Diode effect in Shapiro steps in an asymmetric SQUID with a superconducting nanobridge

TL;DR

The paper tackles the Josephson diode effect in an asymmetric SQUID built from a Bi$_2$Te$_2$Se SNS junction and a Nb nanobridge with a multivalued CPR. It combines dc and ac measurements with a slowly varying phase analysis of RSJ dynamics to show that Shapiro-step asymmetry can be much stronger than the dc critical-current diode effect, due to interference between the sinusoidal CPR of the SNS junction and the linear, multivalued CPR of the nanobridge; heating and the magnetic flux control the sign and strength of the diode effect through the critical phase $oldsymbol{\varphi_c}$. The work identifies two JDE mechanisms—amplitude asymmetry of the first harmonic and phase shifts among harmonics—emphasizing that multivalued CPRs enable pronounced ac-diode behavior even when dc asymmetry is weak. The findings have implications for designing superconducting diodes and for understanding nonlinear dynamics in hybrid Josephson junctions under rf drive, with practical routes proposed to enhance Shapiro-step visibility and diode strength through geometric and thermal tuning.

Abstract

We investigate the Josephson diode effect in an asymmetric SQUID consisting of a sinusoidal Josephson junction formed by a Bi$_2$Te$_2$Se flake and a superconducting Nb nanobridge with a linear and multivalued current-phase relation (CPR). Current-voltage characteristics were measured both in the absence (dc regime) and presence (ac regime) of external microwave irradiation. Our dc measurements reveal only weak critical current asymmetry (i.e. weak Josephson diode effect), while confirming the multivalued behavior of the SQUID. At the same time, the key finding of this work is the observation of strong Shapiro step asymmetry (concerning the dc current direction) in the ac regime at finite magnetic flux. This peculiarity oscillates as a function of magnetic field with the SQUID's periodicity and varies non-monotonically with the increase in microwave power. Our theoretical model shows that the pronounced Shapiro step asymmetry, despite the small diode effect in critical current, arises from the interplay between the sinusoidal and multivalued CPRs of the junctions.

Figures (8)

