# Superconducting proximity effect in $d$-wave cuprate/ graphene
heterostructures

###### Abstract

Superconducting proximity effects in graphene have received a great deal of
attention for over a decade now. This has unveiled a plethora of exotic
effects linked to the specificities of graphene’s electronic properties. The
vast majority of the related studies are based on conventional, low-
temperature superconducting metals with isotropic $s$-wave pairing. Here we
review recent advances made on the less studied case of unconventional high-
temperature superconducting cuprates. These are characterized by an
anisotropic $d$-wave pairing, whose interplay with Dirac electrons yields very
rich physics and novel proximity behaviours. We provide a theoretical analysis
and summarize the experiments reported so far. These unveil hints of
proximity-induced unconventional pairing and demonstrate the gate-tunable,
long-range propagation of high-temperature superconducting correlations in
graphene. Finally, the fundamental and technological opportunities brought by
the theoretical and experimental advances are discussed, together with the
interest in extending similar studies to other Dirac materials.

###### keywords:

Superconductivity, Graphene, Cuprates, Topological Insulators, Proximity
Effect

David Perconte Dario Bercioux Bruno Dlubak Pierre Seneor F. Sebastian Bergeret
Javier E. Villegas*

David Perconte
Institut Néel,
Univ. Grenoble Alpes,
CNRS, Grenoble INP,
25 avenue des martyrs,
38000 Grenoble, France

Dario Bercioux
Donostia International Physics Center, 20018 Donostia-San Sebastián,
Spain &
IKERBASQUE, Basque Foundation for Science,
Euskadi Plaza, 5,
48009 Bilbao, Spain

Bruno Dlubak,
Pierre Seneor,
Unité Mixte de Physique, CNRS, Thales, Université Paris Saclay,
91767, Palaiseau, France

F. Sebastian Bergeret
Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU,
20018 Donostia-San Sebastián,
Basque Country, Spain &
Donostia International Physics Center, 20018 Donostia-San Sebastián,
Spain

Javier E. Villegas,
Unité Mixte de Physique, CNRS, Thales, Université Paris Saclay,
91767, Palaiseau, France
<EMAIL_ADDRESS>
##  1 Introduction

Proximity effects in graphene have been intensively investigated ever since
the early studies evidenced that its electronic properties [1] transform both
the underlying physical mechanisms [2] and the related phenomenology [3]. As
we discuss in Section 1.1, most of the specificities are linked to the fact
that graphene is a two-dimensional (2D) Dirac material, which in many
situations can be described as a gapless semiconductors in which carriers
exhibit a linear dispersion around zero energy. In addition, because it
presents a relatively low carrier concentration, electrostatic gating allows
large modulations of the electrochemical potential. This has a strong impact
on the mechanism underlying the propagation of superconducting correlations
into graphene, the Andreev reflection, and thus provides a knob for tuning the
proximity effect. Beyond this facet, other characteristics such as the
possibility to induce a transition from bulk to edge transport [4, 5, 6] have
fostered significant research efforts. The vast amount of work has been based
on conventional $s$-wave superconductors and, because it is out of the scope
of this review, it will be only briefly recalled hereafter. On the other hand,
superconducting cuprates present many properties that make then attractive in
the framework of superconducting proximity in graphene. One can start by
mentioning the most obvious one, which is that their high critical temperature
is appealing for technological applications. For example, gate-tunable
Josephson coupling across graphene, which has been demonstrated with
low-$T_{\text{C}}$ superconductors, would be a game-changing asset in the
growing field of high-temperature superconducting electronics [7, 8, 9, 10].
Beyond that, arguably one of the most appealing properties of the cuprates is
that they present anisotropic $d$-wave pairing. This means that the amplitude
of the superconducting gap and the superconducting phase vary in the momentum
space, particularly in the ($k_{x}$-$k_{y}$)-plane of graphene’s 2D Dirac
electrons. As discussed in Section 2.3, this has a strong influence on the
proximity effect, for example on the Andreev reflection process and on the
conductance across the cuprate/graphene interface. Along with their interest
and the unique opportunities they offer, cuprate superconductors bring
complexity, particularly regarding the fabrication of solid-state devices,
which is much more challenging than with conventional superconductors.
However, as we detail below, _ad hoc_ techniques have recently allowed
relieving some of those constraints, enabling the experiential demonstration
of the proximity effect in graphene in contact with superconducting cuprates
[11, 12, 13] as well as proximity between graphite and BSCCO [14].

The above ideas are unfolded hereafter, the review being organized as follows:
the specificities of graphene and $d$-wave superconductors are summarized in
Section 1.1 and 1.2, where in addition the motivation for mixing both types of
materials in proximity devices is rationalized. A theoretical section follows
(Section 2), which first focuses on the particularities of the proximity
effect between $d$-wave superconductors and trivial metals (Section 2.1) and
then on those of the proximity between Dirac materials and conventional
superconductors (Section 2.2). These two theoretical sections are introductory
to that devoted to the main topic of the review, which is $d$-wave/Dirac
materials interfaces (Section 2.3), with a stronger focus on the case of
graphene and an extension to the surface state of 3D topological insulators.
Section 3 is devoted to the experiments. After briefly summarizing some of the
work based on $s$-wave superconductors (Section 3.1), we focus on the scanning
tunneling microscopy studies and on devices based on $d$-wave/graphene
heterostructures (Sections 3.2 and 3.3). A brief discussion of experiments
involving $d$-wave cuprates and other Dirac materials is presented thereafter
(Section 3.4), which is followed by the concluding remarks and outlook
(Section 4).

###  1.1 Two-dimensional Dirac materials: graphene and beyond

Graphene, an atom thin hexagonal crystal of carbon, is the most archetypal
two-dimensional (2D) material [15], see Fig. 1(a). It presents remarkable
structural, optical and electronic properties [16]. Thanks to its unique
characteristics, graphene has become a superlative material: when compared to
its size it is probably the strongest material per unit size (tensile strength
in the 100 Gpa range [17], on par with diamond and far greater than steel)
while keeping striking elastic properties (Young’s Modulus in the TPa range
[18]), it offers the highest thermal conductivity at room temperature (in the
order of $5000$ W/mK [19]), it has a large surface area of $2630$ m2/g [20],
absorbs only 2.3 % of light [21] and is impervious so that even helium can’t
pass through [22]. But it is its remarkable electronic and transport
properties that have initially drawn the most attention [1, 23]. As a matter
of fact, the Dirac band structure is one of the striking features of graphene,
it was early unveiled by Wallace sixty years ago [24] to present a linear
dispersion crossing at the so-called Dirac points — see Fig. 1(b). This has
led to impressive mobilities, robustness of its 2D electron gas and exotic
transport phenomena such as massless quasiparticles, chirality, uncommon
Landau levels splitting and the Klein tunneling paradox [25].

Figure 1:  (a) Real-space structure of the graphene lattice, the gray and the
black spheres represent the two inequivalent carbon atoms in the unit cell of
graphene denoted by A and B (yellow region). The carbon-carbon distance is
indicated via $a_{0}$. (b) The band structure of the graphene. The hexagon in
the yellow plane represents the first Brillouin zone. The “Dirac” points are
given by the touching points of the valence and conduction bands in the six
corner of the hexagon, by symmetry only two of those belong to the first
Brillouin zone and are usually named $\mathbf{K}$ and $\mathbf{K^{\prime}}$.

In terms of electronic structure, the carbon atoms in the honeycomb structure
of graphene form the $\sigma$-bonds with neighboring carbon atoms through
$sp_{2}$ orbitals while the remaining $p_{z}$ ones overlap to form a band of
filled $\pi$ orbitals and a band of empty $\pi^{*}$ orbitals that give rise to
respectively the valence band and the conduction band. Thanks to this
electronic configuration, graphene can be thought as a two-dimensional zero-
gap semimetal with in-plane conductivity — see Fig. 1(b). The electronic band
structure of graphene directly arises from the two inequivalent carbon atoms
of the graphene crystal structure, see Fig. 1(a). This also gives rise to a
key feature of graphene for electronic transport: its low-energy
quasiparticles formally described by the Dirac-like Hamiltonian
$\mathcal{H}_{\text{G}}=v_{\text{F}}\bm{\sigma}\cdot\bm{p}$, where
$v_{\text{F}}=10^{6}m/s$ is the Fermi velocity,
$\bm{\sigma}\equiv(\sigma_{x},\sigma_{y})$ are the Pauli matrices
corresponding to a so-called pseudo-spin associated to the two carbon atoms in
the lattice unit cell, and $\bm{p}\equiv(p_{x},p_{y})$ is vector of the
momenta operators in the graphene plane. The two energy bands ($\pi,\pi^{*}$)
intersect at the corner of the hexagonal first Brillouin zone yielding the
conical energy spectrum at the $\mathbf{K}$ and $\mathbf{K}^{\prime}$ “Dirac”
points. Hence, quasiparticles in graphene exhibit the linear dispersion
relation $E=\hbar v_{\text{F}}k$ as if they were massless relativistic
particles, only with a Fermi velocity $v_{\text{F}}$ approximately 300 times
slower than the speed of light. The fact that charge carriers in graphene are
described by the Dirac-like equation (_i.e._ linear dispersion relation)
rather than the usual Schrödinger equation (_i.e._ free electron with
parabolic-like dispersion relation) has direct consequence on the transport
properties when contacted with superconducting electrodes, as discussed in
Section 2.2. The difference is particularly striking when considering effects
with energies close to the crossing point of the linear dispersion -the charge
neutrality point. Only in this regime, a clear distinction between linear and
parabolic energy dispersion plays a significant role.

Still, while theoretically unveiled more than sixty years ago, experimentally,
graphene has been first isolated in 2004 [26]. By leveraging on the
exfoliation technique, it has been possible to access relatively easily high-
quality crystals and explore their unique properties. As early as in 2004, the
mobility of graphene was measured to be exceptionally high. For mechanically
exfoliated graphene, mobilities $\mu>10000$ cm2/Vs at room temperature are now
currently reported, for carrier concentrations (electrons or holes) that can
be modulated by a standard gate up to high values of $10^{13}$cm-2 [27, 28,
29, 26]. The intrinsic (low-temperature) limit of graphene mobility was
revealed by Landau spectroscopy measurements on high purity monolayers on the
surface of a graphite crystal: the reported mobility values exceeded $10^{7}$
cm2/Vs [30].

