Supercurrent reversal in two-dimensional topological insulators

We calculate a supercurrent across a two-dimensional topological insulator subjected to an external magnetic field. When the edge states of a narrow two-dimensional topological insulator are hybridized, an external magnetic field can close the hybridization gap, thus driving a quantum phase transition from insulator to semimetal states of the topological insulator. We find a sign reversal of the supercurrent at the quantum phase transition revealing intrinsic properties of topological insulators via the Josephson effect.

Figure 1

Schematics of the point contact superconductor–2D-topological-insulator–superconductor junction (SC stands for superconductor). The topological insulator strip has a width $d$ and the junction plane resides in the $z=0$ plane so that the superconducting contacts are located at ${\mathbf{r}}_T=(x_T,+d/2,0)$ and ${\mathbf{r}}_B=(x_B,-d/2,0)$. The external magnetic field $\mathbf{H}=(0,0,H)$ is oriented along $z$, normal to the plane of the TI. Top and bottom edge states (shown by parallel arrow lines) are hybridized while left and right edge channels are strongly gapped compared to the region of contact. The top and bottom superconducting electrodes have different macroscopic phases marked by ${\varphi }_T$ and ${\varphi }_B$, respectively. The typical tunneling process of Cooper pairs between the superconductors through the edge channels is also illustrated.

Figure 2

Band structure, Eq. (6), of the 2D TI for different values of the magnetic energy ${\varepsilon }_H$ and the tunneling amplitude $t$. Arrows indicate the spin projection of an electron in the corresponding subband. (a) The hybridization between the helical edge states with the same spin opens a gap of $2t$ at zero momentum. The subbands are doubly spin degenerate in the absence of an external magnetic field. (b) The external magnetic field removes the spin degeneracy and closes the gap. (c) Critical point ${\varepsilon }_H=t$ where two bands touch at $k=0$. (d) Subbands with opposite spins cross at two points when ${\varepsilon }_H>t$.

Figure 3

Normalized critical supercurrent $J_c({\varepsilon }_H,\delta \mu )/J_c(0,0)$ through the point contact configuration proposed where $\left|\mathrm{\Delta }\right|/t=6$. (a) Critical current as a function of magnetic energy normalized by the tunneling amplitude at different values of chemical potential: $\delta \mu /t\in [0,0.6,1,1.4]$. (b) Critical current as a function of chemical potential $\delta \mu$ normalized by the tunneling amplitude $t$ for different values of magnetic energy: ${\varepsilon }_H/t\in [0,0.5,1,1.5]$.