Andreev reflection without Fermi surface alignment in high-T_c van der Waals heterostructures

Potential novel optical devices [1, 2] and non-abelian anyons [3–8] have reinvigorated interest in the superconducting proximity effect. Various approaches to high-temperature superconducting proximity have claimed success [9–17], including the recent report of a proximity effect between Bi₂Sr₂CaCu₂O${}_{8+\delta }$ and Bi₂Se₃ or Bi₂Te₃ [18, 19], via the mechanical bonding technique. Using thin films and ARPES, another group has claimed such interfaces result in an s-wave superconductor in the surface states [20]. One theoretical study has suggested that this is due to a mismatch in the crystal symmetries [21], though another finds  that the d-wave channel is dominant [22]. Two other thin film/ARPES studies suggest the proximity effect is not possible due to the Fermi surface mismatch, which suppresses the interface transparency [23, 24].

We test this hypothesis with tunneling experiments on junctions between Bi₂Sr₂CaCu₂O${}_{8+\delta }$ (Bi-2212) and Bi₂Te₃ or graphite in a magnetic field. These van der Waals materials are chosen as they form mechanical junctions with different Fermi surfaces. The Bi₂Te₃ is hole-doped with a Fermi surface close to the Γ point [25], whereas graphite is a semimetal with pockets close to the zone boundary (see figure 1(c)) [26]. By establishing Andreev reflection and proximity, we show that our original efforts, which focused on similar materials (Bi₂Te₃ and Bi₂Se₃), are not a special case. The ability to obtain Andreev reflection despite the Fermi surface mismatch is explained by considering the extended zone scheme, which is where the Fermi surfaces meet.

Zoom In Zoom Out Reset image size

The heterostructure is probed using differential conductance (dI/dV) to look for Andreev reflection, which is the process where a quasiparticle converts into a Cooper pair and travels from the normal material into the superconductor (see figure 1(a)). Andreev reflection is responsible for the proximity effect and is a stringent test of a transparent interface [27–30]. In metals with no attractive potential, one expects [31] and observes [32] a pair amplitude that can result in a minigap, whose size depends on the device geometry and superconducting phase of the contacts. In confined devices, with two superconductors separated by a normal material, Andreev reflection can produce an Andreev bound state (nABS) inside the normal region [32–36]. The nABS can ultimately lead to a supercurrent inside such devices. We do not expect the nABS to be relevant here as we form planar junctions with only one superconductor. Furthermore, since both Bi₂Te₃ and graphite have been shown to superconduct with applied pressure or doping [37, 38], they must have existing pairing potentials, and thus can hold a proximity effect without the need to invoke the nABS. As described later, the magnetic field dependence of our spectra is inconsistent with previous measurements of nABS or induced pair amplitudes without a corresponding proximity effect (i.e. a superconducting order parameter).

We measured numerous devices with either a high or low barrier. In high-barrier devices we observe tunneling spectra that are consistent with those measured in previous planar junctions, point contacts or STM experiments (see figure 2(b)) [ 39–46]. In addition, they are well described by an accepted theory of tunneling into the c-axis of a d-wave superconductor [30, 42], with the correct value of the superconducting gap (40 meV). In both Bi₂Te₃ and graphite heterostructures with low barriers, the conductance is enhanced below T_c, which is consistent with Andreev reflection in a proximity-induced region. This interpretation is further confirmed by the application of a magnetic field, which generates supercurrents. Tunneling into these supercurrents shifts the momentum of the quasiparticles and ultimately their energy. Since this only occurs for superconducting quasiparticles, the observation of the Doppler effect confirms the presence of Andreev reflection, as established in numerous theoretical and experimental works [44, 45, 47–57].