The main difficulty linked to mechanically exfoliated graphene has however
concerned the small size of the obtained sheets and their random distribution
on the substrate surface [31]. An additional but fundamental constrain has
been the necessity of relying on a specifically tailored substrate (usually
Si/SiO2) leading to interferences of Fabry-Pérot type and allowing their
observation through contrast enhancement [32]. Those limitations, which would
be prohibitive for any exploitation that requires a large-scale integration of
components, have also made any systematic study and integration with specific
substrates more difficult. As such, the implementation of graphene has been a
challenge and the most important problem has been lying in the preparation of
high-quality and well-defined graphene over large surfaces. Thus, many
material studies have aimed at producing large area graphene (_i.e._ substrate
size surfaces) [33, 34]. In this direction, chemical vapor deposition (CVD)
has quickly appeared as a very relevant solution towards the production of
monolayers on wafer scales [35, 36, 37, 38]. Layers of high-quality over large
surfaces are now achieved and easily transferred to any substrate of choice in
soft ambient conditions [39, 40]. This has allowed to envision the integration
of graphene in many systems, even the most delicate or specific ones and has
been key to the integration of graphene with advanced complex oxides such as
high-$T_{\text{C}}$ superconductors [12].

The broader Dirac-material family of three-dimensional topological insulators
(3DTI) shares part of the electronic properties of graphene. These materials
are characterized by a strong spin-orbit coupling, they present an insulating
bulk, but have conducting surface states [41]. As in standard insulators, in
bulk, a finite size energy gap separates their valence and conduction bands.
However, their surface or any boundary with a normal insulator or the vacuum
is characterized by surface states that can be described by an effective
Dirac-like Hamiltonian. These surface states live inside the bulk energy gap
and are topologically protected unless time-reversal symmetry is broken. For
example, strained HgTe represents an important platform for 3DTI.
Additionally, 3DTIs are hosted in several layered structures based on Sb, Bi,
Se and Te, as Bi2Se3, Bi2Te3 and Sb2Te3. These are multi-layers systems that
crystallize in tetradymite structure. Thus, they consist of covalently bonded
quintuple layers (_e.g._ , Se?Bi?Se?Bi?Se) that are stacked in ?A?B?C?A?B?C?
manner and are weakly interacting with van der Waals force [42].

The effective Hamiltonian describing these systems is similar to the graphene
one but without the valley degree-of-freedom. Additionally, the pseudo-spin
$\bm{\sigma}$ is substituted by the fermionic spin degree-of-freedom
$\bm{s}\equiv(s_{x},s_{y})$, thus their effective low-energy Hamiltonian is
$\mathcal{H}_{\text{3DTI}}=v_{\text{F}}(\bm{s}\times\bm{\nabla})_{z}$.
Additional cubic terms in momentum to this effective Hamiltonian can describe
the high-energy hexagonal warping of the Dirac cone of these layered 3DTI
systems [43]. This effective Hamiltonian encodes a peculiar characteristic of
3DTI that is the _spin-momentum locking_ , namely the spin degeneracy is
lifted off for the surface states and the direction of the spin is always
perpendicular to the motion direction [44], this property being similar to the
photon helicity. The spin-momentum locking is typical of all the systems with
a strong spin-orbit interaction [45]. The effective Hamiltonian
$\mathcal{H}_{\text{3DTI}}$ describes the electron gas on a single 3DTI
surface, a similar Hamiltonian describes the opposite surface with carriers
with opposite helicity. However, in transport experiments only one of these
surfaces is contacted, thus a model accounting for a single Dirac cone is
sufficient for our purposes.

Since 2010, many new families of 3DTI have been uncovered and studied,
expanding the available exotic transport properties of graphene [42, 46]. The
advantage of the binary compounds as Bi2Se3 is that they can be easily
prepared under several forms: bulk single crystals, thin films, nanobelts, or
nanoplates. However, they all presents a high density of defects. From the
experimental point of view, one of the major challenges for designing quantum
transport experiments is related to address only the charge carriers on a
specific surface. The problem arises from a small but finite coupling between
the surface and the bulk states of the 3DTI materials. This problem is usually
addressed by introducing more complex _ad hoc_ two-band models, one describing
the surface state, similar to $\mathcal{H}_{\text{3DTI}}$, and one accounting
for the electronic bulk states [42]. A way out from avoiding the influence of
the bulk states on the topological surface modes is to move from binary
compounds (_e.g._ Bi2Se3) to ternary compounds [42, 47, 48]. For example, we
can consider (Bi1-xSbx)2Te3 where the chemical potential can has been tuned
from bulk to valence conduction changing the Sb concentration [49, 50]. There
is another property of interest of 3DTIs is connected to the spin-momentum
locking: the surface states of 3DTIs present a particular photo-response to
circularly-polarized light. However, the associated spin-polarized current is
not detected so easily. The problem arises because spin-momentum locking makes
the charge and the spin life-times identical, as a consequence, the mean-free-
path and the spin diffusion length are of the same magnitude. Therefore, in
diffusive transport regimes, the spin polarization is decreased. The only way
to prevent this problem is to perform experiments in the ballistic regime [51,
52].

###  1.2 $d$-wave cuprates: unique properties and opportunities for proximity
effects in Dirac materials

Cuprates constitute a large family of superconducting oxides that have
remained a central problem in condensed-matter physics since their discovery
over three decades ago. The origin of superconductivity, which in these
materials is far from being understood [53], is connected to the presence of
competing electronic ground-states [54] that stem from strong correlations and
yield a very rich phase diagram [55]. These matters are the focus of intense
research efforts that are out of this review’s scope.

Figure 2: Archetypal $d$-wave superconductor. (a) Unit cell of the high-
temperature superconducting cuprate YBa2Cu3O7-δ [56]. (b) Comparison between
conventional $s$-wave pairing and the $d$-wave pairing characteristic of most
cuprate superconductors. In the latter, the order parameter is anisotropic,
its amplitude being dependent on the direction as shown in the right picture,
and the superconducting phase is shifted by $\pi$ between $d$-wave lobes [57].

Along with those electronic specificities, cuprates also present very unique
superconducting properties. For many of the compounds the superconducting
critical temperature $T_{\text{C}}$ exceeds the normal boiling point of
nitrogen which, accompanied by large critical currents and fields, confers to
the cuprates much potential for a variety technological applications [58].
These span from high-current coated conductors [59], which can be used for
example for the generation of high magnetic fields [60], to high sensitivity
sensors such as bolometers [61, 62] and transition edge detectors [63] or
quantum interference devices [8, 64] such as quantum antennas [9] and
magnetometric sensors [65]. Another key property most cuprates share is that
they exhibit a $d$-wave pairing. This means that their superconducting gap is
anisotropic, with respect to the direction of the momentum and also has a
change of sign, see Fig. 2(b), Refs. [66, 67, 68]. This results in a high-
density of zero-energy quasiparticle (Andreev) surface bound-states whose
density depends on the orientation of the superconductor’s surface relative to
the crystallographic axes [69, 57]. The surface Andreev bound states result in
a zero-energy peak in the tunneling conductance that sharpens upon decreasing
the temperature [69, 57]. As it will be discussed in Section 2, this has
strong effects on the conductance across $d$-wave/metal interfaces in general,
and across $d$-wave/graphene interfaces in particular. The characteristic
$\pi$ phase shift between $d$-wave lobes [70, 71, 68] ( Fig. 2(b)), which
contrast with the single valued phase of $s$-wave superconductors, has other
important consequences. This was early shown by Tsuei  _et al._ , who built a
tricrystal of yttrium barium copper oxide YBa2Cu3O7-x (YBCO) which naturally
enclosed a phase gradient and thus showed spontaneous magnetization at low
temperature [71], and by the work of Il’ichev _et al._ based on a YBCO
bicrystal to realize exotic Josephson junctions with an anomalous magnetic
field periodicity [72]. Indeed, the phase phase shift between $d$-wave lobes
has strong implications when it comes to realizing $d$-wave based Josephson
junctions and devices [73, 74, 75, 76]. One of the outstanding possibilities
is the realization of junctions with a built-in $\pi$-shift between the
electrodes, allowing for half-quantum superconducting loops [77] that enable
novel rapid single flux quantum logic devices architectures with enhanced
capabilities [7]. Furthermore, the combination of high-$T_{\text{C}}$ and the
$\pi$ phase shift between $d$-wave lobes can be used for designing topological
Josephson junction hosting Majorana bound states [78, 79, 80]. Although
extended to different $d$-wave/$s$-wave Josephson junctions [81], so far these
$\pi$ junctions have not been explored using Dirac materials (and particularly
graphene) as Josephson barrier. Yet, these devices would be endowed with an
interesting property, namely tunability, by replacing the usual passive weak
link by a gate-tunable Dirac material. Graphene stems out as a material of
choice in that prospect in that it is widely available, well characterized and
has been implemented in low-temperature superconductor junctions. Finally,
another interesting property of all superconducting cuprates regarding
superconducting proximity effects with Dirac materials is that their
high-$T_{\text{C}}$ is accompanied by a superconducting gap whose amplitude
can reach tens of meV, that is, one order of magnitude larger than that of
conventional low-$T_{\text{C}}$. This is important because, as it will
described below, many of the distinctive features of proximity effects in
these materials are linked to the doping level, and particularly to the height
of the Fermi level relative to the size of the superconducting gap. Yet, as
discussed is Section 1.1, in real Dirac samples, including in single layer
graphene, the doping (and thus the Fermi level) is often inhomogeneous at the
nano-scale, with a variability that generally can reach $\sim$ 10 meV unless
special, practically restrictive measures are taken [82]. Thus, one of the
advantages of high-$T_{\text{C}}$ cuprates over conventional superconductors
in the context of this review’s matter is that they offer a platform in which
the possible electronic inhomogeneities of the graphene (and other 2D Dirac
materials) can often be neglected when compared to the size of the
superconducting gap.

