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9.7 RATIO OF EMISSION TO ABSORPTION In the previous section, the importance of the carrier energy for emission has been demonstrated. Converse to this, for the case of carrier scattering by absorption of a phonon, it is not the energy of the carrier that is the important issue but rather the number of phonons available the more phonons, then the more likely an absorption process. The phonon density, given in equation (9.2), increases as the temperature rises, thus increasing the probability of an absorption. Fig. 9.13 displays the results of calculations of the ratio of the emission to the electron-LO phonon absorption rate, for the same series of quantum wells as in the previous section. It can be seen that the emission rate is always larger than the absorption rate; thus given a carrier population in an excited subband, then when left to reach equilibrium the carriers will always emit more phonons than they absorb and hence scatter down to the ground state.
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Figure 9.13 The ratio of the mean electron-LO phonon emission(l/T2i)to absorption (l/t12) rate, as a function of the subband separation (left) and a more detailed view around the LO phonon energy (right)
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The linearities of the graphs shown in Fig. 9.13 hints at a relationship of the form:
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and in fact, a numerical analysis of the data shows that the 'constant' is equal to 1/kT, i.e. This simple relationship between the ratio of the emission to the absorption scattering rates, the subband separation and the temperature is helpful in summarizing the data presented in Fig. 9.13. For a fixed temperature, this ratio increases as the energy separation between the two levels increases, while for a given subband separation, increasing temperature leads to a decrease in the ratio.
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Figure 9.14 The ratio of the mean electron-LO phonon emission (l/T2i) to absorption (1 /m) rate, as a function of temperature T for a fixed subband separation E21 equal to the LO phonon energy of 36 meV
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These conclusions are not that obvious a priori, and go beyond the simple intuitive picture that the ratio of the scattering rates is controlled by the ratio of the phonon densities, i.e. (N0 + 1 ) / N 0 . For a more detailed investigation into the temperature dependency of the ratio of the emission to absorption scattering rates, consider the data shown in Fig. 9.14, which corresponds to the fixed subband separation of E21 equal to the LO phonon energy, which in this case is 36 meV. At room temperature, the ratio is equal to the well-known result of 4; however, as the temperature decreases, the ratio of these mean scattering rates increases very rapidly and by 77 K emission is more than two orders of magnitude more likely than absorption. 9.8 SCREENING OF THE LO PHONON INTERACTION The longitundinal optical phonon interaction is a polar interaction, thus it can be influenced by the presence of other charges. In particular, in a doped semiconductor there can be many charges which are free to move in an electromagnetic field and like Lenz's law in electromagnetic induction they move to oppose any change. This idea is known as 'screening' and its effect is to reduce the scattering rate due to LO phonons. The screening model of Park et al. [181] can be implemented in a simple way in the formalism here by making the substitution:
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in equation (9.151) except in the form factor Gif(K Z }. The quantity As is known as the 'inverse screening length' and for systems with a majority carrier type this would
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where the index 'j' includes all occupied subbands. The unusual parameters employed in Figs 9.15 and 9.16 were chosen to give direct comparisons with Figs 1 and 2 of Park et al. and the effects of introducing screening are very similar.
Figure 9.15 The electron-LO phonon scattering rate from the first excited to the ground subband as a function of well width for a GaAs quantum well surrounded by 60 A Ga0.7 Al0.3As barriers. The lattice temperature was taken as 15 K, the electron temperature as 100 K and the carrier density as 1011 cm 12.
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The peaks in the curves in Fig. 9.15 occur when the energy separation between the subbands is in resonance with the LO phonon energy, in this case 36 meV. Around this energy screening at this carrier density can reduce the scattering rate by a factor of 2 or 3, though further from resonance screening has a reduced effect. Fig. 9.16 shows the effect of the temperature of the electron distributions in the subbands on the lifetime for two different carrier densities. Such elevated electron temperatures occur commonly in electronic and optoelectronic devices which are all driven by energy some of which is absorbed directly by the electron gas. 9.9 ACOUSTIC DEFORMATION POTENTIAL SCATTERING Combining Lundstrom's [177] equations (2.56) and (2.59), summing over all bulk phonon wave functions which are of the form:
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