions of Fermi energy (115) and Fermi speed ~ ∗ we finally get an equation often used in applications: µ ¶ 2 = ∗ (460) In the case of transition metals (where there is a superposition between bands and , and the Fermi level is within the band) it is possible to use the previous formula in an additive way (460) for the two different types of electrons, with very different values for effective masses and relaxation times. In the case of noble metals (where there is a superposition between bands and , and the Fermi level is above band within a band) wave vectors on the Fermi surface are very near the edge of the first Brillouin zone producing a strong deviation of the Fermi surface from the spherical shape with some band regions having a negative curvature and, thus, implying conduction due to holes. Also in this situation an additive (two bands) model with two types of charges, with opposite signs, is used.
7.4
Electronic thermal conductivity
Here we start with the thermodynamic relation for the differential of specific entropy: 1 (461) = − where = is the internal energy per unit volume and = is the number of electrons per unit volume. is the chemical potential, which
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can be approximated by the Fermi energy . By analogy we can make the assumption that the entropy flux is connected to energy and electrons fluxes through the following equation: j =
1 j − j
(462)
Then irreversible thermodynamics gives the following formula for entropy production (eq. 435): µ ¶ ³´ 1 · j + ∇ · j [] = ∇ (463) Remembering that the heat current density vector is defined as q = j , we obtain: q = j − j = v − v =v − v = (−)j
(464)
We approximate by , and then, using again the method of microscopic kinetic theory (see previous section), we can compute q as: Z 2 (r k)(− )v(k)k q = (465) (2)3 using now the following expression: (r k) = (k)v(k) ·
0 ( − ) r
(466)
which is valid in the absence of external fields. Remembering Fourier law, we can finally write thermal conductivity as follows: µ ¶ Z 2 0 ((k) − )2 − k (467) = (k)v(k)v(k) (2)3 In this case we cannot simply approximate −0 with ( − ), since now we would obtain a null result. Instead we must use the better Sommerfeld approximation (eq. 122). Using, as usual, the result k =
E |∇k (k)|
(468)
and again defining the density of states on the Fermi surface as =
1 2 , (2)3 |∇k (k)|
(469)
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after an integration over energy , we finally get Z 1 2 2 (k) (k)v(k)v(k) = 3
122
(470)
But we have already demonstrated that the above integral coincides with the electrical conductivity divided by the square of the electron charge (eq. 456). Thus: 2 2 = (471) 3 2 The above equation is called Wiedemann-Franz law. If we make use of this result, knowing the trend of electrical conductivity, we can get the trend of thermal conductivity with respect to temperature. Figures are shown in subsection Phenomenology. We have assumed that all relaxation times (scattering processes) involved in heat transport in metals coincide with those involved in charge transport. This is not exact and can induce a deviation from Wiedemann-Franz law, mainly at intermediate temperatures.
7.5
Lattice thermal conductivity (insulators)
In insulating crystals heat conduction is due to the non-equilibrium kinetics of lattice waves packets in presence of a temperature gradient. In this situation the carrier particles are phonons packets moving with the group velocity: v (q) =
(q) q
(472)
and carrying the energy quantum ~ (q). The distribution function in thermodynamic equilibrium is the Bose-Einstein factor (mean occupation number of a normal mode with wave vector q): 0 (q) =
1
(473)
~ (q)
−1 The phonons chemical potential is zero, since the number of phonons is not conserved. The heat density flux vector can be written as: Z 1 X q = (474) (r q)~ (q)v (q)q (2)3
Thus in equation (450) we can write, in the present case:
~ (q)
0 (q) 0 (q) 1 ~ (q) = = µ ¶2 ~ (q) r r 2 r −1
(475)
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which can be written as: (q) 0 (q) = r ~ (q) r
(476)
introducing the specific heat of each phonon (q) (eq. 234): ~ (q)
1 ~2 2 (q) (q) = µ ¶2 ~ (q) 2 −1
(477)
The heat density flux can thus we written as (see (450) with q instead of k): Z 1 X q = − (478) (q)v (q)v (q) (q) q 3 r (2) And then, using Fourier law, the lattice thermal conductivity tensor can be expressed as: Z 1 X (q)v (q)v (q) (q)q. k = (479) (2)3 Sometimes the phonon mean free path is introduced as Λ (q) = v (q) (q). Using a simplest treatment, strictly inspired by the kinetic theory of gases, we could immediately write 1 k = Λ 3
(480)
where is the total specific heat per unit volume and and Λ are opportune average values of v (q) and Λ (q). The above equation can be derived by the following elementary considerations. Phonons propagating a distance ∆ = = Λ corresponding to a temperature drop ∆ , carry a quantity of heat per particle = ∆ , being the contribution of a single particle (phonon) to specific heat. Then the heat flux is p = ∆ where is the number of particles per unit volume and = h2 i. If the temperature 2 gradient is we can write p = − ( ) = − Λ ( ) and = 3, being = h2 i. In a Debye model, neglecting the contribution of optical phonons (which have nearly zero group velocities), the velocity of acoustic phonons is constant and is equal to the mean speed of sound. For phonon-phonon interactions (anharmonic effects) we can consider that Λ( ) ∝ ( )−1 . At high temperatures ∝ and the specific heat is constant. At low temperatures the specific heat follows 3 ,
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while the mean free path is limited by the crystal size Λmax = . From a detailed microscopic point of wiew, it can be shown that only phonon-phonon Umklapp scattering guarantees local thermodynamic equilibrium and is responsible for a finite thermal conductivity.Without the contribution given by scattering events, the thermal conductivity would diverge, as proved by Peierls. Anyhow, without the size limitation to mean free path, at low temperatures k would diverge as ( )−1 ≈ exp(~ min ), where min is the minimum frequency of phonons which may encounter a scattering process. Two phonons can participate to a scattering process when the sum of their wavevectors goes beyond the first Brillouin zone. An important role is also played by elastic scattering off lattice defects. Whenever the wavelength of a phonon is greater than the size of a defect a Rayleigh scattering occurs, for which Λ ∝ |q|−4 .
8 8.1
Appendix: Quantum approximate methods Time-dependent perturbations
A given system, initially isolated, since time instant 0 is exposed to a dynamic perturbation (i.e. to the time-dependent interaction with a second coupled system) until time instant 0 + . The response of the first system (induced quantum transitions) can be obtained solving Schroedinger equation: ´ ³ b ~ = + b ()(0 ) (481) Function (0 ) is 1 when 0 ≤ 0 + and 0 otherwise. b () is the b is the unperturbed Hamiltonian of time dependent perturbation operator. the initially isolated system the spectrum of which is assumed to be known. () Let us first assume that the unperturbed energy levels are discrete and non-degenerate. b = () (482) If at time 0 the system is in the stationary state (it is not necessarily the ground state), then: (0 ) = (483) Following Dirac, we look for a solution of the Schroedinger equation (481) such as: () X () − ~ (484) () =
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That is, we use a generalized form of superposition principle with timedependent coefficients (). At time 0 + the perturbation ceases. In view of the chosen initial condition, the square of the absolute value of (0 + ) represents the transition probability → ( ) from initial state | i = |i to final state | i = |i during a time (by now on we will consider 0 = 0): → ( ) = | ( )|2
(485)
Substituting the wave function (484) in the Schroedinger equation and using R the orthonormality of the set of all (h|i = ∗ r = ), we get a system of ordinary differential equations for the coefficients () ~
() X b = h| ()|i () 6=
where we have defined:
()
(486)
()
− = ~
(487)
and with the initial conditions: (0) = The matrix elements of the perturbation operator are defined as: Z b () = h| ()|i = ∗ b () r
(488)
(489)
and we assumed that all the diagonal elements are zero, as it is true in many important cases (see below). At first we integrate each equation of the system with respect to time obtaining (for 6= ): Z 0 1 X b 0 0 0 h| ( )|i ( ) (490) () = ~ 0 6=
Then we proceed with an iterative solution method assuming that all the zero-th order solutions coincides with the initial conditions: (0) () = (0) =
(491)
Then the first order solutions are obtained substituting the zero-th order solutions in the right side of equations(490) : Z 0 1 X b 0 0 (0) 0 (1) h| ( )|i ( ) (492) () = ~ 0 6=
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thus obtaining (1) ()
1 = ~ 1 ~
Z X 0 0 0 h|b ( )|i = 0 6= Z 0
0
(493)
h|b ( )|i 0
0
The iteration proceeds substituting (0) ()
+
(1) ()
1 = + ~
Z
0
0
h|b ( )|i 0
0
again in the right side of equations (490): Z i 0 0 h 1 X b 0 (2) (0) 0 (1) 0 () = ( ) + ( ) h| ( )|i ~ 0 6=
(494)
(495)
and then obtaining: Z 0 1 b 0 0 (2) () = h| ( )|i + (496) ~ 0 µ ¶2 X Z Z 0 0 00 00 1 0 00 0 h|b ( )|i h|b ( )|i ~ 6= 0 0 and so on, after substitutions: Z i 0 0 h 1 X b 0 (+1) (0) 0 (1) 0 () 0 ( ) + ( ) + + ( ) h| ( )|i () = ~ 0 6=
(497) We consider now the first order approximation in the situation in which c0 . From (493 485) we get: b () = ¯ ¯2 ¯ c ¯ ¡ ¢ 2 h| |i ¯ ¯ 0 (1) 2 sin 2 → ( ) = ¡ (498) ¢ 2 ~2 2
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y
127
1
0.75
0.5
0.25
0 -20
-10
0
10
20 x
2
(2) Function y= sin(2) 2 ; =
If À 1 we obtain that the following function ¡ ¢ 2 2 sin 2 ¡ ¢ 2
(499)
2
can be approximated with an isosceles triangle with base 4 and height 2 : the area under the principal maximum is thus approximately equal to 2 . In this limit ´ can be approximated by a Dirac delta function ³ the function () () ( ) = ~ − . Summarizing, in the limit À 1 ( → ∞) we can write the transition probability as: ¯2 ¡ ¢ 2 ¯¯ c ¯ (1) () h| → ( ) ≈ |i (500) ¯ () − ¯ 0 ~
From an experimental point of view, it turns out that if the initial state |i is an excited one it is not a truly stationary state and it tends to spontaneously decay also in the absence of a any perturbation (spontaneous decay) following an exponential law (in the simplest cases): | ()|2 = −
(501)
where the mean lifetime of state |i depends on the experimental energy () () uncertainty ∆ , around the mean value , with which the energy of the state is known through the empirical inequality: () ≥ ~ ∆
(502)
From a physical point of view, then, the 500 can be used when −1 ¿ ¿ .
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Harmonic perturbation
For the time dependence: c− − c+ + b () =
(503)
´† ³ − + c c , and in the limit → ∞ ( −1 where = ¿ ¿ ), with a procedure similar to that just used, we get the transition probability per unit (1) time e→ in the form: ¯2 ¡ ¢ 2 ¯¯ c± ¯ (1) () h| ≈ |i ± ~ (504) e→ ¯ ¯ () − ~ In this case the delta function tells us that transitions can happen only if the total energy of the two coupled systems is conserved: () ± ~ = 0 () −
(505)
in agreement with the Bohr postulate of old quantum theory when ~ is the energy of a light quantum perturbing an electron system. Using equations (496), instead, we would obtain the second order result (useful to describe e.g. non linear optics): ¯2 ¡ ¢ 2 ¯¯ c± ¯ (2) () e→ ( ) ≈ ± ~ + (506) ¯h| |i¯ () − ~ ¯ ¯2 c± |ih| c± |i ¯¯ ¡ ¢ 2 ¯¯ X h| () () − ± ~ ± ~ ¯ ¯ () ~ ¯6= () − ± ~ ¯
Note that the number of energy quanta ~ of the second system involved in the process is equal to the order of the perturbation. Previous equations describe the absorption or the stimulated emission of one or two energy quanta ~, respectively. At the second order the transition can be seen as the sequence of two transitions through intermediate states |i. Intermediate transitions do not conserve energy. If the final state is a quasi-stationary state the singularity of the Dirac delta function can be removed. Introducing the density of final states () and integrating the (504) we get: Z ¯2 ¡ ¢ 2 ¯¯ c± ¯ (1) () e ± ~ (() )() =(507) → ≈ ¯h| |i¯ () − ~ ¯2 2 ¯¯ c± ¯ () ∓ ~) ¯h| |i¯ ( ~ usually written as:
2 ¯¯ c± ¯¯2 (1) e → ≈ ¯h| |i¯ ( ) ~
(508)
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the above equation is called Fermi’s golden rule, where = ∓~. Studyc± |i we get the so called selection rules, strictly ing the matrix elements h| connected to the symmetry of the system. 8.1.2
Photon-electron interactions: electric dipole selection rules
The theory outlined in the previous paragraph can be applied to the interaction between an electromagnetic wave and the electrons of an atomic system (atoms, molecules, solids, in particular crystals...); it becomes particularly simple in case of a plane, monochromatic, linearly polarized wave of wavelength À , where is the characteristic linear dimension of the elementary volume occupied by electrons (in the case of crystals it is the volume of the primitive cell). For a more general treatment see Optical properties. If the wave is polarized along axes and propagates along axis For the sake of simplicity we refer to the single electron of a hydrogen atom, is the atomic radius and we set the origin at the nucleus. The wave electric field can be written as follows: E = u cos( − ) (509) where = (2) is the wave vector. Thus (||) ≈ = 2 ¿ 1, and we are lead to the simpler form: ¡ ¢ 1 E ≈ u cos() = u + − 2
(510)
As far as this field does not appear anymore as a propagating wave, we can neglect magnetic effects and consider the atom as an electric dipole μ = −r ( is the absolute value of the electron charge) interacting quasi-statically with the instantaneous electric field. Thus we assume as perturbation operator 1 1 (511) + b − b = −b μ·E = b 2 2 c− = 1 b c+ = = 12 . In this b is then written in the form (503) whith 2 c± |i = situation the matrix elements of the perturbation operator are h | 1 h ||i. In order to see when they vanish we must consider the following 2 integral: Z (512) h ||i = ∗ r In case of an hydrogen atom the orbital part of the 22 wavefunctions belonging to the same energy level () ≈ −
136 eV; ( = 1 2 ) 2
(513)
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can be written in spherical coordinates ( ) as: ( ) =
() ( )
(514)
where () is the radial wavefunction, ( ) is a spherical harmonic, is the principal quantum number, is the orbital quantum number and the magnetic one. The parity of these states with respect to a coordinate inversion is defined and depends exclusively on ( ); it is (−1) (when r → −r, = |r| remains the unchanged). In order to have a nonzero integral, since is an odd function, it should be − = ∆ = ± , i.e. the parity of the final state should be different from that of the initial state. A further consideration about the angular momentum of the electron-photon system imposes the stronger limitation: ∆ = ±1 (optical selection rule). In an atom with atomic number in the mean field approximation, the single electron energy levels depend on () and the degeneracy reduces to 2(2 + 1). However in the (514) while the () change as functions of , the ( ) remain the same: the selection rules don’t change. In a crystal instead the initial and final states are Bloch waves normally belonging to different bands and one must consider the total fluctuating electric dipole of the crystal basis. This leads to specific selection rules which can also involve phonons when the adiabatic approximation is not applicable.
8.2 8.2.1
Static perturbation theory Adiabatic switching on
Consider a perturbation operator (without the (0 ) modulation) such as: c b () = lim (515) →0
where 0 and ¿ 1. Setting 0 = −∞ and 0 + = 0, equations (493) can be written as: Z 0 0 1 0 (1) c |i 0 =(516) (0) = h| ~ −∞ Z 0 c|i c|i 0 0 h| h| 0 c|i h| = () = () ~ ~ + −∞ − +
Notice that lim→0 has not been taken yet. Taking the limit now, it is the c, we obtain: adiabatic switching on of the static perturbation (1) (0) =
c|i h| ()
()
−
(517)
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and substituting this result in (484), for a generic perturbed eigenfunction (1) we get: (1) = +
X h| c|i ()
()
6= −
(518)
Using these perturbed eigenfunctions we can now compute the perturbed energy levels as: ³ ´ b c ≈ h(1) | + |(1) (519) i
Up to second order in and considering the necessary condition for series convergence ¯ ¯ ¯ () ¯ c|i¯¯ ¯ − () ¯ À ¯¯h| (520) after long calculations we get:
() c|i + 2 = + h|
¯2 ¯ ¯ ¯ c X ¯h| |i¯ ()
6=
()
−
(521) (1)
In many applications keeping in the final formulae for the coefficients (0) the initial expression: c|i h| () (522) () − + it is possible to account for spontaneous deacay and for other dissipative phenomena provided the value of is adjusted in order to have a better match with experimental data. Finally, with an infinitesimal it is possible to prove that (PP = Cauchy principal part, see also Kramers and Kronig relations): 1 ()
()
− +
≈ PP
1 ()
()
−
¡ () ¢ − − () .
(523)
This last equation and the following one, a new representation of Dirac delta function coming from the Green functions formalism, are very commonly used. µ ¶ 1 1 1 () ≈ − Im (524) = 2 + + 2
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Degenerate energy levels
When we use formulae such as (521) and n (518) owe are assuming that degen() erate levels in the unperturbed spectrum do not exist. As a matter ()
of fact if to a given level there correspond more than one eigenfunction | i in the perturbation series there would be divergent terms such as: c| i h |
(525)
= 1 + 2
(526)
()
()
−
We can overcome this problem as follows: we assume that to the degenerate energy level () there correspond the two unperturbed eigenfunctions 1 = |1i e 2 = |2i. To a first approximation we write the perturbed wavefunction as:
substituting now in the equation for perturbed stationary states ´ ³ ¡ ¢ c b + (1 + 2 ) = () + ∆ (1 + 2 )
(527)
c|1i = h1| c|2i∗ , the solubility condition is: Considering that h2| ¯2 ¯ ³ ´³ ´ c|2i¯¯ = 0 c|1i − ∆ h2| c|2i − ∆ − 2 ¯¯h1| h1|
(529)
and using the orthonormality condition for functions 1 and 2 we get the homogeneous linear algebraic system: Ã !µ ¶ µ ¶ c|1i − ∆ c|2i 0 h1| h1| (528) = c|1i c|2i − ∆ 0 h2| h2|
The two roots are: # " ¯2 ¯ ´2 ³ ´ 1 r³ 1 c|2i¯¯ c|1i − h2| c|2i + 4 ¯¯h1| c|1i + h2| c|2i ± ∆ ± = h1| h1| 2 2 (530) When the diagonal matrix elements are zero, we simply have: ¯ ¯ ¯ c ¯ ± (531) ∆ = ± ¯h1| |2i¯
The unperturbed degenerate level () so splits into the two perturbed nondegenerate levels: ¯ ¯ ¯ c ¯ ± () = ± ¯h1| |2i¯ (532)
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¯ ¯ ¯ c ¯ |2i¯. Using the first of Between them there is an energy gap = 2 ¯h1| the (528) c|2i± = 0 (533) −∆ ± ± + h1|
and the normalization condition
c|2i 0, we get: when h1|
||2 + ||2 = 1
(534)
1 1 ± = √ ; ± = ∓ √ 2 2
(535)
To the two levels + and − (532) correspond respectively the two eigenfunctions + and − : 1 1 (536) ± = √ 1 ∓ √ 2 2 2 After having used this procedure for all unperturbed degenerate levels, we can use the (521) in order to have more accurate level evaluation. The above procedure succeeds, that is we get two different solutions, when the perturbation breaks the symmetry of the unperturbed system.
8.3
Dirac delta function
The Dirac delta function () is defined by the following integral properties: +∞ Z
()( − ) = ()
(537)
−∞
+∞ Z
() = 1
(538)
−∞
where () is any normal function. Strictly speaking no function with these properties can exist and () should be better called a distribution, a new mathematical object clarified by the French mathematician Schwartz in the 1950’s. In physical applications the above properties can be exhibited up to a good approximation by series of functions depending on a parameter. For example the Gaussian functions 2
− 22 ∆(|) = √ 2
(539)
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satisfy exactly the second condition for any and very well the first in the limit → 0. In fact lim
→0
+∞ Z
()∆( − |) ≈ () lim
→0
−∞
+∞ Z
∆( − |) = ()
(540)
−∞
In quantum mechanics (scattering theory and dynamic perturbation theory) one meets integrals of the type +∞ Z
(541)
−∞
where both and are real. We can write +∞ Z −∞
= lim
→∞
+ Z
= lim
→∞
−
µ
− −
¶
= lim 2 →∞
sin() = 2()
(542) has a very sharp maximum In fact for very big values the function (equal to 2) for = 0 and becomes null about the origin at = ±. Then for larger || values it has symmetric vanishingly small damped oscillations. Thus in the limit of big values we can approximate the plot of the function by only the (extremely narrow) isosceles triangle with huge height 2 and vanishing basis 2This narrow triangular peak has area 2 and the function (in the limit → ∞) behaves similarly to 2∆(|) in the limit → 0. In 3D we have Z (r)(r − a)r = (a) (543) 2 sin()
with (r − a) = ( − )( − )( − ). Even though this last notation is not correct (as the product of distributions is not defined), it has an operational meaning thinking of each unidimensional delta function as the limiting value of a series of functions such as ∆(|). Thus we can write Z 1 k·r r =(k) (544) 3 (2) Another useful representation is 1 →0 2 + 2
() = lim
(545)
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Often the property
135
X ( − ) ¯¡ ¢ ¯ ¯ ¯ ¯ = ¯
[()] =
(546)
is used, where the are the roots of equation () = 0.
8.4
Green functions
Referring to the time independent Schroedinger equation: ³ ´ b − =0
(547)
0
the solution (r r ) of the following equation is called Green function ³ ´ 0 0 b − (r r |) = (r − r ) (548) 0
We obtain the (r r |) as a function of eigenvalues and eigenfunctions b (spectral representation). Since the (r) are a set of complete (r) of and orthonormal functions we can write: X 0 0 (r |) (r) (549) (r r |) =
and thus:
³ ´X X 0 0 b − (r |) (r) = ∗ (r ) (r)
8
(550)
. By multiplying both sides of equation by ∗ (r) and integrating over the whole space we get: 0 0 (r |) ( − ) = ∗ (r ) (551) and then 0
(r r |) =
X ∗ (r0 ) (r) −
(552)
The local density of states (r|E) is defined as: X | (r)|2 ( − ) (r|E) =
(553)
8
P
0
0
0
Condition ∗ (r ) (r) =(r − r ) can be written expanding the (r − r ) as a series of eigenfunctions and determining the coefficients using the orthonormality and the R 0 0 definition of : (r)(r − r )r = (r ).
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Using the following identity (see Kramers and Kronig relations): µ ¶ 1 1 = () + −
136
(554)
where is infinitesimal, and introducing the complex energy → − , we then get: 1 (555) (r|E) = lim Im (r r| − ) →0 and the density of states (): Z 1 (556) (E) = lim Im (r r| − )r →0 R 0 (r r|)r is called trace of the Green function (r r |) considered as a whose elements are labelled with the pair of continuous indexes ¡ matrix 0¢ r r .
8.5
Selfconsistent mean field
The band structure of electronic levels in a crystal has been considered within the independent electron approximation in a periodic potential equal for any electron. We thus assumed that the quantum motion of each electron was described by a single-electron wavefunction (r s ) depending only on the coordinates and the spin of the th electron. Here represents the set of quantum numbers defining the stationary Bloch state of the th electron: the wavevector k and the branch index (which depends on the non translational symmetry). Generally the periodic potential is not known a priori (as instead we have always assumed) and should be determined together with functions (r s ). Historically the problem was solved for the first time by Hartree with the heuristic solution we are going to explain, then it was considered by Fock in a variational scheme which included the exchange symmetry of the many body wavefunction (r1 s1 r2 s2 r s )9 in a system with interacting electrons, leading to the Hartree-Fock mean field. We first outline the second more rigorous method. In order to simplify the treatment we consider a many electrons atom in which the nuclear potential (attractive for electrons) exhibits a spherical symmetry. So the Hamiltonian operator describing the motion of electrons ( = 1 2 ) in the atom is: 2 X X 2 2 1X b =−~ ∇2r − + (557) 2 4 |r | 2 4 |r − r | 0 0 6= 9
With the notation used in section 4 it is the (r|0), where corresponds to the ground state with minimum energy.
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and can be rewritten as a sum of single particle Hamiltonians plus an interaction Hamiltonian: X X b + 1 b= (558) 2 6= 2 2 b = − ~ ∇2r − 2 40 |r |
(559)
2 (560) 40 |r − r | Instead, to study the crystal case, we should consider only valence electrons in a periodic potential, as stated in section 4. In the Hamiltonian (557) we neglected the spin-orbit interactions, the spin-spin interactions and all other relativistic effects. The expectation value of the electrons total energy should be calculated as: Z b = ∗ (r1 s1 r2 s2 r s )(r b || 1 s1 r2 s2 r s )Π r =
(561)
The Hartree equations (see below) may be derived assuming that: (r1 s1 r2 s2 r s ) = Π (r s )
(562)
from the following variational condition: Ã ! X b (r s ) − Π (r s )||Π (r s )| (r s ) = 0
(563) where the Lagrange multipliers are needed to define the necessary condition b conditioned by the normalizafor minimum of the functional || XR tion conditions (r s )| (r s ) = ∗ (r s ) (r s )r = 1. s
We look for the specific functional form of each (r s ) which minimizes the energy of the electron system. It is, however, necessary to notice that the form (562) of the many body wave function does not have the necessary exchange symmetry. Fock solved this problem using a Slater determinant wavefunction ¯ ¯ ¯ (r1 s1 ) (r2 s2 ) (r s ) ¯ 1 1 1 ¯ ¯ 1 ¯¯ 2 (r1 s1 ) 2 (r2 s2 ) 2 (r s ) ¯¯ (r1 s1 r2 s2 r s ) = √ ¯ ¯ ! ¯ ¯ ¯ (r1 s1 ) (r2 s2 ) (r s ) ¯ (564)
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in the same variational scheme considering, always by means of Lagrange multipliers, the orthonormality conditions (r s )| (r s ) = . The Slater wavefunction guarantees the exchange antisymmetry (if two columns are exchanged the determinant changes its sign) and the Pauli exclusion principle (if two electronic states are identical, there are two identical rows and then the determinant is zero). Now instead we will deduce Hartree equations following an heuristic method based on a electrostatic statistical approximation for the mean field. If the many body wave function is the product (562), thus implying the statistical independence of the single electronic motions, we can assume that the total charge density at position r, due to all the electrons except , is: ¯2 X ¯¯ ¯ (r) = − (565) ¯ (r)¯ 6=
for the sake of simplicity we have not considered the spin. This mean density distribution creates a mean electrostatic potential (r) obeying the Poisson equation: (r) ∇2 (r) = − (566) ◦ The solution of this equation can be written as the Coulomb integral: ¯ ¯2 ¡ 0¢ Z Z ¯¯ (r0 )¯¯ X r 1 0 0 (567) r (r) = 0 r = − 0 4◦ |r − r | 4◦ 6= |r − r | If we now consider electron , its wave function satisfies the Schroedinger equation: ∙ 2 2 ¸ ~∇ 2 − (568) − − (r) (r) = (r) 2 4◦ |r|
We added together the Coulomb attractive potential energy of nucleus and the mean repulsive Coulomb potential energy (r) = − (r) which takes into account the screening of the nuclear charge due to all other electrons with respect to th electron. Actually this situation can be considered as 2 that of a single attractive force field − 4(r)|r| 2 u due to the interaction with the pointlike charge in a medium where the dielectric function depends on position as: ◦ (r) = (569) 4◦ 1 + |r| (r) Since (r) 0, (r) ◦ . Using the explicit expression of (r) we get the
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2 2 2 ⎢ ~2 ∇ − + ⎣− 2 4◦ |r| 4◦
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¯ ¯2 0 ¯ ¯ Z X ¯ (r )¯ 0
6=
|r − r |
⎤
0⎥ r ⎦ (r) = (r)
139
(570)
This nonlinear system of integro-differential equations can be solved in an it(0) erative way, starting from the hydrogen-like expressions (r) for the (r) in the square bracket corresponding to (r) =0 At any iteration () we get (+1) a better approximation (r) for the (r) and (+1) (r) for the mean field (r) through equation (567). Once the convergence is obtained, we have the selfconsistent mean field and the associated single-electron wave functions which minimize the system total energy.