Unfortunately, all of the interesting properties of cuprates are accompanied
by a drawback: there are delicate materials for which the fabrication of
solid-state devices to study proximity and Josephson effects in combination
with 2D materials is often much more complicated than for conventional
low-$T_{\text{C}}$ superconductors. The latter are usually metals (for example
Al or Nb), whose growth techniques and conditions (_e.g._ evaporation or
sputtering at room temperature and inert atmosphere on many different
substrates, epitaxial growth being unnecessary) make them compatible with
standard nanofabrication approaches (_e.g._ e-beam lithography combined with
lift-off). This allowed the early fabrication of low-$T_{\text{C}}$ graphene-
based Josephson devices [3]. However, cuprates have a complex crystal
structure and composition [56] — see Fig. 2(a) — and the growth of thin films
requires epitaxy on specific substrates at high temperatures (hundreds of
degrees) in oxygen-rich atmosphere [83, 10]. This makes them incompatible with
the approaches traditionally used with low-$T_{\text{C}}$ metals. Furthermore,
the cuprate’s superconducting properties are extremely sensitive to the
presence of structural defects [84, 85] and to the oxygen stoichiometry [54,
55] —which has made the demonstration of superconductivity in single layer
BSCCO challenging [86, 87], for example. This sensitivity limits the
conditions (temperature, chemicals) under which conventional nanofabrication
approaches can be applied to combine $d$-wave cuprates with 2D materials. As
discussed in Section 3.3, this has motivated the development of _ad hoc_
fabrication techniques.

##  2 Theory

###  2.1 The normal/$d$-wave superconductor junction

In this Section we provide a brief review on the superconducting proximity
effect from a theory perspective. It is well-know that metals and
semiconductors in contact with superconductors acquire superconducting
correlations due to the so-called proximity effect [88]. The microscopic
origin of such induced correlations is the process known as the Andreev
reflection: an incident electron forms a Cooper pair in the superconductor.
Charge conservation at the interface results in the reflection of a hole [88].
In a ballistic system, the created electron-hole pair remains coherent over
distances of the order of $v_{\text{F}}/E$, where $E$ is the energy of the
electron measured from the Fermi-level, and $v_{\text{F}}$ the Fermi velocity.
In metallic systems, the velocity of reflected holes is the opposite of one of
the incoming electrons. In Dirac materials, like graphene and three-
dimensional topological insulators, besides this so-called Andreev retro-
reflection, Andreev reflection can be specular as well [89, 2]. Manifestation
of specular Andreev reflection has been detected experimentally in bilayer
graphene [90]. The type of Andreev reflection can be changed from retro to
specular by tuning the electrochemical potential $\mu$ of the normal region:
it is of specular type when $\mu$ is close to the charge neutrality energy
[89], and its of pure retro-type when $\mu$ is much larger than the
superconducting gap energy [91, 92]. In the following, we review the
theoretical approach to Andreev reflection for both, metals and Dirac
materials. Importantly, throughout this Section, we assume that the
superconducting pair potential is spatially constant in the superconducting
regions, and we ignore for simplicity the self-consistency of its spatial
distribution.

The Andreev reflection affects drastically the transport properties of hybrid
superconducting systems. In this review, we focus on lateral structures, i.e.
systems on which superconducting electrodes are grown on top of a 2D or
quasi-2D non-superconducting material (N), see Fig. 3(a). In such a setup, the
proximity effect in N leads to finite superconducting correlations with the
same symmetry as the superconducting lead. The main transport features are
then obtained by analyzing a planar hybrid structure between superconducting
(those proximitized by the superconducting leads) and normal regions — Fig.
3(a).

Figure 3: (a) Sketch of the N/S interface for the case of a parabolic metallic
systems or for the case of graphene. The thick white line separating the N/S
region represent the BTK $Z$ parameter. (b) Sketch of the angles relative to
the normal reflection (blue circle) and the Andreev reflection (red circle).
The incoming carriers arrive with an angle $\theta$ with respect to the normal
to the N/S interface, and are normal reflected with an angle $\pi-\theta$.
Andreev reflected carriers propagate with an angle $\theta^{\prime}$.

We focus here on clean ballistic systems and introduce the Bogoliubov-de
Gennes (BdG) equation

$\displaystyle\mathcal{H}_{\text{BdG}}\circ{\bf\Psi}=E{\bf\Psi}\;,$ (1a) where
$\displaystyle\mathcal{H}_{\text{BdG}}(\bm{r},\bm{r^{\prime}})=\begin{pmatrix}[H(\bm{r})-\mu(\bm{r})]\delta(\bm{r}-\bm{r^{\prime}})&\Delta(\bm{r},\bm{r^{\prime}})\\\
\Delta^{*}(\bm{r},\bm{r^{\prime}})&-[\bar{H}(\bm{r})-\mu(\bm{r})]\delta(\bm{r}-\bm{r^{\prime}})\end{pmatrix}\;.$
(1b)

and the $\circ$ in Eq. (1a) denotes convolution in real space. Because our
focus is on $d$-wave superconductors, we need to keep the two components
dependence in the effective superconducting pairing potential induced in the N
region, $\Delta$ [93]. The above Hamiltonian is a matrix in the particle-hole
space. Each of its components are in the general also matrices whose structure
depends on the specific material. The term $H$ describes the normal state
Hamiltonian, whereas $\bar{H}=\mathcal{T}H\mathcal{T}^{-1}$ is its time-
conjugated, here $\mathcal{T}$ denotes the time-reversal symmetry (TRS)
operator. In a hybrid structure $\Delta$ and the chemical potential $\mu$ in
Fig. 3(a) are step-like functions.

Because of the anistropy in $\Delta$, the search for an exact solution of Eqs.
(1) is a difficult task. Indeed, the spinors $\bm{\Psi}$ oscillates over a
scale of the inverse Fermi wave vector $k_{\text{F}}^{-1}$, whereas the
superconducting correlations usually vary over a much longer scale
$\xi_{\text{S}}$. In this case one can single out the fast oscillations by the
substitution
$\bm{\Psi}\rightarrow\text{e}^{-\text{i}\bm{k}_{\text{F}}\cdot\bm{r}}\bm{\Psi}$.
In leading order in $k_{\text{F}}^{-1}$, Eqs. (1) reduce to the well-known
Andreev equations [93, 94]:

$\begin{pmatrix}-\text{i}\frac{\bm{k}_{\text{F}}}{m}\cdot\bm{\nabla}&\Delta(\bm{k}_{\text{F}},\bm{R})\\\
\Delta^{*}(\bm{k}_{\text{F}},\bm{R})&\text{i}\frac{\bm{k}_{\text{F}}}{m}\cdot\bm{\nabla}\end{pmatrix}{\bf\Psi}=E{\bf\Psi}\;,$
(2)

where $\Delta(\bm{k}_{\text{F}},\bm{R})$ is the Wigner transform of
$\Delta(\bm{r},\bm{r^{\prime}})$ evaluated at the Fermi level, and $\bm{R}$ is
the center of mass coordinate.

For the geometry shown in Fig. 3(a), the N/S interface is located at $x=0$.
The pair potential is only finite at $x>0$. If we denote with $\bm{k}_{\pm}$
the incident and reflected wavevectors then the pair potential of a
$d_{x^{2}-y^{2}}$ is given by
$\Delta(\bm{k}_{\pm})=\Delta_{0}\cos\left[2(\theta\mp\alpha)\right]$, where
$\theta$ is the incident angle of an electron from the normal conductor, and
$\alpha$ the angle between the normal to the interface and the crystalline
axis along which the pair potential reaches its maximum value. As in the
original work by Blonder Tinkham Klapwijk (BTK) [95], an imperfect N/S barrier
is modeled by a delta-like potential, $U(x)=\mathcal{U}_{0}\delta(x)$.

Figure 4: The normalized conductance $G_{\text{NS}}$ as a function of the
energy of injected electrons for (a) $\alpha=0$ and (b) $\alpha=\pi/4$. For
both panels, we have used three different values of the $Z$ parameter [94].

This term leads, to complement the continuity of the wave function with the
following boundary condition for Eq. (2):

$\displaystyle\partial_{x}{\bf\Psi}^{\text{S}}(0^{+})-\partial_{x}{\bf\Psi}^{\text{N}}(0^{-})=2m\mathcal{U}_{0}{\bf\Psi}(0)\;.$
(3)

The boundary problem Eqs. (2) and (3) can be solved straightforwardly. The
electronic transport properties through the N/S interface are determined by
the Andreev, $r_{\text{A}}$, and normal, $r_{\text{N}}$, reflection
amplitudes, of the electrons and hole injected from the normal metal. From the
BTK theory the conductance at low temperatures can be written as [96, 97]:

$G_{\text{NS}}(V)=\frac{e^{2}N_{\text{F}}v_{\text{F}}w}{\pi}\int_{-\pi/2}^{-\pi/2}d\theta\left(1+|r_{\text{A}}(V,\theta)|^{2}-|r_{\text{N}}(V,\theta)|^{2}\right)\;,$
(4)

where $N_{\text{F}}$ is the (2D) density of states at the Fermi level and $w$
the width of the junction. The conductance of a N/$d$-wave superconductor was
first discussed and calculated in Refs. [98, 94]. The main results of those
theories are summarized in Fig. 4 where we show the dependence of the
normalized conductance ($G_{\text{NS}}=G_{\text{S}}/G_{\text{N}}$ with respect
to the normal value $G_{\text{N}}$) through the N/S interface as function of
the injection energy (applied voltage). Different curves correspond to
different values of the BTK parameter $Z=2m\mathcal{U}_{0}/k_{\text{F}}$. In
the left panel the angle $\alpha$ between the normal to the interface and the
direction along which the gap is maximum, is zero. In this case
$\Delta(\bm{k}_{+})=\Delta(\bm{k}_{-})=\Delta_{0}$, and the behaviour of the
conductance resembles the one of an isotropic N/S junction. The zero-bias
conductance reaches its maximum, $G_{\text{NS}}(0)=2$ value when the interface
is transparent ($Z=0$). For a low-barrier transmission, $Z\gg 1$, the
conductance approaches the tunneling behaviour. If the interface is
transparent then the conductance do not depend on the angle $\alpha$ (blue
solid lines in both panels of Fig. 4). The situation changes drastically when
both $\alpha\neq 0$ and $Z$ are finite, as shown in the right panel of Fig. 4.
In this case the zero bias conductance may largely exceed the value of $2$ and
it diverges as $G_{\text{NS}}(0)\sim Z^{2}$ when $Z\rightarrow\infty$. This is
due to the fact that, whereas the normal conductance, $G_{\text{N}}$ is
monotonically reduced by increasing $Z$, $G_{\text{S}}$ remains equal to $2$
for certain values $E_{0}$ of the energy where resonance tunneling through
bound-states takes place. Such bound-states are formed, in analogy to
conventional S-S ballistic junctions, because normal reflected quasiparticles
from the superconductor experiences pair-potentials with different phases,
before and after the reflection [99, 57].