Zoom In Zoom Out Reset image size

The Doppler effect manifests itself as a shift in the Andreev features to lower voltages. This is seen in Bi₂Te₃ and graphite junctions, confirming that they are not due to the normal field dependence of the material. Due to the large size of our devices and the application of the field in the ab-plane, we expect effectively random or no observable shift in ABS. All field-dependent spectra are well described by including the Doppler shift in a previously established model of the Andreev reflection for proximity junctions. Furthermore, no hysteresis or splitting of any features is observed, as had been seen in previous measurements of ABS due to the sign change at the 110 surface of Bi₂Sr₂CaCu₂O${}_{8+\delta }$ (dABS) [40, 42]. Taken together, our results confirm the Andreev reflection between Bi₂Sr₂CaCu₂O${}_{8+\delta }$ and the normal materials, implying the relative alignment of the Fermi surface is not crucial in these heterostructures. This likely occurs due to the different lattice symmetries, which allow the Fermi surfaces to overlap in the extended zone (see figure 1(c)).

To form planar junctions between the normal material and Bi-2212 (T_c ∼ 90 K) we employed the mechanical bonding method as described in references [18, 19]. Both materials were cleaved in an inert glove box, then the Bi-2212 was placed on top of the Bi₂Te₃ or graphite, and GE varnish was applied to the corners of the Bi-2212 (figure 2(c)). Four-point transport measurements were performed at various temperatures ranging from 290 K–10 K. To further clarify the nature of the interface we performed extensive AFM on the cleaved surfaces. The Bi-2212 had extremely large, flat areas, whereas the Bi₂Te₃ and graphite produced step edges with atomically flat regions which were typically tens of microns across (figure 2(e)). Some regions showed mesas jutting out (figure 2(d)). Given the geometry of our devices, this suggests that the tunneling is along the c-axis of both materials, and occurs at planar junctions formed by these mesas touching the Bi-2212. The dI/dV were highly reproducible regardless of the field, temperature or voltage approach taken. To confirm that the dI/dV spectra originate from the junction, for every device we checked different sets of contacts and different combinations of them. Devices where the spectra were not reproducible between different contact sets were not considered. This confirms that the tunneling directly probes the interface between the materials, whereas previous experiments probed the top surface [20, 22–24].

The features we observed might not have arisen from tunneling given the large contact area. However, it is well established that superconductivity occurs within the Cu-O plane, while the outer layers of the Bi₂Sr₂CaCu₂O${}_{8+\delta }$ insulate Bi-O and Sr-O, enabling interlayer Josephson junctions and tunneling within a single crystal [39]. Thus we speculate that the Bi-O layer forms a tunnel barrier with the normal material enabling the dI/dV to provide spectra. The large contact area also suggests scattering on either side could occlude the observation of tunneling. This is not the case for our junctions, as shown by the high-barrier device (figure 2(a)). Consistent with previous experiments and the d-wave gap [39–46], we see a v-like shape at a low T that is gradually filled as the temperature is raised. Furthermore, the data is well described by the standard approach to d-wave tunneling on the c-axis, producing a proper gap at 40 mV [30, 41, 42]. The fit required a broadening parameter ${\rm{\Gamma }}\approx 1\to 4\,\mathrm{meV}$, which was much smaller than any of the features observed. This is likely due to the large mobilities of the normal materials used and the c-axis nature of the tunneling, reducing the likelihood of scattering.

Confirmation that none of the measured features arise from within the materials is provided by extensive measurements of various contact configurations and application of the magnetic field. Four-point and two-point measurements on only one side of the junction were independent of voltage, and the reported features were only observed when the current and voltage were measured across the interface. Two-point measurements on one side of the junction resulted in resistances $1\to 10\,{\rm{\Omega }}$ at low T. Swapping the current and voltage leads on the normal material only produced a slight voltage-independent offset of a few Ω, while high-barrier devices had resistances $\gt 1\,{\rm{k}}{\rm{\Omega }}$ and the low-barrier junctions had ${\rm{d}}{V}/{\rm{d}}{I}\approx 100\,{\rm{\Omega }}$, confirming that the voltage drop primarily occurs at the interface. Further confirmation of the voltage independence of the contribution of the material is shown in figure 2(b), where we measured the response from another high-barrier junction between graphite and Bi-2212 at 10 K in various applied magnetic fields (similar results were observed with Bi₂Te₃). Consistent with the STM measurements, we find no field dependence of the spectra [41], and a slight voltage-independent offset due to the graphite magneto-resistance.

