Dressel.Gruner.Electrodynamics of Solids.2003

[Fin89] J. Fink, Recent Developments in Energy-loss Spectroscopy, in: Advances in Electronics and Electron Physics 75 (Academic Press, Boston, MA, 1989)

[Fis88] Z. Fisk, D.W. Hess, C.J. Pethick, D. Pines, J.L. Smith, J.D. Thompson, and J.O. Willis, Science 239, 33 (1988)

[Gol89] A.I. Golovashkin, ed., Metal Optics and Superconductivity (Nova Science Publishers, New York, 1989)

[Kau98] H.J. Kaufmann, E.G. Maksimov, and E.K.H. Salje, J. Superconduct. 11, 755 (1998)

[Mit79] S.S. Mitra and S. Nudelman, eds, Far Infrared Properties of Solids (Plenum Press, New York, 1970)

[Mot79] N.F. Mott and E.H. Davis, Electronic Processes in Non-Crystalline Materials, 2nd edition (Clarendon Press, Oxford, 1979)

[Mot90] N.F. Mott, Metal–Insulator Transition, 2nd edition (Taylor & Francis, London, 1990)

[Ord85] M.A. Ordal, R.J. Bell, R.W. Alexander, L.L. Long, and M.R. Querry, Appl. Optics 24, 4493 (1985)

[Shk84] B.I. Shklovskii and A.L. Efros, Electronic Properties of Doped Semiconductors, Springer Series in Solid-State Sciences 45 (Springer-Verlag, Berlin, 1984)

13.1.1 Single-particle direct transitions

Optical processes associated with the semiconducting state are fundamentally different from the processes which determine the response of conduction electrons. For semiconductors optical transitions between bands are responsible for the absorption of electromagnetic radiation, and thus for the optical properties. These transitions have been treated in Chapter 6 under simplified assumptions about both the transition rates and the density of states of the relevant bands.

Let us first comment on the notation vertical transitions, which reflects the observation that the speed of light is significantly larger than the relevant electron (and also phonon) velocities in crystals. We can estimate the difference in the wavevectors kl − kl = k involved in interband transitions as the momentum of the photon |q| = ω/c – assuming that the refractive index n = 1 and h¯ ω is the energy difference between the two states. The bands are separated by the

single-particle gap Eg, and kmax = π/a is the wavevector at the Brillouin zone; therefore

With a ≈ 3 A˚ and Eg = 1 eV the ratio is about 5 × 10−4, and, as assumed, the change in the momentum during the course of the optical absorption can indeed be neglected.

In Fig. 13.1 we display the optical parameters, the reflectivity R(ω), and the components of the complex dielectric constant 1(ω) and 2(ω), as well as the loss function Im{1/ ˆ(ω)} for the intrinsic semiconductor germanium measured up to very high energies [Phi63]. There are several features of interest: first there is a broad qualitative agreement between the findings and what is predicted by the Lorentz model, which has been explored in Section 6.1. The frequency dependent response of the various parameters which follow from the model are displayed in Figs 6.3–6.7: the reflectivity is finite – but smaller than unity – as we approach zero frequency, and rises with increasing frequency, reaching a plateau before rolling off at high frequencies – features prominently observed in germanium and also in other semiconductors. Both the real and the imaginary parts of the dielectric constant of germanium are also close to those of a harmonic oscillator (if we neglect the sharp changes in the reflectivity and also in the dielectric constant), as the comparison with Fig. 6.3 clearly indicates. Finally, the broad peak of the loss function at around 15 eV – just as in the case for metals – indicates a plasma resonance in the spectral range which is similar to that observed for simple metals. There are however important differences: we observe considerable structure, shoulders, and peaks in the optical properties, reflecting band structure effects (as we will discuss). Also, the frequency which we would associate with the characteristic frequency of

dc conductivity. For an intrinsic semiconductor the dc conductivity reads

where Eg is the so-called thermal gap, the energy required to create an electron in the conduction band by thermal excitation, leaving a hole in the valence band. The pre-factor σ0 contains the number of carriers and their mobility, and we assume for the moment that they are only weakly temperature dependent. Dc transport experiments on germanium give a thermal gap of Eg = 0.70 eV, significantly smaller than the energy associated with the frequency ω0. In fact, as we can see in Fig. 13.1a, the onset of absorption is not as smooth as the Lorentz model predicts. A closer inspection of 2(ω) reveals a sudden onset of the absorption processes – note that the relation (2.3.26) between the imaginary part of the dielectric constant and the absorption coefficient is

where n is the real part of the refractive index (which is only weakly frequency dependent in the relevant frequency range). This onset of absorption can be associated with the single-particle gap, and we obtain for the optical gap Eg = 0.74 eV

