Giant anomalous Josephson effect as a probe of spin texture in topological insulators

TL;DR

This work demonstrates a giant anomalous Josephson effect in HgTe-based weak links, exploiting the spin-momentum locked surface states of a 3D topological insulator to realize a large $\varphi_0$ shift under in-plane magnetic fields. By combining a carefully engineered asymmetric SQUID with rigorous out-of-plane field compensation and detailed modeling, the authors show that the single-Fermi-contour Dirac-like dispersion of HgTe yields a remarkably large magnetochiral response, with the slope $\partial \varphi_0/\partial B_{\text{ip}}$ reaching $\sim 110\times 2\pi$ rad/T in a particular orientation. Tight-binding simulations and a symmetry-driven spin texture analysis indicate that a non-perpendicular spin-to-momentum angle (approximately $\theta\approx 71^{\circ}$) is responsible for the enhanced $\varphi_0$, consistent with the (013) HgTe surface symmetry. The results establish the anomalous Josephson effect as a sensitive probe of spin texture in chiral 2D systems and offer a transport-based route to quantify spin-momentum locking in topological materials. The approach and findings have broad relevance for identifying materials with strong magnetochiral responses and for advancing topological superconducting platforms.

Abstract

Surface states of topological insulators feature chiral spin-momentum locking. When such states are used as weak link between two superconductors, their spin texture gives rise to the anomalous Josephson effect, i.e., to a $\varphi_0$ shift in the current phase relation. In this work, we explore the anomalous Josephson effect in junctions where the weak link is a HgTe nanowire. We observe a giant anomalous $\varphi_0$-shift of the current-phase relation, which we attribute to the fact that HgTe surface states feature a single Fermi contour. Moreover, by varying the orientation of the in-plane magnetic field, we obtain information about the spin texture in momentum space. In particular, we found that the spin is not exactly perpendicular to the momentum, but shows a significant deviation of 19 degrees. Our results establish the anomalous Josephson effect as a sensitive tool to probe the spin texture of chiral 2D systems.

Figures (18)

• Figure 1: a, Scheme of supercurrent transport mediated by Andreev bound states. The graph in gray (red) shows the situation where counterpropagating quasiparticles have the same (different) momentum, namely, $k_e = k_h$ ($k_e \neq k_h$). For finite accumulated phase $(k_e-k_h)L$ over the junction length $L$, the current-phase relation is shifted by an anomalous phase $\varphi_0$ and it is no longer antisymmetric [$I(\varphi) \neq -I(-\varphi)$]. b-e, Dispersion relation (top) and Fermi surfaces (bottom) for a parabolic (b,d) 2D electron system and a Dirac cone of a single surface of a 3D TI (c,e). The arrows in the Fermi surfaces indicate the spin direction with respect to the momentum, which in the absence of bulk-inversion asymmetry (as for the sketched cases) is $\gamma=90^{\circ}$. If the spin is locked perpendicular to the momentum, then an in-plane field $\vec{B}$ directed along $y$ (d,e) shifts the Fermi contours along the current direction $\hat{x}$, as in panel a, inducing a $\varphi_0$-shift.
• Figure 2: a, Scheme of our asymmetric SQUID device consisting of a reference Al/AlOx/Al tunnel junction (top) and a Nb/HgTe/Nb SNS junction (bottom). A magnetic field applied along the $z$-direction, $B_z$, produces the flux $\Phi$, which modulates the critical current of the SQUID. The red rectangle indicates the surface states wrapping around the HgTe wire. b, Measured current-phase relation (CPR) of the HgTe junction for two distinct samples (see text). c, Modulus of the $n$-th Fourier components (dots) for the two CPRs in b. The exponential decay agrees with the Furusaki-Beenakker predictions (Eq. \ref{['eq:furusaki']}, crosses) in the 1D limit and makes it possible to deduce the average transmission $\tau$. d, CPRs of sample A measured at different temperatures $T$. When increasing the temperature from 100 mK to 700 mK, the relatively stronger suppression of the higher harmonics renders the CPR more sinusoidal.
• Figure 3: Anomalous $\varphi_0$-shift as a function of the in-plane field:a, Anomalous phase shift $\varphi_0$ measured on sample A and normalized to $2\pi$ for the indicated orientation (angle $\gamma =102^{\circ}$) of the in-plane field $\vec{B}_{\text{ip}}$ with respect to the current, namely, the $x$-axis (see inset). b, Slope $\partial \varphi_0/\partial B_y$ measured near $B_y=0$ as a function of the angle $\gamma$, together with a sine fit (fit parameters: horizontal offset $-19^{\circ}$ and amplitude $0.16\cdot 2\pi$/mT). Data in a and b are obtained from different cool-downs. c, Anomalous $\varphi_0$-shift measured on sample B for $B_\text{ip}$ parallel to the $y$-axis. The different symbols refer to eight different measurement runs (see text). d, Exemplary CPRs measured on sample B for four values of $B_\text{ip}$ indicated with arrows in c: $3.48$ mT (blue), $3.72$ mT (green), $3.92$ mT (orange), and $4.16$ mT (red). e, Tight-binding simulations of the $\varphi_0$-shift in an effectively 2D SNS nanowire junction using a linear TI (gray) or a parabolic (orange) single-electron dispersion and applying the in-plane magnetic field $B_y$ perpendicular to the current direction; see text for details. f, Computed CPRs for the linear TI electronic dispersion and magnetic field $B_y$ ranging from $0$ to $95.4 \, \mathrm{mT}$ in steps of $15.9 \, \mathrm{mT}$.
• Figure S1: (a) - Heterostructure layers grown on (013)-GaAs and 4 $\mu$m CdTe containing the 88-nm-thick strained HgTe film, barrier layers of 30-nm-thick Cd$_{\rm 0.65}$Hg$_{\rm 0.35}$Te below and 30-nm-thick Cd$_{\rm 0.65}$Hg$_{\rm 0.35}$Te above HgTe and the 40 nm thick CdTe cap layer. The topological surface states in HgTe are shown by the magenta lines. (b) Simplified electronic band structure of tensile-strained HgTe around the $\Gamma$-point as a function of an in-plane wave vector. The conduction band with bottom $E_c$ is shown in blue, the valence band with top $E_v$ in red and the topological surface states in magenta. The bulk gap $E_g$, opened by strain, is of order 15 meV. The Fermi level position is expected to lie in the valence band near $E_v$.
• Figure S2: (a) SEM image showing sample A after the Nb contact lead lift-off. False color highlights Nb in cyan, and the remaining HgTe layer forming the wire in purple. (b) Zoom SEM image of the contact area, taken at a tilted angle.