• Figure 1: Asymmetric SQUID structure. (a) False-colored Scanning Electron Microscopy (SEM) image of the sample. The top junction is a Superconductor/Normal metal/Superconductor (SNS) Josephson junction formed by a Bi$_2$Te$_2$Se flake, and the bottom one is a niobium nanobridge (NB). (b) An equivalent scheme of the SQUID with two different Josephson junctions. The insets show their current-phase relations (CPRs): sinusoidal function for the SNS junction and linear multivalued function for the nanobridge, where the critical current $I_{\mathrm{nb}}$ corresponds to the critical phase $\varphi_c > \pi$. In the nonstationary state with finite average voltage (positive or negative), the nanobridge CPR takes a sawtooth form, shown by red and blue segments, respectively. External radio-frequency ($rf$) irradiation can be applied to the SQUID.
• Figure 2: Direct current measurements of the sample. (a) CVC of the SQUID measured at temperature $T = 50$ mK and in an external magnetic field $B = 0.8$ G. The black arrows indicate the current sweep direction. The hysteretic behavior of the curve is associated with local overheating. The inset shows a zoomed area of the IV curve around the critical currents in absolute values of current and voltage. The critical currents differ in opposite directions by around 4 $\mu$A. (b) Dependence of $I_{c}$ of the SQUID on the external magnetic field in positive and negative current directions. The critical currents in both directions slightly differ from each other, which is a manifestation of the diode effect. A set of CVCs was measured at a fixed value of magnetic field, and the dots indicate the critical current obtained from each CVC. As can be seen, at a given magnetic field, the dots concentrate around several values of the critical current, demonstrating the multivalued CPR of the SQUID. Different critical currents correspond to switching to the resistive state from different branches of the CPR. The dashed lines represent fits to the two branches of the multivalued $I_{c}(B)$ curve for the positive critical currents with Eqs. \ref{['eq: sharp drops']} and \ref{['eq: smooth segments']}. The line labeled $I_{\text{c max}}$ corresponds to the highest branch (with the largest critical current value), while $I_{\text{c2}}$ represents the lower (second) possible critical current branch at a fixed magnetic field. (c) Dependence of the highest branch of the positive critical current over a wide range of fields. The small-scale oscillations of $I_c (B)$ (because of the flux through the whole SQUID) are modulated by the Fraunhofer dependence of the critical current through the SNS junction, shown by the dashed line. The right panels demonstrate the temperature dependence of (d) the SNS critical current $I_{\mathrm{sns}}$, (e) nanobridge critical current $I_{\mathrm{nb}}$ and (f) critical phase $\varphi_c$. Theoretical fits at high temperature are shown by dashed lines.
• Figure 3: SQUID measurements under radio-frequency (rf) irradiation at $\Phi = 3\Phi_0/4$. (a) Comparison between IV curves of the SQUID with applied microwave irradiation at frequency $f_{\text{rf}} = 2.888$ GHz (solid red) and without it (blue dashed line). (b) Zoomed area of the backward branch of the IV curve near $I_r$ with applied microwave irradiation. The lines bend slightly at the voltages $V_n = (n-1/2)V_0$ with integer $n$ and $V_0 = \Phi_0 f_{\text{rf}}$, shown by dashed gray lines. (c) The same data as in panel (b) but with the horizontal axis representing the calculated differential resistance $R = dV/dI$. The voltage is now expressed in units of $V_0$. (d) Dependence of the differential resistance on the backward branches vs current through the sample and rf power near $I_{r\pm}$.
• Figure 4: Demonstration of the diode effect in Shapiro features (thermally smeared steps) in magnetic field, which reveal themselves as minima of $R = dV/dI$. (a) Five plots of the differential resistance $R$ dependence as a function of voltage at different magnetic fields. The data are obtained by numerically differentiating experimentally measured IV curves with applied microwave irradiation at $f_{\mathrm{rf}} = 3.754$ GHz and power $P_{\mathrm{rf}} = 1$ mW. The red and blue lines correspond to the positive and negative current $I_{\text{dc}}$ through the SQUID, respectively. The dashed gray lines indicate the reference value $R_0$. (b) Results of the theoretical calculations within the RSJ model [Eqs. \ref{['eq:J1beg']} and \ref{['eq:J2beg']}] with $T = 1.8$ K, $\varphi_{c} = 30$, $I_{\mathrm{sns}} = 0.3\,\mu$A, $I_{\mathrm{nb}} = 12\,\mu$A, $I_{\mathrm{ac}} = 12\,\mu$A. The theory qualitatively reproduces the experimentally observed features. However, the horizontal axis in the theoretical subplots had to be manually shifted by $1/2$, see Appendix \ref{['appendix:Shapiro steps shift']} for discussion.
• Figure 5: Quantitative analysis of the Shapiro features and comparison between the experimental data, shown in the top row, and the theoretical results [obtained from Eq. \ref{['eq:R_diff']}], shown in the bottom. (a) and (d) Dependence of the Shapiro features depths $\Delta R_{\pm}$ on the magnetic field at $|V|/V_0 = 6.5$ (corresponding to $n = \pm 7$ in the theoretical model). The dots represent experimental data for positive (red) and negative (blue) current directions, collected with an applied microwave signal at a frequency $f_{\text{rf}} = 3.754$ GHz and a generator power $P_{\text{rf}} = 1$ mW. The solid lines indicate fits with a sinusoidal function, which show good agreement with the experimental points. The theoretical results are also close to a sinusoidal function. (b) and (e) Fraunhofer envelope of the Shapiro features depths asymmetry $\Delta R_{+} - \Delta R_{-}$ over a wide range of the fields. The oscillations of the diode effect strength with SQUID periodicity at small magnetic fields and Fraunhofer envelope at large magnetic fields demonstrate that the asymmetry of the Shapiro features arises from the interplay of the two junctions. (c) and (f) Dependence of $\Delta R_{\pm}$ on microwave power at a fixed magnetic field $B = 0.8$ G, which corresponds to a flux $\Phi/\Phi_0 \approx 0.77$. The curves exhibit a nonmonotonic dependence with periodic changes in the sign of the diode effect. We attribute this behavior to the heating effect of microwave irradiation, which manifests itself through the temperature dependence of $\varphi_{c}$. The crossings of the curves correspond to the condition $\sin \varphi_{c}(T) = 0$.