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Figure 9.20 Illustration of various 'intrasubband' carrier carrier scattering mechanisms in a two-level system
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where the subband indices of the initial states are labelled 'i' and 'j' and those of the final states 'f' and 'g'. The decoupled form of the wave functions, with a component of the motion confined along the z-axis and an in-plane (x-y) travelling wave, suggests that the integrals should be evaluated across the plane and along the growth axis, and that the separation of the carriers be expressed as:
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Goodnick and Lugli [187] (later re-iterated by Smet et al. [188]), followed earlier methods for carrier carrier scattering in bulk (see for example, Ziman [189], p. 170 and Takenaka et al [190]), and took the two-dimensional Fourier Transform of the Coulombic potential to give
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where qxy = |ki kf|, and A\$g is a form factor, i.e.
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Using the form for the matrix element shown in equation (9.204) and then substituting directly into Fermi's Golden Rule (equation (9.1)) gives the lifetime of a carrier in subband 'i' as follows:
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Converting the summations over both final-state wave vectors into integrals introduces a factor of L/(2TT) per dimension, thus giving a factor of A2/(2TT)4 in total (where the general area A = L2) (see Section 2.3), therefore:
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Integrating over all of the states of the second carrier (given by kj) and introducing Fermi-Dirac distribution functions to account for state occupancy, then obtain:
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which is equation (49) in Smet et al. [188]. Following their notation, collect the distribution functions together and label them as Pj,f,g (kj, kf, kg). The first 6-function summarizes in-plane momentum conservation and limits the integral over kg to a contribution when kg = kj + kj kf. In addition, the total energy of the carriers, Eit, etc. are equal to the energy of the relevant subband minima, Ei say, plus the in-plane kinetic energy; thus:
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where kg is known in terms of the other three wave vectors. It is the assumption of parabolic subbands in this last step that will be the limiting factor for the application of this method to hole-hole scattering a point mentioned earlier in the context of the carrier LO phonon scattering rate derivation. Now equation (9.211) represents the scattering rate of a carrier at a particular wave vector ki averaged over all of the other initial particle states kj, and hence the only unknown in this remaining 6-function is the wave vector kf. Contributions to the integral over kf occur when the argument of the 6-function is zero, and indeed given the form for this argument, it is clear that the solutions for kf map out an ellipse. The standard procedure [187,188] is then to introduce relative wave vectors:
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and replace the integration over kf in equation (9.211) with an integration over kf g . Since:
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i.e.
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It follows that:
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as pointed out by Mosko [191], which should then be substituted into equation (9.211). In order to perform the integration over kfg it is converted to plane polar coordinates with dkfg = kfgdkfgd9, where 0 is an angle measured from kij and the trajectory in the kfg-0 plane is deduced from the condition that the argument of the 6-function in equation (9.211) must be zero. The conservation of momentum diagram therefore looks like Fig. 9.21, where:
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In this work, the occupancy of the final states will be assumed to be small, such that the distribution functions dependent upon kf and kg can be ignored. The effect of putting JfFD(kf) and fgFD(kg) to zero can be seen to place an upper limit on the scattering rate. The physical interpretation of this approximation is that final-state blocking is ignored, i.e. the process by which a scattering event is prevented because the required final state is already occupied. This is a common simplification [192]
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