Figure 1

(a) Schematic of a cylindrical 3D topological insulator with a hole drilled through the center. The blue line represents a flux tube threaded through the cylindrical hole. (b) Schematic of a heterostructure of a topological insulator thin film sandwiched between two $s$-wave superconductors. The thin blue lines represent $h/2e$ flux tubes, which generate vortices in the superconductor layers.Reuse & Permissions

Figure 2

Probability distribution for the lowest energy states corresponding to different superconducting penetration depths $\lambda =$ (a) 0.001, (b) 0.2, (c) 0.3, (d) 0.4, (e) 0.5, and (f) 1.5. Note that $\lambda$ is in units of the lattice constant $a$ and the entire flux is spread out in a region with a radius of roughly $5\lambda .$ The pairing potential $\Delta \left(z\right)$ decays from $0.5$ on the surface to 0 within five layers. As $\lambda$ is increased, we see that the states move from being delocalized along the flux tube penetrating the bulk of the topological insulator to being pinned at the surface. The inset shows a schematic of the spatial variation of the superconducting mass and the time-reversal breaking mass associated with the magnetization as $\lambda$ varies. (g) The energies of the lowest energy states as a function of $\lambda .$ There are clear zero modes forming as $\lambda \to \infty .$ Our numerical results show that in the $\lambda \ll a$ regime, the lowest energy linearly decays as the height of the sample increases; in the $\lambda \gg a$ regime, the lowest energy exponentially decays as the height of the sample increases.Reuse & Permissions

Figure 3

Ratio of spin-up to spin-down composition of a Majorana bound state vs flux penetration depth $\lambda .$ The different traces represent the use of different superconducting-pairing potential strengths ($\Delta$). In the thick-flux limit, a larger $\Delta$ corresponds to a smaller ratio of spin up and down, leading to spin-polarized Majorana bound states.Reuse & Permissions

Figure 4

Probability distribution of a Majorana bound state in the topological-insulator–$s$-wave-superconductor heterostructure, where we have used realistic material parameters. For the topological insulator, we have used the parameters of ${\text{Bi}}_2{\text{Se}}_3$, and for the superconducting films on the top and bottom surfaces, we have used the material parameters of a niobium superconductor with a single vortex.Reuse & Permissions