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In essence, the result here can be summarized by saying that, the effective mass of a particle in a quantum well is a function of the well width. The fact that a constant effective mass predicts too high a confinement energy allows the further deduction that the effective mass must be increased as the well width decreases in order to produce agreement with the empirical pseudopotential calculations. In conclusion, the microscopic nature of the pseudopotential calculation gives more detail and allows for more complexity in the eigenstates of heterostructures than methods based on the envelope function/effective mass approximation. In particular, use of the constant-effective-mass approximation has been shown to breakdown for short-period superlattices. 12.4 IN-PLANE DISPERSION In addition to computing the dispersion along the line of symmetry, i.e. the growth (z-) axis, the large-basis method, when applied to superlattices as in Section 12.2, can be used to calculate the in-plane (x-y) dispersion. Such knowledge is fundamental for describing the transport properties of electronic devices which exploit in-plane transport for their operation, such as High Electron Mobility Transistors (HEMTs), as well as optical devices which are influenced by the carriers populating the in-plane momentum states, such as intersubband lasers (see Section 9.22).
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Figure 12.14 The in-plane [100] dispersion curve around the centre of the Brillouin zone for a (GaAs)10(AlAs)10 superlattice (top) in comparison with that for bulk (bottom)
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Fig. 12.14 displays the results of calculations of the in-plane dispersion (top curve), in this case along the [100] direction, for the (GaAs)10(AlAs)10 superlattice of the previous section. On the same axes the bulk dispersion curve (bottom) is also shown for comparison. The subband minima are both measured from the top of the valence band and hence the minimum of the bottom curve represents the band gap of the bulk material, while the difference in the minima represents the quantum confinement energy of this, i.e. the lowest confined state in the superlattice.
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Remembering that for low electron momenta the E-k dispersion curves are parabolic, then as before the effective mass is given by:
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The fitting of parabolas (the solid curves) to the data points in Fig. 12.14 thus allows the effective masses to be calculated (as before in 11). Using the potentials of Mader and Zunger [253], this procedure gives the effective mass along any of the (100) directions in the bulk crystal as 0.082m0. In contrast to this, the in-plane electron effective mass for the superlattice is 0.15mo, which is obviously quite different. This would seem to be a general result, at least for relatively short period superlattices; note the period for this example is l0A0 = 56.5 A. A priori, it might be expected that as there is no confinement in the x-y plane (i.e. parallel to the layers), that the dispersion curves would resemble that of the bulk. In fact, as there is confinement along the z-direction, which leads to a shift in the band minimum the band structure around the minima is clearly different from that of the bulk. The immediate consequence of which is that the effective mass increases. 12.5 INTERFACE COORDINATION In bulk zinc blende material, each anion, e.g. As in GaAs, is bonded to four Ga cations. It has been shown that the atomic pseudopotential of this As is different from the As" (say) in AlAs, which is merely a reflection of the different chemical nature of the bonding between Ga atoms and Al atoms arising from their different electronegativities. Although disappointing in that universal atomic potentials cannot be deduced, the calculations thus far have shown how to deal with such a problem. Consider now a heterojunction between two compound semiconductors which share a common species, e.g. GaAs/AlAs, as illustrated in Fig. 12.15. The As anion at the interface, as indicated, is bonded to two Ga atoms and two Al atoms, and hence they have neither the character of a fully tetrahedrally coordinated As or As". In fact, they have a character which is intermediate between the two, which can be described by the mean: and hence the interface properties can be described better by the mean in the atomic pseudopotentials, i.e.
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12.6 STRAIN-LAYERED SUPERLATTICES As already mentioned, the more recently deduced pseudopotentials generally take account of a much greater diversity of experimental information than just the main
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