Specular Andreev reflection and the ballistic gate dependence of multiple Andreev reflections and critical current in superconductor-graphene-superconductor junctions

Superconductor-graphene-superconductor (SGS) Josephson junctions fabricated on SiO2 in 2008 were shown to be diffusive, with electron mean free paths far below the junction length, leaving open how to reach the ballistic regime and observe the relativistic specular Andreev reflection (SAR) predicted for graphene when the Fermi energy lies within the superconducting gap, together with the strong gate dependence of multiple Andreev reflections (MAR) and of the critical current predicted for ballistic SGS junctions. Later work achieved ballistic graphene Josephson junctions by substrate removal (suspended graphene) and hexagonal boron nitride encapsulation with one-dimensional edge contacts, observed specular interband Andreev reflection at single graphene-NbSe2 van der Waals interfaces, and observed MAR in micrometer-long ballistic graphene junctions at high doping. The surviving open core is the conjunction of these results: observing specular (interband) Andreev physics in the MAR subgap spectrum and in the gate dependence of the critical current of a ballistic two-interface SGS junction with the Fermi energy within the superconducting gap, including the fabrication and noise conditions that make this regime reproducibly accessible.

Background

A superconductor-graphene-superconductor (SGS) junction is a Josephson junction in which a monolayer graphene sheet bridges two superconducting leads: a dissipationless supercurrent (the Josephson current) flows across the graphene, with a maximum value $I_c$ (the critical current), and $R_n$ denotes the normal-state resistance; the product $I_c R_n$ is a standard figure of merit. Transport in such a junction is called ballistic when the carrier mean free path $l$ exceeds the junction length $L$, and diffusive otherwise. At a normal-metal-superconductor (NS) interface an incident electron below the superconducting gap $\Delta$ is reflected as a hole while a Cooper pair enters the superconductor (Andreev reflection). In an ordinary metal, whose Fermi energy is much larger than $\Delta$, the reflected hole retraces the path of the incident electron (retro-reflection). Graphene's carriers are described by a massless Dirac spectrum with conduction and valence bands meeting at the charge-neutrality (Dirac) point; Beenakker predicted in 2006 that when the Fermi energy $E_F$ is smaller than $\Delta$ the reflected hole originates in the valence band and leaves the interface at the mirror-image angle, a distinctive relativistic process called specular Andreev reflection (SAR), unique to systems with a Dirac spectrum. In a two-interface superconductor-normal-superconductor junction, subgap transport proceeds by repeated Andreev reflections between the two superconducting electrodes (multiple Andreev reflections, MAR), producing a ladder of conductance maxima ('subharmonic gap structure') at bias voltages $V_n = 2\Delta/(ne)$. Theory predicts that in ballistic SGS junctions both the MAR spectrum and the gate-voltage dependence of $I_c$ and $I_c R_n$ should show a strong, qualitatively unusual dependence on doping whenever SAR participates. Du, Skachko, and Andrei (Phys. Rev. B 77, 184507 (2008)) measured graphene SGS junctions with Al/Ti (aluminium/titanium) superconducting leads on Si/SiO2 substrates and found, contrary to expectation, that all such devices were diffusive: normal-state transport gave mean free paths of only 10-50 nm, far below the lead separation, the ballistic model overpredicted $I_c$ by more than an order of magnitude, the diffusive junction model agreed quantitatively, and the MAR gate dependence showed no SAR signature. They traced the absence of SAR to charge inhomogeneity (puddles of trapped charges near the Dirac point, mainly substrate-related): observing SAR requires the Dirac-point spread to be smaller than the superconducting gap, corresponding to a gate control of order $0.1\ \mathrm{mV}$ for Al leads, a charge uniformity not met by SiO2-based fabrication. They also found that the measured $I_c$ was reduced below its intrinsic value by 'premature switching' driven by thermal and electromagnetic noise, which they modeled with an effective noise temperature $T_{EM}\sim300\ \mathrm{mK}$. They concluded that improving the contact design and the lead and substrate materials 'may in the future give access to the SAR', noting that graphene devices without superconducting leads had reached mean free paths up to an order of magnitude longer, including in the authors' own suspended-graphene experiments. The subsequent decade transformed exactly these fabrication conditions: suspended graphene (removing the SiO2 substrate); encapsulation in hexagonal boron nitride (hBN), an atomically flat insulating crystal that replaces the SiO2 substrate and strongly reduces trapped-charge disorder, combined with one-dimensional metallic edge contacts (electrodes deposited onto the freshly cut graphene edge rather than its top surface, which gives low-resistance, low-disorder interfaces); and van der Waals superconductors (superconductors exfoliated as thin crystalline flakes and stacked directly onto graphene, such as NbSe2, with gaps several times larger than Al's). Within this now-standard toolbox, ballistic graphene Josephson junctions, specular interband Andreev reflection at a single graphene-superconductor interface, and MAR in ballistic junctions have each been demonstrated separately; what has not been demonstrated is their conjunction in one two-interface device with $E_F$ within the gap.

Problem Statement

The source paper asked how ballistic superconductor-graphene-superconductor (SGS) junctions can be achieved in practice, what Dirac-point charge uniformity is required to observe relativistic specular Andreev reflection (SAR; an incident electron below the superconducting gap is reflected as a hole from the valence band, on the opposite side of the Dirac point) and the predicted ballistic gate dependence of the multiple Andreev reflection (MAR; repeated electron-hole conversion between the two superconducting electrodes) spectrum and of the critical current, how electromagnetic and thermal fluctuations can be suppressed to reach the intrinsic critical current $I_c$, and under what conditions ballistic SGS behavior and SAR signatures can be reproducibly observed. Ballistic SGS junctions, single-interface SAR, ballistic MAR, and intrinsic-$I_c$ measurement have since each been demonstrated separately, so the surviving question is their conjunction: can specular (interband, valence-band) Andreev reflection physics be observed directly in a ballistic two-interface SGS junction whose Fermi energy $E_F$ is within the superconducting gap $\Delta$? Concretely: (i) fabricate an encapsulated or suspended graphene junction with mean free path $l$ exceeding the junction length $L$, transparent superconducting contacts, and Dirac-point charge inhomogeneity small enough that the Fermi-energy spread $\delta E_F \lesssim \Delta$ (as with a large-gap superconductor such as NbSe2, since the approximately 0.1 mV gate-control condition of Al leads remains out of reach); (ii) measure the MAR subgap spectrum and the gate dependence of $I_c$ and of $I_c R_n$ (with $R_n$ the normal-state resistance) from high doping through the Dirac point with fluctuation levels low enough that the intrinsic $I_c$ is not masked; and (iii) establish, by comparison with theory that includes both retro and specular Andreev processes, whether and how the interband (specular) contribution distinguishes the ballistic SGS response from the retro-Andreev, Fabry-Perot-interference (resonant transport of carriers bouncing between the two contacts), and charge-puddle explanations, thereby answering under what conditions the ballistic SGS and SAR signatures predicted in the source paper are reproducibly observable.

References

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