###  2.2 Proximity effects $s$-wave/Dirac material: Specular Andreev
reflection

####  2.2.1 Graphene junctions

Figure 5: (Color online) (a) Electron and hole excitations in the conical band
structure of system for electrons with graphene energy dispersion, red and
blue sphere at energy states at $\mu\pm\epsilon$ are converted into each other
by Andreev reflection at the normal/superconductor interface. Panel (b) and
Panel (c): Sketch of the excitation spectrum for electron in the conduction
band (blue solid line) and holes in the conduction band (red solid line) and
valence band (red dashed line) for the case in which
$\mu\gg\text{Max}[\Delta,\epsilon]$ and $\mu\ll\text{Max}[\Delta,\epsilon]$,
respectively [2, 100].

In this Section, we show how the process of Andreev reflection changes when
considering a Dirac-like Hamiltonian in the BdG Eq. (1b). First, we consider
the case of graphene proximitized with a $s$-wave superconductor [89]. In the
Section 2.3 we generalize the results to the case of a $d$-wave
superconductor.

As mentioned in the introduction, the energy spectrum of graphene is almost
linear around the two Dirac points, $\mathbf{K}$ and $\mathbf{K}^{\prime}$, of
the first Brillouin zone [24], see Fig. 5. Here, we can perform the so-called
low-energy long-wavelength approximation (LWA), and introduce the following
effective Hamiltonian describing the electron behaviour around Dirac points
[23]:

$\displaystyle\mathcal{H}_{\text{LWA}}=v_{\text{F}}\tau_{0}(\sigma_{x}p_{x}+\sigma_{y}p_{y})$
(5)

where $v_{\text{F}}\sim 10^{6}$ m/s is the effective Fermi velocity of the
electron, $\bm{p}$ is the electron momentum operator,
$\tau_{0}=\mathbb{I}_{2}$ represents the identity matrix associated to the two
Dirac cones, and $\bm{\sigma}=(\sigma_{x},\sigma_{y})$ are the Pauli matrices
associated to the so-called pseudo-spin connected to the two carbon atoms A
and B in the unit cell of graphene, see Fig. 1. We disregard the fermionic
spin degree-of-freedom since we only consider situations in which time-
reversal symmetry (TRS) is preserved. The Hamiltonian in Eq. (5) acts on the
following spinor

$\Psi=(\psi_{\text{A}},\psi_{\text{B}},-\psi_{\text{B}}^{\prime},\psi_{\text{A}}^{\prime}),$

where the first two components are associated to the valley $\mathbf{K}$ and
the last two to $\mathbf{K}^{\prime}$, and the indices A and B are associated
to the pseudo-spin. In order to construct the BdG Hamiltonian (1b), for
proximitized graphene it is important to notice that according to Eq. (5) the
electrons in valleys $\mathbf{K}$ and $\mathbf{K}^{\prime}$ are related to
each other via TRS [2]. When considering graphene in the LWA, this symmetry
operator can be expressed as
$\mathcal{T}=-(\tau_{y}\otimes\sigma_{y})\mathcal{K}$, where
$\tau_{y}(\sigma_{y})$ are the $y$ Pauli matrices for the valley(pseudo-spin)
degree-of-freedom, and $\mathcal{K}$ is the complex conjugation operator
[101]. In other words, within the LWA, time-reversed carriers are associated
to the two different valleys. When constructing the BdG matrix Hamiltonian, we
can then split the $8\times 8$ problem into two independent $4\times 4$
problems. Importantly, we assume that all the spatial modulation involved in
the problem takes place on a length scale much larger than the graphene
lattice spacing $a_{0}$. This results in the following BdG Hamiltonian for
graphene,

$\displaystyle\mathcal{H}_{\text{BdG}}^{\text{LWA}}=\begin{pmatrix}v_{\text{F}}\bm{\sigma}\cdot\bm{p}-\mu&\Delta(\bm{r})\\\
\Delta(\bm{r})^{*}&\mu-v_{\text{F}}\bm{\sigma}\cdot\bm{p}\end{pmatrix}.$ (6)

Within this model the pairing potential does not mix the component of the
spinors, thus $\Delta(\bm{r})=\Delta_{0}\sigma_{0}$, this correspond to
consider only intraband superconducting pairing on each sublattice A and B
[102].

The chiral semimetallic nature of graphene enriches the physics of the
electron scattering at a N/S interface. An electron from the conduction band
impinging on the interface with an angle $\theta$ is normal reflected within
the same band with opposite angle $\pi-\theta$ or can be Andreev reflected as
hole either in the conduction or in the valence band. The former process is a
retro-reflection whereas the latter is a specular-reflection [89].

In the practice we are dealing with a genuine two-dimensional scattering
problem with two free parameters, the injection energy $\epsilon$ and the
momentum parallel to the N/S interface $q$ — see Fig. 3(b). Considering an
incoming electron in the conduction band, the incidence angle $\theta$ in
polar coordinate reads:

$\displaystyle\theta$ $\displaystyle=\arcsin\left[\frac{\hbar
v_{\text{F}}q}{\epsilon+\mu}\right],$ (7)

where $(\epsilon+\mu)(v_{\text{F}}\hbar)^{-1}=|\bm{k}|$ is the modulus of the
momentum for the conduction band. For a hole, the value of the reflection
angle can be obtained by simple kinematic considerations: conservation of the
two free parameters in the scattering process — see Fig. 3(b). Thus the angle
of propagation of the hole is

$\displaystyle\theta^{\prime}$ $\displaystyle=\arcsin\left[\frac{\hbar
v_{\text{F}}q}{\epsilon-\mu}\right].$ (8)

We can also treat along the same line of reasoning the case of a hole injected
state. The propagation direction of the hole strongly depends on the value of
the injection energy compared to the chemical potential. In Fig. 3(b) we show
a sketch of the the Fermi circles for electrons and holes in the normal
region; if the momentum $q$ exceeds the radius of the Fermi circle for holes,
the corresponding Andreev reflected state is suppressed. Thus, we introduce a
critical injection angle defined as:

$\theta_{\text{c}}=\arcsin\left[\frac{|\epsilon-\mu|}{\epsilon+\mu}\right]\,.$
(9)

When treating this scattering process, the main difference compared to the
case of electrons in metals arises in the boundary conditions, c.f. Eq. (3).
For graphene it is sufficient to require the continuity of the wave function
at the interface. We can solve this system of equations analytically and
obtain an expression for the normal $r_{\text{N}}$ and Andreev $r_{\text{A}}$
reflection [89, 100]. These reflection amplitudes read:

$\displaystyle r_{\text{A}}$
$\displaystyle=\begin{cases}\text{e}^{\frac{\text{i}}{2}(\theta+\theta^{\prime}-\pi)}\Xi^{-1}\sqrt{\cos(\theta)\cos(\theta^{\prime})}&\text{if}~{}|\theta|<\theta_{\text{c}}\\\
0&\text{if}~{}|\theta|>\theta_{\text{c}}\end{cases}\,,$ (10a) $\displaystyle
r_{\text{N}}$
$\displaystyle=\text{e}^{\frac{\text{i}}{2}(2\theta-\pi)}\Xi^{-1}\left[\cos\beta\sin\left(\frac{\theta+\theta^{\prime}}{2}\right)+\text{i}\sin\beta\sin\left(\frac{\theta-\theta^{\prime}}{2}\right)\right]\,,$
(10b) $\displaystyle\Xi$
$\displaystyle=\cos\left(\frac{\theta+\theta^{\prime}}{2}\right)\cos\beta+\text{i}\cos\left(\frac{\theta+\theta^{\prime}}{2}\right)\sin\beta\;,$
(10c)

where we have introduced the function $\beta(\epsilon)$ defined as

$\displaystyle\beta(\epsilon)=\begin{cases}\arccos\left(\frac{\epsilon}{\Delta}\right)&\epsilon<\Delta\\\
-\text{i}\,\text{arccosh}\left(\frac{\epsilon}{\Delta}\right)&\epsilon>\Delta\end{cases}\,.$
(11)

It is important to note that in the previous derivation, following Ref. [89],
we have performed the so-called Andreev approximation [88] only in the
superconducting region, where we assume that the local chemical potential
$\mu_{\text{S}}$ is the dominant energy scale:
$\mu_{\text{S}}\gg\text{Max}[\Delta,\epsilon]$.

We analyse now the amplitudes in Eqs. (10) similarly to Ref. [89]. The regime
of Andreev retro- or specular-reflection can be addressed by considering the
chemical potential in the normal region $\mu$. If the chemical potential is
the dominating energy scale $\mu\gg\text{Max}[\Delta,\epsilon]$, then the hole
generated by an Andreev refection results in an empty states in the conduction
band with a propagation angle opposite to the one of the incoming electron
$\theta^{\prime}=-\theta$. This is the limit of Andreev retro-reflection, here
we can write a simplified expression for the reflection amplitudes [c.f. Fig.
5(b)]:

$\displaystyle r_{\text{A}}$
$\displaystyle=\frac{\text{i}\cos\theta}{\zeta+\frac{\epsilon}{\Delta}\cos\theta}\,,$
(12a) $\displaystyle r_{\text{N}}$
$\displaystyle=\frac{\text{i}\text{e}^{\text{i}\theta}\zeta\sin\theta}{\zeta+\frac{\epsilon}{\Delta}\cos\theta}\,.$
(12b)

In the opposite regime, $\mu\ll\text{Max}[\Delta,\epsilon]$, the hole
generated by the Andreev reflection is in the valence band and
$\theta^{\prime}=\theta$ — this is the case of Andreev specular-reflection
[89]. Also in this case, we can simplify Eqs. (10) for the normal and Andreev
reflection amplitudes [c.f. Fig. 5(c)]:

Figure 6: (a) Normal (solid-line) and Andreev (dashed-line) reflection as a
function of the injection energy $\epsilon$, for various values of the
chemical potential $\mu$. (b) Differential conductance as a function of the
injection energy $\epsilon$ for the same values of the chemical potential as
in (a). In panel (b), the dotted-dashed line corresponds to the differential
conductance at zero applied voltage, whereas the dashed line represents the
conductance of the normal state. In both panels, the grey area represents
injection energy smaller than the superconducting energy gap $\Delta$, this
represents the scale of references for the energies. The results in the two
panels are in agreement with Refs. [89, 100].