8.6
Semiclassical dynamics in a conservative force field
In a conservative force field weakly depending on position through the potential energy (r), an electron with large momentum behaves almost like a particle obeying Newtonian classical mechanics. In one-dimension, the Hamiltonian operator is: 2 b = b + () (571) 2 We use the equation of the time derivative for the expectation value of a physical quantity = ( ) ( * + Z ) h i b b 1 h b bi 1 h i b = + = ∗ + b (572) ~ ~
and we apply this equation to both position and momentum: hi 1 Dh b iE hi = b = (573) ~ ¿ À hi 1 Dh b iE = b = − (574) ~ Differentiating the first equation with respect to time and substituting the expression of the derivative of the momentum into the second equation, we get: ¿ À 2 hi =− (575) 2 If the dynamics was be completely at any instant of time ¡ we¢ would ® ¡ classical ¢ have = hi and − = − =hi = (). () = − is the
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classical force acting on the particle. Yet in quantum mechanics position and momentum obey Heisenberg uncertainty principle: ~2 2® 2® ( − hi) ( − hi) ≥ 4
(576)
In the hypotesis that q ( ) is a wave packet with a small (but not zero) ® uncertainty of position ( − hi)2 about the mean value hi we can derive ® in a power series of ( − hi) about hi expanding an approximate up to second order : ) À Z ¶ µ 3 ¶ ¿ Z (µ 1 = ∗ ( ) ( ) ≈ + ( − hi)2 |( )|2 =hi 2 3 =hi (577) ³ 2 ´ 2 We neglected 2 ( − hi) |( )| because its integral is identically =hi
zero. So the previous equation can be written as: ¶ ¿ À µ ¶ µ 1 3 2® ≈ + ( − hi) =hi 2 3 =hi
(578)
If the following inequality is satisfied ¯ ¯ ¯µ ¯µ ¶ ¶ ¯ ¯ ¯ ¯ 3 ® 1¯ ¯ ¯ ¯ 2 ¯ ( − hi) ¿ ¯ ¯ ¯ 3 ¯ ¯ ¯ 2 =hi =hi ¯
(579)
® − is very similar to the classical force (). We write again the latter condition as: ¯¡ ¢ ¯ ¯ ¯ 2 ¯ =hi ¯ 2® ¯ (580) ( − hi) ¿ ¯¯¡ 3 ¢ ¯ ¯ 3 =hi ¯
Given the uncertainty of the position, respecting the above inequality is easier if () depends weakly on . Now, using the uncertainty principle, we can write: ® ~2 ® ( − hi)2 ≥ (581) 4 ( − hi)2
But the mean kinetic energy is:
® ( − hi)2 hi2 h2 i = + 2 2 2
(582)
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Figure 5:
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For kinetic energy the condition to heve semiclassical motion is: ® (583) hi2 À ( − hi)2 ® ® The figure shows in which regions of the ( − hi)2 ( − hi)2 plane the conditions for a semiclassical motion are verified. The region below the hyperbola is forbidden since there Heisenberg uncertainty principle would be violated; this zone is labelled as unphysical (UP UnPhysical). In the region above the hyperbola (QM Quantum Mechanics) there is a curvilinear triangle inside which the semiclassical conditions are verified. This area is wider if the potential energy depends weakly on position and in case of large mean linear momentum (in this case the de Broglie wavelength associated with the electron is very small, having introduced a mean de Broglie wavelength = hi associated to the wave packet). In general this graphical representation could change with time. The following case of a simple periodic potential is particularly simple and significative: 2 ) (584) Here, considering both semiclassical conditions and uncertainty principle, we obtain at the end the unique condition ¿ which guarantees that the wave packet representing the electron does not exhibit significant diffraction. In a crystal, let be the lattice parameter, this condition is never verified, since it would imply a wave vector À 2 greater than the edge of the first Brillouin zone. Yet the relevance of semiclassical dynamics applyed to crystals is well known. This apparent contradiction is solved considering the effective mass theorem: when we consider the equivalent semiclassical dynamics in a crystal, () does not include the periodic potential (which is lumped in the electron effective mass) but, instead, just represents the external force field (generally not periodic in space) which weakly depends on position. Unfortunately the considerations shown in this paragraph are not completely general since they require that () is a continuous and differentiable function up, at least, to third derivative and that none among the derivatives is identically zero. In conclusion we cannot apply this method to an harmonic oscillator where () = 12 2 2 . () = 0 sin(
9 9.1
Appendix: Elasticity and Elastic Waves Stresses, strains and elastic constants
As for all other physical properties, the mechanical response of materials depends on the scale at which it is measured. For 3D compact materials (e.g. di-
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amond), when the length scale is much bigger than the interatomic distance, the mechanical behaviour is conveniently modelled by standard continuum mechanics. Yet there exist special materials with a complex mesostructure for which this approach may not be appropriate. Aerogels and some types of cluster assembled carbon films belong to this cathegory. In the following we shall concentrate on elasticity, neglecting more complex phenomena like plasticity and fracture. Engineers usually deal with polycrystalline or multi-phase solids which can be characterized by macroscopic elastic parameters such as the bulk modulus and the shear modulus . Such materials are elastically isotropic at the macro-scale and two parameters are sufficient to describe their behaviour. Physicists are traditionally more interested to anisotropic single crystals, like perfect diamond (cubic) and perfect graphite (hexagonal). The scalar variables volume and pressure are not adequate to define the deformed state of solids: tensor quantities are needed. To describe the kinematics of the deformation process, let a vector field r spans all points within the material volume in an undeformed equilibrium state. If now a material deformation is produced by some external and/or internal agents, the old (undeformed) material positions are mapped into the new ones as r0 = r + u(r) by the displacement vector field u(r), describing both deformations (volume and shape variations) and rigid rotations. In the small strain regime, the symmetric part of the gradient of the displacement field u(r) is the strain tensor ¶ µ 1 (585) = (∇u) = + 2
This tensor describes how infinitesimal cubic volume elements = change their volume and/or their shape. The diagonal components represent 0 tensile strains (e.g., = ( − )), while the off-diagonal components represent shear strains. More precisely, assuming in the following summation from 1 to 3 over repeated vector and tensor suffixes, the relative volume 0 variation ( − ) is equal to , the trace of . is either a pure dilation ( 0) or a pure hydrostatic compression ( 0). Instead, = − 13 is a pure shear strain (a strain deviator, where is the unit tensor): in fact = 0, proving is a pure shear. In the undeformed state the angle, e.g., between x and y is 90 . After a shear deformation the angle is reduced of = = 2 = 2 . Any deformation is the sum of a pure shear and a dilation (compression) as it is evident from the identity = + 13 . Provided the produced strains are small and reversible, a generalized Hooke’s law describes the linear elastic material response:
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(586)
=
In the above equation is the stress tensor. is the −component of the force per unit area acting on a face of a cubic element whose outgoing normal is the unit vector of the −th cartesian axis. The diagonal components represent tractions, while the off-diagonal components represent shear stresses. The elastic material properties are embodied in the fourth-rank tensor the elastic constant tensor with 34 = 81 elements. Because of the symmetry of (see below) and , the number of independent components of diminishes to 36. This number is further lowered to 21 by energetic considerations. Using then Voigt’s contraction scheme
or 11 22 33 23 or 32 13 or 31 12 or 21 or 1 2 3 4 5 6 eq. 586 can be given a simpler matrix form (587)
=
where the elastic constant 6 × 6 matrix is symmetric. The material symmetry produces further reduction. In the case of cubic crystals, for instance, the independent constants are 3 and only 2 in a isotropic material, as anticipated. In the simplest case of isotropic elasticity eq. 587 becomes: ⎛
⎞
⎛
⎞⎛
⎞ 1 ⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎟=⎜ ⎟ ⎟ ⎜ 4 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎠ ⎝ 5 ⎠ ⎝ ⎠ ⎝ 6 (588) Uniaxial stress. In engineering two different elastic constants, namely the Young modulus and the Poisson’s ratio are usually introduced. If a uniaxial stress = 1 = is applied (with all other = 0) only strains = 1 = , = 2 and = 3 = 2 are non-zero. Then is defined as and as −2 = −3 . Since there are only two independent elastic 1 2 3 4 5 6
+ 43 − 23 − 23 0 0 0 − 23 + 43 − 23 0 0 0 − 23 − 23 + 43 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
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constants, and can be obtained from and by the following formulae: 9 3 + 3 − 2 = 2(3 + )
(589)
=
(590)
The meaning of bulk modulus appears clearly in hydrostatic pressure conditions: 1 = 2 = 3 = − and 4 = 5 = 6 = 0. In this case 1 = 2 = 3 = −3 and 4 = 5 = 6 = 0. Thus the relative volume variation = 1 + 2 + 3 = −. From the last equation it follows that the isothermal compressibility , whose positiveness is a condition for material thermodynamic stability, is just the inverse of µ ¶ 1 −1 0 (591) = = − Since too is always positive (see eq. 592), can vary between −1 (when = 0) and 12 (when = 0). In practice there are no materials known for which 0. Shear: In the case of simple shear, e.g. 4 = and all other = 0, then 4 = 2 and all other = 0 If one introduces the angular shear strain measure 4 = 24 = , the last equation reads = , which illustrates the physical meaning of . Cubic lattices have three independent elastic parameters 11 , 12 and 44 : in this case (as for lower symmetry crystals) and are direction dependent and can exceed the limits of isotropic materials. As the cubic case reduces to the isotropic case when 11 − 12 = 244 , a possible measure of the degree of anisotropy is (11 − 12 − 244 ) (11 − 44 ). Hexagonal lattices have five independent elastic parameters 11 , 12 13 , 33 and 44 .
solid 11 12 silicon (cubic) 165.7 63.9 diamond (cubic) 1076.0 125.0 graphite (hexagonal) 1060.0 180.0
13
33
44 79.6 2.33 575.8 3.51 15.0 36.5 4.5 2.26
The elastic moduli are in and the density in 3
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Graphite is globally highly anisotropic but completely isotropic for any rotation around the c axis perpendicular to a graphene plane. Its extremely low 44 value explains its use as a solid lubricant material. Instead the inplane moduli are of the same order of those of diamond, the hardest known material. The elastic energy density E of a strained isotropic material is the quadratic form: 1 E = ( )2 + (592) 2 In this formula spherical symmetry (isotropy) has been fully taken into account. In fact and are invariants of the strain tensor with respect to rotations. The limitation to second order in the strains leads to linear elasticity ( = E ), as stated by eq. 588), with both and positive for E must be minimum in the undeformed state (stability). In mechanical equilibrium the internal stresses in every volume element must balance. This is realized when the stress tensor field obeys the following equations + = = 0 and = (593) where = is an external body force. If external surface forces act on the points of the outer surface bounding the volume , also equilibrium boundary conditions must be taken into account. They read ( )r∈ =
(594)
where n is the outgoing normal of the surface element centered at r ∈ . The boundary conditions will remain the same even in the dynamic case.
9.2
The acoustic waves and their phonons
Though the traditional ways to measure the mechanical properties of materials and, in particular, the elastic constants are based on quasi-static deformation processes (e.g. the tensile test to measure the Young modulus), many important methods are acoustic in nature or make use of acoustic phenomena (ultrasound propagation, acoustic microscopy, acoustic emission, Brillouin scattering, laser induced surface acoustic waves). In the simplest case one measures the time required for a longitudinal ultrasonic pulse to travel back and forth inside a cylindrical sample along its axis. Knowing the length of the cylinder theppulse velocity is obtained. If the cylinder is a long very thin rod, = , being the rod density. To face less naive methods and more complex geometries, some basic results of classical elastodynamics must be employed.
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The elastodynamic equation substituting eq. 593, written in terms of the dynamic displacement field u = u(r ), reads: 2u = 2 ∇2 u + (2 − 2 )∇(∇ · u) (595) 2 q¡ p ¢ where = + 43 is the longitudinal sound velocity and = is the transverse sound velocity. Remembering that the most general deformation process is the superposition of a simple dilation and of a simple shear, one is tempted to try a solution of the type u = u + u with ∇ × u = 0 and ∇ · u = 0. The second condition is identical to that holding for electromagnetic waves in vacuum. This trick perfectly works (instead anisotropic materials require a more complex treatment) leading to two decoupled wave equations for u and u : 2 u 2 u 2 2 = ∇ u and = 2 ∇2 u 2 2
(596)
The fundamental bulk solution of eq. 595 is then the superposition of three independent monochromatic plane waves, one longitudinal () and two (mutually perpendicular) transverse ( 1 2 ), of the type: © ª (597) u = < q eq [q·r− (q))]
where q is the wavevector, is a branch index ( = 1 2 ), q is the complex amplitude of the normal coordinate (q) = q − and eq a polarization unit vector (eq ||q; eq ⊥q). Moreover: ω (q) = |q| and ω1 2 (q) = |q|. The above classical description can be translated into the language of quantum mechanics of the systems of independent harmonic oscillators: the quanta of the fields u are the long-wavelength acoustic phonons whose possible energies are: ¶ µ 1 ~ω (q) (598) (qα) = q + 2
1
Contents 1 Solid bodies 1.1 Order and symmetry . . . . . . . . . . . . . . . . . . . 1.1.1 Simple Crystals: lattices and translational order 1.1.2 Complex crystals: lattice and basis . . . . . . . 1.2 Crystal binding . . . . . . . . . . . . . . . . . . . . . . 1.3 Tensorial observables . . . . . . . . . . . . . . . . . . .
. . . . .
5 6 7 13 13 15
2 Scattering theory 2.1 Elementary theory of elastic scattering . . . . . . . . . . . . . 2.2 Quantum particle scattering amplitude: Born approximation 2.3 Elastic scattering - Bragg law . . . . . . . . . . . . . . . . . . 2.4 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . .
16 17 19 22 24
3 Electronic States 3.1 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Reduction to the first Brillouin zone . . . . . . . . . . . . . . 3.3 Born-von Karman boundary conditions . . . . . . . . . . . . 3.4 Free electron gas . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Plane waves - Fermi energy - Density of states . . . . 3.4.2 Chemical potential and electronic specific heat . . . . 3.4.3 Quasi-particles and Fermi liquid theory (a qualitative introduction) . . . . . . . . . . . . . . . . . . . . . . 3.5 Band structure: nearly free electron approximation . . . . . 3.5.1 Free electron diffraction . . . . . . . . . . . . . . . . 3.5.2 Fermi surface and density of states . . . . . . . . . . 3.6 Band structure of tightly bound electrons . . . . . . . . . . . 3.6.1 LCAO Method . . . . . . . . . . . . . . . . . . . . . 3.7 Electron dynamics in external fields . . . . . . . . . . . . . . 3.7.1 Equivalent Hamiltonian. Effective mass theorem . . . 3.7.2 Impurity levels . . . . . . . . . . . . . . . . . . . . . 3.7.3 Semiclassical dynamics in a crystal . . . . . . . . . . 3.7.4 Limitations in the effective mass description . . . . . 3.7.5 Electric current - Electrons and holes . . . . . . . . . 3.7.6 Excitons . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
29 31 35 37 39 39 41
. . . . . . . . . . . . .
43 44 44 46 47 50 51 52 55 55 58 59 64
4 Adiabatic theorem and vibrational motions
. . . . .
. . . . .
. . . . .
64
2 5 Lattice dynamics 5.1 Lattice specific heat: Einstein model . . . . . . . . . . . . . 5.2 Lattice dynamics: Born - Von Karman model . . . . . . . . 5.3 Linear chain with two atoms per cell: acoustic modes and optical modes . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simple 1D crystal: acoustic phonons and elastic waves . . . 5.5 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Specific heat: Debye model . . . . . . . . . . . . . . . . . . . 5.7 Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
76 78 80 82 88 93
6 Optical properties 6.1 Kramers e Kronig relations . . . . . . . . . . . 6.2 Probability per unit time of optical transitions 6.3 Interband optical transitions . . . . . . . . . . 6.3.1 Direct transitions . . . . . . . . . . . . 6.3.2 Indirect transitions . . . . . . . . . . . 6.4 Intraband transitions and plasmons . . . . . .
. . . . . .
. . . . . .
98 100 103 105 105 108 111
. . . . . .
112 . 112 . 116 . 118 . 119 . 120 . 122
7 Transport phenomena 7.1 Phenomenology . . . . . . . . . . . . . . 7.2 Kinetic theory . . . . . . . . . . . . . . . 7.2.1 Appendix: Liouville theorem . . 7.3 Electrical conductivity . . . . . . . . . . 7.4 Electronic thermal conductivity . . . . . 7.5 Lattice thermal conductivity (insulators)
. . . . . .
. . . . . .
. . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
8 Appendix: Quantum approximate methods 8.1 Time-dependent perturbations . . . . . . . . . . . . 8.1.1 Harmonic perturbation . . . . . . . . . . . . 8.1.2 Photon-electron interactions: electric dipole rules . . . . . . . . . . . . . . . . . . . . . . 8.2 Static perturbation theory . . . . . . . . . . . . . . 8.2.1 Adiabatic switching on . . . . . . . . . . . . 8.2.2 Degenerate energy levels . . . . . . . . . . . 8.3 Dirac delta function . . . . . . . . . . . . . . . . . . 8.4 Green functions . . . . . . . . . . . . . . . . . . . . 8.5 Selfconsistent mean field . . . . . . . . . . . . . . . 8.6 Semiclassical dynamics in a conservative force field
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
67 . 67 . 69
124 . . . . . . 124 . . . . . . 128 selection . . . . . . 129 . . . . . . 130 . . . . . . 130 . . . . . . 132 . . . . . . 133 . . . . . . 135 . . . . . . 136 . . . . . . 139
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9 Appendix: Elasticity and Elastic Waves 142 9.1 Stresses, strains and elastic constants . . . . . . . . . . . . . . 142 9.2 The acoustic waves and their phonons . . . . . . . . . . . . . . 146
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Foreword To make a proper use of these notes, students should carefully read this foreword. The present text is the English translation of an original Italian version used till the second semester of a.y. 2011/12. The translation was quickly accomplished to be ready for the 2013/14 course of Solid State Physics and was far from being completely satisfactory. A better edition is now available for the 2017/18 course. Concerning the content, although I have taken the occasion to introduce some minor changes, the nature of the notes remains the same: a short and synthetic outline of the theoretical part of the lectures on Solid State Physics I gave at Politecnico di Milano in the last eight years. The contents of more practical lectures (very specific examples, exercises and description of experiments), which also belong to the course program and are possible objects of the final exam, are almost not included. Anyhow this notes, initially born as a personal notebook of the teacher and not intended for student studying, should be read only as an additional didactic aid and should never be used in place of a good textbook of Solid State Physics, which remains an indispensable tool for any student. For my lectures I have taken inspiration from many famous textbooks like: J.M. Ziman, Theory of solids (2nd edition), Cambridge University Press (1995); C. Kittel, Introduction to Solid State Physics (8th edition), John Wiley &Sons (2004); N.W. Ashcroft and N.D. Mermin, Solid State Physics, HRW International Editions (1981); W.A. Harrison, Solid State Theory, Dover Publications (1979); F. Bassani e U.M. Grassano, Fisica dello Stato Solido, Bollati Boringhieri (2000). Students should choose one of them and study it carefully. A further reason not to use these notes as a unique reference text is their intrinsic degree of difficulty: many comments, clarifications and examples I always make during lectures are not reported here. Without them understanding very synthetic written considerations may be rather hard. Yet I hope these notes will be of some help for those students who mainly like the very few fundamental principles upon which every physical theory is based: a dry skeleton of fundamental concepts. A last comment: understanding these notes requires a good knowledge of basic quantum mechanics even though the appendices introduce some more advanced topics, e.g. approximate methods like perturbation theory.
4
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Figure 1: Crystals
1
Solid bodies
From a phenomenological and macroscopic point of view a solid body is characterized by both a proper shape and a proper volume (while a liquid owns only a proper volume and both gases and vapours neither). All solids mechanically resist actions tending to varying their shape and/or their volume. The mechanical response, described by a stress-strain law1 , is reversible (often linear) as long as the applied stresses do not exceed critical values: the so called elastic regime. In this last condition shape and volume are thermodynamic state variables like temperature. Beyond such critical stress values solid bodies undergo irreversible processes, e.g. plastic deformation and fracture. Crystalline solids become liquid at a characteristic temperature (melting point) with a first order phase transition. Disordered solids instead exhibit different irreversible processes leading anyhow to a fluid state beyond specific temperature values. Some solid bodies (for instance tungsten oxide 3 ) directly vaporize without passing through the liquid state: this phase transition is called sublimation. From a microscopic (atomistic) point of view solid bodies can be roughly classified as crystalline (monolithic ordered atomic structures, see below), polycrystalline (bodies assembled by many individual crystalline portions called grains, joined together through disordered transition regions or grain boundaries) or amorphous (disordered atomic structures). In the last twenty years, with the advent of nanotech1
More precisely one should speak of stress tensor and strain tensor. See Appendix on elasticity and elastic waves
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nology in the realm of materials science, also the terms nanocrystalline and cluster assembled have been introduced, making the classification less sharp and more complex. In principle the main goal of solid state physics should be the forecast of the atomic structure of solid bodies (and thus of all their properties) solely on the base of quantum mechanics and statistical thermodynamics. This objective is impossible to be reached. The reason is that the underlying quantum many body problem is far too difficult to be solved. In the case of crystals the atomic structure is usually assumed as given (most often based on experimental results like X-Ray diffraction) and then quantum mechanics is used to compute the properties of that particular structure. E.g. we are not able to understand from first principles why solid copper is a face centered cubic crystal with a cubic cell of 3.61 Å edge but, assuming this structure as true, we understand why copper is a noble metal, we can compute its specific heat, electrical and thermal conductivities, optical reflectivity, thermal expansion and many other properties in good agreement with experimental data. This is possible because, given the reference structure, many properties can be reconducted to elementary quantum excitations that, to a first approximation, beahave as ensambles of independent particles which are born, move and die within the body. There exist two types of elementary excitations: quasi-particles (quasi-electrons, holes, polarons) and collective excitations (phonons, plasmons, polaritons, magnons). All these excitations are deeply connected with the real particles like electrons and nuclei which, in the normal energy range of interest for solid state physics, are stable particles whose number is conserved. In the following we will deal mainly with quasi-electrons (which will be simply called electrons), holes and phonons.
1.1
Order and symmetry
Among solid bodies crystals are particularly important. Such bodies are ideally characterized by long range order of atomic positions, which gives crystals a unique property: translational symmetry. The term atomic position used in the crystallographic language should be understood, from a more general point of view, as mean atomic position in conditions of thermodynamic equilibrium and not as instantaneous atomic position. In this respect symmetry itself is a thermodynamic property of crystals. In real crystals atoms vibrate around their stable mean positions (stable provided the temperature is far from the melting point) due to thermal agitation. Furthermore lattice defects (vacancies, interstitials, dislocations ecc), breaking the translational symmetry, are unavoidably present in real crystals. Lattice dynamics studies atomic vibrational motions. For didactic reasons we start
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describing the properties of ideal crystals, free of both lattice defects and thermal motions. First let us consider the so called simple crystals in which atomic positions coincide with the points of a Bravais lattice. 1.1.1
Simple Crystals: lattices and translational order
A Bravais lattice is defined as the discrete infinite set of points (vectors) given by the following formula: n = 1 a1 + 2 a2 + 3 a3 1 2 3 being a triplet of integer numbers and a1 a2 a3 being the primitive vectors (not all in the same plane) of the primitive cell. The primitive cell, which contains only one lattice point, is the volume of space that, when translated through all the vectors in a Bravais lattice, just fills all of space without either overlapping itself or leaving voids. The vectors a are said to generate or span the lattice. Given a Bravais lattice, the primitive vectors are univocally identified (primitive unit cell ) provided the volume of the cell is minimal and the three vectors connect the origin of the cell to nearest neighbors (that is, to lattice points of minimal distance). Alternatively the Wigner and Seitz method can be used. The lattice point taken as origin is connected to all equivalent nearest neighbors generating a set of segments. Then every segment is bisected by a normal plane. The set of all such intersecting planes defines a polyhedron: the Wigner-Seitz cell, containing only one Bravais lattice point, whose volume is identical to that of the primitive unit cell. The advantage of using the Wigner-Seitz cell as the primitive one is that all the symmetry properties of the Bravais lattice (also the non-translational ones, see below) can be directly seen by inspection of it. The figures also illustrate different kinds of cells: one instance of elementary cell (minimal volume, but the connected points are not all nearest neighbors) and one instance of conventional cell (non minimal volume; moreover the one to one correspondence between lattice point n and cell is broken).
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A lattice translation Tnn0 is the vector difference between two vectors n 0 and n and can thus be expressed by the same type of formula representing a 0 0 lattice point. If one assumes n = 0, Tn0 = Tn = n. Any two points r and r within the crystal volume (in general neither coinciding with any lattice point) are physically equivalent if they are connected by a lattice translation 0
r = r + Tn = r+1 a1 + 2 a2 + 3 a3 A physical observable of the crystal, e.g. the electric charge density, turns out to be invariant with respect to any Tn : 0
(r ) = (r + Tn ) = (r)
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0
is then a multiply periodic function of position. In one dimension = + and (+) = (). Thus () is expressible as a Fourier series +∞ X
() =
2
(1)
=−∞
whose Fourier coefficients are computed as 1 =
Z
2
()−
(2)
0
2
The reciprocal lattice vectors = can be written again as
+∞ X
() =
are then introduced and the series
(3)
=−∞
The last is surely a periodic function because 2
= = = 2() = 1
(4)
In three dimensions, for a lattice generated by periodic translation of a primitive cell in the shape of a right-angled parallelepiped, the reciprocal lattice vectors can be written in the simple form g =
2 2 2 u + u + u 1 2 3
and the density can be expressed as X (r) = g ·r
(5)
(6)
=
1
Of course it turns out that
Z
(r)−g ·r r
(7)
g ·T = 1
(8)
In the general case the following more complex definition of the primitive vectors of the reciprocal lattice is needed: a2 × a3 |a1 · a2 × a3 | a3 × a1 = |a1 · a2 × a3 | a1 × a2 = |a1 · a2 × a3 |
b1 = b2 b3
(9) (10) (11)
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then one writes g = 2(b1 + b2 + b3 )
(12)
being the volume of the primitive cell given by ( ) = |a1 · a2 × a3 |
(13)
In this way it is always true that a · b = and the (8) is authomatically satisfied. Fundamental theorems 1. Every reciprocal lattice vector g is perpendicular to a family of (direct) lattice planes with Miller indices2 . 2. If the components of g have no common factors, then in general the first neighbour planes distance is related to g by =
2 |g |
(14)
and, for cubic crystals (s.c., b.c.c., f.c.c.), with a conventional cubic cell of edge , by (15) = √ 2 + 2 + 2 3. The volume of the primitive cell of the reciprocal lattice can be computed as ( ) =
8 3 8 3 = ( ) |a1 · a2 × a3 |
(16)
4. The direct lattice is the reciprocal of its reciprocal lattice. For the primitive cell of the reciprocal lattice the most common definition is the Wigner-Seitz one: the corresponding cell is called first Brillouin zone. The zone center is called Γ point. Besides from its translational symmetry each Bravais lattice is characterized by its point symmetry. The point group is the set of symmetry operations which leave a lattice point (taken as origin) fixed. A point group contains rotations of an angle 2 (with = 2 3 4 6) about crystal 2
(hkl) denotes a plane that intercepts the three points 1 , 2 , and 3 , or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane, in the basis of the lattice vectors. If one of the indices is zero, it means that the planes do not intersect that axis (the intercept is "at infinity").
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Figure 2: axes passing through the fixed point, the inversion (r → −r) and the rotoinversions (equivalent to reflections across crystal planes passing through the fixed point) consisting of a rotation followed by the inversion (in the group algebra the corrisponding operation is the product of the two distinct operations). All Bravais lattices own the inversion symmetry with respect to any lattice point. The space group of the lattice contains both the symmetry translations and the symmetry operations belonging to the point group. Groups theory shows that (in three dimensions) do exist only fourteen (14) different Bravais lattices (result obtained by the French physicist Auguste Bravais in the year 1845) each charatterized by a specific primitive cell (a specific triplet of vectors a1 a2 a3 ). The 14 Bravais lattices distribute themselves among the seven (7) crystal systems (syngonies), illustrated in the figure, for there are only 7 point groups compatible with translational symmetry.
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Solid State Physics Lecture Notes
Figure 3:
12
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Complex crystals: lattice and basis
Figure 4: In the most general case a crystal can be conceptually built decorating every primitive cell of a Bravais lattice with a specific set of atoms called the basis of the crystal. The basis is then the elementary building block which, periodically translated along the space directions defined by vectors a1 a2 a3 , reproduces the whole (in principle infinite) crystal structure. The possible point symmetry (compatible with the translational one) in the presence of a basis is defined by the thirtytwo (32) crystallographic point groups. The overall set of symmetry translations and of non-translational symmetry operations3 which leave the crystal invariant can be further classified among two hundred and thirty (230) space groups. In the following we often shall refer only to simple crystals, unless the discussed physics does require the presence of a basis.
1.2
Crystal binding
A further different classification of crystals hinges on the type of (chemical/physical) binding responsible for their cohesion. Thinking a solid body as a complex many body system made of "valence" electrons and "lattice" ions (the ions in turn being made of both nuclei and "core" electrons), we give the following definition of cohesion energy: the energy needed to disassemble the crystal into a set of neutral atoms positioned each other an 3
Combining the symmetry of a Bravais lattice with that of the basis, new symmetry operations can arise such glide planes and screw axes.
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infinite distance apart (without loosing any original particle). Now thinking to the reverse process (at zero temperature), starting from the disassembled state, the formation of the stable final bound state of the crystal, occupying a finite volume, must correspond to an energy gain (lowering of the total energy) equal to (defining) the cohesion energy. The main acting forces are of electrostatic origin (repulsive between charges of same sign: electron-electron, nucleus-nucleus; attractive between charges of opposite sign: electron-nucleus). Magnetic forces, associated with elementary magnetic moments, connected with electronic and nuclear spins, are not equally important, with some exceptions. Gravitational forces are negligible (except for oriented crystal growth). Yet, only in the case of ionic solids (see below) the role of electrostatic interactions is fully understandable in terms of classical physics. To understand both crystal binding and properties in general a quantum approach is needed. Within this quantum description key items are the overlapping of atomic orbitals (to create extended states), the exchange interactions (related to exchange symmetry of the electrons wave function and thus, in a single particle mean field approximation, to Pauli exclusion principle) and, in some cases, the zero point energy (a consequence of Heisenberg uncertainty principle). An essential role is played by the charge spatial distribution depending on individual presence probability densities of valence electrons and of lattice ions. In this respect metal crystals and ionic crystals constitute two different extreme cases. In the former case the wave functions of valence electrons fill all crystal volume with a weak periodic inhomogeneity of the probability density (quasi-plane wave states) within simple lattices with the highest packing index, mainly face centered cubic (f.c.c.), e.g. , , , and hexagonal closed packed (h.c.p.), e.g. , , , but also body centered cubic (b.c.c.), e.g. , , . Here lattice ions are treated, to a first approximation, as hard spheres embodied in an almost uniform valence electrons cloud. In the latter case (perfect ionic binding) the crystal is a lattice of positive and negative ions (e.g. the salt + · − ) and extended (Bloch) states for valence electrons do not exist, being all electrons in localized atomic states. In the case of single element insulating crystals, e.g. diamond or semiconducting, e.g. Si and , binding is covalent. Extended states of valence electrons exhibit a strong interionic localization (bonds) from which strong bonding anisotropy follows in rather open structures with atomic tetrahedral coordination based on 3 atomic orbitals hybridization. III-V semiconductors, e.g. , or II-VI, e.g. , show a situation intermediate between covalent and ionic binding (there are extended states with strong bonding directionality and partial charge accumulation- of opposite sign- on pairs of nearest lattice ions). At very low temperature noble gases crystallize in a lattice of spherical neutral atoms bonded together because of
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the weak interaction between instantaneous atomic dipoles, fluctuating with zero average (van der Waals forces). Even in this case there are no extended stationary electronic states. Lastly we find layered crystals composed by stacking individual atomic planes (quasi 2D crystals), with strong covalent in-plane bonds and weak interplanar van der Waals interactions. graphite belongs to this class. Graphite has a hexagonal Bravais lattice with a basis of four carbon atoms. In plane bonding is due to overlapping of 22 hybrid atomic orbitals (with a trigonal coordination) while charge fluctuations in the cloud of 2 electrons, originating from atomic orbitals perpendicular to covalent graphenic planes, provides weak van der Waals interactions between pairs of such planes. Thus graphite is a solid lubricant due to the easy glide between adjacent planes. 2 orbitals give rise to extended interplanar quasifree electronic states responsible for electrical conductivity and most optical properties of this crystal. Graphite can be conceived as a sort of molecular crystal, being the graphenic planes the composing molecules. Recently single graphenic planes have been isolated as true 2D crystals (graphene) with exotic optoelectronic properties deriving from electronic bands with linear dispersion associated with the so called massless Dirac fermions.
