The data shown in this Letter were taken from one single-layer graphene device (sample A) and one multilayer device (sample B, approximately ten layers thick). The spectral pattern of these peaks as a function of gate and bias voltage is consistent with a simple theoretical model of ABS spectra presented below, and can be accurately fitted with a more detailed microscopic calculation. We focus on the two lowest-energy conductance peaks that occur below the superconducting gap, and show that they are a signature of transport by means of ABS. In the work described in this Letter, sharp subgap conductance peaks are obtained by tunnelling into a proximity-coupled QD formed within graphene, a high-mobility zero-gap semiconductor 17. However, the ABS peaks in previous SC–QD experiments were strongly broadened, either by the large lead density of states 15 or by the lack of a tunnel barrier 16. SC–QD hybrids in graphene have not been studied, although recent work has predicted 11, 12, 13, 14 and demonstrated 15 that ABS can be isolated by coupling them to discrete QD energy levels. In superconductor (SC)–graphene structures, most previous work has focused on the nature of the supercurrent in well-coupled Josephson junctions 7, 8, 9, 10. Owing to the low density of states of graphene and the sensitivity of the QD levels to an applied gate voltage, the ABS spectra are narrow and can be continuously tuned down to zero energy by the gate voltage.Īlthough signatures of Andreev reflection and bound states in conductance have been widely reported 6, it has been difficult to directly probe individual ABS. Individual ABS form when the discrete QD levels are proximity-coupled to the superconducting contact. The QD is formed in graphene beneath a superconducting contact as a result of a work-function mismatch 4, 5. Here, we report transport measurements of sharp, gate-tunable ABS formed in a superconductor–quantum dot (QD)–normal system realized on an exfoliated graphene sheet. In confined geometries, this process can give rise to discrete Andreev bound states (ABS), which can enable transport of supercurrents through non-superconducting materials and have recently been proposed as a means of realizing solid-state qubits 1, 2, 3. Such an energy would correspond to a “scattering resonance.When a low-energy electron is incident on an interface between a metal and superconductor, it causes the injection of a Cooper pair into the superconductor and the generation of a hole that reflects back into the metal-a process known as Andreev reflection. It would be reasonable to refer to such magic energies as “states” of the well, even though they are unbound.įor a three-dimensional finite square well, such “unbound states” would correspond to energies where the scattering length becomes very large see this hand-waving derivation of the scattering length for an approach which might let you get the idea by sketching some wavefunctions rather than getting lost in a bunch of mathematics. For a one-dimensional well there might be a magic unbound energy where the reflection coefficient goes to zero or to one, or where the phase shift of the transmitted wave vanishes. The approximation is better for the most deeply bound states, where the probability of finding the bound particle outside of the well is the smallest.įor the unbound states of a finite square well, you could reasonably expect some funny business to happen with reflection/transmission coefficients if the wavelength within the well happens to be near a half-integer multiple of the well’s size. Such bound states satisfy the condition that the well size is a half-integer multiple of the wavelength.įor a finite square well, the $E<0$ bound states still fit approximately a half-integer number of wavelengths.
For an infinite square well, the bound states are those which satisfy $\psi(x)=0$ at the edges of the well, so that the wavefunction goes continuously to zero probability of detecting the particle outside of the well.