Diode effect in microwave irradiated Josephson junctions with Yu-Shiba-Rusinov states

TL;DR

Microwave irradiation of Josephson junctions hosting Yu-Shiba-Rusinov states can induce a diode-like nonreciprocity in the critical currents when both particle-hole-normal symmetry and inversion symmetry are broken. The authors develop a Floquet-Keldysh framework to reveal a phase-independent current component $I_{con}$ that shifts the current-phase relation, enabling $I_c^+ \neq I_c^-$ and tunable diode behavior via the microwave amplitude $α$ and frequency $ω_r$, with strongest effects near YSR resonances. A perfect diode is theoretically achievable when $|I_{con}|=|I_{sin}|$, particularly close to resonances $ω_r ≈ ε_{0,T} ± ε_{0,S}$. The results highlight YSR platforms as robust, tunable routes to field-free Josephson diodes and provide practical guidance for experimental realization and interpretation in related setups.

Abstract

We investigate the critical current in microwave-irradiated Josephson junctions hosting Yu-Shiba-Rusinov states due to magnetic impurities. Under two conditions, namely, (i) the breaking of particle-hole symmetry in the normal sense by non-zero potential scattering, and (ii) the breaking of inversion symmetry either by unequal magnitudes of potential scattering and/or magnetic moments, microwave irradiation induces an additional phase-independent contribution to the current. This leads to asymmetric critical currents for opposite current polarities, an effect absent in the same junction without microwave irradiation. The asymmetry is highly tunable via the microwave amplitude and frequency, and we may even achieve perfect asymmetry where the critical current vanishes for one polarity, akin to a perfect diode. While Yu-Shiba-Rusinov states provide the ideal platform for a pronounced asymmetry, we find that as long as the two conditions (i) and (ii) above are met, our proposal does not necessarily depend upon them.

Figures (5)

