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Effect of disorder on the far-infrared conductivity and on the microwave conductivity of two-band superconductors

We consider the far-infrared and the microwave conductivities of a two-band superconductor with non-magnetic impurities. The strong coupling expressions for the frequency and temperature dependent conductivity of a two-band superconductor are developed assuming isotropic bands and interactions. Our numerical results obtained using realistic interaction parameters for MgB$_{2}$ are compared with experiments on this compound. We find that the available experimental results for the far-infrared conductivity of MgB$_{2}$ are consistent with multi-band superconductivity in the presence of a sufficiently strong interband impurity scattering. On the other hand, our numerical results for the microwave conductivity in the superconducting state indicate that the experimental results obtained on samples with the highest transition temperature $T_{c}$ are consistent with a low interband impurity scattering rate but depend sensitively on the ratio of the total scattering rates in the two bands. For the $π$-band scattering rate $γ_π$ not greater than the $σ$-band scattering rate $γ_σ$ there is a single, broad, low-temperature (at about 0.5$T_{c}$) coherence peak in the microwave conductivity. For $γ_π/γ_σ$=4--7 a high-temperature (at about 0.9$T_{c}$) coherence peak is dominant, but there is also a low-temperature peak/shoulder resulting from the contribution of the $π$-band carriers to the microwave conductivity. For $γ_π/γ_σ\gg$1 only the high-temperature coherence peak should be observable.