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For finite kx and/or ky, however, the system of linear equations is generally full, which indicates that a hole state has a finite contribution from all the basis states. This may seem puzzling, in view of the fact that there is essentially no difference between the x and z directions in the bulk crystal. However, this difference arises because of the choice of basis states: each of them has a definite projection of momentum and spin along the z-axis, and therefore (as is known from quantum mechanics) does not have the same property along the x-axis. 10.4 COMPLEX BAND STRUCTURE While equation (10.8) and its solutions in equation (10.9) give the possible values of hole energy for a particular wave vector k, it is interesting to consider the reverse problem: what values of kz may holes have if their energy E and the other two components of the wave vector (kx and ky) are specified For sake of simplicity ky is set to zero, while kx and E are generally non-zero and are real. Equation (10.8) in its expanded form then reads:
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which is, of course, quadratic in k2, with the two solutions:
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each of which delivers two kz's as a positive and a negative root of k2. In real semiconductors the values of the Luttinger parameters 71 and 72 are such that A is always positive, while B and C may be of either sign, depending on E and kx, as well as the material parameters. Consider the case of 4 AC < 0, i.e. C < 0. From the form used to write C it is clear that this will happen when kx and E are such that is satisfied. The is then larger than B, and regardless of the sign of B one of the roots will be positive and the other one negative. Hence there will be a pair of real and a pair of purely imaginary kz values. Consider now the case of 4AC > B2: equation (10.12) then contains the square root of a negative number, which will imply that both roots will be fully complex numbers (with both the real and imaginary parts non-zero). Finally, for 0 < 4AC < B2 both values of are either positive or negative, depending on the sign of B, i.e. all roots are either real or imaginary. Real-valued solutions for kz imply conventional plane-wave envelope wave functions, which is an allowed state in an infinite bulk. Complex kz implies an 'evanescent' wave, which decays in one direction and increases in the opposite direction, and which may simultaneously oscillate (if the real part of kz = 0). Because of the infinite length in the z-direction, and the inability to normalise the wave function, such states are
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not allowed in an infinite bulk crystal, but are perfectly allowed in finite regions, just as is the case with electrons, and can therefore appear in structures of finite extent. The above considerations were presented for the 4x4 model, however the general conclusions about the possible types of evanescent waves remain for more elaborate models as well. The list of all possible states at a particular energy E, that behave exponentially is called the 'complex band structure'. The wave functions vary along a particular direction, say z, as exp(ikzz}, where kz is a real or complex wave vector. The case of electrons in the conduction band is quite different as they have a scalar effective mass and really a very simple complex band structure: their wave vector can be only real or imaginary: but never fully complex. For holes this situation only occurs at kx = ky = 0, when = E / ( j 1 2y2) and = E/(7i + 272), so all kz's are real for E > 0 and imaginary for E < 0. 10.5 BLOCK-DIAGONALISATION OF THE HAMILTONIAN
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There is an elegant method which allows one to simplify the 6x6 Hamiltonian, by recasting it into a block-diagonal form: it then has two 3x3 matrices as its diagonal elements and the remaining two off-diagonal elements are zero-matrices. This is achieved by a suitable change of basis, i.e. by creating a new basis from linear combinations of the existing basis states. Essentially this means that there are two completely independent sets of states, which are eigenstates of either one or the other 3x3 block. Since this block-diagonalisation reduces the size of the system to be considered at any time, it is useful when attempting to obtain a result in analytical form. Block-diagonalisation can also be useful in cases which are more complicated than straightforward bulk material, for example, strained bulk or two-dimensional quantum-well systems. The derivation of the transformation of the Hamiltonian into a block-diagonal form will not be reproduced here, the result will merely be stated, i.e. upon introducing the new basis:
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