CARRIER SCATTERING

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Figure 9.24 The electron electron scattering rate as a function of the initial electron energy, with and without screening, for a carrier density of 1 x 1010 cm 2 (left) and 100x1010 cm 2 (right)

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Figure 9.25 The mean 22-11 electron electron scattering rate as a function of the subband separation, including screening and with a carrier density of 1 x 1010 cm 2 in each level

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AVERAGING OVER AN INITIAL STATE POPULATION

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The method documented so far gives the carrier carrier scattering rate for a particular carrier energy averaged over another initial state distribution; as before for the carrier* Thanks to Jim McTavish for pointing this out.

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AVERAGING OVER AN INITIAL STATE POPULATION

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LO phonon case, it is usually more instructive to average this quantity over the FermiDirac distribution of carriers in the initial state. This could be achieved by using the following mean:

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where the superscript k has been introduced to indicated that this is the in-plane kinetic energy associated with the wave vector ki. By using the result of equation (2.50) in Section 2.4, the denominator is then given by:

As the carrier's in-plane energy is expressed in terms of a wave vector and:

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Hence the above mean can be expressed as:

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Figure 9.26 The mean 22 11 electron electron scattering rate as a function of the carrier density in level |2), for a 300 A infinitely deep GaAs quantum well at 300 K

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Fig. 9.25 shows the electron electron scattering rate averaged over the initial carrier distribution, as given by equation (9.245), for the 22 11 mechanism in a infinitely deep quantum well. The subband separation, AE21 = E2 E1, was varied by adjusting the quantum well width. It can be seen that the electron electron scattering

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CARRIER SCATTERING

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rate increases as the subband separation decreases and that in this particular approach with the low carrier density of 1x1010 cm 2 the rate is only a weak function of temperature.

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Figure 9.27 The mean 22 11 electron electron scattering rate as a function of temperature, for a 300 A infinitely deep GaAs quantum well, with a carrier density of 10x 1010 cm 2

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The effect of increasing carrier density is shown in Fig. 9.26. The range of carrier densities was chosen to span roughly those found in intersubband devices. It can be seen that with this model the intersubband scattering is almost proportional to the carrier density, although it must be noted that in this work any effects due to Pauli exclusion are being ignored this took the form of assuming that the final-state populations were small, as discussed earlier. Inclusion of significant populations in a final-state could lead to final-state blocking being significant, and hence interfere with the proportionality, but at this stage of gaining an intuitive feel for carrier carrier scattering, such investigations necessitate the introduction of yet more variables and unnecessary complication. As in the previous example, it was found that these intersubband scattering rates were quite insensitive to the temperature. This is substantiated further in Fig. 9.27, which shows the temperature dependence of the 22 11 intersubband scattering rate for 10x 1010 cm 2 electrons in the same 300 A infinitely deep quantum well. The rate peaks at around 10 % more than its low temperature value before decreasing towards room temperature. The temperature dependency within the model comes from the screening term as well as both of the initial-state distribution functions. 9.16 INTRASUBBAND VERSUS INTERSUBBAND So far, only the results of calculations of intersubband carrier carrier scattering rates have been presented, but as mentioned before intrasubband events are also possible.

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INTRASUBBAND VERSUS INTERSUBBAND

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The latter are characterised by no change in the numbers of carriers in each subband. Considering only the most simplistic mechanisms of the type 'ii-ii', then Fig. 9.28 plots the carrier density dependency and Fig. 9.29 the temperature dependency of the 22-22 electron electron scattering rate, again for a typical 300 A infinitely deep GaAs quantum well. These figures are the intrasubband equivalents of the intersubband cases of Figs 9.26 and 9.27, respectively, in the last section. Note that as all of the initial and final states are within the same subband, no dependence on an intersubband energy separation is required.

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Figure 9.28 The mean 22-22 electron electron scattering rate as a function of the carrier density, for a 300 A infinitely deep GaAs quantum well

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Therefore, comparing Fig. 9.28 with Fig. 9.26 it can be seen that the dependence of the scattering rate on the carrier density is quite different in this the intrasubband case. The 'almost' proportionality is only evident at the lowest of the three temperatures, and for 77 and 300 K there is a clear non-monotonic behaviour. Fig. 9.29 shows the temperature dependence of this 22-22 rate; this is much stronger than in the intersubband case of Fig. 9.27 and indeed seems to have a linear dependence starting from the low-temperature value. Overall, the intrasubband scattering rates are around one to two orders of magnitude higher than the closest intersubband equivalent. This is a general result which is observed in a range of semiconductor heterostructures. In fact, in this idealistic infinitely deep quantum well, the intersubband rates represent something of a maximum as the overlap of the carrier wave functions is complete, and hence the ratio of the intrasubband to intersubband rates deduced here is probably conservative.

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