Introduction

Diverse nanometre-scale hybrid superconducting devices1,2, that is, two superconductors (S) coupled by a normal-conducting (N) insert, have been utilized for field-effect supercurrent transistors3,4,5, quantum electron pumps6, gate-tuneable quantum systems7 and, more recently, hybrid topological-insulator systems in search for the Majorana fermionic excitation states8,9. However, these efforts have often been hindered by non-ideal S–N contact characteristics and weak superconducting coherence, attributed to the short electronic mean free path (l) and/or a short superconducting coherence length (ξ), compared with the channel length (L) between the two superconducting electrodes.

A promising scheme for realizing short (L<ξ) and ballistic (L<l) Josephson junctions (JJs) is combining superconductors with the two-dimensional electron gas (2DEG) in semiconductor heterostructures with remarkably long l (=10–100 μm). However, detrimental etching processes, required to access the 2DEG layer imbedded deep inside the heterostructure or to remove the native surface oxides, seriously reduces l of the contact10 down to ~10 nm. This results in the diffusive behaviour of the JJs, although the junction channel retains the ballistic character11. Recently, graphene has been adopted for JJ fabrication5,7,12, where charge carriers behave as massless chiral relativistic particles with Cooper pair formation between time-reversal symmetric carriers in opposite valleys. Even for graphene, with high carrier mobility and good superconducting contacts, however, a graphene-based JJ remains diffusive in nature. The major obstacle is that the junction channel length (L=30–300 nm), in conventional planar geometry, cannot be arbitrarily shortened but is limited by the resolution of the electron (e)-beam nanofabrication method3,4,5,7,13. In atomic break junctions14,15, L can be reduced to the atomic scale. But the limited number of conducting channels in the break junction leads to substantially weakened Josephson coupling, making the junction prone to enhanced fluctuations14, both quantum and classical. In addition, independent control over multiple atomic constrictions on a single substrate would be unviable, making it difficult to scale up the devices for practical applications.

Here we demonstrate the fabrication and transport characterization of highly transparent vertical graphene-based JJs with atomic-scale channel length and the scalability along lateral directions. Our devices show the short ballistic Josephson nature with strong Josephson coupling reaching the theoretical limit, the temperature dependence of the Josephson critical current and the phase dependence of the Josephson current that are consistent with the theoretical prediction.

Results

Vertical graphene Josephson junctions

In this study, we used defect-free and single-atom-thick exfoliated graphene as a normal-metallic insert for a graphene JJ (Fig. 1c) in a vertical geometry (Fig. 1b) to overcome both the lithographical limitations in planar geometry (Fig. 1a) and poor scalability of the atomic break junctions (see Methods for fabrication details). Scanning transmission electron microscopy (Fig. 1d) and electron energy loss spectroscopy (Fig. 1e) images of the same junction cross-section confirmed that the monolayer graphene formed an atomically sharp interface with the Ti adhesion layers. Indeed, vertical graphene JJs (vGJJs) have an atomically short channel length of L=0.4 nm, corresponding to the thickness of a single graphene layer, with scalable lateral dimensions (Supplementary Fig. 1). In addition, the excellent chemical inertness of graphene and e-beam nanofabrication for direct deposition of metallic electrodes on both sides of the graphene, combined with the ‘flip-transfer’ technique16, allowed achieving almost perfect transparency of the S–N contacts.

Figure 1: Vertical graphene Josephson junction.
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(a) In a planar-junction geometry, the reduction of the junction length, L, is limited by the roughness of the electrode edges. (b) In a vertical-junction geometry, L is replaced by the thickness of a single graphene layer, even with rough electrode edges. (c) Scanning electron microscopy image of four nominally identical vertical graphene Josephson junctions (vGJJs). The behaviour of the junction JJ2 was described in detail in the text; MAR was measured in the other junctions (JJ1 and JJ3) to estimate superconducting gap. Monolayer graphene, whose boundary is denoted by a green-dashed line, is sandwiched between the top and bottom Ti/Al/Au superconducting electrodes. In a four-probe measurement setup, the current was biased between I+ and I−, along with simultaneous measurements of the voltage drop between V+ and V−. Scale bar, 5 μm. (d) High-resolution bright-field spherical-aberration-corrected scanning transmission electron microscopy (STEM) image of the cross-section of a vGJJ. The monolayer graphene sheet was atomically in contact with the titanium layer. The highest-intensity peak at the interface corresponded to a width of ~0.44±0.01 nm, which was identical to the thickness of monolayer graphene. Scale bar, 5 nm. (e) Electron energy loss spectroscopy image of the same area as the STEM image in d. Red (yellow) colour denotes the titanium (carbon) element. The monolayer graphene consisting of carbon atoms was sandwiched by two 8-nm-thick titanium adhesion layers. Scale bar, 5 nm.

