Tuning Locality of Pair Coherence in Graphene-based Andreev Interferometers

We report on gate-tuned locality of superconductivity-induced phase-coherent magnetoconductance oscillations in a graphene-based Andreev interferometer, consisting of a T-shaped graphene bar in contact with a superconducting Al loop. The conductance oscillations arose from the flux change through the superconducting Al loop, with gate-dependent Fraunhofer-type modulation of the envelope. We confirm a transitional change in the character of the pair coherence, between local and nonlocal, in the same device as the effective length-to-width ratio of the device was modulated by tuning the pair-coherence length ξT in the graphene layer.

A Estimation of V bg dependence of l e and ξ T in graphene.
The right panel of Fig. 2(c) shows the V bg dependence of the normal state resistance R of the TGB. From this, we estimated the carrier mean-free path l e = (W/L 2 R)(π/n) 1/2 /e 2 as in Fig. S1 (red curve). Near the CNP, i.e., |V bg − V CN P | < 5 V, l e shows the unphysical divergence due to the remnant charge carrier density due to the presence of e-h puddles in graphene. Excluding the unphysically diverging region of l e near the CNP, our GAI device was in the diffusive transport regime as L 1 was always larger than l e . The V bg dependence of the pair coherence length ξ T (blue curve) also shows the unphysically diverging behavior near the CNP, where we assumed that ξ T was comparable to the average size of e-h puddles 1 . C Single-particle phase coherence length.
The single-particle phase coherence length l ϕ of the graphene in the normal state can be extracted from the MC measurements. The MC of the graphene layer was taken in a region of 0.5 µm in width and 8 µm in length, which was prepared on the same piece of graphene as the Andreev interferometer used in the study. symbols and lines represent the data and the best fits, respectively. l ϕ was obtained from fitting to the weak-localisation-induced conductivity correction with three parameters (l ϕ , l i , and l * ) where is the flux quantum, l i is the elastic intervalley scattering length, and l * is the  Figure S3: (a) The change in the conductivity, ∆σ, as a function of the magnetic field B for V bg = −19, −10 and 50 V. The dot symbols represent the measured data (each set shifted by 2e 2 /h for clarity) and the lines are best fits to Eq. (S1). (b) The single-particle phase coherence length l ϕ as a function of V bg at 1 K.
scattering length for other elastic scattering sources 4 . Best-fit values of l ϕ at 1 K is shown in Fig. S3(b) as a function of V bg , with the dotted lines as a guide to eyes. Because ξ T no longer grew below T = 600 mK due to the lowtemperature-limit value of T ef f ∼ 600 mK while l ϕ was almost temperature independent below 1 K 5 , l ϕ was always larger than ξ T for the carrier density from the CNP to the highly doped regime and at any temperatures between 50 mK and 1 K. Thus, the carrier motion in graphene was always phase coherent in this study. The values of l * and l i , obtained from the best fits, were insensitive to V bg and were ∼ 100 nm and ∼ 10 nm, respectively.  On the other hand, in the case of α W (or 2ξ T L 1 ), Eq. (1) is expressed phase change of π is also induced depending on the locality of the electronhole pair phase coherence between the G/Al interfaces. Figure  E Temperature dependence of conductance oscillations.
Development of the Fraunhofer-type variation of conductance with temperature was also examined. Figures S6(a), (b), (c), and (d) show the Fraunhofer-type variation of conductance as a function of B, taken at T = 50, 400, 700, 870 mK, respectively, for V bg = 50 V. These data sets were taken after a thermal recycling of the dilution fridge, during which a little change took place in the properties of the GAI. Thus, a small discrepancy in the 50 mK data is seen between the data set in Fig. 2(a), taken before the thermal recycling of the fridge, and that in Fig. S6(a). One should note that the critical field of Al decreased along with the increase of temperature, which limited the clear observation of the local-to-nonlocal crossover in the Fraunhofer-type conductance envelope with temperature. In addition, as discussed below, the high effective electron temperature (T ef f ∼ 600 mK) led to the temperature insensitivity of   The averaged-out MC maintains the Fraunhofer-type conductance modulation, clearly indicating that, at least in this local pair-coherence regime, the observed MC modulation was the unique characteristics originated from the phase-coherent transport in our GAI rather than the sample-specific impurity configurations. Figure S7(b) shows the amplitude of the fast Fourier transformation (FFT) of the second envelope. The amplitude does not decay with N , which is in sharp contrast with the sample-specific MC modulation in the nonlocal pair-coherence regime in previous reports 6,7 , where the amplitude decays linearly with N −0.5 .