# Superconductivity and correlated phases in non-twisted bilayer and trilayer
graphene

Pierre A. Pantaleón<EMAIL_ADDRESS>$Imdea\ Nanoscience,\ Faraday\ 9,\
28049\ Madrid,\ Spain$ Alejandro Jimeno-Pozo  $Imdea\ Nanoscience,\ Faraday\
9,\ 28049\ Madrid,\ Spain$ Héctor Sainz-Cruz  $Imdea\ Nanoscience,\ Faraday\
9,\ 28049\ Madrid,\ Spain$ Võ Tiến Phong Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia PA 19104, USA  Tommaso Cea
$Department\ of\ Physical\ and\ Chemical\ Sciences,\ University\ of\
L^{\prime}Aquila,\ via\ Vetoio,\ Coppito,\ 67100\ L^{\prime}Aquila,\ Italy$
Francisco Guinea<EMAIL_ADDRESS>$Imdea\ Nanoscience,\ Faraday\ 9,\
28049\ Madrid,\ Spain$ $Donostia\ International\ Physics\ Center,\ Paseo\
Manuel\ de\ Lardizabal\ 4,\ 20018\ San\ Sebastian,\ Spain$

(August 28, 2024)

###### Abstract

The discovery of a very rich phase diagram in twisted bilayer graphene [1; 2]
renewed the interest into the properties of other systems based on graphene.
An unexpected finding has been the observation of superconductivity in non-
twisted graphene bilayers and trilayers [3; 4; 5]. In this perspective, we
give an overview of the search for uncommon phases in non-twisted graphene
systems. We first describe results related to the topic before the
aforementioned experiments [3; 4; 5] were published. Then, we address the new
experimental findings which have triggered the recent interest in the problem.
Lastly, we analyze the already numerous theory works studying the underlying
physical processes [6].

##  I Unusual phases in non-twisted graphene stacks. Early results.

Graphite is considered a semimetal, with a highly complex Fermi surface, and a
low density of states at the Fermi level. To a first approximation, graphite
can be seen as a stack of two dimensional graphene layers. The low energy
electronic properties of each graphene layer are well described by the two
dimensional Dirac equation [7; 8]. This model shows that the density of states
at the Fermi level vanishes, suggesting that electron interactions do not play
a major role.

The isolation, and posterior detailed study [9; 10], of individual graphene
layers lead to a search for effects of electronic interactions in single layer
graphene, and also in bilayer graphene and in graphene multilayers. One of the
most striking effects of the electronic interactions in single layer graphene
is the renormalization of the Fermi velocity which characterizes the Dirac
cones [11; 12], similarly to the renormalization processes in Quantum
Electrodynamics. The degeneracy between Landau levels of monolayer graphene is
also broken by interactions [13; 14].

The generalization of the Dirac equation to graphene (Bernal) bilayers [15;
16] replaces the two touching Dirac points by two touching parabolic bands.
This structure leads to a logarithmically divergent susceptibility [17]. In
the renormalization group language, this result implies that the screened
Coulomb interaction is a marginally relevant perturbation, which leads to a
variety of instabilities [18; 19; 20; 21], including superconductivity.
Superconductivity can also arise at high electron dopings, when the Fermi
energy is close to a van Hove instability [22; 23]. Other graphene multilayers
show interesting electronic bands, see, for instance [24]. In particular, an
infinite rhombohedral stack is a Dirac semimetal [25], which implies that flat
two-dimensional bands exist at the top and bottom surfaces, which can lead to
correlated states [26]. Localized states at surfaces are likely in finite
rhombohedral stacks [27; 28]. Finally, infinite graphene stacks can show
stacking disorder (defected graphite), and, at the position of the defects,
quasi-two-dimensional behavior may emerge [29; 30; 31; 32; 33].
Superconductivity in rhombohedral graphite has been predicted theoretically
[34; 35; 36].

On the experimental side, superconductivity was observed in the so called
graphite intercalation compounds, made up of metallic and graphene layers [37;
38; 39], with critical temperatures up to $\sim 10-12$K. Superconductivity at
higher temperatures, $\sim 30$K has been found in fullerenes, $C_{60}$,
crystals doped with alkaline ions [40; 41; 42; 43; 44]. In both cases, the
amount of charge transferred to the carbon systems is considerable, $\sim
0.1-1$ $e$/atom, much larger than typical electrostatic dopings achieved in
few layer graphene stacks. The electronic bands of graphite intercalation
compounds and alkali doped fullerenes are quite different from the bands of
graphene and graphite, and there is no clear evidence for unconventional
superconductivity, although the effect of electron-electron interactions has
been invoked in the doped fullerenes [45; 46; 47; 48; 49; 50; 51].