$\displaystyle r_{\text{A}}$
$\displaystyle=\frac{\text{i}\text{e}^{\text{i}\theta}\cos\theta}{\frac{\epsilon}{\Delta}+\zeta\cos\theta}\,,$
(13a) $\displaystyle r_{\text{N}}$
$\displaystyle=\frac{\text{i}\text{e}^{\text{i}\theta}\frac{\epsilon}{\Delta}\sin\theta}{\frac{\epsilon}{\Delta}+\zeta\cos\theta}\,.$
(13b)

In Eqs. (12) and (13) we have introduced the following function:

$\displaystyle\zeta=\begin{cases}\text{i}\sqrt{1-\left(\frac{\epsilon}{\Delta}\right)^{2}}&\epsilon<\Delta\\\
\sqrt{\left(\frac{\epsilon}{\Delta}\right)^{2}-1}&\epsilon>\Delta\end{cases}\,.$
(14)

In Fig. 6(a), we present the normal and the Andreev reflection as a function
of the injection energy and for various values of the chemical potential of
the normal region $\mu$ to show the cross-over from specular to retro Andreev
reflection. For injection energies $\epsilon$ smaller that the superconducting
gap $\Delta$, we observe that the Andreev reflection goes to zero when the
injection energy is equal to $\mu$, whereas at these energies the normal
reflection equals one. For injection energies larger than the superconducting
gap, the normal reflection is finite whereas the Andreev reflection goes to
zero by increasing the energy.

The differential conductance for the case of graphene is obtained by Eq. (4),
using $N_{\text{F}}=\frac{W}{\pi}\frac{\mu+\epsilon}{\hbar v_{\text{F}}}$ for
the graphene density of states, where $W$ corresponds to the finite transverse
dimension of the graphene sample. As a consequence of the crossover between
specular and retro-reflection of the Andreev channel, the differential
conductance of the N/S interface is zero when the injection energy coincides
with the chemical potential of the normal region $\mu$, this is shown in Fig.
6(b). An additional consequence of the graphene linear dispersion and of
having performed the Andreev approximation only in the superconducting region
is in the value of the differential conductance at zero bias. This is
independent of the chemical potential and equal to $4/3G_{0}(eV)$, where
$G_{0}(eV)=2e^{2}/hN_{\text{F}}$. Another interesting limit studied in Ref.
[89] is the limit of large applied bias where, for any chemical potential, we
find the universal value of the differential conductance of
$(4-\pi)G_{0}(eV)$. These limiting values are consequence of having imposed
the Andreev approximation only in the superconducting region. However, if this
condition is relaxed, the limiting value of the normal conductance goes back
to $G_{0}$, whereas the zero bias conductance equals $2G_{0}$ as shown in
Refs. [91, 92].

####  2.2.2 3DTI junctions

The results obtained so far are not bounded to graphene as a Dirac material
but are strongly connected to its 2D linear dispersing kinetic term.
Therefore, they can be extended to the case of 3DTIs proximitized by a
$s$-wave superconductor. For these systems, the main difference arises in the
structure of the superconducting pairing potential in Eq. (6). This is due to
a different basis describing the spinfull Hamiltonian of a 3DTI, that in this
case reads
$\Psi=(\psi_{\uparrow},\psi_{\downarrow},\psi_{\uparrow}^{\dagger},\psi_{\downarrow}^{\dagger})$
[103, 104]. The pairing potential for the case of a proximitized 3DTI is

$\hat{\Delta}=\begin{pmatrix}0&\text{i}\sigma_{2}\Delta(\bm{r})\\\
-\text{i}\sigma_{2}\Delta^{*}(\bm{r})&0\end{pmatrix}\,.$ (15)

Utilizing this pairing potential and the kinetic term of a 3DTI, the results
for the normal and Andreev reflection are identical to the ones described by
Eqs. (10). As a consequence, the physics of the Andreev specular- and retro-
reflection also appears in 3DTIs [105, 100]. Importantly, as a consequence of
the non-trivial spin structure associated to the spin-momentum locking, the
proximitized structure behaves as an effective $p$-wave superconductor when
the proximitizing superconductor is of $s$-wave type [106]. In this case the
spin pairing has a triplet structure. A magnetic field can modify the nature
of the proximity effect by mixing singlet pairing arising from the
proximitizing $s$-wave superconductor and the $p$-wave from the 3DTI. This
mixing will result in different differential conductance of hybrid structure
of 3DTI/F/S, where F denotes a ferromagnet with an exchange field or a Zeeman
field associated to the magnetic field [107]. Triplet order parameter has been
suggested for many applications including the proposal for a perfect spin-
triplet filter [108]. This type of heterostructure can host zero-energy
Majorana bound states [109, 110, 111, 112]. Hybrid 3DTI/$s$-wave exhibit a
cooling power that is also depending on the two possible regime of the Andreev
reflection, larger for specular and smaller for retro-reflection [100].

####  2.2.3 Interface transparency

When considering the hybrid junction between a normal metal and a
superconductor, we have seen that it is possible to model the transparency of
the interface via a single parameter $Z$ [95]. On the contrary, a similar
procedure is not permitted in a Dirac system, since the only boundary
condition is the continuity of the wave function at the interface [2, 113,
114, 115]. In non-Dirac systems, the main role played by a finite
$Z$-parameter is to increase the magnitude of normal reflection channel at the
N/S interface. We see from the results in Fig. 6(a), that in a two-dimensional
Dirac system, a channel of normal reflection is intrinsically present and that
it can be modulated by tuning the chemical potential of the normal region.
Thus, a strategy to _mimic_ a $Z$-parameter in a Dirac system consists in
considering a finite size electrostatic barrier of width $w$ and height
$\mathcal{U}$. Similarly to the case of charge carriers with parabolic
dispersion, we can obtain the limiting case of a delta-like barrier by sending
$w\to 0$ and $\mathcal{U}\to\infty$ keeping finite the ratio
$\mathcal{U}w/(\hbar v_{\text{F}})\equiv Z$. This approach has used in Refs.
[116, 91, 92], and the approach can be recast into a boundary problem for
scattering matrices [117]. Following this approach, a periodic oscillation of
differential conductance as a function of the injection energy was found, this
result is entirely associated to the Dirac-like dispersion. The period of the
oscillations is proportional to the parameter $Z$ [91, 92, 116]. When graphene
is paired with a $s$-wave superconductor, the differential conductance
evaluated this way and with equal chemical potential in the normal and
superconducting region, resemble the one obtained for electrons with a
quadratic kinetic term, however, for large $Z$ it is not possible to find a
complete suppression of the differential conductance [92], as a consequence of
the Klein tunneling taking place at the interface [92, 113].

###  2.3 Proximity effect $d$-wave/Dirac material

Figure 7: (a) Differential conductance of the hybrid graphene/$d$-wave
superconductor interface as a function of the applied voltage for different
orientations of the superconducting order parameter $\alpha$. The vertical
dashed lines represents the value of the reduced gap for the corresponding
value of $\alpha$ accordingly to the expression
$\Delta(\alpha)=\cos(2\alpha)$. The peaks of the differential conductance
coincide with the gap value only for large values of $\mu_{\text{S}}$,
otherwise, the peaks gets smoother and depart from the value of the gap [91,
92]. (b) Zero-bias conductance peak at $\alpha=\pi/4$ as a function of the
applied voltage for various values of the chemical potential $\mu$ in the
normal region. In both panels we have set the chemical potential of the
superconducting region to $\mu_{\text{S}}=10^{4}\Delta$. The results in the
two panels are in agreement with Refs. [91, 92].

####  2.3.1 Graphene junctions

We consider a $d_{x^{2}-y^{2}}$-wave superconductor with an order parameter
described by the function $\Delta(\bm{k})=\Delta_{0}\cos[2(\theta-\alpha)]$,
where $\theta=\arctan(k_{y}/k_{x})$ parametrize the injection direction, and
$\alpha$ describes different orientation of the gap in the $\bm{k}$-space with
respect to interface [91, 92] — see Fig. 2(b). The differential conductance
can be obtained following the same procedure we used for the case of $s$-wave
pairing, but accounting for a momentum dependent order parameter [92]. This
can change in magnitude with $\alpha$ being larger for $\alpha=0$ and zero for
$\alpha=\pi/4$ and with opposite signs of the different lobes [69, 118]. We
assume, as we did for the case of proximity with a $s$-wave superconductor,
that the order parameter induced in graphene is of intraband type, _i.e._
pairing sites of the same sublattice [89, 91]. The behaviour of the hybrid N/S
junction for a graphene proximitized with a $d$-wave superconductor, differs
considerably compared to the case of electron with a quadratic kinetic term
[91, 92] that we considered at the beginning of this Section. Here we
summarize the main differences and the possible causes:

  * •

The zero-bias conductance peak (ZBCP) is observed in the graphene/$d$-wave
hybrid junction only if $\alpha=\pi/4$ and in the case of neglecting the
Andreev-approximation in the normal part of the junction. This can be
explained by the fact that only for finite $\mu$ in the normal region and for
$\mu\neq\mu_{\text{S}}$, it is possible to achieve a finite channel for normal
reflection. This channel is essential to form the zero energy bound state [94,
98].