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Strong Josephson coupling in vertical graphene Josephson junctions

Below the superconducting critical temperature of the electrodes (Tc,b=0.75 K and Tc,t=1.00 K for the bottom and top electrodes, respectively), the proximity effect1,2 induces superconductivity in the graphene layer along with Josephson coupling, represented by the current–voltage (I–V) characteristics of vGJJ (JJ2) in Fig. 2a. As the bias current increased, the zero-resistance supercurrent state abruptly jumped to the resistive state at the junction critical current of Ic=13.3 μA. Above the critical currents of both electrodes (Ic,b and Ic,t for the bottom and top electrodes, respectively), the I–V curves became linear crossing the origin with the normal-state resistance of RN=21.4 Ω (inset of Fig. 2a). We will see in Fig. 5 that, according to the first-principle calculation for the atomic structure of vGJJ, a potential barrier emerges at a graphene/Ti interface, which makes a main contribution to RN. Whereas, Ti/Al metal-to-metal interfaces are highly transparent and lead to a very low RN (~0.1 Ω) in an Al/Ti/Al junction of a control experiment (see Methods).

Figure 2: Josephson coupling through graphene.
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(a) Current–voltage (I–V) characteristics of the junction JJ2, measured at the base temperature of 50 mK with a current sweep from negative to positive (blue dots) and vice versa (red dots), exhibited hysteretic behaviour, which may have been of thermal origin38. Critical currents of the JJ (Ic), the bottom electrode (Ic,b), the top electrode (Ic,t) and the retrapping current (Ir) are denoted by arrows. Ic and Ic,b coincide with each other at the base temperature. (Inset) I–V characteristics using an expanded scale show the critical current of the top electrode (Ic,t), above which the I–V characteristics exhibited linear behaviour, represented by the red line passing through the origin with normal-state resistance, RN. Ic,b and Ic,t may have been decreased by self-heating in the junction, because they appeared after the JJ switched to the resistive state, whereas Ic itself was free from self-heating. (b) Shapiro steps under microwave exposure of frequency fmw=17 GHz and amplitude P1/2=1.8 (a.u.), occurring in steps of 35.8 μV. (c) Measured ΔV under microwaves of various fmw (symbols) showed good agreement with the ac Josephson relation of ΔV=(h/2e)fmw (line). (d) Colour-coded plot of dV/dI as a function of the bias current and P1/2 at a fixed frequency of fmw=17 GHz; higher-order Shapiro steps were observed.

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Figure 5: Calculation of interfacial potential barriers.
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(Upper panels) Electrostatic potential ‹V›(z) averaged over the xy plane (a) for graphene (G)/titanium (Ti)/aluminium(Al), and (b) for G/Al. The Fermi level is adjusted to zero. (Lower panels) Atomic structure of G (yellow)/Ti (blue)/Al (purple). (c) ‹V›(z) at Ti/G/Ti structure with varying distance between G and Ti layers (upper panel) for Δd=0–1.0 Å in steps of 0.1 Å and (lower panel) for Δd=1.0~5.0 Å in steps of 0.5 Å. (d) Numerical calculations of quantum tunnelling probability P through the potential barrier at G/Ti interface as a function of Δd. Red dashed line represents P through the potential barrier of direct interface between G and Al layer.

