Localized versus Extended Holes in a Two-Band Model for Semiconductor-Metal Transitions

We consider essentially the two-band Hubbard-type model of a nonmagnetic semiconductor studied recently by Falicov and Kimball (FK), but we allow the bandwidth ${\Delta }_v$ of the valence band to be nonzero and the Coulomb repulsion $U_{11}$ between holes to be noninfinite. FK treated this model in an approximation based to some extent on the free-energy variational principle, and they used one-electron wave functions which are localized (Wannier) functions for the valence band, and extended (Bloch) functions for the conductior band. This problem is formulated within the framework of the recently introduced thermal single-determinant approximation. The limit $U_{11}\to \infty$, ${\Delta }_v\to 0$ then provides a strictly variational derivation of the FK results. We then show that for ${\Delta }_v\ne 0$, $U_{11}<\infty$, and temperature sufficiently small ($T<\mathrm{}{T}_0$), a lower free energy is obtained when valence-band Bloch functions are substituted for the Wannier functions. That $T_0$ can be appreciable even when $\frac{U_{11}}{{\Delta }_v}\gg 1$ and the possibility of a transition to the localized picture at $T_0$ are pointed out. It is shown that magnetic inelastic neutron scattering (arising from the magnetic dipolar interaction between neutrons and electrons) and distinguish, at least in principle, between the extended and the localized pictures. The fact that band gaps occur in the thermal-neutron range for interesting materials is noted.