Revealing THz optical signatures of Shiba-state-induced gapped and gapless superconductivity

TL;DR

This work addresses how magnetic impurities generate Shiba states in conventional $s$-wave superconductors and can drive a transition to a gapless superconducting state while preserving superconducting order. It develops a fully self-consistent renormalization framework solved via a sixth-order polynomial root-finding, enabling the complete $T$-$n_i$ phase diagram and equilibrium Green functions. The authors compute both linear and nonlinear THz optical responses: linear absorption remains finite in the gapless phase due to impurity bands, while the nonlinear Higgs-mode dynamics remain coherent and dominate the response even with gapless quasiparticles. The study provides a quantitative, generalizable framework for diagnosing gapless superconductivity and suggests THz spectroscopy as a bulk probe of Shiba states and related condensate dynamics.

Abstract

We report a fully self-consistent calculation of the complex renormalization by exchange interactions and hence the complete phase diagram of conventional $s$-wave superconductors with magnetic impurities as well as the related physical properties including the optical response. We show that a small amount of magnetic disorder can drive the system into a gapless superconducting state, where the single-particle excitation gap vanishes whereas the superconducting order parameter $Δ_0$ remains finite. In this phase, the linear optical conductivity exhibits a finite absorption over the low-frequency regime, particularly for photon energies below the conventional threshold $2|Δ_0|$, even at low temperatures, in sharp contrast to the gapped state. The nonlinear response, however, remains coherent and is dominated by the Higgs-mode dynamics rather than gapless quasiparticle background. These findings reveal a fundamental distinction between dissipative single-particle excitations and coherent collective dynamics of the condensate, a feature likely general to other gapless superconductors, and introduces a fundamentally different detection scheme, using THz spectroscopy to probe the signatures of Shiba states.

Figures (7)

• Figure 1: (a) Temperature dependence of SC gap, (b) single-particle-excitation density of states, and (c) temperature dependence of superfluid density, at different impurity concentrations $n_i$. The inset in (b) provides an enlarged view of the data at $n_i/n_0=0.1,0.4,0.7$. (d) The calculated $T$-$n_i$ phase diagram, including the gapped SC [$\Delta_0\ne0$ and $\rho(0^+)=0$] and gapless SC [determined by $\Delta_0\ne0$ and $\rho(0^+)\ne0$] states as well as the normal state ($\Delta_0=0$). The orange chain curve in (d) denotes the critical temperature $T_c(n_i)$ obtained from the critical theory by Abrikosov and Gorkov osti_4097498gulian2002nonequilibrium, while the light-blue dashed curve in (d) denotes the temperature $T^*$ at which $\gamma_s(n_i)=0.85\Delta_0(T^*,n_i)\exp(-\pi/4)$. Here, $n_0/(2\pi{D})=2.14$ meV. Other used parameters including pairing potential and Debye cutoff frequency are addressed in Table \ref{['parameter']}.
• Figure 2: (a) Zero-temperature optical absorption at different impurity concentrations $n_i$. (b) Schematic illustration of the dominant interband transitions (gray arrows), which are present at all temperatures below $T_c$, including $T=0$, and the additional thermally activated transitions that appear only at finite temperatures (red arrows). The interband transition between impurity bands is usually weak and not shown here. (c) and (d) Optical absorption of gapped and gapless states at different $T$, respectively. The inset in (c) provides an enlarged view of the data at 4 K. Here, $n_0/(2\pi{D})=2.14$ meV. Other used parameters are addressed in Table \ref{['parameter']}.
• Figure 3: (a) Dynamic simulation of gap dynamics $\delta\Delta(t)$ and (b) diagrammatic formulation of the pole structure $|\Pi^{-1}(\Omega)|$ in Higgs-mode Green function as a function of $\Omega$, at different $n_i$ for $T=0$. Inset of (a): the used THz field in the dynamic simulation. (c) Oscillatory component $\delta\Delta_{\rm oc}(t)$ in the gap dynamics $\delta\Delta(t)$ at several $n_i$. (d) $|\Pi^{-1}(\Omega)|$ at fixed driving frequency $\Omega = 2\Omega_o$ and the maximum of $|\delta\Delta_{\rm oc}(t)|/\Delta_0$ (maximum magnitude of oscillation) during the temporal evolution, both plotted as functions of $n_i$ and normalized for comparison. The oscillatory part $\delta\Delta_{\rm oc}(t)$ is isolated by directly removing the slowly varying background component of $\delta|\Delta(t)|$ in the simulation. We approximate response coefficient $\lambda$ as a constant via setting the dimensionless driving strength to $\lambda e^2A_0^2/(mD)=0.01$. Other used parameters are addressed in Table \ref{['parameter']}.
• Figure 4: Feynman diagram of the calculation of the Higgs-mode Green function. The dashed lines represent the SC pairing interaction $g$. The first diagram corresponds to the bare Higgs-mode Green function, i.e., the lowest-order contribution without interaction corrections. The second diagram depicts the first-order correction due to the pairing interaction, illustrating how the Higgs-mode propagator is renormalized by the interaction. The full Higgs-mode Green function requires the resummation of an infinite series of pairing interaction diagrams. This Dyson-type resummation leads to the full expression of the Higgs-mode propagator $D_{\rm H}(\Omega)$, from which the energy dispersion and damping of the mode can be extracted.
• Figure 5: (a): Temperature dependence of the spatially averaged SC gap. (b)–(i): Spatial maps of the SC gap $\Delta({\bf R})$ (in meV) at different temperatures. The simulation incorporates 600 randomly distributed magnetic impurity domains on a $61\times61$ two-dimensional Cartesian grid with the periodic boundary conditions. In our simulation, we set the constant ${\lambda_d}/(4Dm_e)=(0.12)^2~\mu\text{m}^2$, which is a reasonable value for many disordered or thin-film $s$-wave superconductors.