Majorana bound states in double nanowires with reduced Zeeman thresholds due to supercurrents

We study the topological phase diagram of a setup composed of two nanowires with strong Rashba spin-orbit interaction subjected to an external magnetic field and brought into the proximity to a bulk $s$-wave superconductor in the presence of a supercurrent flowing through it. The supercurrent reduces the critical values of the Zeeman energy and crossed Andreev superconducting pairing required to reach the topological phase characterized by the presence of one Majorana bound state localized at each system end. We demonstrate that, even in the regime of the crossed Andreev pairing being smaller than the direct pairing, a relatively weak magnetic field drives the system into the topological phase due to the presence of the supercurrent.

Figure 1

Schematics of the double-NW setup consisting of two semiconducting Rashba NWs (blue strips), aligned along $x$ axis, placed in proximity to a bulk $s$-wave superconductor (green strip). The two NWs are labeled by the index $\tau =1\left(\overline{1}\right)$ corresponding to the upper (lower) NW. An external magnetic field $\mathit{B}$ is applied along the $x$ axis and the SOI vectors ${\mathit{\alpha }}_{\tau }$ point along the $z$ axis in both NWs. The supercurrent ${\mathit{I}}_s$ along $x$ axis is generated in the bulk $s$-wave superconductor, which causes a spatial gradient in the phases of the proximity-induced direct and crossed Andreev pairing amplitudes.

Figure 2

Phase diagram of the double-NW system as function of Zeeman energy ${\mathrm{\Delta }}_Z/\mathrm{\Delta }$ and crossed Andreev pairing amplitude ${\mathrm{\Delta }}_c/\mathrm{\Delta }$. The phase boundaries between different gapped phases in the absence of supercurrent ($\delta =0$, black dashed line) and in the presence of supercurrent ($\delta /\mathrm{\Delta }=0.04$, blue solid line) are found from Eq. (12). The number of BSs is indicated directly in the plot and refers to a given end of the double NW. For finite supercurrent, the topological phase characterized by the presence of one MBS extends to weaker magnetic fields and weaker crossed Andreev pairings. For $\delta =0$, the one-FBS phase becomes a two-MBS phase, which is also the case for ${\mathrm{\Delta }}_c=0$ and ${\mathrm{\Delta }}_Z>{\mathrm{\Delta }}_Z^{+}$ (green solid line) and finite $\delta$. In the absence of supercurrent, $\delta =0$, and in the absence of magnetic field (crossed Andreev pairing), the bulk gap at $k=0$ closes at ${\mathrm{\Delta }}_c=\mathrm{\Delta }$ (${\mathrm{\Delta }}_Z=\mathrm{\Delta }$). For finite $\delta$ and in the absence of magnetic field (crossed Andreev pairing), the bulk gap at $k=0$ closes twice at ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_c^{\pm }$ (${\mathrm{\Delta }}_Z={\mathrm{\Delta }}_Z^{\pm }$). In the presence of supercurrent, the system is in the gapless phase at ${\mathrm{\Delta }}_Z=0$ and ${\mathrm{\Delta }}_c^{-}\le {\mathrm{\Delta }}_c\le {\mathrm{\Delta }}_c^{+}$ (orange solid line). The size of the bulk gap for all parameter values is discussed in Appendix pp2. Other parameters are fixed as ${\mu }_1={\mu }_{\overline{1}}=0$, $\mathrm{\Gamma }=0$, $E_{\text{so},1}/\mathrm{\Delta }=6.25$, $\beta =1.4$.

