Andreev spectrum with high spin-orbit interactions: Revealing spin splitting and topologically protected crossings

In order to point out experimentally accessible signatures of spin-orbit interaction, we investigate numerically the Andreev spectrum of a multichannel mesoscopic quantum wire (N) with high spin-orbit interaction coupled to superconducting electrodes (S), contrasting topological and nontopological behaviors. In the nontopological case (square lattice with Rashba interactions), we find that the Kramers degeneracy of Andreev levels is lifted by a phase difference between the S reservoirs except at multiples of $\pi$, when the normal quantum wires can host several conduction channels. The level crossings at these points invariant by time-reversal symmetry are not lifted by disorder. Whereas the dc Josephson current is insensitive to these level crossings, the high-frequency admittance (susceptibility) at finite temperature reveals these level crossings and the lifting of their degeneracy at $\pi$ by a small Zeeman field. We have also investigated the hexagonal lattice with intrinsic spin-orbit interaction in the range of parameters where it is a two-dimensional topological insulator with one-dimensional helical edges protected against disorder. Nontopological superconducting contacts can induce topological superconductivity in this system characterized by zero-energy level crossing of Andreev levels. Both Josephson current and finite-frequency admittance carry then very specific signatures at low temperature of this disorder-protected Andreev level crossing at $\pi$ and zero energy.

Figure 1

Lifting of spin degeneracy of phase-dependent Andreev levels by SOI for ballistic wires ($L\ll l_e$) in the long-junction limit. Upper panel: Schematic tight-binding band structure of a two-channel ballistic wire in the presence of Rashba spin-orbit interactions. The transverse Rashba coupling opens a gap at the two band crossings leading to nonparabolic asymmetric dispersion relations. This distortion is at the origin of different velocities $v^{+}$ and $v^{-}$ when the Fermi energy lies just below this gap. Lower panel, left: $N_y=1$. Andreev spectrum is not modified by SOI. Lower panel, right: $N_y=2$. The Fermi energy is taken at 1/4 of the tight-binding lower 1D band (${\epsilon }_F=-t/2=-2\mathrm{\Delta }$), which corresponds to the bottom of the upper band. Note the breaking of spin degeneracy. (Parameters are $N_x=50,\lambda =3\mathrm{\Delta }$. The number of S sites are, respectively, $N^S=100\times 1$ and $N^S=100\times 2$).

Figure 2

Effect of SOI on the Andreev spectrum of a diffusive ribbon: $\lambda =3\mathrm{\Delta }$ (left), $\lambda =0$ (right). Other parameters are $N_x^N=50\times 20$ normal sites with on-site disorder $W_d/t=1$ corresponding to $L/{ł}_e\simeq 2.5,\mathrm{\Delta }=t/4,N^S=30\times 20$. Note the lifting of spin degeneracy as well as the larger and sharper phase dependence of Andreev levels in the presence of spin-orbit interactions.

Figure 3

Topological Andreev spectrum of a ribbon built with a hexagonal lattice with next-nearest-neighbor spin-orbit interactions. The ribbon is connected to nontopological superconducting electrodes with a square lattice. The Andreev spectrum is shown for $N_x=N_y=20$ with on-site disorder $W_d=t$, corresponding to the diffusive regime in the absence of SOI. The amplitude of SOI is equal to the superconducting gap. Fermi energy is chosen to be ${\epsilon }_F=-0.33t$ and sits in the spin-orbit gap. The spectrum consists of two chiral Andreev levels corresponding to the two edges of the sample (short-junction limit). These states exhibit a linear phase dependence and cross at zero energy at phase $\pi$ in the limit of very wide ribbons. Inset: exponential dependence of the residual gap at $\varphi =\pi$ as a function of the sample width $W=N_{y}a$ for two different values of the superconducting gap $\mathrm{\Delta }=1$ (circles) and $\mathrm{\Delta }=0.5$ (diamonds).

Figure 4

Effect of S contact asymmetry on the topological Andreev spectrum. Bottom panel: Andreev level spectrum of a hexagonal lattice ribbon in the quantum spin-Hall phase with asymmetrical contacts on the edges, showing two distinct sets of Andreev levels corresponding to the two edges. Top panel: Expanded view of the spectrum close to zero energy showing unavoided level crossings between levels of opposite spin and avoided crossings between levels of the same spin corresponding to opposite edges. Parameters are $N_x=10,N_y=60,W_d=t/2=2\mathrm{\Delta },\lambda =2\mathrm{\Delta },{\epsilon }_F=-\lambda /2,t_{SN}^{up}=1,t_{SN}^{lw}=0.1$.

