Long-Range Propagation and Interference of d-Wave Superconducting Pairs in Graphene.

Recent experiments have shown that proximity with high-temperature superconductors induces unconventional superconducting correlations in graphene. Here, we demonstrate that those correlations propagate hundreds of nanometers, allowing for the unique observation of d-wave Andreev-pair interferences in YBa_{2}Cu_{3}O_{7}-graphene devices that behave as a Fabry-Perot cavity. The interferences show as a series of pronounced conductance oscillations analogous to those originally predicted by de Gennes-Saint-James for conventional metal-superconductor junctions. The present demonstration is pivotal to the study of exotic directional effects expected for nodal superconductivity in Dirac materials.

The superconducting proximity effect in graphene has attracted much interest since the pioneering experiments [1]. This roots down to the graphene's electronic structure, which strongly affects the underlying mechanisms: Andreev reflection and coherent propagation of electron-hole pairs [2]. A distinctive feature is the strong dependence of the proximity behavior on the graphene?s doping level, which dramatically changes the Andreev reflection [3,4] as compared to metals. Other unique features include gate [5] or magnetic-field [6,7] driven transitions from bulk to edge transport. Studies on graphene have also paved the way for understanding the proximity effect in other Dirac materials, such as topological insulators [8][9][10][11].
Experiments have mostly focused on conventional low critical temperature (T C ) superconductors with swave pairing. Despite early theoretical studies showing that d-wave (high-T C ) superconductors should lead to novel directional effects [12,13] and exotic pairing [14,15], evidence for unconventional superconductivity in graphene has been found only recently [16,17]. Scanning electron tunneling microscopy (STM) of graphene on Pr 2−x Ce x CuO 4 (PCCO) revealed a superconducting gap [16] and spectral features suggesting (p-wave) superconductivity induced in graphene. Experiments on YBa 2 Cu 3 O 7 (YBCO) and chemical-vapor-deposited (CVD) graphene devices by some of us [17] showed trans- parent superconductor-graphene interfaces and clear evidence of Andreev reflection. Interestingly, we also found that the Andreev electron-holes pair transmission can be modulated by a back-gate voltage, through a mechanism analogous to the Klein tunneling [18]. However, neither the STM experiments nor ours on solid-state devices probed the length scale over which the unconventional correlations penetrate into graphene.
Here we demonstrate the long-range propagation of unconventional superconductivity into graphene via the observation of d-wave Andreev-pair interferences. These manifests themselves as conductance oscillations in devices that allow confining Andreev pairs within a graphene "cavity" whose length L is up to a few hundreds of nanometers. Predicted for proximitized graphene homojunctions [12,13], the oscillations are analogous to the de Gennes-Saint-James [19] and McMillan-Rowel resonances [20] in the electronic density of states of ultrathin normal-metals backed by superconductors. Figure 1(a) displays a cartoon of the underlying mechanism. Electrons injected in the cavity are Andreev-reflected as holes at the interface with the superconductor, retrace their path to the opposite cavity's end where they are normalreflected, and travel bain ck towards the superconductor, where they are again Andreev-reflected (now as electrons). This results in destructive/constructive interferences dictated by the energy-dependent phase accumulated along the loop. This phenomenon shows up as conductance oscillations as a function of the bias voltage V BIAS . Because Andreev pairs stemming from YBCO have d-wave symmetry, they decay over distances comparable to the mean free path l [21,22], the observation of Andreev-pair interferences implies ballistic or quasiballistic transport. Consistently, the Andreev-pair interferences are accompanied by normal-electron resonances that result from normal-reflections at the two cavity's ends [sketch in Fig. 1(b)] and commensurability between L and the electrons' wavelength [18,[23][24][25][26]. These different resonances result in oscillations as a function of both the gate V G and bias V BIAS , and can be distinguished from Andreev-pairs related ones by their distinct periodicity and voltage regime in which they dominate. To our knowledge, the concurrent observation of both resonances had been restricted to junctions between swave superconductors and topological insulators [9,10], in which Andreev-pair interferences manifest at very lowtemperatures as a function of the gate voltage V G . Here they can be observed at higher temperatures and show in a rich series of oscillations as a function of V BIAS because of the high-T C and large gap of YBCO (tens of meV) which, contrary to low-temperature superconductors [27], may enclose many orders of interference. The planar devices [ Fig. 2(a)] consist of four superconducting (SC) c-axis oriented YBCO 50nm /Au 4nm electrodes disposed within an insulating (INS) YBCO matrix and separated by a gap of length 100 nm< L < 800 nm. As-grown YBCO films showed T C ∼ 90 K while T C ∼ 70 K for lithographed devices, indicating moderate deoxygenation during fabrication. The ultrathin Au interlayer is deposited in situ to preserve the superconducting properties and improve the interfacial transparency [17]. The Au interlayer is crucial: none of the effects reported here could be observed in its absence. Notice that the Au layer thickness it is one order of magnitude thinner than the mean free path in Au, and conse- quently d-wave correlations expectedly propagate undisturbed across it [21,22]. A single-layer CVD grown [28] graphene bridge (G) connects the four electrodes, each of which terminates in a gold pad [1-4 in Fig. 2(b)] used for wire-bonding. The transparency of the electrical contact between each SC electrode and the graphene varied largely, even within a single device. Here we report on effects that arise only at high transparency contacts, which is often the case for just one out of four contacts per device. The structure is covered by a 45 nm thick AlO x layer (gate dielectric) [29]. As reported earlier [30], with this encapsulation the carrier mobility in the used CVD graphene reaches up to µ ∼ 7000 cm 2 V −1 s −1 . The top-gate is made of Au entirely [Fig. 2(c)] and the gate capacitance as measured over insulating YBCO is C G ∼ 5 × 10 11 e cm −2 V −1 (with e the electron charge). Further details are reported elsewhere [17].