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The IcRN product is an important junction parameter that represents the characteristic quality of a JJ, without the sample-specific geometrical factors2. The IcRN product reaches 285 μV, corresponding to 2.59Δ0/e, with superconducting gap Δ0=110 μeV determined by measurements of the multiple Andreev reflection (MAR; see Methods). Here, Δ0 (=2ΔtΔb/(Δt+Δb)) is the combined superconducting gap energy17 for the unidentical gap energy of top (Δt) and bottom (Δb) aluminium electrodes. The normalized value of eIcRN/Δ0 (=2.59) exceeds the theoretical limit [(eIcRN/Δ0)max=2.07] for a JJ in the short (L<ξ) and diffusive (L>l) regime1,18, despite the fact that the observed Ic may have been underestimated owing to the quantum and thermal fluctuations2, and the instrumental noise. The normalized IcRN product is also in sharp contrast to the maximal values reported for superconductor–normal conductor–superconductor (S–N–S) proximity JJs incorporating 2DEG (ref. 19) (~0.9), single-crystalline nanowires20 (~1.5), graphene5 (~1.0), suspended graphene12 (~0.3) and graphene-based superconducting quantum interference device (SQUID)21 (~1.0), all of which are substantially below the upper limit (2.07) for a short diffusive JJ. The large IcRN product of our vGJJ, in sharp discord with the short diffusive character, is explicable by the short ballistic character1,22 (L<ξ, l) of the junction, where the value of eIcRN/Δ0 can reach π maximally but is reduced depending on the Ic-reduction factor α introduced below. It would be a subtle issue to consider the atomic vertical transport of vGJJ in terms of the mesoscopic parameters such as l and ξ. vGJJ turns out to be a unique system in which its vertical transport is microscopic in nature while the in-plane dimensions are in mesoscopic scales. Here, we extend the well-established mesoscopic theory of proximity JJs to describe vGJJ as an extreme case of atomically short L. As shown later, this approach successfully describes the observed features of vGJJs with short ballistic characters of JJs. We consider the Ti layers as parts of the superconducting electrodes with the proximity-induced gap that is the same as that of adjacent Al layers, since Ti layers are much thinner than ξ in Ti, ~140 nm (see also the solutions for the Usadel equation in Discussion). The dependence of IcRN product on sample-specific contact characteristics is also discussed in Supplementary Fig. 2.

Applying microwave irradiation of frequency fmw results in a series of quantized voltage plateaus (known as Shapiro steps2) at Vn=nhfmw/2e in the I–V curves, as shown in Fig. 2b (n is an integer and h is Plank’s constant). The appearance of Shapiro steps by the ac Josephson effect rigorously confirms that the supercurrent originated from Josephson coupling, rather than artefacts such as electrical shorting across the junction. In addition, the in-plane magnetic field dependence of Ic, known as the Fraunhofer pattern2, further supports the establishment of genuine Josephson coupling (Supplementary Fig. 3). In Fig. 2c, the voltage step height, ΔV, and fmw in the range of 5–25 GHz showed good agreement with the expected linear relationship, ΔV=(h/2e) fmw. Shapiro steps, represented as zero differential resistance (dV/dI=0) in Fig. 2d, exhibited quasi-periodic oscillations with increasing microwave amplitude P1/2 in the current range bounded by Ic,b (<Ic,t), above which the bottom electrode becomes normal, losing Josephson coupling along with disappearance of the Shapiro steps. Figure 2c also reveals fractional Shapiro steps for n*=n/2 and n/3, which implies a nonsinusoidal current–phase relation (CPR) for the junction, which will be further discussed later (see also Supplementary Fig. 4).

Temperature dependence of the critical current

Having established strong Josephson coupling in the vGJJ, we now discuss its temperature (T) dependence in detail. The T dependence of Ic, identified as the bright local maximum curves of dV/dI in Fig. 3a, showed two uncommon features. One is that Ic appeared immediately below the electrode critical temperatures (in this case, Tc,b), that is, the junction critical temperature (Tc,JJ) is identical to Tc,b, indicating the establishment of strong Josephson coupling in the vGJJ. In ordinary proximity JJs, Tc,JJ is noticeably below the critical temperature of the electrodes (ΔT~0.3 K, ~0.1 K and ~1.6 K for Nb-2DEG-Nb (ref. 19), Al-nanowire-Al20 and Pb-graphene-Pb JJs23, respectively) because sufficiently strong superconductivity of the electrodes is required for discernible Josephson coupling to be established. Another uncommon feature is that Ic decreases with a convex-shaped T dependence, that is, d2Ic(T)/dT2<0, up to Tc,JJ. Whereas, diverse proximity JJs studied to date have shown a concave-shaped or exponentially decaying Ic(T) for T close to Tc,JJ, which is a typical long-junction behaviour. As L is increased with thicker graphite flakes, the convex-shaped Ic(T) gradually changed to a concave-shaped or exponential decay (Fig. 3b and Supplementary Fig. 5). The convex-shaped Ic(T), especially close to Tc,JJ, was predicted uniquely for JJs in the short-junction limit22 in the 1970s; however, its experimental observation has seldom been reported except for a few recent reports in high-Tc edge JJs, although values of the IcRN product of the junctions fall far below the gap value.

