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Next: Remarks about the cellular Up: Methods for band-structure calculations Previous: Empirical pseudo-potentials

The cellular (Wigner-Seitz) method

The TB model is too crude to be useful in calculations of actual bands, which are to be compared with experimental results. Now we shall consider some of the common methods employed in calculations of actual bands. The cellular method was the earliest method employed in band calculations by Wigner and Seitz. It was applied with success to the alkali metals, particularly to Na and K.

The method begins by observing that because of the Bloch relation, if we solve the Schrödinger equation in one unit cell, we know thie solution in the entire solid. However, we need to impose the boundary conditions that the function, and its derivative should vary continuously at the boundary between two neighboring cells.

In order to find the solution of the Schödinger equation, we assume that the electron, when in a particular cell, say A, is influenced by the potential of the ion in that cell only. The ions in other cells have a negligible effect on the electron in cell A because each of these cells is occupied, on the average, by another conduction electron which tends to screen the ion, thereby reducing its potential drastically. To ensure that the function $\psi_k$ satisfies the Bloch form $\psi_k = e^{ikr} u_k$ , it is necessary that $u_k$ be periodic, i.e. $u_k$ be the same on opposite faces of the cell.

The procedure is now clear in principle: We attempt to solve the Schrödinger equation in a single cell, using for $V(r)$ the potential of a free ion, which can be found from atomic physics. In Na, for instance, $V(r)$ is the potential of the ion core Na$^+$. It is still very difficult, however, to impose the requirements of periodicity on the function for the actual shape of the cell, and to overcome this difficulty Wigner and Seitz replaced the cell by a WS sphere of the same volume as the actual cell. The reason why this method is suitable for Na, is precisely because body-center cubic and face-centered cubic structures have a WS cells that are polyhedra that resemble spheres.

Using these simplifying assumptions concerning the potential and the periodic conditions, one then solves the Schrödinger equation numerically, since an analytical solution cannot usually be found.

Since the potential is spherically symmetric we write the wave function as

\psi_{lm}({\bf r}) = Y_{lm}(\theta,\phi)R_l(r)
\end{displaymath} (236)

where $Y_{lm}$ are spherical harmonics and the radial part satisfies the usual differential equation
R_l''(r) + \frac{2}{r}R_l'(r)+\frac{2m}{\hbar}\left( \epsilon-V(r)-\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}\right) R(r) = 0
\end{displaymath} (237)

Given the potential $V(r)$ and any value of $\epsilon$ there is a unique $R_l$ that solves thsi equations and it is regular at the origin. These functions can be calculated numerically. Next, we write the wavefunction as:

\psi({\bf r},\epsilon) = \sum_{lm} A_{lm} Y_{lm}(\theta,\phi) R_{l}(r,\epsilon).
\end{displaymath} (238)

Now, we need to impose the following boundary conditions:

$\displaystyle \psi({\bf r})$ $\textstyle =$ $\displaystyle e^{-i{\bf k}.{\bf R}}\psi({\bf r}+{\bf R}),$ (239)
$\displaystyle {\bf n}\cdot {\vec \nabla}\psi({\bf r})$ $\textstyle =$ $\displaystyle - e^{-i{\bf k}.{\bf R}}\cdot {\vec \nabla}\psi({\bf r}+{\bf R}).$ (240)

where both ${\bf r}$ and ${\bf R}$ are points on the surface of the cell, and ${\bf n}$ is the outward normal to the face of the WS cell. These boundary conditions introduces ${\bf k}$ into the equations and determine the discrete values of the energies for which these equations have a solution, i.e the energy bands $\epsilon=\epsilon({\bf k})$.

Its is in the impositioon of these conditions that we make the major approximation. First, we take only as many terms in the expansion $\sum_{lm}$ as we are able to handle. Since there is only a finite number of coefficients in the expansions , we can only fit the boundary condition for a finite numbers of points on the cell. This leads to a set of $k$-dependent linear homogeneouos equations for the coefficients $A_{lm}$, that yield the wanted energies $\epsilon_n({\bf k})$.

There are two ways to solve the $k$ dependence:

- We fix $\bf k$ and we do a search to find the energies that correspond to zeroes in the determinant.

- We solve the differential equation for a given value of the energy $\epsilon$ and we look for the vector ${\bf k}$ at which the determinan vanishes. Provided that we have not chosen a value of $\epsilon$ in the middle of the gap, we can always find a solution.

next up previous
Next: Remarks about the cellular Up: Methods for band-structure calculations Previous: Empirical pseudo-potentials
Adrian E. Feiguin 2009-11-04