Dressel.Gruner.Electrodynamics of Solids.2003

48 3 General properties of the optical constants

vector potential (A = AL + AT) can also be decomposed into components parallel and perpendicular to the direction q of the propagating wave. We use the Coulomb gauge and assume · A = 0. Then q · (AT + AL) vanishes. As q points in the direction of propagation, AL must be zero; i.e. in this gauge the vector potential has only a transverse component.

First we notice that in the case of free space EL = BL = 0, i.e. the electromagnetic wave is entirely transverse.1 From Eq. (2.1.15a) we already know that the two components of the electromagnetic radiation are normal to each other:

ET BT q.

If electrical charges and current are present, we can employ the above relations to reduce the electrical continuity equation (2.1.9) to −∂ρ/∂t = ·(JL +JT) = ·JL.

From the previous two relations we see that the longitudinal current density JL is only connected to the scalar potential , and the transverse current density JT is solely determined by the vector potential A. Similar relations can be obtained for the electric field. Since A has no longitudinal component as just derived, Eq. (2.1.17) can be split into two components: the longitudinal part

which vanishes if ρext = 0 as a consequence of Poisson’s equation (2.1.18), and the transverse electric field

Again, the longitudinal component of E is related to the electrical potential , whereas the vector potential A determines the transverse part of the electric field. Looking at two of Maxwell’s equations (2.2.7a and d)

1This statement is valid in the absence of boundaries, and it is not valid, for example, at the surface of a metal. There the propagation of electromagnetic waves with longitudinal electric field or magnetic field components is possible; these, for example, are the TE and TM modes which can propagate in waveguides and will be discussed in Section 9.1.

it becomes clear that the longitudinal component corresponds to the rearrangement of the electronic charge, whereas the transverse component is related to the induced electrical currents.

3.1.2 Material parameters

Our next goal is to describe the longitudinal and transverse responses in terms of the parameters which characterize the medium: the dielectric constant and the magnetic permeability. Let us confine ourselves to isotropic and homogeneous media, although some of the relations are more general. The longitudinal and transverse dielectric constant can be described using a tensor notation:

with 1L(q, ω) the diagonal and 1T(q, ω) the off-diagonal components of the dielectric tensor. The dyad which projects out the longitudinal component is defined

and we find the following relationship between the electric field strength E and electrical displacement D for the two components:

i.e. for an isotropic and homogeneous medium the longitudinal and transverse components of the dielectric constant are independent response functions which do not mix. From Eq. (3.1.2) we see that the longitudinal dielectric constant describes the response of the medium to a scalar potential . This in general arises due to an additional charge, leading to the rearrangement of the initial electric charge

independently at certain q and ω values. Thus we find the frequencies ωT and ωL from the two conditions

At these two frequencies, oscillations in the transverse and in the longitudinal directions are sustained. The consequence of these equations will be discussed in detail in Section 5.4.4, where we see that the vanishing 1(ω) can lead to collective excitations. The characteristic frequencies ωT and ωL are called transverse and longitudinal plasma frequencies, respectively, because they describe the resonance frequency of the electronic charges moving against the positive ions.

To find expressions for the longitudinal and transverse components of the optical conductivity, we separate Ampere’s` law (3.1.9c) into transverse and longitudinal components. Due to Eq. (3.1.9b) the magnetic induction is purely transverse (BL = 0); hence, the longitudinal part simplifies to

by using the Fourier transform of Eq. (3.1.1). Rearranging the terms and using Eqs (2.2.8) and (3.1.2), this eventually leads to

Similar considerations can be made for the transverse direction; however, BTtotal = 0, and the solution to Eq. (3.1.9c) is more complicated. According to Eq. (3.1.12) we eventually arrive at

For µ1 = 1 and q = 0 these two equations can be combined in the general form

In the limit of long wavelengths, the behavior for transverse and longitudinal responses becomes similar. We can also define a complex conductivity σˆ = (iω/4π )(1 − ˆ) in agreement with Eq. (2.2.12), so that

The current density as a response to an electric field is described by a transverse or a longitudinal conductivity. Our goal for the chapters to follow is to calculate this response.