Figure 3: Temperature dependence of the junction critical current.
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(a) Colour-coded plot of dV/dI, measured with a current sweep from negative to positive as a function of temperature T. Tc,b(t) was determined as the temperature where the critical current Ic,b(t) vanished at the interface between the bottom (top) electrode and graphene. Above the critical temperatures of Tc,b and Tc,t at the bottom and top electrodes, respectively, dV/dI became equal to the normal-state resistance, RN. (b) Experimentally measured Ic (blue symbols) of monolayer graphene vGJJ (JJ2), along with the best-fit curve to the short ballistic junction characters (blue line). Temperature dependences of Ic for vGJJs made of five-layer graphene (red symbols) and 43-nm-thick graphite (green symbols). Lines are provided as guides.

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For more compelling evidence for the short ballistic character of vGJJ, we performed a quantitative analysis on Ic(T). From a microscopic viewpoint, the Josephson current in the ballistic regime is carried by discrete energy states of Andreev-reflected coherent electron–hole pairs, referred to as an Andreev-bound state (ABS)24. This ABS has recently been experimentally demonstrated as a new test bed for quantum information devices, that is, the Andreev-level qubit25. For L<ξ, Josephson coupling is established by a single pair of ABS per conducting channel in the graphene layer, with an energy of , where τ is the junction transparency, and δ is the macroscopic quantum phase difference between the two superconducting electrodes. Here, τ is the ensemble-averaged transparency out of ~RQ/RN~103 conducting channels, representing the overall behaviour of the junction (RQ=h/e2 is the quantum resistance). In thermal equilibrium at T, the Josephson current is given by , where is the Josephson current carried by an ABS pair and is the Fermi–Dirac distribution function (ħ=h/2π and kB is Boltzmann constant). Consequently, the CPR of a short ballistic JJ is given by:

In a current-biased configuration, δ can have an arbitrary value, where Ic corresponds to the maximum value of IJ with respect to δ, that is, . As shown in Fig. 3b, the short ballistic character (black solid curve), obtained from equation (1), is in excellent agreement with the experimental data (symbols), showing a convex-shaped Ic(T) with two best-fit parameters: τ (=0.98) and α (=0.93). High transparency almost reaching the ideal value of τ=1 reflects the high quality S–N interfaces in our device. Here, α parameterizes the reduction of measured Ic compared with the theoretical limit. The 7% reduction can be accounted for by premature switching due to fluctuations2 and/or instrumental noise, the Fermi-velocity mismatch at the S–N interfaces or the inverse proximity effect on the electrodes by graphene26.

Current-phase relation measurements

For a rigorous confirmation of the short ballistic nature of the vGJJ and its potential application to Andreev-level qubits, it is essential to directly measure the CPR, IJ(δ), which is closely related to the energy spectrum of the ABS. According to equation (1), a short ballistic JJ with τ=1 has a highly skewed nonsinusoidal CPR (red curve in Fig. 4c), distinctively different from the CPR of a short diffusive JJ (blue curve; calculated from the Usadel equations given in ref. 18) and the sinusoidal CPR of a conventional tunnelling JJ (tJJ) in the limit of τ=0 (green curve). Since the current-bias measurement is not sufficient for the CPR measurements, we performed phase-sensitive SQUID interferometry with varying δ, which captured the essence of the microscopic processes related to Josephson coupling15. The CPR measurement setup was based on an asymmetric dc-SQUID (Fig. 4a), which consisted of an Al/AlOx/Al tJJ of phase difference γ as a reference junction and a vGJJ as the subject junction under investigation of CPR [IJ,vGJJ(δ)], imbedded in a superconducting loop. The phase difference in the vGJJ, imposed by the relationship δ=γ+2πΦa/Φ0, was controllable by an external magnetic flux, Φa, threading the SQUID loop (see Methods for the screening effect correction). Here, Φ0=h/(2e) is the magnetic flux quantum. In the limit of a much larger critical current for the tJJ (Ic,tJJ), compared with that of the vGJJ (), the critical current of the SQUID, Ic,SQ=max[Ic,tJJsinγ+IJ,vGJJ(δ)], is dominated by the tJJ for γ~π/2, such that Ic,SQ(Φa)~Ic,tJJ+IJ,vGJJ(2πΦa/Φ0+π/2). Thus, in principle, the magnetic flux dependence of Ic,SQ directly represents the CPR of the vGJJ.