  * •

For angles different from $\alpha=\pi/4$, the peak sits to a different energy,
reflecting the reduction of the energy gap expressed by the function
$\Delta(\bm{k})$. This behaviour depends on the strength of the chemical
potential in the superconducting region, the peaks appear exactly at the
position of the gap for large $\mu_{\text{S}}$, whereas they get smoother and
have a maximum for a lower energy compared to the gap for smaller
$\mu_{\text{S}}$ [91, 92].

  * •

Contrary to the case of quadratic kinetic terms, the ZBCP does not diverge
because the conductance of the normal state $G(\epsilon\to 0)\neq 0$,
therefore the ZBCP is always bound to $2G_{0}$.

These features are summarized in the two panels of Fig. 7, where the
differential conductance $G_{\text{NS}}(V)$ is calculated using Eq. (4). For
equal chemical potential in the normal and superconducting graphene regions,
we recover a result similar to the one obtained for electrons with quadratic
dispersion when $Z=0$ [94, 91, 92] — see Fig. 4. This is shown in Fig. 7(b)
for the case $\mu\equiv\mu_{\text{S}}$.

We note in passing that the assumption of intraband constraint for the
proximity effect can be relaxed. It was proven that choosing a pairing beyond
the intraband constraint can lead to a $d$-wave pairing in intrinsic graphene
[119, 120]. Additionally, self-consistent calculations have shown that the
induced pairing by a $d$-wave superconductor leads to an effective $p_{x}$\-
or $p_{y}$-wave pairing symmetry depending on the orientation of the boundary,
_i.e._ armchair or zigzag [121]. However, the authors note that this effective
$p+\text{i}p$ wave pairing does not break neither TRS nor produce spin-triplet
states [121].

####  2.3.2 3DTI junctions

In the case of a 3DTI as Dirac material, we have to include the momentum
dependent order parameter as we have done in Eq. (15). Here we account for the
anisotropy of the order parameter with $\Delta\to\Delta(\bm{k})$ [122].
Interestingly, because of the spin-momentum locking, the ZBCP we have seen for
the case of graphene or in a $d$-wave superconductor, moves from being two-
fold degenerate to non-degenerate. As a consequence, this state describes a
zero energy Majorana fermion with peculiar dependence on external
magnetization [122, 123, 124]. Here, it was shown that the ZBCP splits into
two separate peaks by introducing a Zeeman field perpendicular to the N/S
interface, this behaviour is opposite to what expected for the case of
proximization with a $s$-wave superconductor. Additionally, the ZBCP is
independent from a Zeeman field applied perpendicular to the N/S interface and
disappear when the Zeeman field is parallel to the interface [122]. The
authors of Ref. [122] note that the Majorana fermions spread along the
interface separating the normal and the superconducting region.

By analysing the cooling power and of a Josephson system based on
3DTI/$d$-wave junctions, it was found that the thermal conductance is
profoundly sensitive to the components of the pairing state and that this can
be varied by changing the length of the 3DTI region of the junction [117,
125].

##  3 Current experimental status

###  3.1 Experiments $s$-wave/graphene as the starting point

The first graphene-based Josephson junction was realized fourteen years ago
[3] with an exfoliated graphene flake on top of SiO2 and contacted by two Al
electrodes. Since then, many groups realized and improved the quality of
graphene Josephson junctions — namely the contact transparency, and the
graphene mobility. One crucial improvement is to use $h$-BN to encapsulate
graphene which result in a higher graphene mobility [126]. This allowed to
image the supercurrent distribution in the graphene layer close and far from
the charge neutrality [4], revealing guided electronic mode along the edge
close to charge neutrality. A second improvement is to use superconducting
electrodes with high critical magnetic field such as Molybdenum Rhenium [6] or
Niobium Nitride [127]. This allowed to couple quantum Hall integer edge
channels to superconducting leads, a promising step towards the realization of
Majorana modes [110] or parafermions [128], predicted to appear from the
coupling of fractional quantum Hall edge channels to superconducting leads.
Whereas high-quality contacts between graphene and $s$-wave superconductors
have been achieved, the techniques used are of little help to reach the same
contact quality between graphene and a $d$-wave superconductors. As discussed
earlier it is not possible to deposit a $d$-wave superconductor by evaporation
on a $h$-BN graphene stack. Indeed, $d$-wave superconductor require specific
growth conditions: specific substrate, high temperature, O2 pressure, which
are not compatible with the presence of graphene. The existing works combining
graphene with $d$-wave superconductor [11, 12, 13] have therefore used a wet
transfer process to deposit a sheet of CVD grown graphene on top of a
superconducting oxide. Although it achieves lower graphene mobility, the
strength of this technique is its scalability. An unexplored way, so far,
would be to use an exfoliable $d$-wave superconductor to realize a $d$-wave
superconductor graphene stack by the pick-up technique. This technique has
been successfully applied to bilayer graphene with low-temperature
superconductor [90] and high-temperature superconductor with topological
insulator as detailed below.

###  3.2 STM studies of $d$-wave superconductor/graphene proximity effect

Figure 8: STM studies of proximity induced superconductivity in graphene on
top of a high-$T_{\text{C}}$ superconductor. (a) Sketch of the investigated
samples, that consist of a single layer graphene sheet deposited on a 200 nm
thick PCCO superconducting film that is grown on a STO substrate. The
tunneling spectra is measured at different points by positioning the STM tip
above the graphene layer. (b) Theoretical expectations for the tunneling
conductance spectra in case of $p$-wave induced symmetry. The different
projections of the p-wave symmetry result in characteristic spectral features
that are correlated with the crystal orientation of the underling PCCO, as
discussed in the main text. (c)-(e) show experimental tunneling conductance
spectra which, depending on the tip position, can have a V-shape (a), a zero-
bias conductance peak (d) or split ZBCPs. The inset show the topographic image
of the areas where spectra are taken. Adapted from Ref. [11].

The first reports of proximity induced superconductivity in graphene using
cuprate superconductors were based on scanning tunneling microscopy (STM)
experiments [11]. The superconductor was Pr2-xCexCuO4 (PCCO), whose critical
temperature, $\sim 20$ K is lower than the $T_{\text{C}}$ archetypal cuprate
YBa2Cu3O7 (YBCO) for which $T_{\text{C}}\sim 90$ K. The main characteristic of
the studied electron-doped PCCO films, in addition to a relatively long
coherence length $\sim$ 30 nm, is that the $d$-wave features are allegedly
absent from the density of states because samples were in the dirty limit,
that is, the mean free path is shorter that the coherence length. As shown in
Fig. 8(a), the proximity effects were studied in bilayers consisting of
chemical vapor deposited (CVD) single-layer graphene (SLG) that is transferred
onto 200 nm PCCO film using standard techniques [129]. STM was used to locally
measure the differential conductance _vs._ voltage bias V in areas where the
surface topography made sure the presence of SLG as shown in Figs. 8(c)-8(e).
These experiments revealed gap-related spectral features that were only
present when the critical temperature was well below the $T_{\text{C}}$ of the
PCCO and gradually disappear as temperature was increased. Three main
different types of behavior were observed, which are illustrated by the
measurements displayed in Fig. 8(c)-8(e). The most common spectra (45 % of
them) contained V-shaped features, see Fig. 8(c), that resemble those expected
in bare PCCO. Other spectra presented very distinct features such as Zero-Bias
Conductance Peaks (30 % of the spectra as in Fig. 8(d)) and split-ZBCPs (25 %
of the spectra as in Fig. 8(e)) were observed that could not be found in bare
PCCO. The authors ascribed the different spectral features to the emergence of
$p$-wave superconductivity in the SLG due to proximity with PCCO. This
conclusion is supported by theoretical calculations of the density of states
induced in graphene due to proximity with $d$-waves superconductors [130,
121]. These theories suggest that the different types of experimentally
observed conductance curves can be explained if one considers that different
projections of the $p$-paring were probed by the STM tip. This scenario was
ascribed to the surface roughness of the PCCO film, which lead to local
variations of the crystal orientation, each of them associated with a
different $p$-wave projection in the SLG plane [see sketch in Fig. 8(b)].

###  3.3 Devices based on $d$-wave superconductors and graphene

If STM experiments, as those discussed in the previous section, are a
convenient approach to explore the proximity-induced superconducting
correlations and density of states in graphene, they do not provide direct
access to the conductance across the cuprate/graphene interface. In addition,
the existing STM experiments could not probe the length-scale over which the
unconventional correlations penetrate into graphene [11]. Moreover, the
geometry needed for STM does not allow for an easy gating of the graphene
layer, and hence it prevents for a control of the Fermi level. Yet, as
discussed in Section 2.3, those aspects are crucial for understanding,
tailoring and eventually exploiting high-$T_{\text{C}}$ superconducting
proximity effects in graphene. A way of tackling these issues is by
fabrication of solid-state devices, similar to those used with conventional
low-$T_{\text{C}}$ superconductors (see Section 3.1). Yet, as explained in
Section 1.2, the growth of cuprates is incompatible with some of the standard
lithography techniques. Because of the need to grow them on specific
substrates, at very high deposition temperatures in an oxygen-rich atmosphere,
cuprates cannot be grown on graphene. Thus, the standard approach consisting
of the deposition of low-$T_{\text{C}}$ superconducting electrodes on top of a
graphene flake lying on a substrate, cannot be applied for the case of
cuprates. Instead, the cuprate film has to be epitaxially grown first on an
appropriate substrate, _e.g._ SrTiO3 (STO), and then lithographic technique
have been used to define the superconducting electrodes’ geometry, before a
graphene sheet can be transferred onto them. Early attempts, by Sun _et al._
[131], to implement such procedure via conventional photolithography and dry
etching techniques to define the YBCO electrodes on which graphene was
subsequently transferred, demonstrated that the cuprate/graphene interfaces
obtained in that way generally presented very low transparency, thus
precluding proximity effects. The experiment suggested that exposure of YBCO
to chemicals and resists degrades its superconducting and normal-state
properties on the $c$-axis surface, leading to an insulating layer. The
experiments by Sun _et al._ nevertheless showed hints of induced
superconductivity in graphene in some of the measured devices. This was
ascribed to a direct coupling between graphene and the YBCO’s CuO2 planes at a
local scale, which could be randomly achieved due to very rough surface of the
used films.

