Imaginary part of the optical conductivity of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\mathrm{Bi}{\mathrm{O}}_3$

The frequency dependence of the imaginary part of the infrared conductivity is calculated for a superconductor. Sharp structure, characteristic of superconductivity with an order parameter with $s$-wave symmetry, appears in the BCS limit as a minimum at a frequency equal to twice the gap value. This structure scales with temperature but gets progressively smeared and shifted as impurity scattering is increased. The relationship between low-frequency results and the zero-frequency limit is investigated. Experimental results on ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\mathrm{Bi}{\mathrm{O}}_3$ are also presented. The low-frequency imaginary part of the conductivity displays a minimum at $2\Delta \approx 12$ meV, and provides unequivocal evidence of an $s$-wave superconducting order parameter. Strong-coupling (Eliashberg) results show similar trends. Using this formulation we find that the electron-phonon coupling in ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\mathrm{Bi}{\mathrm{O}}_3$ must necessarily be small, with coupling constant $\lambda \approx 0.2$, in agreement with conclusions drawn from measurements of the real part of the conductivity. Thus ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\mathrm{Bi}{\mathrm{O}}_3$ is an $s$-wave superconductor that is not driven by the conventional electron-phonon interaction.