1.3
Tensorial observables
The properties of a given crystal depend from both its structure (including defects) and chemical composition. The physical, chemical... observables can be classified as scalar (rank 0 tensors), vectorial (rank 1 tensors) and tensorial (rank ≥ 2 tensors). Without pretending any generality, we examine by means of an example the role of symmetry in determining the crystal properties. Here we consider only spatially homogeneous observables (the local quantities are averaged over a primitive cell). In particolar we examine dielectric susceptibility χ representing the response of an insulating crystal to an external electrostatic field E. Introducing the polarization vector P (mean electric dipole moment per unit volume) it turns out that (for not too intense fields and excluding ferroelectric crystals4 ) = ◦
(17)
In this formula, as in all tensorial equations, the sumPfrom 1 to 3 over all pairs of repeated indexes is understood (in our case =13 ) and ◦ is the vacuum dielectric constant. χ = is a rank 2 tensor. In isotropic solids the e (as a function of temperaferroelectric crystals can own a permanent polarization P ture) indipendently from the application of an external field. In this case the law must be modified as = ◦ + e 4
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tensor is diagonal: = . In this case the nine components of χ collapse into a unique scalar and the vector P is always parallel to the electric field E. In general there is a variable degree of crystalline anisotropy. Let us now consider the effect of a coordinates transformation, for instance a rotation. After this geometric operation the new coordinates (e.g. of a lattice point) can be expressed in terms of the old ones as: 0
= (18) ³ ´ 0 where the matrix = cos d is made by the cosines of the angles between each new coordinate axis and each old one. Then linear algebra gives us the tensor transformation formula: 0
(19)
=
If the considered rotation is a symmetry operation of the crystal (that is, 0 it belongs to the crystal space group) then χ = χ and the constraints 5 − = 0 lower the number of independent components of tensor χ. The number of independent tensor components can be reduced further by other physical laws not directly related with the space symmetry of the crystal. For example, the stress tensor is symmetric = (it has only 6 independent compenents) because of the requirements of static equilibrium with respect to rigid rotations. The strain tensor, related to the displacement vector u by definition as = (12)( + ), is symmetric too by its definition itself. It follows that the elastic constants tensor (a 4 rank tensor), expressing the elastic linear response of a crystal in the generalized Hooke’s law = , consists of no more than 36 independent components (in front of a total number of components 34 = 81). Different considerations of energetic character lower that number further to 21. Every residual reduction is caused by symmetry, down to cubic lattices (s.c., b.c.c. and f.c.c.) described by only 3 independent elastic constants and to the limiting case of isotropic solids (e.g. polycrystalline bodies) described by only 2 independent elastic constants. A complete description of the role of symmetry on tensor observables requires the use of group theory.
2
Scattering theory
To ascertain the static (average) structure of solids elastic scattering of probe particles is utilized. Using undulatory terms one can say that the structural 5
0
These conditions can be written in compact form as χ = aχa−1 = χ, introducing the matrix a−1 , the inverse of matrix a.
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information is extractable from diffraction of the wave associated with the particle used as probe. For X-rays the probe particle is a photon and the wave is of electromagnetic nature. When electrons or neutrons are employed as probes, the wave is their quantum wave function. In the case of crystals, from the measurement of the intensity of the scattered wave as a function of scattering angle it is possible to reconstruct both the Bravais lattice and the atomic positions of the basis. In the following mainly the quantum scattering of non-zero rest mass particles, like electrons and neutrons, will be treated. Because of different interactions (electrons interact with internal charges through electromagnetic forces and have a small penetration power; neutrons interact only with nuclei through strong forces at very short distances and also sense nuclear spins) electrons are suitable to explore surface crystallography or that of thin films, while neutrons give us bulk information and are very sensitive probes of light elements solids, in particular if solids contain hydrogen and its isotopes. Even X-Rays can provide bulk information. For X-Rays the interaction takes place mainly with valence and core electrons. In this case, beyond elastic scattering, also Compton scattering is relevant.
2.1
Elementary theory of elastic scattering
Some general theoretical quantities involved in the explanation of elastic scattering/diffraction experiments off crystals can be introduced in the following semi-phenomenological way. Let us consider a crystal hit by a monochromatic plane wave. In the case of an electromagnetic wave we could write its incident electric field as E◦ exp(k · r). Let us ussume that the pointlike atoms of the crystal, positioned at the points n of its Bravais lattice, react to the incident field each emitting in turn a spherical wave E◦ exp(k · n)n exp( |r − n| ) |r − n| whose amplitude is proportional to the incident field in n and to an atomic scattering amplitude n . Here = |k | = 2, being the wavelength of the incident wave. We define the scattered wave vector as k = r |r| = r in such a way that |k | = |k | (elastic scattering). The angle between k and k is the scattering angle . We set our detector of scattered waves atq the point r very far from the (finite) ¡ ¢2 p crystal in such a way that |r − n| = 1 − 2 r·n + ' 1 − 2 r·n ' ¡ ¢ √ r·n 1 − ' − k · n, where we have used 1 − ' 1 − 2 valid if ¿ 1 (far field approximation). Then the total waveX amplitude at E◦ exp(k · the detector can be written as: E(r) ≈ E◦ exp(k · r) + n
n)n exp() exp(−k · n)Now, if the incident beam is collimated and if
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the angle is not too small, in the simplest Xcase of identical atoms (n = ) exp [− (k − k ) · n], or, introwe can write: E(r) ≈ (E◦ exp()) n X exp (−Q · n). ducing the transferred wave vector Q = k −k , E(r) ≈ (E◦ exp()) n
The measured intensity of the scattered wave is thus proportional to ¯ ¯2 ¯X ¯ ¯ ¯ 2 exp (−Q · n)¯ (Q) ∝ |E(r)| ∝ ¯ ¯ n ¯
which is a function of the scattering angle as |Q| = 2 sin(2)In the case X of a cubic lattice n = u + u + u , the sum exp (−Q · n) is
the product of three unidimensional sums of the type
n −1 X
exp (− ) =
=0
1−exp(− ) 1−exp(− )
(a partial sum of the geometric series
∞ X
exp (− ) =
=0 1 ). 1−exp(− )
In the crystal there are 3 primitive cells. We can trans-
form this last result, without changing its squared modulus, as this way (Q) sin2 ( 2) sin2 ( 2)
2 ( 2) sin2 ( 2) sin2 ( 2) ∝ sin . sin2 ( 2) sin2 ( 2) sin2 ( 2)
sin( 2) sin( 2)
In
The study of the function
shows that,
I(x) 100
75
50
25
0 -15
-10
-5
0
5
10
15 x
= = 10 for big enough N, it is almost zero everywhere except where = (2) ,
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being any integer (positive or negative, including zero). There the function (an extremely sharp maximum) is equal to 2 Remembering the expression for the reciprocal lattice vectors of a cubic lattice, we conclude that the condition to have maxima in the scattering intensity (constructive interference) is Q = g for any triplet also corresponding to a family of crystal planes of Miller indeces . In the next subsection this selection rule is further examined in a more rigorous way for particle scattering and shown to be fully general.
2.2
Quantum particle scattering amplitude: Born approximation
Let us consider a steady flow of fast particles of mass , initially free and all with the same kinetic energy ~2 |k |2 2, which interact with a crystal ideally without thermal motions. This may be a beam of electrons or neutrons. The stationary state of a single particle that interacts with the crystal is determined by the mean potential energy field (r) which has the same symmetry of the direct lattice, and the wave function of a scattered particle is governed by the time-independent Schroedinger equation. ~2 2 − ∇ (r)+ (r)(r) =(r) 2 If |k |2 = |k|2 =
2 = 2 ~2
(20)
(21)
the equation can be written as: ¡ 2 2¢ 2(r) (r) ∇ + (r) = ~2
(22)
The initial state of incoming particles is described by the plane wave (r) =k ·r
(23)
p = ~k = ~k
(24)
where is the linear momentum of the particles and k the incident wave vector. In general an incident probability current density vector is collectively associated to states like this j =
~ ~k (∗ (r)∇ (r) − (r)∇∗ (r)) = 2
(25)
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The plane wave satisfies the homogeneous equation (empty lattice): ¡ 2 2¢ (26) ∇ + (r) =0 ¡ ¢ 0 The Green function (r r ) of operator ∇2 +2 is, by definition, the solution of the non homogeneous equation ¡ 2 2¢ 0 0 (27) ∇ + (r r ) = (r − r ) 0
where (r) is the Dirac delta function. Using (r r ) the more complex equation ¡ 2 2¢ ∇ + (r) = (r) (28) can be solved by means of a superposition integral: Z 0 0 0 (r) = (r)+ (r r )(r )r
(29)
The explicit form of the Green function turns out to be (see e.g. Davydov, Quantum Mechanics, Pergamon Press): 0
(r r ) = −
0 r−r
0 4 |r − r |
(30)
The initial Schroedinger equation transforms then into the integral equation: (r) = (r)−
2~2
Z
0 r−r
0 0 0 0 (r ) (r )r |r − r |
(31)
¯ 0¯ If ¯r ¯ 1 (far field approximation) the spherical wave emitted from 0 point r within the scattering volume can be approximated by the product of a spherical wave emitted from the origin (always taken within the scattering volume) and of a plane wave emitted in the direction of wave vector k = r 0 r−r
−k ·r0 ≈ 0 |r − r |
(32)
being k = r the scattered wave vector 6 . The angle between k and k is called scattering angle. The direction of k can be given by the angles r q ¯ ¯ ³ 0 ´2 ³ 0 0 0¯ ¯ ¯r − r ¯ = 1 − 2 r·r + ' 1 − 2 r·r ' 1 − √ where we have used 1 − ' 1 − 2 valid if ¿ 1 6
0
r·r
´
0
' − k · r
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and of spherical coordinates and the infinitesimal solid angle about this direction is Ω = sin . The radial probability current density vector µ ¶ ~ (r) ∗ (r) ∗ (r) − (r) (33) j (r) = 2 is associated to the scattered particle beam. The integral equation can be written again as k ·r + (r) = (34) Seen from afar, the set of scattering centers (contained within the scattering volume ), when hit by a monochromatic plane wave, generates a modulated spherical wave characterized by the scattering amplitude Z 0 0 0 0 −k ·r = − (r ) (r )r (35) 2 2~
Within Born approximation potential energy is considered as a perturbation with respect to kinetic energy. Consequently under the integration symbol 0 0 one writes (r ) ≈k ·r . In this way multiple scattering is neglected and the first order scattering amplitude is obtained as: Z 0 0 0 = − (r )−Q·r r = (36) 2 2~
k + Q|(r)|k = − 2~2 where Q = k − k
(37)
is the transferred wave vector. Within this approximation the scattering amplitude is proportional to the Fourier transform of index Q of interaction potential energy. Perturbations produced by the corresponding force field generate transitions between initial states |k and final states |k + Q = |k both described by plane waves (far from the scattering volume). The quantities k |(r)|k are called matrix elements of perturbation operator (r). The differential cross section (Ω) = (number of scattered particles per unit time within the solid angle Ω)/(flux of incident particles) = ( 2 Ω ) turns out to be: (38) (Ω) = | |2 Ω
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2.3
Solid State Physics Lecture Notes
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Elastic scattering - Bragg law
In case of elastic scattering initial kinetic energy is conserved and thus |k| = |k| Furthermore, in crystals the potential energy owns the lattice periodicity and can be represented as the Fourier series X g ·r (39) (r) =
g being the reciprocal lattice vectors, with Fourier coefficients Z 1 = (r)−g ·r r Then the scattering amplitude can be written as: Z X 0 0 0 g ·r −Q·r = − r = 2 2~ Z 0 X 0 (g −Q)·r r = − 2 2~
(40)
(41)
If the scattering volume is much bigger than |Q|−3 and also, as it is generally true in most scattering experiments, than , and if the far field approximation is still valid, in practice the integration can be extended over all space ( → ∞), so obtaining = −
4 2 X (Q − g ) ~2
(42)
From this last result it is seen that the scattering amplitude is different from zero (in the real case, it is maximum) if and only if Q = g . Formula (42) has a simple physical meaning thanks to the properties of reciprocal and direct lattices. A particular family of direct lattice planes owns as normal vectors the reciprocal lattice vectors labelled by the triplet of least integers () (identical to crystallographic Miller indices of that family of planes) and by all tripletys {} that are obtained from () multiplying every index by the same integer . E.g. in a simple cubic lattice: (110),(220),(330),..... Vectors g220 ,g330 ,..... share the same direction with vector g110 and for the moduli we have: |g220 | = 2 |g110 |, |g330 | = 3 |g110 |,.....√ The first neighbour planes distance of this family is 110 = 2 |g110 | = 2 and represents the space period of the Bravais lattice along direction 110 perpendicular
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to all considered planes. Let us consider the necessary condition to have maximum scattering amplitude: Q = k − k = g
(43)
|k − g |2 = |k |2
(44)
|k |2 − 2k · g + |g |2 = |k |2
(45)
and write it as that is For elastic scattering and introducing the Bragg angle as the complement to 900 of the angle between k and g , one gets |g | − 2 |k | sin = 0
(46)
This law tells us that maximum scattering is obtained when the projection of particle wave vector onto the vector g equals half modulus of g (see Umklapp below). For = 2 |g | and = 2 |k | is the de Broglie wavelength of incident particles, one eventually obtains 2 sin =
(47)
the well known Bragg law (which gives the angular positions of Bragg peaks, being the Bragg angle one half of scattering angle: consider the vector diagram of equation k − k = g ). To arrive at (42) we used the result: Z 0 0 (48) (g −Q)·r r = (2)3 (g − Q) easily obtainable in 1D as Z
+∞
= 2()
(49)
−∞
considering that (see Appendix) Z +∞ 1 = 2 −∞
Z + 1 lim = →∞ 2 − sin() = () = lim →∞
(50)
is anyhow useful to describe the interaction with small crysThe form sin() tals and is the basis for the experimental determination of crystalline domains size analyzing the shape of diffraction peaks. If a family of lattice
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planes {} satisfies Bragg law g − Q = 0, the peak intensity (area) is proportional to the squared modulus of and dipends on the matter distribution (distribution of scattering centers) within a single primitive cell, that is on the basis containing atoms. In the case of neutron scattering, due to the extremely small interaction range,Pin (40) one can write (being the integral limited to only one cell) (r) = =1 (r − r ). The r are the positions of the basis nuclei of primitive cell and all coordinates refer to the local origin of the cell itself. This form of interaction energy between probe particles and crystal atoms is called Fermi pseudo-potential. The are the scattering amplitudes of nuclei. Then it turns out: Z X 1 = (r − r )−g ·r r = =1 1 X −g ·r (51) = =1 The quantity =
X
−g ·r
(52)
=1
is called the geometric structure factor. The intensity of a single Bragg peak () is then proportional to the squared modulus of . Sometimes, due to a particular basis structure generating distructive interference, may be null and the corresponding Bragg peak disappears (see below). In the case of both X-Rays and electrons is given by the same formula but with electronic quantities (Q = g ) substituting the nuclear constants . The (Q) is an individual atomic property mainly related to core electrons. Quantities (Q) are called atomic form factors (see below).
2.4
X-Ray Diffraction
For X-Ray elastic scattering (XRD: X-Ray Diffraction ) most considerations already introduced for particle scattering remain valid. In particular similar formulae for both scattering amplitude and differential cross section are obtained, provided the role of (r) is played by the electronic charge density (r). In fact the wave equation, in the monochromatic case, for the electric field propagating in a medium of index of refraction (r) can be
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Solid State Physics Lecture Notes
reconducted to the vector Helmholtz equation7 µ ¶ 2 2 2 ∇+ 2 E(r) = 0
25
(53)
Taking into account that (above plasma frequency, see eq. 428), for X-Rays we can write (r) 2 = 1 − (54) 2 0 one gets the equation ¶ µ 2 (r) 2 E(r) (55) ∇ + 2 E(r) = 2 0 with 2 2 = 2 . This equation is quite similar to the scalar equation 22. In our analogy the role of is then played by : Z 1 = (r)−g ·r r (56) if (r) =
X
=1
(r − r )
(57)
(in the assumption, not applicable to covalent solids, that the electron charge density is the sum of the electron densities of each atom ):
∝ = = =
1 1 1 1
Z
X
=1
Z
X
=1
X
=1
X
(r − r )−g ·r r = (r − r )−g ·(r−r ) −g ·r r =
−g ·r
Z
(r)−g ·r r =
−g ·r (g )
(58)
=1
where 7
The propagation of an electromagnetic wave in a mean with refractive index is de2 2 scribed by equation ∇2 E = 2 E 2 , where E is the electric field. When E = E(r) exp(−ω), the Helmholtz equation is obtained.
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Solid State Physics Lecture Notes
(Q) =
Z
(r)−Q·r r
26
(59)
once that the diffraction condition is set Q = g . In case of X ray diffraction we define the structure factor as: X = −g ·r (g )
(60)
=1
where depends on the nature of the pth atom.
Monoatomic lattice with a basis: X −g ·r = (g ) = (g )
(61)
=1
P We can calculate = =1 −g ·r in case of a bcc lattice considered as a simple cubic lattice with a basis: = 1 2 r1 = 0 r2 = (2)(x + y + z) g = (2)(x + y + z) = 1 + exp[−g · ((2)(x + y + z))]½= ++
= 1+exp(−(++)) = 1+(−1)
=
2 when + + is even 0 when + + is odd
¾
in reciprocal space this converts the simple cubic lattice with side 2 (the reciprocal of a simple cube with side ) in a fcc lattice with a conventional cell with side 4, i.e. exactly the reciprocal of a bcc with side , for which is 1 (see picture 6.11 Ashcroft-Mermin). P Then we calculate = =1 −g ·r in case of a fcc considered as a simple cubic lattice with a basis: = 1 2 r1 = 0 r2 = (2)(x + y) r3 = (2)(y + z) r4 = (2)(x + z) g = (2)(x + y + z)
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= 1 + exp[−g · ((2)(x + y))] + exp[−g · ((2)(x + z))] + exp[−g · ((2)(y + z))] = =½ 1 + exp(−( + )) + exp(−( + )) + exp(−( + )) = ¾ 0 if in : 2 are even and 1 is odd or 2 odd and 1 even = 4 if in : 3 are even or 3 are odd in reciprocal space this converts the simple cubic lattice with side 2 (the reciprocal of a simple cube with side ) in a bcc with a conventional cell with side 4, i.e. exactly the reciprocal of a fcc with side , for which is 1 (see picture 6.11 Ashcroft-Mermin). P Finally we calculate = =1 −g ·r in case of diamond lattice considered as a fcc with a basis: = 1 2 r1 = 0 r2 = (4)(x + y + z) g = (b1 + b2 + b3 ) b1 = (2)(y + z − x) b2 = (2)(z + x − y) b3 = (2)(x + y − z) ⎧= 1 + exp(− 2 ( + + )) = ⎫ ⎨ 2 if + + is two times an even number ⎬ 1 ± , if + + is odd = ⎩ ⎭ 0 if + + is two times an odd number
Bragg formulation. As has already been discussed in the particle scattering, Q = g is equal to 2 sin =
(62)
where is the spacing between to planes in the family of crystal planes with indices (), is the angle between k (and k ) and the plane, is the wave length of radiation X, in an integer number. Q = g is equal Q = k − k = 2 2 sin = 2 = g . Von Laue formulation. Since |k | = |k | = , then Q = k − k = g k = k − g by squaring both sides of this (and since = |k − g |) we obtain:
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Solid State Physics Lecture Notes
k · Powder diffraction:
g 1 = |g | |g | 2
28
(63)
= 2 where is the angle of the diffraction (angle between k and k ). 2 sin
(64)
OBSERVATIONS: The intensity of diffraction is proportional to the Fourier transform of the static autocorrelation function: (Q) is the scattering amplitude, (Q) is the intensity experimentally measured with scattering wave vector Q = k −k defined by experimental conditions.
(Q) ∼
Z
(r)−Q·r r Z
−Q·r0
Z
(Q) ∼ |(Q)| ∼ (Q) (Q) ∼ (r ) r ∗ (r)Q·r r = Z Z Z ∗ −Q·R = (r + R) (r) rR = (R)−Q·R R (65) 2
∗
0
0
(R) is called static autocorrelation function or Patterson function.
(R) = x-rays: neutrons:
Z
(r + R) ∗ (r)r Z (R) = (r + R)∗ (r)r 2
(R) =
Z X
= 2
X
(r + R − r )
(R − (r − r ))
X
(r − r )r = (66)
The formula for neutrons refers to a monoatomic material, with equal for any atom.
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Liquids (neutron scattering):
2
(Q) ∼
= 2
Z X
X
(R − (r − r ))−Q·R R =
−Q·(r −r ) = 2
2
à X
1+
2
= +
Z X
X X
6=
−Q·(r −r )
!
=
(R)−Q·R R =
µ ¶ Z 2 −Q·R = 1 + (R) R
(67)
where (R) is the pair-correlation function, i.e. (R)R gives the mean numbers of atoms in a volume R around distance R from one atom in the liquid ( (R) = (R): static non local homogeneity hypothesis). In the hypothesis of statistical isotropy ((R) = ()): µ ¶ Z −Q·R (Q) ∼ 1 + () R = µ ¶ Z 2 − cos 2 2 sin = = 1 + () ¶ µ Z sin 2 2 = 1 + 4 () 2
(68)
where () is the radial distribution function, i.e. 42 () is the mean number of atoms located in a spherical shell with thickness centered on an atom in the liquid. () exhibits characteristic oscillations with maxima in values of corresponding to a short range order (dynamic statistical order: first neighbours, second neighbours, etc.). A monoatomic gas doesn’t exhibit characteristic oscillations in ().
3
Electronic States
Physical and chemical properties in solids strictly depend on electronic motion, both stationary and not. Considering valence electrons motion as independent from motion of ions in the lattice (or from motion of nuclei, since
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core electrons can be considered strictly bounded to them) provides just a good approximation that will be examined in detail at the beginning of chapter 4. See chapter 4 also for a better comprehension of this complex issue. However in this chapter we will assume this independency and, moreover, we will assume nuclei as stationary in their equilibrium positions. If these equilibrium positions form a Bravais lattice (in case with a basis) the reduced many-electron problem can be simplified again and considered as a one-electron problem in a mean periodic potential (see also paragraph immediately after equation (220)). We start trying to solve the equation for stationary states for an isolated multielectron system:
where and
b (r) = () (r) H
b=H b0 = b + (r) + (r) H b =
X |pb |2
2
=−
~2 X 2 ∇r 2
(69) (70) (71)
is the total kinetic energy operator for valence electrons, (r) =
2 1X 2 6= 40 |r − r |
(72)
is the Coulomb potential energy associated with repulsive interactions between electrons, XX 2 (73) (r) = − 4 |r − l| 0 l
is the Coulomb potential energy associated with attractive interactions electronion. For simplicity we considered a simple monoatomic crystal: l indicating the positions of nuclei/ions and their bare charge. r = (r1 s1 r2 s2 r s r s )
represents the set of dynamic coordinates (position and spin) of all valence electrons. We neglect relativistic effects (including spin-orbit interactions and direct magnetic spin-spin interactions). With a procedure similar to the Hartree or Hartree-Fock self-consistent mean-field approximation used for many-electron
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atoms (see appendix), we simplify the previous many-body problem to an effective one-electron problem b (r ) = (r ) H
(74)
|pb |2 b + h (r )i + (r ) (75) H = 2 hi representing the average value of electron-electron interactions. Here we do a further approximation: we assume that the mean periodic potential h (r )i + (r ) = (r ) is the same for every electron (a good approximation for macroscopic crystals): for this reason by now on we drop the subscript . From a physical point of view the sum of h (r)i and of the bare periodic potential (r) is equivalent to the electrostatic screening of the latter. In other words: the generic valence electron moves in an attractive periodic potential (r) resulting from the interaction with lattice ions, whose positive charge is partially screened from the mean negative charge distribution of all other valence electrons. This mean field approach is static in nature and neglects the dynamic correlation effects present within the screening electrons cloud surrounding ions (and the screening holes cloud surrounding every generic electron, see below the introduction to the quasiparticle concept). In a Fourier representation of periodic potential this Hartree-Fock approximation can be described by a dielectric function depending on wave vector (see Bloch’s theorem). This dielectric function substitutes 0 in a formula in which the Fourier transform of (r) appears.
3.1
Bloch’s Theorem
b n is defined as follows: The translation operator T
b n (r) = (r + Tn ) = (r+1 a1 + 2 a2 + 3 a3 ) T
(76)
b n () = (+Tn ) = (+) T
(77)
In a one dimensional space:
We consider now the problem of determining stationary electronic states in the lattice within an independent electron approximation in a mean periodic potential: (78) (+Tn ) = (+) = ()
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We have to solve the eigenvalue equation: b =
where
2 b = b + () 2 b is invariant under translations such as T b n: Since
b n] = 0 b T [
we can actually write
(79)
(80)
(81)
b n ()() b n () b b + )( + ) = () b T T = (
From a general principle of quantum mechanics, if the commutator between two operators vanishes, then the two operators share the same set of eigenb n commute, it is possible to write: b and T functions. In this case, since b n () = (n)() T
b n is an additive operator, which means: T
b n2 () = (+1 + 2 ) = b n1 T T
(82)
(83)
b n1 +n2 () = (+(1 + 2 )) = T
Using now the eigenvalue equation we obtain:
(n1 )(n2 )() = (n1 + n2 )()
(84)
(n1 )(n2 ) = (n1 + n2 )
(85)
This equation is satisfied by (n) =
(86)
for any choice of the complex number , provided that it has dimensions of reciprocal length. Exploiting periodicity we apply the normalization condition within a single primitive cell of volume : Z ¯ ¯2 ¯b ¯ |()| = 1 = ¯Tn ()¯ = Z 2 2 = |(n)| |()| Z
2
(87)
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i.e. |(n)|2 = 1 = | |2
(88)
=
(89)
b n () = (+Tn ) = (+) = () T
(90)
() = ()
(91)
(+Tn ) = (+) = ()
(92)
then, necessarily with real. Therefore:
This property is satisfied by any () (Bloch wavefunction):
in which is a periodic function in . From a physical (intuitive) point of view it is important to consider that Bloch functions are plane waves modulated by a periodic factor. b ( () is a stationary state of an electron in To any eigenfunction of a periodic potential) b () = () (93)
a wave vector is associated such that equation (90) holds, and therefore also equations (91) and (92) are true. Stationary energy levels associated to the electron depend on wave vector too. In 3D we generalize the above result using the vector notation: (r + T ) = k·T k (r)
(94)
k (r) = k (r)k·r
(95)
k (r + T ) = k (r)
(96)
where and
b = −~∇ to a Bloch wave and to a Applying the momentum operator p plane wave leads to two different result. For a Bloch wave: bk (r) 6= ~kk (r) p
(97)
This means that a Bloch wave is not an eigenfunction of momentum operator as instead a simple plane wave is. Thus the term ~k is called
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quasimomentum or crystal momentum. The mean momentum (the quantum expectation value of momentum in a Bloch state), instead, is equal to (for a complete derivation see Appendix E, Ashcroft and Mermin): p =
k =v ~ k
(98)
k = Normally Bloch stationary states are propagating waves unless k 0 (In which case we have a standing wave. This can happen at specific points: see below other typical bands structures k ). The existence of stationary electronic propagating waves in a periodic accelerating potential is a purely quantum effect. Since in real crystals the electron motion usually encounters some resistance, there must be something that alters the perfect periodic structure of the crystal, a periodicity defect. Insertion of the Bloch wave function in the form 91 into 3D version of eq. 93 yields the following eigenvalue problem:
− id est:
¢ ~2 ¡ 2 ∇ k + 2k · ∇k − 2 k + (r)k = k k 2 "
# (b p + ~k)2 + (r) k = k k 2
b = −~∇ is the momentum operator. where p
(99)
(100)
For any fixed k, the eigenvalue problem (100) defines a spectrum of discrete eigenvalues k = k . Every energy level k is labelled with a branch index which stands for the set of quantum numbers needed to label the eigenvalue spectrum at fixed k. Usually these k are degenerate energy levels: all states belonging to the same branch can be written as different k·r Bloch functions , where now represents the subset of k () = k () quantum numbers required to span all states. depend on the nontranslational part of the space group of the crystal. By fixing and letting k vary we obtain a hypersurface (a line in 1D, a surface in 2D): a specific electronic branch of the band structure of the crystal. Letting both and k vary we obtain the entire band structure. In this context a specific band is the whole energy sub-interval spanned by all k values. Sometimes the word band is used to indicate a branch, generating some confusion. In particular, adopting our nomenclature, we can say that, in some crystal structures, different energy bands can be (partially) superimposed. As far as the physical meaning of the single electron energy levels k is concerned, if the periodic potential is the self consistent one obtained by the Hartee-Fock method
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(see appendix on self consistent mean field) assuming a Slater determinant crystal wavefunction, describing all itinerant electrons, built by all occupied spin-orbitals, with orbital parts constituted by the one-electron Bloch states k () = k ()k·r , we can state two relevant results. First (Koopman’s theorem): −k equals the ionization energy of a single electron initially in state k () being the global initial state the minimum energy one (ground state). Second: k − k ≈ ∆ approximately equals the transition energy between the initial state k (), belonging to the global ground state, and the final state k (), belonging to an excited global state. The two results are tenable provided all other spin-orbital are unchanged in the involved process (no relaxation). The fact that both the Coulomb and the exchange integrals present in Hartee-Fock theory scale as 1 where is the number of primitive cells make the above statements particularly good for itinerant states such as Bloch waves (see e.g. the tight binding method). For a proof (including the definitions of Coulomb and exchange integrals) see the advanced textbook G. Grosso & G. Pastori Parravicini Solid State Physics (2-nd ed.) Academic Press (2014), pag. 146. If instead the present version of the Density Functional Approximation is used, the k have generally no direct physical meaning (ibidem pag. 163).
3.2
Reduction to the first Brillouin zone
In one dimension. Let us first consider = ( being any reciprocal lattice vector). In this case (), associated to the eigenvalue , is a periodic function with the same space periodicity of the lattice. ( + ) = ( + )(+) =
= ()
(101)
= ()
We can generalize that to a 3D situation as: g (r + T ) = g (r)
(102)
We consider now, for the sake of simplicity, a generic Bloch wavefunction = in a one dimensional situation; let , and , be fixed. () b and related to the eigenvalue/energetic The state , eigenfunction of level , is also eigenfunction of bn and thus related to the eigenvalue :
bn =
Since there is a one-to-one correspondence between the eigenvalue and the energy level , and since is a periodic function of with
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periodicity = 2, also will be a periodic function of with the same periodicity. The representation of the states as functions of , with −∞ +∞ (repeated-zone scheme), is so highly redundant, since the same branch segment is shown in all infinitely many equivalent wave vector periods. Fixing univocally the corresponding to a specific energy level/state is now necessary. This is achieved using the reduced-zone scheme where we consider only the wave vectors belonging to the first Brillouin zone, in such a way to associate to any state of branch a single wave vector . In one dimension this corresponds to limiting the vector within the interval: − ≤ (103) A third representation is the extended-zone scheme in which in any Brillouin zone there is just one electronic branch. Starting in the first zone and increasing, in any direction, the modulus of the wave vector, we will assign to any zone only one specific branch , in such a way to progessively increase the index (the first branch in the first zone, the second branch in the second zone and so on).
