The optical constants of amorphous Ge are determined for the photon energies from 0.08 to 1.6 eV. From 0.08 to 0.5 eV, the absorption is due to k-conserving transitions of holes between the valence bands as in p-type crystals; the spin-orbit splitting is found to be 0.20 and 0.21 eV in non-annealed, and annealed samples respectively. The effective masses of the holes in the three bands are 0.49 m (respectively 0.43 m); 0.04 m, and 0.08 m. An absorption band is observed below the main absorption edge (at 300 °K the maximum of this band is at 0.86 eV); the absorption in this band increases with increasing temperature. This band is considered to be due to excitons bound to neutral acceptors, and these are presumably the same ones that play a decisive role in the transport properties and which are considered to be associated with vacancies. The absorption edge has the form: ω2ϵ2∼(hω−Eg)2 (Eg = 0.88 eV at 300 °K). This suggests that the optical transitions conserve energy but not k vector, and that the densities of states near the band extrema have the same energy-dependence as in crystalline Ge. A simple theory describing this situation is proposed, and comparison of it with the experimental results leads to an estimate of the localization of the conduction-band wavefunctions.

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Optical Properties and Electronic Structure of Amorphous Germanium

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Temperature dependence of the energy gap in semiconductors

Band structure of indium antimonide

A numerical Kramers-Kronig analysis is used to predict the refractive-index perturbations produced in crystalline silicon by applied electric fields or by charge carriers. Results are obtained over the 1.0-2.0 \mu m optical wavelength range. The analysis makes use of experimental electroabsorption spectra and impurity-doping spectra taken from the literature. For electrorefraction at the indirect gap, we find \Delta n = 1.3 \times 10^{5} at \lambda = 1.07 \mu m when E = 10^{5} V/cm, while the Kerr effect gives \Delta n = 10^{-6} at that field strength. The charge-carrier effects are larger, and a depletion or injection of 1018carriers/cm3produces an index change of \pm1.5 \times 10^{-3} at \lambda = 1.3 \mu m.

Electrooptical effects in silicon

http://nano-bio.ehu.es/files/articles/Gonze_CPC_2009_590.pdf

ABINIT: First-principles approach to material and nanosystem properties

By using reflectivity and absorption measurements in the region 5 to 35 micron, the effect of free carriers on the optical constants has been determined for $n$- and $p$-type germanium, silicon, and indium antimonide, and for $n$-type indium arsenide. The contribution of the free carriers to the electric susceptibility is obtained from the optical constants. A carrier effective mass, ${m}_{s}$, is defined in terms of the susceptibility, and the significance of ${m}_{s}$ is considered for four different types of energy band structure. The experimental values of ${m}_{s}$ are compared with those calculated by using data from other experiments. Good agreement was found for $n$- and $p$-type silicon, $n$-type germanium, and $p$-type indium antimonide. In $p$-type germanium, the susceptibility due to transitions between the overlapping bands in the valence band was taken into account. However, the resulting ${m}_{s}$, for a sample of \ensuremath{\sim}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ impurity concentration, is larger by a factor of 1.8 than that calculated by using cyclotron resonance data. In $n$-type indium antimonide ${m}_{s}$ increases with carrier concentration. If one assumes ${m}_{s}$ to be energy-dependent, the shape of the conduction band calculated is consistent with previously reported measurements of the shift of the intrinsic absorption edge with electron concentrations. In the case of $n$-type indium arsenide, ${m}_{s}$ differs from the effective mass reported from thermoelectric measurements but gives good agreement with the values determined from the shift of the intrinsic absorption edge for an impure specimen.

Determination of Optical Constants and Carrier Effective Mass of Semiconductors

The problem treated is the effect of lattice vibrations in producing a shift of the energy levels which results in a temperature dependent variation of the energy gap in semiconductors. Calculations for silicon and germanium give results of the same order of magnitude as the observed temperature dependent shift of the absorption band edge. The effect of lattice vibrations on the energy gap has been treated previously on the basis of broadening rather than shifting of the energy levels. The effect was found to be negligible for non-polar crystals, whereas according to our treatment it should be much larger. For polar crystals our result turns out to be essentially the same as was given by the previous treatment.

