Spin-Orbit Splitting of Andreev States Revealed by Microwave Spectroscopy

We perform microwave spectroscopy of Andreev states in superconducting weak links tailored in an InAs-Al (core-full shell) epitaxially grown nanowire. The spectra present distinctive features with bundles of four lines crossing when the superconducting phase difference across the weak link is 0 or $\pi$. We interpret these features as arising from zero-field spin-split Andreev states. A simple analytical model, which takes into account the Rashba spin-orbit interaction in a nanowire containing several transverse subbands, explains these features and their evolution with magnetic field. Our results show that the spin degree of freedom is addressable in Josephson junctions and constitute a first step towards its manipulation.

Popular Summary

Atomic spectra contain fine structure—split spectral lines that arise from the coupling of the spin of the electron with its orbital motion around the nucleus. Here, we show analogous fine structure in the quantized excitation spectrum of a superconducting electrical circuit that includes a semiconducting nanowire with strong spin-orbit coupling. Remarkably, the spin state of a single electron in the nanowire has a measurable impact on the electrical properties of the circuit, which contains over a trillion electrons.

The circuit consists of a submicron indium arsenide (InAs) nanowire enclosed by a superconducting aluminum loop. Discrete localized states, known as “Andreev bound states,” form in the nanowire as a result of coupling to the superconductor. When absorption of a photon induces a transition between two of these states, the loop inductance changes. We measure the absorption spectrum of the circuit by monitoring the resulting frequency shift of a microwave resonator inductively coupled to the loop. The spectrum shows a fine structure of spin-split Andreev states, well accounted for by a simple model with spin-orbit coupling as the key ingredient.

In the longer term, the challenge will be to manipulate an individual spin in a superconductor—the spin of a single quasiparticle that is physically located at the same place as the many electrons forming the superconducting ground state. Here, we have done the first step: showing that there are spin-resolved states.

Figure 1

Effect of the Rashba spin-orbit coupling (RSO) on Andreev levels. (a) Weak link of length $L$ between superconductors with phase difference $\delta$. Blue star symbolizes a scatterer at position $x_0$. (b) Dispersion relation for a purely one-dimensional weak link in the presence of RSO (green solid lines; labels $\uparrow \downarrow$ indicate spin in the $y$ direction). Density of states of superconducting electrodes is sketched at both ends of the wire. (c) Andreev reflections (AR) at the superconductors couple electrons (full circles) with holes (open circles) of opposite spins and velocities, leading to the formation of ABS. Blue arrows indicate reflections due to a scatterer. (d) Energy of ABS (excitation representation). Thin lines in (c) and (d): transmission $\tau =1$, ABS formed from right-moving electrons and left-moving holes (solid), or the opposite (dashed). Backscattering ($\tau \ne 1$) leads to opening of gaps at the crossings highlighted with blue circles in (d). Resulting spin-degenerate Andreev levels are shown with thick solid lines. (e)–(g) Effect of RSO in the presence of two transverse subbands, only the lowest one being occupied. (f) Dispersion relation (subband spacing and superconducting gap are in a ratio that roughly corresponds to our experiments). Gray solid lines labeled $1\uparrow \downarrow$ and $2\uparrow \downarrow$ are dispersion relations for uncoupled subbands. RSO couples states of different subbands and opposite spins leading to hybridized bands (green solid lines) with energy-dependent spin textures. Fermi level $\mu$ is such that only the lowest energy bands $m_1$ and $m_2$ are occupied. AR couples, e.g., a fast electron from $m_2$ to a fast hole (in black) and a slow electron from $m_1$ to a slow hole (in red). (g) Construction of ABS: Black and red loops are characterized by different absolute velocities. Spins pointing in different directions symbolize spin textures of the bands. Thin red and black lines, solid and dashed in (e),(g): ABS at $\tau =1$ associated with different spin textures. Thick black lines in (e): ABS when crossings highlighted with blue circles are avoided due to backscattering.

