Transport signatures of topological superconductivity in a proximity-coupled nanowire

We study the conductance of a junction between the normal and superconducting segments of a nanowire, both of which are subjected to spin-orbit coupling and an external magnetic field. We directly compare the transport properties of the nanowire assuming two different models for the superconducting segment: one where we put superconductivity by hand into the wire and one where superconductivity is induced through a tunneling junction with a bulk $s$-wave superconductor. While these two models are equivalent at low energies and at weak coupling between the nanowire and the superconductor, we show that there are several interesting qualitative differences away from these two limits. In particular, the tunneling model introduces an additional conductance peak at the energy corresponding to the bulk gap of the parent superconductor. By employing a combination of analytical methods at zero temperature and numerical methods at finite temperature, we show that the tunneling model of the proximity effect reproduces many more of the qualitative features that are seen experimentally in such a nanowire system.

Figure 1

Model geometry. A 1D nanowire (of infinite length) with spin-orbit coupling is placed on top of a junction between an insulator ($x<0$) and a superconductor ($x>0$). A magnetic field ${\mathbf{B}}_{\text{ext}}$ is applied perpendicularly to the effective Rashba field. We chose the latter to be along the $z$ axis and ${\mathbf{B}}_{\text{ext}}$ to be along the wire.

Figure 2

(a) Energy spectrum of a one-dimensional nanowire with Rashba spin-orbit coupling. The Fermi energy $\mu$ is measured from the degeneracy point at $k=0$. The spin-orbit coupling strength is parameterized by the spin-orbit energy $E_{so}=m{\alpha }^2/2$ and the spin-orbit wave vector $k_{so}=m\alpha$. (b) Energy spectrum of a Rashba nanowire in the presence of an external magnetic field applied parallel to the wire. Magnetic field opens a gap of size $2J$ at $k=0$.

Figure 3

Bogoliubov-de Gennes excitation spectra of normal (left) and superconducting (right) segments of one-dimensional nanowire with Rashba spin-orbit coupling in the absence of a magnetic field. Colors denote different spin states. An incident electron from the normal segment of the wire ($i_{\sigma }$) can be Andreev reflected ($a_{\sigma }$), normally reflected ($r_{\sigma }$), or transmitted to the superconducting segment ($t_i$). Spectra shown here taking $\mu =0$.

Figure 4

Bogoliubov-de Gennes excitation spectra of normal (left) and superconducting (right) segments of nanowire in the presence of an external field in the intrinsic pairing model. Chemical potential is fixed at $\mu =0$ and $E_{so}\gg J,\mathrm{\Delta }$. The spectra are shown here for $J/\mathrm{\Delta }=0.25$.

Figure 5

Conductance spectra in strong spin-orbit limit ($E_{so}\gg J,\mathrm{\Delta }$), plotted for several values of $Z$ with fixed $J/\mathrm{\Delta }$ at zero temperature. (a) Nontopological phase ($J=0.5\mathrm{\Delta }$). (b) Topological phase ($J=1.5\mathrm{\Delta }$). Chemical potential is fixed at $\mu =0$.

Figure 6

Conductance spectra in strong spin-orbit limit ($E_{so}\gg J,\mathrm{\Delta }$), plotted for several values of $J/\mathrm{\Delta }$ with fixed $Z$ at zero temperature. (a) Nontopological phase ($J<\mathrm{\Delta }$) with $Z=0.3$. (b) Topological phase ($J>\mathrm{\Delta }$) with $Z=1$. Chemical potential is fixed at $\mu =0$.

Figure 7

Conductance deep in the topological phase ($J\gg E_{so},\mathrm{\Delta }$), plotted for several values of barrier strength $Z_{\text{eff}}=Z\sqrt{E_{so}/J}$. The zero-bias conductance is fixed at $2e^2/h$ and the width of the zero-bias peak is controlled by $Z_{\text{eff}}$.

Figure 8

Zero-bias conductance $G\left(0\right)$ as a function of temperature deep in the topological phase ($J\gg E_{so},\mathrm{\Delta }$), plotted for several values of $Z_{\text{eff}}=Z\sqrt{E_{so}/J}$. At $T=0$ the zero-bias conductance is $2e^2/h$, while finite temperature more significantly alters the conductance as the effective barrier strength is increased.

Figure 9

(a) Conductance as a function of energy $E$ and Zeeman splitting $J$ at finite temperature within the intrinsic pairing model. Plotted with $\mu /\mathrm{\Delta }=1.2,E_{so}/\mathrm{\Delta }=1.5,Z=3$, and $T/T_c=0.1$. (b) Line cuts of plot in (a) for different values of $J$, ranging from $J/\mathrm{\Delta }=0$ to $J/\mathrm{\Delta }=3$ in steps of $J/\mathrm{\Delta }=0.1$. Plots offset by $0.1\times e^2/h$ for clarity.

Figure 10

Conductance obtained within the tunneling model in the limit $(\mu +E_{so})\gg \mathrm{\Delta }$ and $J=0$ at zero temperature. Plotted for $Z=3$ and $\gamma =1.5$, corresponding to an induced gap of $E_g\approx 0.7\mathrm{\Delta }$.

Figure 11

Bulk excitation spectrum in a nanowire with tunneling-induced superconductivity for various field strengths at zero temperature. (a) In the absence of the field, the lower branch of the spectrum has an excitation gap $E_g$ and the spectrum is degenerate at $k=0$. (b) Application of a field lifts the degeneracy and reduces the gap on the lower branch at $k=0$. (c) At the critical field strength $J_c=\sqrt{{\mu }^2+{\gamma }^2}$, the gap closes. (d) The gap is reopened in the topological phase as the field is increased beyond $J_c$. In all cases, there are no bulk states at energies $E>\mathrm{\Delta }-(g_S/g)J$, as the energy acquires an imaginary part. This signifies that these states are lost to the superconducting substrate. All plots shown for $\mu /\mathrm{\Delta }=1.2,E_{so}/\mathrm{\Delta }=1.5,\gamma /\mathrm{\Delta }=1.5$, and $g_S/g=0.05$.

Figure 12

Zero-bias conductance deep in the topological phase $J\gg \gamma$, plotted as a function of tunneling strength $\gamma$ (in units of the bulk superconducting gap $\mathrm{\Delta }$) for several temperatures $T$ (in units of ${\mathrm{\Delta }}_{\text{eff}}=\mathrm{\Delta }\sqrt{E_{so}/J}$). All curves plotted with $Z_{\text{eff}}=3$.

Figure 13

(a) Conductance as a function of energy $E$ and Zeeman field $J$ at finite temperature within the tunneling model. Plotted with $\mu /\mathrm{\Delta }=1.2,E_{so}/\mathrm{\Delta }=1.5,\gamma /\mathrm{\Delta }=1.5,Z=3,T/T_c=0.1$, and $g_S/g_N=0.05$. (b) Line cuts of plot in (a) for different values of $J$, ranging from $J/\mathrm{\Delta }=0$ to $J/\mathrm{\Delta }=3$ in steps of $J/\mathrm{\Delta }=0.1$. Plots offset by $0.1\times e^2/h$ for clarity.