Patterson, Bailey - Solid State Physics Introduction to theory

For the high-frequency or optic case r¨ → ∞, and r → 0 because the ions cannot follow the high-frequency fields so

We can use the above to get an expression for the polarization, which in turn can be used to determine the electron–phonon interaction. First we need to evaluate P.

We work out the polarization for the longitudinal optic mode, as that is all that is needed. Let

by definition since q is the direction of motion of the vibrational wave and is perpendicular to rT. There is no free charge to consider, so

236 4 The Interaction of Electrons and Lattice Vibrations

which is known as the LST (for Lyddane–Sachs–Teller) equation. See also Born and Huang [46 p. 87]. This will be further discussed in Chap. 9. Continuing, by (4.66),

from which we determine by (4.60), (4.77), (4.78), (4.80), and (4.81)

238 4 The Interaction of Electrons and Lattice Vibrations

The Electron–Phonon Interaction due to the Polarization (A)

In the continuum approximation appropriate for large polarons, we can write the electron–phonon interaction as coming from dipole moments interacting with the gradient of the potential due to the electron (i.e. a dipole moment dotted with an electric field, e > 0) so

Since P = ArL and we have determined A, we need to write an expression for rL. In the usual way we can express rL at lattice position Rn in terms of an expan-

sion in the normal modes for LO phonons (see Sect. 2.3.2):

4.3 The Electron–Phonon Interaction 239

or in the continuum approximation

(compare with Eqs. (2.140), (2.141)). Substituting and making a change in dummy summation variable:

Energy and Effective Mass (A)

We consider only processes in which the polarizable medium is at absolute zero, and for which the electron does not have enough energy to create real optical phonons. We consider only the process described in Fig. 4.6. That is we consider the modification of self-energy of the electron due to virtual phonons. In perturbation theory we have as ground state |k, 0q with energy

and no phonons. For the excited (virtual) state we have one phonon, |k – q, 1q . By ordinary Rayleigh-Schrödinger perturbation theory, the perturbed energy of the ground state to second order is:

240 4 The Interaction of Electrons and Lattice Vibrations

But

k

Fig. 4.6. Self-energy Feynman diagram (for interaction of electron and virtual phonon)

4.3 The Electron–Phonon Interaction 241

Thus the self-energy is increased by the interaction of the cloud of virtual phonons surrounding the electrons.

Experiments and Numerical Results (A)

A discussion of experimental results for large polarons can be found in the paper by Appel [4.2, pp. 261-276]. Appel (pp. 366-391) also gives experimental results for small polarons. Polarons are real. However, there is not the kind of comprehensive comparisons of theory and experiment that one might desire. Cyclotron resonance and polaron mobility experiments are common experiments cited. Difficulties abound, however. For example, to determine m** accurately, m* is needed. Of course m* depends on the band structure that then must be accurately known. Crystal purity is an important but limiting consideration in many experiments. The chapter by F. C. Brown in the book edited by Kuper and Whitfield [4.23] also reviews rather thoroughly the experimental situation. Some typical values for the coupling constant αc (from Appel), are given below. Experimental estimates of αc are also given by Mahan [4.27] on p. 508.

Table 4.4. Polaron coupling constant

Material αc

KBr 3.70

GaAs 0.031

InSb 0.015

CdS 0.65

CdTe 0.39

242 4 The Interaction of Electrons and Lattice Vibrations

4.4Brief Comments on Electron–Electron Interactions

(B)

A few comments on electron–electron interactions have already been made in Chap. 3 (Sects. 3.1.4 and 3.2.2) and in the introduction to this chapter. Chapter 3 discussed in some detail the density functional technique (DFT), in which the density function plays a central role for accounting for effects of electron–electron interactions. Kohn [4.20] has given a nice summary of the limitation of this model. The DFT has become the traditional way nowadays for calculating the electronic structure of crystalline (and to some extent other types of) condensed matter. For actual electronic densities of interest in metals it has always been difficult to treat electron–electron interactions. We give below earlier results that have been obtained for high and low densities.

Results, which include correlations or the effect of electron–electron interactions, are available for a uniform electron gas with a uniform positive background (jellium). The results given below are in units of Rydberg (R∞), see Appendix A. If ρ is the average electron density,

is the average distance between electrons. For high density (rs << 1), the theory of Gell-mann and Bruckner gives for the energy per electron

For low densities (rs >> 1) the ideas of Wigner can be extended to give

In the intermediate regime of metallic densities, the following expression is approximately true:

E = 2.21 − 0.916 + −

0.031ln rs 0.115 (R∞) ,

N rs2 rs

for 1.8 ≤ rs ≤ 5.5. See Katsnelson et al [4.16]. This book is also excellent for DFT. The best techniques for treating electrons in interaction that has been discussed in this book are the Hartree and Hartree–Fock approximation and especially the density functional method. As already mentioned, the Hartree–Fock method can give wrong results because it neglects the correlations between electrons with antiparallel spins. In fact, the correlation energy of a system is often defined as the

244 4 The Interaction of Electrons and Lattice Vibrations

described) should be a sudden transition. By now, many examples are known, NiO was one of the first examples of “Mott–Hubbard” insulators – following current usage. Anderson has predicted another kind of metal–insulator transition due to disorder.6 Anderson’s ideas are also discussed in Sect. 12.9.

Kohn has suggested another effect that may be due to electron–electron interactions. These interactions cause singularities in the dielectric constant (see, e.g., (9.167)) as a function of wave vector that can be picked up in the dispersion relation of lattice vibrations. This Kohn effect appears to offer a means of mapping out the Fermi surface.7 Electron–electron interactions may also alter our views of impurity states.8 We should continue to be hopeful about the possibility of finding new effects due to electron–electron interactions.9

4.5 The Boltzmann Equation and Electrical Conductivity

4.5.1 Derivation of the Boltzmann Differential Equation (B)

In this section, the Boltzmann equation for an electron gas will be derived. The principle lack of rigor will be our assumption that the electrons are described by wave packets made of one-electron Bloch wave packets (Bloch wave packets incorporate the effect of the fields due to the lattice ions which by definition change rapidly over inter ionic distances). We also assume these wave packets do not spread appreciably over times of interest. The external fields and temperatures will also be assumed to vary slowly over distances of the order of the lattice spacing.

Later, we will note that the Boltzmann equation is only relatively simple to solve in an iterated first order form when a relaxation time can be defined. The use of a relaxation time will further require that the collisions of the electrons with phonons (for example) do not appreciably alter their energies, that is that the relevant phonon energies are negligible compared to the electrons energies so that the scattering of the electrons may be regarded as elastic.

We start with the distribution function fkσ(r,t), where the normalization is such that

dkdr fkσ (r,t) (2π)3

is the number of electrons in dk (= dkxdkydkz) and dr (= dxdydz) at time t with spin σ. In equilibrium, with a uniform distribution, fkσ → fk0σ becomes the Fermi–Dirac distribution.

7See [4.19]. See also Sect. 9.5.3.

8See Langer and Vosko [4.24].

9See also Sect. 12.8.3 where the half-integral quantum Hall effect is discussed.