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Remembering that the upper sign describes phonon absorption then this would imply:
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which are clearly the desired results, hence finally:
The case for a negative A follows a similar route and leads to the same end result. Therefore, equation (9.151) represents the final form for the lifetime ri of a carrier in a subband 'i' with an in-plane wave vector ki before scattering by an LO phonon. The information regarding whether such an event is with respect to absorption of a phonon or emission, and the final state of the carrier, is incorporated within the variables A and P' (which is within y"). This result is a particularly powerful expression because it is applicable to all two-dimensional carrier distributions, regardless of the particular form for the wave functions. Such information is wrapped up in the form factor Gif(Kz), and thus the carrier-LO phonon scattering rate can be calculated for any semiconductor heterostructure simply by evaluating a one-dimensional integral. 9.5 APPLICATION TO CONDUCTION SUBBANDS In the previous section, the carrier-LO phonon scattering rate, which is the reciprocal of the lifetime, was derived for a two-dimensional distribution, as found in the subbands formed in quantum well systems. In this present section, this result, as summarized in equation (9.151), will be applied to a variety of examples in order to gain an intuitive understanding of this important phenomenon. Fig. 9.8 displays the intersubband scattering rate as a function of the total initial energy Eit, as defined in equation (9.97), for an electron in the second subband of an infinitely deep quantum well with respect to LO phonon emission and scattering into the ground state. (Note here the initial carrier energy domain is from the subband minimum upwards.) The scattering rate increases as the carrier approaches the subband minimum, which in this case is around 220 meV. This is a general result and occurs for subband separations greater than the LO phonon energy. If the quantum well width is allowed to increase, then the energy separation between the initial and final subbands decreases. Fig. 9.9 illustrates the effect that such a series of calculations has on the scattering rate. It can be seen that in all cases the scattering rate increases; however, for quantum well widths greater than 300 A, there
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Figure 9.8 The scattering rate via LO phonon emission for an electron initially in the second subband and finally in the ground state, of a 100 A GaAs infinitely deep quantum well at a lattice temperature of 77 K
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Figure 9.9 indicated
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The scattering rate as given in Fig. 9.8, but for a variety of well widths (in A), as
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is a small region where the scattering rate falls very rapidly to zero. This is indicative of the subband separation, E2 E1, being less than the LO phonon energy. The 'cut-off' in the scattering rate occurs because the electrons in the upper subband have not sufficient energy to emit an LO phonon and hence are unable to scatter. This feature is illustrated schematically in Fig. 9.10. Moving from the upper right down the curve, electrons have sufficient kinetic energy which, when combined with the potential energy from being in the upper subband, allows them to emit a phonon; however, the third electron represents the minimum kinetic energy for scattering.
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Below this point, the electrons are less than an LO phonon away from the energy minimum of the complete system and hence can not scatter.
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Figure 9.10 The effect of the LO phonon energy 'cut-off' on intersubband electron scattering
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Fig. 9.11 shows the corresponding intrasubband scattering rate for an electron in the second subband. It can be seen that the behaviour of the rate with the initial energy is qualitatively similar to the intersubband case shown in Fig. 9.8. However, in addition there is a cut-off energy at 260 meV, which is an LO phonon energy above the subband minima. This is always the case for intrasubband scattering when a carrier is within a phonon energy of the subband minima it can not emit a phonon. Note that the scattering rate is almost an order of magnitude higher for the intrasubband case than for the intersubband case. This actually represents something of a minimum, as generally the intrasubband rate is between one and two orders of magnitude higher than the intersubband rate. This is because the overlap of the wave function with itself is always complete, whereas the overlap of two distinct wave functions is often only partial. In this case, with this hypothetical infinitely deep quantum well, the overlap of the wave functions for the intersubband case is higher than the usual situation.
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9.6 AVERAGING OVER CARRIER DISTRIBUTIONS The formula shown in equation (9.151) gives the lifetime of a carrier in a particular subband with a definite in-plane wave vector ki with respect to scattering with an LO phonon into another subband. In real situations, there isn't just one carrier in the initial subband, plus an empty final subband; in fact, there are generally Fermi-Dirac distributions in both of the subbands (see Section 2.4). It is then more useful to know the mean scattering rate (or lifetime) of a carrier.
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