Umklapp of free electron parabola . As a limiting case, we consider now an empty (periodic!) lattice. This physically corresponds to a vanishingly small periodic potential. The Bloch states reduce to plane waves −12 k·r (they describe the motion of a free electron) and the energy levels associated to them are given by the following formula, which can be considered also as a valid representation in the extended-zone scheme with an arbitrary space period. ~2 |k|2 k = 2
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The picture shows the result of this fictitious periodicity of an empty lattice. In order to obtain the reduced-zone scheme we must back translate (umklapp) of reciprocal lattice vectors, the different pieces of the parabola from any higher order Brillouin zone into the first Brillouin zone. A real periodic potential would introduce a distortion of the parabola and, generally, the opening of a prohibited energy gap between two consecutive bands. Writing the Bloch functions in their characteristic expressions can definitely persuade us that the states described with wave vector k and with wave vector k0 = k − g are exactly the same state. 0
k ·r −g·r k·r k·r (r) = = = = k (r) k (r) k0 (r) k0 (r) k0 −g·r , since −g·r and (r) In the formula we defined k (r) = k0 (r) k ¯0 ¯ ¯ ¯2 in the direct lattice have the same periodicity. Notice that ¯k (r)¯ = ¯ ¯2 ¯ ¯ ¯k−g (r)¯ .
3.3
Born-von Karman boundary conditions
Since real crystals are finite, they are not perfectly periodic. Yet, by neglecting surface states and introducing appropriate boundary conditions compatible with translational invariance, Bloch’s theorem still holds and provides a suitable approximation for the real case. In one dimension, considering N (N1) primitive cells in a crystal of length = we can write the periodic boundary conditions as: ( + ) = () = ()
(104)
which requires = 1This last equation implies that the possible (allowed) wave vectors belong to the discrete set of values: =
2
(105)
In the (− ≤ ) we have exactly N independent corresponding to integer varying between − 2 and 2 : −
≤ 2 2
(106)
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Since ∆ = +1 − = 2 2, wave vector nearly form a continuous set of values. The situation in 3D is very similar to this one. We consider = 1 2 3 cells, in the first Brillouin zone the allowed wave vectors are given by the following formula: ¶ µ 1 2 3 k(1 2 3 ) = 2 (107) b1 + b2 + b3 1 2 3 where −
≤ 2 2 = 1 2 3
(108)
Taking into account spin degeneracy and Pauli principle, it follows that a single branch k of the band structure (in the first Brillouin zone) can guest up to 2 electronic states (Bloch waves). As a result, knowing the band structure of a crystal, including the existence and width of forbidden energy gaps, it is possible to foresee whether the material is a conductor (metal) or a semiconductor/insulator. As an example we consider a monovalent element which crystallizes as a simple crystal, such as (a body centered cubic structure). In this case, in the ground state, the set of valence electrons occupies just the first half of k1 , leaving the second half empty. If accelerated by an external electrical field (see below "Single electron Dynamics") there will be a net current generated by the highest energy electrons making transitions to close empty energy states belonging to the same band: such a crystal is a conductor. Following similar lines of reasoning, to get a more general rule of thumb, for the sake of simplicity we assume that contiguous allowed bands, spanned by the different branches k , are separated by forbidden energy intervals called band gaps, i.e. ranges of energy that an electron in the solid may not have (see e.g. "Nearly-free electron bands"). Let be the atoms valence and let be the number of atoms in the basis; if is odd, the crystal should be a conductor, otherwise if is even, the crystal should be an insulator (or a semiconductor, if the width of the first prohibited band is less than or equal to ). However some real situations are more complicated and there are a lot of important counterexamples of our rough rule. For example all of the elements in the second column of the periodic table ( = 2) form metallic crystals, even though they are simple crystals (one atom per primitive cell). In these cases we should consider the complex topology of the real band structure in three dimensional space. Different branches with energy k can exhibit a partial superposition of the relative energy bands, along different directions in space. This is true for the above case (e.g for ): as a consequence the first band is not full
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and the second is also partly occupied giving rise to a metallic behavior. We will see that the empty states of the first band seem to behave as positive charge carriers (holes). In one dimension superposition of bands is not allowed because it would imply branch crossing: in a 1D world would be an insulator (or a semiconductor) according to our rule of thumb.
3.4 3.4.1
Free electron gas Plane waves - Fermi energy - Density of states
If we neglect completely the periodic potential, the one electron Hamiltonian coincides with the kinetic energy operator: 2
b = − ~ ∇2 2 and the equation for stationary states becomes
(109)
~2 2 − ∇ k (r) = k k (r) (110) 2 We already know that eigenvalues (degenerate energy levels) in this case are given by the following expression ~2 (2 + 2 + 2 ) ~2 |k|2 = (111) 2 2 and that the corresponding eigenfunctions are the plane waves 1 (112) k (r) = √ k·r where is the crystal volume. Vectors k are the discrete ones given by the periodical boundary conditions. If we consider the set of allowed wave vectors (corresponding to a lattice of points in space) we can notice that there is a volume (2)3 corresponding to any vector/point/orbital state (plane wave). The dispersion relation (111) shows that the √ locus of points of constant energy is a spherical surface of radius () = 2~. The number of states with energy between 0 and is then k =
4 ()3 () = 2 3 3 (2)
(113)
the factor 2 is introduced to take into account the spin degeneracy. The density of states thus assumes expression µ ¶3 2 2 √ (114) () = = 2 2 ~2
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The free electron gas ground state (at absolute zero) is determined by ordering the states of the system (112) by increasing energy, and by consecutively filling up the unoccupied quantum states with the lowest energy, according to the Pauli exclusion principle. When all the electrons have been put in, the Fermi energy is the energy of the highest occupied state. The corresponding spherical surface has a radius of = ( ). The Fermi wave vector is obtained from (113) by equalizing ( ) to the number of electrons contained in the volume µ ¶ 13 2 = 3 ¡ ¢ 23 ~2 3 2 (115) = 2 Both Fermi wave vector and Fermi energy depend on valence electron density (that is the number of valence electrons per unit volume). The probability that, at a temperature , a state with energy will be occupied is given by the Fermi-Dirac distribution (| ) (| ) =
1 exp( − )
(116)
+1
where is the chemical potential of an electron. Since depends weakly on the temperature (see "Electronic specific heat"), in most of the cases the approximation ( ) ≈ (0) = can be used. Wecan now introduce now the density of occupied states D(| ) = () (| ) = 2 2
µ
2 ~2
¶ 32
√ exp( − )+1
Obviously for an isolated conductor we have Z Z ∞ D(| ) = () = 0
(117)
(118)
0
When the Fermi wavelength = 2 has a value comparable with the interatomic distances (in 1 when = 2) we expect a Bragg reflection (electron diffraction, see "Scattering theory") to occur off a family of lattice planes for electrons on the Fermi surface. In this case the free electron model fails.
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Chemical potential and electronic specific heat
We start with a property (pointed out by Sommerfeld) of Fermi-Dirac distribution related to the fact that by differentiating the (116) with respect to energy we obtain a function similar (but not identical) to the Dirac delta function (red curve showed in the figure, with a negative sign). The width at half maximum of this function is ∼ ¿ ≈ . Thus the exact delta function behaviour is exhibited only at → 0 f
3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2
2.5
3 E
In this picture chemical potential is 2 a.u.
To make the above assumption stronger, we observe that: Z ∞ (| ) = (0| ) − (∞| ) = 1 − 0
The property is the following Z Z ∞ ()(| ) ≈ 0
0
2 () + 6
µ
¶
( )2
(119)
(120)
=
For a proof see, e.g., Ziman, Principles of the theory of solids 2-nd ed., Cambridge University Press. There, to derive the previous formula, the following identity (obtained by integration by parts) is used: µ ¶ Z ∞ Z ∞ (| ) () (| ) = () − (121) 0 0 R 0 0 where () = 0 ( ) . Then (120) can be written in another form (useful in applications) as: ¶ ¶ µ µ Z ∞ 2 2 (| ) ≈ () + () − ( )2 (122) 2 6 0 =
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When the temperature dependent term can be neglected it is possible to write −(| ) ≈ ( − ). We can now use eq. (120) to determine the dependence of the chemical potential on temperature. We write (118) as µ ¶ Z Z 2 () ≈ () + ( )2 (123) = 6 0 0 Since
Z
Z
() =
0
0
() +
Z
()
(124)
It is possible to obtain µ ¶ Z 2 () ≈ ( )2 ≈ ( ) ( − ) 6
(125)
Resolving the previous formula with respect to the chemical potential we get µ ¶ 2 ≈ − ln () ( )2 (126) 6 The apparently rough approximation ≈ is usually acceptable even beyond room temperature. The electronic contribution to specific heat could also be found by differentiating the internal energy with respect to temperature, being Z ∞ = ()(| ) (127) 0
Here we use a more direct approach. Because of the exclusion principle, at temperature , only electrons =
Z
+
−
2
2
() ≈ ( )
(128)
are thermally excited. Each of them contributes to the part of internal energy depending on the temperature = with an individual contribution Then ≈
¤ £ = ( ) ( )2 = 22 ( )
(129)
The exact calculation would give us the constant 2 3 instead of the prefactor 2 of the previous formula. In conductors this contribution to specific heat is important at low temperature and gives us a way to measure ( ).
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Quasi-particles and Fermi liquid theory (a qualitative introduction)
The dependence of density of occupied states on temperature shows us that, due to the exclusion principle, just a tiny amount of electrons with energy close to Fermi energy (in an interval ± 2 ) can be thermally excited. In the scheme of single particle states in a mean field, we can say that the excited state is defined both by the wave functions of the (few) electrons with an energy bigger than the Fermi one (more precisely, bigger than the chemical potential) and by the wave functions of the (many) remaining electrons with an energy smaller than the Fermi one. In the ground state all states up to are occupied and all states above it are empty. is of the order of some (for example in the value is = 31 ), while at room temperature we have ≈ 25×10−3 . The above "electron representation" of the excited state is very redundant. In order to get closer to the (quasi)electrons and holes representation we will use later, we introduce here, in a qualitative way, some important concepts due to Lev Landau. We linearize the dispersion relation about the energy point ≈ ¶ µ ≈+ ( − ) = ( − ) where the Fermi velocity is = }−1 () and where = } is the momentum of the quasi particles. If we consider as the new zero of the energy scale, the dispersion relation becomes = ( − ). This is valid only for excited electrons with . In the limit of validity of the linearization we have unoccupied states if . Afterwards we will see that the excited state can be described by few particles with negative charge (electrons) for which the previous dispersion relation is valid with = 0 and = and by equally few particles with positive charge (holes) for which is valid the dispersion relation obtained by the previous formula multiplied by −1: = − = ( − ) with . For the holes this transformation is equivalent to a time reversal transformation, as originally shown by Feynman. Electrons attract holes and holes attract electrons. Electrons with a cloud of holes and holes with a cloud of electrons are called quasi-particles and acquire an effective mass. Furthermore, since the dynamic cloud is not perfectly stationary, the quasi-particles have a finite life time () beyond which they are annihilated by collisions. Even neglecting this electron-electron effects (the cloud), just introducing the periodic potential, the velocity and the mass of electrons and holes can be changed.The above considerations will help us to explain why the mass of an electron in a crystal lattice is different from that of the electron in vacuum. Moreover
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two quasi-electrons repel each other less than two electrons separated by the same distance in vacuum. This can be represented by introducing a dielectic constant different from the vacuum one. All the same, mutatis mutandis, is valid for interactions between two holes and for electron-hole interactions. The number of quasi-particles is not constant (as is the number of electrons) but depends on temperature.The complex dynamic system described in this paragraph is called Fermi liquid. The theoretical framework needed to take into account many body effects in condensed matter is a particular aspect of Quantum Field Theory. We will not enter this complex field. The interested reader is referred to R.D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2-nd ed. Dover 1992.
3.5 3.5.1
Band structure: nearly free electron approximation Free electron diffraction
If we allow alkaline atomic elements to crystallize, due to their weak atomic first ionization potential, we can consider a correspondingly weak periodic potential () and treat it as a static perturbation (in comparison to kinetic energy). Then standard 2-nd order perturbation theory (see Appendix on "Approximate Methods") gives us the following expression for the perturbed energy levels (): ¯ ¯2 0 ¯ ¯ 2 k| (r)|k 2 ¯ ¯ X ~ |k| ³ ´ (k) ≈ + k|(r)|k + (130) 2 0 2 ~2 2 |k| − |k | k 6=k 2 Here we have adopted the Dirac notation. The unperturbed states | are the plane waves in the vacuum lattice (see above). We have to examine the matrix elements Z 0 0 1 1 √ −k·r (r) √ k ·r r = k| (r)|k ≡ (131) Z 0 1 −(k−k )·r (r)r =
The periodic potential energy () can be represented as a Fourier series X (r) = g g·r (132) g
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where the are the reciprocal lattice vectors and where the Fourier coefficients g can be written as: Z 1 (r)−g·r r (133) g = Thus: Z X 0 1 k| (r)|k = g g·r −(k−k )·r r = g Z 0 1 X = g (g−k+k )·r r =g k0 k−g g 0
(134)
In order to obtain (134) we used the periodic boundary conditions which give us a discretized set of wave vectors (105). A similar treatment can be found above in "Scattering theory" where a continuous set of wave vectors 0 is used. So, unless = − , the elements of the matrix are zero and the R 1 diagonal element |()| = () represents the mean value of potential energy. If we set this value to zero we can write the expression for perturbed energy levels as: ~2 |k|2 X + (k) ≈ 2 g6=0
|g |2 ¡ 2 ¢ ~2 |k| − |k − g|2 2
(135)
This representation makes sense for any with the exception for the ones that satisfy the degeneracy condition (525) (0) (k) = (0) (k − g)
(136)
|k| = |k − g|
(137)
|k|2 = |k|2 −2k · g + |g|2
(138)
i.e., using the (111): This condition is equivalent to
and consequently to k·
|g| g = |g| 2
(139)
For a better understanding of this equation in terms of Bragg reflection read the subsection Elastic Scattering - Bragg law. This condition in one dimension is satisfied by = ± = ± (). In this case the perturbation theory
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for degenerate energy levels (see "Approximate methods") must be applied first, using only two basis wave functions as 1 1 ± = √ (140) ∓ √ − 2 2 ¡ ¢ ~2 2 so splits into two The unperturbed degenerate level (0) (± ) = 2 non-degenerate energy levels
¯ ~2 ³ ´2 ¯¯ ¯ = ± ¯ 2 ¯ 2 ±
between which a forbidden energy gap is generated ¯ ¯ ¯ ¯ = 2 ¯ 2 ¯
¯ ¯2 If we examine the probability densities ¯± ¯ we can find that ¶ µ ¯ + ¯2 2 ¯ ¯ = sin2 ( ) µ ¶ ¯ − ¯2 2 ¯ ¯ = cos2 ( )
(141)
(142)
(143) (144)
¯ − ¯2 ¯ ¯ has its maximum values where = (we can find the minima of the potential in correspondence with the ions: lattice points), while ¯ + ¯periodic ¯ ¯2 has its maximum values at interstitial positions between ions, i.e. at ¡ ¢ + 12 (maxima of the periodic potential). This explains the difference of energy between the two different perturbed states. Since the degeneration has been removed it is now possible to use the general formula for non-degenerate energy levels for all wave vectors in the first Brillouin zone, including the ones at the zone boundaries, to further improve the accuracy of the method. 3.5.2
Fermi surface and density of states
The Fermi surface is the set of surfaces characterized by equations = (k) for each partially occupied zone . The bigger is the periodic potential, the stronger is the deviation of the shape of this surface from that of the free electron sphere. The density of states as a function of energy for a given branch is derived by integrating the density of states 2 (2)3 in space k over the infinitesimal volume bounded by the two surfaces with equation = (k) and + = (k) and dividing the result by :
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Z 2 () = k = () = (2)3 Z Z 2 = ⊥ = 3 3 4 |∇k | (2)
47
(145)
where is the infinitesimal element of the first surface while ⊥ represents the local distance between the two surfaces. We used the following formula: = |∇k | ⊥ . Thus the total density of states is: Z X X (146) () = 3 () = 4 |∇k |
If we plot the density of states as a function of energy, where the mean velocity of a Bloch state k = (∇k ~) is zero (this is a critical point in the first Brillouin zone) we find discontinuities in () called Van Hove singularities (the virtual discontinuities in () are smoothed out by integration). Since (−k) =(k), as stated by the Kramers theorem (symmetry − ), Γ [(k = 0)] is always a critical point. In lattices such as f.c.c., because of symmetry reasons, also [k =(2)(100)] and [k =(2)(12 12 12)] are critical points. Segment Γ is called Λ while segment Γ is called ∆. For a given branch , close to a critical point it is possible to write = (k) as a quadratic form; using the directions of the local principal axes we get: =0 + 2 + 2 + 2
(147)
The generic critical point is indicated with where represents the number of negative coefficients in the above expression. As an example we consider the density of states near 0 in an isotropic situation: = = = 0. Using polar coordinates |k| we obtain: Z p |k|2 sin = 2 32 −0 (148) () = 3 4 |k|= − 2 |k| 4 0 Using the effective mass ∗ approximation (see below) in this case we would get = ~2 2∗ . Similar considerations may be applied to the study of phonon dispersion relations (see below) in order to obtain their density of states in terms of frequency.
3.6
Band structure of tightly bound electrons
While in the interstitial space between two lattice ions, where the periodic potential is nearly constant, the Bloch wave functions are very similar to
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plane waves, close to the positions where lattice ions are located Bloch waves resemble localized atomic orbitals. In other words we assume that in the vicinity of each lattice point in position n the periodic potential can be approximated by the atomic potential of a single ion located in the same position n (r − n). In order to describe electronic bands in crystals with strongly bounded electrons we try to develop a Bloch state as an appropriate sum, over all positions in the crystal, of atomic orbitals (r − n) centered on ions themselves ( is the set of quantum numbers defining one of the possible stationary state for an electron in the mean field of central forces of a single atom). Thus satisfies the stationary Schroedinger equation (we put = 0): ∙ ¸ ~2 2 − ∇ +0 (r) (r)= (r) (149) 2 Then a trial Bloch function of the whole crystal can be written as: 1 X k·n (r − n) (150) k (r) = √ n
Computing k (r + l), being l too a lattice translation like n, and multiplying the right side of the equation by k·l −k·l we get : 1 X k·(n−l) (r − (n − l)) = k (r + l) = k·l √ n 1 X k·h (r − h) = k·l k (r) = k·l √ h Since both l and n − l = h are lattice translations, we have just verified that (150) satisfies Bloch’s theorem. Neglecting superposition integrals, function (150) satisfies the normalization condition, i.e.: Z
∗ k (r) k (r)r
Z 1 X X k·(n0 −n) 0 = 1+ ∗ (r − n) (r − n )r ≈1 n 0 n 6=n
The expectation value of energy in state k will be then: ∙ ¸ Z ~2 2 ∗ (k) = k (r) − ∇ +(r) k (r)r 2
(151)
(152)
where (r) is the exact periodic potential. Introducing ∆(r) = (r)−0 (r), a function with maximum at r = 0 (we set lattice point n as origin of coordinates), and again neglecting some superposition integrals, we get the final result:
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(k) =
X
Eh k·h
49
(153)
h
with:
E0 ≈ Z Eh6=0 ≈ ∗ (r + h)∆ (r) (r)r
(154)
Equation (153) is the exact Fourier representation of a periodic function of k (see below Wannier theorem); here we use for it the approximate expression (154) of Fourier coefficients valid within the tight binding approximation. Because of the nature of both ∆ (r) and (r), we expect Eh to be very small, with the exception of first neighbors. In one dimension ( = ), considering that E± is negative due to the attractive nature of the periodic potential with respect to the vacuum level, it is possible to write: () ≈ − 2 |E± | cos()
(155)
So we have a band with a minimum min = − 2 |E± | at = 0 and a maximum max = + 2 |E± | at the zone boundary edges where = ±: therefore the bandwidth is 4 |E± |. In case of a strong bond the hopping probability from one site to the next (proportional to Eh ) is small and the band is quite flat (localized atomic level). E(k)
ak
"tight binding" band The black curve represents (), the red curve the atomic level, the green curve the parabolic approximation. We have set min = 0. For small values of we can approximate the band as: ~2 2 ()≈ ∗ 2
(156)
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Comparing with (155) we obtain for the effective mass: ∗ =
~2 22 |E± |
(157)
So, electrons in bands of strongly localized states have a big effective mass. Remembering the discrete allowed wavevectors of the first Brillouin zone, in a crystal (a polystable system with ions - i.e. potential wells) the original atomic level splits in a band of levels () as a result of the above hopping events (transitions) between different wells. Jumps are promoted by ∆ , acting as a perturbation operator in (154) which resemble a matrix element of a transition in perturbation theory (see Appendices). 3.6.1
LCAO Method
A "tight binding" energy band built with a given atomic state can exhibit a partial energy superposition with another one built with a different state (corresponding to a different atomic energy level ). Along a given direction this generally happens when |k| is greater than a threshold value in the first zone. In this case there is a loss of significance in the concept of distinct band and band (take the example of band 4 and band 3 in transition elements as shown, e.g., in Ziman, Theory of Solids (2-nd ed.) pag. 113). This inconvenient can be eliminated using a linear combination of all (or almost all) different atomic orbitals (and not only one) corrisponding approximately to the same atomic energy ( = within the same atomic schell, for example the valence shell 22 of Carbon, containing four valence electrons and eight possible atomic orbitals, some filled and some empty) and so obtaining a generalization of the above trial tight binding wave functions, that is the LCAO Bloch functions: 1 X X k·n (r) = √ (r − n) k n
The variational coefficients can be determined minimizing ∙ ¸ Z ~2 2 ∗ (k) = k (r) − (r)r ∇ +(r) k 2
(158)
(159)
In interstitial regions atomic states (r − n) and Bloch states are very different one another; this happens because of energy barriers between pairs of ions which appear in the periodic potential and which alter the potential wells produced by isolated ions. Besides bound atomic states are an incomplete set for the description of a Bloch state with energy greater than periodic
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potential maxima (non tunneling states), where the energy spectrum is basically continuous (see unbound states with positive energy in isolated atoms). In the figure below the function ∆ is shown together with two different energy levels greater and smaller than periodic potential maxima. Thus it was necessary to look for better approximate methods based on basis functions closer to exact Bloch functions, but the description of such methods is beyond the scope of these lecture notes.
∆() and periodic potential
3.7
Electron dynamics in external fields
Being a consequence of translational invariance, equation (153) has a general validity and gives us the possibility to prove a theorem which is very crucial in the study of electron dynamics, when the crystal is plunged in an external field. The statement of Wannier theorem is contained in the following identity (−∇)k (r) = (k)k (r)
(160)
b = (−∇) is the operator obtained from function (k) in which where the following substitution is made: k → − ∇
(161)
We prove the theorem dropping subscript and considering only one band in a one dimensional situation where the lattice positions are given by = . b = (−∇), built starting from (k) expressed as We apply operator the Fourier series (153), to Bloch wave function k () and perform just a few steps:
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X )k () = E k () = (162) X 2 1 E (1 + + ()2 2 + )k () = = 2 X X E k (+) = E k () = ()k () =
c.v.d. In the last step we used Bloch’s theorem. 3.7.1
Equivalent Hamiltonian. Effective mass theorem
We consider now the motion of an electron in an external field (for example an electric field); let us assume that the electron initially occupies a Bloch state in a partially occupied band. Let (r) be the extra potential energy (to be added to the periodic potential (r)) gained by the electron plunged in the external field. The wave function of the electron obeys Schroedinger equation: b0 + (r) ~ = (163) where 2 b0 = − ~ ∇2 + (r) (164) 2 and the isolated crystal problem b0 k (r) = (k)k (r)
(165)
has already been solved. We look for a solution of (163) as a wave packet of Bloch states with time dependent coefficients: X k ()k (r) (166) (r ) = k
where the sum is over an appropriately limited (with respect to zone boundary) interval of values around a specific mean k. Since the set of allowed k is almost continuous in a macroscopic crystal, it is even possible to write: Z (r ) = k ()k (r)k (167)
This dynamic description of an accelerated wave packet is substantially a particle description (within the entire crystal volume). In fact in this way the
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uncertainty in the instantaneous position of the electron is greater than the lattice parameter but less than the minimum linear size of the crystal volume (yet, see below Limitations in the effective mass description). Introducing (167) in (163) we get: Z b 0 k (r)k + (r) (168) ~ = k ()
b0 and the Wannier theorem, it is possible Using the eigenvalue equation of to write: Z ~ = k ()(−∇)k (r)k + (r) = (169) Z = (−∇) k ()k (r)k + (r) = = (−∇) + (r)
i.e.:
= [(−∇) + (r)] We can now define the equivalent Hamiltonian operator as: ~
b = (−∇) + (r)
(170)
(171)
It is equivalent to the former Hamiltonian operator: 2
b = − ~ ∇2 + (r) + (r) 2
(172)
Formally (−∇) can be viewed as the kinetic energy operator associated with a particle immersed in an external field (r). Operator (−∇) contains both the inertial properties of the particle and the effect of the periodic potential. The result in (170), together with eqs. (174) and (176), is called effective mass theorem for a Bloch-waves packet. If we assume that k correspond to the minimum of a partially occupied band (it could be the bottom (0) of the conduction band in a semiconductor), in 1 it is possible to write: ~2 2 ()≈ (0) + (173) 2∗ where ~2 ´ (174) ∗ = ³ 2 () 2
0
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So the corresponding equivalent Hamiltonian is: 2 b = − ~ ∇2 + (0) + (r) 2∗
(175)
If we count energy from (0), we get:
2 b = − ~ ∇2 + (r) 2∗
(176)
This Hamiltonian leads the motion of an electron which in the absence of an external force F = −∇ (r) (177) would be free but with a mass different from the one in vacuum: it is just the effective mass ∗ , a concept we have already introduced in qualitative terms. In this way the periodic potential has been included in the equivalent 2 ~2 kinetic energy operator − 2 ∗∇ . If we, e.g., carefully observe the trend of LCAO bands around the top of the valence band and around the bottom of the conduction band in the first Brillouin zone and if, in particular, we apply the previous procedure to the top of the valence band, we will notice that in this case electrons have a negative effective mass, whose absolute value is generally different from that of electrons at the bottom of conduction band. E 8
6
4
2 0 -2.5
-1.25
0
1.25
2.5 k
The implied negative kinetic energy is difficult to conceive, unless we remove this oddness reversing the energy axis, counting energy downwards from the top of the valence band. This seems not to make any sense for valence band electrons, but will be useful, and physically sound, for holes.
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Impurity levels
If a tetravalent semiconductor such as is doped with substitutional atoms with five valence electrons such as , the exceeding electron can be bound near the donating atom in a localized state belonging to an energy level within the forbidden energy gap just below the conduction band. Due to thermal agitation this electron may gain enough energy to occupy a new (propagating) Bloch state in the conduction band and potentially become a charge carrier under the effect of an external field; in this case the initial localized state has been thermally ionized. In a situation like this one the semiconductor is said to be doped and the arsenic atom is called a donor. The energy levels of this impurity can be described as follows. The exceeding electron coming from atom , in a crystal, is plunged in an effective electric field generated by the positive ion acquiring an additional Coulomb-like potential energy: 2 (r) = − (178) 4 where is the distance between the electron and the ion and is the electric permittivity of . The wave function of the exceeding electron obeys the effective Schroedinger equation: µ ¶ ~2 2 2 − ∗∇ − = (179) 2 4 The energy eigenvalues are the hydrogen-like levels: = = −
∗ 4 82 2 2
(180)
These levels lie just below the conduction band. In fact in ∗ = 10 and = 10 0 and the ionization energy will be 1 ≈ 10−3 rydberg (1 rydberg 4 = 8 2 2 ≈ 136 eV) which is an amount of energy lower than at room 0 temperature (25 × 10−3 eV). This means that doping is an effective way to increase the electron density in the conduction band of . Since the Bohr 2 ∗ radius in ∗ = ∗ 2 is a factor ( 0 ) = 100 greater than the radius of hydrogen atom, the effective mass theorem can be used in this situation (the radius is greater than the lattice constant). 3.7.3
Semiclassical dynamics in a crystal
If (r) depends weekly on position, it is possible that the mean de Broglie wavelength = 2 of electrons, conceived as Bloch waves packets, is short enough not to produce strong electron-wave diffraction. In this case we cope
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with a semiclassical situation and it isn’t necessary to solve equation (170), but we can instead start with an equivalent classical Hamiltonian. Here we follow the most straightforward path to obtain the result. Let us begin with the original classical Hamiltonian: |p|2 = + (r) + (r) 2
(181)
using the Jordan quantization rules (first quantization) r→ r p→ −
~∇
(182) (183)
we obtain the Hamiltonian operator 2
b = − ~ ∇2 + (r) + (r) 2
(184)
Applying the Wannier theorem to the band structure = (k)
(185)
k → − ∇
(186)
and using the substitution we get the equivalent Hamiltonian operator: b = (−∇) + (r)
(187)
In a semiclassical situation it is possible to build the equivalent classical Hamiltonian by reversing the quantization rules (quantization dismounting) r→ r
(188)
p −∇ → ~
(189)
and so obtaining = Now, using Hamilton’s equations
³p´ ~
+ (r)
r = p p = − r
(190)
(191)
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we get ¡ ¢ p~ r = p (r) p = − r
(192)
i.e.: r 1 (k) = ~ k (r) k = − ~ r
(193) (194)
Differentiating the first equation with respect to time and substituting in it k from the second equation we obtain the Newton’s law: µ ¶ (r) 1 2 (k) 2 r − (195) = 2 2 ~ kk r This defines the tensor: µ
1 ∗ (k)
¶
=
1 2 (k) ~2
(196)
In 1 the effective mass is thus defined as: ∗ () =
~2
(197)
2 () 2
and it is a function of v,m
2
1
0 -2.5
-1.25
0
1.25
2.5 k
-1
-2
Speed and effective mass vs.