Figure 9: Fabrication of solid-state devices for measuring the conductance
across the $d$-wave/graphene interface. (a) The fabrication is based on 4 main
steps. First, an Au (4 nm)/YBCO (50 nm) film is grown on SrTiO3. In step 1,
photolithography is used to define a resist mask through which the film is
irradiated with 110 KeV O+ ions at fluence $5\times 10^{14}$ cm-2. The
irradiation makes the unprotected YBCO insulating, while the protected YBCO
keeps superconducting (S) properties. In step 2, Au is removed with a soft Ar+
ion etching and the resist mask is desolved in acetone, after which a planar
device is obtained in which four S electrodes (bright color) are separated by
insulating YBCO (dark). In step 3, a single-layer graphene sheet is
transferred onto the electrodes and subsequently patterned using
photolithography and oxygen plasma to define a bridge between the S
electrodes. Removal of the remaining resist yield the final device. (b) Shows
a typical conductance spectrum in a device in which the YBCO/Au/graphene
interface is highly transparent, which is characterized by a zero bias peak
that grows as temperature is decreased below $T_{\text{C}}$. The inset shows a
typical microscopy image on an actual device (scale bar is 10 $\mu$m). Adapted
from Ref. [12].

An alternative fabrication method to improve the coupling between graphene and
$d$-wave superconductivity was developed by Perconte  _et al._ [12] and it is
schematically shown in Figure 9 (a). The first feature of the new approach is
that YBCO films are covered _in situ_ (that is, right after its growth in a
pulsed laser deposition system and without breaking the vacuum of the
deposition chamber) by a few nm thick Au capping layer. This layer protects
the YBCO surface during the subsequent fabrication steps, preserving its
superconducting properties. Importantly the Au/cuprate transparency is very
high as demonstrated in Ref. [132] for Au/BSCCO interface and [12] for Au/YBCO
interface. Moreover, because the capping layer of Au is one order of magnitude
thinner than the mean free path in this material, one expects $d$-wave
superconducting correlations to propagate essentially undisturbed across it
[133, 134]. A second feature of this fabrication method is that, in order to
define the device geometry, in particularly the YBCO electrodes, masked ion
irradiation is used instead of etching. Irradiation turns the exposed YBCO
amorphous, and thus insulating, while the masked YBCO areas keep
superconducting properties. After removal of the thin Au layer from the areas
exposed to the ion bombardment (using a mild Ar+ plasma) and of the resist
mask, a pattern of superconducting YBCO/Au electrodes embedded in an
insulating YBCO matrix is obtained. The device obtained following this
procedure is therefore nearly planar (only a 4 nm topography step due to the
Au thickness) which contrasts with the case of etched electrodes that yield
large steps comparable with the YCBO thickness. Thus, in the irradiated
devices, graphene lies on a flatter surface. The rest of the fabrication
process is standard and includes, after the transfer of CVD graphene, a
lithography and O2 plasma etching to shape graphene into a bridge connecting
the YBCO/Au electrodes. In the device shown in Fig. 9, the graphene bridge is
contacted with a total of four electrodes. This allowed two, three or four
probe measurements. Electrostatic doping of the graphene can be done either by
using the high-$k$ dielectric STO substrate as a backgate, or an AlOx top-gate
that is deposited after completing the lithography steps 1-4, as shown in Fig.
9(a). Figure 9(b) shows a three-probe conductance measurement obtained in this
type of device. One observes that, as temperature is decreased below
$T_{\text{C}}$, the low-bias conductance is gradually enhanced to form a peak
that extends over an energy scale comparable to the expected superconducting
gap of YBCO. The authors of [12] explained this behavior by assuming a highly
transparent Au/graphene interface, which allows superconductivity to be
proximity-induced in the graphene that lies directly on the superconducting
YBCO/Au contact. In this scenario, as theoretically explored [92], the
measured conductance essentially corresponds to the conductance across the
interface that separates superconducting graphene (that lying directly on the
YBCO/Au electrodes) and normal graphene (the one lying on insulating YBCO).

Figure 10: Superconducting Klein-like tunneling. (a) Schematic of the model
used to describe the conductance across the interface between proximitized
graphene (lying on superconducting YBCO/Au) and graphene lying on insulating
YBCO, which is back gated across the dielectric STO substrate as shown in the
lower cartoon. A potential step of width w and height $U_{0}$ defines a region
of normal graphene (N′) that separates superconducting graphene with Fermi
energy $E_{\text{F}}^{\prime}$ from normal graphene (N) with gate-tunable
Fermi energy $E_{\text{F}}$($V_{\text{G}}$). (b) Contour plot of the
normalized conductance (colour scale) measured as a function of the bias $V$
across the junction and gate voltage $V_{\text{G}}$. (c) Conductance as a
function of $V_{\text{G}}$ at fixed bias $V$. These profiles correspond to the
‘cuts’ of the plot in (b), and show the oscillatory behavior of the
conductance as a function of the gate voltage. Adapted from Ref. [12].

The above assumption was confirmed via backgating experiments [12], with the
geometry shown in Fig. 10(a). The electrostatic doping only occurs in the
portion of graphene lying on insulating YBCO because the electric field is
screened elsewhere by the (super)conducting YBCO/Au. In the reported
measurements, upon application of a gate voltage, the spectra is strongly
modified as compared to the case shown in Fig. 9(b). This can be seen in Fig.
10(b), which shows a contour plot of the conductance (color code) _vs._ bias
($x$-axis) as a function of the applied gate voltage ($y$-axis). In
particular, the relative height of low-bias conductance enhancement (red area
in the color plot), as well as the background conductance level, is
periodically modulated by the gate voltage, as shown in Fig. 10(c).

Figure 11: Long-range propagation and interference of $d$-wave Andreev pairs.
(a) Microscope image of a graphene cavity formed in between two
superconducting electrodes (SC) bridged by an overlaying single-layer graphene
sheet on an insulating substrate. The scale bar is 10 $\mu$m. The scheme is a
cartoon of the cavity. The graphene doping and thus its Fermi energy depends
on whether it lies on insulating (brown) or superconducting (yellow) YBCO,
which creates the cavity in which the electron and Andreev pair interferences
occur. The YBCO/Au/graphene interface measured in series with the cavity is
characterized by a barrier strength $Z$. (b) Typical experimental conductance
vs. bias across a cavity, which shows oscillations with two different periods
as shown by the black (long period) and red (short period) arrows. (c)
Numerical simulation of the cavity’s conductance using the model described in
the main text. (d) Oscillations period extracted from the modelling as a
function of the inverse of the cavity’s length. The ratio between the long
(black) and short (red) oscillations period is 2. (e) Scheme of the different
types of interferences responsible for the conductance oscillations.
Interferences in a cavity in proximity with a superconductor. In the left
case, an electron is Andreev-reflected at the SC/cavity interface as a hole
propagates back to the other end. There, it is normal-reflected towards the
SC/cavity interface, where it undergoes again Andreev-reflection. This allows
the interference that yield the short period oscillations. In the right case,
the electron is normal-reflected at the SC/cavity, travels back to the other
cavity interface where it is also normal-reflected- which produces the long
period resonances. The predominance of one or the other type of resonances
depends on the ratio between Andreev reflection and normal reflection at the
SC interface and thus, for a given interface transparency, on the energy of
the involved electrons. Adapted from Ref. [13].

This behavior can be explained considering doping profile across the interface
between superconducting and non superconducting graphene, as sketched in the
top part of Fig. 10(a). The Fermi level in the superconducting graphene is
pinned because here gate effect are screened by the YBCO/Au. Contrarily, in
normal graphene (on top of insulating YBCO) the Fermi level varies upon
gating, and consequently the Fermi vector $k_{\text{F}}$ of the carriers
flowing across the interface does too. The authors of Ref. [12] assumed a
step-like profile of the Fermi energy, as shown in Fig. 10(a) due to edge
effects (_e.g._ fringe fields associated to the backgate geometry, gradient of
electronic properties between irradiated and non/irradiated YBCO…). Such a
profile results in a potential energy step $U_{0}$ for electrons between
superconducting and normal graphene. This situation is analogous to that in
graphene $n–p–n$ or $p–n–p$ junctions, in which Klein tunneling is observed
[113, 2, 135, 114]. But now, because of the presence of superconducting
correlations, transport at low-bias involves Andreev pairs. In this scenario,
interference between Andreev particles (or electrons for bias above the
energy-gap) are expected in the central (normal) region [92], whose width $w$
was estimated at $w\sim 60$ nm, as particles undergo reflections and travel
back and forth between its two ends. This leads to a modulation of the
conductance that depends on the phase picked by the charge carriers when
travelling across that central region, which is $\chi=-wk_{\text{F}}$. Thus,
upon gating, the induced change of $k_{\text{F}}$ leads to a periodic
modulation of the conductance, which explain the experiments in Fig. 10(b) and
10(c). All these findings were supported by a numerical calculation of the
conductance using the model of Ref. [92]. A good agreement between experiment
and theory was found [12].

The above experiments demonstrated the superconducting proximity effect across
the $d$-wave/graphene interface and its modulation via gate voltage. However,
from those experiments one could not determine the (lateral) length scale over
which the induced superconducting correlations propagate into graphene. To
investigate this issue, similar devices were fabricated but with a much
shorter distance between the superconducting electrodes [13]. Such device is
shown Fig. 11(a). The distance between the superconducting electrodes is of
the order of a few hundred nm. If this distance is shorter than the coherence
length for Andreev pairs and mean-free path of the electrons, then both
Andreev pairs and normal electrons can coherently propagate and suffer
reflections at both ends of the normal graphene. In such a case the device
behaves as a “cavity” in which reflections at its ends leads to Andreev pair
and electron confinement, and ultimately to interference effects. Such
interferences, illustrated in the schemes of Fig. 11(e) are respectively
analogous to the De Gennes-Saint James [136] or McMillan-Rowell oscillations
[137] for Andreev pairs, and to Fabry-Pérot interferences for electrons [138,
139, 114, 140, 141]. The existence of these interference effects was confirmed
in the experiment of Ref. [13]. They manifest in the conductance measurements
via two distinct families of oscillations as a function of the bias voltage
across the cavity [Fig. 11(b)]. The shorter period oscillations correspond to
Andreev pair interference effects, and are dominant at low bias. The
oscillations stemming from electron interference present a doubling of the
period and are dominant at bias above the superconducting energy gap. To
support this interpretation of the experiments, numerical simulations of the
conductance were performed based on Blonder-Tinkham-Klapwijk model generalized
to $d$-wave superconductors [99, 142] (see Section 2) to deal with
YBCO/Au/graphene interfaces of finite transparency and on the proximitized
graphene homojunction [91, 92] to model the cavity’s conductance in series.
This approach allowed reproducing numerically the main features of the
experimental curves, including not only the oscillations, but also the
background conductance shape, which is determined by the transparency of the
YBCO/Au/graphene interface (described by the barrier strength parameter $Z$ of
the BTK theory). The experiments were carried in devices with different
cavity’s lengths, up to 800 nm, which allowed verifying the consistency of the
analysis as the oscillations periods extracted from the analysis of the
experimental curves scaled with the inverse of the cavity’s length as expected
from the theory, see Fig. 11(d). In summary, these experiments demonstrated
the long-range (hundreds of nm) of the superconducting correlations induce in
graphene by proximity with a $d$-wave superconductor. The model used
considered that the induced correlations retain the original $d$-wave symmetry
of the cuprate, and no indication of a different symmetry could be found from
the analysis.

