VALENCE BAND STRUCTURE AND THE 6x6 in Java

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10.2 VALENCE BAND STRUCTURE AND THE 6x6
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HAMILTONIAN
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It is beyond the scope of this book to derive the Hamiltonian that describes the valence band states, so its form will be stated and it will be employed in the calculation of quantised states within the valence band of nanostructures in order to illustrate how it is used. In many cases good accuracy can be obtained by using the so-called 6x6 Hamiltonian, although its shortened version, the 4x4 Hamiltonian is frequently just as good it all depends on the energy range of interest. As mentioned above, a valid basis set can be the atomic p-like orbitals with x, y, and z-like spatial symmetry, here denoted as | X } , |Y}, | Z ) , each also having the spin projection along the z-axis equal to either +1/2 or 1/2, denoted as | or | respectively. However, common practice
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MULTIBAND ENVELOPE FUNCTION (K.P) METHOD, Z. IKONIC
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is to change from the |X T), |Y T), |Z |), |X j), |Y j}, |Z |) basis into another one, such that its member functions are simultaneously the eigenstates of the angular momentum operator (with eigenvalues J equal to 3h/2 or h/2), and of its projection along the z-axis mj (with eigenvalues equal to 3/2 or 1/2). This is achieved by making appropriate linear combinations of the atomic basis states. The list of this new set of | J, mj) basis states is given below in equation (10.1). The precise form of the Hamiltonian depends on the pre-factors in equation (10.1), for example the presence of the imaginary number i or 1 does not change the state properties, and it also depends on how these states are ordered in the list. There is no unique choice that is universally accepted in the literature, however one of the frequently used possibilities, see for example [228], which will be adopted here, reads:
Just as with the original (|X T), |Y T), |Z T), |X j}, |Y j}, |Z | states, the new basis states are all orthogonal to each other. In this | J, mj) basis the 6 x 6 Hamiltonian that describes the HH, LH and SO bands for the bulk reads:
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where k = kx iky, 71,2,3 are the Luttinger parameters, and ASO is the spin orbit splitting the spacing between the HH (or LH) band and the SO band at the (k = 0)
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VALENCE BAND STRUCTURE AND THE 6 X 6 HAMILTONIAN
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F-point. The values of 71,2,3 and ASO in some common semiconductors are given in Table 10.1. In writing equation (10.2) the convention has been used that the hole energy is measured from the top of the valence band downwards (the inverted energy picture). This is because it is usually easier to look at the hole band structure in the same manner as that for electrons, and this is possible if only holes are considered. For the true energy picture (as in Fig. 10.1), all the terms in the Hamiltonian should be multiplied by -1. Furthermore, it is important to note that the coordinate system in which this Hamiltonian is written is not oriented arbitrarily the axes x, y, and z are aligned along the edges of the crystalline cubic unit cell. The Schrodinger equation corresponding to this Hamiltonian may still be written as
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but w = w(r) is a vector (a 6-component array). Note that the Hamiltonian is a Hermitian matrix with simple scalars as its elements (which depend on the material parameters) and wave vector components kx,ky, and kz. If the eigenenergies and the corresponding eigenvectors are found, the latter will obviously be lists of (possibly complex-valued) scalar constants. What would these mean It is implicitly assumed that the wave function has a plane wave form, i.e. all the 6 components of w have the common, plane wave type of spatial behaviour, i.e. ~ exp(ik.r). This makes them the components of the envelope wave function and, if interested in a more detailed form, each of them multiplies a corresponding basis state from the list in equation (10.1) and can be added together to construct the 'true' microscopic wave function of a state with energy E. The eigenvectors are generally 'full', i.e. their entries usually have non-zero values, which means that a plane-wave state has all the 6 basis states admixed. This is in contrast to the conduction band, where one usually deals with 'pure' spin-up or spin-down states. To have the usual meaning of a wave function, an eigenvector w has to be normalised to unity, i.e. all the components of a vector may have to be multiplied by a suitable constant so that:
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is satisfied. The 't' symbol means the Hermitian conjugate, i.e. the transpose (a column-vector becomes a row-vector) followed by the complex conjugate (take the complex conjugate of all the elements). In order to find eigenenergies for a specified k, solutions can be sought to det|H E\ = 0, which delivers a sixth-order polynomial in the energy E. This could only be solved numerically, unless it is noted that it can be factored into two identical thirdorder polynomials thus allowing for analytic, though lengthy, solutions. However, the situation is quite simple at the zone centre, i.e. if we set kx = ky = kz = 0 the
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