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Analyzing the figure we can understand some interesting applications. Imagine that an electron in a LCAO conduction band at a time = 0 has = 0. The electron is then accelerated by a constant electric field E = −u () ( 0). The electron feels the force F = − u = u . Integrating the (194) we get: 1 = (198) } As we can see the wave vector grows linearly with . In the reduced zone scheme, when = , a Bragg reflection happens; instantaneously the 0 wave vector becomes = − = − 2 = − and then it starts growing linearly again till the new reflection at the edge of the Brillouin zone occurs. In the figure are shown in black the trend of the mean velocity of the electron and his effective mass as functions of (red curves refer to the valence band). At first ( = 0 and = 0) the mass is constant and positive and the velocity grows linearly. At = (2) the mass becomes infinite, initially positive then negative, and in a while it stabilizes on a negative value; the velocity, instead, has already gained its maximum so it decreases until its value becomes zero at the edge of the Brillouin zone. These anomalies in the Bloch waves packet show that this description of the electron motion isn’t completely classical. After the Bragg reflection everything starts all over again: thus a constant force would produce a periodic motion! In the real situation (see Transport properties) due to scattering off phonons and lattice imperfections the electron wave vector has always limited to small values close to = 0 being cutt off by scattering processes and the periodic motion can be hardly observed experimentally. 3.7.4
Limitations in the effective mass description
Under too either intense or fast-varying fields, interband transitions become possible (tunneling under the energy barrier of hight ) and the concept of effective mass cannot be applied any more, at least in clear rigorous terms (see Optical properties). It is possible to define a upper circular frequency limit as: ~ and a upper limit on the applied electric field as: ≤
(199)
(200) where is the lattice parameter. Actually these conditions are neither restrictive enough nor unique. In a device the particle description (wave packet) |E| ≤
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needs the charge carrier localization ∆ to be appropriate with respect to the overall linear dimension: ∆ should be should be small enough so that the charge carriers could be considered as particles with respect to the size of the device itself, but, at the same time, big enough to have a quick reaction ∆ to external stimuli. The uncertainty ∆ of wave vector of a Bloch waves packet corresponds to an energy uncertainty of thermal origin: ~2 (∆)2 = 2∗ Using the Heisenberg’s uncertainty principle ∆ =
~∆∆ ≥ ∆∆ ≥
(201)
(202) (203)
we get ∆ ≥
∆ ≥ √ ∗ 2
(204) (205)
At 300 with = 1 (199) gives us a bandwidth of 2.4 x 1014 Hz while (204) gives 6.3 x 1012 Hz. At 300 (205) gives r ∆ ≥ 76 (206) ∗ In a device with an effective mass of 0.067 the smallest dimension should be about 29.4 . 3.7.5
Electric current - Electrons and holes
We want now to deepen former considerations considering a semiconductor with a completely occupied valence band and a completely empty conduction band.
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If at = 0 we apply force F, the response of each electron in the valence band can be treated as in the previous chapter. The trend of dynamic variables is shown in the first column of the following picture.
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At any instant in time the sum of the wave vectors is X = 0
61
(207)
and the total electric current is proportional to (which we are going to call current for the sake of simplicity): =
X X 1 k =0 (−) h i = (−) ~ k
(208)
We have so obtained the crucial result that a completely occupied band cannot conduct current. Let us assume now that, initially, a single electron occupies the conduction band, thus leaving an empty state in the valence band.
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At instant 1 we have, in the conduction band, X = 5 0
62
(209)
and the current is µ ¶ X X 1 k 1 k = = (−) h5 i = (−) (−) h i = (−) 6= 0 ~ k ~ k 5 (210) While, in the valence band, X = −5 = 3 0 (211)
with a current equal to µ ¶ X X 1 k 1 k = (−) h3 i = (−) (−) h i = (−) = = ~ k ~ k 3 (212)
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Therefore the total current is = + . Considering a subsequent instant = 2 it is possible to notice that the sum of wave vectors associated to all occupied states in the valence band decreases (it increases as the electric field increases) and that the total energy increases. The energy of the empty state decreases while the wave vector and the energy associated to the electron in the conduction band increase. The set of electrons in the valence band behaves as a single positive particle (with a wave vector opposite to the one of the empty state) whose energy increases, if we reverse the energy axes. This particle has a positive effective mass, at least in correspondence with small wave vectors, while electrons would have a negative effective mass. The picture below represents the two possible representations: the electron representation is shown in the left column while in the right one the electron-hole representation is depicted.
Basically this is the concept of hole in a dynamic band representation. Since it is a positive particle, the hole could also be considered as spatially localized describing it as an appropriate Bloch waves packet in the electron-hole representation. The electron-hole representation is a quite general concept that can be applied to all systems of Fermions in the framework of second quantization (see e.g. R.D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2-nd ed. Dover 1992). In doped with trivalent impurities (acceptors), such as , in order
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to complete tetrahedral bonding the substitutional atom captures a valence electron from , thus becoming a negative ion. Thereby in valence band remains an empty state that could be described by a hole. This hole is attracted by the impurity. As long as it is in a localized state, the hole could be described as stated in section Impurity levels and its energy levels are situated in the gap just above the valence band. If then the hole is thermally ionized and submitted to an external electric field, it can carry charge and contribute to the total current, as already shown for electrons. In this case we have a p-type semiconductor. 3.7.6
Excitons
In an intrinsic semiconductor each electron-hole pairs could also be generated by the absorption of a photon with energy greater than (see Optical properties). The couple is in a bound metastable state whose energy levels can be calculated using the same method presented in section Impurity levels introducing the reduced mass of the system, a sort of hydrogen atom where the nucleus is the hole (Wannier exciton). In optical absorption experiments peaks related to transitions between exciton levels can be measured in the vicinity of the absorption edge.
4
Adiabatic theorem and vibrational motions
Modelling a crystal as a set of independent valence electrons moving in a mean perfectly periodic potential generated by the interaction with a rigid lattice of ions does not explain many important physical facts. Among these: the temperature dependence of specific heat, thermal expansion, melting (thermodynamic properties), the temperature dependence of electrical resistivity of conductors and of thermal conductivity of insulators, low temperature superconductivity (transport properties), optic absorption of ionic crystals in the infrared, Brillouin and Raman scattering of light (optic properties). In this section we discuss an indispensable approximation for a feasible theory of all the above phenomena: the Born-Oppenheimer approximation or adiabatic theorem (principle). Before starting the treatment, we warn the reader that this approximation does not explain other important physical facts, like superconductivity, which require an "ad hoc" approach. In the previous sections we assumed that each independent valence electron in a solid moved in a mean periodic field caused by the lattice of ions and the other electrons (the symmetry of this field is the same as that of the space group of the considered crystal). Now we want to consider both the thermal
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motion of ions about the official atomic points (at a given temperature) and the motion of valence electrons in the crystal volume; the instantaneous position of a generic ion with charge is the sum of a vector of the Bravais lattice l and a displacement ul () (for the sake of simplicity, for the moment we consider a simple crystal, a crystal with a basis will be considered later). The dynamic variables of this quantum many-body problem are the displacements ul and the instantaneous positions r of all valence electrons. Thus the crystal Hamiltonian is: b = b + b + (r) + (r u) + (u) H
(213)
where r is the set of all electronic coordinates and u is the set of all ionic coordinates. ~2 X 2 ∇r (214) b = − 2 is the total kinetic energy operator of electrons; ~2 X 2 b = − ∇ul 2 l
(215)
is the total kinetic energy operator of ions;
2 1X (r) = 2 6= 40 |r − r |
(216)
is the repulsive Coulomb potential energy of electrons; (r u) = −
XX
l
2 40 |r − l − ul |
(217)
is the attractive Coulomb potential energy of ions and electrons; (u) =
1X 2 2 2 lh6=l 40 |l + ul − h − uh |
(218)
is the repulsive Coulomb potential energy of ions. First step. Since the electron mass is much smaller than the mass of an ion (if we consider a single proton, we have: = 1836 ) as a first approximation we can assume that the electrons wavefunction (r|u) has only a parametric dependence on displacements u. In particular, considering stationary states () (r|u) exp(− }), the (r|u) can be thought as the eigenfunctions of the reduced Hamiltonian (in which the term b has been neglected) :
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b0 = b + (r) + (r u) H
66
(219)
b0 (r|u) = () (u) (r|u) H
(220)
b=H b0 + b + (u) H
(221)
()
Electron energy levels (u) have a parametric dependence on the instantaneous positions of ions l + u. The wavefunctions (r|u) are many-electron functions (states). However it is possible to solve problem (220) using a mean field approximation (for example Hartree’s approximation) and obtain a set of single electron states (independent particles or quasi-particles, see above). Setting u = 0 in the single-electron mean-field Schrödinger equation, the total mean potential energy h (r)i + (r 0) exhibit the translational invariance of the Bravais lattive and it is possible to use Bloch’s theorem to calculate monoelectronic states (band structure). As a second step, we look b now for eigenfunctions of the full Hamiltonian H b HΨ(r u) = Ψ(r u)
where we use the superposition principles of states as: X Ψ(r u) = () (u) (r|u)
(222)
(223)
Multiplying the eigenvalue equation (222) by ∗ (r|u) and integrating over the electronic coordinates r ( (r|u) form a set of orthonormal functions), we get: h i () () b + () (u) + (u) () (224) (u) = (u)
where we have neglected several non-adiabatic terms which are listed below. In this approximation for each electronic stationary state (r|u) we find a spectrum of vibrational states () (u) corresponding to vibrational energy () () levels . In equation (224) the approximate total electronic energy (u) has the role of that part of potential energy which depends on ions motion. In order to obtain (224) we neglected the following terms: ¶ ´ µZ ~2 X ³ () ∗ − 2∇u (u) · (r|u)∇u (r|u)r + (225) 2 µZ ¶ ~2 X () ∗ 2 (u) − (r|u)∇u (r|u)r 2
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Solid State Physics Lecture Notes
The matrix indexed by and Z ∗ (r|u)∇u (r|u)r exhibits diagonal elements equal to zero, since Z Z ∗ (r|u)∇u (r|u)r = ∇u ∗ (r|u) (r|u)r
67
(226)
(227)
is the gradient of the (constant) total number of electrons with respect to ion displacements. Diagonal elements of the other matrix: Z −~2 (228) ∗ (r|u)∇2u (r|u)r 2 are multiples of the total kinetic energy of electrons by a factor and, thus, negligible. In fact in the worst case, i.e. in case of very strong electronion interaction, it is possible to write (r|u) = (r − u) and diagonal elements become: Z
5 5.1
−~2 2 ∇u (r − u)r = − u) 2
Z
−~2 2 ∇ (r − u)r 2 r (229) Generally, instead, all non-diagonal elements are different from zero thus possibly leading to transitions between electronic states while ions are moving (electron-phonon interaction). ∗ (r
∗ (r − u)
Lattice dynamics Lattice specific heat: Einstein model
In solid insulators and at high temperatures the Dulong and Petit’s law is valid. This law states that the molar specific heat is constant and equal to 3 where = ' 831 −1 −1 is the ideal gas constant, is the Avogadro constant and the Boltzmann constant. In contrast with what classical statistical mechanics states (equipartition theorem), experimentally tends to decrease towards zero at low temperatures. Einstein gave a first explanation of this behaviour developing a quantum oversimplified model of vibrational motion of nuclei in crystals. In this model Einstein assumes that every nucleus performs harmonic oscillations about its equilibrium position; all atoms oscillate with the same frequency . In light
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of modern quantum lattice dynamics, this assumption provides satisfactory results only at high temperatures, when the motions of nuclei can be considered as independent. However Einstein model explains in a qualitative way the dependence of on temperature when → 0. Each nucleus is considered a quantum harmonic oscillator with oscillation frequency equal to ; the spectrum of energy levels is then: µ ¶ 1 = + (230) ~ 2
Neglecting the zero-point energy, with a simple derivation we can obtain the mean energy associated to each oscillator: ~
=
~
= ~
(231)
−1 it is the energy associated to a single quantum of vibrational energy ~ multiplied by the oscillator occupation number given by the Bose-Einstein distribution.
=
1
(232)
~
−1 Now it is possible to conclude that the internal energy of the crystal (we are considering one mole) is: = 3 (233)
Thus is equal to: =
µ
¶
~
3 (~ )2 = µ ~ ¶2 2 −1
(234)
showing a temperature trend qualitatively similar to the experimentally observed one: lim ( ) = 0
(235)
lim ( ) = 3
(236)
→0
→∞
Around the Einstein temperature = ~ the curve exhibits a very sudden exponential change in the slope. Examining more carefully the experimental results we can notice that, when , tends to 0 more slowly than as it is stated by the exponential decay in the Einstein model. The low
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temperature empirical trend is ∝ 3 . In addition, at high temperatures the experimental is not exactly constant but it slowly rises as the temperature increases. This last effect would require the inclusion of anharmonic corrections (see below) and, being almost negligible, will not be considered further. In order to improve the agreement between theory and experiment at low temperatures, it is necessary to leave behind the Einstein model and to develop a more realistic model, able to better describe the true lattice dynamics. In particular at low temperatures motions of two near nuclei are generally correlated and this lowers the intensity of internal forces within the couple.
5.2
Lattice dynamics: Born - Von Karman model
On the basis of the adiabatic principle a full quantum treatment of lattice dynamics could be built. An alternative approach, phenomenological lattice dynamics, can be started with an initially classical description. The reason for the "quasi-classical" vibrational behaviour of nuclei in many circumstances is related to their very short thermal De Broglie wavelength. Let us consider a particle of mass accomplishing an oscillatory motion. The quantum mechanical version of virial theorem states that: 2 h i = hi where is the kinetic energy and a potential energy of the type ∝ . Thus for a harmonic oscillator h i = h i like in classical mechanics and h2 i h + i = 2 h i. Then h i can be estimated as 2 = 32 and an average De Broglie wavelenth of thermal origin as = √ 2 = √3 . Inserting h i
˚ which is the mass of the proton at room temperature one finds ≈ 145 shorter than the typical lattice spacing. This is even more true for heavier nuclei. According to adiabatic approximation we assume the existence of a vibrational potential energy which is a function of displacements of nuclei with respect to their official positions in the crystal when considered with no thermal motions (from now on we will use the Born and Huang notation) µ ¶ = Rn + r = r (237) n = 1 a1 + 2 a2 + 3 a3 + r µ ¶ is the position, with respect to the origin 0, of nucleus with r n mass situated in the primitive cell (which contains atoms) identified by lattice point Rn . r is the position of nucleus with respect to vector Rn (internal origin of primitive cell n). In the crystal, ideally with no
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thermal motions, each nucleus is located exactly in its official position and the potential energy attains its minimum value. We introduce now the displacements of atomic nuclei: µ ¶ µ ¶ µ ¶ 0 (238) =r −r u n n n 0
r indicates the instantaneous position of that nucleus ( n ), which fluctuates due to vibrational motions. If thermal motions slightly perturb the periodic structure of the crystal (in thermodynamic equilibrium this means that crystal temperature is much less than melting temperature) a series expansion of potential energy is feasible and can be arrested to the second order (the minimum value is put equal to zero): 1
13
1 XXX = Φ 0 2 0 0 0 nn
µ
0
0 n n
¶
µ
n
¶
0
µ
where we introduced the interatomic force constants tensor ∙ µ µ 0 ¶ 0 ¶¸ = Φ 0 Φ 0 0 n n n n defined as: Φ 0 0
µ
0
0 n n
¶
=
µ
2 ¶ µ 0 ¶ 0 0 n n
0
0 n
¶
(239)
(240)
(241)
The double sum nn is to be extended to all crystal cells for both n and 0 () n . Physically is the energy (u) + (u) = (u) which appears in eq. (224), including here the general situation of crystal with a basis, and () it is related to the electronic state (u) (for example the ground state) being a perturbation of it. This approximation is called harmonic approximation. Higher order terms in the series expansion of potential energy, the anharmonic terms, should be introduced in order to explain melting, thermal expansion and thermal conductivity. Harmonic approximation explains essential aspects of lattice vibrations-photon scattering (Brillouin and Raman scattering), the dependence of specific heat on temperature, infrared optical properties of ionic crystals, insulators and semiconductors and electrical resistivity in conductors (electron-lattice vibrations scattering). Remembering that a quantum treatment will be finally necessary, we start developing first a classical dynamics description. The vibrational kinetic energy of the
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crystal can be then written as: ¯ µ ¶¯2 1 ¯ 1 XX ¯¯ = ¯¯u˙ n ¯ 2 n
(242)
Introducing the Lagrangian = − , the Lagrange equations
µ ¶− µ ¶ =0 u˙ u n n
(243)
give birth to a system of classical (Newtonian) dynamic equations, a system µ ¶ of 3 equations in the 3 variables n u ¨
µ
n
¶
=−
1 XX 0
n
Φ
0
µ
0
0 n n
¶
:u
µ
0
0 n
¶
(244)
where : is a rows by columns product of the square matrix corresponding to 0 tensor Φ = [Φ 0 ] multiplied by vector u =[ 0 ] ( = 1 2 3). F
µ
n
¶
=−
1 XX 0
n
0
Φ
µ
0
0 n n
¶
:u
µ
0
0 n
¶
(245)
it is the total elastic force, due to displacements of all the other nuclei, applied to nucleus situated in cell n. Consequently it is clear the operational physical meaning of µ µ 0 ¶ 0 ¶ ˜ 0 ≡ −Φ 0 0 0 n n n n which is the component of the force applied on nucleus in cell caused by 0 0 0 a unitary displacement along direction of nucleus in cell .The number of independent components in tensor Φ = [Φ 0 ] depends on the point symmetry of the crystal. If tensor Π =[Π 0 ] represents a symmetry operation of the point group of the crystal (for example a rotation of order around an axis which passes through a fixed point) then (due to the transformation law of 0 tensors after a change in coordinates given by Π : r = Π : r) it must be 0
Φ = ΠΦΠ−1 = Φ
(246)
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where Π−1 is the inverse of matrix Π (Π is a unitary matrix: det Π = 1, and Π = Π−1 ). Applying a relation like that to any operation in the group we obtain some constraints determining the independent components of Φ. Now, due to the translational symmetry of the crystal it is possible to write 0 n = n + h, where h ≡ Th is a lattice translation (in the same way in this paragraph we will write Rn as n). So we can write: µ ¶ µ 0 ¶ µ 0 ¶ 0 Φ =Φ (247) =Φ 0 n n+h h n n 0
Notice that Φ depends only on the difference n − n = h. We can eventually write the dynamic equations as u ¨
µ
n
¶
=−
1 XX h
0
Φ
µ
0
h
¶
:u
µ
0
n+h
¶
(248)
Where the sum over h involves all the lattice translations which join cell n with the other ones (including h = 0). Applying periodic boundary conditions, as in the case of electrons, and thus imposing that µ ¶ µ ¶ (249) u =u n + ( − 1)a n ( = 1 2 3; = 1 2 3 total number of cells) the dynamic equations (248) maintain a translational invariance, although they refer to a finite number of nuclei. The fundamental solutions (lattice waves) should then satisfy Bloch theorem in its discrete formulation, id est: µ ¶ µ ¶ u =u q·n (250) n 0 To every lattice wave at least one wave vector q is associated such that the previous equation is satisfied. As in the case of electrons (k vectors), the set of q vectors is that composed by the vectors contained in the first Brillouin zone. Bloch theorem helps us to reduce the set of 3 dynamic equations to a set of 3 equations which describes the vibrations of the nuclei contained in cell 0. Dropping subscript 0 and considering explicitly the dependence on q, the new reduced set of equations becomes à ! 1 ³ 0 ´ X µ 0 ¶ X q·h : u q u ¨ ( q) = − Φ (251) h 0
h
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We introduce now the dynamic matrix 0 ¶ µ 0 ¶ XΦ h = q·h D √ q 0 h
µ
(252)
as the discrete Fourier Transform of interatomic force constant tensor. It is then convenient to write system (251) as: µ 0 ¶ 1 ³ 0 ´ √ Xp u : u q ¨ ( q) = − 0 D (253) q 0
We now look for elementary harmonic solutions as: 1 (q)e ( q) −(q) u ( q) = p
(254)
where we introduced the polarization unit vector e ( q) and the scalar complex amplitude (q). Substituting this type of solution into the dynamic equations we obtain the homogeneous system of linear algebraic equations (eigenvalue problem): µ 0 ¶ ¾ ³ 13 1 ½ ´ X X 0 2 0 − (q) 0 0 0 q = 0 (255) q 0 0
from which it is clear that, for any q, the squares of the natural frequencies are the eigenvalues of the dynamic matrix while the polarization unit vectors are its eigenvectors.The eigenvalues are computed solving the equation (solution condition of the above system): ½ µ 0 ¶ ¾ 2 − (q) 0 0 = 0 det 0 q For any value of vector q (there are independent q vectors in the first Brillouin zone, as just remembered) the equation has 3 countable solutions 2 (q) = 2 (q) characterized by a branch index . The dynamic matrix is self-adjoint: thus we have real eigenvalues 2 (q) and a full set of orthonormal eigenvectors: 1 13 X X
X
³ ´ 0 = 0 ∗ ( q ) q
³ 0 ´ ∗ ( q ) 0 q = 0 0
(256) (257)
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Solution functions = (q) (only positive roots have physical significance as vibrational frequencies) are called dispersion relations. Considering vibrational motions of nuclei in a primitive cell we cope with 3 degrees of freedom. Among them 3 refer to the center of mass of the basis (acoustic branches), while the other 3 −3 refer to internal motion of the basis (optical branches). The term acoustic comes from the fact that, when q → 0, acoustic motions (if coherently, and not thermally, excited) coincide with the macroscopic elastic waves characterized by (q) = (q|q|)q, where (q|q|), which depends only on the direction of the propagation q|q|, is the speed of sound and corresponds to a quasi-longitudinal wave (1 = ) or to two different quasi-transverse waves (2 = 1 , 3 = 2 ). These three speeds of sound are generally different one another. The term quasi indicates that the polarization of these waves is mainly longitudinal or mainly transverse. Along particular symmetry directions and in specific crystals (the cubic ones, as an example) these waves are completely longitudinal or completely transverse. The term optical derives from the fact that the corresponding modes can have a fluctuating electric dipole determining the optical infrared absorption in the crystal or can have associated a fluctuation of electrical polarizability determining, in the crystal, the vibrational Raman scattering of photons (in the infrared, visible and ultraviolet spectrum). If the crystal is a Bravais lattice with a basis limited to a single atom per primitive cell, only acoustic modes can exist. In silicon (face centered cubic lattice with 2 atoms per primitive cell: diamond structure) in addition to the 3 acoustic branches there are 3 × 2 − 3 = 3 optical branches: one of them is longitudinal, the other two are transverse. Using again Bloch theorem it is now possible to express the most general solution of the dynamic equations as the following linear combination of all lattice waves:
u
µ
n
¶
13 X X 1 =p (q)e ( q ) q·n − (q) q∈BZ
(258)
Introducing the 3 normal coordinates (q) = (q)− (q) ((q) are the amplitudes of normal coordinates) it is possible to write for the real displacements: ! Ã µ ¶ 13 X X 1 1 p = u (q )e ( q ) q·n + n 2 q∈BZ
Substituting these solutions in the vibrational Hamiltonian
c °2017-2018 Carlo E. Bottani
=
1 XX n
Solid State Physics Lecture Notes
µ ¶ u˙ u˙ n
µ
n
¶
75
−
and, finally, using the properties of polarization unit vectors and the fact that ions displacements (although they have been written so far as complex quantities) components are real quantities, we get the most important issue in the harmonic approximation: 13 ª 1 X X© | (q )|2 + 2 (q) |(q )|2 = 2 q∈BZ
where (q ) =
˙ ) (q
(259)
(260)
are the kinetic momenta conjugated to normal coordinates (q ). We have thus proved that the vibrational behaviour of the crystal is equivalent to that of a set of 3 independent harmonic oscillators with frequencies 2 (); there is a one to one correspondence between oscillators and lattice waves (normal modes) (q ). Collective motions (q ) are harmonic motions, while the motion of a single nucleus is not. For this reason the (q ) are called collective coordinates and their energy quanta } () behave as Bose particles, or quasi-particles (see below); as a matter of fact they are not associated to individual particles (which would be individually localized in the limit of uncertainty principle). The introduction of normal coordinates decouples the system of harmonic dynamic equation describing the motion of nuclei and produces 3 independent elementary equations: ¨(q ) + 2 (q)(q ) = 0
(261)
which can be obtained from using the Hamilton equations: ˙ ) = (q (q) − = ˙ (q ) (q)
(262) (263)
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5.3
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Linear chain with two atoms per cell: acoustic modes and optical modes
Let us consider a 1 crystal with lattice parameter and a basis with two ions of mass 1 and 2 (1 2 ) located in the official (frozen) positions µ ¶ 1 = µ µ ¶ ¶ 1 2 = + 2 and take only into account the interactions between them and with their first neighbours. It is not necessary for the second ion to be located in the middle of the cell. However we made this choice in order to use a unique interatomic force constant . Considering the interaction between ions inside the cell = 0 and with ions in the adjacent cells = ±1, we find nonzero force constants which determine the dynamics of ions in the cell = 0 (we drop subscripts 11 of Φ) ¶ ¶ µ ¶ µ µ 21 12 11 = −; = −; Φ = 2; Φ Φ 0 0 0 ¶ ¶ µ ¶ µ µ 12 21 22 = − = −; Φ = 2; Φ Φ −1 1 0 by means of which we can build the dynamic matrix : ! µ 0 ¶ à 2 − √ − (1 + ) 1 1 2 = 2 − √1 2 (1 + + ) 2 The eigenvalues of this matrix are the solutions of the following equation (solubility conditions of system(255)): µ ¶µ ¶ ³ ´ 2 2 42 2 2 − − − cos2 =0 1 2 1 2 2 i.e.: 212 ()
= ∓
where =
r ³ ´ ³´ sin2 1−4 2
1 2 and = 1 + 2
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are, respectively, the reduced mass and the total mass (mass of the center of mass) of the basis. When → 0, we asymptotically have r || 1 ≈ 2 r µ ¶ 2 2 2 2 ≈ 1− (264) 8 while near the edges of the first Brillouin zone (where = ±) we get r 2 1 = r 1 2 2 = (265) 2 The first branch of eigenvalues has an acoustic nature (see next section) while the second one an optical (see Optical properties below) nature. Moreover between 1 () and 2 () pthere is a gap of prohibited frequencies. Frequencies greater than 2 (0) = 2 are not possible. 1.5 1.25 y 1 0.75 0.5 0.25 0 -2.5
-1.25
0
1.25
2.5
x
Dispersion relations in the linear biatomic chain (y=
p
)
In figure the dispersion relations in the first Brillouin zone ( = ≤ ) with 1 = 22 are shown; the minimum forbidden frequency interval in correspondence with the edges of the zone is clearly visible.
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5.4
Solid State Physics Lecture Notes
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Simple 1D crystal: acoustic phonons and elastic waves
This case is even simpler than the previous one: the linear atomic chain is a 1D simple crystal with only one lattice ion (nucleus) of mass per primitive cell of size (the 1D Bravais lattice positions are = ). We still consider only first neighbors interactions of strength . The problem is so simple that we can solve it directly using Newton’s law for the motion of a generic nucleus interacting with nuclei − 1 and + 1. In this way we get the system of ( = 0 1 2 − 1) coupled equations: ¨ = −2 + −1 + +1
(266)
Since we cope with a periodic lattice (1D crystal), Bloch theorem (in its discrete form) can be used: (267) = 0 in this way we get a single equation which describes the motion of the nucleus located in the first cell ( = 0): µ ¶ µ ¶ ³´ ³´ 2 2 − ¨0 = − 0 + 0 0 = − (1 − cos())0 + (268) We now look for a harmonic solution such as − 0 () = √ and we get the solubility condition: ³ ´ ³´ 2 2 sin =4 2
i.e.
r ¯ ³ ´¯ ¯ ¯ =2 ¯sin ¯ 2
(269)
(270)
(271)
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1
y 0.75
0.5
0.25
0 -2.5
-1.25
0
1.25
2.5
p x Dispersion relation y= 2
In figure the dispersion relation in the first Brillouin zone (− = ≤ ) is shown. In the limit → 0 r () ≈ || (272) while
r (± ) = 2
(273)
is the crystal cutoff frequency for acoustic modes. When → 0, = (2) and the lattice wave feels the crystal as a continuous elastic medium. This directly follows from the dynamic equation rewritten as ¶ µ ³´ −2 + −1 + +1 (274) ¨ = 3 2 which, for long wavelengths, can be well approximated by the wave equation
2 2 = 2 2
(275)
where we have introduced = 3 (mass density), = (Young’s modulus) and ( ) (continuous field of elastic displacements). The solutions of this last equation are elastic propagating waves ( ) = (−) with speed of sound = =
s
(276)
(277)
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and linear dispersion relation () = ||
(278)
In the discrete (atomic) case, comparing the limits of the dispersion relations when → 0, we notice that r (279) =
In other words in a crystal it is possible to identify the acoustic modes with long wavelength with the elastic waves in the same crystal considered as a continuous medium (see below Debye model).
5.5
Phonons
Referring to the situation of the previous case, we now introduce the most general solutions of dynamic equations as a linear combination of all lattice waves: X 1 () −() () = √ ∈BZ
(280)
The N discrete allowed wavevectors are those imposed by the periodic boundary conditions −1 = 0 (281) for we can write: − +1 2 2
() =
X
√ − 2( )
(282)
We define: = −
(283)
as the normal coordinates and r ¯ µ ¶¯ ¯¯ ¯¯ sin = 2 ¯ ¯
(284)
as the corresponding proper frequencies. We can now write the vibrational Lagrangian as: 0 −1 0−1 1 X 1 X 2 = ˙ − (+1 − )2 2 2
(285)
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and the Hamiltonian as =
0 −1 0−1 1 X 1 X 2 + (+1 − )2 2 2 2
where =
= ˙ ˙
(286)
(287)
are the conjugated momenta. Substituting (282) in the Lagrangian and in the Hamiltonian, after some lenghty computation we omit here (imposing () to be real), we can finally get: ¾ X ½¯¯ ¯¯2 2 2 ˙ ¯ ¯ − | |
− +1 2 2
1 = 2
1 = 2 where
(288)
− +1 2 2
X © 2 ª | | + 2 | |2
(289)
=
˙
(290)
are the canonical conjugated momenta. As in the general 3D case, the periodic monoatomic linear chain can be considered as a set of N independent harmonic oscillators, each oscillator being represented by a normal coordinate. Introducing now the Hamiltonian operator ˆ = 1 2 where
¯ ¯2 ¾ X ½¯¯ ¯¯2 ¯ ¯ 2 ¯ˆ ¯ + ¯ˆ ¯
− +1 2 2
(291)
ˆ = −~
(292)
it is immediately possible to switch to a quantum description of lattice vibrations: the crystal is, from a vibrational point of view, equivalent to a system of quantum independent harmonic oscillators. In the 3D general case each oscillator with normal coordinate (q ) owns a frequency (q ) and a polarization unit vector e(q ) and can assume the discrete energy levels ¶ µ 1 (q ) = (q ) + ~(q ) (293) 2
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where (q ) = 0 1 2 3 The total vibrational energy is thus equal to: ¶ 13 µ X X 1 (q ) + ~(q ) = 2 q∈BZ
(294)
(q ) represents the quantum number -(phonon population number) associated to the mode (normal coordinate) (q ). The set of 3 numbers (q ) completely determines the vibrational quantum state of the crystal at zero temperature. In thermal equilibrium at temperature , the vibrational contribution to the internal energy of the crystal is: ¶ 13 µ X X 1 h(q )i + ~(q ) = 2 q∈BZ where h(q )i =
1 ~ (q)
(295)
(296)
−1 is the Bose-Einstein distribution function. Thus, within the harmonic approximation, the vibrational properties of the crystal is described by an ideal Bose-Einstein gas of phonons. Sometimes the term phonon is used to indicate the collective coordinates (and the normal modes) (q ) and not only their quantum vibrational energy ~(q ). Also in interactions with electrons and photons (emission, absorption and scattering events), the quantized lattice vibrations behave in the same way as particles associated to collective coordinates characterized by a crystal momentum ~q (in the reduced zone scheme or in the repeated zone scheme) and by an energy ~(q ) (which is a periodic function of q in the repeated zone scheme); lattice waves can be distinguished in acoustic and optical phonons. When, in an interaction process, only phonons with wave vectors situated in the first Brillouin zone are involved, the process is called N process (normal), otherwise, if some wavevectors exceed the the first Brillouin zone, we have a U process (umklapp).
