The Absorption Edge for 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 

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. 

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  

we have to introduce new proportionality factors in expression (4.35) to take this effect into account  

we can rewrite expression (4.35) in a more general way, which takes into account the temperature dependence of rjs.  

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