Indirect gap

The occupied bands are called valence bands the empty bands are called conduction bands. The top of tire valence band is usually taken as energy zero. The lowest conduction band has a minimum along the A direction the highest occupied valence band has a maximum at F. Semiconductors which have the highest occupied k -state and lowest empty state at different points are called indirect gap semiconductors. If k = k, the semiconductor is call direct gap semiconductor. Gennanium is also an indirect gap semiconductor whereas GaAs has a direct gap. It is not easy to predict whether a given semiconductor will have a direct gap or not. 

Figure C2.16.3. A plot of tire energy gap and lattice constant for tire most common III-V compound semiconductors. All tire materials shown have cubic (zincblende) stmcture. Elemental semiconductors. Si and Ge, are included for comparison. The lines connecting binary semiconductors indicate possible ternary compounds witli direct gaps. Dashed lines near GaP represent indirect gap regions. The line from InP to a point marked represents tire quaternary compound InGaAsP, lattice matched to InP.

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. 

As described in the chapter on band structures, these calculations reproduce the electronic structure of inhnite solids. This is important for a number of types of studies, such as modeling compounds for use in solar cells, in which it is important to know whether the band gap is a direct or indirect gap. Band structure calculations are ideal for modeling an inhnite regular crystal, but not for modeling surface chemistry or defect sites. 

Indigoid soluble dyes, 7 373t Indigo vat dye, 9 181 Indirect-arc furnaces, 12 297—298 Indirect coal liquefaction, 6 858-867 Indirect cooler evaporators, 21 537 Indirect extrusion, copper, 7 693 Indirect food additives, 12 29, 34 categories of, 12 31 Indirect-gap semiconductors, 14 837 ... 

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 ... 

Figure 4.10(b) shows the temperature dependence of the absorption spectrum expected for an indirect gap. It can be noted that the contribution due to becomes less important with decreasing temperature. This is due to the temperature dependence of the phonon density factor (see Equation (4.37)). Indeed, at 0 K there are no phonons to be absorbed and only one straight line, related to a phonon emission process, is observed. From Figure 4.10(b) we can also infer that cog shifts to higher values as the temperature decreases, which reflects the temperature dependence of the energy... 

From optical spectra, a bandgap of 2.5 eV was estimated (based on an indirect gap), and this increase from the negative bandgap of bulk HgSe (see earlier) was attributed to size quantization. 

Reference 59 provides a comprehensive explanation of the optical spectra and extracted bandgaps. The direct bandgap of ca. 2.36 eV is compared to the literature value of ca. 2.2 eV and explained by size quantization in the fairly small (20 nm) crystals. An indirect bandgap of 1.9 eV was measured (literature value < 1.4 eV), but it was stressed that this provided an upper limit only, since the absorption in this region was dominated by free-carrier absorption, which masked the indirect absorption. Annealing decreased the conductivity and the free-carrier absorption and changed the indirect gap to > 1.3 eV. 

The estimated bandgaps for the two materials were 1.17 eV indirect (M0S2) and 1.14 eV direct (MoSc2). The latter is unusual, since this is the approximate value of the indirect gap of MoSc2 the direct gap is substantially higher. 

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. 

Concerning the electronic minimum gap (which is direct or quasi-direct in all the studied wires, see Ref. [121,122,149,154] for details) at the DFT level (see Table 9) we find that it decreases monotonically with the wire diameter. The calculated values are larger than the electronic bulk indirect gap, thus reflecting the quantum confinement effect. This effect, which has been recently confirmed in STM experiments [37,143], is related to the fact that carriers are confined in two directions, being free to move only along the axis of the quantum wires. Clearly we expect that, increasing the diameter of the wire, such an effect becomes less relevant and the electronic gap will eventually approach the bulk value. 

Experimentally, a metal-insulator transition has been observed between 8 and 12 kbar in Rb4C60 through NMR [44]. This is illustrated in Fig. 17 by the temperature dependence of 1IT1 for different pressures. While the behavior at 1 bar is dominated by an activated component, very similar to the one of K4C60 in Fig. 14, it gradually evolves towards a linear behavior, which is characteristic of a metal. This is the so-called Korringa law where the slope is proportional to n(E )2. Remarkably, the pressure needed to close the gap is quite modest, consistent with the idea that the indirect gap in the band structure could indeed be quite small. To date, this is the only report about such a transition. 

NiO 3.47 A p-type semiconductor with indirect gap optical transition. 371, 372... 

The distinction between a direct and an indirect gap is absent in a-Si H because the momentum is not conserved in the transition. 

Notice first that the minimum gap between the valence and conduction bands in Fig. 4-4 occurs at F this is the minimum gap for the entire Brillouin Zone. The corresponding energy difference, in this case less than 1 eV, is called Eq and, as indicated in E ig. 4-3, is the minimum energy at which absorption occurs. It is the same Eq evaluated in Eq. (3-39) and discussed there. The situation has a complication in the homopolar semiconductors in that the minimum energy in the conduction band does not occur at F where the valence-band maximum is. The difference is called an indirect gap and absorption cannot occur at that energy in the absence of other perturbations such as thermal vibrations. For this reason we ghose InAs as better suited for discussion than silicon. 

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