Josephson diode effects in twisted nodal superconductors

Recent Josephson tunneling experiments on twisted flakes of high-$T_c$ cuprate superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ revealed a nonreciprocal behavior of the critical interlayer Josephson current, i.e., a Josephson diode effect. Motivated by these findings we study theoretically the emergence of the Josephson diode effect in twisted interfaces between nodal superconductors, and highlight a strong dependence on the twist angle $\theta$ and damping of the junction. In all cases, the theory predicts diode efficiency that vanishes exactly at $\theta ={45}^{\circ }$ and has a strong peak at a twist angle close to $\theta ={45}^{\circ }$, consistent with experimental observations. Near ${45}^{\circ }$, the junction breaks time-reversal symmetry $\mathcal{T}$ spontaneously. We find that for underdamped junctions showing hysteretic behavior, this results in a dynamical Josephson diode effect in a part of the $\mathcal{T}$-broken phase. The direction of the diode is trainable in this case by sweeping the external current bias. This effect provides a sensitive probe of spontaneous $\mathcal{T}$ breaking. We then show that explicit $\mathcal{T}$-breaking perturbations with the symmetry of a magnetic field perpendicular to the junction plane lead to a thermodynamic diode effect that survives even in the overdamped limit. We discuss an experimental protocol to probe the double-well structure in the Josephson free energy that underlies the tendency towards spontaneous $\mathcal{T}$ breaking even if $\mathcal{T}$ is broken explicitly. Finally, we show that in-plane magnetic fields can control the diode effect in the short junction limit, and predict the signatures of explicit $\mathcal{T}$ breaking in Shapiro steps.

Figure 1

(a) Two flakes of a $d$-wave superconductor form a twisted $c$-axis Josephson junction. In the absence of time reversal $\mathcal{T}$ and inversion $\mathcal{P}$, the critical Josephson currents for the two current directions are allowed to be unequal. (b) Schematic phase diagram implied by the GL theory. Dynamical Josephson diode effect (d-JDE) is found to occur inside the spontaneously $\mathcal{T}$-broken phase.

Figure 2

Schematic of the phase dynamics for $-J_0<J_c$ and ${\beta }_c\gg 1$. (a), (b) for $2J_{c2}>J_{c1}cos\left(2\theta \right)>0.79J_{c2}$ [see Eq. (16)]; (c), (d) for $0.79J_{c2}>J_{c1}cos\left(2\theta \right)$. In (a) and (b) one observes that $\varphi$ can relocate to the other local potential minimum at $J^{\prime }$, when the original minimum becomes unstable. However, energy conservation allows unbounded motion only in case (d).

Figure 3

(a) Characteristic currents for the dynamical diode effect from the RCSJ model (11) as a function of $J_{c1}\left(\theta \right)$ for the inital state of $\varphi$ taken to be at the right potential energy minimum (see Fig. 2). $J_{c1}\left(\theta \right)=0$ corresponds to $\theta ={45}^{\circ }$. Dashed black line shows the $J_0^{\prime }$ value, where the initial potential minimum (taken to be at $\varphi ={\varphi }_0>0$) becomes unstable [see Fig. 2]. The black dotted and solid lines show the normalized critical current for positive and negative current bias [see Figs. 2 and 2] in the ${\beta }_c\to \infty$ limit. Diode effect ($I_c^{+}\ne I_c^{-}$) is present for $J_{c1}$ less than a critical value (16). Colored lines represent $I_c^{-}$ from the numerical solution of Eq. (11) for finite ${\beta }_c$ ($I_c^{+}$ remains the same as for ${\beta }_c\to \infty$). The transition from diode to nondiode regime remains abrupt for all ${\beta }_c$, but shifts to lower $J_{c1}\left(\theta \right)$ values. (b) Diode efficiency ${\eta }_{\text{dyn}}$, Eq. (17), as a function of twist angle $\theta$ for $J_{c1}(\theta =0)=0.1J_{c2}$. Note that the system is in a TRSB state already for $\theta >39.2^{\circ }$ for these values.

Figure 4

Critical value of the first-harmonic critical current for the onset of the diode effect. For large McCumber parameter ${\beta }_c$, the result agrees with the analytical ${\beta }_c\to \infty$ limit (16). Inset shows the behavior near the onset of hysteretic behavior of the RCSJ model ${\beta }_c^{\mathrm{hyst}}$.