Figure 4: Measurements of the current-phase relation.
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(a) Schematic diagram of a dc-superconducting quantum interference device (SQUID) containing a tJJ of phase difference γ and a vGJJ of phase difference δ in a superconducting loop. (b) Optical micrograph from the SQUID interferometer used in this study. The tJJ and monolayer graphene are denoted by black- and red-dotted lines, respectively. Scale bar, 5 μm. (c) A highly skewed CPR for an ideal short ballistic JJ with τ=1 (δmax=π; red line), a skewed CPR for a short diffusive JJ (δmax=0.63π; blue line), and a sinusoidal CPR for tJJs with τ=0 (δmax=π/2; green line). δmax, denoted by an arrow for each case, represents the phase difference at which IJ is maximized. (d) Experimentally measured magnetic field dependence of the critical current of the SQUID at the base temperature, Ic,SQ (symbols), was in very good agreement with the expected variation calculated by the short ballistic theory of a JJ (red line) with fitting parameter τ=0.99. The theory for a short diffusive JJ (blue line) cannot account for the highly skewed experimental data.

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In the SQUID interferometer shown in Fig. 4b, the tJJ was designed to have a much larger Ic,tJJ (=44.1 μA) than Ic,vGJJ (=5.1 μA). Ic,SQ as a function of Φa (symbols in Fig. 4d) exhibited a highly asymmetric, skewed CPR of the vGJJ, which was well described by the best-fit Ic,SQ(Φa) for a short ballistic JJ calculated using equation (1) (red curve), with the fitting parameter τ=0.99. However, the short diffusive character (blue curve) could not account for the high skewedness of the observed Ic,SQ(Φa). Thus, our CPR measurements provided irrefutable evidence for the short ballistic nature of vGJJs and the spectrum of the ABS. Qualitatively, the same behaviour was consistently obtained from other vGJJs prepared in a similar fashion (Supplementary Fig. 6).

Calculation for the potential barriers at the interfaces

To examine the characteristics of the interfaces, we performed the first-principles calculations using Vienna Ab Initio Simulation package27. We adopted the generalized gradient approximation28 augmented with the Tkatchenko–Scheffler van der Waals correction29 for the exchange-correlation of electrons. The cutoff energy for the plane wave-basis expansion was set to be 400 eV. The atomic relaxation was continued until the Helmann–Feynman forces acting on the atoms were <0.01 eV Å−1 (ref. 30). Here, to investigate potential barriers, if any, that emerge at graphene (G)/Ti and Ti/Al interfaces, we modelled atomic structure of vGJJ that consisted of 2 × 2 unit cell of graphene/eight-atom-thick Ti(0001)/three-atom-thick Al(111) slabs stacked along the z axis (visualized in the lower panel of Fig. 5a), and the vacuum region between the slabs was set to 15 Å. The grid for Brillouin zone sampling was set to 15 × 15 × 1 for relaxation and to 31 × 31 × 1 for charge density calculation. Calculated electrostatic potential ‹V›(z) averaged over the xy plane is shown in the upper panel of Fig. 5a with the same length scale as in the lower panel. There appears a narrow potential barrier (red dotted circle) above the Fermi level with height (U) of ~170 meV and width (s) of 0.22 Å. The barrier is located between the first and the second Ti layers. This is mainly owing to the expansion of Ti atomic distance as a result of the charge transfer from Ti to G. On the other hand, at the Ti/Al interface (blue dotted circle), there is no potential barrier, which implies an ideal contact between two metallic materials. In the case of G/Al interface in Fig. 5b, however, there appears a much larger and wider potential barrier than that in G/Ti. This justifies that Ti constitutes an adhesion layer between graphene and aluminium electrodes. For a more detailed analysis of the potential barrier at G/Ti interface, we investigated dependence of ‹V›(z) on the distance (d) between G and Ti slabs in terms of Δd≡d–deq with deq (=2.24 Å), the equilibrium distance (Fig. 5c). As Δd increases, the potential barrier between the first and the second Ti layers gradually decreases and vanishes above Δd~0.3 Å, but a new barrier appears at the G/Ti interface with U saturating to the workfunction of Ti (~4.3 eV). At sufficiently low temperature (kBT~4 μeV≪U) and low bias voltage (Vb~10 μeV≪U), electronic transport though the potential barrier solely arises from the quantum tunnelling. With slowly varying potential U(z), tunnelling probability P can be numerically calculated using Wentzel–Kramers–Brillouin (WKB) approximation as:31

