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Dissipative conductivity of a dirty superconductor with Dynes subgap states under a dc bias current up to the depairing current density

We study the dissipative conductivity $σ_1$ of a dirty superconductor with a finite Dynes parameter $Γ$ under a dc-biased weak time-dependent field. The Usadel equation for the current-carrying state is solved to calculate the pair potential, penetration depth, supercurrent density, and quasiparticle spectrum. It is shown that, while the depairing current density $j_d$ for $Γ=0$ is coincident with the Kupriyanov-Lukichev theory, a finite $Γ$ decreases the superfluid density, resulting in a reduction of $j_d$. The broadening of the peaks of the quasiparticle density of states induced by a combination of a finite $Γ$ and a dc bias can reduce $σ_1$ below that for the ideal dirty BCS superconductor with $Γ=0$, while subgap states at Fermi level proportional to $Γ$ results in a residual conductivity at $T\to 0$. We find the optimum combination of $Γ$ and the dc bias to minimize $σ_1$ by scanning all $Γ$ and all currents up to $j_d$. By using the results, it is possible to improve $j_d$ and reduce electromagnetic dissipation in various superconducting quantum devices.