Dressel.Gruner.Electrodynamics of Solids.2003

displacement D and the electric field E are conveniently expressed. According to Eq. (2.1.10) with ψ = 0 the electric field reads E = E0 sin{q · r − ωt}, and we can write for the displacement

to demonstrate that 1 and 2 span a phase angle of π/2. Here 1 is the in-phase and2 is the out-of-phase component; we come back to the loss angle δ in Eq. (2.3.25). The notation accounts for the general fact that the response of the medium can have a time delay with respect to the applied perturbation. Similarly the conductivity can be assumed to be complex

to include the phase shift of the conduction and the bound current, leading to a more general Ohm’s law

and we define5 the relation between the complex conductivity and the complex dielectric constant as

Besides the interchange of real and imaginary parts in the conductivity and dielectric constant, the division by ω becomes important in the limits ω → 0 and ω → ∞ to avoid diverging functions. Although we regard the material parameters 1 and σ1 as the two fundamental components of the response to electrodynamic fields, in subsequent sections we will use a complex response function, in most cases the conductivity σˆ , when the optical properties of solids are discussed.

A number of restrictions apply to this concept. For example, the dielectric constant 1 introduced by Eq. (2.2.5) is in general not constant but a function of both spatial and time variables. It may not just be a number but a response function or linear integral operator which connects the displacement field D(r, t) with the electric field E(r , t ) existing at all other positions r and times t earlier than t

The consequence of this stimulus response relation is discussed in more detail in

5Sometimes the definition σ2 = −ω 1/(4π ) is used, leading to ˆ = 4π iσˆ /ω. At low frequencies the difference between the two definitions is negligible. At high frequencies σ2 has to be large in order to obtain 1 = 1 in the case of a vacuum.

Section 3.2. In the general case of an anisotropic medium, the material parameters1, µ1, and σ1 may be direction dependent and have to be represented as tensors, leading to the fact that the displacement field does not point in the same direction as the electric field. For high electric and magnetic fields, the material parameters 1, µ1, and σ1 may even depend on the field strength; in such cases higher order terms of a Taylor expansion of the parameters have to be taken into account to describe the non-linear effects.6 The dielectric properties ˆ can also depend on external magnetic fields [Lan84, Rik96], and of course the polarization can change due to an applied magnetic field (Faraday effect, Kerr effect); we will not consider these magneto-optical phenomena. We will concentrate on homogeneous7 and isotropic8 media with µ1 = 1, 1, and σ1 independent of field strength and time.

2.2.3 Wave equations in the medium

To find a solution of Maxwell’s equations we consider an infinite medium to avoid boundary and edge effects. Furthermore we assume the absence of free charges

(ρext = 0) and external currents (Jext = 0). As we did in the case of a vacuum we use a sinusoidal periodic time and spatial dependence for the electric and magnetic

waves. Thus,

describe the electric and magnetic fields with wavevector q and frequency ω. We have included a phase factor φ to indicate that the electric and magnetic fields may be shifted in phase with respect to each other; later on we have to discuss in detail what this actually means. As we will soon see, the wavevector q has to be a complex quantity: to describe the spatial dependence of the wave it has to include a propagation as well as an attenuation part. Using the vector identity (2.1.4) and with Maxwell’s equations (2.2.7a) and (2.2.7d), we can separate the magnetic and electric components to obtain

law (2.2.7c), we arrive at × B = ( 1µ1/c)(∂E/∂t) + (4π µ1σ1/c)E. Combining

6A discussion of non-linear optical effects can be found in [Blo65, But91, Mil91, She84].

7The case of inhomogeneous media is treated in [Ber78, Lan78].

8Anisotropic media are the subject of crystal optics and are discussed in many textbooks, e.g. [Gay67, Lan84, Nye57].

20 2 The interaction of radiation with matter

this with Eq. (2.2.15) eventually leads to the wave equation for the electric field

displacement current; the last term is due to the conduction current. Similarly we obtain the equation

describing the propagation of the magnetic field. In both equations the description contains an additional factor compared to that of free space, which is proportional to the first time derivative of the fields. Of course, we could have derived an equation for B. As mentioned above, we neglect magnetic losses. From Eq. (2.2.7b) we can immediately conclude that H always has only transverse components. The electric field may have longitudinal components in certain cases, for from Eq. (2.2.7d) we find · E = 0 only in the absence of a net charge density.

