Ballistic Graphene Josephson Junctions from the Short to the Long Junction Regimes

We investigate the critical current $I_C$ of ballistic Josephson junctions made of encapsulated graphene–boron-nitride heterostructures. We observe a crossover from the short to the long junction regimes as the length of the device increases. In long ballistic junctions, $I_C$ is found to scale as $\propto \exp (-k_{B}T/\delta E)$. The extracted energies $\delta E$ are independent of the carrier density and proportional to the level spacing of the ballistic cavity. As $T\to 0$ the critical current of a long (or short) junction saturates at a level determined by the product of $\delta E$ (or $\mathrm{\Delta }$) and the number of the junction’s transversal modes.

Figure 1

(a) Map of differential resistance vs current $I$ and gate voltage $V_G$. The data are shown for junction A and taken at a temperature $T=1.5\mathrm{K}$. The superconducting region of zero resistance can be observed around $I=0$. The current through the junction is swept from negative to positive; therefore, the transition at the negative $I$ corresponds to the retrapping current $I_R$, while the transition at the positive $I$ corresponds to the switching current $I_S$. (b) Differential resistance vs bias voltage ($V_B$) for junction A taken at the Dirac point. Several multiple Andreev reflection (MAR) peaks are observed: $2\mathrm{\Delta }$, $\mathrm{\Delta }$, $2/3\mathrm{\Delta }$; with $\mathrm{\Delta }\approx 1.2\mathrm{meV}$. (c) The critical current $I_C$ (top) and the normal conductance of the junction (bottom) plotted vs gate voltage $V_G$ in the hole conduction regime. Both quantities demonstrate Fabry-Perot oscillations and are roughly proportional to each other.

Figure 2

Critical currents $I_C$ plotted on a semi-log scale vs temperature $T$ for junctions A–D. $I_C$ at several gate voltages are presented for each junction; the values of $V_G$ are shown relative to the Dirac point. (a) The data for the shortest junction, A ($L=200\mathrm{nm}$). The gray lines are fitted according to Eq. (1), using the superconducting gap $\mathrm{\Delta }$ extracted in Fig. 1. (b)–(d) $I_C$ vs $T$ for junctions B–D, respectively (see Supplemental Material [16] for junctions E, F, G). The slope of $\log (I_C)$ vs $T$ is independent of $V_G$. In the case of long ballistic graphene junctions, the inverse slope $\delta E$ is expected to be independent of the carrier density and inversely proportional to $L$.

Figure 3

(a) Normal conductance of junctions A–D normalized by the width of the junctions $G_N/W$. Even though the device lengths are different by up to a factor of 10, the three curves are very close to each other, thus proving the ballistic nature of these junctions. At positive $V_G$, $G_N$ of all junctions is found to approach $G_0=N{e}^2/h$ (gray dashed line), indicating consistently high contact transparency for $n$ doping in these devices. Inset: Transmission coefficient $\tau$ of junction A. $\tau$ is calculated via two methods: comparing the normal conductance $G_N$ to the ballistic limit $G_0=N{e}^2/h$ (blue), and fitting the critical current $I_C$ vs temperature $T$ (red). Both methods provide consistent results and indicate high contact transparency for $n$ doping. (b) Energy $\delta E$ extracted from the slope of $\log (I_C)$ vs $T$ for junctions (B–G). As expected in the long junction regime, $\delta E$ depends only on device length $L$ and is almost density independent through both the electron and hole doping. (c) Length dependence of $\delta E$ showing a linear $1/L$ dependence.

Figure 4

(a) The ratio $h{I}_C/Ne{\mathrm{\Delta }}_0$ measured on junction A at $T=300\mathrm{mK}$ as a function of $V_G$. The number of modes $N$ is $W(4\sqrt{n/\pi })$, where $n$ is the carrier density as determined from $V_G$. As the gate voltage increases and the transmission of the graphene-MoRe interfaces approaches 1, the plotted ratio saturates. Inset: Differential conductance vs bias voltage $V_B$ for the 650 nm long junction F, gated to the $p$-doped regime ($V_G=-4.2\mathrm{V}$). The period of the Fabry-Perot oscillations yields the level spacing, $E_0\approx 2\mathrm{meV}$, which is consistent with the expected $E_0=2{\pi }^2\delta E$ ($\delta E\approx 0.1\mathrm{meV}$ for this junction.) (b) The ratio $h{I}_C/Ne\delta E$ for junctions in the long regime (B,C,E,G), measured as a function of $V_G$ at 60 mK. (See Supplemental Material [16] for junction F. Junction D does not yet saturate at the base temperature.) At higher $V_G$ the ratio converges toward a constant value in all junctions.