Dressel.Gruner.Electrodynamics of Solids.2003

Fig. 5.15. (a) Real part and (b) imaginary part of the dielectric constant ˆ(q, ω) of a freeelectron gas as a function of frequency and wavevector at T = 0 in three dimensions calculated using Eq. (5.4.21). The frequency axis is normalized to the Fermi frequency ωF = vFkF/2, and the wavevector axis is normalized to the Fermi wavevector kF.

Fig. 5.16. Frequency and wavevector dependence of (a) the real and (b) the imaginary part of the conductivity σˆ (q, ω) of a free-electron gas at T = 0 in three dimensions after Eq. (5.4.25). The frequency axis is normalized to the Fermi frequency ωF = kFvF/2 = EF/h¯ , and the wavevector axis is normalized to the Fermi wavevector kF.

constant has the form

the zeros of which lead to the ultraviolet transparency of metals and the longitudinal plasma oscillations with the dispersion relation ωp(q) given above.

In the opposite, quasi-static, limit for qvF ω screening becomes important, and indeed in this limit the transverse and longitudinal responses are fundamentally different. Here we find

and the response is primarily real; we recover dissipation in the small q (but still qvF ω) limit. In contrast,

is primarily imaginary; there is no dissipation, and we also recover the Thomas– Fermi screening in this limit.

All this is valid for a three-dimensional degenerate electron gas. Both the single-particle and collective excitations are somewhat different in twoand onedimensional electron gases for various reasons. First, phase space arguments, which determine the spectrum of single-particle excitations, are somewhat different for a Fermi sphere in three dimensions, a Fermi circle in two dimensions and two points (at +kF and −kF) in one dimension. Second, polarization effects are different in reduced dimensions, with the consequence that the frequency of the plasmon excitations approach zero as q = 0. These features are discussed in Appendix F.

A final word on temperature dependent effects is in order. Until now, we have considered T = 0, and this leads to a sharp, well defined sudden change in the occupation numbers at q = kF. At finite temperatures, the thermal screening of the Fermi distribution function leads to similar smearing for the single-particle excitations. With increasing temperature, we progressively cross over to a classical gas of particles, and in this limit classical statistics prevails; the appropriate expressions can be obtained by replacing mvF2 by 3kBT .

References

[Abr72] A.A. Abrikosov, Introduction to the Theory of Normal Metals (Academic Press, New York and London, 1972)

6

Semiconductors

The focus of this chapter is on the optical properties of band semiconductors and insulators. The central feature of these materials is the appearance of a single-particle gap, separating the valence band from the conduction band. The former is full and the latter is empty at zero temperature. The Fermi energy lies between these bands, leading to zero dc conduction at T = 0, and to a finite static dielectric constant. In contrast to metals, interband transitions from the valence band to the conduction band are of superior importance, and these excitations are responsible for the main features of the electrodynamic properties. Many of the phenomena discussed in this chapter also become relevant for higher energy excitations in metals when the transition between bands becomes appreciable for the optical absorption.

Following the outline of Chapter 5, we first introduce the Lorentz model, a phenomenological description which, while obviously not the appropriate description of the state of affairs, reproduces many of the optical characteristics of semiconductors. The transverse conductivity of a semiconductor is then described, utilizing the formalisms which we have developed in Chapter 4, and the absorption near the bandgap is discussed in detail, followed by a summary of band structure effects. After discussing longitudinal excitations and the q dependent optical response, we briefly mention indirect transitions and finite temperature and impurity effects; some of the discussion of these phenomena, however, is relegated to Chapter 13.

Optical properties of semiconductors are among the best studied phenomena in solid state physics, mainly because of their technological relevance. Both experimental data and theoretical description are quite advanced and are the subject of many excellent textbooks and monographs, e.g. [Coh88, Hau94, Yu96].

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6.1The Lorentz model

6.1.1Electronic transitions

Before discussing optical transitions induced by the electromagnetic fields between different bands in a solid, let us first examine a simple situation: the transitions between a ground state and excited states of N identical atoms, the wavefunctions of which are 0(r) and l (r). The electrodynamic field is assumed to be of the form

and leads to the admixture of the excited states to the ground state. This can be treated by using the time dependent Schrodinger¨ equation with the perturbation given for an electric field by

H = eE · r .

Because of this admixture, the resulting time dependent wavefunction is

with El denoting the various energy levels. The coefficients al (t) are obtained from the solution of the Schrodinger¨ equation

where H0 l = El l describes the energy levels in the absence of the applied electromagnetic field. Inserting the wavefunction into Eq. (6.1.3) we find that the coefficients are

Here

l er 0 dr = erl0

is the matrix element of the dipole moment P = −er. The induced dipole moment

(6.1.5)