We performed three-probe measurements, with the current I injected e.g. from 1 to contact 4 and the V BIAS measured between 1 and 3. Thus we probe the YBCO-Au-graphene interface in series with the graphene "cavity" formed between the SC electrodes, as sketched in in Fig. 2(d). The wiring as well as Au/YBCO contact resistances are Ohmic and negligible [17]. The differential conductance G(V BIAS ) = dI/dV BIAS is measured using a Keithley 6221 current source coupled to a Keithley 2182.
Examples of low-temperature (3.2 K) differential conductance measurements are shown in Fig. 3(a), 3(d) and Fig. 4(a). The differential conductance G(V BIAS ) shows a low-bias feature that stands out from the high-bias conductance level. That is either a conductance decrease   Fig. 4(a)] within a typical bias range |V BIAS | < δ exp ∼ 20 − 70 mV highlighted by the grey shade. Those features are reminiscent of the conductance decrease/increase observed for e|V BIAS | < ∆ in superconducting/normal-metal junctions depending on the interface transparency [31]. However, here they extend over a bias range δ exp that clearly exceeds the superconducting energy-gap expected for moderately-underdoped to optimally-doped YBCO, which is reported in the range ∆ YBCO ∼ 15−30 meV [32][33][34][35]. Furthermore, details such as the sharp zero-bias peak observed for some devices (see e.g. B4D and E3D in the Supplemental Material [36]) are characteristic of junctions involving d-wave superconductors, and appear when the d-wave nodes form a certain angle with respect to the junction interface [33,37,38].
In addition to the central features, an oscillation pattern appears in Fig. 3(a) and 3(d), this extends over a bias range well above δ exp . Indeed, two types of oscillations exist: the ones with longer period V long , predominant for |V BIAS | > δ exp , and the other with shorter V short which are more prominent at |V BIAS | < δ exp . Figure 4 show an example of gating effects. Figure 4(a) displays G(V BIAS ) without gate voltage (V G = 0), and shows the general features discussed above. Figure 4(b) is a color plot of a series of G(V BIAS ) at constant temperature and varying V G . The central (red) feature cor-responds to the zero-bias conductance peak. The conductance is periodically modulated by V G : a pattern of oblique lines (light-blue/green) indicates that V G gradually "shifts" the oscillations observed as a function of V BIAS . The oblique lines' slope gradually varies with V G , which results in pronounced curvature over the plot periphery (low V G or high |V BIAS |).
We discuss now a model that explains the main observations, namely (i) the increase or decrease of the zerobias conductance; (ii) the origin and period of conductance oscillations. The model is based on the Blonder-Tinkham-Klapwijk (BTK) formalism [31] extended to junctions between d-wave superconductors and normal metals [39] and to proximitized graphene homojunctions [12,13]. A scheme is shown in Fig. 2(d). First, we consider the YBCO-Au-graphene interface. Because the Au thickness (∼ 5 nm) is well-below the mean free path l Au ∼ 40 nm [40] and the low-T coherence length ξ Au = v F l/(6πKT ) ∼ 30 nm [41], we characterize that interface [black in Fig. 2(d)] via a single Blonder-Tinkham-Klapwijk barrier-strength parameter Z. If Z = 0, the transmission between YBCO and graphene is only mediated by the Andreev reflection. By increasing Z, Andreev reflection turns less dominant, normal reflection is enhanced, and transmission is dominated by tunneling. We model the graphene "channel" where Andreev pairs and normal electrons propagate, considering three different Here the Fermi energy E F,B is expectedly different from E F,A , E F,C as the graphene's doping depends on the substrate's electronic properties [42]. Within this model, the structure is a graphene homojunction with a gate-tunable Fermi energy step, as sketched in the upper Fig. 2(d), and behaves as a resonant cavity for electrons and Andreev pairs.
