Optical and Transport Properties of Solids

We now start on a discussion of the optical properties of solids. In optical experiments in which we measure reflectivity, transmission coefficient, luminescence, or light scattering, we can learn a great deal about energy band structure, impurities, phonons, defects and magnetic excitations. In particular, optical measurements are used to determine the complex dielectric or complex conductivity, which is directly related to band-structure.

The propagation of electromagnetic waves in materials is described by the Maxwell’s equations, complemented by appropriate constitutive equations of the specific system. In this section we will summarize the optical phenomenological constants, more commonly investigated in experiments. The considerations are rather general and may apply to homogeneous materials of different nature.

11.1 Macroscopic Theory of Optical Constants in Homogeneous Materials

We study for simplicity non-magnetic media (B = H; μ = 1) in the absence of external charges and currents (ρext = 0; Jext = 0). The Maxwell’s equations for electromagnetic waves propagating in a medium and interacting with the internal charge density ρ(r, t) and internal current density J(r, t ) are (in Gauss units).

We consider the propagation, along say the z-direction, of transverse electromagnetic waves of given angular frequency ω in isotropic materials, and specify the Maxwell’s equations in the geometry of Figure 11.1.

We take the electric field E in the x-direction, the magnetic field B in the y-direction, and the current density J parallel to the electric field; we can write:

Figure 11.1 Geometry chosen for the description of transverse electromagnetic fields in isotropic materials. The electric field and the internal density current are in the x-direction, the magnetic field is in the y-direction, the wave propagation is along the z-direction.

This is the basic information, provided by the Maxwell’s equations, on the electric field and current density; to proceed further we must establish the relationship between them phenomenologically or microscopically.

Local and Non-Local Constitutive Relationship for the Conductivity

The above Eq. (11.4) can be closed by appropriate assumption and elaboration of the conductivity response function σ, connecting the internal current density J to the electric field E in the specific medium. We could use, for instance, the constitutive equation linking the polarization P to the electric field E via the polarizability response function, or the dielectric response function linking the induction electric field D = E + 4πP to the electric field. Within the linear response approximation, we can write quite generally that the α-component of the current density is related to the β-components of the electric field by the relation:

In isotropic media, the conductivity tensor takes the form σαβ = σδαβ , and can be treated as a scalar quantity. We assume that the scalar conductivity σ is a homogeneous function both of time coordinates and spatial coordinates.

With the adopted assumptions of time and space translational invariance, Eq. (11.5) becomes:（ comment)

According to the causality principle,

The homogeneous approximation in eq. (11.6) suggests to express it in terms of the Fourier transforms of the quantities of interest, the Fourier transform of the conductivity function is defined as

Equation (11.6) in real space-time domain, or equivalently Eq. (11.7) in momentum-frequency domain, entail a non-local spatial relationship between current density and electric field. (In fact,…)

In the approximation of local response, it is assumed that J(r, t ) depends only on the electric field E(r, t’) in the point r, at any previous time t’< t . This assumption entails that the conductivity function in real space takes the form,

The Fourier transform becomes independent of the wave vector and we have

where σ(ω) is the complex conductivity function of the medium.

Thus, the local-response regime corresponds to neglecting the wave vector dependence of the response function. In this situation the constitutive equation, linking the current density to the electric field takes the form,

where the assumed time dependence of the fields J and E is specified through the angular frequency ω of the conductivity. According to Eq. (11.8), the current density at a given point in the material is proportional to the value of the electric field at the same point: this is the so-called local-response regime. A local relationship between current and field can be justified when the average distance travelled by the carriers is small with respect to the length of spatial variation of the electric field; otherwise, one should follow the trajectory of the carriers to find the effect of the spatially varying electric field: this is the so-called non-local-response regime.

Local Relationship for the Conductivity and Expression of the Optical Constants

In the rest of this section, we start from the local constitutive equation (11.8) between the current density and the electric field; then we work out the other frequency-dependent optical constants which are commonly used for the description of the optical properties of matter. The basic equation (11.4) joined with the local assumption gives,

The solution of this equation inside the material is a damped (or undamped) wave, which can be written in the form

where the complex refractive index N(ω) is given by

The magnetic field associated to the electric field E(z) is given by

It is customary to write the complex refractive index in the form N = n+ik, where n is the ordinary refractive index and k the extinction coefficient; Eq. (11.10a) can be re-written in the form,

From Eq. (11.11), we see that the velocity of the electromagnetic wave in the medium is c/n; the penetration depth (classical skin depth, defined as the distance at which the field amplitude drops of 1/e) is

the attenuation coefficient is also referred to as absorption coefficient.

spectroscopic ellipsometer

From Eq. (11.10b), and N2 = ε, it is seen that the conductivity and the dielectric function satisfy the relation

Separation of the real and imaginary part gives

These equations link the real and imaginary part of the dielectric function to the imaginary and real part of the conductivity.

