Majorana splitting from critical currents in Josephson junctions

A semiconducting nanowire with strong Rashba spin-orbit coupling and coupled to a superconductor can be tuned by an external Zeeman field into a topological phase with Majorana zero modes. Here we theoretically investigate how this exotic topological superconductor phase manifests in Josephson junctions based on such proximitized nanowires. In particular, we focus on critical currents in the short junction limit ($L_{\mathrm{N}}\ll \xi$, where $L_{\mathrm{N}}$ is the junction length and $\xi$ is the superconducting coherence length) and show that they contain important information about nontrivial topology and Majoranas. This includes signatures of the gap inversion at the topological transition and a unique oscillatory pattern that originates from Majorana interference. Interestingly, this pattern can be modified by tuning the transmission across the junction, thus providing complementary evidence of Majoranas and their energy splittings beyond standard tunnel spectroscopy experiments, while offering further tunability by virtue of the Josephson effect.

Figure 1

Sketch of a SNS junction with S and N regions of finite length $L_{\mathrm{S}}$ and $L_{\mathrm{N}}$. The transmission across the junction can be controlled by the parameters ${\tau }_{\mathrm{L}\left(\mathrm{R}\right)}\in [0,1]$, which parametrize the coupling between neighboring sites at the interfaces. (b) For $B>B_{\mathrm{c}}$, the S regions become topological and the system hosts four MBSs close to zero energy at $\varphi =\pi$: two outer ${\gamma }_{1,4}$ and two inner ${\gamma }_{2,3}$ ones. The MBSs are localized at the ends of the S regions and their wave functions exhibit an oscillatory exponential decay into the bulk of S with a localization length ${\xi }_{\mathrm{M}}$. (c) Relation between normal transmission $T_{\mathrm{N}}$ and coupling parameter ${\tau }_{\mathrm{L}\left(\mathrm{R}\right)}=\tau$ for different values of the Zeeman field. Parameters: $\mathrm{\Delta }=0$ and $\mu =0.5$ meV, $L_{\mathrm{N}}=20$ nm, and ${\alpha }_{\mathrm{R}}=20$ meV nm.

Figure 2

Low-energy spectrum as a function of the superconducting phase difference in a short SNS junction for $L_{\mathrm{S}}<2{\xi }_{\mathrm{M}}$ (a) and $L_{\mathrm{S}}\gg 2{\xi }_{\mathrm{M}}$ (b) at $B=2B_{\mathrm{c}}$. The two lowest levels, almost insensitive to phase, correspond to the two outer Majoranas ${\gamma }_{1,4}$. The next two levels below the topological minigap (green dashed line), are strongly dispersive with phase, and come from the hybridization of the inner Majoranas ${\gamma }_{2,3}$, which anticross the former at $\varphi =\pi$. Parameters: $L_{\mathrm{N}}=20$ nm, (a) $L_{\mathrm{S}}=2000$ nm, (b) $L_{\mathrm{S}}=10000$ nm, ${\alpha }_{\mathrm{R}}=20$ meV nm, $\mu =0.5$ meV, and $\mathrm{\Delta }=0.25\mathrm{meV}$.

Figure 3

Supercurrent in a short SNS junction as a function of $\varphi$ at $B=2B_{\mathrm{c}}$ for $L_{\mathrm{S}}<2{\xi }_{\mathrm{M}}$ (a) and $L_{\mathrm{S}}\gg 2{\xi }_{\mathrm{M}}$ (b). Different curves correspond to different values of $\tau$. A sawtooth profile in (b) is developed when MBS wave functions within each superconducting section do not overlap, Fig. 1. This feature remains robust as $\tau$ is reduced, but not so with finite temperature (see dashed purple curve, with ${\kappa }_{B}T=0.02$ meV and $\tau =1$). Parameters: $L_{\mathrm{N}}=20$ nm, (a) $L_{\mathrm{S}}=2000$ nm, (b) $L_{\mathrm{S}}=10000$ nm, ${\alpha }_{\mathrm{R}}=20$ meV nm, $\mu =0.5$ meV, $\mathrm{\Delta }=0.25\mathrm{meV}$, and $I_0=e\mathrm{\Delta }/\hslash$.

Figure 4

Critical current $I_{\mathrm{c}}$ in a short and transparent SNS junction as a function of the Zeeman field $B$. In (a) different curves represent different values of the superconducting region length $L_{\mathrm{S}}$. As $B$ increases towards $B_{\mathrm{c}}$, $I_c$ decreases, approximately following the evolution of the gap. At the gap closing $B=B_{\mathrm{c}}$ (red vertical dashed line), $I_{\mathrm{c}}$ exhibits a nontrivial feature but remains finite. For $B>B_{\mathrm{c}}$, the critical current develops an oscillatory behavior that is related to the finite inner-outer Majorana splitting and disappear as the length $L_{\mathrm{S}}$ is increased. Notice that the magnitude of $I_{\mathrm{c}}$ for $B>B_{\mathrm{c}}$ increases with increasing $L_{\mathrm{S}}$ while its value for $B<B_{\mathrm{c}}$ remains unchanged. This unconventional dependence is a direct signature of nontrivial topology of the junction. (b) $I_c$ for fixed $L_{\mathrm{S}}=2000$ nm and increasing SO strength ${\alpha }_{\mathrm{R}}$. Observe that at large fields and in the very strong SO regime the critical current saturates to $1/2$ of its $B=0$ value. Parameters: $L_{\mathrm{N}}=20$ nm, ${\alpha }_{\mathrm{R}}=20$ meV nm, $\mu =0.5$ meV, $\mathrm{\Delta }=0.25\mathrm{meV}$, and $I_0=e\mathrm{\Delta }/\hslash$.

