*In the plane of the quantum well.

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of the type7k2zbecome k z Yk z , while terms of the type jkz become (jkz + kzj)/2 (note that the 7's are position dependent in a heterostructure). Furthermore, the diagonal elements of the Hamiltonian are amended with the potential V, which could have contributions from the valence band offset in the particular material, the potential from an external electrostatic field, or the self-consistent space-charge electrostatic potential. Following this it was shown by Foreman [228] that further modifications are necessary which improve the accuracy of the method, and under these developments the Hamiltonian now reads:

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where:

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It is possible to show that equation (10.29) reduces to equation (10.2) if the Luttinger parameters (71, 72 and 73) are set as constants. As for the case of the bulk Hamiltonian, equation (10.29) can also be blockdiagonalised into two 3x3 blocks, which read:

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where P and Q are the same as above, but R, 5, S and C now read:

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It would now appear unclear what values of 72 and 73 should be used in 7o, however it is a good approximation to use their average values across the structure. For a layer-type structure (i.e. with a constant composition inside any of the layers) use equation (10.29), or equation (10.31), to find the boundary conditions for the wave function at interfaces. This is achieved by formal integration across the interface, in the same way as with the effective mass Schrodinger equation (resulting in the conclusion that w, as well as ( l / m * ) d w / d z , is conserved across the interface). Such integration of equations (10.29) or (10.31) shows that the amplitudes Fi in the wave function vector are individually conserved, and also that there are particular linear combinations of both the derivatives and the amplitudes of all Fi components that are conserved across the interface (see later). Methods of calculating the eigenstates of heterostructures may be divided in two groups. One of them, which is practical only for one-dimensional heterostructure potentials (like quantum wells), uses a 'layer approach' and first finds the relevant properties of each single layer in the structure, before proceeding to find its 'global' properties in particular, the eigenstates. The other approach considers directly the heterostructure as a whole. It is computationally more demanding than the layer approach, but is equally applicable to quantum wells, wires and dots. In either case, the final ingredient we need for the calculation is the position-dependent potential V(z) (or V(r] in multidimensional structures) that is to be used in the Hamiltonian equations (10.29) or (10.31). 10.8 VALENCE BAND OFFSET In case of an unstrained system the valence band offset has a meaning analogous to that in the conduction band: V here shows the valence band edge (of both the HH and LH branches) in any layer. This is precisely how it enters the Hamiltonian. The valence band offsets at heterointerfaces are generally available from the literature, but there is a considerable amount of scatter in this data. In Table 10.1 values are given of the valence band offset (VBO) with respect to vacuum, for a few common semiconductors. For alloys it is usual to use linear interpolation to approximate.

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Having found the VBO values for two materials of interest, we subtract the two in order to get the relative offset. The VBO data in Table 10.1 are given in the real energy scale, hence the material with higher VBO will be the quantum well in other words, the relative band offset should be multiplied by 1 before being used in either of the Hamiltonians in equation (10.29) or (10.31), which are written in the inverted energy picture. It should be noted, however, that the valence band offsets determined in this way are only approximate (the exception being GaAs/AlAs), and for more accurate values the literature should be consulted. The case of strained structures is more complicated, and requires some care in using the available data on valence band offsets. The potential V which should be inserted into the Hamiltonian is not any of the band edges, defined by equation (10.28), because these expressions already include the effects of strain and cannot be filtered out of the Hamiltonian. Instead, V in the Hamiltonian is the potential before the addition of strain. In many cases data can be found on the so-called average valence band energy Eav, which is the the weighted mean of the three valence bands. In an unstrained material, where HH and LH band edges are degenerate, the weighted mean (in the inverted energy picture!) is Aso/3 above the valence band edge. Alternatively, the interface of two semiconductors may be characterised by the average valence band offset, A.Eav, which is the difference of E av 's in the two materials. Since ASO is material dependent, the AEav is not the same as the valence band offset, even in an unstrained material. It has been established that AE av is roughly constant with strain, and can therefore be used as a single parameter to describe the interface. If a strained semiconductor layer is grown on an unstrained substrate (made of a different material), and the values of AEav and ASo in the epitaxial layer material are known, then the difference AE av Aso/3 clearly gives the energy of the HH/LH valence band edge in this material without strain, measured from the Eav value in the substrate. In the same manner the valence band edge can be obtained in all layers of a multilayer structure, measured from the same reference point, regardless of whether a particular layer is in direct contact with the substrate or not. This is the potential that should be used in the Hamiltonian. It can be subtracted from any desired reference energy, if it is preferred to have output energies measured from that point. Since the valence band edge in a layer before strain has no physical significance in the strained system, one reasonable choice for a reference point might be the lowest valence band edge in the quantum well either HH or LH, whichever came out to be lower (if there are different quantum wells, the deepest one could be chosen). Another choice might be the valence band edge in the substrate, which is unstrained, and still has its HH and LH band edges degenerate. In some cases, for a particular materials interface and particular strain conditions, data may be quoted like 'AB on CD has the valence band offset of AEV (eV)'. This means that strained material AB grown on unstrained substrate CD has such an offset between its valence band edge (either HH or LH, whichever is the lower in this case) and the valence band edge in the substrate. To use this data for a calculation within

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