Dressel.Gruner.Electrodynamics of Solids.2003

4.1.2 Longitudinal conductivity: the response to the scalar field

The above considerations led to the response of the electron gas to a transverse electromagnetic perturbation. Longitudinal currents and electric fields which occur due to the interaction of the charge density ρ and scalar external potential (which we assume to be also time dependent) can be discussed along similar lines. Within the framework of linear response the interaction between this potential and the charge density is

HintL = ρ(r) (r, t) dr , (4.1.22)

with

N

ρ(r) = −e δ{r − ri }

i=1

the summation over all electrons at sites ri . Here (r, t) is the selfconsistent potential1 including the external perturbation and the induced screening potential:= ext + ind. The interaction form is similar to Eq. (4.1.8). By replacing

the calculation of the absorbed power proceeds along the lines we performed for the vector potential and the current density. We find that

in analogy to Eq. (4.1.15). The Fourier transform of the longitudinal electric field is given by Eq. (3.1.2):

−iq (q) = EL(q) ,

and on utilizing this we find that the longitudinal conductivity per unit volume is

The continuity equation (2.1.9) connects the density fluctuations and the longitudinal current density. With both J and ρ described by the spatial and time variation as exp {i(q · r − ωt)}, the continuity equation becomes

iq · J(q)L = iωρ(q) .

1Note that in textbooks on solid state theory often the particle density N and the potential energy V are used, quantities which are related to the charge density and potential by ρ = −eN and V = −e , respectively. To follow common usage, we switch to this notation in Section 4.3.

Consequently, the conductivity per unit volume can also be written in the form which includes the longitudinal current density

This expression can be compared with Eq. (4.1.16) derived earlier for the transverse conductivity.

Since the longitudinal and transverse conductivities are described by identical expressions, we can formally combine the two and in general write

As implied by the notation, the longitudinal components of the currents lead to the longitudinal conductivity and the transverse components to the transverse conductivity.

4.2 The semiclassical approach

The electric current and electronic charge density induced by external electromagnetic fields can also be discussed using second quantization formalism, with the applied potentials either described as classical fields or also quantized. The so-called semiclassical approximation, where we treat both A(r, t) and (r, t) as classical fields, is valid for field strengths and frequencies for which the quantization of the electromagnetic radiation can be neglected and linear response theory applies; this is the case for all methods and optical experiments discussed in this book.

The second quantized formalism, as utilized here, is appropriate for materials where the electronic states can be described as plane waves; this is valid for free electrons or for electrons in a crystalline solid when they are described in terms of Bloch wavefunctions. While this is certainly a restriction, when compared with the applicability of the Kubo formula, the formalism has been utilized extensively to describe the response of Fermi liquids, including their broken symmetry ground states. The formalism can be used to derive the conductivity, although here we merely write down the appropriate expressions for the currents and charge densities, and these can simply replace the currents and charge densities in the expressions we have derived above. An introduction to second quantization can be found in [Hau94, Kit63, Mah90, Sch83] and similar textbooks; here we merely quote the final results.

We write the field operator (r) which obeys the equation of motion

The appropriate expressions of the charge density and current given above can be inserted into the Kubo formula (4.1.16) and into the expression which gives the conductivity in terms of the density fluctuations; with these substitutions these are cast into a form using the second quantization formalisms.

4.3 Response function formalism and conductivity

Next we relate the dielectric constant and the conductivity to the changes in the electronic states brought about by the electromagnetic fields; for simplicity we restrict ourselves to cubic crystals for which the conductivity and dielectric constant are scalar. The formalism is applied for a collection of electrons which are described by Bloch wavefunctions, and as such is applied to crystalline solids. We will make use of what is usually called the selfconsistent field approximation as follows. Starting from the considerations of Section 3.1, we first write the total potential as

and define the dielectric constant ˆ(q, t) as

The potential (q, t) leads to changes in the electronic density of states; these changes in turn can be obtained using the Heisenberg picture by treating (q, t) as a perturbation. The variations in the density of states cause an induced potentialind(q, t) through Poisson’s equation, and, through the selfconsistent equations above, this finally yields the expression of the dielectric constant in terms of the changes in the electronic states.

The procedure we describe eventually leads to expressions for the longitudinal response; the transverse response can be arrived at by similar arguments, but calculating the modification of the electronic states brought about by the vector potential A(q, t).

4.3.1 Longitudinal response: the Lindhard function

The derivation is somewhat tedious but the end result is transparent and easily understood. As usual, we derive the expressions for the complex frequency and wavevector dependent dielectric constant; the conductivity can be subsequently written in a straightforward fashion. For the derivation of the response functions we begin with the Heisenberg equation, and write the time evolution of the electron

Let us introduce the unperturbed Hamiltonian H0

H0|kl = Ekl |kl ,

and describe the interaction with the local electromagnetic potential by the interaction energy V (r, t) with HL = H0 + V . The particle density is decomposed into the unperturbed density N0 and the density fluctuation induced by the interaction N (r, t) = N0 + δ N (r, t). Now we obtain

where the second order terms in the perturbation (e.g. V δ N ) have been neglected. In what follows we use Bloch wavefunctions to describe the electron states

where l is a band index, is the volume of the system, and ukl (r + R) = ukl (r), with R a Bravais lattice vector. These states are eigenstates of both H0 and N0, with eigenvalues Ekl and f 0(Ekl ), respectively.

f 0(Ekl ) = [exp{(Ekl − EF)/ kBT } + 1]−1

is the Fermi–Dirac distribution function, EF is the Fermi energy, and kB is the Boltzmann constant. Ekl indicates the energy of the |kl state; note that this energy depends on the wavevector: Ek = E(k).

