Patterson, Bailey - Solid State Physics Introduction to theory

316 6 Semiconductors

where n(E) is the number of states per unit volume of real space with energy E and dVk is the volume of k-space with energy between E and E + dE. Since we have derived (see Sect. 3.2.3)

for

2

E = 2me k 2 ,

with a spherical energy surface,

Vk = 43 πk3 ,

so we get (6.131).

We know that an ellipsoid with semimajor axes a, b, and c has volume V = 4πabc/3. So for Si with an energy represented by ((6.110) with origin shifted so k0 = 0)

for the electron density of states effective mass.

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The average energy (over a cycle) dissipated per unit volume is thus

where g = ωτand gc = ωcτ. We get a peak when g = gc. If there is more than one resonance there is more than one maximum as we have already noted. See Fig. 6.7.

Fig. 6.8. Band structures for Si and Ge. For silicon two results are presented: nonlocal pseudopotential (solid line) and local pseudopotential (dotted line). Adaptation reprinted with permission from Cheliokowsky JR and Cohen ML, Phys Rev B 14, 556 (1976). Copyright 1976 by the American Physical Society

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in getting an idea about the complexities involved. As discussed by Cohen and Chelikowsky [6.8], optical properties have been very useful in obtaining experimental results about actual band structures.

For very complicated cases, models are still useful. A model by Kane has been found useful for many II-VI and III-V semiconductors [6.16]. It yields a conduction band that is not parabolic, as well as having both heavy and light holes and a split-off band as shown in Fig. 6.10. It even applies to pseudobinary alloys such as mercury cadmium telluride (MCT) provided one uses a virtual crystal approximation (VCA), in which alloy disorder later can be put in as a perturbation, e.g. to discuss mobility. In the VCA, Hg1–xCdxTe is replaced by ATe, where A is some “average” atom representing the Hg and Cd.

Fig. 6.10. Energy bands for zincblende lattice structure

If one solves the secular equation of the Kane [6.16] model, one finds the following equation for the conduction, light holes, and split-off band:

6.2 Examples of Semiconductors 321

where is a constant representing the spin-orbit splitting, Eg is the bandgap, and P is a constant representing a momentum matrix element. With the energy origin chosen to be at the top of the valence band, if >> Eg and Pk, and including heavy holes, one can show:

In the above, m is the mass of a free electron (Kane [6.16]).

Knowing the E vs. k relation, as long as E depends only on |k|, the density of states per unit volume is given by

Finally, for the conduction band, if ħ2k2/2m is negligible compared to the other terms, we can show for the conduction band that

This clearly leads to changes in effective mass from the parabolic case (E k2).

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Brief properties of MCT, as an example of a II-VI alloy, [6.5, 6.7] showing its importance:

1.A pseudobinary II-VI compound with structure isomorphic to zincblende.

2.Hg1–xCdxTe forms a continuous range of solid solutions between the semimetals HgTe and CdTe. The bandgap is tunable from 0 to about 1.6 eV as x varies from about 0.15 (at low temperature) to 1.0. The bandgap also depends on temperature, increasing (approximately) linearly with temperature for a fixed value of x.

3.Useful as an infrared detector at liquid nitrogen temperature in the wavelength 8–12 micrometers, which is an atmospheric window. A higher operating temperature than alternative materials and MCT has high detectivity, fast response, high sensitivity, IC compatible and low power.

4.The band structure involves mixing of unperturbed valence and conduction band wave function, as derived by the Kane theory. They have nonparabolic bands, which makes their analysis more difficult.

5.Typical carriers have small effective mass (about 10−2 free-electron mass), which implies large mobility and enhances their value as IR detectors.

6.At higher temperatures (well above 77 K) the main electron scattering mechanism is the scattering by longitudinal optic modes. These modes are polar modes as discussed in Sect. 10.10. This scattering process is inelastic, and it makes the calculation of electron mobility by the Boltzmann equation more difficult (noniterated techniques for solving this equation do not work). At low temperatures the scattering may be dominated by charged impurities. See Yu and Cardona [6.44, p. 207]. See also Problem 6.7.

7.The small bandgap and relatively high concentration of carriers make it necessary to include screening in the calculation of the scattering of carriers by several interactions.

8.It is a candidate for growth in microgravity in order to make a more perfect crystal.

The figures below may further illustrate II-VI and III-V semiconductors, which have a zincblende structure. Figure 6.11 shows two interpenetrating lattices in the zincblende structure. Figure 6.12 shows the first Brillouin zone. Figure 6.13 sketches results for GaAs (which is zincblende in structure) which can be compared to Si and Ge (see Fig. 6.8). The study of complex compound semiconductors is far from complete.4

4 See, e.g., Patterson [6.30].

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GaAs Eg

Fig. 6.13. Sketch of the band structure of GaAs in two important directions. Note that in the valence bands there are both light and heavy holes. For more details see Cohen and Chelikowsky [6.8]

Density of States for Effective Hole Masses (A)

If we have light and heavy holes with energies

each will give a density of states and these density of states will add so we must replace in an equation analogous to (6.131),

Alternatively, the effective hole mass for density of states is given by the replacement of

6.2.2 Comments about GaN (A)

GaN is a III-V material that has been of much interest lately. It is a direct wide bandgap semiconductor (3.44 electron volts at 300 K). It has applications in blue and UV light emitters (LEDs) and detectors. It forms a heterostructure (see