Topological superconductivity in Rashba semiconductors without a Zeeman field

In this manuscript, I present new hybrid devices based on multiwire/channel Rashba semiconductors, which harbor Majorana fermions (MFs) without a Zeeman field. In contrast, magnetic fluxes, supercurrents, or electric fields can be employed, yielding an enhanced device manipulability. The generic topological phase diagram for two-nanowire/channel systems exhibits features of quantum criticality and a rich interplay of phases with 0, 1, or 2 MFs per edge. The most prominent and experimentally feasible implementation relies on the already existing platforms of InAs two-dimensional electron gas (2DEG) on top of a Josephson junction. Appropriate design of the latter device allows phases with 1 or 2 MFs, both detectable in zero-bias anomaly peaks with a single or double unit of conductance.

Figure 1

(a) and (b) Side and full views, respectively, of two Rashba NWs and (c) side view of a Rashba semiconducting film, on top of a Josephson junction. The interface is interrupted by an insulating loop ($\mathrm{AB}\mathrm{\Gamma }\mathrm{\Delta }$). Threading magnetic flux (${\mathrm{\Phi }}_{\mathrm{flux}}$) through the loop, leads to Majorana bound states extended along the $y$ axis and localized at the edges along the $x$ axis. The required magnetic flux can be generated by (i) a supercurrent flow ($J_{\mathrm{sc}}=2e{\mathrm{\Phi }}_{\mathrm{flux}}/\hslash$) through the junction or (ii) by placing the semiconductor in an electric field (${\mathcal{E}}_{AB}={\dot{\mathrm{\Phi }}}_{\mathrm{flux}}$), produced here by side gates (sg).

Figure 2

(a) Cross-section of Fig. 1 with couplings NW-NW ($t_{\perp }$), SC-SC (${\tilde{t}}_{\perp }$), and NW-SC ($\mathrm{T}$). Each cross-section behaves as a superconducting quantum interference device and the total system transits to a TSC phase for a window of values for the inserted flux (${\mathrm{\Phi }}_{\mathrm{flux}}$) in the loop ($\mathrm{AB}\mathrm{\Gamma }\mathrm{\Delta }$). (b) Induced multicomponent superconducting gap on the NWs, when flux is indirectly generated by a supercurrent flow. For $J_{\mathrm{sc}}=\pi$, time-reversal symmetry ($\mathcal{T}$) is violated, since the intra- and inter-NW gaps exhibit a $\pi /2$-phase locking.

Figure 3

Topological phase diagrams for InSb and InAs ($\mu =0$, $\mathrm{\Delta }=250\mu \mathrm{eV}$). (a) ${\mathrm{\Delta }}_{\perp }=90\mu \mathrm{eV}$ and $\varphi =1/2$ ($J_{\mathrm{sc}}=\pi$). Four distinct phases appear: $\mathcal{N}=0,\pm 1,2$, with $\left|\mathcal{N}\right|$ the number of MFs per edge. The 1MF-phases (blue & yellow) are enclosed by the critical lines ${\mathrm{\Delta }}^2=t_{\perp }^2+{({\mathrm{\Delta }}_{\perp }\pm v_{\perp })}^2$, defined by the bulk gap closings at $k=0$, as in (c) for $(v_{\perp },t_{\perp })=(300,136)\mu \mathrm{eV}$. The $\mathcal{N}=-2$ phase is protected by chiral symmetry and arises from gap closings at $\pm k_{*}$, as in (b) for $(v_{\perp },t_{\perp })=(300,233)\mu \mathrm{eV}$. The $\mathcal{N}=+1$ and $\mathcal{N}=-2$ phases, overlap, yielding the 1MF-phase with $\mathcal{N}=-1$. The latter appears between the parallel lines: $t_{\perp }=\sqrt{{\mathrm{\Delta }}^2-{\mathrm{\Delta }}_{\perp }^2}$ and $t_{\perp }\simeq \mathrm{\Delta }$. Two quantum tricritical points emerge: $P_1\simeq ({\mathrm{\Delta }}_{\perp },\mathrm{\Delta })$ and $P_2=(2{\mathrm{\Delta }}_{\perp },\sqrt{{\mathrm{\Delta }}^2-{\mathrm{\Delta }}_{\perp }^2})$, where the 0,1, and 2 MF phases meet. (d) ${\mathrm{\Delta }}_{\perp }=50\mu \mathrm{eV}$ and $\varphi =1/2$. Results similar to (a), but with the window for the 1MF-phase suppressed as it depends on ${\mathrm{\Delta }}_{\perp }$. For ${\mathrm{\Delta }}_{\perp }=0$, the 1MF phase disappears, since Kramers degeneracy is restored and only MF pairs are allowed. (e) ${\mathrm{\Delta }}_{\perp }=50\mu \mathrm{eV}$ and $\varphi =0.45$. Further away from the $\pi$ junction, the critical points $P_{1,2}$ vanish.

Figure 4

Full view of a Rashba semiconducting film, on top of a Josephson junction. The interface is interrupted by an insulating loop enclosed by the yellow lines. Threading magnetic flux through the loop, for instance, by effecting an electric field using the side gates, leads to Majorana bound states extended along the $y$ axis and localized at the edges along the $x$ axis. The top gate can be employed for tuning the chemical potential of the semiconducting film and thus controlling the topological phase diagram.

Figure 5

Full view of a three-dimensional Rashba semiconducting nanowire with square cross-section ($yz$ plane) on top of a Josephson junction. The interface is interrupted by an insulating part (depicted with black) which partially blocks the proximity effect and renders the proximity induced superconductivity inhomogeneous along the $y$ axis. Using the side gates, one can induce an electric field in the semiconductor, which in the steady state can support Majorana bound states extended along the $y$ axis and localized at the edges along the $x$ axis.