In addition, some early results suggested the existence of correlated phases
in Bernal bilayers [52; 53; 54; 55; 56; 57] and also in $ABC$ trilayers [58;
59; 60]. Some observations of electronic gaps in transport measurements can be
due to domain walls in the bilayers [61]. Unusual phases have also been
reported in other graphene trilayers [62; 63; 58], and also in thicker
multilayers [64; 65; 66].

Finally, it is worth remarking that a number of works report phenomena
compatible with superconductivity in bulk graphite [67; 68; 69; 70; 71; 72;
73; 74]. It is unclear the role that two dimensional electron gases trapped at
stacking defects can play in these observations [29; 30; 31; 70; 73; 32; 33].

##  II Superconductivity in Bernal bilayer and rhombohedral trilayer graphene

###  II.1 Bilayer graphene

Superconductivity in Bernal bilayer graphene was discovered by Zhou $et.\ al.$
[3]. This material is the most common allotrope of graphene because its
stacking is the structural ground state of two graphene layers [75; 76]. The
experiments in Ref. [3] consist of inverse compressibility and magneto-
resistance measurements on bilayer graphene encapsulated in non-aligned hBN
and graphite gates. Upon applying a displacement field perpendicular to the
layers, a gap opens at charge neutrality, while the bands develop a ‘Mexican
hat’ dispersion, leading to van Hove singularities that promote strong
correlations. In this setup, Ref. [3] shows a map of the inverse
compressibility $\kappa=\partial\mu/\partial n_{e}$, where $\mu$ is the
chemical potential, as a function of hole doping $n_{e}$ and displacement
field $D$. The map reveals a series of dips in $\kappa$, which delineate
different regions in the phase space defined by $n_{e}$ and $D$, see Fig.
1(a). Furthermore, measuring the magneto-resistance yields the frequency of
quantum oscillations normalized to $n_{e}$, $f_{\nu}$, which is the fraction
of the Fermi surface enclosed by a cyclotron orbit. The results show that
$f_{\nu}$, and hence the number of pockets in the Fermi surface, is different
in each region. Together, these phenomena indicate a cascade of transitions
between flavour-symmetry-breaking metallic phases, which resemble those
observe in twisted graphene stacks [77; 78; 79] and RTG [80], as well as
Lifshitz transitions which change the topology of the Fermi surface.
Subsequent experiments have studied in great detail the correlated phases of
bilayer graphene [81; 82]. In particular, on top of the flavour polarized
metals, Ref. [82] reports the observation of several new phases, such as
correlated metals with linear-in-temperature resistivity, a Wigner crystal,
and a novel Wigner-Hall crystal, an electron crystal which has a quantized
Hall conductance.

Figure 1: Superconductivity in bilayer graphene. (a) Inverse compressibility
without magnetic field, as a function of electron doping $n_{e}$ and
displacement field $D$. (b) Longitudinal resistivity ($R_{xx}$) at $T=10$ mK
and $B_{\parallel}=165$ mT, showing a superconducting sleeve in bright cyan.
(c) Fast Fourier transform of the magneto-resistance versus frequency
normalized to $n_{e}$, measured at circled points in (a). The insets shows the
inferred Fermi surfaces for each phase. (d) $R_{xx}$ versus $B_{\parallel}$
and $n_{e}$ at $D=1.02$ V/nm. Figures extracted from Ref. [3].

