Magneto-Josephson effects in junctions with Majorana bound states

We investigate 1D quantum systems that support Majorana bound states at interfaces between topologically distinct regions. In particular, we show that there exists a duality between particle-hole and spin degrees of freedom in certain spin-orbit-coupled 1D platforms such as topological insulator edges. This duality results in a spin analog of previously explored “fractional Josephson effects”—that is, the spin current flowing across a magnetic junction exhibits $4\pi$ periodicity in the relative magnetic field angle across the junction. Furthermore, the interplay between the particle-hole and spin degrees of freedom results in unconventional magneto-Josephson effects, such that the Josephson charge current is a function of the magnetic field orientation with periodicity $4\pi$.

Figure 1

(a) Phase diagram for 1D system: gapless ($B^2\le {\mu }^2$ and ${\Delta }^2\le b^2)$, $\Delta$ (${\Delta }^2-b^2>\max [B^2-{\mu }^2,0]$), and B ($B^2-{\mu }^2>\max [{\Delta }^2-b^2,0]$) phases. Both $\Delta$ and B phases are gapped. (b) The $\Delta$-B-$\Delta$ junction supports Majorana bound states at the domain walls.[4] (c) The dual configuration of B-$\Delta$-B junction that also supports Majorana bound states at the domain walls.

Figure 2

(a) The scheme to measure unconventional magneto-Josephson effect. Josephson currents are measured for the B-$\Delta$-B junction. In the right region, the transverse magnetic field winds at rate ${\omega }_L=\gamma b_r$, which modulates the Josephson current at half the frequency, ${\omega }_L/2$. (b) Comparison between analytical expressions and numerical results for $j_M$ and $j_Z$. The parameters are ${\mu }_{l/m/r}=0$, $b_{l/m/r}=E/2$, ${\Delta }_m=2.5E$, ${\Delta }_{l/r}=E$, $B_{l/r}=2E$, $B_m=E$. For $E=0.1$ meV and $v={10}^4$ m/s, the length unit is $\xi =66$ nm and the current unit is $j_0=50$ nA. The superconducting angles are fixed ${\varphi }_{l/r}=\pi /2$, ${\varphi }_m=0$.

Figure 3

Contour plot of (a) spin current $j_r^S$ and (b) charge current $j_r^Q$, both of which are $4\pi$ periodic in ${\theta }_r$ and $2\pi$ periodic in ${\varphi }_r$. The other angles are fixed ${\varphi }_l=\pi /2$, ${\varphi }_m={\theta }_{l/m}=0$. The parameters are the same as in Fig. 2.