To understand the dI/dV from the low-barrier junctions, we review what is expected, observed in proximity devices by tunneling [18, 19, 29, 58–61] and confirmed with ARPES [62]. The superconducting order parameter is induced into the normal material by the conversion of a quasiparticle current into a supercurrent via Andreev reflection. This involves an electron crossing the interface by forming a Cooper pair, resulting in the doubling of conductance (figure 1(a)). In less transparent interfaces, the conductance at zero bias is smaller than two and the shape at finite voltage is altered [30, 60]. The width of the zero-bias feature reflects the full gap of the superconductor. However, if superconductivity is induced in the normal material, the gap of the superconductor is reduced at the surface. Thus we define an induced (${{\rm{\Delta }}}_{i}$) and reduced gap (${{\rm{\Delta }}}_{r}$) at the interface (figure 1(a)).

Since the Andreev reflection into the induced gap occurs in the normal material without a physical barrier, for $V\lt {{\rm{\Delta }}}_{i}$ the normal quasiparticle is converted to a Cooper pair at the interface between the normal and induced superconducting regions, at which point it can easily travel into the superconductor. Thus ${\rm{d}}{I}/{\rm{d}}{V}(V\leqslant {{\rm{\Delta }}}_{i})$ will resemble what is expected for a low-barrier contact. For ${{\rm{\Delta }}}_{i}\lt V\lt {{\rm{\Delta }}}_{r}$ the normal quasi-particles now make it all the way to the real barrier, where they will Andreev reflect into the reduced gap region of the superconductor—albeit with a finite barrier—producing peaks at $V\approx {{\rm{\Delta }}}_{r}$. If the region over which the reduction in the gap of the superconductor is small, then normal quasiparticles may also tunnel into the full gap of the superconductor (${{\rm{\Delta }}}_{0}$), resulting in a peak in dI/dV at $V={{\rm{\Delta }}}_{0}$. It should be noted that in a real system, one must also account for the evolution of the wavefunctions and potential multiple reflections. Numerical calculations in s-wave [63] and in d-wave [64] proximity junctions confirm this picture. Thus the induced and reduced gaps produce Andreev features in the dI/dV that are tell-tale signs of the proximity effect [18, 19, 29, 58–62].

The observed features are reproduced by modifying a standard approach to tunneling into d-wave superconductors [30, 60, 63, 64]. Specifically, the differential conductance below T_c [dI/dV]_S, divided by the normal state conductance [dI/dV]_N is given by the half-sphere integration over the solid angle Ω:

$$\begin{eqnarray}&&\sigma (E)=\displaystyle \frac{\int {\rm{d}}{\rm{\Omega }}\cos {\theta }_{N}{\sigma }_{S}(E)}{\int {\rm{d}}{\rm{\Omega }}{\sigma }_{N}\cos {\theta }_{N}}\end{eqnarray} \tag{ 1 }$$

where E is the quasiparticle energy and θ is the incidence angle (relative to the interface normal) in the normal material, ${\sigma }_{N}$ is the conductance from normal to normal material with the same geometry, and

$$\begin{eqnarray}&&{\sigma }_{S}=\displaystyle \frac{{\sigma }_{N}(1+{\sigma }_{N}| {k}_{+}{| }^{2}+({\sigma }_{N}-1)| {k}_{-}{k}_{+}{| }^{2})}{1+({\sigma }_{N}-1)| {k}_{-}{k}_{+}{| }^{2}\exp ({\rm{i}}{\phi }_{-}-{\rm{i}}{\phi }_{+})}\end{eqnarray} \tag{ 2 }$$

