Model study of the semiconductor-metal transition in ${\mathrm{BaBi}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Pb}}_{\mathrm{x}}$${\mathrm{O}}_3$ by use of a spatially inhomogeneous order-parameter approximation

Phase transitions in a solid state are often described by a self-consistent single order-parameter approximation. Taking as an example the high-temperature superconducting alloys ${\mathrm{BaBi}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Pb}}_{\mathrm{x}}$ ${\mathrm{O}}_3$, I show that such models can lead to unrealistic numerical results. These difficulties are removed by considering a spatially inhomogeneous order-parameter approximation. The presented model study shows that a local charge-density-wave instability preserves the semiconducting behavior of ${\mathrm{BaBiO}}_3$ up to 65% Pb composition. Thus, the main results of a previous work of Jurczek and Rice are reinforced and agree with recent optical conductivity measurements of Tajima et al.