Supercurrent in the Presence of Direct Transmission and a Resonant Localized State

We study the current-phase relation (CPR) of an InSb-Al nanowire Josephson junction in parallel magnetic fields up to 700 mT. At high magnetic fields and in narrow voltage intervals of a gate under the junction, the CPR exhibits $\pi$ shifts. The supercurrent declines within these gate intervals and shows asymmetric gate voltage dependence above and below them. We detect these features sometimes also at zero magnetic field. The observed CPR properties are reproduced by a theoretical model of supercurrent transport via interference between direct transmission and a resonant localized state.

Figure 1

(a) False-colored SEM image of the nanowire SQUID (left, scale bar $1\mathrm{\mu }\mathrm{m}$) and an enlargement of one arm (right, scale bar 100 nm). Dielectric shadow walls (yellow) define the two Josephson junctions (JJ1 and JJ2, critical currents $I_{c1}$ and $I_{c2}$) in two InSb nanowires, and the superconducting Al (blue) loop on the substrate. White arrows indicate the current directions along the two SQUID arms between the source and drain connected in the four-terminal setup. JJi is coupled to three underlying gates at voltages $V_{Li}$, $V_{Gi}$, and $V_{Ri}$ ($\mathrm{i}=1$,2). Magnetic fields $B_z$ and $B_y$ are applied along JJ1 and out of plane, respectively. (b) $V$ as a function of $I_b$ and $B_y$ at $B_z=0\mathrm{mT}$ and $B_z=600\mathrm{mT}$, overlapped by $I_{sw}$ traces (red). The gate settings are $V_{G1}=2.36\mathrm{V}$, $V_{L1}=V_{R1}=1.5\mathrm{V}$, $V_{G2}=2.88\mathrm{V}$. and $V_{L2}=V_{R2}=1.75\mathrm{V}$.

Figure 2

(a) $dI/d{V}_b$ as a function of $V_b$ and $V_{G1}$ at $B_z=0\mathrm{mT}$ and $V_{L1}=V_{R1}=1.5\mathrm{V}$. JJ2 is pinched off by setting $V_{G2}=V_{L2}=V_{R2}=0\mathrm{V}$. A line cut (gray) with sharp coherence peaks and broad resonance (red dashed line) peaks is shown below. (b) $I_{sw}$ as a function of $B_y$ and $V_{G1}$ at $B_z=0\mathrm{mT}$ and $V_{L1}=V_{R1}=1.5\mathrm{V}$. JJ2 is turned on by setting $V_{G2}=2.88\mathrm{V}$ and $V_{L2}=V_{R2}=1.75\mathrm{V}$. Horizontal line cuts (red, blue, and black) are shown below and a vertical line cut (purple) is displayed on the right. (c) Same as (b) but for $B_z=600\mathrm{mT}$.

Figure 3

$I_{sw}$ as a function of $B_y$ and $V_{G1}$ at (a) $B_z=0\mathrm{mT}$ and (b) $B_z=490\mathrm{mT}$ (left) and $B_z=720\mathrm{mT}$ (right). The other gate voltages are $V_{L1}=V_{R1}=1.75\mathrm{V}$, $V_{G2}=2.88\mathrm{V}$, and $V_{L2}=V_{R2}=1.75\mathrm{V}$. Horizontal line cuts (red and blue) are shown below and a vertical line cut (purple) is shown on the right.

Figure 4

(a) Schematic of the electron distribution in a nanowire JJ (top). Schematic of the two-dot model: the red dot (energy $E_1$, charging energy $U$) and the black dot (energy $E_2$, no charging energy) (bottom). Tunneling couplings to the leads (left, right) and between the dots are taken in ratios ${\mathrm{\Gamma }}_1^L:{\mathrm{\Gamma }}_1^R:\kappa =0.001:0.001:1$ and ${\mathrm{\Gamma }}_2^L:{\mathrm{\Gamma }}_2^R:E_2=0.33:0.67:1.2$. (b) $G$ as a function of $E_1$ for $B=0$ (black) and $B=2\mathrm{\Delta }$ (red). (c) $I_c$ (units $2e\mathrm{\Delta }/\hslash$) as a function of $\varphi$ and $E_1$ for $B=0$. Horizontal line cuts (blue, red, and black) are shown below, and a vertical line cut (purple) is shown on the right. (d) Analogous to (c), but for $B=2\mathrm{\Delta }$. (e) $E_{\pi }$ as a function of $E_1$ for different $B$. In (b)–(e), the charging energy is neglected by taking $U=0$.