Indirect-gap materials

The situation is very different in indirect gap materials where phonons must be involved to conserve momentum. Radiative recombination is inefficient, resulting in long lifetimes. The minority carrier lifetimes in Si reach many ms, again in tire absence of defects. It should be noted tliat long minority carrier lifetimes imply long diffusion lengtlis. Minority carrier lifetime can be used as a convenient quality benchmark of a semiconductor. 

In materials with a band structure such as that sketched in Figure 4.8(b), the bottom point in the conduction band has a quite different wave vector from that of the top point in the valence band. These are called indirect-gap materials. Transitions at the gap photon energy are not allowed by the rule given in Equation (4.29), but they are still possible with the participation of lattice phonons. These transitions are called indirect transitions. The momentum conservation rule for indirect transitions can be written as... 

Indirect transitions are much weaker thau direct trausitious, because the latter do uot require the participation of photons. However, many indirect-gap materials play an important role in technological applications, as is the case of silicon (band structure diagram iu Figure 4.7(a)) or germanium (baud structure diagram shown later, in Figure 4.11). Hereafter, we will deal with the spectral shape expected for both direct and indirect transitions. 

For indirect-gap materials, all of the occupied states in the valence band can be connected to all the empty states in the conduction band. In this case, the absorption coefficient is proportional to the product of the densities of initial states and final states (see Eqnation (4.27)), bnt integrated over all the possible combinations of states separated by bro being the energy of the phonon involved). This... 

Table 4.3 The frequency dependence expected for the fundamental absorption edge of direct- and indirect-gap materials...

It should be noted that the frequency dependence is different to those expected for direct-gap materials, given by Equations (4.33) and (4.34). This provides a convenient way of determining the direct or indirect nature of a band gap in a particular material by simply analyzing the fundamental absorption edge. Table 4.3 summarizes the frequency dependence expected for the fundamental absorption edge of direct- and indirect-gap materials. 

The general shape of the absorption edge for an indirect-gap material has been sketched in Figure 4.10(a). In this figure (a plot of versus >), two different linear regimes are clearly observable. The straight line at lower frequencies shows an absorption threshold at a frequency of a>i= cog — 12, which corresponds to a process... 

Because of the involvement of phonons in indirect transitions, one expects that the absorption spectrum of indirect-gap materials must be substantially influenced by temperature changes. In fact, the absorption coefficient must be also proportional to the probabihty of photon-phonon interactions. This probabihty is a function of the number of phonons present, t]b, which is given by the Bose-Einstein statistics ... 

In discussing deep levels in wide band-gap semiconductors, the first requirement is to define deep and wide. The latter can be done relatively easily, although arbitrarily. We list in Table I the 4°K band gaps of the various III-V semiconductors, based on the tabulation by Strehlow and Cook (1973). We shall call those with Eg> 1.5 eV the wide band-gap ones. In practice, our review will present data only on GaAs and on GaP as prototypes of direct and indirect gap materials of this class. These are also the only two materials of this class that have been extensively studied and that are in common use. Discussion of deep levels in ternary and quarternary alloys of III-V semiconductors are omitted since treating these in detail might well have doubled the size of this chapter. 

The direct transitions dominate only in much smaller crystallites [13]. As the size of the crystallites covered in the current experiments is 2 nm or more, the porous and nanocrystalline light emitting silicon is "indirect" gap material. The calculated radiative life time decreases with decreasing crystallite size for both the phonon assisted and direct transitions. The predicted dependence [13] correlates reasonably with... 

The dependence of the optical absorption in small isolated silicon nanociystals on the photon energy shows a slow increase above the band gap energy which is typical for an indirect gap material [11]. A sharp increase is observed only when the photon energy reaches the value of the direct gap transition (about 3.4 eV) at the F-point k = 0). With decreasing crystallite size below about 4 nm the band gap increases [38] and the PL shows a pronoimced blue shift [39], The results reproduced fi-om ref. [38] and [39] are shown in Fig. 3. 

T. Takagahara, K. Takeda, Theory of the Quantum Confinement Effect on Excitons in Quantum Dots of Indirect-Gap Materials. Physical Review B 1992,46,15578-15581. 

According to theory, BeS is an indirect-gap material with the highest valence band edge at F and the lowest conduction band edge at X. 

Indirect-gap materials are the materials for which the top of the VB and the bottom of the CB are not the same value of k (e.g.. Si, Ge, GaP). In the case of an indirect-gap SC, the transition of an electron between the VB and CB involves a substantial change in the momentum of the electron. Therefore, silicon, for instance, in the bulk form is not a luminescent light emitter, in contrast to direct-gap materials, which are mostly efficient emitters [68]. This difference between direct and indirect band structures is very important for LEDs, SC lasers, and PV cells. 

Because in indirect-gap semiconductors generation and recombination of carriers are equally difficult, even in indirect-gap materials the distribution of electrons and holes is governed by the Fermi function. Because the Fermi energy can never be farther than half the energy gap from one band edge or the other, the density of carriers in an indirect-gap material is still determined by the minimum energy gap in spite of the difference in momenta of the band minima in indirect-gap materials. 

The consequences of the need for a phonon to permit an electron and a hole to interact in indirect-gap materials include ... 

The maximum recombination time for electrons and holes is much longer (of the order of microseconds) in an indirect-gap material than in a direct gap semiconductor (nanoseconds). 

Theoretically, the opposite dependence of direct and indirect gaps on pressure could be used to convert indirect gap materials to direct gaps. However, a negative pressure (tensile stress) would have to be applied to achieve this conversion. Ceramics, including semiconductors, tend to be weaker in tension than in compression. Even by placing the indirect material in a strained-layer superlattice (see Chapter 7), which can achieve the highest tensile stress levels, it has been impossible to convert indirect semiconductors to direct gaps before the stress is relieved by formation of dislocations or by fracture. 

The differences in lattice plane stacking have a direct effect on the electronic properties of SiC. For example, the energy gaps for the various polytypes are 3.33, 3.26, 3.02, and 2.39 eV for the 2H, 4H, 6H, and 3C respectively. [14] All four polytypes are indirect gap materials with a corresponding direct gap in excess of 4 eV. 

Carrier mobilities are higher and absorption coefficients lower in nanocrystalline indirect-gap materials. 

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