If Faraday’s law is expressed in q space, Eq. (2.1.15a), we immediately see that for a plane wave both the electric field E and the direction of the propagation vector q are perpendicular to the magnetic field H, which can be written as

c

(2.2.17)

µ1ω

E, however, is not necessarily perpendicular to q. Without explicitly solving the wave equations (2.2.16), we already see from Eq. (2.2.17) that if matter is present with finite dissipation (σ1 = 0) – where the wavevector is complex – there is a phase shift between the electric field and magnetic field. This will become clearer when we solve the wave equations for monochromatic radiation. Substituting Eq. (2.2.14a) into Eq. (2.2.16a), for example, we obtain the following dispersion relation between the wavevector q and the frequency ω:

where nq = q/|q| is the unit vector along the q direction. Note that we have made the assumption that no net charge ρext is present: i.e. · E = 0. A complex wavevector q is a compact way of expressing the fact that a wave propagating in the nq direction experiences a change in wavelength and an attenuation in the medium compared to when it is in free space. We will discuss this further in Section 2.3.1. The propagation of the electric and magnetic fields (Eqs (2.2.16)) can now be written in Helmholtz’s compact form of the wave equation:

It should be pointed out that the propagation of the electric and magnetic fields is described by the same wavevector q; however, there may be a phase shift with respect to each other (φ = 0).

In the case of a medium with negligible electric losses (σ1 = 0), Eqs (2.2.16) are reduced to the familiar wave equations (2.1.19) describing propagating electric and magnetic fields in the medium:

There are no variations of the magnitude of E and H inside the material; however, the velocity of propagation has changed by ( 1µ1)1/2 compared to when it is in a vacuum. We immediately see from Eq. (2.2.18) that the wavevector q is real for non-conducting materials; Eq. (2.2.17) and the corresponding equation for the electric field then become

indicating that both quantities are zero at the same time and at the same location and thus φ = 0 as sketched in Fig. 2.1. The solutions of Eqs (2.2.20) are restricted to transverse waves. In the case σ1 = 0, both E and H are perpendicular to the direction of propagation nq; hence, these waves are called transverse electric and magnetic (TEM) waves.

2.3 Optical constants

2.3.1 Refractive index

The material parameters such as the dielectric constant 1, the conductivity σ1, and the permeability µ1 denote the change of the fields and current when matter is present. Due to convenience and historical reasons, optical constants such as the real refractive index n and the extinction coefficient k are used for the propagation and dissipation of electromagnetic waves in the medium which is characterized by the wavevector (2.2.18). Note that we assume the material to extend indefinitely, i.e. we do not consider finite size or surface effects at this point. To describe the optical properties of the medium, we define the complex refractive index as a new response function

where the real refractive index n and the extinction coefficient (or attenuation index) k are completely determined by the conductivity σ1, the permeability µ1, and the dielectric constant 1: 9

These two important relations contain all the information on the propagation of the electromagnetic wave in the material and are utilized throughout the book. The optical constants describe the wave propagation and cannot be used to describe the dc properties of the material. For ω = 0 only 1, σ1, and µ1 are defined. The dielectric constant, permeability, and conductivity are given in terms of n and k:

where the approximation assumes | 1| 1. In Table 2.1 we list the relationships between the various response functions. In the following chapter we show that the real and imaginary components of Nˆ , ˆ, and σˆ , respectively, are not independent but are connected by causality expressed through the Kramers–Kronig relations (Section 3.2). If Nˆ is split into an absolute value |Nˆ | = (n2 + k2)1/2 and a phase φ according to Nˆ = |Nˆ | exp{iφ}, then the phase difference φ between the magnetic and dielectric field vectors introduced in Eqs (2.2.14) is given by

In a perfect insulator or free space, for example, the electric and magnetic fields are in phase and φ = 0 since k = 0. In contrast, in a typical metal at low frequencies σ1 |σ2|, leading to n ≈ k and hence φ = 45◦.

Propagation of the electromagnetic wave

Let us now discuss the meaning of the real part of the refractive index n attributed to the wave propagation; the dissipations denoted by k are then the subject of the

9We consider only the positive sign of the square root to make sure that the absorption expressed by k is always positive.

Table 2.1. Relationships between the material parameters and optical constantsˆ, σˆ , and Nˆ .

The negative sign in the time dependence of the traveling wave exp{−iωt} was chosen (cf. Eqs (2.2.14)).

following subsections. If the wavevector given by Eq. (2.3.2) is substituted into the equations for the electromagnetic waves (2.2.14), we see the real part of the wavevector relate to the wavelength in a medium by Re{q} = q = 2π/λ. If µ1 = 1, the wavelength λ in the medium is given by

except in the vicinity of a strong absorption line (when n < 1 is possible in a narrow range of frequency), it is smaller than the wavelength in a vacuum λ0, and the reduction is given by the factor n.

