  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.