###  3.4 Experiments on topological insulators and d-wave superconductors

Figure 12:  Examples of hybrid structures between 3D topological insulators
and $d$-wave superconductors. (a) Bi2Se3 is grown on top of BSCCO by molecular
beam epitaxy, (b) the corresponding ARPES spectra shows a Dirac like energy
dispersion, adapted from Ref. [143]. (c) Conversely BSCCO islands have been
grown on top of Bi2Te2Se (d) the heterostructure conductance show
superconductivity related features indicating proximity effect, adapted from
Ref. [144].

The quest to evidence and manipulate Majorana fermions is a very hot topic
[145, 146] motivating most of the study of superconductor/topological
insulator devices — although other prompt applications of these devices
exists, for example for enhanced photoluminescence [147, 148, 149]. Like for
graphene, experiments between topological insulator and $s$-wave
superconductor have been focused on the transport properties of Josephson
junctions [48]. These include HgTe/CdTe in contact with Al leads which showed
missing Shapiro steps interpreted as induced superconductivity in the quantum
spin Hall edges [150]. Both 2D and 3D topological insulators were studied in
this work, and the differences between both cases was discussed extensively in
Ref. [151]. The three-dimensional topological insulator Bi2Se3 has been
contacted with Al leads [152] showing gate tunable supercurrent as well as Nb
superconducting leads showing multiple Andreev reflections [153]. More
recently, Josephson junction with Al superconducting electrodes and a Bi2Se3
spacer showed a critical current minimum at zero magnetic field, possibly
indicating a $p$-wave induced superconductivity [154, 155, 156]. In a recent
paper on arxiv [157], the group of David Goldhaber-Gordon reported a new
technique yielding transparent interface between the 3D topological insulator
(Bi0.4Sb0.6)2Te3 and the superconducting alloy PdTe, as the authors note,
their results are compatible with Majorana fermions but are not an evidence
for their existence. Dirac semimetals have also been used as spacers in
$s$-wave Josephson junctions, Cd3As2 [158, 159], Bi1-xSbx [160, 161], with
evidence for topological superconductivity such as $4\pi$ periodic
supercurrent or a Zeeman field transition between a normal and a $\pi$
Josephson junction.

he work on $d$-wave superconductors combined with topological insulators is so
far scarce. Theoretically, it has been proposed that the elementary
excitations at the interface between a $d$-wave superconductor and a
topological insulator are Majorana fermions [122]. The few experiments
combining topological insulators with $d$-wave superconductors are summarized
in Fig. 12. The two techniques exploited are point contact spectroscopy
measurements [162] and Angle-Resolved Photoemission Spectroscopy (ARPES)
[143]. These two approaches led to contradicting results [163, 143, 144]
concerning the possibility to induce $d$-wave superconductivity in topological
insulators. Using topological insulators grown on a $d$-wave superconductor
and ARPES the groups of T. Valla and Z. Hasan [163, 143] concluded that the
proximity effect is prohibited in these structures due to Fermi surface
mismatch. Burch’s group, based on exfoliated d-wave/TI devices, transport
experiments [164] and theory [14] showed the opposite. To our knowledge, no
other topological insulator in proximity with $d$-wave superconductors has
been studied so fat, which might be due to the growth conditions making it
difficult to combine both class of materials. Among the different topological
insulators WTe2 [165] seems a very suitable candidate to be brought into
contact with a $d$-wave superconductor, because it can be exfoliated and
transferred onto different materials [166]. Furthermore, monolayer WTe2 has
recently been shown to display superconductivity [167] and the pairing
symmetry is so far unknown. One can only speculate that in the case of a
$d$-wave pairing symmetry, one would be able to realize WTe2 (SC)/WTe2
junction by local gating [168]. On a final note, it is worth pointing out that
in, general, the smoking gun signatures of superconductivity mediated by
topological edge states channels did not reach a consensus yet [169], and that
these difficulties will of course apply to the case of unconventional
superconductors.

##  4 Summary and Outlook

The combination of $d$-wave cuprate superconductors with Dirac materials bears
great fundamental interest and potential for technological applications. As we
discussed in the theory section, the isotropic pairing results in a large
density of zero-energy superconducting quasiparticles and shows a non-uniform
superconducting phase. This results in directional effects, dictated by the
orientation between the cuprate’s and the Dirac material’s crystalline axes
within the $x$-$y$ plane. Unique features are expected, such as the generation
of unconventional pairing symmetries, the large dominance of low-energy
Andreev bound states in the conductance across the cuprate/Dirac material
interface, or the tantalizing possibility of realizing $\pi$ Josephson
junctions with present gate tunability, potentially at relatively high
temperatures of the order of tens of K. So far, very few experimental works
have been conducted, to a great extent because the realisation of junctions
between cuprates and Dirac materials which are sufficiently transparent to
allow for the proximity effect is far more challenging than with conventional,
low-$T_{\text{C}}$ superconductors. Despite this, graphene as a prototypical
Dirac material has been successfully combined with different cuprates.
Spectroscopic measurements of graphene on PCCO using scanning tunneling
microscopy found signatures of proximity-induced unconventional $p$-wave
pairing in graphene [11]. The development of ad hoc nanolithography approaches
enabled the fabrication of YBCO/graphene devices with transparent interfaces
[12] in which the tunneling of Andreev pairs can be modulated by an
electrostatic gate, analogously as in the Klein tunneling of normal electrons.
The same type of devices have allowed demonstrating that $d$-wave
superconducting correlation can coherently propagate into graphene over
distances of the order of hundreds of nm [13]. These studies are promising
steps towards the realization of $d$-wave Josephson junction with a Dirac
material spacer [92], which in practical terms would require further enhancing
and reliably controlling the interfaces’ transparency. Further prospects
include the demonstration of high-temperature Josephson effects across
topological insulators and other materials from the ever-growing library of
exfoliated 2D materials [170, 171]. As described above, a few promising
results already exist based on experiments with bilayer heterostructures that
combine cuprates with topological insulators [163, 143, 144], and novel
effects have been found in junctions based on the exfoliated cuprate bismuth
strontium calcium copper oxide (BSCCO) [172, 173], which opens worthwhile
avenue towards BSCCO–graphene and BSCCO-3DTI devices.

Acknowledgements

Work at Unité Mixte de Physique CNRS/Thales supported by ERC grant N. 647100
”SUSPINTRONICS”, French ANR grant ANR-17-CE30-0018-04 ”OPTOFLUXONICS”, COST
action ”Nanocohybri”, European Union’s H2020 Excellent Science Grant N. 881603
”GRAPHENE CORE 3”, Labex NanoSaclay ANR-10-LABX-0035 and JSPS core-to-core
program ”A. Advanced Research Networks”. The work of S.B. and D.B. is support
from Ministerio de Ciencia e Innovación (MICINN) through Project N.
PID2020-114252GB-I00 (SPIRIT) and PID2020-120614GB-I00 (ENACT), and by the
Transnational Common Laboratory $Quantum-ChemPhys$. Further, D.B. and S.B.
ackowledge the funding from the Basque Government’s IKUR initiative on Quantum
technologies (Department of Education). F.S.B thanks Prof. Björn Trauzettel
for his hospitality at Würzburg University, the A. v. Humboldt Foundation for
financial support, and EU’s the Horizon 2020 Research and Innovation Program
fro funding under Grant Agreement No. 800923 (SUPERTED).

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David Perconte is a postdoctoral Researcher at the institut Néel lab in
Grenoble. He received his PhD from the University Pierre et Marie Curie in
2018 working on graphene d-wave superconductor junction. He later did a
postdoctoral stay in the Autonoma university of Madrid working in the group of
Isabel Guillamon and Hermann Suderow. Dario Bercioux is an Ikerbasque Research
Associate at the Donostia International Physics Center (DIPC) in Donostia-San
Sebastián. He received is PhD from the University of Naples in 2005. Afterword
he was a postdoctoral fellow in Regensburg, Freiburg and at the FU Berlin. He
joined the DIPC in the 2014 as Ikerbasque Research Fellow. Bruno Dlubak is a
CNRS Researcher at the Unité Mixte de Physique CNRS-Thales lab, associated to
the Paris-Saclay University. He received his PhD from the University of Paris-
Sud in 2011 working on graphene spintronics. After a postdoctoral appointment
at the University of Cambridge working in John Robertson’s group, he became a
CNRS Researcher in 2014. F. Sebastian Bergeret is a Researcher at the
Materials Physics Center (CSIC) and head of the Mesoscopic Physics Group where
he leads research on quantum transport, superconductivity and magnetism. He
earned his PhD at the Bochum University (Germany) in 2002. Javier E. Villegas
is a Research Director at the CNRS. He leads research on superconducting
hybrids and oxides the CNRS/Thales joint lab (France). He earned his PhD at
the University Complutense of Madrid (Spain) in 2004, where he also worked as
adjunct professor until 2005. Following a postdoctoral research period in the
group of I.K. Schuller at the University of California-San Diego (USA), he
became CNRS permanent staff in 2007.