where ${k}_{\pm }=\tfrac{E-\sqrt{| {E}^{2}| -| {{\rm{\Delta }}}_{\pm }^{2}| }}{| {{\rm{\Delta }}}_{\pm }| }$ and ${{\rm{\Delta }}}_{\pm }=| {{\rm{\Delta }}}_{\pm }| \exp ({\rm{i}}{\phi }_{\pm })$, and the electron-like and hole-like quasiparticles are effective pair potentials with the corresponding phases ${\rm{i}}{\phi }_{\pm }$. For simplicity we assume that the induced order has the same d-wave k-dependence as the bulk Bi-2212. Thus for all calculations we set the hole-like and the electron-like quasiparticles transmitted into the superconductor to experience the same effective pair potentials. In particular, for the c-axis tunneling that is relevant here, they have similar dependence on the azimuthal angle α in the ab-plane ${{\rm{\Delta }}}_{+}={{\rm{\Delta }}}_{-}={{\rm{\Delta }}}_{0}\cos (2\alpha )$. Scattering-induced energy broadening (Γ) is included in the calculation by adding an imaginary term to the energy of the quasiparticles. The real part of the resulting ${\sigma }_{S}$ then gives the differential conductance (G) with broadening from the normal material included via the Γ term ($G({\rm{\Delta }},{\rm{\Gamma }},V)={\rm{Re}}[\sigma ]$, where V is the applied bias) [65, 66].

The total Andreev reflection spectrum at a zero magnetic field is obtained by calculating the reflection and the transmission in the proximity region, followed by reflection at the interface between the two materials [67]. Incoming quasiparticles with energies smaller than ${{\rm{\Delta }}}_{i}$, Andreev-reflect at the first interface. This gives rise to a $G({{\rm{\Delta }}}_{i},{\rm{\Gamma }},V)$, which is expected to consist of a central peak with a width that is typically much smaller than the bulk gap of Bi-2212, corresponding to the induced gap in the normal material ($G({{\rm{\Delta }}}_{i},{\rm{\Gamma }},V)$). Quasiparticles with higher energies do not Andreev-reflect off the first interface, and instead transmit as normal particles. The transmission rate is 2-$G({{\rm{\Delta }}}_{i},{\rm{\Gamma }},V)$, where the 2 accounts for the fact that Andreev reflection involves a charge of −2e due to the conversion of two normal quasiparticles into a Cooper pair (figure 1 B). Quasiparticles with energies between ${{\rm{\Delta }}}_{i}$ and ${{\rm{\Delta }}}_{r}$ Andreev-reflect at the second interface and give rise to a term $G({{\rm{\Delta }}}_{r},{\rm{\Gamma }},V)$. We expect $G({{\rm{\Delta }}}_{r},{\rm{\Gamma }},V)$ to consist of peaks at an energy that is smaller than the bulk gap of Bi-2212 (${{\rm{\Delta }}}_{r}$), due to the suppression of superconductivity at the interface. Lastly, an additional term is included, $G({{\rm{\Delta }}}_{0},{\rm{\Gamma }},V)$, which can arise due to the inhomogeneity in the tunnel junction (a few high-barrier junctions in parallel with the low-barrier proximity junction) or from a small reduced gap region. The total dI/dV is calculated using ${G}_{{\rm{total}}}={f}_{1}\times [G({{\rm{\Delta }}}_{i},{\rm{\Gamma }},V)+(2-G({{\rm{\Delta }}}_{i},{\rm{\Gamma }},V))G({{\rm{\Delta }}}_{r},{\rm{\Gamma }},V)]+{f}_{2}\times G({{\rm{\Delta }}}_{0},{\rm{\Gamma }},V)$. The parameters f₁ and f₂ account for the different relative areas of the proximity and high-barrier junctions. Specifically, the two types of junctions are in parallel such that their total conductance is equal to their relative volume fractions (${f}_{\mathrm{1,2}}$) times their intrinsic conductivity. The calculated spectra in this model with the barrier strength (Z), scattering-induced energy broadening (Γ), and the superconducting gap (Δ) used as fit parameters, show excellent agreement with the experimental conductance measurements (purple lines in figures 2(a), 3(c) and (d)). Typically we find ${f}_{1}\approx 0.85$, $5\,\mathrm{meV}\leqslant {{\rm{\Delta }}}_{{\rm{induced}}}\leqslant 10\,\mathrm{meV}$, $15\,\mathrm{meV}\leqslant {{\rm{\Delta }}}_{{\rm{reduced}}}\leqslant 35\,\mathrm{meV}$, $1\,\mathrm{meV}\leqslant {\rm{\Gamma }}\leqslant 4\,\mathrm{meV}$, ${Z}_{{\rm{low}}}\approx 0.1$ and ${Z}_{{\rm{high}}}\geqslant 3$.

