Dressel.Gruner.Electrodynamics of Solids.2003

The region qωvF + 2qkF < 1, also indicated in Fig. 5.11, is called the quasi-static limit; for small q values this indicates the region where, due to the low frequency of the fluctuations, screening is important. For small q values, series expansion gives in the τ → ∞ limit

where we have retained an extra term compared with the Boltzmann equation (5.2.22).

5.4 Longitudinal response

So far the discussion in this chapter has focused on the transverse response, i.e. the response of the medium to a vector potential A(q, ω). Now we discuss the response to static (but later also time dependent) charges. First, the low frequency limit (ω → 0) is analyzed; this leads to the well known Thomas–Fermi screening of the electron gas. Next, the response as obtained from the selfconsistent field approximation (Section 4.3.1) is discussed in detail; this gives the response functions appropriate for arbitrary q values. Finally, we examine the collective and single-particle excitations of an electron gas at zero temperature. Although we confine ourselves to the framework of the classical electrodynamics, and first quantization is used, the problem can also be treated in the corresponding second quantized formalism [Hau94, Mah90].

5.4.1 Thomas–Fermi approximation: the static limit for q < kF

If we place an additional charge −e into a metal at position r = 0, the conduction electrons will be rearranged due to Coulomb repulsion; this effect, called screening, can be simply estimated by the so-called Thomas–Fermi method.

Starting from Poisson’s equation (2.1.7) in Coulomb gauge

2 ind(r) = −4π δρ(r) = −4π [ρ(r) − ρ0] = 4π e[N (r) − N0] , (5.4.1)

where δρ(r) = ρ(r) − ρ0 is the deviation from the uniform charge density, δ N (r) = N (r) − N0 is the deviation from the uniform particle density, and (r) is the electrostatic potential at position r. We assume that (r) is a slowly varying function of distance r, and that the potential energy of the field V (r) = −e (r) is simply added to the kinetic energy of all the electrons (the so-called rigid band approximation); i.e.

The carrier density can be expanded around the Fermi energy EF in terms of (r)

Interestingly, the screening does not depend on the mass of the particles or other transport properties because it is calculated in the static limit. Note that without assuming a degenerate free-electron gas we can write more generally

which in the classical (non-degenerate) limit leads to the Debye–Huckel¨ screening

λDH = 4π N0e2/kBT 1/2

since ∂E/∂ N0 = 3kBT /2N0. For an isotropic metal the solution

is called the screened Coulomb or Yukava potential. It is appropriate only close to the impurity; at distances larger than 1/2kF the functional form is fundamentally

different [Kit63]. The screening length 1/λ is about 1 A˚ for a typical metal, i.e. slightly smaller than the lattice constant, suggesting that the extra charge is screened almost entirely within one unit cell.

The Fourier transform of Eq. (5.4.8) can be inverted to give the screened potential

For q → 0, the dielectric constant diverges; at distances far from the origin the electron states are not perturbed.

5.4.2 Solution of the Boltzmann equation: the small q limit

The Thomas–Fermi approximation is appropriate for static screening (ω = 0). If we are interested in the frequency dependence, we have to utilize the Boltzmann equation (5.2.7) for the case of a longitudinal field; from there we obtain the longitudinal current, in a fashion similar to Eq. (5.2.13). Assuming quasi-free electrons or a spherical Fermi surface, after some algebra we arrive at the following expression for the longitudinal dielectric constant:

Eq. (5.2.19). If q is small, utilizing the Taylor expansion in q yields the expression

Eq. (5.2.21) obtained for the transverse response and resembles the Drude formula (5.1.9) as the q = 0 case. This is not surprising, for at q = 0 the distinction between longitudinal and transverse response becomes obsolete. For finite q, the expression tends to the static limit

As the plasma frequency ωp2 = 4π N e2/m, the first term is identical to that derived within the framework of the Thomas–Fermi approximation.

5.4.3 Response functions for arbitrary q values

The semiclassical calculation of the q dependent conductivity is limited to small q values since it does not include the scattering across the Fermi surface; the appearance of vF in appropriate expressions merely reflects through the quantum statistics the existence of the Fermi surface. If we intend to take into account these processes, we must consider the time evolution of the density matrix as in Section 4.3.1. A more detailed discussion of the problems can be found in various textbooks [Cal91, Mah90, Pin66, Zim72].

The response functions to be evaluated are

as derived in Eq. (4.3.21). Following a traditional treatment of the problem we evaluate the dielectric constant, but we also give the appropriate expressions for the conductivity. We will evaluate these expressions in three dimensions: the discussion of the one-dimensional and two-dimensional response functions can be found in Appendix F.

For T = 0, the Fermi–Dirac distribution fk0 = 1 for k < kF and 0 for k > kF, which allows the integral to be solved analytically,3 and we find:

where D(EF) is the density of states of both spin directions at the Fermi level,

3 Following the method given in footnote 2 on p. 111, the steps are as follows:

The first part is solved in a similar way by using k + q instead of k.

the well known Lindhard response function. Again, Ln denotes the principal part of the complex logarithm. The frequency and wavevector dependence of the real and imaginary parts of the dielectric response function χˆ (q, ω) are calculated by utilizing Eq. (3.2.7). In the 1/τ → 0 limit,