Transcendental momentum quantization in semiconducting Rashba nanowires and zero energy states in their normal and superconducting phase

Abstract

We study finite system properties of the canonical low energy model for a semiconducting nanowire with Rashba spin-orbit coupling. The case of an isolated wire as well as of one proximitized by an s-wave superconductor are considered. Already for the normal wire, the presence of spin-orbit coupling leads to eigenstates of the finite system composed of more than two momentum eigenstates. The quantization condition for the wavevectors is not that of a quantum box, but given instead by a transcendental equation linking the involved wavevectors. For the wire with superconducting pairing, the presence of electron and hole channels complicates the composition of the eigenstates. In this case we derive an approximate quantization condition close to the phase boundary, and a condition for the appearance of exact zero energy states. It can be satisfied both in the topological and in the trivial phase. Both the trivial and topological zero energy states contribute to the linear transport through Andreev reflection and direct transmission processes, with their relative importance depending on the degree of the states' localization at the boundary.

Figures (16)

• Figure 1: Rashba nanowire. (a) General setup, and (b) Topological phase diagram, where $\nu$ denotes the winding number topological invariant. (c) Energy and modified inversion symmetry of the lowest excitation in the numerically solved discretized nanowire with toy model parameters (cf. Tab. \ref{['tab:parameters']}) and open boundary conditions. Red (blue) colour scale applies when the lowest excitation is even (odd) under $\tilde{I}$, respectively. White dotted line marks the boundary of the topological phase for $L\rightarrow\infty$. Note the asymmetry with respect to $\mu_c = 0$, due to the asymmetry of the normal wire's dispersion with respect to $\mu_c$.
• Figure 2: Robustness of the zero energy states to disorder in a toy wire. On each site a random potential is added, $\varepsilon_i \in [-W,W]$. The points correspond to $(\text{$V_\mathrm{Z}$},\mu_c)$ parameters where the lowest excitation has energy lower than $2\%$ of $\Delta$. In grey are shown the results from the clean wire, in color those for $W=2\Delta$ (a) or $W=4\Delta$ (b). While the locations of zero energy states change, they remain stable both in the topological and in the trivial phase. Disorder even widens the region with zero energy states in the trivial phase.
• Figure 3: Configuration of the solutions of $E^2(k)=0$ in the topologically trivial and non-trivial phase.
• Figure 4: Zero energy lines obtained numerically and with several analytical approaches. The model parameters are given in Table \ref{['tab:parameters']}. The orange lines in (a) and (b) are obtained with Eq. \ref{['equation: zero energy lines continuum']} for $k_\Sigma = \pi n/L_+$ ($n=1,\ldots,N$). Purple lines are the solutions of an analogous equation $\text{$V^2_\mathrm{Z}$} = (\varepsilon_f - \mu_c)^2 - 2E_{so}(\varepsilon_f +\mu_c) + \Delta^2$ from the supplementary material of Ref. DasSarma2012, plotted for comparison. In (c)-(d) we use Eq. \ref{['equation: lattice vz constraint, first']} from the lattice model, together with $k_f = \pi n/L_+$.
• Figure 5: Sketch of the band structure of a normal wire with magnetic field applied in the $z$ direction. In the region I four Bloch states are available at each energy, all from the $-$ band. This region shrinks fast with increasing $\text{$V_\mathrm{Z}$}$. In the Zeeman gap (region II) there are only two Bloch states at each energy, and in the region III again four, one pair from branch $+$ and one from $-$.