5.6
Specific heat: Debye model
As in the case of electronic states, in a macroscopic crystal the phonons wave vectors (where coincides with the total number of primitive cells) quasi-continuously occupy the first Brillouin zone in reciprocal space. This suggests to approximate this quasi-continuous filling with a truly continuous one and to introduce the concept of density of vibrational normal modes (if
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instead we considered a nanometric crystal, this continuous approximation could not be used). In general in a crystal of volume containing primitive cells, the elementary volume of reciprocal space associated to a single wave vector is equal to: 83 8 3 8 3 = = |a1 · a2 × a3 |
(297)
as turns out from periodic boundary conditions. Furthermore if we consider a monoatomic simple crystal, to any wave vector there correspond three normal acoustic modes (one longitudinal and two transverse). For any polarization (phonon branch) the above quantity may be defined density of normal modes: it is constant in reciprocal space but, in general, it is frequency dependent in frequency space. To get this frequency dependent density, we start with the quantity () which measures how many normal modes with a given polarization have frequency within 0 and . Fixing the dispersion relation = (q) defines a surface in reciprocal space and this surface encloses a volume () such that: () (298) () = ¡ 83 ¢
Now the density of modes as a function of can be computed as the derivative of (): () (299) () = In 1 the constant frequency surface degenerates into two discrete points and the volume () into a linear segment () bounded by these points. For example, in the monoatomic 1 chain treated above, for a given the equation r ¯ ³ ´¯ ¯ ¯ =2 (300) ¯sin ¯ 2
has the two roots
⎛ ⎞ µ ¶ 1 2 arcsin ⎝ q ⎠ ± () = ± 2 1
(301)
⎞ ⎛ µ ¶ 1 4 arcsin ⎝ q ⎠ () = 2+ () = 2 1
(302)
which defines a segment of length:
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thus we have:
⎞
⎛
1 () 2 arcsin ⎝ q ⎠ () = ¡ 2 ¢ = 2 1
84
(303)
to which there corresponds the 1D density: ⎞⎤ ⎡ ⎛ 1 ⎣ 2 1 arcsin ⎝ q ⎠⎦ = q q () = 2 1 1 1−
(304) 1 2 4
In one dimension the density of acoustic p modes diverges in correspondence of the maximum possible frequency 2 at zone boundary, where the phonon group velocity is zero, while it is nearly constant for small wave vectors, where the group velocity is constant and equal to the speed of sound. In view of the difficulties of generalization of this procedure when applied to a complex 3 crystal whose lattice dynamics is governed by realistic atomic force constants, Peter Debye introduced an approximate model for the derivation of () considering the crystal as an isotropic elastic continuous body. In this way only non dispersive acoustic modes (elastic waves) do exist. The discrete nature of the real finite crystal is lumped into a maximum allowed frequency, called the Debye frequency . In a continuous and isotropic elastic solid body (see Appendix Elasticity and Elastic Waves) a dynamic deformation process can generally be described as the superposition of a simple dilatation (compression) field and by a simple shear strain field. This corresponds to a total displacement field u u = u + u
(305)
∇ × u = 0 ∇ · u = 0
(306) (307)
where
Displacement fields u and u obey these two decoupled wave equations: 2 u = 2 ∇2 u 2 2 u = 2 ∇2 u 2
(308) (309)
where =
s
+ 43
(310)
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is the longitudinal speed of sound and =
r
(311)
is the transverse speed of sound. In previous equations is the mass density, is the bulk modulus and is the shear modulus. These two elastic constants can be expressed by a function of the couple of technical constants (Young’s modulus) and (Poisson’s ratio) using the following formulae: 3(1 − 2) = 2(1 + )
(312)
=
(313)
For thermodynamic stability reasons it is always . In case of a simple uniaxial deformation in the direction of axis : = and = = − . is the axial stress while , , are the three diagonal components of the strain tensor. Moreover it is possible to write: + + =
+ + =
(314)
We call = − 13 ( + + ) the hydrostatic pressure and we notice that = −
(315)
If we consider, for example, the longitudinal acoustic branch, the dispersion relation is: q = |q| = 2 + 2 + 2 (316)
and, fixing , the (316) represents in space q the equation of a spherical surface with radius . Then it is: 4 () = 3
() =
4 3
() =
³ ´3
¡ 83 ¢
µ
µ ¶3
3 6 2 3
= ¶
(317)
3 6 2 3
=
2 22 3
(318) (319)
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i.e. we have a parabolic density of modes. Repeating all this for the two transverse branches (degenerate in the continuous and isotropic elastic mean) we get the total density of modes: 2 () = 2 3 2
(320)
1 2 1 = 3+ 3 3
(321)
where
The Debye frequency is then obtained from condition: Z () = 3
(322)
0
as:
µ
¶ = 18 With the previous formula it is possible to write: 3
2 3
() =
92 3
(323)
(324)
Now we can generalize the specific heat expression in the Einstein model (see above). We consider the Einstein formula still valid for the single mode with frequency but now we sum up over all the modes using () as a weighting function: =
µ
¶
=
Z
0
2
~
(~) µ ~ ¶2 () 2 −1
(325)
This expression for the specific heat fits rather well most experimental data. Every solid is characterized by its Debye temperature: Θ =
~
(326)
Setting: ~ (327) we can rewrite the expression of the specific heat as (assuming = ): µ ¶3 Z Θ 4 = 3 3 (328) Θ ( − 1)2 0 =
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At low temperatures (Θ → ∞) the integral tends to a constant and the specific heat tends to zero as 3 . At high temperatures Θ tends to zero, thus the integral can be approximated as: µ ¶3 Z Θ Z Θ 4 Θ 2 3 ≈ 3 = (329) ( − 1)2 0 0 and the specific heat obeys the classical Dulong and Petit law: ≈ 3 = 3
(330)
Thus a crystal behaves in a classical way when Θ and in a quantum way if Θ . For example diamond (one of the two allotropes of carbon in the solid state) with Θ = 1860 and silicon with Θ = 625 both exhibit a quantum thermodynamic behaviour at room temperature (300 ).
List of Debye temperatures
The figure below shows the universal trend of Debye specific heat (328) as a function of the reduced temperature = Θ :
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1
Cv/3R
0.75
0.5
0.25
0 0
0.5
1
1.5
2 x
Debye specific heat The success of the Debye model finds a simple physical justification in the behaviour of Bose-Einstein factor as a function of frequency at low tempera−
~ (q)
tures. As a matter of fact when → 0 we have that (q) ≈ : normal modes with a lower frequency has a higher occupational factor. These modes are just the acoustic modes about the center of the Brillouin zone. For the contribution of these very modes the Debye model provides a very good approximation. Simple crystals have only acoustic modes, while more complicated complex crystals need a better theoretical description including the optical modes.
5.7
Polaritons
The denomination optical modes originates from the fact that these modes determine the linear optical properties of polar (ionic) crystals in the infrared. These crystals are non monoelemental crystals in which the chemical bond is, at least partially, of ionic character. The extreme case is constituted by alkali metals halides (for example + − ) in which the interatomic forces have a purely electrostatic nature and the electron states are localized on individual ions. However also the cases of III-V compounds, such as and II-VI compounds, such as (zincblende), should be considered. In these crystals, where the bonds are mixed covalent/ionic, the electrons eigenstates are Bloch waves and the usual band structure picture holds. All these structures share the absence of a center of symmetry and, as a consequence, in every primitive cell they exhibit an electric dipole moment (we have a total electric dipole of ions forming the crystal basis). A characteristic crystallografic structure is zincblende: its Bravais lattice is f.c.c. (as in the case of − or Si); the basis is formed by two atoms located in (0,0,0) and in (1/4,1/4,1/4), all along the principal diagonal of the conventional cubic
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cell, in units of (cubic lattice parameter). We now imagine that in (0,0,0) the positive ion is situated, while in (1/4,1/4,1/4) the negative one. Considering the propagation of electromagnetic waves along direction (111), we can approximate the lattice dynamic as the one of a linear chain with two atoms per cell (see above), as long as the branch 2 () of the dispersion relations is concerned. We study the forced response of the chain under the action of the (local) electric field of an electromagnetic wave E = E0 (−)
(331)
which propagates in the same direction (111). The corresponding 3 problem is very difficult to be analyzed since we should consider both the longitudinal and transverse polarizations, for both the field of the wave in the crystal (within matter also longitudinal electromagnetic waves may exist) and for the field of atomic displacements. Using a simplified model, regardless of the polarization we can write directly the equations of the forced motion of ions as ( is the absolute value of an ion charge) ¶ µ µ ¶ µ ¶ µ ¶ 2 2 1 1 + + + 1 ¨ = −2 −1
2 ¨
µ
2
¶
1
1
2
2
((
+
)−)
+0 µ ¶ µ ¶ µ ¶ 2 1 1 = −2 + + + +1 ((
−0
+
)−)
(332)
(333)
Assuming that the wavelength = 2 is much greater than , we apply Bloch theorem to previous equations (as already done for the chain with only one atom per cell) in the hypothesis that = (this could be justified by quantum theory). In this way it is possible to considerably simplify the equations of motion: 2 (1 − 2 ) + 1 2 = + (1 − 2 ) − 2
¨1 = − ¨2
0 − 1 0 − 2
(334)
Subtracting the second equation from the first, and introducing the reduced mass , we get: 2 0 − 2 (1 − 2 ) + (1 − 2 ) = 2
(335)
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Introducing now the normal coordinate = 1 − 2 and the characteristic frequency r 2 = = 2 ( = 0) (336)
we obtain:
0 − 2 + 2 = 2
The forced solution is: =
0 − − 2
2
(337)
(338)
Thus the associated electric dipole moment is: = =
2 0 − 2 − 2
(339)
and the polarizability of the single cell (of the basis) = (0 ) may be written as: 1 2 = (340) 0 2 − 2
Neglecting local-field effects, if there are cells per unit volume, the dielectric susceptibility of the crystal can be written as: = =
1 2 0 2 − 2
Consequently the dielectric constant becomes: µ ¶ 2 = 0 (1 + ) = 0 1 + 2 − 2 where we introduced the plasma frequency of lattice ions: s 2 = 0
(341)
(342)
(343)
q Note that when the frequency is equal to = 2 + 2 we have () = 0. Moreover it is (Lyddane-Sachs-Teller relation): (0) 2 = 2 (∞)
(344)
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¶ µ 2 (0) = (∞) 1 + 2
91
(345)
epsilon rel. 3.75
2.5
1.25
0 0
0.5
1
1.5
2
2.5
omega/omegaT -1.25
Relative dielectric constant of GaAs As shown in the picture (black curve), in the frequency interval ∆ between and the dielectric constant is negative. Since the speed of propagation of the electromagnetic wave is: r 0 = ≈ (346) where is the refractive index, we have that in the interval ∆ the electromagnetic wave cannot propagate in the crystal. An e.m. wave like that would be totally reflected from the surface of the crystal (Restrahlen effect). Thus, independently from periodicity effects, in a ionic crystal there is an energy gap prohibited to photons ∆ = ~∆. If in the model we include dissipative interactions, the trend of the dielectric function (the real part) is the one without divergences shown in the picture (red curve: absorption and dispersion in infrared). The second figure shows both the real and the imaginary part of the refractive index, including dissipation (cfr eq. 347).
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2.5
2
1.5
1
0.5
0 0
0.5
1
1.5
2
omega/omrgaT
Complex index of refraction (GaAs) In order to describe the absorption (attenuation of the electromagnetic wave) we introduce a complex wave vector. Thus the dispersion relation of electromagnetic waves in the crystal can by described as: r 0 = = = r³ (347) ´ 2 1 + 2 −2
Solving with respect to we get the complex wave vector: sµ ¶ 2 1+ 2 = − 2 ck
(348)
5
3.75
2.5
1.25
0 0
0.5
1
1.5
2
omega/omegaT
Polariton dispersion relations In the last figure (relative to not including dissipation) the real part of the product of the wave vector and the speed of light in vacuum as a function of the frequency is shown in black. The imaginary part, corresponding to the attenuation of the electromagnetic wave, is shown in red. The group
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velocity can never exceed . The asymptotic phononic and photonic trends are evident. The mixed photon-optical phonons excitations introduced in this section are quantized (their energies are ~()) and they are called polaritons. In a more realistic 3 model, in which transverse optical phonons are distinguished from longitudinal optical phonons (and transverse e.m. wave are distinguished from longitudinal ones) we have = (0) and = (0). Moreover the e.m. transverse waves couple exclusively with transverse optical phonons and they can propagate outside the interval ∆. When = mixed longitudinal optical phonon/longitudinal e.m. wave, stationary modes exist. In the previous picture the relative dispersion relation would be represented as a vertical line. Generally and lie in the infrared (in = 51 × 1013 s−1 and = 55 × 1013 s−1 ). In a ionic crystal a moving electron carries with itself a cloud of optical phonons which changes its mass (the mass increases): this quasi-particle is called polaron. Also in non-polar crystals, such as silicon, optical modes interact with photons, but this interaction causes nonlinear optical properties (Raman scattering) and the effect is evident also in the visible and in the ultraviolet. From a quantum point of view the description of scattering processes needs a second order correction in dynamic perturbation theory (see Appendix: Quantum approximate methods).
5.8
Inelastic scattering
Let us assume that the interaction potential in a crystal can be described as a sum of atomic potentials (for the sake of simplicity consider a simple crystal in which the atoms occupy the nodes of the Bravais lattice): X (r) = (r − rn ) n
As we already learned in the theory of scattering, the scattering amplitude is given by the following expression: Z Z X −Q·r (Q) ∼ (r) r ∼ −Q·r (r − rn )r = = =
Z X
X
n
−Q·(r−rn ) −Q·rn (r − rn )r =
−Q·rn
n
∼
n
(Q)
X n
Z
−Q·(r−rn ) (r − rn )r ∼
−Q·rn
(349)
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P Where n −Q·rn is the structure factor of the whole crystal, and the atomic shape factor is: Z −Q·R (R)r
(Q) =
When atoms occupy equilibrium positions (there are no lattice vibrations) rn = n =1 a1 + 2 a2 + 3 a3 it is
X n
−Q·n = (Q − g)
where g is a vector of the reciprocal lattice, and Bragg law for elastic diffraction is recovered. In fact for any vector m in the lattice: X X X −Q·n = −Q·(n+m) = −Q·m −Q·n (350) n
n
n
P
i.e. is either n −Q·n = 0 or, if Q = g (a vector of the reciprocal lattice), P it−Q·n = . n Otherwise, if atoms vibrate about their equilibrium positions, we can write their displacements as a superposition of all normal modes summing over index q. For the sake of simplicity we drop branch index ; moreover there is not any index since we are considering just one atom per cell: X rn = n+ (un (q) + u∗n (q)) (351) q
where we added the complex conjugate in order to have real displacements. The sum is confined in the first Brillouin zone and the sum over is implied. Using Bloch theorem we again write un (q) = u0 (q)q·n and u0 (q) = (q)e(q)−(q) where e(q) is the polarization vector of the normal mode q, and (q) √ is the amplitude of the corresponding normal coordinate, equal to (q) in the notation used in the description of lattice vibrations. Thus considering equations (351) and (349) we have: X X ∗ −Q·rn = −Q·(n+ q (un (q)+un (q))) = (Q) ∼ n
=
X n
−Q·n
Y q
n
−Q·(un (q)+u∗n (q))
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Considering only small lattice displacements ∗
−Q·(un (q)+un (q)) ≈ 1 − Q · (un (q) + u∗n (q)) − |Q · u0 (q)|2 + and just the linear terms, i.e.: X Y ∗ −Q·(un (q)+un (q)) ≈ 1 − Q · (un (q) + u∗n (q)) q
(352)
(353)
q
we get:
(Q) ∼ =
n
Ã
n
−Q·n −
X
X
−Q·n
1−
X
= (Q − g) − −
X q
X q
−Q·n
n
X q
Q · (un (q) + X q
!
u∗n (q))
Q · (u0 (q)q·n +u∗0 (q)−q·n ) =
(Q · e(q)) (q)−(q) (q)
(Q · e(q)) (q)
=
X
X −Q·(n−q) + n
−Q·(n+q)
n
where normal coordinates have real amplitudes. We consider now the Bravais lattice inversion symmetry, in this case the lattice sum over q is equal to the one over −q, and we have: X (Q) ∼ (Q − g) − (Q · e(q)) (q)−(q) (Q − q − g)+ −
X q
q
(Q · e(q)) (q)(q) (Q − q − g)
(354)
We now apply the Fermi’s golden rule (where is the transition probability per unit time), id est (Q) ∼ = 2 |h1|0 |2i|2 (2 − 1 ± ~), ~ in the form valid for interaction with a time harmonic potential 0 ± (in this case we have a term ±~ in a Dirac delta function which represents the conservation of energy). We consider as initial state |1i ∼ k ·r , and as final state h2| ∼ k ·r (initially free particles with a defined wavevector interacting with a crystal). InR this case the matrix element h1|0 |2i is equal to the scattering amplitude 0 (r)−Q·r r , where Q = k − k . Thus the first term in equation (354) gives us the elastic contribution in the expression of the intensity and is proportional to (Q − g)(2 − 1 ) | (Q)|2 . In this way we find the Bragg law, with the condition that scattered particles and incident particles share the same energy. The second and the third term give us an inelastic contribution in the expression of intensity, corresponding to a first
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P order inelastic scattering, and proportional to | (Q)|2 q |Q · e(q)|2 2 (q) (Q − q − g)(2 − 1 ± ~(q)). This means that there is an active energy exchange mechanism between the particle (a neutron, for example) or between a photon of the electromagnetic field (light or X rays, for example) and the quanta of vibrational energy. A vibrational quantum with energy ~(q) can be created or destructed. Factor (Q − q − g) imposes the relation: Q = k − k = q + g (355) If the equality is satisfied when g = 0, id est Q is in the first Brillouin zone of the reciprocal lattice, we have a Normal or N process, otherwise, if Q is outside the first Brillouin zone, id est equation (355) is satisfied when g 6= 0, we have a Umklapp or U process. In the first situation equation (355) can be seen as a principle of conservation of quasi-momentum (exchange between the crystalline momentum ~q of a phonon and the momentum of the particle): ~Q = ~k − ~k = ~q In the second situation the additional term ~g corresponds to a term of variation of the momentum associated to the motion of the center of mass of the entire crystal. The Umklapp process can be considered as the creation (or destruction) of a phonon associated to a Bragg reflection. In a quantum description of lattice vibrations, the mean value (it is a thermal mean) of h 2 (q)i depends on the population of the vibrational mode with wave vector q, described by the Bose-Einstein distribution. Actually, if we considered the quadratic terms in the expansion (352), both the elastic term and the inelastic one in the expression of intensity should be multiplied by a factor of: Y 2 (1 − |Q · u0 (q)|2 ) = − q |Q·u0 (q)| = −2 q
where −2 is the Debye-Waller factor, where = =
1X 1X |Q · u0 (q)|2 = |Q · e(q)|2 2 (q) = 2 q 2 q 2 (q) 1X |Q · e(q)|2 2 q
√ In the last expression we wrote the normal coordinate (q) with the notation used in the description of lattice vibrations ((q) shouldn’t be confused with the absolute value of Q = k − k , which is a scattering wave vector). When lattice vibrations are considered, the intensity of Bragg peaks
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(and the one of inelastic peaks) is thus decreased due to a disorder which introduces a "noise" in the crystal translational symmetry. In a D quantum E 2 (q) description of lattice vibration, the mean value (thermal mean) of ¡ ¢ ~ is equal to (q) + 12 , where (q) is the occupation number for bosons in the quantum harmonic oscillator corresponding to the mode q: (q) =
1 exp( ~(q) )
−1
The Debye-Waller factor depends on the temperature, and in particular is proportional to at high temperatures (above the Debye temperature), while it assumes a constant value, still different from zero (due to the vibrational zero-point energy), at low temperatures. If we decide to consider also the nonlinear terms, which contain products of more than one term such as Q · (un (q) + u∗n (q)), in the expression of intensity we would have also terms corresponding to second order scattering (with a smaller intensity with respect to the first order), id est corresponding to an exchange of energy and momentum with two or more phonons (multiple phonons processes). Space-time autocorrelation function: In a "static" crystal we have that: Z (Q) ∼ (Q) ∼ (R)−Q·R R which is the static structure factor, is equal to the spatial Fourier transform of the static autocorrelation function: Z (R)= (r + R) ∗ (r)r In a crystal or in a mean with vibrational motions we have that: Z (Q ) ∼ (Q ω) ∼ (R)−Q·R R which is the dynamic structure factor of the crystal, it is equal to the spacetime Fourier transform of the dynamic autocorrelation function (where the time integration is done on time intervals larger than characteristic times in vibrational motions, the integral corresponds to a thermal average operation):
(R) ∼
Z Z
∗
(r + R + ) (r )r ∼
¿Z
À (r + R) (r 0)r ∗
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Optical properties
Generally the linear optical properties in solids (dispersion, absorption, reflection, refraction) depend on macroscopic interaction processes between photons, electrons and phonons. In the figure below the qualitative trend of the absorption coefficient (358) in a doped polar semiconductor is shown where it is possible to see all possible microscopic mechanisms.
Absorption coefficient In this section we will analyze explicitly just the principal band and the continuous background (in the previous section we introduced the polaritonic absorption), however the principles already treated would give us the possibility of understanding from a quantum point of view all the peaks in the figure. To start we build a general connection between the macroscopic optical parameters (refractive index and extinction coefficient) and the transition probabilities between different quantum states in a solid excited by an electromagnetic radiation. The electric field of a monochromatic plane wave, with polarization unit vector e, which propagates in a solid along the axis is described, in the typical complex form, by expression:
E = 0 e(−) (356) n o ( −) Thus the effective electric field will be e Re 0 . Macroscopically the dispersion and absorption (extinction) processes can be described introducing a complex wave vector e = + and we can write: E = 0 e− (−)
(357)
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Id est, we have a wave attenuation. The intensity of the wave (averaged ³ n o´2 = 12 |E |2 , exhibits an over a period), proportional to Re 0 (−) exponential decay in space: = 0 −2 = 0 −Σ
(358)
Introducing the dispersion relation e e
= with complex refractive index: 0
(359)
00
e = +
we have
Σ=2
³ ´
(360)
”
(361) 0
”
We now the complex relative dielectric constant e = +√ = ¡ introduce 0¢ 1 + + ” ( is the complex dielectric susceptibility). Since e ≈ e , it is: ³ 0 ´2 ³ 00 ´2 0 0 = 1 + = − (362) ´ ³ ´ ³ ” 0 00 = ” = 2 Using the second equation in (362) we can thus write: Σ=
³ ´ ”
0
But, since the imaginary part in the refractive index is always less than the real part, using the first equation in(362) we get: Σ=
³ ´
”
p ' 1 + 0
³ ´
”
the approximation being valid especially about the absorption frequencies, 0 where ' 0. Now we want to underline the connection between the extinction coefficient and the microscopic transition probabilities (per unit time) induced by the coupling between the solid and the electromagnetic field of the wave. We consider now the electromagnetic energy density associated to the wave (averaged on a period):
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1 ³ 0 ´2 = 0 |E |2 2 and the Poynting vector (averaged overe a period too): 1 0 = 0 |E |2 2
100
(363)
(364)
The conservation equation for the electromagnetic energy density in the stationary state is: =− (365) where is the power density absorbed by the solid; the same power is lost by the wave. So it is: = 0 (366) Since the intensity of the wave is proportional to , decreases in space as stated by the law (358) thus it is: ³ ´ ” 1 ³ 0 ´2 = − 0 Σ = − 0 2 0 |E |2 2
(367)
1 = 0 ” () |E |2 2
(368)
Using, again, the second equation in (362), we finally get:
Now, introducing the probability per unit volume and time of a single microscopic event which leads to the extinction of a photon to happen, we get: X 1 0 ” () |E |2 = ~ (369) 2 The quantity can be estimated using the Fermi’s golden rule (see Appendix: Quantum approximate methods). Solving with respect to ” () we get: µ ¶ 2~ X ” () = (370) 0 |E |2
6.1
Kramers e Kronig relations ”
Once the ” () = () has been calculated with (369), it is possible to 0 0 obtain () = () − 1 using an integral relation (and vice versa). It is
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a general equation which comes both from the linear relation between the polarization vector P() and the electric field () and from the principle of causality (linear response theory). In presence of delay effects (due to dissipation and dispersion), the most general linear relation between P() and () can be written as: Z ∞ P() = 0 ( )( − ) (371) 0
If (− ) = (− ) it is P() = 0 (); which gives () its own meaning. In particular () is equal to zero when 0 (principle of causality: effects cannot precede causes): so we integrate between 0 and ∞. Equation (371) implies also the time invariance (the hereditary integral is a convolution: id est ( ) = ( − )). If the polarizing field is harmonic: () = exp(−) we have µZ ∞ ¶ ( ) − (372) P() = 0 0
So P() = P exp(−) and we can write the important relation:
being
e() P = 0 e() =
Z
∞
( )
(373)
(374)
0
from which it results immediately that: e(−) = e∗ ()
(375)
More generally e() is a tensor e () which depends on the symmetry of the crystal. In cubic crystals it reduces to a scalar quantity. Considering the principle of causality, in equation (374) we could integrate between −∞ and +∞ thus concluding that e() is the Fourier transform of (). We can make now an analytic continuation of e() in the upper semiplane, introducing 0 0 ” the complex frequency = + ” . Since lim” −→∞ − = 0, e() is analytic in the upper semiplane. As a consequence (using Cauchy theorems): I e() = 0 (376) −
for any circuit in the upper semiplane which does not contain . Let us assume ≥ 0 on the real axis. We chose as a circuit = 1 ∪ 2 ∪ 3 ∪ 4 0 where 1 (− ≤ ≤ −; ” = 0), 2 ( = + = − 0), 3 (+ ≤
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0
≤ ; ” = 0) and 4 ( = = 0 − ). Solving the four line integrals in the limit −→ 0 and −→ ∞, we get from (376): +∞ Z −∞
0
e( ) 0 − e () = 0 0 −
(377)
where we should consider the integral as : Cauchy principal value: lim→0 We can rewrite the previous equation as: 1 e() =
+∞ Z
hR −
0
e( ) 0 0 −
−∞
(378)
Introducing again the real and the imaginary part of the susceptibility, we obtain the Kramers e Kronig relations: 1 () = 0
+∞ Z
0
” ( ) 0 0 −
−∞ +∞ Z
1 () = − ”
−∞
0
(379)
0
( ) 0 0 −
(380)
0
() is the Hilbert transform of ” () and vice versa. Since from (375) we 0 get that () is an even function, while ” () is an odd function, multiplying 0 the numerator and denominator by + , we can rewrite equations (379) as: 2 () = 0
+∞ Z 0
2 ” () = −
0
0
” ( ) 0 0 2 − 2 +∞ Z 0
0
(381)
0
( ) 0 02 2 −
(382)
From the first equation of (381) we obtain the static relative dielectric constant as: +∞ Z ” 0 ( ) 0 2 0 0 (383) (0) = 1 + (0) = 1 + 0 0
−∞
+
R +∞ i . +
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The (383) is called sum rule. As a direct application of (381), we assume that, in (369), ∝ ( − 0 ) (384) So from (370) it is ” () = ( − 0 ), where is a constant. Using the (383) we notice that = 0 [ (0) − 1] 2. Now, using the first equation in (381), we finally get: ¤ £0 20 (0) − 1 0 0 (385) () = 1 + () = 1 + 20 − 2 Tthis equation can be compared with the one obtained in the polariton theory.
6.2
Probability per unit time of optical transitions
When a crystal is perturbed by an electromagnetic wave whose electric field in vacuum is 1 E = e0 (k·r−) + (386) 2 in (370) the transition probability per unit time (and volume) may be evaluated by means of the Fermi’s golden rule (see Appendix: Quantum approximate Methods). If we consider only the first order absorption processes in the perturbation theory (the initial state is the ground state with single electron () energies ) the rule is: ´ 2 ¯¯ c ¯¯2 ³ () () = ¯h | |i¯ − − ~ ~
(387)
where the matrix elements of the time independent part of the perturbation operator can be written as: c|i = h |
0 h|k·r e·b p|i 2
(388)
b = −~∇ is the momentum operator and is the electron mass in where p vacuum. In order to explain the (388) we should remember that, using the vacuum gauge, the electromagnetic field can be derived using just the vector potential A (with divergence equal to zero: ∇ · A = 0) as: A B = ∇×A E = −
(389)
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In this situation the Hamilton function of a particle with charge and inertial momentum p =v, moving in an external electromagnetic field, can be written as: |p − A|2 (390) = 2 If the charge is also under the effect of an additional electrostatic field (due to other fixed charges) with potential (r), and thus it has a potential energy of (r) = (r), the total Hamilton function becomes: |p − A|2 + (r) (391) 2 Developing the square of the absolute value, and substituting to p the momentum operator, we obtain the quantum Hamiltonian operator as: =
b = b0 + b
where
(392)
|b p|2 b + (r) (393) 0 = 2 |A|2 b = − A·b p+ ≈ − A·b p (394) 2 In the last approximation we neglected the quadratic term, according to first order perturbation theory. This is consistent with the linear response theory we are developing, see (371). In this way we cannot take into account nonlinear optical phenomena. This treatment of e.m. radiation-matter interaction in which particles obey the Schroedinger nonrelativistic quantum mechanics and e.m. fields obey Maxwell equations is called semiclassical. This semiclassical treatment does not explain spontaneous emission and high energy relativistic processes. If the wave electric field has an expression such as (386) then, using (389), the expression of the vector potential becomes: 0 (k·r−) + e 2 So, if the charged particle is an electron with = −, it is: A=
(395)
c− + b = 0 k·r e·b p− + = (396) 2 We neglect the c.c., accounting for stimulated emission in which the initial state is an excited state whose energy is higher than that of the final state, and get the (387) and the (388) in the form: ³ ´ ¯2 2 2 |0 |2 ¯¯ () () k·r ¯ = h| e·b p|i (1 − ) − − ~ (397) ~ 4 2 2
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Using the (370) we get: ” () =
³ ´ ¯2 2 X ¯¯ k·r ¯ (1 − ) () − () − ~ (398) e·b p |i h | 0 2 2
where we introduced the probability that the initial state is occupied and the probability 1 − that the final state is unoccupied. Using the first of 0 0 the Kramers and Kronig equations we get () = 1 + (). Id est the unity plus a sum of other terms such as the one in the right side of equation (385): one term for each possible transition between the initial and the final state of each electron in the solid (assuming unitary volume).
