Josephson Inductance as a Probe for Highly Ballistic Semiconductor-Superconductor Weak Links

We present simultaneous measurements of Josephson inductance and dc transport characteristics of ballistic Josephson junctions based upon an epitaxial Al-InAs heterostructure. The Josephson inductance at finite current bias directly reveals the current-phase relation. The proximity-induced gap, the critical current and the average value of the transparency $\overline{\tau }$ are extracted without need for phase bias, demonstrating, e.g., a near-unity value of $\overline{\tau }=0.94$. Our method allows us to probe the devices deeply in the nondissipative regime, where ordinary transport measurements are featureless. In perpendicular magnetic field the junctions show a nearly perfect Fraunhofer pattern of the critical current, which is insensitive to the value of $\overline{\tau }$. In contrast, the signature of supercurrent interference in the inductance turns out to be extremely sensitive to $\overline{\tau }$.

Figure 1

(a) Schematic of the Josephson junction array. The actual array is made of 2250 Al islands. (b) Scanning electron micrograph of a portion of the array, taken prior to the deposition of the gate dielectric and of the global top gate. (c) Sequence of the topmost layers for the heterostructure under study. The Al oxide and the Au layer have been lithographically deposited after the wafer growth. (d) RLC resonance spectra for different values of the dc current through the array of Josephson junctions, measured at $T=500\mathrm{mK}$. (e) Circuit scheme of the cold RLC resonator used in this work.

Figure 2

(a) Josephson inductance $L$ versus current bias $I$ measured for different temperatures from 100 to 900 mK (solid lines). Dashed lines show $L$ computed from Eq. (3) with parameters $I_0=5.882\mu \mathrm{A}$, $\overline{\tau }=0.94$, ${\mathrm{\Delta }}_{\mathrm{Al}}=180\mu \mathrm{eV}$, and ${\gamma }_B=1.0$ (see text). (b) Zero-bias inductance $L(0)$, normalized to ${\mathrm{\Phi }}_0/(2\pi I_0)$, plotted versus temperature (symbols), together with the prediction from Eq. (3) for (red curve) ${\mathrm{\Delta }}_{\mathrm{Al}}=180\mu \mathrm{eV}$, ${\gamma }_B=1.0$, and (blue curve) ${\mathrm{\Delta }}_{\mathrm{Al}}=220\mu \mathrm{eV}$, ${\gamma }_B=1.7$. (c) CPR curves obtained by integrating data in panel (a) using Eq. (2). (d) Symbols show a normalized representation of data in panel (a) (see text). Lines show the prediction of the $T=0$ limit of Eq. (3) for selected values of the transparency $\overline{\tau }$. (e) $L(I)$ at $T=100\mathrm{mK}$ plotted for different gate voltage values $V_g$ (solid lines), together with the computed $L(I)$ from Eq. (3) (dashed lines). The number of supercurrent-carrying channels is deduced from the number $N$ in Eq. (4) that best fits the data.

Figure 3

(a) Color plot of the differential resistance plotted as a function of perpendicular magnetic field $B_{\perp }$ and current bias. The dashed yellow line shows the expected critical current $I_c(B_{\perp })$ for a rectangular junction with effective length $a=960\mathrm{nm}$ and width $w=3.15\mu \mathrm{m}$, see inset. (b) Zero-bias Josephson inductance $L$ as a function of $B_{\perp }$ for the central lobe in the diffraction pattern (symbols) together with the curves deduced from Eq. (3) for $\overline{\tau }=0.94$ (red), $\overline{\tau }\to 1$ (blue), and $\overline{\tau }\to 0$ (green). For the latter curve, the parameter $I_0$ has been rescaled by a factor 2.06 to match the measured zero-field inductance [31].