We provide a comparison of accurate full potential band calculations of the Fermi surfaces areas and masses of ${\mathrm{MgB}}_2$ and ${\mathrm{ZrB}}_2$ with the de Haas–van Alphen data of Yelland et al. and Tanaka et al., respectively. The discrepancies in areas in ${\mathrm{MgB}}_2$ can be removed by a shift of $\sigma$ bands downward with respect to $\pi$ bands by 0.24 eV. Comparison of effective masses lead to orbit averaged electron-phonon coupling constants ${\lambda }_{\sigma }=1.3\mathrm{}$ (both orbits), ${\lambda }_{\pi }=\mathrm{0.5.}\mathrm{}$ The required band shifts, which we interpret as an exchange attraction for $\sigma$ states beyond local density band theory, reduces the number of holes from 0.15 to 0.11 holes per cell. This makes the occurrence of superconductivity in ${\mathrm{MgB}}_2$ a somewhat closer call than previously recognized, and increases the likelihood that additional holes can lead to an increased $T_c.$

• Received 22 February 2002

DOI:https://doi.org/10.1103/PhysRevB.66.024521