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and as these integrals are independent of each other then finally the optical deformation scattering rate for a carrier with an initial wave vector ki into all final states of a given subband is given by:
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9.12 CONFINED AND INTERFACE PHONON MODES Forming quantum wells or superlattices clearly changes the electronic energy levels of a crystal from what they are in an infinite bulk crystal, which is the main subject of this book. In fact, all of the crystal properties are changed to a greater or lesser extent. Perhaps of secondary importance to the effect on the electronic energy levels, and the
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subsequent changes that this induces in scattering rates, exciton energies, impurity energies, etc. are the fundamental changes introduced to the phonon modes. It can be appreciated that such changes are likely as the bulk LO phonon in GaAs has an energy of around 36 meV while in AlAs it is closer to 50 meV. Hence, forming a superlattice with alternating layers of GaAs and AlAs is going to have some effect on the phonon energies. Various models have been put forward to account for this change in symmetry. At one end of the scale, some models consider each semiconductor layer as a continuum of material with macroscopic-like properties, namely the Dielectric Continuum model, (see for example [178]), or the Hydrodynamic model, (see for example [182]). Alternative approaches have considered the allowed vibrational modes calculated directly from the viewpoint of individual atomic potentials (see for example, [183]). See Adachi [14], p. 70, for an introduction. Such improved models for phonons in heterostructures lead to modes which are confined to the individual semiconductor layers confined modes while some propagate along boundaries between the layers the so-called interface modes. Recent work has shown that while the electron-phonon scattering rates from these individual modes are quite different, the total rate from all of the modes collectively is quite similar to that from bulk phonons (see Kinsler et al. [184]). However, this is still a very active area of research and future developments will need to be monitored. 9.13 CARRIER-CARRIER SCATTERING Fermi's Golden Rule describes the lifetime of a particle in a particular state with respect to scattering by a time-varying potential. For phonon scattering, this harmonic potential is derived from the phonon wave function, which is itself a travelling wave. For the case of one carrier scattering against another due to the Coulomb potential, there appears to be no time dependency. The Born approximation is often cited in the literature when discussing carrier carrier scattering; this is just a way of working scattering from a constant potential into Fermi's Golden Rule. This is achieved by considering that the perturbing potential is 'switched on' only when the particle reaches the same proximity. For an excellent introduction to the Born approximation, see Liboff ([185], p. 621). Therefore, the perturbating potential appearing in Fermi's Golden Rule for the interaction of two isolated carriers is the Coulombic interaction, i.e.
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where E = ErEo is the permitivity of the material and r is the separation of the electrons. Now the initial and final states, |i) and |f), respectively, of the system both consist of two electron (or hole) wave functions, as carrier carrier scattering is a two-body problem, and thus there is a much greater variety of scattering mechanisms possible than in the essentially one-body problem encountered in phonon scattering. Fig. 9.19
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illustrates all of the possible mechanisms in a two-level system, where at least one of the carriers changes its subband, these are usually referred to as intersubband transitions. However, the distinction now is not quite so clear (see below). The central diagram in Fig. 9.19 illustrates the symmetric intersubband event, '22 11' which moves two carriers down a level. The left and right figures show Auger-type intersubband scattering, where one carrier relaxes down to a lower subband, giving its excess energy to another carrier which remains within its original subband
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Figure 9.19 Illustration of various intersubband carrier carrier scattering mechanisms in a two-level system
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In addition to the above, there are also scattering events where the number of carries in each subband does not change. Some of these are illustrated schematically in Fig. 9.20. Clearly the first of these, '22 22' is an intrasubband event; however, the second and third events are more difficult to categorise precisely because although the number of carriers in each subband remain the same, the interaction itself is between carriers in different subbands. Pauli exclusion prevents carriers with the same spin occupying the same region of space, which therefore lowers their probability of scattering; in this work attention will be focused on collisions between particles with anti-parallel spins. Such considerations of spin-dependent scattering are often referred to by the term exchange [186]. Given that there are four possible carrier states involved, then in a N-level system there are 4N different scattering events. In this two-level system these are as follows: 11 11, 11-12, 11-21, 11-22, 12-11, 12-12, 12-21, 12-22, 21 11, 21-12, 21-21, 21-22, 22-11, 22-12, 22-21, and 22-22. Note that completely different events of the type 'ij-fj' are possible in quantum wells with three or more subbands, and interactions of this type have been shown to be important in optically pumped intersubband lasers [186]. Therefore, taking a heterostructure wave function, of the form shown in equation (9.80), then the matrix element in Fermi's Golden Rule (equation (9.1)) becomes: <f|H|i> = Web Pages 1d implementation in .net
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