Model of the Electron-Phonon Interaction and Optical Conductivity of ${\mathrm{Ba}}_{1-x}{\mathbf{K}}_x{\mathrm{BiO}}_3$ Superconductors

We investigate the physical properties of the ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{BiO}}_3$ compounds with a focus on the optical properties. Results from the simple Holstein model, describing a single band coupled to an oxygen breathing mode with parameters derived from first principles calculations, are in excellent agreement with a broad range of experimental information. It accounts for an insulating parent compound at $x=0$, with a direct (optical) and an indirect gap, and a metal insulator transition around $x=0.38$. Strong electron-phonon coupling leads to spectral weight redistribution over a frequency scale much larger than the characteristic phonon frequency and to strongly anharmonic phonons. We find that the metallic phase in the vicinity of phase boundary is close to the polaronic regime.

Figure 1

Panel (a): Normalized interacting OSW as a function of bare $e$-ph coupling, ${\lambda }_0$, at half-filling ($x=0$) for a particle-hole symmetric model in the normal and CDW phases. Panel (b): Normalized Drude weight and inverse effective mass as a function of ${\lambda }_0$. The tight binding parameters for this figure are $t_1=0.3926\mathrm{eV}$ and $\hslash {\omega }_0=0.08\mathrm{eV}$.

Figure 2

(a) Calculated and measured OC at half-filling ($x=0$). Measured OCs of ${\mathrm{BaBiO}}_3$ are reproduced from 7, 8, labeled by (1) and (2), respectively. Inset: The renormalized phonon density of states. (b) Calculated and measured effective carrier number per Bi ion participating in the optical transitions. (c) Electron DOS in the charge ordered phase with an indirect gap of the order of $\Delta \approx 0.5\mathrm{eV}$. $A$ and $B$ indicate the majority and minority sublattices for electron density, respectively.

Figure 3

(a) Calculated and experimental [8] conductivity for the indicated doping $x$ at zero temperature. Inset: Effective carrier number participating in the OC as a function of frequency. (b) The phonon density of states at $x=0.5$. (c) Normalized interacting OSW by noninteracting value as a function of doping. (d) Drude weight, $W_{\mathrm{D}}$, of the OC, normalized by total weight $W_{\mathrm{opt}}$, as a function of the doping level. In panels (c) and (d), system at $x=0$ is in CDW phase, while other data points show normal metallic phase.