Dressel.Gruner.Electrodynamics of Solids.2003

138 6 Semiconductors

results after some calculations [Woo72], and we define a polarizability

which is purely real as no absorption has been considered. We use h¯ ωl0 = El − E0 for the energy difference between the two states. The dielectric constant, for N atoms, is then

The polarizability can be regarded as a parameter accounting for the effectiveness of the transition, and the fraction of energy absorbed can be expressed in a dimensionless unit as

This is called the oscillator strength, which was introduced in Eq. (5.1.32) in terms of the momentum matrix element |pl0|2:

both can readily be converted by using the commutation relation [p, x] = −ih¯ . The oscillator strength is related to the power absorbed by the transmission, which

As discussed in Appendix D in more detail, the sum of all transitions should add up to unity; this is the so-called oscillator strength sum rule

Fig. 6.1. Frequency dependence of the dielectric constant ˆ(ω) for a transition between well defined energy levels with energy separation ω0/(2π c) = ν0 = 100 cm−1, together with the dielectric constant for a transition between energy levels of width 1/(2π cτ ) = γ = 50 cm−1; the spectral weight is given by the plasma frequency ωp/(2π c) = νp =

500 cm−1.

and for the induced dipole moment pˆ (ω) = −erˆ(ω) we obtain

where αˆ a(ω) is the atomic polarizability. The macroscopic polarizability is the sum over all the atoms N per unit volume involved in this excitation: P = N pˆ = N αˆ aE = χˆeE. Since the dielectric constant is related to the dielectric susceptibility via ˆ(ω) = 1+4π χˆe(ω), we obtain for the frequency dependent complex dielectric constant

We assume here that each atom contributes one electron to the absorption process.

Fig. 6.2. Real and imaginary parts of the conductivity σˆ (ω) versus frequency calculated after the Lorentz model (6.1.16) for center frequency ν0 = ω0/(2π c) = 100 cm−1, the width γ = 1/(2π cτ ) = 50 cm−1, and the plasma frequency νp = ωp/(2π c) = 500 cm−1.

where ω0 is the center frequency, often called the oscillator frequency, 1/τ denotes the broadening of the oscillator due to damping, and ωp = 4π N e2/m 1/2 describes the oscillator strength, and is referred to as the plasma frequency. It is clear by comparing this expression with Eq. (5.1.4) that the Drude model can be obtained from the Lorentz model by setting ω0 = 0. This is not surprising, as we have retained only the inertial and damping terms to describe the properties of the metallic state.

6.1.2 Optical properties of the Lorentz model

The frequency dependence of the various optical constants can be evaluated in a straightforward manner. Fig. 6.2 displays the real and imaginary parts of the

of 1 are at ±γ /2 around the oscillator frequency ν0; 1(ω) crosses zero at approximately ν0 and νp. The inset shows the real and imaginary parts of the dielectric constant on a logarithmic frequency scale.

as a function of frequency. We have chosen the center frequency ν0 = ω0/(2π c) = 100 cm−1, the width γ = 1/(2π cτ ) = 50 cm−1, and the plasma frequency νp =

Fig. 6.4. Refractive index n and extinction coefficient k versus frequency calculated using the Lorentz model for ν0 = 100 cm−1, γ = 50 cm−1, and νp = 500 cm−1. Absorption (corresponding to large k) occurs mainly in the range ±γ around ν0.

and

are displayed in Fig. 6.3 on both linear and logarithmic frequency scales. The refraction coefficient Nˆ can be calculated using Eqs (2.3.3) and (2.3.4); the frequency dependence of the real and imaginary parts of Nˆ are shown in Fig. 6.4. Using these parameters, we can, by employing Eqs (2.4.15) and (2.4.14), calculate the reflectivity R(ω) and the phase shift φr(ω), and the results are displayed in Fig. 6.5. With the help of Eqs (2.3.34) the real and imaginary parts of the complex surface impedance, RS(ω) and XS(ω), are calculated and displayed in Fig. 6.6. The electronic loss function 1/ ˆ of a Lorentz oscillator is shown in Fig. 6.7.

Three spectral ranges, the Hagen–Rubens, the relaxation, and the transparent regime, could be distinguished in the case of the Drude model; for the Lorentz model here four regimes with distinctively different spectral characteristics are of importance: a low frequency range at which the material does not absorb; the

Fig. 6.5. Reflectivity R and the phase angle φr as a function of frequency in the Lorentz model. The reflectivity shows a strong increase at the center frequency of the oscillator ν0 = 100 cm−1, has a plateau between ν0 and νp, and drops at νp = 500 cm−1.

spectral range close to the center frequency ω0 at which electrons are excited and thus absorption dominates; a range of high reflectivity for ω0 < ω < ωp; and the transparent regime for large frequencies ω > ωp.

Low frequency range

At low frequencies ω < (ω0 − 1/τ ) the real part of the conductivity σ1 is small and there is little absorption. The real part of the dielectric constant saturates at a constant value 1(ω → 0) as the frequency is reduced. It can be evaluated by the Kramers–Kronig relation (3.2.31) with 1(ω → ∞) = 1:

In this region, the behavior of the refractive index n(ω) follows the dielectric constant. Depending on its value in the low frequency range the reflectivity can

Fig. 6.6. Frequency dependence of the complex surface impedance Zˆ S(ω) = RS(ω) + iXS(ω) calculated after the Lorentz model. There is a peak in both the real and imaginary parts at the plasma frequency νp = 500 cm−1.

be large

according to Eq. (2.4.16), as illustrated in Fig. 6.5. However, the light which is not reflected due to the impedance mismatch – caused by the step in the dielectric constant or refractive index – is transmitted with almost no attenuation: α → 0 and k → 0 for ω → 0. In contrast to XS(ω), which increases linearly with frequency up to ω0, for ω → 0 the surface resistance RS(ω) approaches the constant value Z0/n according to Eq. (2.3.34a).

Absorption range

As soon as the frequency approaches the center frequency, electrons can be excited across the bandgap. Thus the vicinity of the oscillator frequency ω0 is characterized by strong absorption due to the large conductivity σ1 (although the loss function − Im{1/ ˆ} does not indicate any change by entering this spectral range). The width

Fig. 6.7. Real and imaginary parts, Re{1/ˆ} and − Im{1/ˆ}, of the frequency dependent loss function. The imaginary part peaks at νp and Re{1/ˆ} shows an anti-resonance-like shape. The inset shows the behavior on linear scales.

of the absorption range is characterized by the damping rate: (ω0 − 1/2τ ) < ω < (ω0 + 1/2τ ); it is the full width at half the conductivity maximum. As seen in Fig. 6.3 the real part of the dielectric constant crosses zero at ω0 with negative slope, and it comes back to positive values at ωp. Between the two extrema of

the dielectric constant at ωextr = ω0 ± 1/2τ (in the case of small damping) the dispersion is negative. As displayed in the figures, most optical quantities, such as

the refractive index n, the reflectivity R(ω), or Re{1/ ˆ(ω)}, exhibit strong changes in this range.

Reflection range

The third frequency range, between (ω0 + 1/τ ) and ωp, is characterized by a high and almost frequency independent reflectivity; the phase angle, however, varies strongly, as displayed in Fig. 6.5. At the oscillator frequency ω0 and at the plasma frequency ωp the curvature of the reflectivity changes. We have already mentioned that the Drude model is a special case of the Lorentz model with ω0 = 0; hence the reflection range corresponds to the relaxation regime in the former limit. The