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the same material system, but under different strain conditions, it will be necessary to find whether the HH or LH band in the strained AB material makes its valence band edge (which depends on the sign of the strain), then find Pe and Qe and then use the appropriate expression from equation (10.28) to get the AB valence band edge without strain.
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10.9 THE LAYER (TRANSFER MATRIX) METHOD In order to describe the procedure of finding the quantised state energies and wave functions, the block-diagonal form of the Hamiltonian will be used, i.e. attention will be focussed on just one of the 3 x 3 blocks. However, the method is straightforwardly applicable to any other size of the Hamiltonian, e.g. the 2 x 2, or one of the non-blockdiagonalised 6x6 or 4x4 forms, or in fact any other form, see [232]. Consider a structure modulated along one dimension (i.e. a quantum well) with arbitrarily varying material composition and potential (see Figs. 1.9- 1.13 for a few examples). For the purpose of finding its bound states, or the tunnelling probability, the heterostructure of interest is subdivided into a number (Nz, number of coordinate points) of thin layers, and within each layer the potential (which includes any selfconsistent potential, if such a calculation was performed) is taken to be constant, as are the values of the Luttinger parameters (note that under these conditions S = E and C = 0 in equations (10.31-10.32)). If the structure is step-graded, comprising some number of layers of different material composition and width (perhaps like the structure in Fig. 3.15), and provided any continuously varying potentials (e.g. selfconsistent potential) are absent, the computational layers in this calculation coincide with actual material layers, and need not be very thin. Similarly to the case of quantised states of electrons, the aim is to describe the wave function of a quantised state in terms of its form within each layer. These are then joined at the interfaces using appropriate boundary conditions. Within such an approach the need is not to generate all the energy eigenstates of the bulk Hamiltonian for a specified wave vector, but rather all the solutions that correspond to a definite energy, i.e. the complex band structure has to be found. In order to find the complex band structure in the valence band, consider a structure grown in the [001] direction, which it is convention to define as the z-axis. Since the structure composition and the potential are modulated (varied) along z, but are constant along the x- and y-axes, the wave function in the x-y plane must behave like a plane wave, hence kx and ky must be real, while kz is arbitrary. The Hamiltonian is written as (to make the writing shorter, the factor h2/2mo is taken to be absorbed into the 7 parameters):
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where HO, H1, and H2 are the 3 x 3 matrices that are associated with the corresponding powers of the wave vector kz, that is:
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At any specified value of the energy E, the values of the complex wave vector kz may be viewed as eigenvalues of the 3 x 3 non-linear eigenvalue problem [H(kz ) E] [F] = 0, where [F] is the eigenfunction vector of length 3. This eigenproblem is non-linear because eigenvalues (kz, not E!) appear in powers of both 1 and 2, in contrast to the standard linear eigenproblem where the energy appears only linearly on the diagonal of the matrix. Non-linear eigenproblems of this type are solved by a trick which converts them to a doubled-in-size, 6x6 linear eigenvalue problem, for which well-developed techniques exist (the method is readily generalised to handle any polynomial-type non-linear eigenproblem). The linear problem to be solved thus reads:
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where 0 and 1 are the 3 x 3 null and unity matrices. Note that the first row in equation (10.35) is just an identity. The solution of this non-Hermitian but linear matrix eigenproblem, by standard diagonalisation routines [233], delivers the six, generally complex-valued wave vectors kz, and the corresponding six eigenstates. These states, denoted as [u], are expressed in the basis F1 -F3 for the upper block. The [u] -states are thus particular linear combinations of F1-F3 that, at energy E, behave as plane waves, i.e. exp(ikzz), within a layer. It should be noted that the first three components of an eigenvector of equation (10.35) are the amplitudes of the basis states, and the other three, when multiplied by i, will be their derivatives. In further considerations the notation [u] will denote the vector of length 6, with the amplitudes and derivatives. For convenience, these states may then be divided into two groups, as follows: a state which has purely real kz may be classified according to the sign of kz+, if it is positive the state is 'forward', otherwise it is 'backward'; if kz is complex, including purely imaginary, it is classified as forward if the wave function decays to the right,
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Strictly speaking, it should be tested for the current it carries, where the current density operator is (2H2Kz + H 1 )/h, but for the purpose here, the sign test suffices.
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