Figure 3

Bulk energy spectrum $E/\mathrm{\Delta }$ as a function of momentum $k/k_{\text{so},1}$ for ${\mathrm{\Delta }}_Z=0$ and different values of ${\mathrm{\Delta }}_c$. In the trivial phase, ${\mathrm{\Delta }}_c<{\mathrm{\Delta }}_c^{-}$ (${\mathrm{\Delta }}_c/\mathrm{\Delta }=0.1$), the spectrum is gapped for all momenta (green dot-dashed line). For ${\mathrm{\Delta }}_c^{-}<{\mathrm{\Delta }}_c<{\mathrm{\Delta }}_c^{+}$ (${\mathrm{\Delta }}_c/\mathrm{\Delta }=0.9$), the spectrum is gapped at $k=0$ but there is no gap at some finite momentum (red solid line). Thus, the system is in the gapless phase without any BSs in the spectrum. In the one-FBS phase, $\mathrm{\Delta }>{\mathrm{\Delta }}_c^{+}$ (${\mathrm{\Delta }}_c/\mathrm{\Delta }=1.7$), again, the energy spectrum is gapped for all values of $k$ (blue dashed line). Other parameters are the same as in Fig. 2.

Figure 4

Phase diagram of the double NW as function of Zeeman energy ${\mathrm{\Delta }}_Z/\mathrm{\Delta }$ and phase gradient $\delta /\mathrm{\Delta }$ for fixed crossed Andreev pairing (${\mathrm{\Delta }}_c/\mathrm{\Delta }=0.5$). The number of BSs in the corresponding phases is indicated directly in the figure. If the supercurrent is too strong $\delta >{\delta }_g$, the system is in the gapless phase (orange area). Generally, the supercurrent helps to enter the topological phase hosting one MBS at lower values of ${\mathrm{\Delta }}_Z$. If ${\delta }_c<\delta <{\delta }_g$, a relatively weak magnetic field is already sufficient to achieve the topological phase; however, if ${\mathrm{\Delta }}_Z=0$, the system is in the gapless phase (orange line). The phase boundaries between gapped phases (blue solid lines) are given by Eq. (12). In the special case $\delta =0$, a zero-energy FBS is composed of two MBSs (green line) [78]. Other parameters are the same as in Fig. 2. The size of the bulk gap for all parameter values is discussed in Appendix pp2.

Figure 5

Phase diagram as function of Zeeman energy ${\mathrm{\Delta }}_Z/\mathrm{\Delta }$ and crossed Andreev pairing amplitude ${\mathrm{\Delta }}_c/\mathrm{\Delta }$. The phase boundaries between different gapped phases for ${\mu }_{\tau }=\mathrm{\Gamma }\equiv 0$ (${\mu }_{\tau }=\mathrm{\Gamma }\equiv 1.1\mathrm{\Delta }$) are given by Eq. (12) [Eq. (17)] and shown by black dashed line (blue solid line) for two cases: (a) $\beta =1.4$ and (b) $\beta =1$. The effect of interwire tunneling $\mathrm{\Gamma }$ can be compensated by tuning ${\mu }_{\tau }$ to a sweet spot, ${\mu }_{\tau }=\mathrm{\Gamma }$, for small values of ${\mathrm{\Delta }}_Z$. Note that the phase transition from the topological phase hosting one MBS at each system end to the trivial one-FBS phase is pushed to higher values of the Zeeman energy. For ${\mu }_{\tau }=\mathrm{\Gamma }$ and ${\mathrm{\Delta }}_c=0$, the bulk gap at $k=0$ closes at ${\mathrm{\Delta }}_Z={\overline{\mathrm{\Delta }}}_Z^{\pm }$. In the finite $\mathrm{\Gamma }$ regime [panel (a)], the gapless phase (orange area) appears at large values of ${\mathrm{\Delta }}_c^g$, while it is absent in the $\mathrm{\Gamma }=0$ case in the same parameter range. In both cases, there is a gapless phase along ${\mathrm{\Delta }}_Z=0$ axis (orange line). In the case $\beta =1$ [panel (b)], a gapless phase (orange area) appears in the system both for zero and finite values of $\mathrm{\Gamma }$ and extends over a large range of magnetic field. In contrast to the case described in panel (a), there are no BSs in the system for ${\mathrm{\Delta }}_Z<{\tilde{\mathrm{\Delta }}}_{Z,-}{|}_{\beta =1}$ given that ${\mathrm{\Delta }}_c>{\mathrm{\Delta }}_c^{+}{|}_{\beta =1}$. Other parameters are the same as in Fig. 2.

