Microwave Signature of Topological Andreev level Crossings in a Bismuth-based Josephson Junction

Demonstrating the topological protection of Andreev states in Josephson junctions is an experimental challenge. In particular the telltale $4\pi$ periodicity expected for the current phase relation has remained elusive, because of fast parity breaking processes. It was predicted that low temperature ac susceptibility measurements could reveal the topological protection of quantum spin Hall edge states by probing their low energy Andreev spectrum at finite frequency. We have performed such a microwave probing of a phase-biased Josephson junction built around a bismuth nanowire, a predicted second order topological insulator, and which was previously shown to host one-dimensional ballistic edge states. We find absorption peaks at the Andreev level crossings, whose temperature and frequency dependencies point to protected topological crossings with an accuracy limited by the electronic temperature of our experiment.

Figure 1

Phase-dependent diagonal susceptibility as a signature of Andreev level crossings. (a) Derivative of the Fermi function entering in the expression of ${\chi }_D^{\prime \prime }\propto -i^2(\phi )\partial f/\partial \epsilon$. (b) $S$-Bi ring threaded by a dc plus ac flux. (c) Topologically trivial Andreev spectrum displaying an avoided crossing at $\phi =\pi$. This leads to a peak splitting of ${\chi }_D^{\prime \prime }$ at $\pi$ since $i(\phi =\pi )=0$. (d) Topologically protected crossing at $\phi =\pi$ with nonzero $i(\phi =\pi )$, leading instead to a peaked ${\chi }_D^{\prime \prime }$ at $\pi$. The variations of ${\chi }_D^{\prime \prime }$ with phase are obtained from the tight binding computation of phase dependent Andreev bound states in an SNS junction on a hexagonal lattice with on site disorder: without SOI in (c) and with next-nearest-neighbor SOI in (d) at temperatures $T=0.01\mathrm{\Delta }$ (blue) and $T=0.1\mathrm{\Delta }$ (red), with $\mathrm{\Delta }$ the superconducting gap (see Ref. [19] for details).

Figure 2

(a) and (b) Principle of the experiment: the Bi nanowire is modeled by an inductance $L_J$ in parallel with a resistance $R_r=1/G_{qp}$ and a $W$ wire symbolized by an inductance $L_W$. The SQUID is inserted in the resonator modeled by a $R$,$L$,$C$ circuit measured in transmission with an inductive coupling to the microwave generator and a capacitive coupling to the cryogenic amplifier. (c) Scanning electronic micrograph of the Bi SQUID sample. (d) Field induced variations of the quality factor and frequency of the resonator’s third eigenmode (average over 50 curves). We have arbitrarily fixed $\delta f=\delta Q=0$ at $B=0$. Note the sharp periodic absorption dips on $\delta Q(B)$ due to the Bi junction whereas the smooth parabolic shift $\delta f(B)$ is characteristic of the field dependent penetration depth of the resonator’s Nb meander lines.

Figure 3

(a) Temperature dependence of the dissipation peak, ${\chi }^{\prime \prime }(\phi )$, around $\pi$ at $f=3.9\mathrm{GHz}$ and $T=0.1$, 0.15, 0.21, 0.27, 0.33, 0.39, 0.54 K from top to bottom (curves are shifted for clarity). Points are experimental data, solid line are fits of Eq. (2) to data taking ${\epsilon }_T$ as an adjustable parameter which is found to vary between 3 and 4 K. (b) and (c) Temperature dependence of the amplitude ${\delta }_{\pi }({\chi }^{\prime \prime })\propto {\delta }_{\pi }1/Q$ and width $\delta \mathrm{\Phi }/{\mathrm{\Phi }}_0$ of the dissipation peaks around $\pi$, for different frequencies (diamonds 0.47 GHz, circles 1.4 GHz, squares 3.9 GHZ, crosses 4.5 GHz). The solid line in (c) corresponds to $T/{\epsilon }_T$ with ${\epsilon }_T=3.5K$, the average extracted Thouless energy from our experimental data. (d) Main panel: relaxation rate $\gamma$ deduced from the experiments of panel b. Inset: frequency dependence of the absorption peak area measured at 100 mK. A reasonable fit with Eq. (3) is obtained taking $\gamma =3\times 10^{10}{\mathrm{s}}^{-1}$, despite the dispersion in the data, mostly due to uncertainties in frequency-dependent calibration parameters.