The conductance is calculated numerically by considering the YBCO-Au-graphene interface [33,39] and the proximitized graphene homojunction [12,13] in-series (see Supplemental Material for details [36]). The input parameters are Z, the reduced Fermi energies F,i ≡ E F,i /∆ (with i = A, B, C and ∆ the energy-gap induced in A by proximity effect), the reduced cavity's length Λ ≡ L/λ F,A (with the Fermi wavelength λ F,A , chosen as a reference following the original model [12,13]), and α the effective angle between d-wave anti-nodes and the homojunction interfaces. The key parameters are Λ and F,A : they determine V long and V short . The parameters α and Z determine the conductance background shape, and particularly whether there is a conductance decrease or enhancement around zero bias. We stress that the sharp zero-bias peak observed in some devices cannot be reproduced by the simulation unless we consider α = 0 (see Supplemental Material [36] Sec. 7). This is a consequence of the d-wave symmetry of the correlations propagating across the different interfaces. The parameters F,B and F,C only change the oscillations' phase and amplitude, but do not affect their period [13]. To choose the sim-ulation parameters that yield the best agreement with the experimental curves we proceed as detailed in the Supplemental Material in Sec. 2-4 [36]. Simulations examples are shown in Fig. 3(b), 3(e) and Fig. 4(c) for the experiments in Fig. 3(a), 3(d) and Fig. 4(a), respectively. The simulations closely reproduce the main experimental features [further examples and the simulations' parameters shown in the Supplemental Material [36] in Secs. 5 and 6].
In order to illustrate the different contributions to the conductance, Figs. 3(c) and 3(f) display the conductance of a proximitized graphene homojunction [13] alone, i.e. without a finite-Z junction in series. The same F,i , Λ and α as for the full-model calculations are used.
Here the conductance shows a background dependence on V BIAS very different from the experiments, which evidences that it is strongly influenced by the YBCO-Augraphene interface. In particular, the zero-bias "dip" only emerges if a finite-Z junction is considered. However, Figs. 3(c) and 3(f) do show the short and long period oscillations observed experimentally. This demonstrates that they originate within the graphene homojunction. Notice that the ratio between the short and long periods is different from that in the experiments (and full model simulations): in Fig. 3(c) and 3(f), V short is clearly shorter (relative to V long ) than in Fig. 3(a)-3(b) and 3(d)-3(e). This is because in experiments (and full-model) V BIAS is divided between the graphene homojunction and the YBCO-Au-graphene interface, and the strongly non-linear conductance leads to a voltage distribution that varies depending on the V BIAS . In particular, for eV BIAS < ∆ the conductance across the YBCO-Augraphene interface decreases (this is more pronounced for higher Z values) while the homojunction conductance increases [see Figs. 3(c), 3(f)]. Consequently, V short (observed for eV BIAS < ∆) appears "stretched" relative to the V long (observed for eV BIAS > ∆).
The oscillations' physical meaning can be understood from Fig. 5. This displays the short (red circles) and long (black squares) oscillation periods as a function of the inverse of the cavity length L −1 device defined upon devices fabrication. Specifically, we plot the period V = v∆ , where ∆ is estimated as discussed in the Supplemental Material [36] and v is the period obtained from our model once the contribution of the YBCO-Au-graphene interface has been removed -as in Fig. 3(c) and 3(e) -to avoid the aforementioned V BIAS division artifacts.
The long-period oscillations follow V th, long = hv F /2L device . This is as expected from the interference between electrons travelling back and forth from one cavity side to the other after normal reflections [ Fig. 1(b)]. This period results [5,18,[23][24][25] from the interference condition 2L device k = 2nπ (with n an integer and k the electron wavevector) and the graphene's linear dispersion, which yields V = v F k [43]. These interferences imply normal-electron coherence over ∼ L device . Their amplitude is greater for 100 nm < L device < 300 nm but drastically diminishes beyond that. This suggests that the mean free path is above 100 nm but clearly below 1 µm, in agreement with a rough estimate based on the carrier mobility [44] which yields l 100 − 260 nm . Notice that the scaling of the period with 1/L device rules out Tomasch resonances [45][46][47] within YBCO, which should scale with its thickness and should therefore show the same period for all devices.