Figure 6

Phase diagram of the double NW as function of chemical potential ${\mu }_{\overline{1}}/\mathrm{\Delta }$ and Zeeman energy ${\mathrm{\Delta }}_Z/\mathrm{\Delta }$ for different values of the phase gradient: $\delta =0$ (green dot-dashed line), $\delta /\mathrm{\Delta }=0.01$ (blue dashed line), and $\delta /\mathrm{\Delta }=0.04$ (red solid line). All lines are obtained analytically from the gap closing condition at $k=0$ given by Eq. (22). The area corresponding to the topological phase increases with the corresponding increase in the supercurrent strength $\delta$, thus allowing one to apply weaker magnetic fields to achieve the one-MBS phase. However, the topological phase at ${\mathrm{\Delta }}_Z=0$ remains out of reach as the gapless phase (orange) (shown here for $\delta /\mathrm{\Delta }=0.04$) appears at high values of $\delta$. The number of BSs is indicated in the plot. Other parameters are fixed as ${\mu }_1=0$, $\mathrm{\Gamma }=0$, $E_{\text{so},1}/\mathrm{\Delta }=6.25$, $\beta =1.4$, and ${\mathrm{\Delta }}_c/\mathrm{\Delta }=0.5$.

Figure 7

Phase diagram of the double NW as function of chemical potential ${\mu }_{\overline{1}}/\mathrm{\Delta }$ and crossed Andreev pairing potential ${\mathrm{\Delta }}_c/\mathrm{\Delta }$. (a) For ${\mathrm{\Delta }}_Z=0$, the phase boundary between the trivial and two-MBS phase for $\delta =0$ given by Eq. (25) is shown by the black dashed line. In the presence of supercurrent, $\delta /\mathrm{\Delta }=0.04$, a gapless phase emerges (orange region). The one-MBS topological phase can be achieved only for finite magnetic fields. (b) At ${\mathrm{\Delta }}_Z=0.5\mathrm{\Delta }$, the topological phase emerges at lower values of the crossed Andreev pairing in the presence of $\delta$ [here, $\delta /\mathrm{\Delta }=0.04$; blue solid line is determined from the gap closing condition at $k=0$] compared to the case $\delta =0$ (black dashed line). The number of BSs is indicated in the plot. Other parameters are chosen as ${\mu }_1=0$, $\mathrm{\Gamma }=0$, $E_{\text{so},1}/\mathrm{\Delta }=6.25$, $\beta =1.4$, ${\mathrm{\Delta }}_c/\mathrm{\Delta }=0.5$.

Figure 8

Bulk gap $E_g/\mathrm{\Delta }$ as function of Zeeman energy ${\mathrm{\Delta }}_Z/\mathrm{\Delta }$ and crossed Andreev pairing amplitude ${\mathrm{\Delta }}_c/\mathrm{\Delta }$ for two cases: (a) $\delta =0$ and (b) $\delta /\mathrm{\Delta }=0.04$. The bulk gap closing at $k=0$ given by Eq. (12) is shown in black dashed line. In both cases, the size of the energy gap increases as we move further away from the gap closing points. Other parameters are the same as in Fig. 2.

Figure 9

Bulk gap $E_g/\mathrm{\Delta }$ as function of Zeeman energy ${\mathrm{\Delta }}_Z/\mathrm{\Delta }$ and applied supercurrent $\delta /\mathrm{\Delta }$. All parameters are the same as in Fig. 4. (a) If the supercurrent is weak, which is our regime of interest, the zero bulk gap corresponds to the topological gap closing at $k=0$. If the supercurrent exceeds the critical value ${\delta }_g$, superconductivity is destroyed and the system becomes gapless. (b) Fixing ${\mathrm{\Delta }}_Z/\mathrm{\Delta }=0.6$, we observe that the bulk gap, which determines the localization length of MBSs, grows as a weak supercurrent is applied.