Upon applying an in-plane magnetic field, a superconducting sleeve appears
near the boundary between two of these phases, as shown in Fig. 1(b), for
dopings near where a van Hove singularity is expected. One of these phases,
dubbed ‘Sym12’, appears to have twelve Fermi pockets (four flavours with three
pockets each due to trigonal warping effects, see Fig. 5(f)) and the other,
‘PIP2’, a symmetry-broken phase with two large and two small pockets, one per
flavour, which may be nematic. Superconductivity manifests as vanishing
resistivity, and a Fraunhofer pattern in the presence of a magnetic field,
conclusive evidence of phase coherence. The critical temperature, deduced from
non-linear transport and Berezinskii-Kosterlitz-Thouless theory, is
$T_{c}\approx 26$ mK. At this temperature, a magnetic field of $40$ mT would
break the pairs of a conventional spin-singlet superconductor, since the
Zeeman effect would overcome the superconducting gap. Superconductivity in
bilayer graphene survives at fields over ten times larger than this Pauli
limit, of $B\gtrsim 500$ mT, which is strong evidence of spin-triplet pairing.

The emergence of superconductivity is better understood by comparing transport
experiments at zero and finite magnetic field, throughout the region of phase-
space that becomes superconducting. At zero magnetic field, the sleeve between
PIP2 and Sym12 is a state distinct from both, with higher resistance, and,
most likely, insulating, see Fig. 1(d). Its resistance has non-linear
behaviour and drops abruptly at a threshold in the applied current. Turning on
and increasing a magnetic field (in- or out-of-plane) reduces this threshold
until it vanishes, giving rise to superconductivity. This suggest that the
resistive state is spin-unpolarized and that the superconducting state emerges
from it when the magnetic field polarizes the spin.

A relevant measure of the level of disorder in the material is the ratio
between the superconducting coherence length, estimated to be $\xi\approx 250$
nm and the electronic mean free path, of $l_{M}\approx 5$ $\mu$m, comparable
to the dimensions of the device. The ratio is $d={\xi}/{l_{M}}<0.05$, which
implies an ultra-clean regime of superconductivity, in which pair-breaking
effects due to non-magnetic impurities are weak, and which is therefore
compatible with the unconventional spin-triplet pairing inferred from the rest
of the data. A similar ratio is estimated in the experiments described next.

Figure 2: Superconductivity in bilayer with WSe2. (a) $R_{xx}$ versus hole
doping and displacement field, at zero magnetic field, showing a
superconducting sleeve in dark blue. (b) Fast Fourier transform of the
magneto-resistance versus $n_{e}$. (c) Cut of the resistivity at $D=1$ V/nm,
insets show the Fermi surfaces inferred from (b). (d) $R_{xx}$ as a function
of doping and temperature for $D=0.9$ V/nm. (e) $R_{xx}$ versus doping and in
plane magnetic field, at $D=1.1$ V/nm. Red circles indicate the zero-field
$T_{c}$. Figures extracted from Ref. [5].

The second set of ground-breaking experiments on bilayer graphene have been
done by Zhang $et.\ al.$ [5]. The main idea was to stabilize superconductivity
by placing tungsten diselenide (WSe2) on top of the bilayer, a strategy that
has already proved fruitful in twisted bilayer graphene, yielding
superconductivity at record low twist angles [83]. For bilayer graphene, Ref.
[5] reveals that the interplay with WSe2 leads to a significant, global
enhancement of superconductivity: it appears even in the absence of a magnetic
field, its critical temperature increases by an order of magnitude, to
$T_{c}\approx$ 260 mK and the filling range of the superconducting sleeve is
similarly extended, compare Fig. 2(a) to Fig. 1(a). There is compelling
evidence that these effects are due to Ising spin-orbit coupling (SOC) induced
by WSe2. Note that in the absence of WSe2 the SOC in bilayer graphene is
negligible [84].

A key aspect of Fig. 2(a) is the asymmetry with respect to the $D$ field
direction. In particular, superconductivity appears just at positive $D$.
Theory predicts that Ising SOC only couples to the top layer, which is next to
WSe2 [85; 86]. For hole doping, a positive $D$ field pushes the wavefunction
towards WSe2, letting the carriers benefit from Ising SOC. For similar values
of $n_{e}$, the phase at negative $D$ fields is a resistive state that
disappears upon applying magnetic fields, which is most likely the same phase
from which superconductivity emerges in Ref. [3]. It appears that this is also
the phase that here, at positive $D$ field, competes with superconductivity,
cutting through the superconducting region and splitting it in two, see Fig.
2(d). A resistive state, flanked by superconducting domes, is reminiscent of
the situation observed in e.g. twisted bilayer graphene [2]. These
observations support the idea that Ising SOC contributes to superconductivity
in the system.

