Graphene-based tunable SQUIDs

When two superconductors are connected by high-quality graphene, the proximity effect allows a dissipationless current to flow through the graphene, from one superconductor to the other, over distances up to several microns.^1–12 The maximum supercurrent depends on the carrier density of the graphene, which can be controlled using the electric field effect. With appropriate fabrication techniques, it is possible to create high-quality graphene-based Josephson junctions with a tunable critical current density that can reach ∼3 μA/μm.¹³ Such junctions have potential advantages in addition to their tunability: the two-dimensional nature of graphene may allow the junctions to operate in large in-plane magnetic fields; ballistic transport in the graphene can produce a non-sinusoidal current-phase relationship and a stronger response to changes in flux;^14–20 and the cleanliness of encapsulated graphene could result in lower intrinsic noise than tunnel-barrier devices. Graphene junctions have recently been used to build simple superconducting circuits including superconducting quantum interference devices (SQUIDs).^21,22 Here, we report the fabrication and characterisation of DC SQUIDs with pairs of encapsulated graphene Josephson junctions that can be tuned individually using local gates.

The layout of the devices is illustrated in Fig. 1(a). The superconductor-graphene-superconductor (SGS) junctions are formed from monolayer graphene encapsulated between two ∼30 nm thick crystals of hexagonal boron nitride (hBN). The stack is fabricated on a quartz substrate by a dry-peel transfer technique²³ and then annealed at 300 °C in Ar H₂ for 3 h. A PMMA resist layer is patterned by electron-beam lithography, and a reactive ion etch (a mixture of CHF₃ and O₂) removes the un-masked areas of the stack. The selectivity of the etch causes narrow strips of graphene to be exposed along the contact edges. A niobium film of 50 nm thickness is then deposited through the same mask to contact the exposed graphene edges and to form the SQUID loop. The graphene in each junction is approximately 400 nm long (in the direction of current flow) and 3 μm wide. In our previous work, junctions fabricated using this technique were shown to display low contact resistance, high mobility, and ballistic transport over several microns.¹³ By contrast, the junctions used in previous graphene-based SQUIDs were in the diffusive transport regime.^21,22 In the final lithography step, aluminium field-effect gates are deposited on top of the hBN covering the junctions, through a second e-beam patterned PMMA resist layer. Two SQUIDs with nominally identical junctions were fabricated and measured for this work: one with a loop area ≈ 210 μm² and the other with a loop area ≈ 2900 μm². Both SQUIDs are fabricated simultaneously using the same hBN/graphene stack. Measurements were made at 20 mK using a dilution refrigerator system equipped with an 8T superconducting solenoid and bottom loading sample exchange mechanism.²⁴ 

Figures 1(b)–1(d) show the characteristics of the SQUID with a loop area of  ≈ 210 μm² over a range of magnetic fields. Figure 1(b) shows the superposition of the small scale modulation of the critical current I_C, due to interference around the SQUID loop, and larger-scale modulation of the critical current I_C, which is the Fraunhofer pattern arising from interference of different current paths across the finite area of the junctions. The period of the Fraunhofer pattern is 0.6 mT, which is consistent with the junction area. The smaller-scale oscillations, magnified in Fig. 1(c), correspond to an area of ≈ 230 μm², similar to the loop. Small differences may be attributed to flux focussing due to the finite width of the superconducting wire. Figure 1(d) shows the DC voltage V across the SQUID at several different values of the drive current I. At higher I, it displays the familiar response of a DC SQUID and transitions to superconductivity in the graphene junctions below V ≈ 110 μV. From this, we estimate the characteristic energy $e I C ( J ) R N ( J ) ≳ 100 μ eV$ for the junctions, where $I C ( J )$ and $R N ( J )$ are the critical current and normal-state resistance of a single junction, respectively. This is consistent with our previous measurements of ballistic Nb/graphene/Nb junctions, where I_CR_N was found to scale as 1/L for L between 150 nm and 2.5 μm.¹³ We have measured the temperature dependence of the critical currents and found that they do not change significantly up to a temperature of 300 mK. The critical temperature of the junctions is $T C = e I C ( J ) R N ( J ) / k B ∼ 1 K$⁠, and hence all the results presented here are in the low temperature limit (T ≪ T_C).

A practical benefit of gated junctions is the ability to tune their critical currents and thereby alter the characteristics of the SQUID. Figure 2(a) shows the change in the critical current in both SQUIDs as a function of their junction gate voltages. The charge neutrality points of all four junctions, identified as the points of minimum I_C, are in the range of – 0.2 V to 0.5 V. At more positive voltages (n-type doping), the total critical current increases significantly in both devices. The behaviour of I_C close to the charge neutrality point can also reveal the quality of the junctions: Figure 2(b) shows evidence of Fabry-Pérot oscillations in the critical current of a junction. We have studied such oscillations previously and determined that they are due to partial reflections from p–n (or n'–n) junctions that form in the graphene due to local doping by the contacts (see supplementary material of Ben Shalom et al.¹³) The presence of Fabry-Pérot oscillations demonstrates that transport in the graphene is ballistic and phase coherent on the scale of the junctions' length.^9,13

We study the effects of tuning the junctions in a SQUID with a 2900 μm² loop, shown in Fig. 3. Here, V_G1 is fixed at 1.00 V and V_G2 is varied in the range of 0.5 V to 2 V. From our estimate of the gate capacitance, this corresponds to n-type doping ∼1.5 × 10¹¹ cm⁻² in junction 1 and up to 4.5 × 10¹¹ cm⁻² in junction 2. The most noticeable effect of gating is that the modulation of I_C(Φ) changes its symmetry with respect to the direction of current flow, as shown in Fig. 3(a). In a SQUID with identical junctions and a symmetric loop, the modulation of the total critical current I_C(Φ) will be symmetric with respect to the direction of current flow. However, if the inductances of the loop arms are not equal or if the junctions do not have the same critical current, then I_C(Φ) will lose this symmetry. This is because self-induced flux in the loop depends on the direction of the current when the arms of the SQUID are not identical.²⁵ The loop in our SQUID has an intrinsic asymmetry: one arm of the loop is approximately twice the length of the other. We find that the modulation of I_C is most symmetric at V_G2 ≈ 1.14 V. At this point, the asymmetry in the junction critical currents cancels the asymmetry of the SQUID loop. Further increasing V_G2 increases the total I_C and also reintroduces an asymmetry in the modulation of I_C, as expected.