Temperature Dependence of the Energy Gap in Semiconductors

The absorption by free carriers in $n$-type silicon has been studied in a spectral region from 1 to 45 microns for samples of various impurity and carrier concentrations. In all samples measured the absorption consists of a band between 1.5 and 5 microns in addition to an absorption which rises smoothly with wavelength. The absorption band is found to be proportional to the carrier concentration in samples doped with different donor impurities. It is suggested that the band is associated with the excitation of carriers to a higher lying energy band. The behavior of the smoothly rising portion of the absorption curve is in agreement with the Fan and Fr\"ohlich theory of free-carrier absorption. Quantitative agreement is obtained with a value of $0.3m$ for the effective mass.

Infrared Absorption in n-Type Silicon

The optical properties of tellurium and of trigonal and amorphous selenium have been investigated at wavelengths extending from the intrinsic absorption edge to about 152 microns using polarized radiation. The refractive indices of tellurium and trigonal selenium have been determined from 4 to 14 microns and from 9 to 23 microns, respectively. For amorphous selenium, the refractive index estimated from measured reflectivity shows no appreciable variation from 30 to 152 microns. The temperature shift and the pressure shift of the intrinsic absorption edge in amorphous selenium are found to be -1.45\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$ ev/\ifmmode^\circ\else\textdegree\fi{}K and -2.0\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ ev/atmos, respectively. Lattice absorption bands have been observed in tellurium and selenium; they are attributed to the excitation of combination modes. In tellurium, a strong absorption band has been observed at 11 microns, which is present only for $E\ensuremath{\parallel}c$ radiation. The band indicates that there are overlapping branches in the valence band which are separated by about 0.11 ev. The effective mass in the lower branch is estimated to be about four times smaller than that in the upper branch. It is possible that the structure of the valence band is responsible for the high-temperature reversal of the Hall effect in tellurium. The usual carrier absorption increasing smoothly with wavelength has been studied for tellurium using polarized radiations. The effective mass of holes in tellurium has been determined from reflectivity measurements: ${m}_{\ensuremath{\perp}}\ensuremath{\sim}{m}_{\mathrm{II}}=0.45 m$ at 300\ifmmode^\circ\else\textdegree\fi{}K, ${m}_{\ensuremath{\perp}}=0.30 m$ and ${m}_{\mathrm{II}}=0.45 m$ at 100\ifmmode^\circ\else\textdegree\fi{}K.

Optical Properties of Tellurium and Selenium

High resistance at crystal grain boundaries in $n$-type germanium is investigated. The resistance is symmetrical with respect to the direct on of the current and resembles the characteristics of a rectifier in the blocking direction. Such barriers are also photosensitive. The barrier is eliminated when the material is converted to $p$-type by nucleon bombardment or heat treatment. A theory is developed assuming the existence of surface states at the boundary. The ability of the barriers to withstand high voltages, around 100 volts, is explained by showing that the surface charge increases with increasing voltage. The dc conductance of the barrier, measured at different temperatures, agrees with theory in the dependence on temperature as well as in the order of magnitude. At sufficiently low temperatures the barriers show a capacitance independent of the frequency, whereas at higher temperatures the barrier admittance is strongly frequency dependent. These results are in agreement with the theory, showing that at low temperatures the current across the boundary is mainly carried by electrons, the hole current becoming increasingly important as the temperature is raised. The height of the potential barrier above the Fermi level is determined and found to be independent of temperature. A small difference in the measured breakdown voltage for the two directions of current is attributed to a difference in impurity concentration on the two sides of the boundary, which is confirmed by the ac measurements. The number of electrons on the boundary states is found to be of the order ${10}^{12}$ ${\mathrm{cm}}^{\ensuremath{-}2}$ at the breakdown, which may be the saturation of the boundary states. However, the field at breakdown is only a few times lower than the critical value for the onset of the Zener current, and this mechanism cannot be definitely ruled out.

H. Y. Fan

Papers

Determination of Optical Constants and Carrier Effective Mass of Semiconductors

Temperature Dependence of the Energy Gap in Semiconductors

Infrared Absorption in n-Type Silicon

Optical Properties of Tellurium and Selenium

Grain Boundary Barriers in Germanium