First, we relate the induced charge density to the total potential energy V (r, t) acting on our electron system. This potential includes both the external and the screening potential; we will evaluate the latter in a selfconsistent fashion. Taking the matrix elements of the Heisenberg equation between states |kl and |k + q, l

Because these states are eigenstates of H0, the first term on the right hand side can be written as

k + q, l |[H0, δ N ]|kl = Ek+q,l − Ekl k + q, l |δ N |kl ,

as the definition of . With these expressions, Eq. (4.3.6) is cast into the following form:

× V (q, t) k + q, l | exp{iq · r}|kl . (4.3.13)

This is a time dependent equation for δ N , the contribution to the density operator resulting from the induced particle density.

Next we use Eq. (4.3.13) to derive an expression for the complex dielectric function ˆ(q, ω). In doing so, we use the so-called adiabatic approximation, where we assume that the perturbation is turned on gradually starting at t = −∞ with a time dependence exp{ηt}, and we will take the limit η → 0 after appropriate expressions for the response have been derived. Consequently we assume that the time dependence of the external scalar potential has the form

Since the Fourier components are independent of each other, it is sufficient to consider only one component. We also assume that the induced screening potential energy, the total potential energy, and the density fluctuations all have the same exp{−iωt + ηt} time dependence. With this we can rewrite Eq. (4.3.13) as

(4.3.15) connecting δ N , the induced density, to V (q, t), the total selfconsistent perturbing potential energy.

What we apply to the system is the external potential ext; however, Eq. (4.3.15) is given in terms of (q, t), the total potential, which includes also the screening

potential ind. Rewriting Eq. (4.3.2) yields

Our task now is to establish a relationship between the induced potential and the

changes in the electronic density. The energy Vind(r, t) = −e ind(r, t) of the induced screening potential is related by Poisson’s equation:

2Vind(r, t) = −4π e2 δ N (r, t) .

The electronic charge density ρ can be written in terms of the particle density operator N as

ρ = −e N = −e Tr {N δ{r − r0}} ,

where Tr indicates the trace. Using the identity #k,l |kl kl| = 1 we obtain

δ N = Tr {δ N δ{r − r0}} = k + q, l |δ N |kl kl|δ{r − r0}|k + q, l ,

k,q l,l

(4.3.16) where r0 indicates the electron positions and l, l are the band indices. Now we replace the states in the second matrix element with Bloch functions and use δ{r − r0} to perform the integration over r. Replacing r0 by r, this yields

Putting this expression for the change in particle density into the Poisson equation gives the response to the change in the potential

We can utilize the periodicity of the function u(r) in the same way as before, and hence convert the integral over the entire system to one over a single unit cell and a sum over the unit cells. Following the same procedure as we employed in deriving Eqs (4.3.9) and (4.3.11), we arrive at the following expression for the induced

86 4 The medium: correlation and response functions

where we have used the definition of Eq. (4.3.12). Substituting Eq. (4.3.15) into Eq. (4.3.18) gives a relationship between the induced and total potential energy:

Now we have prepared all the ingredients necessary to evaluate the dielectric constant or other material parameters as a function of wavevector and frequency. Rearranging Eq. (4.3.2), we have

and the previous two expressions allow us to write the Fourier component of the dielectric constant in terms of Bloch functions as follows:

an expression usually referred to as the Lindhard form for the dielectric constant;ˆ(q, ω) refers to the longitudinal dielectric constant, the component related to the scalar potential. The same information about the longitudinal response is contained in χˆ (q, ω), the so-called Lindhard function defined in Eq. (3.1.32)

(4.3.21) the function which describes the change of particle density δ N due to an external scalar potential energy V .

With the general relation introduced in Eq. (3.2.7), the expression involving the differences in the occupation of the states and their energies, the so-called polarization function found in Eqs (4.3.20) and (4.3.21), becomes

In addition, the summation over all k values in the first term on the right hand side can be written as

This separation allows us to make the substitution k + q → k in the first term on the right hand side; hence, this change does not affect the summation.

Next we calculate the real part of ˆ = 1 + i 2 by substituting the previous displayed expression into Eq. (4.3.20); we find

These expressions for the real and imaginary parts of the dielectric constant include both transitions between bands corresponding to different l and l indices, and also transitions between states within one band. They will be used later for both metals and semiconductors, with intraband excitations more important for the former and interband transitions more important for the latter. As shown in Appendix C, the matrix element

| k + q, l | exp{iq · r}|kl |2

can be evaluated using the so-called k · p perturbation theory for small q values.

4.3.2 Response function for the transverse conductivity

The considerations which lead to the longitudinal response can also be applied to the response to a transverse electromagnetic field defined by A = (−ic/ω)E. As