– a value close to the thermal gap given above. The behavior close to the gap is shown in more detail in Fig. 13.2 for another intrinsic semiconductor PbS, for which detailed optical experiments on epitaxial films have been conducted in the gap region [Car68, Zem65]. It is found that α(ω) increases as the square root of the frequency

since the frequency dependence described by Eq. (6.3.11) dominates the absorption process. The gap here is 0.45 eV, in excellent agreement with what is measured by the dc transport; both of these observations provide clear evidence that this material has a direct bandgap. The dielectric constant 1 – also shown in Fig. 13.2 – is in full agreement with such an interpretation; the full line corresponds to a fit of the data by Eq. (6.3.12) with the gap Eg = 0.47 eV.

While the sharp onset of absorption is found in a large number of materials, confirming the well defined bandgap, often the functional dependence is somewhat different from that expected for simple direct transitions. The absorptivity of InSb displayed in Fig. 13.3 illustrates this point: near the gap of 0.23 eV the absorption has the characteristic square root dependence as expected for allowed direct transitions, see Eq. (6.3.11), but a fit over a broad spectral range requires an additional term which can be described as

Fig. 13.4. (a) Electronic band structure of germanium in various crystallographic directions of high symmetry, and (b) the corresponding density of states D(E), calculated by the empirical pseudopotential method. Some critical points are indicated, which correspond to a flat dispersion of E(k) (after [Her67]).

corresponding density of states obtained by the preceeding equation. Particular symmetry points are labeled corresponding to the common notation found in various books on semiconductors [Coh89, Gre68, Pan71, Yu96], and will not be discussed here. What appears in the formulas for the optical properties in the case of direct transitions is the joint density of states calculated by Eq. (6.3.2)

using the dispersion relations as derived from band structure calculations. If we assume that the transition matrix element is independent of wavevector, the frequency dependence of the absorption is, through Eq. (6.3.4), determined by the frequency dependence of the joint density of states. For particular k values, called critical points, k [El (k) − El (k)] has zeros, and thus the density of states displays

the so-called van Hove singularities at these singularities. In the vicinity of these critical points the energy difference δE(k) = El (k) − El (k) can be expanded as

in three dimensions. Utilizing such an expansion, the joint density of states and, through Eqs (6.3.11) and (6.3.12), the components of the complex dielectric constant can be evaluated; the behavior depends on the sign of three coefficients α1, α2, and α3. In three dimensions, four distinct possibilities exist, which are labeled M j according to the number j of negative coefficients. For M0, for example, all coefficients are positive and

with the resulting 2(ω) shown in Fig. 13.5; the other possibilities are also displayed. Not surprisingly, the real part of the dielectric constant also displays sharp changes at these critical points. In one dimension, the situation is entirely different. First of all

and we have only two possibilities, with α either positive or negative. At the onset of an interband transition, the joint density of states diverges as

at the critical points, for α1 > 0 for example; the behavior of 2(ω) for the two solutions is displayed in the lower part of Fig. 13.5.

After these preliminaries we can understand the features observed in the optical experiments as consequences of critical points and relevant dimensionalities near these critical points. In Fig. 13.6 the measured imaginary part of the dielectric constant 2(ω) of germanium is displayed together with results of calculations [Phi63]. Without discussing the various transitions and critical points in detail, it suffices to note that modern band structure calculations such as the empirical pseudopotential method give an excellent account of the optical experiments and provide evidence for the power and accuracy of such calculations. At high frequencies the behavior shown in Fig. 13.6 gives way to a smooth ω−2 decrease of the conductivity, the Drude roll-off associated with inertia effects. For 1/τ and Eg much smaller than the plasma frequency, the real part of the dielectric constant