Figure 2

Possible parity-conserving transitions in a weak link with spin-split ABS [level positions correspond to the phase difference indicated with an arrow in Fig. 1]. Blue line corresponds to the ground state. (a) Pair transitions. A pair of quasiparticles are created from the ground state, either both in the same manifold (solid arrows) or not (dashed arrows). (b) Single-particle transitions. A quasiparticle already present in one ABS (solid dot) is excited to another ABS, either in the same (dotted arrows) or in another (solid arrows) manifold. (c) Corresponding transition energies as a function of the phase difference $\delta$ across the weak link. (Transitions involving quasiparticles in the continuum are not represented).

Figure 3

Microwave excitation spectrum measured at a gate voltage $V_g=-0.89\mathrm{V}$. The gray scale represents the frequency change $f-f_0$ of a resonator coupled to the weak link when a microwave excitation at frequency $f_1$ is applied as a function of the phase difference $\delta$ across the weak link. In the right half of the figure, some transition lines are highlighted. Red line corresponds to a pair transition; green lines are single-particle transitions.

Figure 4

Experimental setup. (a) False-color scanning-electron-microscope image of the InAs-Al core-shell nanowire. The Al shell (gray) is removed over 370 nm to form the weak link between the superconducting electrodes. A close-by side electrode (Au, yellow) is used to gate the InAs exposed region (green). (b),(c) The nanowire is connected to Al leads that form a loop. This loop is located close to the shorted end of a coplanar wave-guide (CPW) resonator. (d) The CPW resonator is probed by sending through a bus line a continuous microwave tone at its resonant frequency $f_0=3.26\mathrm{GHz}$ and demodulating the transmitted signal, yielding quadratures $I$ and $Q$. Microwaves inducing Andreev transitions are applied through the side gate (frequency $f_1$) using a bias tee, the dc port being used to apply a dc voltage $V_g$.

Figure 5

Excitation spectra at $V_g=0.5\mathrm{V}$. (a) Large-scale spectrum at zero magnetic field. (b) Enlargement of the same data with fits (see text). (c) Dependence of the spectrum with the amplitude $B$ of an in-plane magnetic field applied at an angle of $-45°$ with respect to the nanowire axis. Green lines are fits (see text).

Figure 6

Effect of an in-plane magnetic field on the ABS excitation spectrum around $\delta =0$. The Andreev states correspond to the same gate voltage as in Fig. 5. Field is applied parallel [(a),(c)] or perpendicular [(b),(d)] to the wire. Green lines are the result from the theory using $g_{\perp }=12$ and $g_{\parallel }=8$ (see text).

Figure 7

Effect of an in-plane magnetic field on the band structure (top row), the Andreev levels (bottom row, left) and the excitation spectrum (bottom row, right). (b) Reference graphs at zero field, (a) field applied along the wire axis, (c) field applied perpendicularly to the wire axis. The field effect on the band structure is exaggerated for clarity. The model parameters for the Andreev levels and the excitation spectrum are the same as in Fig. 5 and $B=10\mathrm{mT}$.

Figure 8

Correlation (normalized to maximum value) between finite magnetic field data and theory as a function of the $g$ factor, for field direction parallel (blue) or perpendicular (red) to the nanowire.

Figure 9

(a) Data at $V_g=-0.89\mathrm{V}$, with yellow arrows pointing to transition lines that are replicas of lines appearing exactly 3.26 GHz above. (b) Same data superimposed with predictions of the single-barrier model using parameters corresponding to the spectrum of ABS shown in (c). Single-particle transitions (green lines) between the three manifolds labeled 1, 2, 3 in (c) are visible in the data. Red line in (b) is the pair transition leading to two quasiparticles in manifold 1.

Figure 10

Spectrum as a function of gate voltage at $\delta =\pi$ [right of (a), left of (b)] for two intervals of gate voltage and taken during different cooldowns. (a) Left panel and (b) right panel: Phase dependence at the gate voltages corresponding to $V_g=-0.24\mathrm{V}$ (a) [resp. $V_g=-1.21\mathrm{V}$ (b)], which correspond to the leftmost (resp. rightmost) gate voltages of the panels showing the $V_g$ dependence.