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Figure 9.11 The intrasubband scattering rate via LO phonon emission for the same case as that shown in Fig. 9.8
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A simple weighted mean over a distribution of carriers in the initial subband might look like the following:
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where the subscript 'i' on the distribution functions indicates the subband, i.e. to be evaluated with the 'quasi' Fermi energy of that subband. However, this still disregards the distribution in the final subband; in this case, filled states could prevent carriers from scattering into them, thus reducing the probability of an event. This effect of final-state blocking can be incorporated into the above to give:
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where the double subscript 'if is used to indicate that this scattering rate is an average over the subband populations in the initial and final states. The integrals are evaluated from the subband minimum of the initial state up to some defined maximum. In the calculations that follow, this maximum has been chosen to be the energy of the highest point in the potential profile, which means physically that any carriers above the barrier are assumed to ionise rapidly. An alternative to this may be to choose the Fermi energy plus 10kT, or similar. Equation (9.153) may be simplified slightly, as the denominator is simply the number of carriers in the subband divided by the density of states (see equation (2.48)). Fig. 9.12 shows this calculated mean for the same system as before, i.e. a GaAs infinitely deep quantum well, for the case of scattering via the emission of a LO phonon from the second subband to the ground subband. In this series of calculations the width of the well was varied in order to scan the difference between the energy
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Figure 9.12 The mean scattering rate averaged over distributions in both the initial and final subbands, as a function of the subband separation, shown for three different temperatures
band minima (labelled here as E21 = E2 E1) through the LO phonon energy, and the electron density in each subband was assumed to be 1010 cm 2 . The electron temperature, i.e. the temperature inserted into the Fermi-Dirac distribution function was taken equal to the lattice temperature in systems under excitation where nonequilibrium distributions will be present, this assumption is almost certainly not true. Again, the infinitely deep quantum well is a good illustrative example, as the overlap of the wave functions, which is contained within the form factor Gif, does not change, i.e. the effect on the scattering rate is due entirely to the energy separation. In the long-range scan, Fig. 9.12 (left), it can be seen that as the subband separation decreases, the scattering rate increases up to almost a 'resonance' point and then decreases rapidly. At energies above the resonance, the scattering rate has only a weak temperature dependence, but below it, the dependence is stronger. Figure. 9.12 (right) illustrates this resonance effect more clearly, for a smaller range of energies. As may be expected for this fixed phonon energy, which in GaAs is 36 meV, the peak in the scattering rate occurs when the subband separation is equal to the phonon energy. The right hand figure highlights well the strong temperature dependence of the scattering rate for subband separations below the LO phonon energy. At very low temperatures, the 'cut-off in the scattering is almost as complete as that shown by the single-carrier case in the previous section. However, as the temperature increases the carrier distributions broaden, so although the subband separation remains below the phonon energy, the carriers in the upper level spread up the subband, with a proportion having enough kinetic energy to be able to emit a LO phonon and scatter to the lower level. As the temperature increases, this proportion increases and hence the mean scattering rate also increases.
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