Robustness of Majorana fermions in proximity-induced superconductors

In two-dimensional chiral $p$-wave superconductors, the zero-energy Majorana fermion excitations trapped at vortex cores are protected from the thermal effects by the minigap, ${\Delta }^2/{ϵ}_F$ ($\Delta$: bulk gap, ${ϵ}_F$: Fermi energy), which is the excitation gap to the higher energy bound states in the vortex cores. Robustness to thermal effects is guaranteed only when $T⪡{\Delta }^2/{ϵ}_F\sim 0.1\text{mK}$, which is a very severe experimental constraint. Here we show that when $s$-wave superconductivity is proximity-induced on the surface of a topological insulator or a spin-orbit-coupled semiconductor, as has been recently suggested, the minigaps of the resultant non-Abelian states can be orders of magnitude larger than in a chiral $p$-wave superconductor. Specifically, for interfaces with sufficient barrier transparencies, the minigap can be as high as $\sim \Delta ⪢{\Delta }^2/{ϵ}_F$, where $\Delta$ is the bulk gap of the $s$-wave superconductor.

Figure 1

(Color online) Proximity induced pairing on the TI surface. The (red) region on the left is the TI, and the (blue) region on the right is an $s$-wave superconductor. “Integrating out” the superconducting degrees of freedom produces the self-energy $\Sigma$ on the TI surface, where $\Sigma$ is given by the tunneling Hamiltonian $\mathcal{T}$ and the Green’s function $G_{\text{SC}}^{\left(0\right)}$ of the superconductor (see text for details).

Figure 2

(Color online) A trijunction-pair geometry of superconducting islands deposited on the TI surface (top view) to confine and manipulate Majorana fermions. The superconducting islands are connected strong superconducting loops enclosing fluxes ${\Phi }_{n,m}$. The values of the fluxes ${\Phi }_{1,2}={\Phi }_{3,4}=\frac{\pi }{2}+\delta \varphi$, ${\Phi }_{2,3}=\frac{\pi }{2}-\delta \varphi$, and ${\Phi }_{4,1}=-3\frac{\pi }{2}-\delta \varphi$ which satisfy ${\sum }_n{\Phi }_n=0$ control the phases of the superconducting island. For the given superconducting phase configuration and $\delta \varphi =\frac{\pi }{6}$ the structure contains a vortex with a trapped Majorana state only on trijunction A. By changing $\delta \varphi$ to $-\frac{\pi }{6}$, the discrete vortex together with the Majorana state is transferred to the trijunction B. The Majorana state is transported from A to B by the delocalized Majorana fermion state formed on the one-dimensional line junction (of length $L$) connecting A and B in the intermediate stage with $\delta \varphi =0$.

Figure 3

(Color online) Numerical results for the vortex minigap $\Delta E$ plotted against the renormalized Fermi level $\tilde{U}$ on the TI surface ($\Delta E$, $\tilde{U}$ scaled by $\tilde{\Delta }$). The solid (red) line gives the mini-gap when the vortex core size ${\xi }_v=\xi =\tilde{v}/\tilde{\Delta }=v/\lambda$, as is appropriate in a regular vortex with a continuously varying phase. The dashed (black) line shows that the excitation gap is even larger when the vortex core size is smaller, as is expected in a discrete vortex (see Fig. 1). The inset shows the TI sphere with a vortex and an antivortex (with reduced superconducting amplitudes at the vortex cores) situated at the north and the south poles.