• Figure 1: (a) Illustration of a JJ with arbitrarily oriented spins on the superconducting tip ($T$) and substrate ($S$), with gaps $\Delta_T$ and $\Delta_S$. $J_{T/S,k}$ and $K_{T/S}$ denote the magnetic moment and potential-scattering terms. For simplicity, we set $\Delta_T=\Delta_S$. (b) Illustration of the diode effect. The inset shows the CPR (arbitrary units) without (red) and with (blue) microwave radiation at zero voltage. While the former is centered at zero, the microwave-induced phase-independent contribution shifts it upward, resulting in two different effective critical currents $I_c^\pm$. Note that $I_c^+=I_c^-$ in the absence of microwaves. The main panel shows the current-voltage characteristics (arbitrary units). For currents in the range $[I_c^-, I_c^+]$, the JJ remains voltage-free, while outside this range it switches to a finite-voltage state. We only show the switching branch of the background current, leaving out the Shapiro steps and the retrapping current.
• Figure 2: Numerically obtained currents for $\zeta=5\Delta_T=5\Delta_S$, $\mu=0$, $\mathcal{T}=0.002\zeta$ (normal-state transparency $\sim10^{-5}$Cuevas1996), $\Gamma=2\times10^{-3}\Delta_T$, and radiation strength $\alpha=0.5$. Currents are normalized by the normal-state resistance $R_N$. Left panels show the magnitudes of the critical currents $I_c^{\pm}$ as functions of $\omega_r$. Central panels display the corresponding $I_{\cos}$. Solid lines denote the full microscopic result, while dotted lines show the Tien-Gordon prediction obtained from the IVC [Eq. \ref{['ITG']}]. Right panels show the IVC $I_{\mathrm{dc},V}$, where the dotted curve for negative voltages represents $-I_{\mathrm{dc},V}(-V)$; non-overlapping curves indicate a non-reciprocal response. Dashed lines mark the YSR transition energies $\epsilon_{0,T}+\epsilon_{0,S}$ (gray) and $|\epsilon_{0,T}-\epsilon_{0,S}|$ (green) Huang2021tdnote, where $\epsilon_{0,T/S}$ denotes the YSR energy in lead $T/S$. Panels (a-c) correspond to an inversion-symmetric junction with $K_T=K_S$ and $|\mathbf{J}_T|=|\mathbf{J}_S|$ at a relative angle $\pi/2$, yielding $I_c^+=I_c^-$ and no DE. In (d-f), inversion symmetry is broken by $K_T\neq K_S$ with $\mathbf{J}_T=\mathbf{J}_S$, resulting in a DE driven by a finite phase-independent current. Panels (g-i) show the case $K_T=K_S\neq0$ with unequal magnetic moment magnitudes, which also breaks inversion symmetry and produces a DE.
• Figure 3: Numerically obtained current for the same parameters as in Fig. \ref{['Fig2']}(d-f), but with varying $\alpha$ and a fixed radiation frequency $\omega_r=0.3$, close to the YSR transition resonance at $\omega_r=0.368$ (black dashed line in Fig. \ref{['Fig2']}(d-f)). The currents are normalized by the numerically obtained normal-state resistance $R_N$. (a) shows the radiation-dressed CPR. For $\alpha=1.548$, $I_{\text{con}}$ is slightly larger than $I_{\sin}$, almost resulting in a perfect diode with $I_c^-\approx 0$. (b) shows $I_c^{\pm}$. We define $I_c^+=\text{max.}[I(\phi)]$ and $I_c^-=-\text{min.}[I(\phi)]$. The sharp corners occur when $I_{\sin}(\alpha)$ vanishes, which causes one of the two critical currents, $I_c^{\pm}$, to vanish, depending on the sign of $I_{\text{con}}$. E.g., when $I_{\text{con}}<0$ and $I_{\sin}=0$, $I_c^+=0$ whereas $I_c^-=|I_{\text{con}}|$. The inset zooms into the region where a perfect diode is obtained, with $I_c^-=0$. (c) shows $I_{\sin}$ and $I_{\text{con}}$ as a function of $\alpha$. The point where the cross over yields a perfect diode.
• Figure 4: Numerically obtained current for the same parameters as in Fig. \ref{['Fig2']}(d-f), but with varying $\alpha$ as well as radiation frequency. The currents are normalized by the numerically obtained normal-state resistance $R_N$. In all plots, we denote $\omega_r=\epsilon_{0,T}+\epsilon_{0,S}$ and $\omega_r=(\epsilon_{0,T}+\epsilon_{0,S})/2$, corresponding to the two leading resonances (cf. Eq. \ref{['ITG']}) arising from the YSR transition at $\epsilon_{0,T}+\epsilon_{0,S}$, as shown in Fig. \ref{['Fig2']}(d-f). (a) shows $I_{\text{con}}$, revealing the dominant peak at $\omega_r=\epsilon_{0,T}+\epsilon_{0,S}$ for small $\alpha$, followed by a subdominant peak at $\omega_r=(\epsilon_{0,T}+\epsilon_{0,S})/2$ for larger values of $\alpha$. This is because with increasing alpha, the higher order Bessel prefactors in Eq. \ref{['ITG']} start contributing. (b) shows $I_{\sin}$, which is revealing a strong microwave-renormalization with the same resonant features as in (a). The oscillatory dependence on $\alpha$ depends strongly on $\omega_r$, and deviates substantially from the AA ($\sim J_0(\alpha)$). (c) Diode efficiency $\eta_{\text{DE}}$. We obtain high values where $I_{\text{con}}$ is large, and matches $I_{\sin}$ in magnitude. Remarkably, we even achieve $\eta_{\text{DE}}=1$, a perfect diode, near the resonance at $\omega_r=\epsilon_{0,T}+\epsilon_{0,S}$.
• Figure 5: Numerically obtained tunnel IVC, $I_{\text{dc},V}$, in a clean tunnel JJ with $\Delta_T=1.0$, $\Delta_S=1.5$, $\Gamma=0.02$, with varying Fermi-level $\mu$ (equal for both leads $\mu_T=\mu_S=\mu$). The bandwidth is $4\zeta$. The currents are normalized by the numerically obtained normal-state resistance $R_N$. For negative voltages, the dotted red line shows $-I_{\text{dc}, V}(-V)$. When it does not match the blue line, the current is non-reciprocal. (a) shows IVC for $\mu=0$, revealing a perfectly odd IVC as we retain PHN symmetry. (b) IVC for $\mu=\zeta$, and (c) IVC for $\mu=1.5\zeta$. They reveal that the non-reciprocity increases commensurately with $\mu$ (see insets for a zoomed-in view).