Figure 1

(a) Simplest possible version of the proposed setup. A strip of Au(111) thin film (or any other metal with a Rashba split surface state) is deposited on top of a conventional superconductor. An external field is applied parallel to the wire in order to drive the system into a topological SC state. Majorana end states can be detected by tunneling, e.g., with a scanning tunneling microscopy (STM) tip. (b) So long as the surface state survives the deposition of a gate dielectric, the surface-state chemical potential can be controlled by a top gate. (c) Tunneling density of states $N\left(\varepsilon \right)$ as a function of energy $\varepsilon$; the full superconducting gap ${\Delta }_B$ is induced on the bulk states by the proximity effect. The surface gap develops a smaller gap ${\Delta }_S$ due to indirect scattering from disorder and interactions. (d) Sketch of the band structure of a metal with a Rashba spin-orbit split surface band. Bulk states are projected onto the plane of the surface and the nonzero bulk projected density of states is indicated by gray shading. The surface-state band forms within a region momentum space where there is no bulk states. The figure shows a one-dimensional cut through the surface Brillouin zone. The chemical potential $\mu$ is represented by a dashed line. The surface Fermi energy ${\varepsilon }_F$ and spin-orbit splitting at the Fermi surface ${\Delta }_{\mathrm{so}}$ are indicated for the surface bands.Reuse & Permissions

Figure 2

(a) Numerical phase diagram for a 40-site-wide wire in a perpendicular field ($\mathbf{B}\sim \widehat{z}$) as a function of chemical potential $\mu$ and magnetic field ${\mu }_{0}B$. Black lines indicate subband bottoms in the normal state (without superconductivity) and red-filled regions indicate the presence of Majorana end states, which occur when the subband degeneracy is removed and the subband splitting is sufficiently larger than the pairing gap $\Delta$. The subbands are initially degenerate for $B=0$ and split quadratically as $B$ is increased. The blue-shaded region indicates the Zeeman gap for a full two-dimensional sample. In the wire, the topological region extends slightly outside the Zeeman gap for sufficiently large $B$ (for ${\mu }_0{B}^2/{\Delta }_{\mathrm{so}}\gtrsim \Delta )$. The simulation parameters are $t=50$, ${\alpha }_R^2/t=10$, and ${\Delta }_S=1$. (b) Same setup described in (a) but with the magnetic field applied along the wire $(\mathbf{B}\sim \widehat{x})$. Unlike the perpendicular field case, the wire always remains in the topological region so long as ${\mu }_{0}B>{\Delta }_S$ and an odd number of subbands is occupied. Unlike the parallel field case, the black lines that indicate subband bottoms split linearly in the applied field, giving rise to a crisscrossing diamond pattern of topological and nontopological phases. (c) Parallel field phase diagram for a 10-site-wide wire; topological regions occupy a smaller fraction of the phase diagram.Reuse & Permissions

Figure 3

Surface-pairing gap ${\Delta }_S$ for various ${\Delta }_{\mathrm{so}}/{\mu }_{0}B$. The reduction of the induced surface gap due to disorder is very weak for ${\Delta }_{\mathrm{so}}\gg {\mu }_{0}B$. The plot is generated from the calculations of disorder-induced pair breaking from Ref. 16.Reuse & Permissions

Figure 4

Depiction of virtual scattering processes that mix bulk and surface bands and generate surface superconductivity (left column) along with representative Feynman diagrams (right column). In the diagrams, lines labeled by $S$ and $B$ indicate surface-state and bulk-state propagators, respectively; propagators with left (right) arrows are conventional particle (hole) propagators, whereas propagators with both left and right arrows are anomalous propagators due to the Cooper pair condensate. Each process shown in the left column represents half of the corresponding diagram (to complete the diagram, the process is repeated in reverse). The top row depicts elastic scattering from impurities, represented diagrammatically by $\times$'s connected by a dashed line (indicating scattering off of the same impurity). The middle and bottom rows show inelastic processes that generate surface pairing; wavy lines represent either screened Coulomb interactions or phonons. The middle row shows inelastic pair scattering from surface to bulk and the bottom row shows interaction-induced surface-bulk tunneling that is accompanied by the creation of a bulk particle-hole pair.Reuse & Permissions

Figure 5

Electrostatic potential profile from the applied gate voltage (bottom) aligned with the proposed materials stack [top, shown here rotated ${90}^{\circ }$ relative to Fig. 1b]. The surface chemical potential is shifted by $\delta {\mu }_s=-e\phi \left(0\right)$ relative to the bulk chemical potential. Estimates using typical material parameters demonstrate that one can readily tune the chemical potential by $\pm 100$ meV, despite the presence of a large density of states from the metallic bulk.Reuse & Permissions

Figure 6

A log-log plot of scaling of the maximal minigap ${\Delta }_{\mathrm{mg}}$ for the tight-binding model for $p+ip$ superconducting strips from Ref. 14 as a function of hopping strength $t$. The maximal minigap size occurs for strips of width $W\approx {\xi }_0$. Best-fit lines are shown in black. For perfectly rectangular strips with straight ends (squares), ${\Delta }_{\mathrm{mg}}\sim {\Delta }_s{(t/{\Delta }_s)}^{-1.052}$ as reported in Ref. 30. In contrast, for wires with slightly rounded ends the best fit scales as ${\Delta }_{\mathrm{mg}}\sim {\Delta }_s{(t/{\Delta }_s)}^{-0.614}$. The optimal width is almost nearly identical (within a few percent) for both straight and rounded ends. For each value of $t$, the straight and rounded end data were taken from for geometries with the same width.Reuse & Permissions