Josephson diode effect with Andreev and Majorana bound states

TL;DR

This work demonstrates that a Zeeman field component parallel to the spin-orbit axis induces an asymmetric phase-dependent spectrum in short Josephson junctions, yielding nonreciprocal Josephson transport and a Josephson diode effect (JD) across both trivial (ABS) and topological (Majorana bound states) regimes. By modeling a three-region SNS junction with Rashba SOC and homogeneous B field, the authors map how JD efficiency, critical currents, and current–phase curves track the closing and reopening of bulk gaps and the formation of Majorana modes; longer superconductors amplify Majorana nonlocality and strengthen the JD. In the tunneling regime, the JD is Majorana-only, offering a robust signature of topological physics and a route to Majorana-based diodes. The results also show finite-temperature robustness of the JD and discuss multichannel extensions, highlighting practical pathways to identify topological phases and implement Majorana-driven diodes in nanoscale superconducting circuits.

Abstract

Superconductor-semiconductor hybrids are useful for realizing the Josephson diode effect, where nonreciprocity in the supercurrents occurs due to the interplay of the Josephson effect and applied magnetic fields. These junctions can host Andreev and Majorana states with the same ingredients, though their interplay with the Josephson diode effect is unclear. In this work, we consider short Josephson junctions based on superconductor-semiconductor systems under homogeneous Zeeman fields and investigate the Josephson diode effect in the presence of Andreev and Majorana states. Under generic conditions, the Zeeman field component parallel to the spin-orbit axis promotes an asymmetric low-energy spectrum as a function of the superconducting phase, which persists in the trivial and topological phases hosting Andreev and Majorana bound states, respectively. Interestingly, this asymmetry creates supercurrents that are not odd functions of the phase difference, leading to a nonreciprocal behaviour and the Josephson diode effect. We show that the Josephson diode effect is particularly promoted under the presence of both zero-energy Andreev and Majorana bound states, revealing that Josephson diodes can be realized in the trivial and topological phases of superconductor-semiconductor hybrids. We then demonstrate that the Zeeman field evolution of the diode's efficiencies can map the topological phase transition and the formation of Majorana bound states via an oscillatory behavior that becomes more visible in long superconductors. While Josephson diodes generally exist in the trivial and topological phases of Josephson junctions, we discover that in the tunneling regime only a Josephson diode effect in the topological phase remains due to the finite contribution of Majorana bound states. Our findings clarify the Josephson diode effect and aid in realizing Majorana-only Josephson diodes.

Figures (11)

• Figure 1: (a) A JJ formed by a nanowire with SOC and an homogeneous magnetic field. The left and right sides of the nanowire denoted by S$_{\rm L,R}$ (orange regions) are of finite length and contain proximity induced superconductivity from conventional superconductors, while the finite middle light blue region is left without superconductivity and denoted by N. The JJ is formed along $x$, while the SO axis lies along the $y$-direction and is indicated by the blue arrow. The Zeeman field $B$ results from an applied magnetic field throughout the JJ and makes an angle $\theta$ with the $x$-axis, thus containing components that are parallel and perpendicular to the SO axis. (b) Positive and negative energies closest to zero versus momentum for distinct values of $\theta$, indicating $\Delta_{1(2)}$ at zero momentum (positive Fermi momenta $+k_F$). Here, the Zeeman field is $B=1.2B_c$, for which $\Delta_2=0$ at $\theta_c \approx 0.1216\pi$ indicated by red curve. (c,d) Gaps $\Delta_{1,2}$ as a function of $\theta$ and $B$, with $\Delta_{2}$ at $+k_F$ with respect to the Fermi level. Here, $B=B_c$ marks the topological phase transition $\Delta_1=0$. Parameters: $\alpha_R = 20$meVnm, $\Delta = 0.25$meV, and $\mu = 0.5$meV.
• Figure 2: (a-f) Low-energy spectrum as a function of the superconducting phase difference $\phi$ in the trivial (a-c) and topological phases (d-f) at distinct values of $\theta$. The four energy levels closest to zero energy $E=0$ are indicated in orange color. The green and magenta horizontal lines represent the inner ($\Delta_1$) and outer ($\Delta_2$) gaps, respectively, obtained from the bulk model given by Eq. (\ref{['eq:ham_BDG']}). In the bottom panels $\Delta_{1}$, appears at higher energies, that is, $\Delta_{1}>\Delta$. $\Delta_{2}$ in (f) lies below $E=0$. Parameters: $L_{\rm S} = 2\,\mu$m and $L_{\rm N}=20$ nm, $\tau=1$, while the rest of parameters are the same as in Fig. \ref{['Fig1']}.
• Figure 3: (a-f) Low-energy spectrum as a function of the magnitude of the Zeeman field B for distinct values of $\theta$ at $\phi=0$ (a-c) and $\phi \neq 0$ (d-f). The choice of $\phi \neq 0$ in (d-f) corresponds to $\phi$ where the four MBSs split in energy, see Fig. \ref{['Fig2']}(d-f). In (f), $\phi = \pi$ is selected since the exact splitting point of the MBSs is unclear, see Fig. \ref{['Fig2']}(f). The green and magenta curves show the Zeeman dependence of $\Delta_1$ and $\Delta_2$, respectively. (g-i) illustrate the dependence of the low-energy spectra on $L_\mathrm{S}$, with $B=1.8B_c$. Parameters: $L_{\rm S} = 2\,\mu$m and $L_{\rm N}=20$ nm, $\tau=1$, while the rest of parameters are the same as in Fig. \ref{['Fig1']}.
• Figure 4: (a,b) Josephson currents as a function of the superconducting phase difference $I(\phi)$ in the trivial (a) and topological (b) phases for distinct $\theta$. The Josephson currents in (a,b) correspond to the phase-dependent spectrum shown in Fig. \ref{['Fig2']}. (c-e) Contribution of the ABSs ($I_{\rm ABS}$) and quasicontinuum ($I_{\rm quasi.}$) to the total Josephson current $I_\mathrm{total}$ corresponding to (a) in the trivial phase. (f-h) Contribution of the MBSs ($I_{\rm MBS}$) and quasicontinuum ($I_{\rm quasi.}$) to the total Josephson current $I_\mathrm{total}$ corresponding to (b) in the topological phase. Parameters: $L_{\rm S} = 2\,\mu$m and $L_{\rm N}=20$ nm, $\tau=1$, $I_0 = e\Delta/\hbar$, $T=0$, while the rest of parameters are the same as in Fig. \ref{['Fig2']}.
• Figure 5: (a,b) Critical currents $I_{c}^{\pm}$ as a function of the Zeeman field amplitude $B$ for distinct values of $\theta$, and at $L_{\rm S}=2$$\mu$m and $L_{\rm S}=6$$\mu$m. (c,d) The same as in (a,b) but at fixed $\theta=\pi/12$ and $\theta=\pi/6$ for different $L_{\rm S}$. Parameters: $L_{\rm N}=20$ nm, $\tau=1$, $I_0 = e\Delta/\hbar$, while the rest of parameters are the same as in Fig. \ref{['Fig2']}.