Significant contributions of the Higgs mode and impurity-scattering self-energy corrections to the low-frequency complex conductivity in dc-biased superconducting devices

We investigate the complex conductivity of superconductors under a direct-current (dc) bias based on the well-established Keldysh-Eilenberger formalism of nonequilibrium superconductivity. This robust framework allows us to account for the Higgs mode and impurity-scattering self-energy corrections, which are now 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. We begin by nonperturbatively calculating the equilibrium Green’s functions under a bias dc, followed by an analysis of the time-dependent perturbative components. This approach enables us to derive the complex conductivity formula for superconductors ranging from clean to dirty limits, applicable to any bias dc strength. We validate our theoretical approach by reproducing known results and experimentally observed features, such as the characteristic peak in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:msub><a:mi>σ</a:mi><a:mn>1</a:mn></a:msub></a:math> and the dip in <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:msub><d:mi>σ</d:mi><d:mn>2</d:mn></d:msub></d:math> attributed to the Higgs mode. Our calculations reveal that the Higgs mode and impurity-scattering self-energy corrections significantly affect the complex conductivity even at low frequencies (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mi>ℏ</g:mi><g:mi>ω</g:mi><g:mo>≪</g:mo><g:mi mathvariant="normal">Δ</g:mi></g:math>), relevant to superconducting device technologies. Specifically, we find that the real part of the low-frequency complex conductivity, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:msub><k:mi>σ</k:mi><k:mn>1</k:mn></k:msub></k:math>, exhibits a bias-dependent reduction up to <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><n:mi>ℏ</n:mi><n:mi>ω</n:mi><n:mo>∼</n:mo><n:mn>0.1</n:mn></n:math>, 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, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><q:msub><q:mi>σ</q:mi><q:mn>2</q:mn></q:msub></q:math>, 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 oscillating superfluid density (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. These results highlight the importance of considering the Higgs mode and impurity-scattering self-energy corrections in the design and optimization of superconducting devices under a bias dc. Published by the American Physical Society 2024