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Figure 12.20 Comparison of the perturbative approach to the pseudopotential calculation of a superlattice, with Nn = 3 valence bands, with that of the large-basis (LB) method of earlier
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The difference between the energies of Fig. 12.20
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where nz is the period of the superlattice in lattice constants. In comparison, the number of plane waves required in the expansion set for the large-basis calculation was deduced empirically in Section 12.2 as approximately 128nz; hence, for the large basis approach: Thus, it can be seen that the order of both matrices scales linearly with the superlattice period nz. However, it is clear that if the number of bands Nn required in the perturbative calculation is less than 64, then the Hamiltonian matrix will be smaller than
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The computational effort associated with the actual process of diagonalisation of these matrices scales as the square of the order, and hence this part of the perturbative approach will require less computer time, as well as less memory, than the large basis approach. 12.12 SUPERLATTICE DISPERSION AND THE INTERMINIBAND LASER
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As yet, the dispersion curves for electron movement along the growth (z-) axis, i.e., perpendicular transport, have not been calculated by using the perturbative approach to the superlattice.
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Figure 12.22 The two lowest energy dispersion curves (minibands) along the growth (z-) axis of a (GaAs)10(Gao.sAlo.2As)10 superlattice
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Using the virtual crystal approximation to obtain the form factors for the alloy, Fig. 12.22 plots the lowest two conduction bands of the superlattice, for a range of electron wave vectors across the Brillouin zone of a (GaAs)10(Gao.sAlo.2As)10 superlattice. Just as the superlattice Brillouin zone is often referred to as the minizone, these superlattice dispersion curves are called minibands (see Section 2.13). As discovered in Section 2.13, when using the Kronig Penney model, the lowest-energy miniband has its minimum at the zone centre, while the second miniband has its minimum at the zone edges. Thus, if carriers are injected into the upper of these two minibands then they will rapidly lower their energy, via the fast non-radiative intraminiband scattering processes of phonon emission and carrier scattering. These have times similar to those characterising intrasubband processes, perhaps of the order of 100 fs (see 9). Given the functional form of the E k curves within the superlattice Brillouin zone, this energy lowering implies an increasing momentum and the carriers thus move towards the zone edge.
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Figure 12.23 The carrier dynamics within an interminiband laser
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At this point, the carriers have reached the bottom of the miniband and can no longer lower their energy through intraminiband relaxation processes, and they must wait the relatively long time for an interminiband scattering event to occur. Following this, any carrier scattering to the states near the zone edge of the lowest miniband, will scatter rapidly to lower momentum states. Such a scenario of a long lifetime in the upper level, compared to a short lifetime in the lower level, for the selection of electron momenta near the superlattice zone edge, is reminiscent of the requirements of the intersubband laser of 9. Indeed, the combination of these scattering processes within the two lowest minibands of a superlattice forms the basis of a new form of semiconductor laser known as the interminiband laser [41,42]. One exciting feature of this device is that it lases without an overall population inversion: it is only necessary to obtain a localised inversion in the states near the zone edge for stimulated emission to occur.
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An electric field can be considered merely as a perturbation on the semiconductor system, just as the superlattice itself was. However, given the inherent periodic nature of the pseudopotential method, the electric field itself has to be periodic and repeated within each unit cell, as shown in Fig. 12.24. Taking the first and last atomic positions within the unit cell of interest as the extent of the electric field, and defining the zero-field point as the centre of the cell,
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