6.3 6.3.1
Interband optical transitions Direct transitions
As an application of (398) we consider an intrinsic semiconductor at low temperature: initially the valence band is completely occupied and the conduction band completely unoccupied, then an e.m. radiation represented by the plane wave (386) hits the crystal. Transitions between occupied Bloch states in the valence band |i = k (r)k ·r
(399)
with energy (k ) and unoccupied Bloch states in conduction band | i = k (r)k ·r
(400)
with energy (k ) occur, generating electron-hole pairs. Considering the matrix elements in the (388) we notice that they are proportional to: Z ¡ ¢ k·r −k ·r k·r −~h| ∇|i = −~ ∗ ∇ k (r)k ·r r (401) k (r)
where the integral is over the whole crystal volume. Introducing the relative electronic coordinates r0 = r − n (where n is a generic node of the Bravais lattice) and integrating over the volume of a single primitive cell, we can write: Z ³ ´ 0 X 0 0 0 0 k·r −(k −k−k )·n −k ·r k·r k ·r0 ∗ (r ) ∇ (r ) r h | ∇|i = k k n
0
Using (350), we notice that the lattice sum
(402) −(k −k−k )·n is zero, unless: n
P
k − k = k + g
(403)
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where g is a vector of the reciprocal lattice. This conservation principle of the total wave vector in the crystal is a consequence of the translational symmetry. Considering just the first BZ (g = 0) and observing that the photon wave vector k is negligible with respect to the dimensions of the Brillouin zone (electric dipole approximation), the previous relation becomes k = k . So in the band representation (reduced zone scheme) the allowed transitions are only the vertical ones. Since each initial state is orthogonal to each final state we finally get: Z 0 0 0 k·r −~h | ∇|i ≈ ∗ (404) k (r ) (−~∇) k (r )r
0
Using the following relation: h|b p|i =
´ ³ (0) (0) − h|r|i ~
(405)
which comes from the time derivative of operator r: b r p 1 h b i = r0 = ~
we get: k·r
−~h|
∇|i ≈ (k )
where:
Z
(406)
0
0
0
0
0
∗ k (r )r k (r )r
(407)
[ (k ) − (k )] (408) ~ Here we can notice how the transition probability depends on the crystal point symmetry. In case of a crystal with a center of symmetry, a transition can only occur when the parity of the initial state is different from that of the final state. Now we can finally get a more compact form of the imaginary part of the dielectric susceptibility as a sum over all k0 in the first Brillouin zone: 2 X ” () = | (k)|2 [ (k) − (k) − ~] (409) 2 2 0 k (k ) =
where:
(k) = (k )
Z
0
0 ∗ k (r )
³ ´ 0 0 0 e · r k (r )r
(410)
Generally (k) can be considered as almost constant, so: 2 | |2 X () ≈ [ (k) − ~] 0 2 2 k ”
(411)
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where (k) = (k) − (k) is the vertical difference in energy between the conduction and the valence band. The latter equation can be rewritten as: 2 | |2 ~2 ” () ≈ (~) (412) 0 2 (~)2 where: (~) =
X k
[ (k) − ~]
(413)
is by definition the joined density of states. Considering that the density of states in the BZ is 2 (2)3 and considering a unitary volume, the joined density of states can also be written as: Z 2 (~) ≈ [ (k) − ~] k (414) (2)3 and, using (146):
Z 1 k (415) (~) ≈ 3 4 |∇k (k)| where k is the constant energy surface of equation (k) = ~. Equation (414) is equal to (415) due to the following property of the Dirac delta function: P Z ( ) [()] () = 0 (416) | ( )| where are the roots of equation () = 0. The assumption we made that both the band and are doubly degenerate because of the spin, is always valid for centrosymmetric crystals but it is not applicable to, e.g., zincblende structures. We have a critical point whenever ∇k (k) vanishes; the joined density of states exhibist then a Van Hove singularity (147). Assuming parabolic (and isotropic) bands around Γ we can write: ~2 |k|2 (k) = (k) − (k) ≈ + 2∗
(417)
where:
∗ (0)∗ (0) (418) ∗ (0)+∗ (0) is the reduced mass of the electrons and holes effective masses and = (0) − (0). Now, applying the methods learned in subsection Density of states, we get, in the vicinity of the fundamental absorption edge, the approximate form: ∗ =
52
(~) =
2
(∗ )32 p ~− 3
(419)
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. Band structure in Germanium 6.3.2
” of Germanium
Indirect transitions
In the previous subsection the absorption properties of a semiconductor with a direct gap such as , and all the II-VI compound are explained. But, when the maximum of the valence band and the minimum of the conduction band do not share the same k the related direct transition is not allowed not being vertical (see the figure below). In this case to find out the absorption threshold one must consider more complex transition paths involving phonons.
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Indirect interband transitions This is the case, for example, of , , . In silicon, e.g., the transition (Γ) → () is prohibited by the electric dipole selection rules. In presence of phonons an indirect transition can happen in two subsequent steps. These indirect processes are also important in semiconductors with a direct gap when vertical transitions are forbidden by the point symmetry (eq. 407), even though only in Γ, as it happens in 2 . Referring to the adiabatic theorem (eq. 220) the interaction (r u) can be written as the following lattice sum: X X (r u) ≈ (r − n) − un ·∇ (r − n) (420) n
n
where we introduced the screened pair potentials (r − n − un ). Developing un in normal coordinates (258) and considering, for the sake of simplicity, only one active normal mode, the electron-phonon perturbation operator can be written as: Ã ! X (q) b = − √ e (q) ·∇ (r − n)q·n − (q) + (421) 2 n
The term − (q) is responsible for the absorption (destruction) of a phonon, while the c.c. is responsible for the stimulated emission (creation) of a phonon. A deeper analysis of the matrix elements of operator (421) between initial and final states such as |() (u) (r|0) (see adiabatic theorem in order to understand the meaning of symbols) shows that the allowed transitions conserve the total wave vector (see also equation (355) and inelastic scattering theory): k − k = ±q + g (422)
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Normal events correspond to g = 0 and Umklapp events to g 6= 0. In the presence of the perturbation (421) and of the photon-electron perturbation b (396) the indirect transition can occur in two different ways. In the first one there is at first a direct vertical transition (Γ) → (Γ) and then, due to a creation or a destruction of a phonon with wave vector q(), the quasi horizontal transition (Γ) → () occurs. In the second one at first, due to a creation or destruction of a phonon with wave vector q(), the transition (Γ) → () takes place and then there is the vertical transition () → () caused by the absorption of a photon. The probability per unit time of these events can be calculated using the Fermi’s golden rule up to second order: ¯ à !¯2 X ¯¯ 2 X h | b |ih| b |i h | b |i ¯¯ b |ih| = + ¯ ¯ (423) ¯~ − − ~ − ± ~ (q) ¯
k k
× [ (k ) − (k ) − ~ ± ~ (q)]
Since the previous formula contains (q), the amplitude of the phonon normal coordinate, the probability of indirect transitions depends on temperature. (q) is a fluctuating random variable whose quadratic mean value can be calculated by means of the Bose-Einstein distribution. We can actually use the fact that the mean total energy of the oscillator is twice its mean potential energy: ® 2 (q) 2 (q) ∝
~ (q)
~ (q)
(424)
−1
In (423) the factor multiplying the delta function weakly depends on the wave vector. Moreover, since the second denominator is always greater than the first one, the second type of transition can be neglected. In this way, repeating considerations similar to those in the previous subsection, the imaginary part of for an indirect transition reduces to: Z Z 2 ®2 ” [ (k ) − (k ) − ~ ± ~(q)] k k (425) ∝ (q )
Assuming parabolic bands around stationary points we get, near the indirect band gap = () − (Γ), ” ∝ (~ ∓ ~(q) − )2
(426)
when ~ ≥ ± (q) and 0 otherwise. Neglecting the dispersion of phonon bands (as in the Einstein model), we set as mean frequency (q) = (q),
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the frequency corresponding to wave vector q = q . Because the (425) strongly depends on temperature it is possible to experimentally ascertain the existence of an indirect gap measuring the temperature dependence of absorption. The electron-hole pairs with different wave vectors created during indirect processes generally have a longer lifetime than that of the pairs with the same wave vector generated in direct processes.
6.4
Intraband transitions and plasmons
In metals and in extrinsic semiconductors, besides the interband transitions at high frequencies already treated in the previous subsection, there are quasifree charge carriers which populate partially occupied bands and cause the continuous background shown in the introductive figure on optical absorption. A quantum treatment of intraband indirect transitions (vertical intraband transitions do not exist) could proceed exactly as in the previous subsection just considering occupied and unoccupied states in the same band. The states associated to quasi-free electrons are perturbed by both electronphonon and photon-electron interactions. The model adopted to describe this situation is similar to that used in the theory of stationary transport phenomena (see below) in which just the electron-phonon interaction is considered (for the dynamic part) and the states are Bloch wave packets obeying semiclassical equations. Here we limit ourselves to some simple considerations in the free electrons approximation. We proceed in analogy with the coupling between photons and oscillating ions plasma described in subsection polaritons. We can apply the same theory to the jellium: a plasma constituted by free electrons and by a uniform background of positive ions. The main difference lies in the fact that here the resonance frequency corresponding to binding energy is zero: thus wepwrite = 0. Besides the plasma frequency is that of free electrons = 2 . The dielectric constant in the plasma then becomes: µ ¶ 2 = 0 (1 + ) = 0 1 − 2 (427) and the refractive index:
r
2 (428) 2 Consequently the dispersion relation of transverse e.m. waves becomes: sµ ¶ 2 = 1− 2 (429) =
1−
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2.5
2
1.5
1
0.5
0 0
0.5
1
1.5
2 omega
Jellium dispersion relation
A thorough analysis of Maxwell equations in jellium would show that transverse e.m. waves can propagate only if their frequency is above the plasma frequency (real , black line in the figure). Below the plasma frequency waves are damped (imaginary , red in the figure). Exactly at the plasma frequency ( ) = 0 stationary longitudinal waves, associated to a longitudinal oscillating electric field, are established in jellium (mixed collective non propagating modes). These stationary waves are quantized and their energy quanta ~ are called plasmons. In metals the plasma frequency occurs in the ultraviolet. Plasmons are the evidence of collective motions whose understanding requires going beyond the approximation of independent electrons used in normal band theory. The oversimplified model we have used here has classical bases (Drude-Lorentz theory). A quantum description of the free electron gas would lead to a wave vector depending dielectric function (spatial dispersion) as it is introduced in the theory of potential screening (217).
7 7.1
Transport phenomena Phenomenology
In a crystal containing mobile charges (conductor or semiconductor) the presence of an electric field E of external origin causes an irreversible charge transport phenomenon (electric current) removing the body from thermodynamic equilibrium. On the other hand, in all crystals a temperature gradient ∇ causes an irreversible heat flux called conduction. From a macroscopic
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point of view (adopted by the thermodynamics of irreversible processes) when the cause of the flux is small, we are within the linear response regime and two famous phenomenological laws, Ohm’s local law and Fourier’s law, well describe both transport processes: j = E
(430)
q = −k ∇
(431)
where j is the charge current density vector. Its flux through a surface represents the amount of charge per unit time, current , flowing through the surface: Z = j · n (432)
The principle of electric charge conservation can be written as the continuity equation for the charge density (charge per unit volume). + ∇ · j =0
(433)
In the Fourier’s law q is the heat current density vector. Its flux through a surface represents the amount of heat flowing through the surface per unit time: Z † = q · n (434)
q is strictly connected to the entropy current density vector q whose divergence appears in the balance equation for entropy density (entropy per unit volume) ³q ´ + ∇· = [] (435) [] is the entropy production (inherently positive). The latter equation shows, in a local form, the second principle of thermodynamics. For electrical conductivity we use the symbol while we use the symbol k for thermal conductivity. More precisely in crystals both conductivities are second order tensors and the two phenomenological laws should be generalized as: =
13 X
(436)
= −
13 X
(437)
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These new laws show that generally the current density vectors have not the same direction of the electric field and of the temperature gradient. As a matter of fact even more general kinetic coefficients exist: a temperature gradient can generate an electric current and an electric field can create a heat flux. The kinetic microscopic theory (see below) explains these correlated effects called thermoelectric effects (Seebeck and Peltier effects). Furthermore, the Onsager theorem, based on microscopic reversibility of dynamic laws, states that the two conductivities are symmetric tensors: = , = . Since every symmetric tensor, with an appropriate rotation of the coordinate system, can be reduced to the diagonal form: ⎞ ⎛ 1 0 0 σ = ⎝ 0 2 0 ⎠ (438) 0 0 3 it turns out that the number of independent components of both σ and k can be, at most, three. Due to crystal symmetry this number can be further decreased. Actually, let R be a symmetry operation and R the unitary matrix which transforms the atom coordinates (position vectors) according to the symmetry operation, then it should be: 0
σ = RσR −1 = σ
(439)
Repeating this consideration for each symmetry operation, it is shown that triclinic, monoclinic and orthorhombic crystals have three independent components: 1 6= 2 6= 3 ; tetragonal, trigonal and hexagonal crystals have two independent components: 1 = 2 6= 3 ; and cubic crystals only one: 1 = 2 = 3 (isotropy). The situation is analogous for thermal conductivity. As far as the temperature dependence is concerned, the electrical resistivity ( = 1) of metals shows a monotonically increasing trend. At low temperatures it increases as 0 + 5 . 0 is called residual resistivity depending on static lattice defects (vacancies/interstitials, dislocations, packing defects). At high temperatures the resistivity increases linearly: ∝ . Generally, as stated by Matthiessen law, the total resistivity is the sum of both an athermal (0 ) and a thermal contribution.
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Electric conductivity vs T The case of semiconductors is more complex since their resistivity depends also on the possible doping and its treatment needs special considerations. In metals the thermal conductivity at low temperatures linearly increases with , reaches a maximum value and then rapidly decreases until it reaches a constant value at high temperatures. In insulators, instead, at low temperatures the thermal conductivity increases as 3 , reaches a maximum value and then decreases as −1 at high temperatures. In metals, at very low temperatures, the ratio between electrical and thermal conductivity is substantially proportional to temperature and does not depend on the particular metal (Wiedemann-Franz law).
Metals
Insulators
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Kinetic theory
Since transport coefficients refer to irreversible processes in non-equilibrium conditions, for a theoretical treatment we cannot exploit the traditional bridge connecting the microscopic (quantum) dynamic description to equilibrium macroscopic properties (functions of state): the equilibrium statistical mechanics. As long as electrons are concerned, for equilibrium situations major information is given by the Fermi-Dirac distribution (116). In the present non-equilibrium case we should use, instead, the Boltzmann kinetic theory adapted to electrons in the semiclassical dynamic scheme. At first we start with a completely classical description of a gas with non interacting electrons. In the phase space (r p) of one electron we can introduce the single particle distribution function: (r p)rp = representing the number of electrons whose instantaneous state is within a volume Γ = rp. = = (r p)rp represents the probability of occupation of the instantaneous state (r p). If, moreover, the electrons are under the action of an external field of forces F(r) = −∇U (r), the motion is led by the single particle Hamilton function = |p|2 2 + U (r). For a given energy , the dynamic trajectory of an electron in phase space obeys Hamilton equations (191) from which one can demonstrate the Liouville theorem: (r p) (and, consequently, (r p)) is constant along all dynamic trajectories: (r p) =0
(440)
The proof is shown in the appendix below. If now we want to introduce in the simplest way the effect of interactions (collisions) with other particles (electrons and/or phonons) we can correct ad hoc the Liouville theorem as follows: µ ¶ (r p) (r p) (r p) − 0 (r p) = (441) =− (p) The previous equation is based on the heuristic hypothesis that, in the absence of external forces and temperature gradients, thermal equilibrium is restored by scattering between particles in a mean relaxation time (p). In the classical case 0 (r p) is the Maxwell-Boltzmann distribution 0 (r p) = −1 exp(−0 (r p) ) where = is the partition function.
Z
exp(−0 (r p) )rp
(442)
(443)
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Computing the total derivative in equation 441 and using the Hamilton equations we get the Boltzmann equation: ³ p ´ (r p) − 0 (r p) + F(r) · + · =− p r (p)
(444)
Using the effective mass theorem (equivalent Hamiltonian) we now modify the above equation to make it applicable to a quantum electrons system. Electrons will be described by semiclassical Bloch wave packets moving in a partially occupied band = (k) neglecting interband transitions. This is the case of a conductor under the effect of a weak electric field E: F(r) = −E. Employing the results of subsection Semiclassical dynamics, we can now write the Boltzmann equation in the stationary case = 0 as: F(r) (r k) (r k) (r k) − 0 (k) · + v(k) · =− ~ k r (k) where: v(k) =
1 (k) ~ k
(445)
(446)
In the fermion quantum situation 0 (k) ≈
exp
h
1 (k)− (r)
i
(447) +1
is the Fermi-Dirac distribution. The density of occupied states in k space depends also on r and its expression per unit volume is: D(r k| (r)) =
2 (r k) (2)3
(448)
where we have considered the spin degeneracy and assumed a local temperature field (r) with a weak gradient. We can thus apply a linear response approximation. To do that we rewrite equation (445) as: (r k) = 0 (k) − (k)
F(r) (r k) (r k) · − (k)v(k) · ~ k r
(449)
We solve now this equation approximately up to the first step of an iterative procedure: (r k) ≈ 0 (k) − (k)
0 (k) F(r) 0 (k) · − (k)v(k) · ~ k r
(450)
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Computing the partial derivatives 0 (k) 0 = ~ v(k) k µ ¶ 0 (k) 0 0 (k) − = = − r r (r) r we finally get: ∙ ¸ 0 ( − ) v(k) · F(r) − = (r k) ≈ 0 (k) − (k) (r) r = 0 (k) + (r k) 7.2.1
118
(451)
(452)
Appendix: Liouville theorem
The motion of a set of N particles in phase space is equivalent to the motion of a fluid with density = (r p); thus the conservation principle of mass corresponds to the conservation of the number of particles. This principle can be written in the form of a continuity equation: + ∇· (v) = 0 In the phase space of a particle (independent particles gas in an external field): µ µ ¶ X ¶ 3 3 (r p) X + + =0 =1 =1 Using the Hamilton equations we get: (r p) + { } = 0 where: 3 3 X X − { } = =1 =1
is the Poisson bracket. The total derivative of , as already discussed in the derivation of (444), is: (r p) = + { } and, consequently, it is immediately obtained that = 0 (c.v.d.). Note that the last equation is very similar to that for the time derivative of a quantum operator b b b h b bi = + . (453) ~
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Electrical conductivity
Once we have solved the Boltzmann equation (445) in the linear response regime (eq. 452), the current density vector in the local Ohm law (430) can be computed (as a function of the electric field) as the integral over the whole k space : Z Z −2 j = − D(r k| (r))v(k)k = (r k)v(k)k (454) (2)3 To a first approximation, neglecting the thermoelectric effects due to the thermal gradient, we get Z 22 j= (k)( − ) (v(k) · E) v(k)k (2)3 Where, based on Kramers theorem ((−k) =(k) → v(−k) = −v(k)) and 0 (−k) = 0 (k), we have taken into account that Z 2 (455) 0 (k)v(k)k = 0 (2)3 and made the further approximation 0 ≈ −( − ). Thus (see subsection density of states): Z 22 j= (k)( − ) (v(k) · E) v(k) 3 (2) |∇k (k)| and finally 2
j =
Z
(k) (v(k) · E) v(k) (k)
(456)
where the integral is over the Fermi surface (k) = ; we now define the density of states on the Fermi surface as: (k) =
1 2 3 (2) |∇k (k)|
(457)
The electric conductivity tensor can be eventually written in the form: Z 2 σ= (k) (k)v(k)v(k) (458)
Notice that only electrons on the Fermi energy can contribute to the charge transport process. In metals the dependence of conductivity on temperature is lumped in (k) = (k), on the Fermi surface, which is mainly due to
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electron-phonon scattering. The probability of a scattering event per unit time is ( (k))−1 and can be computed by means of the Fermi’s golden rule up to first order using the matrix elements of an appropriate electron-phonon interaction operator. At high temperatures this implies a dependence of ( (k))−1 on h2 (q)i which, in turn, is ∝ and explains the linear trend of resistivity with respect to temperature. At low temperatures, instead, elastic scattering with lattice imperfections prevails, and this explains the athermal residual resistivity. At intermediate temperatures a simple theory combining the different processes in a sufficiently accurate way does not exist. Actually it is very complicated to account for (Umklapp) processes. In simple metals the Fermi surface is spherical (free electrons model), to a first approximation, and all the quantities that have to be integrated are constant. So in this case we get, after a few simple steps: 2 (2∗ )32 p = 3 2 2 ~3 2
(459)
Using the expressions of Fermi energy (115) and Fermi speed ~ ∗ we finally get an equation often used in applications: µ ¶ 2 = ∗ (460) In the case of transition metals (where there is a superposition between bands and , and the Fermi level is within the band) it is possible to use the previous formula in an additive way (460) for the two different types of electrons, with very different values for effective masses and relaxation times. In the case of noble metals (where there is a superposition between bands and , and the Fermi level is above band within a band) wave vectors on the Fermi surface are very near the edge of the first Brillouin zone producing a strong deviation of the Fermi surface from the spherical shape with some band regions having a negative curvature and, thus, implying conduction due to holes. Also in this situation an additive (two bands) model with two types of charges, with opposite signs, is used.
7.4
Electronic thermal conductivity
Here we start with the thermodynamic relation for the differential of specific entropy: 1 (461) = − where = is the internal energy per unit volume and = is the number of electrons per unit volume. is the chemical potential, which
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can be approximated by the Fermi energy . By analogy we can make the assumption that the entropy flux is connected to energy and electrons fluxes through the following equation: j =
1 j − j
(462)
Then irreversible thermodynamics gives the following formula for entropy production (eq. 435): µ ¶ ³´ 1 · j + ∇ · j [] = ∇ (463) Remembering that the heat current density vector is defined as q = j , we obtain: q = j − j = v − v =v − v = (−)j
(464)
We approximate by , and then, using again the method of microscopic kinetic theory (see previous section), we can compute q as: Z 2 (r k)(− )v(k)k q = (465) (2)3 using now the following expression: (r k) = (k)v(k) ·
0 ( − ) r
(466)
which is valid in the absence of external fields. Remembering Fourier law, we can finally write thermal conductivity as follows: µ ¶ Z 2 0 ((k) − )2 − k (467) = (k)v(k)v(k) (2)3 In this case we cannot simply approximate −0 with ( − ), since now we would obtain a null result. Instead we must use the better Sommerfeld approximation (eq. 122). Using, as usual, the result k =
E |∇k (k)|
(468)
and again defining the density of states on the Fermi surface as =
1 2 , (2)3 |∇k (k)|
(469)
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after an integration over energy , we finally get Z 1 2 2 (k) (k)v(k)v(k) = 3
122
(470)
But we have already demonstrated that the above integral coincides with the electrical conductivity divided by the square of the electron charge (eq. 456). Thus: 2 2 = (471) 3 2 The above equation is called Wiedemann-Franz law. If we make use of this result, knowing the trend of electrical conductivity, we can get the trend of thermal conductivity with respect to temperature. Figures are shown in subsection Phenomenology. We have assumed that all relaxation times (scattering processes) involved in heat transport in metals coincide with those involved in charge transport. This is not exact and can induce a deviation from Wiedemann-Franz law, mainly at intermediate temperatures.
7.5
Lattice thermal conductivity (insulators)
In insulating crystals heat conduction is due to the non-equilibrium kinetics of lattice waves packets in presence of a temperature gradient. In this situation the carrier particles are phonons packets moving with the group velocity: v (q) =
(q) q
(472)
and carrying the energy quantum ~ (q). The distribution function in thermodynamic equilibrium is the Bose-Einstein factor (mean occupation number of a normal mode with wave vector q): 0 (q) =
1
(473)
~ (q)
−1 The phonons chemical potential is zero, since the number of phonons is not conserved. The heat density flux vector can be written as: Z 1 X q = (474) (r q)~ (q)v (q)q (2)3
Thus in equation (450) we can write, in the present case:
~ (q)
0 (q) 0 (q) 1 ~ (q) = = µ ¶2 ~ (q) r r 2 r −1
(475)
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which can be written as: (q) 0 (q) = r ~ (q) r
(476)
introducing the specific heat of each phonon (q) (eq. 234): ~ (q)
1 ~2 2 (q) (q) = µ ¶2 ~ (q) 2 −1
(477)
The heat density flux can thus we written as (see (450) with q instead of k): Z 1 X q = − (478) (q)v (q)v (q) (q) q 3 r (2) And then, using Fourier law, the lattice thermal conductivity tensor can be expressed as: Z 1 X (q)v (q)v (q) (q)q. k = (479) (2)3 Sometimes the phonon mean free path is introduced as Λ (q) = v (q) (q). Using a simplest treatment, strictly inspired by the kinetic theory of gases, we could immediately write 1 k = Λ 3
(480)
where is the total specific heat per unit volume and and Λ are opportune average values of v (q) and Λ (q). The above equation can be derived by the following elementary considerations. Phonons propagating a distance ∆ = = Λ corresponding to a temperature drop ∆ , carry a quantity of heat per particle = ∆ , being the contribution of a single particle (phonon) to specific heat. Then the heat flux is p = ∆ where is the number of particles per unit volume and = h2 i. If the temperature 2 gradient is we can write p = − ( ) = − Λ ( ) and = 3, being = h2 i. In a Debye model, neglecting the contribution of optical phonons (which have nearly zero group velocities), the velocity of acoustic phonons is constant and is equal to the mean speed of sound. For phonon-phonon interactions (anharmonic effects) we can consider that Λ( ) ∝ ( )−1 . At high temperatures ∝ and the specific heat is constant. At low temperatures the specific heat follows 3 ,
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while the mean free path is limited by the crystal size Λmax = . From a detailed microscopic point of wiew, it can be shown that only phonon-phonon Umklapp scattering guarantees local thermodynamic equilibrium and is responsible for a finite thermal conductivity.Without the contribution given by scattering events, the thermal conductivity would diverge, as proved by Peierls. Anyhow, without the size limitation to mean free path, at low temperatures k would diverge as ( )−1 ≈ exp(~ min ), where min is the minimum frequency of phonons which may encounter a scattering process. Two phonons can participate to a scattering process when the sum of their wavevectors goes beyond the first Brillouin zone. An important role is also played by elastic scattering off lattice defects. Whenever the wavelength of a phonon is greater than the size of a defect a Rayleigh scattering occurs, for which Λ ∝ |q|−4 .
8 8.1
Appendix: Quantum approximate methods Time-dependent perturbations
A given system, initially isolated, since time instant 0 is exposed to a dynamic perturbation (i.e. to the time-dependent interaction with a second coupled system) until time instant 0 + . The response of the first system (induced quantum transitions) can be obtained solving Schroedinger equation: ´ ³ b ~ = + b ()(0 ) (481) Function (0 ) is 1 when 0 ≤ 0 + and 0 otherwise. b () is the b is the unperturbed Hamiltonian of time dependent perturbation operator. the initially isolated system the spectrum of which is assumed to be known. () Let us first assume that the unperturbed energy levels are discrete and non-degenerate. b = () (482) If at time 0 the system is in the stationary state (it is not necessarily the ground state), then: (0 ) = (483) Following Dirac, we look for a solution of the Schroedinger equation (481) such as: () X () − ~ (484) () =
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That is, we use a generalized form of superposition principle with timedependent coefficients (). At time 0 + the perturbation ceases. In view of the chosen initial condition, the square of the absolute value of (0 + ) represents the transition probability → ( ) from initial state | i = |i to final state | i = |i during a time (by now on we will consider 0 = 0): → ( ) = | ( )|2
(485)
Substituting the wave function (484) in the Schroedinger equation and using R the orthonormality of the set of all (h|i = ∗ r = ), we get a system of ordinary differential equations for the coefficients () ~
() X b = h| ()|i () 6=
where we have defined:
()
(486)
()
− = ~
(487)
and with the initial conditions: (0) = The matrix elements of the perturbation operator are defined as: Z b () = h| ()|i = ∗ b () r
(488)
(489)
and we assumed that all the diagonal elements are zero, as it is true in many important cases (see below). At first we integrate each equation of the system with respect to time obtaining (for 6= ): Z 0 1 X b 0 0 0 h| ( )|i ( ) (490) () = ~ 0 6=
Then we proceed with an iterative solution method assuming that all the zero-th order solutions coincides with the initial conditions: (0) () = (0) =
(491)
Then the first order solutions are obtained substituting the zero-th order solutions in the right side of equations(490) : Z 0 1 X b 0 0 (0) 0 (1) h| ( )|i ( ) (492) () = ~ 0 6=
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thus obtaining (1) ()
1 = ~ 1 ~
Z X 0 0 0 h|b ( )|i = 0 6= Z 0
0
(493)
h|b ( )|i 0
0
The iteration proceeds substituting (0) ()
+
(1) ()
1 = + ~
Z
0
0
h|b ( )|i 0
0
again in the right side of equations (490): Z i 0 0 h 1 X b 0 (2) (0) 0 (1) 0 () = ( ) + ( ) h| ( )|i ~ 0 6=
(494)
(495)
and then obtaining: Z 0 1 b 0 0 (2) () = h| ( )|i + (496) ~ 0 µ ¶2 X Z Z 0 0 00 00 1 0 00 0 h|b ( )|i h|b ( )|i ~ 6= 0 0 and so on, after substitutions: Z i 0 0 h 1 X b 0 (+1) (0) 0 (1) 0 () 0 ( ) + ( ) + + ( ) h| ( )|i () = ~ 0 6=
(497) We consider now the first order approximation in the situation in which c0 . From (493 485) we get: b () = ¯ ¯2 ¯ c ¯ ¡ ¢ 2 h| |i ¯ ¯ 0 (1) 2 sin 2 → ( ) = ¡ (498) ¢ 2 ~2 2
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127
1
0.75
0.5
0.25
0 -20
-10
0
10
20 x
2
(2) Function y= sin(2) 2 ; =
If À 1 we obtain that the following function ¡ ¢ 2 2 sin 2 ¡ ¢ 2
(499)
2
can be approximated with an isosceles triangle with base 4 and height 2 : the area under the principal maximum is thus approximately equal to 2 . In this limit ´ can be approximated by a Dirac delta function ³ the function () () ( ) = ~ − . Summarizing, in the limit À 1 ( → ∞) we can write the transition probability as: ¯2 ¡ ¢ 2 ¯¯ c ¯ (1) () h| → ( ) ≈ |i (500) ¯ () − ¯ 0 ~
From an experimental point of view, it turns out that if the initial state |i is an excited one it is not a truly stationary state and it tends to spontaneously decay also in the absence of a any perturbation (spontaneous decay) following an exponential law (in the simplest cases): | ()|2 = −
(501)
where the mean lifetime of state |i depends on the experimental energy () () uncertainty ∆ , around the mean value , with which the energy of the state is known through the empirical inequality: () ≥ ~ ∆
(502)
From a physical point of view, then, the 500 can be used when −1 ¿ ¿ .
