Experimental and materials considerations for the topological superconducting state in electron- and hole-doped semiconductors: Searching for non-Abelian Majorana modes in 1D nanowires and 2D heterostructures

In proximity to an $s$-wave superconductor, a one- or two-dimensional, electron- or hole-doped semiconductor with a sizable spin-orbit coupling and a Zeeman splitting can support a topological superconducting (TS) state. The semiconductor TS state has Majorana fermions as localized zero-energy excitations at order parameter defects such as vortices and sample edges. Here we examine the effects of quenched disorder from the semiconductor surface on the stability of the TS state in both electron- and hole-doped semiconductors. By considering the interplay of broken time-reversal symmetry (due to Zeeman splitting) and disorder we derive an expression for the disorder suppression of the superconducting quasiparticle gap in the TS state. We conclude that the effects of disorder can be minimized by increasing the ratio of the spin-orbit energy with the Zeeman splitting. By giving explicit numbers we show that a stable TS state is possible in both electron- and hole-doped semiconductors for experimentally realistic values of parameters. We discuss possible suitable semiconductor materials which should be the leading candidates for the Majorana search in solid state systems.

Figure 1

(a) Calculated disordered TS state quasiparticle gap $E_g$ as a function of semiconductor mobility ${\mu }_{sm}$ for electron-doped wires with $E_{\mathrm{SO}}=2m^{*}{\alpha }^2=2$ K (Ref. 34) and $E_{\mathrm{SO}}=4$ K allowing for future electric-field-enhanced spin-orbit coupling (Ref. 37). (b) The values of $V_Z=2\lambda$ realizing $E_g$ in panel (a) plotted with ${\mu }_{sm}$. The gaps in the adjacent bulk superconductor are taken to be ${\Delta }_s=2$ K corresponding to Al and ${\Delta }_s=4$ K corresponding to Nb.

Figure 2

(a) Calculated disordered TS state quasiparticle gap $E_g$ as a function of semiconductor mobility ${\mu }_{sm}$ for hole-doped wires with $E_{\mathrm{SO}}=300$ K (Refs. 22 and 23). (b) The values of $V_Z=2\lambda$ realizing $E_g$ in panel (a) plotted with ${\mu }_{sm}$. The gaps in the adjacent bulk superconductor are taken to be ${\Delta }_s=2$ K corresponding to Al and ${\Delta }_s=4$ K corresponding to Nb.