where the average potential is:

the dimensionless correction factor is:

and m is the electron mass. s1 And s2 represent the start and the end point of the potential, respectively. In Fig. 5d, calculated P exhibits an exponential decay (black symbols) with Δd. However, it is close to unity near the equilibrium point (Δd<0.3 Å), which indicates the ohmic transport through the barrier with high transparency rather than tunnelling conduction. This supports that the vGJJ behaves as a S–N–S JJ, rather than a tJJ. Compared with transparent G/Ti interface, G/Al shows much suppressed P~0.1 (red dotted line in Fig. 5d).

Discussion

We clarify the role of graphene in vGJJ by comparing two different types of vertical JJs with (JJwG in Fig. 6a) and without (JJwoG in Fig. 6a) graphene insertion on the same substrate. Here, the graphene was a bilayer. The JJwG shows typical I–V curves of JJ with finite critical current and normal resistance of RN~30 Ω as shown in Fig. 6b. However, the JJwoG shows an insulating behaviour with much larger RN ~3 kΩ (Fig. 6c). During the device fabrication processes of JJwoG, the part of Ti layer that has not been covered with graphene and exposed to the ambient environment was oxidized as shown in the schematic plot of Fig. 6c and gave degraded interfacial characteristics32,33. This implies that even the holes and defects that may have been accidently present in the graphene could not short the junction. Next, we removed the oxidized titanium layer by in situ Ar ion beam etching and evaporated 70-nm-thick Ti layer and 300-nm-thick Al layer in sequence, fabricating oxide-layer-free Ti-based vertical junction (Ti-vJ), shown in Fig. 7a. At the base temperature, Ti-vJ showed much smaller RN (~0.1 Ω) and much larger Ic (>60 μA) compared with typical values of vGJJs (Fig. 7b), which indicates that the normal-metallic thin Ti layer showed a resistance too low to produce a ‘weak’-link JJ. In other words, the Ti-vJ as a whole constituted a single Al superconductor rather than a weak link JJ. In contrast, a graphene layer, although atomically thin, gave an effective weak link for the short ballistic proximity JJ. This emphasizes the uniqueness of graphene as an insertion material between superconductors for realizing the short ballistic proximity Josephson coupling.

Figure 6: Vertical graphene Josephson junctions with and without graphene insertion.
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(a) Optical image of vGJJ devices with graphene (JJwG) and without graphene (JJwoG). Bottom electrode looks brighter than yellowish-coloured top electrode. Scale bar, 10 μm. (b) I-V curves of the JJwG at various temperatures and the schematic of JJwG. (c) I-V curves of the JJwoG show an insulating behaviour similar to those of a superconductor–insulator–superconductor junction with the Josephson current completely suppressed. Schematic structure of JJwoG is shown.

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Figure 7: Titanium-based vertical junction.
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(a) Optical image of Ti-vJ device is denoted by an arrow. Red dotted line outlines the region of few-layer graphene. Scale bar, 10 μm. (b) Current–voltage (I–V) curves of the Ti-vJ measured in zero magnetic field, B (black curve), and B=150 G, which is larger than the critical magnetic field of aluminium, Bc (red curve). Schematic structure of Ti-vJ is shown on the right panel.