The temperature-dependent dI/dV of the Bi-2212/Bi₂Te₃ junctions is shown in figure 3(a). The dI/dV, when normalized to T_c, reveals a zero-bias peak that emerges just below T_c and eventually evolves into three features at low temperature. The first is an Andreev reflection peak near the zero-bias, whose width is much smaller than the full gap of Bi-2212 (labeled ${{\rm{\Delta }}}_{i}$). This feature could be a dABS [30, 40, 42]. However, at low T, the feature reaches an amplitude of nearly twice the normal state conductance, which is consistent with standard Andreev reflection. Furthermore, ab-plane tunneling only reveals a narrow peak at zero-bias and at the full gap at Bi-2212 (${{\rm{\Delta }}}_{0}$). The full gap is also seen in our data, and its temperature dependence matches well what is observed in high-barrier devices (figure 2) and the established trends for cuprates. Another peak in our data (${{\rm{\Delta }}}_{r}$) around 20 meV also approaches 2. Taken together, these three peaks are consistent with the proximity effect (figure 1(a)). Specifically, we expect perfect Andreev reflection since there is effectively no barrier between the normal material and the induced superconductor, while the inverse proximity effect reduces the size of the gap at the interface resulting in a peak at the reduced (${{\rm{\Delta }}}_{r}$) gap, and at the full gap (${{\rm{\Delta }}}_{0}$). We find similar spectra and temperature dependence from a low-barrier device with graphite as the normal material (figure 3(b)). The fact that we observe a zero-bias feature with a height of nearly 2 and a full gap at 40 meV, adds confidence that these features arise from Andreev reflection and do not require the Fermi surface of the normal material to directly overlap that of the superconductor.

Zoom In Zoom Out Reset image size

To confirm the results of these features from Andreev reflection, we use an in-plane magnetic field $B={\rm{\nabla }}\times A(r)$, which dramatically affects the ABS but causes little magneto-resistance in the normal materials. The superconducting order parameter will acquire an inhomogeneous phase $\left(\phi (r)=-2\tfrac{\pi }{{\phi }_{0}}\int A(r^{\prime} ){\rm{d}}{r}^{\prime} \right)$, where ${\phi }_{0}$ is the flux quantum. This produces a diamagnetic screening current where the Cooper pairs acquire the momentum ${\hslash }{k}_{s}={\rm{\nabla }}\phi$. Since Andreev reflection involves tunneling into this supercurrent, they are Doppler shifted by ${E}_{D}=-\tfrac{{{\hslash }}^{2}{k}_{\perp }{k}_{s}}{2{m}_{e}}$, where k_⊥ is the transverse wavenumber. Thus the magnetic fields shift the Andreev reflection features by ${\rm{\Delta }}E=-{v}_{F}{P}_{S}\sin \theta$ (where v_F, P_S, and θ are the normal material Fermi velocity, Cooper pair momentum, and the angle between the electron trajectory and magnetic field). The superfluid momentum (P_S) is linearly proportional to B and includes a geometric factor whose exact size is difficult to estimate in proximity devices [40, 42, 44, 45, 47–50, 53, 54, 56]. Thus for features involving the tunneling of Cooper pairs, we expect to include into the calculation a Doppler shift: ${\rm{\Delta }}E=D| B|$, where D is a constant and $| B|$ is the magnetic field strength. As discussed in detail later, care must be taken, as the magnetic field changes the energy dependence of the probability of Andreev tunneling. Thus the Doppler shift cannot be directly applied to the positions in the voltage of the features in dI/dV.