The short period oscillations, clearly visible for V < ∆, follow V th,short = hv F /4L device . This period is expected from the interference of Andreev pairs [19,20] -scheme in Fig. 1(a). In this case the resonance condition reads 2L device |k 1 − k 2 | = 2nπ, where k 1 and k 2 are the electron and reflected hole wavevectors, whose difference is established by the applied voltage |k 1 − k 2 | = 2V / v F [20]. These resonances imply that the superconducting coherence is preserved over ∼ L device .
Our model also explains the gating effects. Figure 4(d) displays a simulated conductance as a function of bias and Fermi energy, which reproduces the main experimental features -particularly the pattern of oblique lines that characterize the conductance modulation by by V G [Fig.4(b)]. The correspondence between the y-axis in simulations and experiments implies that the Fermi energy (and vector k F ) is varied about proportionally to V G , in agreement with earlier experiments [17]. Varying k F produces a periodic modulation of the conductance, via the resonance condition 2L device k F = 2nπ, yielding the oblique lines. Note that the model also reproduces the slope gradual change, and the pronounced curvature over the plot's periphery, which is obtained by including in the model F,B (V G ) = F,A (V G ) = F,C (V G ) to account for a partial pining of graphene Fermi energy on conducting YBCO.
In summary, we have realized ballistic planar devices in which graphene is proximitized by a d-wave cuprate superconductor. We find that the d-wave correlations propagate into graphene over hundreds of nanometers at ∼ 4 K. The confinement of Andreev pairs and electrons in graphene homojunctions of submicrometric lateral dimensions produces quantum interferences that show as conductance oscillations. Those involving Andreev pairs, more prominent below the superconducting gap, are analogous to the De Gennes-Saint James oscillations. Although difficult to observe in graphene/low-temperature superconductor devices, here they are very clear due to the large YBCO gap (tens of mV), which fits many orders of interference within. The unusual d-wave Andreev-pair interferences are accompanied by normal-electron ones. The simultaneous observation of both is rare, and stems from the fact that the coherence length of d-wave correlations is limited by the mean free path [22]. Thus, the necessary condition for both types of resonances is satisfied simultaneously here. Extensions of this study to graphene with lower intrinsic doping and higher carrier mobility should allow exploring specular Andreev reflection regimes [3] rand Josephson effects. The latter would also require enhancing the transparency of the graphene/YBCO interface, possibly by replacing the Au interlayer, in order to routinely obtain low-Z junctions. Further, perspectives include novel directional effects [14,15] linked to the d-wave character of the induced correlations, which could be exploited in more sophisticated devices to realize topological states [48]. This work should also further encourage studies in which graphene is replaced by other 2D materials with intrinsically interesting proximity behavior [49,50], eventually combined with exfoliated d-wave superconductors [51].  We reproduce the experimental device conductance via numerical simulations based on combination of the models of Refs. [1] and [2]. The model of Ref. [1] applies to a graphene homojunction proximitized by a d-wave superconductor.
The junction is divided into three different sections A, B, C connected in series. Each of them has a different Fermi energy and together form an energy quantum well which is tuneable upon application of a gate voltage (in the case studied here, we assume that the different gate capacitance on insulating and metallic YBCO lead to different gating effects in A, C and B). In order to allow for Andreev reflection at the A/B interface, the model [1] assumes that the electronic density of states in region A presents a superconducting energy-gap. The homostructure conductance is obtained by matching the electronic wave functions in the A/B and C/B interfaces. Given the transmission T = 1 − |r| 2 and reflection R = |r A | 2 coefficient, the full expressions for r and r A are cumbersome and can be found in the appendix of Ref. [3]. The conductance across the A/B/C structure is given by: and θ the electron angle of incidence with respect to the interfaces.
Both r and r A and thus the conductance G 1 (V 1 ) depend on the following parameters: the reduced channel length The second block of the model takes into account the finite transmission of the YBCO/Au/graphene interface that is model as a d-wave/metal interface as studied in Refs. [2] and [4]. Specifically, the conductance is given by: where Γ ± = (eV 2 /|∆ (θ ± ) |) − (eV 2 /|∆ (θ ± ) |) − 1 and θ + = θ, θ − = π − θ . While ∆(θ) = ∆ cos [2 (θ − α)] and α is the angle between the superconducting order parameter and the interface. The conductance G 2 (V 2 ) thus depends on the parameters α and on the BTK barrier strength Z. As discussed in the main text this block of the model leads to a proper description of the background conductance vs. the voltage bias.
