The band energy model

The band model is applied to perfect crystals with translational symmetry, to find the electronic structure of the system. Following the work of Slater and Koster [90], the electronic wave functions are approximated by linear combinations of atomic orbitals, in term of Bloch sums, used as basis functions. Starting with an atomic orbital $\phi_{i\alpha}({\bf r}-{\bf R}_i)$, located on an atom at ${\bf R}_i$, the Bloch sum is written as

$$\Psi_{{\bf k},i\alpha}({\bf r})=\frac{1}{\sqrt{N}} \sum_{{\bf... ...rm exp}(i{\bf k \cdot R}_i) \phi_{i\alpha}({\bf r}-{\bf R}_i).$$ (7.7)

Here, N is the number of unit cells and $\alpha$ is an atomic orbital index. However, the atomic orbitals $\phi_{i\alpha}({\bf r}-{\bf R}_i)$, and thus the Bloch sums, are not orthogonal to one another. Löwding [94] defined new atomic orbitals $\psi_{i\alpha}({\bf r}-{\bf R}_i)$ as linear combinations of the original ones, and orthogonal to each other. These Löwding orbitals preserve the symmetry of the orbital from which they are derived. The Hamiltonian matrix elements in the basis of the Bloch sum formed from the Löwding functions are thus

$$H_{i\alpha , j\beta} \int \Psi_{{\bf k},i\alpha}^\ast({\bf r}) \hat{H} \Psi_{{\bf k},j\beta}({\bf r}) d{\bf r}$$ = (7.8)

$$\frac{1}{N}\sum_{{\bf R}_i,{\bf R}_j} {\rm exp}[i{\bf k \cdot}({\... ...pha}^\ast({\bf r}-{\bf R}_i) \hat{H} \psi_{j\beta}({\bf r}-{\bf R}_j) d{\bf r}.$$ =

This can be further simplified since one of the two sums gives exactly N. We thus have

$$H_{i\alpha , j\beta} = \sum_{{\bf R}_j} {\rm exp}[i{\bf k \cd... ...\bf R}_i) \hat{H} \psi_{j\beta}({\bf r}-{\bf R}_j) d{\bf r}.$$ (7.9)

Further simplifications are needed to calculate the integrals above. Slater and Koster [90] proposed to neglect the three-center integrals with respect to the two-center integrals. The three-center integrals involve two orbitals located at different atoms, and a potential part of the Hamiltonian due to a third atom. Thus, in the Slater and Koster's approach, only the potential (considered spherical) due to the two atoms upon which the orbitals are located, is taken into account. The integrals can therefore be replaced by parameters which depend on the internuclear distance $\vert{\bf R}_j - {\bf R}_i\vert$, and on the symmetry of the orbitals involved. For example, if $\psi_{i\alpha}$ had the symmetry of a p orbital, it could be expressed as a linear combination of $p\sigma$ and $p\pi$, where $\sigma$ and $\pi$ refer to the component of the angular momentum around the axis between the two atoms.

The Hamiltonian matrix elements can therefore be written

$$H_{i\alpha , j\beta} = \sum_{{\bf R}_j,J} {\rm exp}[i{\bf k \... ...a J}(\vert{\bf R}_j - {\bf R}_i\vert)G_{\alpha\beta J}(k,l,m),$$ (7.10)

where J is the angular momentum of the orbitals about the axis between the two atoms ( $\sigma, \pi, \delta$ and so on), $h_{\alpha\beta J}$ is a constant for a given $\vert{\bf R}_j - {\bf R}_i\vert)$, and $G_{\alpha\beta J}(l,m,n)$ is the angular dependence of the integral, l,m and n being the direction cosines of the vector ${\bf R}_j - {\bf R}_i$. The parametric forms of $h_{\alpha\beta J}(\vert{\bf R}_j - {\bf R}_i\vert)G_{\alpha\beta J}(k,l,m)$ are tabulated by Slater and Koster [90]. The band structure can now be obtained by diagonalizing the Hamiltonian, for different values of $\bf k$.