Consider now an electromagnetic wave that impinges at normal incidence on the surface of an isotropic material, where it is partially transmitted and partially reflected. In the geometry of Figure 11.2 we have for the spatial part,

From Eqs. (11.14) the continuity condition at z = 0 of the electric field E(z) parallel to the surface gives Et = Ei + Er. From Eqs. (11.3a) and (11.14), the continuity condition of the y-components of the magnetic field gives NEt = Ei − Er . The ratio Er /Ei equals (1 − N)/(1 + N);

Figure 11.2 Schematic representation of incident, reflected and transmitted electromagnetic waves at the surface (the z = 0 plane) of an isotropic material; the normal incident situation is considered.

thus the reflectivity R at normal incidence is

Another convenient measurable quantity for describing a surface is the surface impedance Z defined as

From Eqs. (11.3a) and (11.10a) we have also

So, the relationships between optical constants in isotropic media, in the linear and local response regime, are summarized in Table 11.1.

Before closing, it is very instructive to consider the implications of the occurrence ε1(ω) > 0 and ε2(ω) = 0 for some given ω region;

then,

From Eqs. (11.2) and (11.11), it is seen that the medium can sustain undamped transverse waves of frequency ω and propagation vector q, with dispersion relation

In the local-response regime, we are considering, ε1(q, ω) = ε1(0, ω) = ε1(ω). The undamped transverse waves of the system can be interpreted as elementary excitations of energy hω and momentum hq. [for example, the concept has been considered in the study of the infrared dielectric properties of polar crystals and polaritons. In the case of longitudinal waves and longitudinal dielectric function, plasmon excitations are inferred from the vanishing of both ε1 and ε2.]

11.2 The Drude Theory of the Optical Properties of Free Carriers

Consider a free-electron gas with n carriers per unit volume, each with effective mass m and charge (−e); the carriers are embedded in a uniform background of neutralizing positive charge. In the Drude theory, the response of the carriers to a spatially uniform driving electric field of frequency ω and amplitude E0, is described via the classical equation of motion

where u(t) is the displacement of the particle and τ is a phenomenological relaxation time.

In Eq. (11.17), the displacement u(t ) is expected of the type u(t ) = u0exp (−iωt ), with the same direction and frequency of the driving electric field; insertion into Eq. (11.17) gives

The Drude frequency dependent complex conductivity is thus

where σ0 is the static conductivity. The dynamical conductivity (11.18) combines a wealth of relevant physical information with all the merits of simplicity, the Drude model remains a precious tool for the understanding of features of intraband transitions in solids.

The expression (11.18) of the conductivity contains two important parameters, the carrier density n and the relaxation time τ , that set the two energy scales of the phenomena under investigation.

The other scale of energy comes out from the relation connecting the static conductivity to the free-electron plasma frequency ωp via the carrier density:

To estimate the plasma energy, we use the standard expression that links the electron density to the dimensionless parameter rs , and obtain

When a current density J flows in a medium driven by an electric field E, the power dissipated in the unit volume is

where the amplitude E0 is taken as real. Averaging on time, we have that

where σ1 denotes the real part of the conductivity. Thus for ωτ<< 1 the metal is strongly dissipative, while it becomes (almost) lossless if ωτ>> 1.

Other properties of the metal are seen in more direct form elaborating the other optical constants. In particular from Eq. (11.13) the complex dielectric constant becomes

the real and imaginary parts of the dielectric function are

Low-Frequency Region

In the low-frequency region ωτ is negligible with respect to 1, and the material is strongly dissipative. The dielectric function (11.20) becomes

Thus also n(ω) and k(ω) are singular for ω → 0. Since n(ω) >> 1 the reflectivity is almost 1 and the metal is strongly reflecting: metals are excellent mirrors in this frequency region.

The penetration depth is

High-Frequency Region

The high-frequency region, extends approximately from the relaxation frequency to the plasma frequency. When ωτ is much larger than unity, we can neglect unity in the denominators of Eqs. (11.20b) and the dielectric function becomes

The above relations show that k(ω) >> n(ω)>> 1; since the refractive index n(ω) is still much larger than 1, the reflectivity is almost 1 and the metal is strongly reflecting also in this frequency region. The penetration depth, is given by

and is now approximately constant.

Ultraviolet Region

In this region, Eqs. (11.20) for the dielectric function read

Figure 11.3 Behavior of real and imaginary part of the dielectric function of a free-electron gas with the Drude model; we have taken hωp = 10 eV and γ = h/τ = 0.1 eV.

We see that ε1(ω) is positive for ω ≥ ωp; the reflectivity changes from (almost) one to (almost) zero, when ω increases above the plasma frequency, and the metal becomes transparent for ω > ωp. The schematic behavior of the dielectric function of a metal in the Drude model is given in Figure 11.3.

