Fractional Josephson Vortices and Braiding of Majorana Zero Modes in Planar Superconductor-Semiconductor Heterostructures

We consider the one-dimensional (1D) topological superconductor that may form in a planar superconductor-metal-superconductor Josephson junction in which the metal is subjected to spin-orbit coupling and to an in-plane magnetic field. This 1D topological superconductor has been the subject of recent theoretical and experimental attention. We examine the effect of a perpendicular magnetic field and a supercurrent driven across the junction on the position and structure of the Majorana zero modes that are associated with the topological superconductor. In particular, we show that under certain conditions the Josephson vortices fractionalize to half-vortices, each carrying half of the superconducting flux quantum and a single Majorana zero mode. Furthermore, we show that the system allows for a current-controlled braiding of Majorana zero modes.

Figure 1

(a) Planar Josephson junction connecting two SC leads through a 2DEG with Rashba SOC, in the presence of an in-plane and a perpendicular magnetic field. The Josephson vortices in the junction can split into fractional vortices carrying fractions of the SC flux quantum $hc/2e$. (b) Phase diagram of the junction, viewed as a 1D system, as a function of the phase difference $\phi$ between the two SCs and the parallel magnetic field (see Ref. [19]). (c) $\phi (x)$ along the junction for different values of $B_{\parallel }$ close to the critical value, $g{\mu }_B{B}_{\parallel ,c}W/v_F=\pi /4$, at which the ground state of the junction switches from $\phi \approx 0$ (within the topologically trivial phase) and $\phi \approx \pi$ (in the topological phase). (d) Energy-phase relations for the values of $B_{\parallel }$ in panel (c).

Figure 2

[(a)1–3] Proposed setup for braiding. Three Josephson junctions meet at a $T$ junction. The perpendicular magnetic field makes ${\phi }_{1,2,3}$ vary along the junction. Whenever ${\phi }_{1,2,3}(x)$ cross the critical values ${\phi }_{\pm }\text{mod}(2\pi )$, the corresponding junction undergoes a topological transition. Red (white) lines indicate segments in the topological (trivial) phase, respectively; the MZMs ${\gamma }_{a-f}$ are indicated by red circles. The currents $I_{i,i+1}$ shift the superconducting phases, controlling the position of the MZMs. A closed path in the space of the currents implements an exchange of ${\gamma }_a$ and ${\gamma }_b$. The configurations of the junctions at three points along this path are shown in panels 1–3. (b) ${\phi }_{1,2,3}(x)$ corresponding to the configuration shown in panel (a.1). The critical values ${\phi }_{\pm }$ are indicated. (c) The plane ${\phi }_{1,0}+{\phi }_{2,0}+{\phi }_{3,0}=0$ to which the system is confined. ${\phi }_{i,0}$ are controlled by the currents, according to Eq. (2). At the blue, green, and red lines, ${\phi }_{1,0}$, ${\phi }_{2,0}$, or ${\phi }_{3,0}$, respectively, cross one of the transition values (${\phi }_{+}$ or ${\phi }_{-}$), either at the inner or outer edge of the corresponding junction. The transition lines cut the plane into polygons, where each polygon hosts a fixed number of MZMs. The polygons are colored according to that number. The braiding path connects the three configurations 1–3 shown in panel (a). The left (right) white star marks the point where MZMs in junction 2 (1) are both at the outer edge, respectively.