In studies of dABS in Bi-2212, applying a magnetic field splits and/or suppresses the zero-bias peak due to the Doppler effect, except when the field is aligned in the ab-plane [40, 42]. Alternatively, in S/N/S or N/S/N one can observe nABS due to multiple Andreev reflections between the interfaces. These nABS, which can play a crucial role in the proximity effect and supercurrents, have an energy defined by the phase difference across the confined region, ${{\rm{\Delta }}}_{0}$ and the transparency of the barrier [33]. In a confined structure with an induced pair amplitude, the resulting minigap was reduced by a perpendicular magnetic field [32], as expected from the theory [31], due to ϕ between the superconducting contacts.

The situation in our devices is quite different from what is typically observed in low-T_c structures. Due to the large ${{H}}_{c2}\approx 90\,{\rm{T}}$, we do not expect significant shifts in ${{\rm{\Delta }}}_{0}$ for the fields applied here ($\leqslant 7.5\,{\rm{T}}$) [68]. However, if nABS emerge due to the small mesas seen in AFM, then their energy will be periodic in the magnetic field as the phase winds through 2π [33–36]. Using the cross-sectional area of our Bi₂Sr₂CaCu₂O${}_{8+\delta }$ ($\approx \,{10}^{-7}{{\rm{m}}}^{2}$) or accounting for the small size of the mesas, we find that the features will reach zero energy at $B\approx {10}^{-5}\,{\rm{T}}$. Thus if our features arise from nABS, their shifts with the field will essentially be random for the sizes of the fields we apply. Furthermore, nABS should only appear below the induced gap. Hence the feature at 40 meV should result from tunnelling into the full gap of Bi₂Sr₂CaCu₂O${}_{8+\delta }$, whereas the features at lower energy could be nABS. As such, while the field may tune the zero-bias peak and the peak at ${{\rm{\Delta }}}_{r}$, the full gap will not be affected.

In figures 3(c) and (d), we show the dI/dV spectra at 6.5 K with a magnetic field. The size of the dI/dV at zero-bias and 43 mV is shown in figures 3(e) and (f), revealing the systematic suppression of the spectra, suggesting this may be the result of dABS. However, the full gap, reduced gap and width of the induced gap features all move to a zero-bias upon applying the magnetic field. This provides strong evidence against the ABS as the full gap (${{\rm{\Delta }}}_{0}$) should be field-independent, as seen in high-barrier junctions (figure 2(b)). Thus we attribute the magnetic-field-induced changes to the Doppler effect.

This is confirmed by including the Doppler effect in our calculation of the c-axis conductance spectra using the formalism developed for anisotropic superconductors [30]. As described in [18, 19], we modified this formalism to include contributions from the induced, reduced and fully gapped regions. A proper theoretical approach would self-consistently calculate the gap and account for potential multiple reflections due to the gradual change in the gap. However, since we do not investigate confined structures, and our minimal model captures our results, we believe it is appropriate to investigate the effects of the magnetic field. Once the zero-field spectra are captured by our model, we follow the established procedure for the Doppler effect by calculating the Andreev reflection probability $a(\tilde{E})$ with $\tilde{E}=E+{\rm{\Delta }}E$. The entire field dependence is reproduced using only the Doppler factor D = 0.4 for Bi₂Te₃ and D = 0.2 for graphite. The resulting dI/dV and their values at fixed bias are shown in figures 3(c $\to$ f). Despite the linear dependence of the Doppler shift on the field, its effect on the tunneling conductance is non-trivial as it enters indirectly through the probability of Andreev reflection. Thus as expected from equation (2), changes in dI/dV at fixed voltages with applied magnetic field can be non-trivial. Despite the difficulty of reproducing dI/dV with only one fitting parameter (as opposed to the peak position), we observe an excellent agreement. This confirms that the field dependence of the tunneling is governed by Andreev reflection and the Doppler effect.