Figure 11.4 Representation of the dispersion curves of transverse electromagnetic modes and longitudinal plasmons in a bulk metallic system. It is seen that transverse electromagnetic modes in the bulk material are possible for ω > ωp; for high values of q the dispersion relation approaches the light line.

As discussed at the end of Section 11.1, the occurrence ε1(ω) > 0 and ε2(ω) = 0 entails that the medium can sustain undamped transverse waves, whose frequency and propagation vector are linked by the dispersion relation

Illustrative Examples of Application of the Drude Theory

In Figure 11.5, the measured imaginary part of the dielectric function of copper is shown in the wavelength range from 0.365 to 2.5 μm (hω is in the energy range from 3.4 to 0.5 eV). At the low frequencies, ε2(ω) decreases rapidly with increasing ω; then there is a sudden increase for hω ≈ 2.1 eV, which marks the onset of interband electronic transitions; below this threshold, the Drude theory is very useful for an overall understanding of the optical constants. Figure 11.5 Imaginary part of the dielectric

function of copper at different temperatures

The complex dielectric function ε(ω) of a metal, can be expressed as the sum of a Drude free-carrier contribution, given by Eq. (11.21) for ωτ >> 1, and an interband contribution; this latter, for frequencies sufficiently below the onset of interband transitionscan be considered as a constant, here denoted as εinter; we have thus

where λ is the free-space wavelength of light. From the above expressions, we expect that ε1(λ) varies linearly with λ2, and the same occurs for ε2(λ)/λ; we also expect that ε2 is rather sensitive to the temperature and increases with temperature, since the relaxation time is expected to decrease with temperature.

In Figure 11.6, we present the experimental data of ε1 and ε2/λ versus λ2 for copper in the infrared and red region: it can be noticed that the data fall almost exactly on straight lines, and the temperature dependence of ε2/λ is in agreement with what expected.

Figure 11.6 Dielectric function of copper in the infrared and red region at different temperatures. (a) ε1 versus square of wavelength (with λ in micron); (b) ε2/λ versus square of wavelength (with λ in micron).

Range of Validity of the Drude Theory

The Drude theory gives an overall comprehension of the intraband contribution to the optical constants of metals; however it is not sufficient for a quantitative account of experimental. There are several reasons for this. (i) The conduction band structure of the electrons in actual metals may be significantly different from the parabolic energy band assumption (with effective mass m), implicit in the Drude model. ?(ii) The relaxation time appears as an energy independent phenomenological parameter, while an analysis of scattering mechanisms implies an appropriate energy dependence.? (iii) A more basic reason is the assumption that spatial dispersion can be neglected. ?

In the Drude theory the current density at a given point in the material is taken as proportional to the value of the electric field at that same point as summarized in Eq. (11.8). This assumption can be justified when the average distance travelled by the carriers is small with respect to the length of spatial variation of the electric field.

We can estimate the mean free path travelled by the carriers with the following arguments. Consider first the case ωτ<<1, corresponding to a relaxation time τ much smaller than the period T =2π/ω of the electromagnetic field. The displacement of an electron with typical Fermi velocity vF between two collisions is about

Equations (11.22a,b) apply to the low- and high-frequency regions, respectively; they can be more conveniently elaborated into a unique equation by defining the frequency dependent complex mean free path

We can also estimate the distance along which the electromagnetic field changes significantly. Consider Eq. (11.11), and taking into account that n(ω) is much smaller than k(ω) in all the frequency range below the plasma frequency, we can approximate the distance of significant change of the field with the penetration depth δ(ω) of the electromagnetic wave in the metal.

We can now compare the penetration depth δ(ω) with the estimated mean free path |(ω)| of carriers. The region of local regime, also called of normal skin effect is characterized by the condition

The region where the local approximation is no more valid, and rather it holds

is referred to as extreme anomalous region.

In very pure and good conductors at very low temperatures, this extreme anomalous region may extend significantly around the relaxation frequency. The determination of the optical constants when spatial dispersion has to be taken into account, requires more sophisticated semi-classical approaches based on the Boltzmann transport equation which is the subject of the next section.

11.3 Transport Properties and Boltzmann Equation

In the study of intraband transport processes, we focus for simplicity on metals with a unique partially filled conduction band of energy E = E(k). At thermodynamic equilibrium, the probability of occupation of an energy level E(k) is given by the Fermi-Dirac distribution function

When external perturbations (electric fields, magnetic fields, temperature gradients) are applied to the sample, the electron distribution is disturbed from the equilibrium Fermi-Dirac function. In general the disturbed distribution function f (r, k, t), in addition to k, depends also on the real space coordinate r, and on time t;