The response of a SQUID to changes in magnetic flux Φ can also be controlled using the gate voltages. The response is typically quantified by the transfer function V_Φ = dV/dΦ. We measure V_Φ by modulating an external magnetic field at 4 Hz with a magnitude equivalent to 0.007 Φ₀ in the loop and detecting the corresponding AC voltage across the device with a lock-in amplifier. Figure 3(b) shows V_Φ as a function of I and Φ close to a point of maximum critical current. In normal operation as a magnetometer, the drive current I should exceed I_C at all Φ, to avoid sudden transitions to the fully superconducting state. With this restriction, we find a highest transfer function of V_Φ = 250 μV/Φ₀ for the data in Fig. 3(b). Over the range of gate voltages shown in Fig. 3(c), the peak transfer function can be varied by more than 40%. At high gate voltages, the peak transfer function increases to V_Φ = 300 μV/Φ₀, which compares favourably with traditional low temperature DC SQUIDs.²⁵ 

A noticeable feature in Fig. 3(b) is that the magnitude of the transfer function [green trace in Fig. 3(b)] is larger on the negative slope of the voltage modulation (dV/dΦ < 0) than on the positive slope (dV/dΦ > 0). This asymmetry is expected when the two arms of the SQUID are not the same,²⁵ and in Fig. 3(c), we find a minimum asymmetry in V_Φ when the gates are tuned closest to the symmetric point.

The overall sensitivity of a SQUID to magnetic field depends on the transfer function and the noise in the measurement of voltage across the device. In our measurements, the noise is dominated by a room-temperature voltage preamplifier, limiting the sensitivity to $≈ 100 μ Φ 0 / Hz$⁠. However, we see no sign of intrinsic 1/f noise originating from the device down to a noise level of $≈ 30 nV / Hz$⁠, and so it is likely that the intrinsic sensitivity of the device is better. To achieve a sensitivity of $1 μ Φ 0 / Hz$ in the 2900 μm² device, it would be necessary to reduce the noise level below $0.27 nV / Hz$⁠.

The critical currents of the individual junctions can be inferred by comparing the shape of I_C(Φ) in Fig. 3(a) with a numerical simulation of the device.²⁵ In doing so, we find that the depth of the modulation of I_C in Fig. 3(a) appears to be incompatible with the expected inductance of the loop. A loop inductance of L ∼1 nH is required to achieve good agreement with the data, regardless of the asymmetry of the loop, whereas an inductance L ≈ 160 pH is expected from the device geometry (including a kinetic inductance ∼20 pH^25,26) This discrepancy can be due to the current-phase-relation (CPR) of the junctions modifying the inductance of the system.²⁷ When the numerical simulation is modified to use a CPR that has a smooth sawtooth shape, as expected for graphene junctions in the ballistic regime,^14–20 the inductance required to fit the data is significantly reduced. The overlayed lines in Fig. 3(a) show the predicted I_C(Φ) for a SQUID with L = 600 pH, a loop asymmetry of 30%, and a smoothed sawtooth CPR. For comparison, an inductance L ≈ 1.1 nH is required to achieve a similarly good fit if the junctions are modelled with a standard, sinusoidal CPR. From the simulations shown in Fig. 3(a), we estimate that the critical current of junction 1 is $I C ( J 1 ) ≈ 1.1 ± 0.2 μ A$ while $I C ( J 2 )$ varies from 0.7 μA to 2.7 μA as V_G2 is increased. Even when assuming a smoothed sawtooth CPR, there is still a significant discrepancy in the apparent loop inductance and the origin of this discrepancy is not known. However, the presence of a non-sinusoidal CPR is consistent with the observation of ballistic transport in Fig. 2(b) and in other junctions fabricated using the same process.^13,18,20 It also explains why the critical current is never modulated close to zero in either of our SQUIDs. This is normally due to self-induced flux in the SQUID loop; however, our devices do not have a large enough inductance for this explanation to hold (particularly the 210 μm² device).

In conclusion, we have demonstrated graphene superconducting quantum interference devices with independently tunable Josephson junctions. By varying the critical currents of the two junctions, the transfer function of the SQUID can be controlled. We measure a highest transfer function of ≈ 300 μV/Φ₀, which is limited by the voltage that can be safely applied to the gates on our device. The sensitivity of the device as a magnetometer is currently limited by the noise of the room-temperature voltage preamplifier. The sensitivity would likely be improved by incorporating a cryogenic preamplifier into the measurement, and this would also allow the intrinsic noise of the SGS junctions to be investigated. The current results demonstrate the potential of SGS junctions to add functionality to SQUIDs and indicate the future optimisations that are required to approach state-of-the-art magnetic field sensitivity in a graphene-based SQUID magnetometer.

We thank V. I. Fal'ko for useful discussions. This project has received funding from InnovateUK, UK EPSRC and the European Union's Horizon 2020 research and innovation programme under Grant agreement No. 696656. Yu.A.P. acknowledges support by the Royal Society (WM110105). J.R.P. acknowledges support of the Marie Curie People Program (REA Grant agreement No. 618450).