Figure 5

Thermodynamic Josephson diode effect as a function of twist angle $\theta$, obtained from the Ginzburg-Landau description in Eq. (7) (equivalent to the overdamped regime ${\beta }_c\ll 1$ of the RCSJ model). We take $J_{c2}=0.1J_{c1}$, which sets the critical twist angle for the appearance of topological superconductivity to ${\theta }_c=39.2^{\circ }$ through Eq. (3), denoted by vertical black dashed lines. (a) Critical Josephson currents $I_c^{\pm }$ develop an asymmetry in the presence of the $\mathcal{T}$-breaking term $J_m$ (here $J_m=\sqrt{2}{J}_{c2}$). (b) The corresponding difference $\mathrm{\Delta }{I}_c=I_c^{+}-I_c^{-}$ and figure of merit $\eta$, defined in Eq. (19), exhibit a sharp increase upon entering the spontaneous $\mathcal{T}$-breaking phase. The diode effect vanishes at $\theta ={45}^{\circ }$, where the diode polarity reverses. (c) The figure of merit $\eta$ in the $\theta \text{−}{J}_m$ plane. The maximal diode efficiency $\left|\eta \right|=\frac{1}{3}$ lies on the $J_m/J_{c1}=\pm cot2\theta$ curves (the orange dashed line shows the “$-$” solution).

Figure 6

Thermodynamic Josephson diode effect as a function of the $\mathcal{T}$-breaking perturbation $J_m$, obtained from the Ginzburg-Landau description in Eq. (7) (equivalent to the overdamped regime ${\beta }_c\ll 1$ of the RCSJ model). Various curves denote $J_{c2}/J_{c1}$ between 0 and 0.1, corresponding to ${\theta }_c$ ranging from ${45}^{\mathrm{o}}$ (vanishing topological region) to $39.2^{\circ }$. (a) For small twist angles, the efficiency parameter $\eta$ shows a broad and low peak centered around a large optimal value $J_m^{\mathrm{opt}}$. (b), (c) For larger twist angles near ${45}^{\circ }, \eta$ peaks near its theoretical maximum of $\frac{1}{3}$ for the value of $J_{c2}/J_{c1}$ that best approximates ${\theta }_c\approx \theta$ through Eq. (3). The $\eta$ peaks become sharper, and occur at a decreasing optimal value $J_m^{\mathrm{opt}}$, when $\theta$ approaches ${45}^{\circ }$.

Figure 7

Critical value of the first-harmonic Josephson current $J_{c1}\left({\theta }_{cr}^m\right)$ for the emergence of two potential minima in the Josephson energy $U_m\left(\varphi \right)$ (23) for $J_0=0$ (illustrated by insets) as a function of TRSB Josephson current $J_m$.

Figure 8

(a) Illustration of the Josephson potential (23) with TRSB term $J_m<J_{c2}$ in the absence of external current. (b) Four characteristic current values $I_{L,R}^{\pm }$ for ${\beta }_c\to \infty$ and $J_m/J_{c2}=0.5$. The colored lines show the values where the $\varphi$ can actually escape the local minimum, leading to the onset of the voltage. The black points mark all values where one of the local minima ceases to exist, not necessarily leading to the voltage onset. (c) Maximum diode efficiency [obtained by varying $J_{c1}$ for each value of $J_m$] for the thermodynamic and dynamic diode effect.

Figure 9

(a) Three current sweeping protocols [Eq. (24)] for $t_c=2000t_0$. (b), (c) Voltage (averaged over $10t_0$) from numerical solution of Eq. (11) for ${\beta }_c=5$ (b) and ${\beta }_c=80$ (c). Voltage is shown for the second cycle of the current sweeping to suppress the effects of initial conditions. The time at which voltage first appears corresponds to a critical current value in (a). The two half-sweep directions (purple line) lead to identical critical current amplitudes, as is expected in the absence of explicit TRSB. For the full-sweep protocol (green dashed line), on the other hand, the current is sweeped in the opposite direction with respect to the position of the phase at zero current (see Fig. 2). The difference in full-sweep and half-sweep current demonstrates the bistability of the Josephson energy.

Figure 10

(a)–(d) Voltage (averaged over $10t_0$) from numerical solution of Eq. (11) with added TRSB term $J_m$ (values given in the panels) and current driving shown in Fig. 9. Insets in (d) marked by black dashed line show magnified transition region (not to scale). In contrast to Fig. 9, the half-sweep critical current (related to the voltage onset time) depends on the current polarity (red solid line for “$+$,” blue solid line for “$-$”), reflecting the explicit TRSB. The full-sweep critical current (green dashed line) coincides with one of the two half-sweep values when the first harmonic is large (a) or extremely small (d). For other values of $J_{c1}$ (b), (c), the full sweep shows values of critical current distinct from both (c) or at least one (b) of the half-sweep protocols. The latter observation can be used to demonstrate bistability of the Josephson potential even if TRSB is explicitly broken.

Figure 11

Fraunhofer interference patterns, Shapiro steps and the JDE. (a), (b) In the presence of a dominant second harmonic and $J_m=0.25J_{c2}$ (with $J_{c2}=0.2$ in units of $J_{c1}$), the envelopes of the positive and negative critical currents are different. At half-integer flux quanta, the curves intersect and the diode effect vanishes because the contribution from the second harmonic goes to zero due to its $\pi$ periodicity. (c), (d) Close to ${45}^{\circ }$, the normalized Shapiro step heights (indicated with the color scale) are asymmetric about zero voltage.