Phase-Shift Analysis of the Fermi Surface of Copper

Abstract

In an earlier paper a method was described whereby the partial-wave phase shifts that characterize the interaction between the conduction electrons and the lattice in a metal may be derived from experimental Fermi-surface data. In the present paper we apply the method of phase-shift analysis to study the shape of the Fermi surface of copper, which is known to be strongly perturbed by the $d$-like energy bands that lie almost 2 eV below the Fermi level. By adjusting the values of the $s$, $p$, $d$, and $f$ phase shifts, and the Fermi-energy parameter, we construct a model Fermi surface on which the areas of the $〈100〉$ belly and the $〈111〉$ neck and belly orbits, the dog's bone, the four-cornered rosette, and the lemon orbit, are in good agreement with the results of precision measurements of the corresponding de Haas-van Alphen frequencies. The belly anisotropy of the model surface is also in good agreement with the experimental data, and the volume enclosed by the surface does not differ significantly from 1 electron/atom. The radii of the Fermi surface of copper in the (100) and (110) symmetry zones are determined to an accuracy of ±0.1%, and the results are in good agreement with the radii recently deduced by Halse by an independent technique. It is shown that the numerical values of the phase shifts are consistent with the position of copper in the Periodic Table. The local potential of Chodorow for ${\mathrm{Cu}}^{+}$ produces phase shifts that are in substantial agreement with the results of the present work. A simple nonlocal correction to the Chodorow potential is proposed, such that the Fermi surface derived from the modified potential is entirely consistent with the experimental data. The energies associated with certain optical transitions in metallic copper are computed from the modified potential, and are found to agree with the results of recent piezo-optical experiments to better than 0.2 eV. It is concluded that the method of phase-shift analysis is capable of representing accurately the form of the $d$-like electronic energy bands in metals, and that the modified Chodorow potential may well prove to be the best starting point for a full calculation of the band structure of copper in the vicinity of the Fermi level.