Figure 5

$I_{\mathrm{c}}$ for fixed $L_{\mathrm{S}}=2000$ nm and increasing values of $\tau \in [0.6,1]$, in steps of 0.01 (indicated by vertical orange arrow). In the trivial phase, $B<B_{\mathrm{c}}$, the critical current is quickly suppressed as $\tau$ is decreased towards the tunneling limit. In the topological phase the critical current is suppressed much more slowly with $\tau$, which results in a re-entrant $I_{\mathrm{c}}$ at $B_c$ for small $\tau$. For $B>B_{\mathrm{c}}$, the period of the $I_{\mathrm{c}}$ oscillations is doubled as the transparency of the junction is reduced (magenta and blue dashed lines). Parameters are the same as in Fig. 3.

Figure 6

Low-energy Andreev spectrum as a function of the Zeeman field in a short SNS junction at $\varphi =\pi$ for different values of the coupling between N and S regions $\tau$. Note that upon decreasing $\tau$ the low-energy zero-energy crossings for $B>B_{\mathrm{c}}$ merge in pairs, which results in a doubling of the period of oscillation in $I_c$. Parameters same as in Fig. 3 and $L_{\mathrm{S}}=2000$ nm.

Figure 7

Critical currents as a function of the Zeeman field in a short SNS junction. [(a) and (c)] Asymmetry in the couplings to the left and right superconducting leads, ${\tau }_{\mathrm{L}\left(\mathrm{R}\right)}$, for fixed $L_{\mathrm{S}}$. The coupling to the right S region ${\tau }_{\mathrm{R}}$ is fixed and the one for the left one varies from tunnel to the full transparent regime in steps of 0.1. This situation is expected to arise in experiments. Observe that the oscillations for $B>B_{\mathrm{c}}$ remain robust with a clear period doubling for decreasing ${\tau }_{\mathrm{L}}$. [(b) and (d)] Asymmetry in the length of the superconducting regions $L_{\mathrm{S}}=2000$ nm for different values of the coupling parameter ${\tau }_{\mathrm{L}\left(\mathrm{R}\right)}=\tau$ from tunnel to the full transparent regime in steps of 0.1. Parameters: $L_{\mathrm{N}}=20$ nm, ${\alpha }_{\mathrm{R}}=20$ meV nm, $\mu =0.5$ meV, and $\mathrm{\Delta }=0.25$ meV.

Figure 8

Low-energy Andreev spectrum as a function of the superconducting phase difference $\varphi$ at $B=2B_c$ for different values of the coupling between N and S regions $\tau$. Note that upon decreasing $\tau$ the outer (two lowest) MBSs acquire a finite energy at $\varphi =2\pi n$, for $n=0,1,2,\cdots$, while developing a crossing at $\varphi =\pi$ with their inner counterparts. This is the limit of two identical decoupled superconducting nanowires with overlapping Majoranas at each end. Parameters same as in Fig. 3 and $L_{\mathrm{S}}=2000$ nm.

Figure 9

Effect of finite temperature on the critical currents as a function of the Zeeman field in a short SNS junction. (a) Full transparent junction $\tau =1$ and increasing temperature (top-to-bottom curves). From top to bottom: red, yellow, green, blue, indigo, violet, dashed black and dash-dot black curves correspond to ${\kappa }_{\mathrm{B}}T=(0,0.01,0.02,0.04,0.08,0.16,0.2,0.4)\mathrm{\Delta }$. (b)–(d) Different panels correspond to a different temperature value for different cases of $\tau$. Observe that the reentrant behavior, oscillations for $B>B_{\mathrm{c}}$ and period doubling effect remain robust at finite temperatures, but much less than the superconducting gap. Parameters: $L_{\mathrm{N}}=20$ nm, ${\alpha }_{\mathrm{R}}=20$ meV nm, $\mu =0.5$ meV, and $\mathrm{\Delta }=0.25$ meV.

Figure 10

(a) Effect of disorder on the critical currents as a function of the Zeeman field in a short SNS junction. Notice that the oscillations for $B>B_{\mathrm{c}}$ persist even when the strength of disorder is comparable to the value of the chemical potential $\mu$. (b) Majorana localization length as a function of $B$ for different values of the SOC strength. Parameters: $L_{\mathrm{N}}=20$ nm, ${\alpha }_{\mathrm{R}}=20$ meV nm, $\mu =0.5$ meV, and $\mathrm{\Delta }=0.25$ meV.