Dressel.Gruner.Electrodynamics of Solids.2003

erties coming from thermally excited electrons and holes have to be included. The number of these excitations increases exponentially with increasing temperature, and is small even at room temperature in most typical (undoped) semiconductors. If the relaxation time associated with these excitations is long, absorption due to these excitations – often called the free-electron absorption – occurs well below the gap frequency, and, as a rule, a simple Drude model accounts for the optical properties [Ext90]. In Fig. 13.8 the optical conductivity σ1(ω) and σ2(ω) of silicon is displayed for two different doping levels: for an acceptor concentration of 1.1 × 1015 cm−3 leading to a resistivity of 9.0 cm, and a donor concentration of 4.2 × 1014 cm−3 which gives ρdc = 8.1 cm. The frequency dependence measured by time domain spectroscopy at room temperature can be well described for both cases by a simple Drude model with somewhat different scattering rates: 1/(2π τ c) = 40 cm−1 and 20 cm−1, respectively.

The onset of the absorption is different for a highly anisotropic band structure. Of course, strictly one-dimensional materials are difficult to make, but in several so-called linear chain compounds (the name indicates the atomic or molecular arrangements in the crystal) the band structure is anisotropic enough such that it can be regarded as nearly or quasi-one-dimensional. Near to the gap, the optical conductivity of such materials can be described by the one-dimensional form of semiconductors we have discussed in Chapter 6. Expression (6.3.16) assumes that the transition matrix element between the valence and conduction states is independent of frequency; thus the functional dependence of σ1 on ω is determined solely by the joint density of states. In general this is not the case, as can be seen from the following simple arguments. The periodic potential leads to a strong modification of the electronic wavefunctions only for states close to the gap; for wavevectors away from the gap (and therefore for transition energies significantly different from the gap energy Eg) such perturbation is less significant, and the electron states are close to the original states which correspond to a single band. As transitions between states labeled by different k values in a single band are not possible (note that optical transitions correspond to q = 0), the transition probability is presumed to be strongly reduced as the energy of such transitions is increased from the gap energy Eg. Thus the conductivity is expected to display a stronger frequency dependence than that given by Eq. (6.3.16). All this can be made quantitative by considering the transitions in one dimension [Lee74]. One finds that

Energy hω (eV)

Fig. 13.9. Frequency dependence of the conductivity in highly anisotropic semiconductors K2[Pt(CN)4]Br0.3·3H2O (KCP) at 40 K and (NbSe4)3I at room temperature (data taken from [Bru75, Ves00]). The lines correspond to a power law behavior with exponents as indicated. The insets show the same data on linear scales.

the periodic potential the valence and conduction bands have the same curvature and the same mass. Both can be expressed in terms of the gap energy Eg.

A strongly frequency dependent optical conductivity has indeed been found in several one-dimensional or nearly one-dimensional semiconductors, such as K2[Pt(CN)4]Br0.3·3H2O (better known by the abbreviation KCP) [Bru75] and (NbSe4)3I [Ves00]. The conductivity of these much studied compounds is displayed in Fig. 13.9. The full line plotted in Fig. 13.9a is the expected ω−3 frequency dependence; for (NbSe4)3I, however, we find a stronger dependence, presumably due to the narrow band, with the bandwidth comparable to the optical frequencies where the measurements were made.

Electron–electron interactions may modify this picture. Such interactions lead, in strictly one dimension, to an electron gas which is distinctively different from a Fermi liquid. In not strictly but only nearly one dimension such interactions lead to a gap, and the features predicted by theory [Gia97] are expected to be observed only at high energies. These features include a power law dependence of the conductivity, with the exponents on ω different from −3 (see above). Experiments on highly anisotropic materials, such as (TMTSF)2PF6 [Sch98], have been interpreted as evidence for such novel, so-called Luttinger liquids.

A strict singularity at the single-particle gap, of course, cannot be expected for real materials; deviations from one dimensionality, impurities, and the frequency dependence of the transition matrix element involved, all tend to broaden the optical transition. In materials where the band structure is strongly anisotropic, another effect is also of importance. For a highly anisotropic system where the bands do not depend on the wavevector k, but with decreasing anisotropy, the parallel sheets of the band edges become warped, and the gap assumes a momentum dependence. Consequently the gap – defined as the minimum energy separation between both bands – becomes indirect. This can be seen most clearly by contrasting the thermal and optical gaps in materials of varying degrees of band anisotropy with the thermal measurements (i.e. dc resistivity) resulting in smaller gaps as expected and the difference increasing with decreasing anisotropy [Mih97].

13.1.2 Forbidden and indirect transitions

Forbidden transitions, as a rule, have low transition probabilities and therefore are difficult to observe experimentally. A possible example is presented in Fig. 13.2 for PbS, where the characteristic (ω − ωg)3/2 dependence of the absorption is seen, together with the (ω − ωg)1/2 dependence close to the bandgap; this latter gives evidence for direct allowed interband transitions as discussed above.

Less controversial is the observation of indirect transitions; the reason for this is that – in addition to the different frequency dependence – these transitions are

independent absorption at Eg, in contrast to the absorption process which goes to zero for singe-particle transitions as the bandgap is approached from above. Calculations can be performed for transitions with energies near to the gap, and in general one finds a broad peak at Eg due to excitonic absorption which merges with the single-particle absorption as the energy increases [Ell57]. This has been seen in a number of small bandgap semiconductors, and the frequency dependence of the absorption coefficient of GaAs measured at different temperatures is displayed in Fig. 13.12 as an example. The full lines are fits according to an expression derived for the energy dependence of the process [Stu62]; note also the strong temperature dependence and the thermal smearing of the excitonic absorption peak

– both natural consequences of the small energy scales involved.

13.2 Effects of interactions and disorder

Optical transitions discussed previously occur for pure and crystalline semiconductors where the nature of transition and its dependence on the underlying band structure depends essentially on the existence of well defined k states. Now we turn to situations where this strict periodicity is broken, either by the introduction of impurities or by the preparation conditions which prevent the lattice forming. In the former case we talk about impure semiconductors; and in the latter case about amorphous semiconductors.

13.2.1 Optical response of impurity states of semiconductors

Impurity states have been extensively studied by a wide range of optical methods, mainly because of the enormous role these electronic states play in the semiconductor industry. Extrinsic conduction associated with such impurity states is a standard issue for solid state physics, and transport effects which depend on the impurity concentration are also well studied. For small concentrations these impurity states are localized to the underlying lattice, but an insulator–metal transition occurs at zero temperature [Cas89] as the impurity concentration increases.

While there are numerous examples for these impurity effects phenomena, they have been studied in detail for phosphorus doped silicon Si:P. From transport and various spectroscopic studies it is found, for example, that the ionization energy, i.e. the energy needed to promote an electron from the j = 1 level of the donor phosphorus to the conduction band, is 44 meV; via the description (6.5.3) in terms of Bohr’s model this corresponds to the spatial extension of the impurity states of approximately 25 A,˚ significantly larger than the lattice periodicity. It is also known that due to valley–orbital interaction the various s and p impurity levels are slightly split; this splitting is about one order of magnitude smaller than the ionization energy. Localized impurity states can be described by the Rydberg