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We examine whether the Ti adhesion layer in a vGJJ should be considered as a normal metal or a superconductor by numerically calculating the proximity-induced superconducting gap Ti (ΔTi) using quasi-classical Green’s function method. Usadel34 applied Eilenberger’s quasi-classical equilibrium theory35 to dirty superconductors adopting impurity-averaged Green’s functions. For a more convenient interpretation introduced by Nazarov36, these Green’s functions are parameterized by two complex angles, the polar angle and the azimuthal angle ϕ(x, E). Here x and E represent the position and energy, respectively. determines the pair correlation strength and ϕ(x, E) represents the superconducting phase as a function of x and E. Local superconducting properties can be quantified from the solution of the Usadel’s equation:

with proper boundary conditions specified at a given structure of superconducting and normal materials. Here, D=vFl/3~33 cm2 s−1 is the diffusion constant of three-dimensional normal metal with the Fermi velocity of vF~106 m s−1 and the mean free path of l~10 nm for thermally evaporated Ti. Δ(x) represents the position-dependent superconducting gap of the materials themselves. For the S–N–S structure (Fig. 8a) consisting of Al and Ti without graphene layer, Δ(x) vanishes in the Ti layer but remains finite and uniform for the Al layers, that is, Δ(−L/2<x<L/2)=0, Δ(x>L/2 or x<-L/2)=ΔAl. L is the thickness of the Ti layer. By changing the variable , equation (5) transforms into a real second-order differential equation:

Figure 8: Proximity-induced superconducting gap.
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(a) Proximity-induced superconducting gap, Δind, is denoted by the red dashed line for aluminium (Al)/titanium (Ti)/Al JJ. L is the junction length between the two Al layers. (b) Normalized LDOS at the centre of Ti layer calculated for the structure of a. Inset, L dependence of ΔTi=Δind (0). Black and red arrows indicate the case of L=16 and 70 nm, respectively. (c) Schematic configuration of a vGJJ and Δind (red dashed line).

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for the region of Ti layer (−L/2<x<L/2) with boundary conditions at the interfaces as:

The local density of states (LDOS) at position x and energy E is given by

where N0 is LDOS of the normal Ti layer. LDOS at the centre of Ti layer (x=0) is numerically solved with varying L as shown in Fig. 8b. Additional scattering at the Ti/Al interfaces is neglected because of the absence of potential barrier as discussed in Fig. 5a. Superconducting coherence length, , represents the characteristic decay length of pair correlation in the normal metal. ΔTi is defined as a proximity-induced gap in the LDOS at the centre of the Ti layer [Δind (x=0)] and is plotted as a function of L normalized by ξ in the inset of Fig. 8b. For the parameters of the vGJJ discussed in the main text, L=16 nm and ξ~140 nm for ΔAl=110 μeV. L is shorter than ξ by an order of magnitude so that ΔTi is almost identical to ΔAl. This is the case even for the Ti-vJ in Fig. 7 (L=70 nm) as indicated by the red arrow in the inset of Fig. 8b. Schematics of Δind (x) are denoted by the red dashed lines in Fig. 8a,c. Ti layer has Δind comparable to ΔAl, and it would result in an ill-defined ‘weak’-link JJ. This can be interpreted as a Ti layer effectively shorting the two Al superconductors. According to the analysis in Fig. 5, on the other hand, the graphene would act as a major barrier for a vGJJ.

This paper reports the first observation of Josephson coupling in a short and ballistic regime, which is expected to enable studies on exotic, but highly elusive to date, phenomena arising from strong Josephson coupling reaching the theoretical limit. The vGJJ in this study may be suited to scalable quantum devices using ABS, but further work is required to devise a way of incorporating thousands of Andreev channels present. It would also be stressed that the proposed scheme of the vertical structure is not limited to superconducting electrodes and graphene, but is readily applicable to a variety of electrodes (for example, ferromagnets) and cleavable materials with exotic properties, such as a three-dimensional topological insulating phase, layered high-Tc superconductivity of cuprates and iron-pnictides or various collective electronic states in transition-metal dichalcogenides. This will open the pathway to a wide range of research opportunities for the fundamental physics manifested at the atomic-scale interfaces of different materials, as well as the applications for highly coherent and scalable superconducting hybrid quantum devices.