Using I = G (V ) dV we obtain G 1 (I) and G 2 (I) and from these, we calculate the conductance when the two building blocks are connected in-series: Finally, using V = 1 G(I) dI, we obtain G(V ).

II. SIMULATION PROCEDURE
In order to reproduce the experimental curves via simulations, we typically follow the procedure illustrated below.
Blue curves are experimental data and orange curves correspond to the simulations. The examples in the following is is based on device A3D.
We first find Z and α that allow mimicking the conductance background. These parameters respectively characterize the YBCO/Au-graphene interface and the d-wave nature of the superconducting correlations.
Let us first assume that the normalized cavity length Λ → 0 (so that there are no visible oscillations), fix α = 0 and vary Z, the results are presented in the various panels of Fig. S1. We see that one needs relatively large Z = 2 to better reproduce the V-shaped "dip" around zero bias. Considering the above, we would fix Z = 2 and α = 0 for the A3D device. the period is also modified, which is also expected. Indeed, from the definition of the parameters and the graphene's linear dispersion relation [3], the oscillations period is the same for any constant Λ/ F,A . Note however that the choice of F,A is not entirely free because this parameter modifies may introduce an asymmetry in the curve. Globally the best agreement is found for the combination F,A = 20, Λ = 10.
Finally, F,B and F,C are adjusted in order to modulate the oscillations amplitude. As shown earlier for this model [3], the two latter parameters have no impact on the oscillations period, only on their amplitude and phase. This is because the Fermi velocity mismatch modifies the transmission/reflection across the interface. This is illustrated

IV. QUANTITATIVE EVALUATION OF THE SIMULATION PARAMETERS
The simulations are based on dimensionless parameters whose values for each device are given in a table below. It is very important that their consistency can be quantitatively evaluated using fundamental relations and the actual device dimensions. As explained, above the two parameters that determine the oscillations period are Λ are F,A .
Specifically, what defines the oscillations period is the ratio Λ/ F,A .
If we take the definition of the parameters Λ ≡ L/λ F,A and F,A ≡ E F,A /∆, and we consider the dispersion relationship in graphene E F = v F k F = hv F /λ F (with k F and v F = 10 6 m · s −1 the Fermi vector and velocity, and h the Planck constant), it is straightforward to find the following expression for the superconducting gap ∆: Equation (SE4) allows evaluating the simulations parameters quantitatively. If we input in Eq. (SE4) the actual graphene cavity length in the studied devices L device (distance between YBCO electrodes defined lithographically) and the used Λ F,A , we obtain a value for superconducting gap, which is similar for most of the devices varies between 15 meV and 20 meV. This is well within the values expected for the same YBCO films when moderately underdoped [5] and materials in proximity with YBCO [6].
That a strong proof of the consistency of the analysis, from the quantitative point of view.
V. In principle and accordingly to [7], we do not expect the Au layer to significantly disturb the d-wave pairing, because it is one order-of-magnitude thinner than the mean free path in this material.
In our experiments, the key evidence for the propagation of the d-wave correlations into graphene is the observation of spectral features that cannot be explained based on s-wave superconductivity. In particular, the sharp zero-bias peak in some others [ Fig. 3(a)  This is illustrated by the example shown in Fig. S12, which corresponds to device B4D. The best agreement between theory and experiment is obtained with a barrier strength Z = 2.5 and an effective angle between d-wave nodes and junction interfaces α = 0.39.
In Fig. S13, we show simulations based on s-wave superconductivity both at the YBCO/Au-graphene interface and across the proximitized graphene homojunction. One can see that this assumption does not allow reproducing the experimental behaviour, particularly the sharp zero-bias conductance peak, even if the barrier strength Z is diminished to zero. Notice also that if the barrier strength Z is finite, a "dip" emerges around zero bias where s-wave quasiparticle tunnelling is strongly supressed. This is just opposite to the behaviour observed in the experiments.
Finally, we show in Fig. S14 simulations which consider d-wave superconductivity in the YBCO/Au-graphene interface and s-wave superconducting correlations across the graphene homojunction. One can see that under this assumption one cannot reproduce the behaviours observed experimentally.
The next example corresponds to device A5U, results in Fig. S15. We assume s-wave correlations across every junction interface. Again, the zero-bias conductance peak cannot be reproduced by considering only s-wave correlations-wave propagating the junction interfaces.
If we consider d-wave correlation only for the YBCO/Au-graphene interface and s-wave correlations across the tions propagate into the graphene homojunction, and exclude the s-wave pairing component.
Notice finally that the effective angle between the d-wave nodes and the junction interfaces changes randomly from device to device, which implies that it is not fixed by the device geometry. This suggests that the the contact between