The phase velocity is simply the ratio of frequency and wavevector

vph describes the movement of the phase front, vgr can be pictured as the velocity of the center of a wavepackage; in a vacuum vph = vgr = c. In general, vph = c/n(ω) can be utilized as a definition of the refractive index n. Experimentally the wavelength of standing waves is used to measure vph.

Both n and k are always positive, even for 1 < 0; but as seen from Eq. (2.3.3) the refractive index n becomes smaller than 1 if 1 < 1 − (π σ1/ω)2. For materials with σ1 = 0, the wavevector q is real and we obtain q = ωc ( 1µ1)1/2nq, with the

(a real quantity and n ≥ 1). From Eq. (2.3.4) we immediately see that in this case the extinction coefficient vanishes: k = 0. On the other hand, for good metals at low frequencies the dielectric contribution becomes less important compared to the conductive contribution σ1 |σ2| (or | 1| 2) and thus

Attenuation of the electromagnetic wave

Substituting the complex wavevector Eq. (2.3.2) into the expression (2.2.14a) for harmonic waves and decomposing it into real and imaginary parts yields

Now it becomes obvious that the real part of the complex wavevector q expresses a traveling wave while the imaginary part takes into account the attenuation. The first exponent of this equation describes the fact that the velocity of light (phase velocity) is reduced from its value in free space c to c/n. The second exponent gives the damping of the wave, E(r) exp{−αr/2} = exp{−r/δ0}. It is the same for electric and magnetic fields because their wavevectors q are the same. The amplitudes of the fields are reduced by the factor exp{−2π k/n} per wavelength λ in the medium (Fig. 2.2). We can define a characteristic length scale for the attenuation of the electromagnetic radiation as the distance over which the field decreases by the factor 1/e (with e = 2.718):

definition of the skin depth, the field penetration depends on the imaginary part of the wavevector q inside the material; δ0 is therefore also a measure of the phase shift caused by the material. While for metals the expression (2.3.16) is sufficient, there are cases such as insulators or superconductors where the general formulas (2.3.15) have to be used.

Complementary to the skin depth δ0, we can define an absorption coefficient α = 2/δ0 by Lambert–Beer’s law10

to describe the attenuation of the light intensity I (r) = I0 exp{−αr} propagating in a medium of extinction coefficient k:

The power absorption coefficient has the units of an inverse length; α is not a fundamental material parameter, nevertheless it is commonly used because it can be easily measured and intuitively understood. With k = 2π σ1µ1/(nω) from Eq. (2.3.6), we obtain for the absorption coefficient

implying that, for constant n and µ1, the absorption α σ1; i.e. highly conducting materials attenuate radiation strongly. Later we will go beyond this phenomenological description and try to explain the absorption process.

Energy dissipation

Subtracting Eq. (2.2.7c) from Eq. (2.2.7a) after multiplying them by H and E, respectively, leads to an expression of the energy conservation in the form of

in the absence of an external current Jext, where we have used the vector relation· (E × H) = H · ( × E) − E · ( × H). The first terms in the brackets of Eq. (2.3.20) correspond to the energy stored in the electric and magnetic fields. In the case of matter being present, the average electric energy density in the field ue and magnetic energy density um introduced in Eq. (2.1.20) become

10Sometimes the definition of the attenuation coefficient is based on the field strength E and not the intensity I = |E|2, which makes a factor 2 difference.

While in the case of free space the energy is distributed equally (ue = um), this is not valid if matter is present. We recall that the energy transported per area and per time (energy flux density) is given by the Poynting vector S of the electromagnetic wave:

for plane waves the Poynting vector is oriented in the direction of the propagation q. The conservation of energy for the electromagnetic wave can now be written as

the limiting case valid for free space with Jcond = 0 has already been derived in Eq. (2.1.22). Hence, the energy of the electromagnetic fields in a given volume

either disperses in space or dissipates as Joule heat Jcond · E. The latter term may be calculated by

where we have made use of Eqs (2.2.11) and (2.3.7). The real part of this expression, P = σ1 E02, describes the loss of energy per unit time and per unit volume, the absorbed power density; it is related to the absorption coefficient α

The loss tangent denotes the phase angle between the displacement field D and electric field strength E. It is commonly used in the field of dielectric measurements where it relates the out-of-phase component to the in-phase component of the electric field.

Since the real part of the Poynting vector S describes the energy flow per unit time across an area perpendicular to the flow, it can also be used to express the energy dissipation. We have to substitute the spatial and time dependence of the electric field (Eq. (2.3.14)) and the corresponding expression for the magnetic field into the time average of Eq. (2.3.22) which describes the intensity of the radiationS t = (c/16π )(E0 H0 + E0 H0). The attenuation of the wave is then calculated by