Energy Bands of Aluminum

The Green's function method for investigating the band structure of crystals is applied here to aluminum. The potential used in the calculations is close to that employed by Heine, the difference consisting of a correction and a small modification required for the simple application of the Green's function method. Extensive calculations were made in order to study the convergence of the energies and wave functions and the perturbation arising from the use of the "muffin-tin" potential instead of the originally determined $V\left(\mathrm{r}\right)$. The convergence of the energies is shown to be very rapid; the convergence for the eigenfunctions is somewhat slower but is still satisfactory. The perturbations of the eigenvalues are shown to be smaller than the errors expected as a result of the inaccuracies in the present potential, which is known comparatively accurately. The relative separations of the calculated levels are in good agreement with those for Heine's result except for his upper $K_1$ level which is about 0.25 ry too high. Our energies lie about 0.1 ry below Heine's as a result of the correction to the potential. It is shown that the calculated eigenvalues can be fitted remarkably well for filled and low-lying excited states by a nearly free electron (pseudopotential) interpolation scheme. It is also shown that the Bloch functions can be determined in a simple fashion from the nearly free electron scheme. By virtue of the similarity of the present bands with those Harrison obtained by the pseudopotential interpolation procedure, we conclude that the shape of the Fermi surface associated with the present results is consistent with available experimental data.