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Harmonic perturbation
For the time dependence: c− − c+ + b () =
(503)
´† ³ − + c c , and in the limit → ∞ ( −1 where = ¿ ¿ ), with a procedure similar to that just used, we get the transition probability per unit (1) time e→ in the form: ¯2 ¡ ¢ 2 ¯¯ c± ¯ (1) () h| ≈ |i ± ~ (504) e→ ¯ ¯ () − ~ In this case the delta function tells us that transitions can happen only if the total energy of the two coupled systems is conserved: () ± ~ = 0 () −
(505)
in agreement with the Bohr postulate of old quantum theory when ~ is the energy of a light quantum perturbing an electron system. Using equations (496), instead, we would obtain the second order result (useful to describe e.g. non linear optics): ¯2 ¡ ¢ 2 ¯¯ c± ¯ (2) () e→ ( ) ≈ ± ~ + (506) ¯h| |i¯ () − ~ ¯ ¯2 c± |ih| c± |i ¯¯ ¡ ¢ 2 ¯¯ X h| () () − ± ~ ± ~ ¯ ¯ () ~ ¯6= () − ± ~ ¯
Note that the number of energy quanta ~ of the second system involved in the process is equal to the order of the perturbation. Previous equations describe the absorption or the stimulated emission of one or two energy quanta ~, respectively. At the second order the transition can be seen as the sequence of two transitions through intermediate states |i. Intermediate transitions do not conserve energy. If the final state is a quasi-stationary state the singularity of the Dirac delta function can be removed. Introducing the density of final states () and integrating the (504) we get: Z ¯2 ¡ ¢ 2 ¯¯ c± ¯ (1) () e ± ~ (() )() =(507) → ≈ ¯h| |i¯ () − ~ ¯2 2 ¯¯ c± ¯ () ∓ ~) ¯h| |i¯ ( ~ usually written as:
2 ¯¯ c± ¯¯2 (1) e → ≈ ¯h| |i¯ ( ) ~
(508)
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the above equation is called Fermi’s golden rule, where = ∓~. Studyc± |i we get the so called selection rules, strictly ing the matrix elements h| connected to the symmetry of the system. 8.1.2
Photon-electron interactions: electric dipole selection rules
The theory outlined in the previous paragraph can be applied to the interaction between an electromagnetic wave and the electrons of an atomic system (atoms, molecules, solids, in particular crystals...); it becomes particularly simple in case of a plane, monochromatic, linearly polarized wave of wavelength À , where is the characteristic linear dimension of the elementary volume occupied by electrons (in the case of crystals it is the volume of the primitive cell). For a more general treatment see Optical properties. If the wave is polarized along axes and propagates along axis For the sake of simplicity we refer to the single electron of a hydrogen atom, is the atomic radius and we set the origin at the nucleus. The wave electric field can be written as follows: E = u cos( − ) (509) where = (2) is the wave vector. Thus (||) ≈ = 2 ¿ 1, and we are lead to the simpler form: ¡ ¢ 1 E ≈ u cos() = u + − 2
(510)
As far as this field does not appear anymore as a propagating wave, we can neglect magnetic effects and consider the atom as an electric dipole μ = −r ( is the absolute value of the electron charge) interacting quasi-statically with the instantaneous electric field. Thus we assume as perturbation operator 1 1 (511) + b − b = −b μ·E = b 2 2 c− = 1 b c+ = = 12 . In this b is then written in the form (503) whith 2 c± |i = situation the matrix elements of the perturbation operator are h | 1 h ||i. In order to see when they vanish we must consider the following 2 integral: Z (512) h ||i = ∗ r In case of an hydrogen atom the orbital part of the 22 wavefunctions belonging to the same energy level () ≈ −
136 eV; ( = 1 2 ) 2
(513)
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can be written in spherical coordinates ( ) as: ( ) =
() ( )
(514)
where () is the radial wavefunction, ( ) is a spherical harmonic, is the principal quantum number, is the orbital quantum number and the magnetic one. The parity of these states with respect to a coordinate inversion is defined and depends exclusively on ( ); it is (−1) (when r → −r, = |r| remains the unchanged). In order to have a nonzero integral, since is an odd function, it should be − = ∆ = ± , i.e. the parity of the final state should be different from that of the initial state. A further consideration about the angular momentum of the electron-photon system imposes the stronger limitation: ∆ = ±1 (optical selection rule). In an atom with atomic number in the mean field approximation, the single electron energy levels depend on () and the degeneracy reduces to 2(2 + 1). However in the (514) while the () change as functions of , the ( ) remain the same: the selection rules don’t change. In a crystal instead the initial and final states are Bloch waves normally belonging to different bands and one must consider the total fluctuating electric dipole of the crystal basis. This leads to specific selection rules which can also involve phonons when the adiabatic approximation is not applicable.
8.2 8.2.1
Static perturbation theory Adiabatic switching on
Consider a perturbation operator (without the (0 ) modulation) such as: c b () = lim (515) →0
where 0 and ¿ 1. Setting 0 = −∞ and 0 + = 0, equations (493) can be written as: Z 0 0 1 0 (1) c |i 0 =(516) (0) = h| ~ −∞ Z 0 c|i c|i 0 0 h| h| 0 c|i h| = () = () ~ ~ + −∞ − +
Notice that lim→0 has not been taken yet. Taking the limit now, it is the c, we obtain: adiabatic switching on of the static perturbation (1) (0) =
c|i h| ()
()
−
(517)
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and substituting this result in (484), for a generic perturbed eigenfunction (1) we get: (1) = +
X h| c|i ()
()
6= −
(518)
Using these perturbed eigenfunctions we can now compute the perturbed energy levels as: ³ ´ b c ≈ h(1) | + |(1) (519) i
Up to second order in and considering the necessary condition for series convergence ¯ ¯ ¯ () ¯ c|i¯¯ ¯ − () ¯ À ¯¯h| (520) after long calculations we get:
() c|i + 2 = + h|
¯2 ¯ ¯ ¯ c X ¯h| |i¯ ()
6=
()
−
(521) (1)
In many applications keeping in the final formulae for the coefficients (0) the initial expression: c|i h| () (522) () − + it is possible to account for spontaneous deacay and for other dissipative phenomena provided the value of is adjusted in order to have a better match with experimental data. Finally, with an infinitesimal it is possible to prove that (PP = Cauchy principal part, see also Kramers and Kronig relations): 1 ()
()
− +
≈ PP
1 ()
()
−
¡ () ¢ − − () .
(523)
This last equation and the following one, a new representation of Dirac delta function coming from the Green functions formalism, are very commonly used. µ ¶ 1 1 1 () ≈ − Im (524) = 2 + + 2
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Degenerate energy levels
When we use formulae such as (521) and n (518) owe are assuming that degen() erate levels in the unperturbed spectrum do not exist. As a matter ()
of fact if to a given level there correspond more than one eigenfunction | i in the perturbation series there would be divergent terms such as: c| i h |
(525)
= 1 + 2
(526)
()
()
−
We can overcome this problem as follows: we assume that to the degenerate energy level () there correspond the two unperturbed eigenfunctions 1 = |1i e 2 = |2i. To a first approximation we write the perturbed wavefunction as:
substituting now in the equation for perturbed stationary states ´ ³ ¡ ¢ c b + (1 + 2 ) = () + ∆ (1 + 2 )
(527)
c|1i = h1| c|2i∗ , the solubility condition is: Considering that h2| ¯2 ¯ ³ ´³ ´ c|2i¯¯ = 0 c|1i − ∆ h2| c|2i − ∆ − 2 ¯¯h1| h1|
(529)
and using the orthonormality condition for functions 1 and 2 we get the homogeneous linear algebraic system: Ã !µ ¶ µ ¶ c|1i − ∆ c|2i 0 h1| h1| (528) = c|1i c|2i − ∆ 0 h2| h2|
The two roots are: # " ¯2 ¯ ´2 ³ ´ 1 r³ 1 c|2i¯¯ c|1i − h2| c|2i + 4 ¯¯h1| c|1i + h2| c|2i ± ∆ ± = h1| h1| 2 2 (530) When the diagonal matrix elements are zero, we simply have: ¯ ¯ ¯ c ¯ ± (531) ∆ = ± ¯h1| |2i¯
The unperturbed degenerate level () so splits into the two perturbed nondegenerate levels: ¯ ¯ ¯ c ¯ ± () = ± ¯h1| |2i¯ (532)
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¯ ¯ ¯ c ¯ |2i¯. Using the first of Between them there is an energy gap = 2 ¯h1| the (528) c|2i± = 0 (533) −∆ ± ± + h1|
and the normalization condition
c|2i 0, we get: when h1|
||2 + ||2 = 1
(534)
1 1 ± = √ ; ± = ∓ √ 2 2
(535)
To the two levels + and − (532) correspond respectively the two eigenfunctions + and − : 1 1 (536) ± = √ 1 ∓ √ 2 2 2 After having used this procedure for all unperturbed degenerate levels, we can use the (521) in order to have more accurate level evaluation. The above procedure succeeds, that is we get two different solutions, when the perturbation breaks the symmetry of the unperturbed system.
8.3
Dirac delta function
The Dirac delta function () is defined by the following integral properties: +∞ Z
()( − ) = ()
(537)
−∞
+∞ Z
() = 1
(538)
−∞
where () is any normal function. Strictly speaking no function with these properties can exist and () should be better called a distribution, a new mathematical object clarified by the French mathematician Schwartz in the 1950’s. In physical applications the above properties can be exhibited up to a good approximation by series of functions depending on a parameter. For example the Gaussian functions 2
− 22 ∆(|) = √ 2
(539)
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satisfy exactly the second condition for any and very well the first in the limit → 0. In fact lim
→0
+∞ Z
()∆( − |) ≈ () lim
→0
−∞
+∞ Z
∆( − |) = ()
(540)
−∞
In quantum mechanics (scattering theory and dynamic perturbation theory) one meets integrals of the type +∞ Z
(541)
−∞
where both and are real. We can write +∞ Z −∞
= lim
→∞
+ Z
= lim
→∞
−
µ
− −
¶
= lim 2 →∞
sin() = 2()
(542) has a very sharp maximum In fact for very big values the function (equal to 2) for = 0 and becomes null about the origin at = ±. Then for larger || values it has symmetric vanishingly small damped oscillations. Thus in the limit of big values we can approximate the plot of the function by only the (extremely narrow) isosceles triangle with huge height 2 and vanishing basis 2This narrow triangular peak has area 2 and the function (in the limit → ∞) behaves similarly to 2∆(|) in the limit → 0. In 3D we have Z (r)(r − a)r = (a) (543) 2 sin()
with (r − a) = ( − )( − )( − ). Even though this last notation is not correct (as the product of distributions is not defined), it has an operational meaning thinking of each unidimensional delta function as the limiting value of a series of functions such as ∆(|). Thus we can write Z 1 k·r r =(k) (544) 3 (2) Another useful representation is 1 →0 2 + 2
() = lim
(545)
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Often the property
135
X ( − ) ¯¡ ¢ ¯ ¯ ¯ ¯ = ¯
[()] =
(546)
is used, where the are the roots of equation () = 0.
8.4
Green functions
Referring to the time independent Schroedinger equation: ³ ´ b − =0
(547)
0
the solution (r r ) of the following equation is called Green function ³ ´ 0 0 b − (r r |) = (r − r ) (548) 0
We obtain the (r r |) as a function of eigenvalues and eigenfunctions b (spectral representation). Since the (r) are a set of complete (r) of and orthonormal functions we can write: X 0 0 (r |) (r) (549) (r r |) =
and thus:
³ ´X X 0 0 b − (r |) (r) = ∗ (r ) (r)
8
(550)
. By multiplying both sides of equation by ∗ (r) and integrating over the whole space we get: 0 0 (r |) ( − ) = ∗ (r ) (551) and then 0
(r r |) =
X ∗ (r0 ) (r) −
(552)
The local density of states (r|E) is defined as: X | (r)|2 ( − ) (r|E) =
(553)
8
P
0
0
0
Condition ∗ (r ) (r) =(r − r ) can be written expanding the (r − r ) as a series of eigenfunctions and determining the coefficients using the orthonormality and the R 0 0 definition of : (r)(r − r )r = (r ).
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Using the following identity (see Kramers and Kronig relations): µ ¶ 1 1 = () + −
136
(554)
where is infinitesimal, and introducing the complex energy → − , we then get: 1 (555) (r|E) = lim Im (r r| − ) →0 and the density of states (): Z 1 (556) (E) = lim Im (r r| − )r →0 R 0 (r r|)r is called trace of the Green function (r r |) considered as a whose elements are labelled with the pair of continuous indexes ¡ matrix 0¢ r r .
8.5
Selfconsistent mean field
The band structure of electronic levels in a crystal has been considered within the independent electron approximation in a periodic potential equal for any electron. We thus assumed that the quantum motion of each electron was described by a single-electron wavefunction (r s ) depending only on the coordinates and the spin of the th electron. Here represents the set of quantum numbers defining the stationary Bloch state of the th electron: the wavevector k and the branch index (which depends on the non translational symmetry). Generally the periodic potential is not known a priori (as instead we have always assumed) and should be determined together with functions (r s ). Historically the problem was solved for the first time by Hartree with the heuristic solution we are going to explain, then it was considered by Fock in a variational scheme which included the exchange symmetry of the many body wavefunction (r1 s1 r2 s2 r s )9 in a system with interacting electrons, leading to the Hartree-Fock mean field. We first outline the second more rigorous method. In order to simplify the treatment we consider a many electrons atom in which the nuclear potential (attractive for electrons) exhibits a spherical symmetry. So the Hamiltonian operator describing the motion of electrons ( = 1 2 ) in the atom is: 2 X X 2 2 1X b =−~ ∇2r − + (557) 2 4 |r | 2 4 |r − r | 0 0 6= 9
With the notation used in section 4 it is the (r|0), where corresponds to the ground state with minimum energy.
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and can be rewritten as a sum of single particle Hamiltonians plus an interaction Hamiltonian: X X b + 1 b= (558) 2 6= 2 2 b = − ~ ∇2r − 2 40 |r |
(559)
2 (560) 40 |r − r | Instead, to study the crystal case, we should consider only valence electrons in a periodic potential, as stated in section 4. In the Hamiltonian (557) we neglected the spin-orbit interactions, the spin-spin interactions and all other relativistic effects. The expectation value of the electrons total energy should be calculated as: Z b = ∗ (r1 s1 r2 s2 r s )(r b || 1 s1 r2 s2 r s )Π r =
(561)
The Hartree equations (see below) may be derived assuming that: (r1 s1 r2 s2 r s ) = Π (r s )
(562)
from the following variational condition: Ã ! X b (r s ) − Π (r s )||Π (r s )| (r s ) = 0
(563) where the Lagrange multipliers are needed to define the necessary condition b conditioned by the normalizafor minimum of the functional || XR tion conditions (r s )| (r s ) = ∗ (r s ) (r s )r = 1. s
We look for the specific functional form of each (r s ) which minimizes the energy of the electron system. It is, however, necessary to notice that the form (562) of the many body wave function does not have the necessary exchange symmetry. Fock solved this problem using a Slater determinant wavefunction ¯ ¯ ¯ (r1 s1 ) (r2 s2 ) (r s ) ¯ 1 1 1 ¯ ¯ 1 ¯¯ 2 (r1 s1 ) 2 (r2 s2 ) 2 (r s ) ¯¯ (r1 s1 r2 s2 r s ) = √ ¯ ¯ ! ¯ ¯ ¯ (r1 s1 ) (r2 s2 ) (r s ) ¯ (564)
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in the same variational scheme considering, always by means of Lagrange multipliers, the orthonormality conditions (r s )| (r s ) = . The Slater wavefunction guarantees the exchange antisymmetry (if two columns are exchanged the determinant changes its sign) and the Pauli exclusion principle (if two electronic states are identical, there are two identical rows and then the determinant is zero). Now instead we will deduce Hartree equations following an heuristic method based on a electrostatic statistical approximation for the mean field. If the many body wave function is the product (562), thus implying the statistical independence of the single electronic motions, we can assume that the total charge density at position r, due to all the electrons except , is: ¯2 X ¯¯ ¯ (r) = − (565) ¯ (r)¯ 6=
for the sake of simplicity we have not considered the spin. This mean density distribution creates a mean electrostatic potential (r) obeying the Poisson equation: (r) ∇2 (r) = − (566) ◦ The solution of this equation can be written as the Coulomb integral: ¯ ¯2 ¡ 0¢ Z Z ¯¯ (r0 )¯¯ X r 1 0 0 (567) r (r) = 0 r = − 0 4◦ |r − r | 4◦ 6= |r − r | If we now consider electron , its wave function satisfies the Schroedinger equation: ∙ 2 2 ¸ ~∇ 2 − (568) − − (r) (r) = (r) 2 4◦ |r|
We added together the Coulomb attractive potential energy of nucleus and the mean repulsive Coulomb potential energy (r) = − (r) which takes into account the screening of the nuclear charge due to all other electrons with respect to th electron. Actually this situation can be considered as 2 that of a single attractive force field − 4(r)|r| 2 u due to the interaction with the pointlike charge in a medium where the dielectric function depends on position as: ◦ (r) = (569) 4◦ 1 + |r| (r) Since (r) 0, (r) ◦ . Using the explicit expression of (r) we get the
c °2017-2018 Carlo E. Bottani system of Hartree equations: ⎡
2 2 2 ⎢ ~2 ∇ − + ⎣− 2 4◦ |r| 4◦
Solid State Physics Lecture Notes
¯ ¯2 0 ¯ ¯ Z X ¯ (r )¯ 0
6=
|r − r |
⎤
0⎥ r ⎦ (r) = (r)
139
(570)
This nonlinear system of integro-differential equations can be solved in an it(0) erative way, starting from the hydrogen-like expressions (r) for the (r) in the square bracket corresponding to (r) =0 At any iteration () we get (+1) a better approximation (r) for the (r) and (+1) (r) for the mean field (r) through equation (567). Once the convergence is obtained, we have the selfconsistent mean field and the associated single-electron wave functions which minimize the system total energy.
8.6
Semiclassical dynamics in a conservative force field
In a conservative force field weakly depending on position through the potential energy (r), an electron with large momentum behaves almost like a particle obeying Newtonian classical mechanics. In one-dimension, the Hamiltonian operator is: 2 b = b + () (571) 2 We use the equation of the time derivative for the expectation value of a physical quantity = ( ) ( * + Z ) h i b b 1 h b bi 1 h i b = + = ∗ + b (572) ~ ~
and we apply this equation to both position and momentum: hi 1 Dh b iE hi = b = (573) ~ ¿ À hi 1 Dh b iE = b = − (574) ~ Differentiating the first equation with respect to time and substituting the expression of the derivative of the momentum into the second equation, we get: ¿ À 2 hi =− (575) 2 If the dynamics was be completely at any instant of time ¡ we¢ would ® ¡ classical ¢ have = hi and − = − =hi = (). () = − is the
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classical force acting on the particle. Yet in quantum mechanics position and momentum obey Heisenberg uncertainty principle: ~2 2® 2® ( − hi) ( − hi) ≥ 4
(576)
In the hypotesis that q ( ) is a wave packet with a small (but not zero) ® uncertainty of position ( − hi)2 about the mean value hi we can derive ® in a power series of ( − hi) about hi expanding an approximate up to second order : ) À Z ¶ µ 3 ¶ ¿ Z (µ 1 = ∗ ( ) ( ) ≈ + ( − hi)2 |( )|2 =hi 2 3 =hi (577) ³ 2 ´ 2 We neglected 2 ( − hi) |( )| because its integral is identically =hi
zero. So the previous equation can be written as: ¶ ¿ À µ ¶ µ 1 3 2® ≈ + ( − hi) =hi 2 3 =hi
(578)
If the following inequality is satisfied ¯ ¯ ¯µ ¯µ ¶ ¶ ¯ ¯ ¯ ¯ 3 ® 1¯ ¯ ¯ ¯ 2 ¯ ( − hi) ¿ ¯ ¯ ¯ 3 ¯ ¯ ¯ 2 =hi =hi ¯
(579)
® − is very similar to the classical force (). We write again the latter condition as: ¯¡ ¢ ¯ ¯ ¯ 2 ¯ =hi ¯ 2® ¯ (580) ( − hi) ¿ ¯¯¡ 3 ¢ ¯ ¯ 3 =hi ¯
Given the uncertainty of the position, respecting the above inequality is easier if () depends weakly on . Now, using the uncertainty principle, we can write: ® ~2 ® ( − hi)2 ≥ (581) 4 ( − hi)2
But the mean kinetic energy is:
® ( − hi)2 hi2 h2 i = + 2 2 2
(582)
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Solid State Physics Lecture Notes
Figure 5:
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For kinetic energy the condition to heve semiclassical motion is: ® (583) hi2 À ( − hi)2 ® ® The figure shows in which regions of the ( − hi)2 ( − hi)2 plane the conditions for a semiclassical motion are verified. The region below the hyperbola is forbidden since there Heisenberg uncertainty principle would be violated; this zone is labelled as unphysical (UP UnPhysical). In the region above the hyperbola (QM Quantum Mechanics) there is a curvilinear triangle inside which the semiclassical conditions are verified. This area is wider if the potential energy depends weakly on position and in case of large mean linear momentum (in this case the de Broglie wavelength associated with the electron is very small, having introduced a mean de Broglie wavelength = hi associated to the wave packet). In general this graphical representation could change with time. The following case of a simple periodic potential is particularly simple and significative: 2 ) (584) Here, considering both semiclassical conditions and uncertainty principle, we obtain at the end the unique condition ¿ which guarantees that the wave packet representing the electron does not exhibit significant diffraction. In a crystal, let be the lattice parameter, this condition is never verified, since it would imply a wave vector À 2 greater than the edge of the first Brillouin zone. Yet the relevance of semiclassical dynamics applyed to crystals is well known. This apparent contradiction is solved considering the effective mass theorem: when we consider the equivalent semiclassical dynamics in a crystal, () does not include the periodic potential (which is lumped in the electron effective mass) but, instead, just represents the external force field (generally not periodic in space) which weakly depends on position. Unfortunately the considerations shown in this paragraph are not completely general since they require that () is a continuous and differentiable function up, at least, to third derivative and that none among the derivatives is identically zero. In conclusion we cannot apply this method to an harmonic oscillator where () = 12 2 2 . () = 0 sin(
9 9.1
Appendix: Elasticity and Elastic Waves Stresses, strains and elastic constants
As for all other physical properties, the mechanical response of materials depends on the scale at which it is measured. For 3D compact materials (e.g. di-
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amond), when the length scale is much bigger than the interatomic distance, the mechanical behaviour is conveniently modelled by standard continuum mechanics. Yet there exist special materials with a complex mesostructure for which this approach may not be appropriate. Aerogels and some types of cluster assembled carbon films belong to this cathegory. In the following we shall concentrate on elasticity, neglecting more complex phenomena like plasticity and fracture. Engineers usually deal with polycrystalline or multi-phase solids which can be characterized by macroscopic elastic parameters such as the bulk modulus and the shear modulus . Such materials are elastically isotropic at the macro-scale and two parameters are sufficient to describe their behaviour. Physicists are traditionally more interested to anisotropic single crystals, like perfect diamond (cubic) and perfect graphite (hexagonal). The scalar variables volume and pressure are not adequate to define the deformed state of solids: tensor quantities are needed. To describe the kinematics of the deformation process, let a vector field r spans all points within the material volume in an undeformed equilibrium state. If now a material deformation is produced by some external and/or internal agents, the old (undeformed) material positions are mapped into the new ones as r0 = r + u(r) by the displacement vector field u(r), describing both deformations (volume and shape variations) and rigid rotations. In the small strain regime, the symmetric part of the gradient of the displacement field u(r) is the strain tensor ¶ µ 1 (585) = (∇u) = + 2
This tensor describes how infinitesimal cubic volume elements = change their volume and/or their shape. The diagonal components represent 0 tensile strains (e.g., = ( − )), while the off-diagonal components represent shear strains. More precisely, assuming in the following summation from 1 to 3 over repeated vector and tensor suffixes, the relative volume 0 variation ( − ) is equal to , the trace of . is either a pure dilation ( 0) or a pure hydrostatic compression ( 0). Instead, = − 13 is a pure shear strain (a strain deviator, where is the unit tensor): in fact = 0, proving is a pure shear. In the undeformed state the angle, e.g., between x and y is 90 . After a shear deformation the angle is reduced of = = 2 = 2 . Any deformation is the sum of a pure shear and a dilation (compression) as it is evident from the identity = + 13 . Provided the produced strains are small and reversible, a generalized Hooke’s law describes the linear elastic material response:
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(586)
=
In the above equation is the stress tensor. is the −component of the force per unit area acting on a face of a cubic element whose outgoing normal is the unit vector of the −th cartesian axis. The diagonal components represent tractions, while the off-diagonal components represent shear stresses. The elastic material properties are embodied in the fourth-rank tensor the elastic constant tensor with 34 = 81 elements. Because of the symmetry of (see below) and , the number of independent components of diminishes to 36. This number is further lowered to 21 by energetic considerations. Using then Voigt’s contraction scheme
or 11 22 33 23 or 32 13 or 31 12 or 21 or 1 2 3 4 5 6 eq. 586 can be given a simpler matrix form (587)
=
where the elastic constant 6 × 6 matrix is symmetric. The material symmetry produces further reduction. In the case of cubic crystals, for instance, the independent constants are 3 and only 2 in a isotropic material, as anticipated. In the simplest case of isotropic elasticity eq. 587 becomes: ⎛
⎞
⎛
⎞⎛
⎞ 1 ⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎟=⎜ ⎟ ⎟ ⎜ 4 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎠ ⎝ 5 ⎠ ⎝ ⎠ ⎝ 6 (588) Uniaxial stress. In engineering two different elastic constants, namely the Young modulus and the Poisson’s ratio are usually introduced. If a uniaxial stress = 1 = is applied (with all other = 0) only strains = 1 = , = 2 and = 3 = 2 are non-zero. Then is defined as and as −2 = −3 . Since there are only two independent elastic 1 2 3 4 5 6
+ 43 − 23 − 23 0 0 0 − 23 + 43 − 23 0 0 0 − 23 − 23 + 43 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
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constants, and can be obtained from and by the following formulae: 9 3 + 3 − 2 = 2(3 + )
(589)
=
(590)
The meaning of bulk modulus appears clearly in hydrostatic pressure conditions: 1 = 2 = 3 = − and 4 = 5 = 6 = 0. In this case 1 = 2 = 3 = −3 and 4 = 5 = 6 = 0. Thus the relative volume variation = 1 + 2 + 3 = −. From the last equation it follows that the isothermal compressibility , whose positiveness is a condition for material thermodynamic stability, is just the inverse of µ ¶ 1 −1 0 (591) = = − Since too is always positive (see eq. 592), can vary between −1 (when = 0) and 12 (when = 0). In practice there are no materials known for which 0. Shear: In the case of simple shear, e.g. 4 = and all other = 0, then 4 = 2 and all other = 0 If one introduces the angular shear strain measure 4 = 24 = , the last equation reads = , which illustrates the physical meaning of . Cubic lattices have three independent elastic parameters 11 , 12 and 44 : in this case (as for lower symmetry crystals) and are direction dependent and can exceed the limits of isotropic materials. As the cubic case reduces to the isotropic case when 11 − 12 = 244 , a possible measure of the degree of anisotropy is (11 − 12 − 244 ) (11 − 44 ). Hexagonal lattices have five independent elastic parameters 11 , 12 13 , 33 and 44 .
solid 11 12 silicon (cubic) 165.7 63.9 diamond (cubic) 1076.0 125.0 graphite (hexagonal) 1060.0 180.0
13
33
44 79.6 2.33 575.8 3.51 15.0 36.5 4.5 2.26
The elastic moduli are in and the density in 3
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Graphite is globally highly anisotropic but completely isotropic for any rotation around the c axis perpendicular to a graphene plane. Its extremely low 44 value explains its use as a solid lubricant material. Instead the inplane moduli are of the same order of those of diamond, the hardest known material. The elastic energy density E of a strained isotropic material is the quadratic form: 1 E = ( )2 + (592) 2 In this formula spherical symmetry (isotropy) has been fully taken into account. In fact and are invariants of the strain tensor with respect to rotations. The limitation to second order in the strains leads to linear elasticity ( = E ), as stated by eq. 588), with both and positive for E must be minimum in the undeformed state (stability). In mechanical equilibrium the internal stresses in every volume element must balance. This is realized when the stress tensor field obeys the following equations + = = 0 and = (593) where = is an external body force. If external surface forces act on the points of the outer surface bounding the volume , also equilibrium boundary conditions must be taken into account. They read ( )r∈ =
(594)
where n is the outgoing normal of the surface element centered at r ∈ . The boundary conditions will remain the same even in the dynamic case.
9.2
The acoustic waves and their phonons
Though the traditional ways to measure the mechanical properties of materials and, in particular, the elastic constants are based on quasi-static deformation processes (e.g. the tensile test to measure the Young modulus), many important methods are acoustic in nature or make use of acoustic phenomena (ultrasound propagation, acoustic microscopy, acoustic emission, Brillouin scattering, laser induced surface acoustic waves). In the simplest case one measures the time required for a longitudinal ultrasonic pulse to travel back and forth inside a cylindrical sample along its axis. Knowing the length of the cylinder theppulse velocity is obtained. If the cylinder is a long very thin rod, = , being the rod density. To face less naive methods and more complex geometries, some basic results of classical elastodynamics must be employed.
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The elastodynamic equation substituting eq. 593, written in terms of the dynamic displacement field u = u(r ), reads: 2u = 2 ∇2 u + (2 − 2 )∇(∇ · u) (595) 2 q¡ p ¢ where = + 43 is the longitudinal sound velocity and = is the transverse sound velocity. Remembering that the most general deformation process is the superposition of a simple dilation and of a simple shear, one is tempted to try a solution of the type u = u + u with ∇ × u = 0 and ∇ · u = 0. The second condition is identical to that holding for electromagnetic waves in vacuum. This trick perfectly works (instead anisotropic materials require a more complex treatment) leading to two decoupled wave equations for u and u : 2 u 2 u 2 2 = ∇ u and = 2 ∇2 u 2 2
(596)
The fundamental bulk solution of eq. 595 is then the superposition of three independent monochromatic plane waves, one longitudinal () and two (mutually perpendicular) transverse ( 1 2 ), of the type: © ª (597) u = < q eq [q·r− (q))]
where q is the wavevector, is a branch index ( = 1 2 ), q is the complex amplitude of the normal coordinate (q) = q − and eq a polarization unit vector (eq ||q; eq ⊥q). Moreover: ω (q) = |q| and ω1 2 (q) = |q|. The above classical description can be translated into the language of quantum mechanics of the systems of independent harmonic oscillators: the quanta of the fields u are the long-wavelength acoustic phonons whose possible energies are: ¶ µ 1 ~ω (q) (598) (qα) = q + 2
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