Relating Andreev Bound States and Supercurrents in Hybrid Josephson Junctions

We demonstrate concomitant measurement of phase-dependent critical current and Andreev bound state spectrum in a highly transmissive InAs Josephson junction embedded in a dc superconducting quantum interference device (SQUID). Tunneling spectroscopy reveals Andreev bound states with near unity transmission probability. A nonsinusoidal current-phase relation is derived from the Andreev spectrum, showing excellent agreement with the one extracted from the SQUID critical current. Both measurements are reconciled within a short junction model where multiple Andreev bound states, with various transmission probabilities, contribute to the entire supercurrent flowing in the junction.

Figure 1

(a) Schematic representation of the energy spectrum of a planar JJ. Two superconducting leads with a BCS-like DOS and induced gap ${\mathrm{\Delta }}^{*}$ laterally confine a normal region in the 2DEG, in which discrete ABSs form within the induced superconducting gap. (b) Schematic representation of the device under study and its measurement setup. (c) False-color scanning electron micrograph of the device under study. The insulating substrate is gray, the semiconductor is red, and the thin Al film is blue. Top gates, colored yellow, were deposited over an insulating ${\mathrm{HfO}}_2$ layer (not visible). A metallic strip line, not used in this Letter, was deposited on the left-hand side of the loop. Scale bar is $2\mu \mathrm{m}$. (d) Enlargement close to JJ1. A central top gate tuned the density in the normal region of JJ1, two additional top gates confined electrons below the superconducting leads and electrostatically defined a tunneling constriction between the normal region of JJ1 and a superconducting plane. Scale bar (white) is 500 nm. (e) Schematic cross section of JJ1. A negative gate voltage applied to the lateral gates depleted lateral InAs regions, leaving a conducting electron layer solely below the Al contacts and within the normal region of JJ1.

Figure 2

(a) Loop resistance $d{V}_2/d{I}_2$ as a function of out-of-plane magnetic field $B_{\perp }$ and dc current $I_0$. Gate voltage $V_{G2}$ was set to $-3\mathrm{V}$, so that a current can only flow in JJ1. (b) As in (a), but with the supercurrent flowing in both junctions. Notice the different magnetic field scale, highlighting supercurrent oscillations with a characteristic period consistent with one flux quantum impinging the loop area. The vertical dashed line marks the point with zero external magnetic field $B_{\perp }$. (c) Tunneling conductance $d{I}_1/d{V}_1$ as a function of bias voltage $V_{\mathrm{dc}}$, measured for the same magnetic field range as in (b). (d) Density of states within the normal region of JJ1, computed from the data in (c) using a deconvolution algorithm.

Figure 3

(a) SQUID critical current as a function of out-of-plane magnetic field $B_{\perp }$ [blue dots, extracted from Fig. 2] together with the critical current calculated based on the data in Fig. 2 (green) and a fit to the full numerica model (red). The fit used the effective transmissions ${\overline{\tau }}_1$ and ${\overline{\tau }}_2$ as free parameters. Blue data points have been horizontally shifted by $170\mu \mathrm{T}$ in order to match the other curves. The vertical blue marker shows the position where $B_{\perp }=0$ occurred in the raw data. (b) Phase differences ${\phi }_1$ and ${\phi }_2$ across JJ1 and JJ2, respectively. Dashed lines refer to the simplified model, which considers $I_{C}2\gg I_{C}1$ and no loop inductance. Solid lines are calculated using our full model and the fit result of (a). (c) Energy of the most transmissive ABS visible in Fig. 2 (blue dots) together with a fit to Eq. (1) that includes the nonlinear mapping between ${\mathrm{\Phi }}_{\mathrm{ext}}$ and ${\phi }_1$ shown in (b) (red line). The dashed black line is a fit to Eq. (1) using the simplified assumption ${\phi }_1=2\pi {\mathrm{\Phi }}_{\mathrm{ext}}$. In this case, the fit was forced to match the data in the points of highest energy.