Inelastic scattering and phonon effects

The first model is by Mott and Kaveh (1985), who obtain the scaUng theory result by a different approach. A feature of the model is that it starts with a specific physical mechanism, that of multiple scattering, rather than a scaling argument. There is also a direct connection to the earlier model for cr, , . The theory uses a modified form of the scattering equation given by Kawabata (1981), 

Cb is the Boltzmann conductivity given by Eq. (7.51), is the momentum at the Fermi energy and C is a constant estimated to be near unity. This equation is derived as a description of the effects of weak disorder on the conductivity well above E(. and is extended to describe the conductivity at the mobility edge. The first term on the right hand side is the usual Boltzmann conductivity to which a factor g is added for the same reason as in Eq. (7.52). The second term in the bracket describes the effects of multiple scattering on the electron. Briefly, the amplitude of the wavefunction contains a sum over the scattering terms a, such that Za, = 1. Only the first order term a, contributes to the conductivity so that o is proportional to. 

The sum on the right hand side of Eq. (7.67) corresponds to the second term in the bracket of Eq. (7.66). Kawabata s calculation did not include the size of the sample, which Mott and Kaveh (1985) add as the factor (1 — a /L), reasoning that when the elastic scattering length is comparable to L, then the multiple scattering is reduced. Furthermore, in the presence of inelastic scattering, the term L is replaced by L for the same reason as the substitution was made in the scaling theory, that the coherence of the wavefunction is lost. 

The next step in the argument is to extrapolate the conductivity to the mobility edge. In the absence of inelastic scattering (or for an infinite sample size), the conductivity of Eq. (7.66) goes to zero when 

This expression defines the mobility edge. When there is a finite value of the inelastic scattering length, L, the conductivity at the mobility edge is no longer zero but from Eq. (7.66) has the value 

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