Density of states of disordered topological superconductor-semiconductor hybrid nanowires

Using Bogoliubov-de Gennes equations we numerically calculate the disorder averaged density of states of disordered semiconductor nanowires driven into a putative topological $p$-wave superconducting phase by spin-orbit coupling, Zeeman spin splitting, and the $s$-wave superconducting proximity effect induced by a nearby superconductor. Comparing with the corresponding theoretical self-consistent Born approximation (SCBA) results treating disorder effects, we comment on the topological phase diagram of the system in the presence of increasing disorder. Although disorder strongly suppresses the zero-bias peak (ZBP) associated with the Majorana zero mode, we find some clear remnant of a ZBP even when the topological gap has essentially vanished in the SCBA theory because of disorder. We explicitly compare effects of disorder on the numerical density of states in the topological and trivial phases.

Figure 1

Disorder-averaged DOS in the topological phase for the semiconductor nanowire in a magnetic field with Zeeman splitting $V_Z=5$ K, proximity-induced pairing potential of amplitude ${\Delta }_0=3$ K, Rashba SO-coupling strength ${\alpha }_R=0.3$ eV $-$ Å and $\mu =0$. The crosses in the plots show the energy levels for a typical disorder realization for each of these mobililties. For this choice of parameters the clean quasiparticle gap $\Delta =1.3$ K and coherence length $\xi =0.3$ $\mu$m. The difference panels (a–h) correspond to different disorder strengths characterized by $E_s=\hslash /2\tau =0.01$, 0.05, 0.1, 0.2, 0.46, 1.3, 3.3, and 7.3 K. The corresponding mean-free paths are $\lambda =v_F\tau$ in the panels and the self-consistent Born gaps $E_{\mathrm{SCBA}}=1.2$, 1.1, 0.9, 0.8, 0.13, 0, 0, and 0 K. Typical spectra.

Figure 2

Quasiparticle gap $\Delta$ (in the color bar) versus Zeeman potential $V_Z$ and scattering rate $E_s$ in the topological phase. The black region represents the gapless phase and is therefore not topologically robust.

Figure 3

Disorder-averaged DOS in the nontopological phase for the semiconductor nanowire in a magnetic field with Zeeman splitting $V_Z=0$ and 1 K [panels (a) and (b), respectively) at $\mu =0$ for different disorder strengths characterized by scattering rates $E_s=\hslash /2\tau$.

Figure 4

Disorder-averaged DOS in the nontopological phase for the semiconductor nanowire in a magnetic field with Zeeman splitting $V_Z=1$ and 5 K [panels (a) and (b), respectively] at $\mu =5$ K for different disorder strengths characterized by scattering rates $E_s=\hslash /2\tau$.

Figure 5

Quasiparticle gap ($\Delta$) versus Zeeman potential $V_Z$ for different values ($\mu =0$ and 5 K) for the chemical potential $\mu$. The vanishing of the quasiparticle gap marks the topological quantum phase transition from the nontopological phase at small $V_Z$ to the topological phase at large $V_Z$.