Quantum oscillations confirm that the phase diagram observed in [5] at
negative $D$ field is almost identical to the one observed in bilayer graphene
in [3], which shows a transition between the PIP2 and Sym12 phases. In
contrast, for positive $D$ field, the data suggests that, due to band
splitting induced by Ising SOC, the phase adjacent to PIP2 has six large and
six small Fermi pockets, so it is similar to PIP2 but in the trigonal warping
regime, see Fig. 2(b) and (c). Another difference is that superconductivity
emerges from PIP2 after Ising SOC presumably selects a symmetry-broken state
that enables pairing, while in bilayer graphene encapsulated in hBN the parent
state seems to be another one with higher resistance, see Fig. 1(d).

The critical magnetic field differs starkly on both sides of the
superconducting dome, as shown in Fig. 2(e). The low $n_{e}$ side violates the
Pauli limit by a factor of six, while the high $n_{e}$ barely does, perhaps
hinting at a nuanced evolution of the Fermi surface with doping within the
dome. A possibility is that the system develops Ising superconductivity, which
promotes Cooper pairs of type $\ket{K^{+},\uparrow;K^{-},\downarrow}$, locking
the spin and valley degrees of freedom and making the state very resilient to
in-plane magnetic fields. However, as pointed in Ref. [5], the moderate
critical fields observed may be more consistent with an Ising SOC-induced
inter-valley coherent phase.

###  II.2 Trilayer graphene

Figure 3: Superconductivity in RTG. (a) Inverse compressibility as a function
of $n_{e}$ and $D$. (b) $R_{xx}$ versus $n_{e}$ and $D$, showing two
superconducting sleeves SC1 and SC2 in bright cyan. (c) Fast Fourier transform
of the magneto-resistance as a function of $n_{e}$ near SC1 and frequency
$f_{\nu}$. (d) $R_{xx}$ versus $n_{e}$ near SC1, at zero magnetic field. (e)
Magneto-resistance versus $n_{e}$ near SC2 (top) and $R_{xx}$ versus $n_{e}$
near SC2, at zero magnetic field (bottom). (f) Fast Fourier transform of (e),
along the coloured arrows. (g) $R_{xx}$ as a function of $n_{e}$ and
temperature, for SC1. (h) $R_{xx}$ versus $n_{e}$ and in-plane magnetic field,
at the partially flavour polarized phase adjacent to SC1. Figures extracted
from Ref. [80] and Ref. [4].

The experiments by Zhou et. al. [80; 4] in rhombohedral trilayer graphene
(RTG) reported unambiguous observations of correlated behaviour [80] and
superconductivity [4] in a crystalline graphene system. Rhombohedral $ABC$
stacking is a metastable three layer allotrope of graphene, with Bernal $ABA$
being the ground state, although recent results suggest $ABC$ may be stable
[87]. RTG is similar to BBG in that a perpendicular displacement field opens a
gap and flattens the low-energy bands, setting the stage for strong
correlations, although it is worth noting that RTG has flat bands even in the
absence of a $D$ field. In this material, inverse compressibility and magneto-
resistance measurements also uncover a cascade of flavour-symmetry breaking
transitions, see Fig. 3(a), including a spin-polarized half-metal phase and a
spin- and valley-polarized quarter-metal. Ref. [80] uses a Stoner model,
similar to the one in Ref. [77], to explain the cascade at positive filling.
At negative fillings, shown in Fig. 3(a), the situation is more nuanced, due
to the interplay between flavour symmetry breaking and Lifshitz transitions,
which gives rise to several partially flavour polarized phases.
Superconductivity has been found in two regions, SC1 and SC2, of the phase
space defined by $n_{e}$ and $D$, as shown in Fig. 3(b). On top of vanishing
resistivity, non-linear transport in the presence of a magnetic field yields
Fraunhofer patterns, definitive evidence of superconductivity. The estimated
critical temperatures are $T_{c1}=106$ mK for SC1 and $T_{c2}=50$ mK for SC2.
Like in BBG, the superconducting sleeves in RTG appear close to flavour-
symmetry-breaking transitions.