Methods

Device fabrication

VGJJs were fabricated by the ‘flip-transfer’ scheme16 developed in the group, which is an extension of the ordinary graphene-transfer technique37. Monolayer graphene was mechanically exfoliated on a sacrificial substrate covered with a water-soluble poly (4-styrenesulfonic acid) (PSS) layer and an LOR (MicroChem) resist layer. Using standard electron (e)-beam lithography with 950 K poly (methyl methacrylate), the bottom electrode was patterned and developed. A stack of Ti/Al/Au (8/50/5 nm) was directly deposited onto graphene by e-beam evaporation. During the metal lift-off process in an 80 °C hot xylene bath, the PSS and LOR layers remained stable. After dissolving the PSS layer on the water surface to detach the sacrificial substrate, the entire structure of graphene in contact with the bottom electrode was flipped over and transferred to a new substrate. Then, the LOR layer was removed in Remover PG solution, and the top electrode (8/200/5 nm of Ti/Al/Au) was e-beam deposited onto the opposite surface of the graphene.

Measurements

Devices were thermally anchored to the mixing chamber of a 3He/4He dilution refrigerator (Oxford Kelvinox AST) and cooled to a base temperature of T=50 mK. To measure the I–V characteristics, we used a four-probe configuration, shown in Fig. 1c. Current was injected between I+ and I−; the voltage difference across the junction varied from V+ to V−. All measurement lines were electrically filtered by two-stage low-pass RC filters at the mixing chamber, with a cutoff frequency of ~10 kHz, in combination with another set of RC filters and π-type low-pass LC filters, with a cutoff frequency of 10 MHz at room temperature. For the T-dependence measurements of Fig. 3b and the CPR measurements of Fig. 4c, critical switching currents were determined by averaging the critical junction currents measured repeatedly over 104 times using an 18-bit data acquisition board (NI-DAQ 6281) with a linearly increasing bias current.

Determination of the superconducting gap

Superconducting gap for the electrodes in our devices can be determined accurately by analysing the MAR signal. However, Ic of JJ2 (=13.3 μA) is so high that the voltage directly jumps to~200 μV after Josephson current switching (Fig. 2a and Supplementary Fig. 7), and it is not possible to observe the MAR, which appears at the subgap regime (V<2Δ0/e). However, the junctions JJ1 and JJ3 fabricated together with JJ2 using identically prepared top and bottom Al electrodes have Ic’s sufficiently low as to show the MAR. There appear clear differential conductance (dI/dV) peaks at V=2Δ0/e and Δ0/e for JJ1 and at V=2Δ0/e for JJ3 (Fig. 9a,b). Temperature dependence of Δ0 is well described by BCS theory as shown in Fig. 9c,d. Both junctions show the identical Δ0, which ensures the junction-by-junction uniformity of superconducting electrodes and Δ0=110 μeV for JJ2.

Figure 9: Multiple Andreev reflection.
figure 9

Differential conductance (dI/dV) measured at the base temperature, for JJ1 (a) and for JJ3 (b). MAR peaks are denoted by arrows. The Josephson critical current of each junction is Ic,JJ1=0.53 μA and Ic,JJ3=0.19 μA. The differential conductance map corresponding to the MAR as a function of temperature and the bias voltage, for JJ1 (c) and for JJ3 (d) along with the BCS-type temperature dependence of the gap voltages denoted by dotted lines.

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Correction of the screening effect in the CPR measurements

With the finite self-inductance Lind of the SQUID loop, the phase relation is corrected by δ=γ+2πΦt/Φ0, where Φt=Φa+Φs is the total magnetic flux, the sum of the applied magnetic flux threading the SQUID loop Φa and the screening magnetic flux, Φs=LindIcir, induced by the circulating screening current Icir=(Ic,tJJsinγ−IJ,vGJJ(δ))/2 around the loop with self-inductance Lind. Calculation of Ic,SQ for the data fitting is done by maximizing ISQ=Ic,tJJsinγ + IJ,vGJJ(δ) under the constraint of the above phase relation. With a small value of the screening parameter βm≡2LindIcir,max/Φ0~0.19<1, the screening effect manifests itself as only a constant phase shift in the CPR measurements and does not show hysteresis between the curves obtained for opposite directional magnetic field sweeps. The experimental data show no hysteresis, either, in the magnetic field sweep direction. Here, Lind~7.9 pH is calculated using the commercial package COMSOL Multiphysics; Icir,max=(Ic,tJJ+Ic,vGJJ)/2 is the maximum circulating screening current.

Additional information

How to cite this article: Lee, G.-H. et al. Ultimately short ballistic vertical graphene Josephson junctions. Nat. Commun. 6:6181 doi: 10.1038/ncomms7181 (2015).