3.1.3 Response to longitudinal fields

Let us apply an external longitudinal electric field EL and assume that the material equation has the general form given in Eq. (2.2.13), with ˆ the time and space dependent, or alternatively through the Fourier transformation the frequency and wavevector dependent, dielectric constant. The electric field is related to the

external charge by Poisson’s equation (2.2.7d), · DL(r, t) = 4πρext(r, t). The external displacement field DL leads to a rearrangement of the charge density and,

in turn, to a space charge field Epol. The total electric field is then given by

EL(r, t) = DL(r, t) + Epol(r, t) ,

where the space charge field Epol(r, t) is related to the polarization charge density by

· Epol(r, t) = 4πρind(r, t) .

By combining the previous equations, we obtain the Poisson relation

· EL(r, t) = 4π [ρext(r, t) + ρind(r, t)] .

In summary, the spatial and temporal Fourier transforms of the longitudinal displacement, electric field, and space charge field lead to

where ρ(q, ω) = ρext(q, ω) + ρind(q, ω) = ρtotal(q, ω), the total charge density. Within the framework of the linear approximation, we can define a wavevec-

tor and frequency dependent dielectric constant ˆ(q, ω) (which in the case of an

Substituting the material equation (2.2.5), DL(q, ω) = EL(q, ω) + 4π P(q, ω), the polarization P is related to the space charge field Epol by

54 3 General properties of the optical constants

can also express the susceptibility χˆeL(q, ω) in terms of the scalar potential; the

total quantity (q, ω) = ext(q, ω) + ind(q, ω) is often referred to as the screened potential. Poisson’s equation (2.1.7) can be transformed in Fourier form:

−q2 (q, ω) = −4πρ(q, ω) = −4π [ρext(q, ω) + ρind(q, ω)]. Thus we obtain

If we are interested in the screened potential (q, ω), the appropriate response

function2 χˆ L(q, ω) is that which describes the charge ρind(q, ω) induced in response to the screened potential (q, ω):

or in the inverted form ˆL(q, ω) = 1 − (4π/q2)χˆ L(q, ω). In the case of only free charge carriers, as in a metal, the real part χ1 is always positive. The equation

ρind(q, ω) = χˆ L(q, ω) (q, ω) can be compared with Ohm’s law, Eq. (2.2.11): in both cases, the response functions (σˆ and χˆ ) relate the response of the medium

[ρ(q, ω) and J(q, ω)] to the total (i.e. screened) perturbation (E and , respectively). This was first derived by Lindhard [Lin54]; χˆ is consequently called the Lindhard dielectric function.

From Eqs (3.1.27) and (3.1.22b) we can derive an expression for the dielectric constant in terms of the electric field and displacement field:

The significance of the above equations is clear: if the response to an external field E(q, ω) has to be evaluated, 1/ ˆ(q, ω) is the appropriate response function, whereas the response to a screened field is best described by ˆ(q, ω).

We have also seen that the conductivity connects the local current density

· JL(q, ω) + ∂ρind/∂t = 0, and for a current which is described by the usual

2Commonly, the dielectric response function is called χˆ , which must not lead to confusion with the electric susceptibility χˆe or the magnetic susceptibility χˆm. Sometimes χ is called generalized susceptibility.

harmonic wavefunction exp{i(q · r − ωt)}, the above equation can be written as q · JL(q, ω) = ωρind(q, ω). Using Eqs (3.1.33) and (2.2.11) finally leads to

This is the well known relationship between the dielectric constant and the conductivity.

3.1.4 Response to transverse fields

Similarly to longitudinal fields, a simple relationship between the electric field and the induced current can also be derived for transverse fields. In a similar way as the charge density ρ is connected to the scalar potential by the Lindhard function, ρ = χˆ , now the current density J is related to the vector potential A by the conductivity; the proportionality factor is

Thus we can define a dielectric constant for the transverse response according to Eq. (3.1.8); this gives the relationship between the transverse conductivity and the transverse dielectric constant:

the same as obtained previously for the longitudinal conductivity and dielectric constant.

The analogy between longitudinal and transverse fields ends here; it is not possible to define a transverse dielectric constant corresponding to the longitudinal case. Assuming a homogeneous isotropic medium, the divergence of the transverse fields vanishes by definition. With q · ET = q · DT = 0 the denominator in Eqs (3.1.33) and (3.1.34) is zero.