Before closing, let us discuss some alternate possibilities. For example, Andreev bound states formed at the 110 interface between a d-wave superconductor/normal metal can create a peak at zero bias [30, 40, 43]. However, the central peak in our data is very different from that of an Andreev bound state. We observe a reduced gap as well as a central gap, in spite of the fact that the Andreev bound state only shows up as a zero-bias peak. Lastly, the Andreev bound states are expected to split by the application of a magnetic field, while the central peak (the induced gap) in our data, not only does not split, but decreases in width with the field (figure 4 (A) [40, 43].

Zoom In Zoom Out Reset image size

Other bound states, such as geometrical resonances (McMillan–Rowell and Tomasch oscillations) [69, 70], can create peaks in the differential conductance spectra. However, these peaks emerge at certain voltage positions in the Andreev spectra. Tomasch oscillations are due to resonances in the superconductor and create resonance features in the dI/dV at voltages given by: ${{eV}}_{n}=\sqrt{{(2{\rm{\Delta }})}^{2}+{\left(\tfrac{{{nhv}}_{{fS}}}{2{d}_{S}}\right)}^{2}}$, with Δ being the superconducting energy gap, v_fS being the Fermi velocity in the superconductor, d_S being the thickness of the superconductor, and n being the dip number. McMillan–Rowell oscillations occur due to geometrical resonances in the normal material and the voltages of the oscillatory features are linear with n (${\rm{\Delta }}V=\tfrac{{{hv}}_{{fN}}}{4{{ed}}_{N}}$), with v_fN being the Fermi velocity in the normal material and d_N being the thickness of the normal layer at which the reflections occur. Neither of these oscillations agree with the peaks we observe in our data. Furthermore, these oscillations typically create features at finite bias, in contrast to our data where we observe a zero-bias conductance peak [69]. Nonetheless, the magnetic field dependence of the McMillan–Rowell and Tomasch oscillations are different from our data. These bound states are expected and seen to split as well as shift by the application of a magnetic field [71].

A proximity effect along the c-axis of Bi-2212 seems surprising given the short coherence length. However, the proximity effect in the cuprates is governed by their low diffusion coefficient and small density of states [28]. Thus the Andreev reflection and proximity effect observed here could result from the small density of states of Bi₂Te₃/graphite, and their poor c-axis transport. What about the mismatch in the Fermi surfaces between Bi-2212 and the normal materials? Since the lattice symmetries of the normal materials are quite distinct from Bi-2212, the Fermi surfaces touch in the extended Brillouin zone [25, 26, 72]. This is shown in figure 1(c), where the Fermi level of the hole-doped Bi₂Te₃ is around the Dirac point [73], and the graphite is of ABAB stacking [74]. This argument holds for a wide range of Fermi levels.

Strong similarities between Bi₂Te₃ and graphite devices may suggest that our results are intrinsic to Bi-2212. For example, the junctions could lead to strain that produces mechanical breaks. Indeed, some devices have sharp features in the dI/dV as seen in the point contact [75]. These result from reaching the critical current in the inhomogeneous superconductor. However, such devices did not show the features reported here, and the sharp peaks were suppressed much faster in the applied magnetic field than expected from the Doppler. Furthermore, the Andreev features only appear in measurements performed across the interface.

The work at the University of Toronto was supported by the Natural Sciences and Engineering Research Council of Canada, the Canadian Foundation for Innovation, and the Ontario Ministry for Innovation. KSB acknowledges support from the National Science Foundation (grant DMR-1410846). The work at Brookhaven National Laboratory (BNL) was supported by DOE under contract no. DE-AC02-98CH10886. The crystal growth at Princeton was supported by the US National Science Foundation, grant number DMR-0819860.