Figure 5

From Andreev spectrum to Josephson current. Top panel: Phase dependence of the first two levels in the Andreev excitation spectrum of a square-lattice tight-binding wire, with Rashba SOI. Parameters are $N_x=130,N_y=4,W_d=0.75t,\lambda =2\mathrm{\Delta }=t/2$. These two levels correspond to opposite spin states and cross each other at 0 and $\pi$. Middle panel: Currents carried by these levels; one can clearly identify, for these two single-level currents, a contribution which is an even function of phase, of opposite sign for the two levels. Bottom panel: Same quantity in the presence of a small Zeeman field perpendicular to the wire $E_Z=0.02$ (in $\mathrm{\Delta }$ units). Avoided crossings at 0 and $\pi$ lead to discontinuities in the single-level currents, which become odd functions of phase.

Figure 6

Effect of spin-orbit interactions on the phase-dependent Josephson current for the square lattice with nearest-neighbor Rashba SOI interactions. Upper panel, left: Ballistic wire with $N_y=2$ (same parameters as Fig. 1). Upper panel, right: Diffusive wire in the long-junction limit with the same parameters as Fig. 2. The amplitude and skewness of the phase-dependent Josephson current are decreased in the presence of SOI for the ballistic wire, whereas they are increased for the diffusive wire. Lower panel: Comparison of the disorder dependence of the Josephson current amplitude with and without Rashba spin-orbit interactions. One observes a weaker disorder dependence in the presence of spin-orbit interactions. The inset shows the increase of the Josephson current with $\lambda$ up to $\lambda =4=t$ in the diffusive regime at $W_d=4$.

Figure 7

Phase-dependent Josephson current for a topological junction compared to a nontopological one: hexagonal lattice with SOI $\lambda =2\mathrm{\Delta }$ and without SOI for different transmissions (uniform along the NS interface). Other parameters are $N_x=10$ and $N_y=60$. Upper panel: In the presence of SOI, the Josephson current exhibits a sawtooth shape with sharp discontinuities at odd multiples of $\pi$, which is characteristic of a 1D ballistic SNS junction. The amplitude of the sawtooth is decreased for imperfect transmission at the NS interface but the discontinuity at $\pi$ is robust. The curves with diamond points correspond to a small disorder $W_d/t=0.25$, whereas green solid lines correspond to larger values and a perfect transmission at the S/N interfaces, $W_d/t=2$ and $W_d/t=3$. Middle panel: The current phase relation (CPR) without SOI, for $W_d/t=0.25$ for different NS interface transmissions $T$ between 0.1 and 1. The CPR turns harmonic as $\tau$ decreases. Lower panel: Contrast of the disorder dependence of the amplitude of the Josephson current for different transmission coefficients for topological and nontopological junctions.

Figure 8

${\varphi }_0$ junction behavior in a topological wire with out-of-plane Zeeman field. The phase-dependent Josephson current in the Kane-Mele model with asymmetric NS coupling between the two edges is shown for different values of $B_Z$, going from 0 to 0.3 from top to bottom (in $\mathrm{\Delta }$ units). Fermi energy is taken at $-t/4$ in the spin-orbit gap. Other parameters are similar to those in Fig. 4, $N_x=20$, $N_y=60$, $\lambda =2\mathrm{\Delta },{\epsilon }_F=-\lambda /2,t_{SN}^{up}=1,t_{SN}^{lw}=0.1$.

Figure 9

Phase dependence of ${\chi }_D^{″}$ for the square lattice (nontopological regime) at a temperature equal to the superconducting gap. As explained in the text, this quantity is close to the squared single-level current, averaged over the whole spectrum. It is nearly $\pi$ periodic in the absence of spin-orbit interactions (lower panel) and insensitive to a small Zeeman field (blue curve $B_Z=0$, red curve $B_Z=0.02$). The same quantities are shown in the presence of spin-orbit interactions $\lambda =2\mathrm{\Delta }$ in the upper panel. Spin splitting and crossings of the energy levels at $\varphi =0$ and $\pi$ give rise to a very different behavior with broad maxima at 0 and $\pi$ reflecting levels crossings, and sharp dips in a Zeeman field. Numerical simulations correspond to $N_x=100$ and $N_y=4$ with a square lattice. The amplitude of disorder is $W_d=3\mathrm{\Delta }$, yielding $L/l_e=3.7$.

Figure 10

Phase dependence of ${\chi }_D^{″}$ computed for the hexagonal lattice in the nontopological (no SOI, upper panel) and topological regimes (with SOI $\lambda =2\mathrm{\Delta }$, lower panels), at temperatures equal to $0.01\mathrm{\Delta }$ (blue) and $0.1\mathrm{\Delta }$ (red). Other parameters are $N_x=10$ and $N_y=60$. In the topological regime, we also show data for $N_y=20$ (crosses). The peak at $\pi$ carries the signature of the disorder-protected level crossing in the spectrum and is very sensitive to the presence of the coupling between the edges when the width of the sample is of the order of the superconducting coherence length. The lowest panel illustrates the effect of a Zeeman field which splits the zero-energy level crossing into two crossings that are symmetric around $\pi$.