Significant Contributions of the Higgs Mode and Self-Energy Corrections to Low-Frequency Complex Conductivity in DC-Biased Superconducting Devices

We investigate the complex conductivity of superconductors under a DC bias based on the Keldysh-Eilenberger formalism of nonequilibrium superconductivity. This framework allows us to account for the Higgs mode and impurity scattering self-energy corrections, which are known to significantly impact the complex conductivity under a bias DC, especially near the resonance frequency of the Higgs mode. The purpose of this paper is to explore the effects of these contributions on the low-frequency complex conductivity relevant to superconducting device technologies. Our approach enables us to derive the complex conductivity formula for superconductors ranging from clean to dirty limits, applicable to any bias DC strength. Our calculations reveal that the Higgs mode and impurity scattering self-energy corrections significantly affect the complex conductivity even at low frequencies, relevant to superconducting device technologies. Specifically, we find that the real part of the low-frequency complex conductivity exhibits a bias-dependent reduction up to \(\hbar \omega \sim 0.1\), a much higher frequency than previously considered. This finding allows for the suppression of dissipation in devices by tuning the bias DC. Additionally, through the calculation of the imaginary part of the complex conductivity, we evaluate the bias-dependent kinetic inductance for superconductors ranging from clean to dirty limits. The bias dependence becomes stronger as the mean free path decreases. Our dirty-limit results coincide with previous studies based on the so-called slow experiment scenario. This widely used scenario can be understood as a phenomenological implementation of the Higgs mode into the kinetic inductance calculation, now justified by our calculation based on the robust theory of nonequilibrium superconductivity, which microscopically treats the Higgs mode contribution.