3.1.5 The anisotropic medium: dielectric tensor

As mentioned earlier, the strict separation of the longitudinal and transverse response as summarized in Eq. (3.1.20) only holds for non-magnetic (µ1 = 1) and isotropic materials. In anisotropic media the situation is different: the polarization and induced currents can generally flow in any direction that is different from that of the electric field, and it is possible to mix both components to induce, for example, a longitudinal current with a purely transverse electric field. The situation can be described by the dielectric tensor ¯. The nine components of ¯ are not all independent from each other.

In the absence of an external magnetic field, even for anisotropic media, the real part of the complex dielectric tensor is symmetric (the Onsager relation): it is always possible to find a set of principal dielectric axes such that the real dielectric tensor can be diagonalized. The imaginary part of the dielectric tensor (i.e. the conductivity tensor) is also symmetric and can be put into diagonal form. However, the directions of the principal axes of these two tensors are in general not the same. They do, however, coincide for crystals with symmetry at least as high as orthorhombic [Nye57].

A mixing of longitudinal and transverse responses also occurs if the medium is bounded, i.e. in the presence of surfaces. In particular if the frequency ω is comparable to the plasma frequency ωp, longitudinal plasma waves may occur.

3.2 Kramers–Kronig relations and sum rules

The various material parameters and optical constants introduced in the previous section describe the response of the medium to applied electromagnetic radiation within the framework of linear response theory; the frequency dependence of the response is called dispersion. The complex dielectric constant ˆ and the complex conductivity σˆ can be regarded as the prime response functions of the material, describing the electric polarization and current induced in response to the applied electric field. The change of the electromagnetic wave in the material was discussed in terms of the refractive index Nˆ and in terms of the complex surface impedance Zˆ S, for example; these are also complex response functions. For the optical parameters, such as the amplitude of the electromagnetic wave which is transmitted through an interface or which is reflected off a boundary of two materials, the second components are the phase shifts φt and φr, respectively, which are experienced by the electromagnetic fields there; both components constitute a response function. Hence we always deal with complex response functions describing the response of the system to a certain stimulus; it always contained possible dissipation and some phase change. General considerations, involving causality, can be used to derive important relations between the real and imaginary parts of the complex response functions. They were first given by Kramers [Kra26] and Kronig [Kro26], and play an important role, not only in the theory of response functions [Bod45, Mac56].

These relations are also of great practical importance: they allow for the evaluation of the components of the complex dielectric constant or conductivity when only one optical parameter such as the reflected or absorbed power is measured. With R(ω) obtained over a broad frequency range, the dispersion relations can be utilized to evaluate φr(ω). There are two restrictions of practical importance: the data have to cover a wide spectral range and the sample must not be transparent.

If just the transmitted power through a sample of finite thickness is measured, the Kramers–Kronig analysis does not allow the determination of both components without knowing the reflected portion.

These dispersion relations can also be used – together with physical arguments about the behavior of the response in certain limits – to derive what are called sum rules.

3.2.1 Kramers–Kronig relations

For the derivation of the general properties we assume a linear response to an external perturbation given in the form of

This describes the response Xˆ of the system at time t and position r to an external stimulus fˆ at times t and locations r . The function Gˆ (r, r , t, t ) is called the response function, and may be the conductivity, the dielectric constant, the susceptibility, or any other optical constant, such as the refractive index. Since the origin of the time scale should not be of physical significance, Gˆ (t, t ) is a function of the difference of the argument t − t only. In the following we will neglect the spatial dependence of the external perturbation3 and restrict ourselves to the local approximation: we assume that the response at a particular position r depends only on the field which exists at that particular place (described by the delta function: δ{x} = 1 if x = 0):

Gˆ (r, r , t, t ) = δ{r − r }Gˆ (t − t ) .

The validity of this approximation is discussed at length in later chapters. We also assume that the medium is isotropic and homogeneous, and thus Gˆ is a scalar. With these assumptions, Eq. (3.2.1) becomes

where we suppress the spatial dependence for above reasons. If the system is required to be causal, then

which basically means that there is no response prior to the stimulus and we can simplify !−∞∞ dt to !−t ∞ dt . Further analysis is more convenient in Fourier space,

3For a discussion of non-local effects on the derivation of the Kramers–Kronig relations and sum rules, see [Mar67].