Quantum oscillations data near the SC1 superconducting sleeve, shown in Fig.
3(c), reveals that it arises from a symmetric state, Sym4, in which each
flavour has an annular Fermi surface, and which is adjacent to a phase with a
complex Fermi surface, spin-unpolarized and partially valley-polarized,
probably with large and small Fermi pockets, like PIP2 in BBG. On the other
hand, SC2 emerges at the boundary between a spin-polarized, valley-unpolarized
half-metal and a partially-flavour polarized state, which also appears to have
large and small Fermi pockets, see Fig. 3(f). Further differentiating these
two superconducting regions, SC1 shows only a small Pauli limit violation, and
its $T_{c}$ dependence on $B$ is compatible with conventional spin-singlet
pairing. Still, it is worth noting that the critical magnetic field is very
asymmetric with respect to $n_{e}$, like in BBG-$\text{WSe}_{2}$, compare Fig.
3(g) and Fig. 2(e). In stark contrast, SC2 violates the Pauli limit by an
order of magnitude. Moreover, spin-singlet pairing for SC2 is precluded by the
fact that its parent state is spin-polarized, so the two spin orientations are
separated by the exchange interaction energy ($\sim 1-10$ meV), over two
orders of magnitude greater than the superconducting gap. Therefore, there is
strong evidence that SC2 is a spin-triplet superconductor. Just like in BBG,
an estimate of the level of disorder gives $d\sim 0.05$, so the system is
clean enough to support unconventional spin-triplet pairings.

Figure 4: Logarithmic plot of critical temperature $T_{c}$ versus Fermi
temperature $T_{F}$ for various superconductors. Black shapes correspond to
non twisted graphene systems. Adapted from Ref. [2]

###  II.3 Comparison to twisted stacks

It is illustrative to compare the superconductivity and correlated phases
observed so far in twisted and non-twisted stacks. Some of the similar
features shared by both types of stacks are that (i) superconductivity occurs
at very low carrier densities. The density in twisted bilayer graphene is
determined by the number of electrons which can be accommodated into the
central bands, while in non twisted systems the density is fixed by the states
available in the flat regions of the low energy bands. In both cases these
estimates give $n_{e}\sim 10^{12}\text{ cm}^{-2}$. Knowledge of $n_{e}$ allows
for the classification of interactions according to their strength, and
reveals that the long range Coulomb interaction, $V_{C}\approx
e^{2}k_{F}\propto e^{2}\sqrt{n}_{e}$, is larger than the bandwidth and it is
parametrically larger than other interactions [88], which scale as $n_{e}$.
Also, superconductivity: (ii) violates the Pauli limit, indicating that the
pairing is, at least in part, spin-triplet; (iii) is adjacent to correlated
insulators [2; 89; 90; 91; 92; 93; 3; 5] and (iv) is modified when the
graphene stack is in proximity of a WSe2 monolayer [83; 5]. In addition, there
are (v) cascades of flavour-symmetry breaking transitions [77; 78; 80; 4; 81;
82; 3; 5]; (vi) Quantum Anomalous Hall phases [94; 80; 82] and (vii) strange
metals with linear-in-temperature resistivity [95; 82].

The most notable differences include: (i) $T_{c}\sim 1$ K in twisted stacks,
and $T_{c}\sim 10-100$ mK in non-twisted stacks. Since the densities are
similar, this means that twisted stacks are in a regime of stronger coupling,
see Fig. 4. (ii) Superconductivity in non-twisted stacks appears only when
applying a displacement field. (iii) The relevant bands are isolated from the
rest of the spectrum only in twisted stacks with an even number of layers.
(iv) Angle disorder, strain and lattice relaxation have a serious impact on
twisted stacks. (v) In proximity of a WSe2 monolayer, $T_{c}$ is enhanced in
BBG, while in twisted bilayer graphene superconductivity is stabilized at a
lower angle, but $T_{c}$ does not increase. (vi) Some phenomena have only been
observed in twisted stacks, e.g. Chern insulators [96; 97; 98; 99; 100; 101;
102], nematicity [103], Pomeranchuk effect [104; 105] or re-entrant
superconductivity with a magnetic field [91], while others like the Wigner and
Wigner-Hall crystals have only been seen in BBG [82].

##  III Theoretical Models

###  III.1 Non-interacting band structures

Figure 5: Band structure in the full Brillouin zone for: a) Bernal bilayer
graphene and d) rhombohedral trilayer graphene. The large hexagon near zero
energy is the corresponding Brillouin zone. The small hexagon is the Dirac
point where the low energies of the corresponding system are calculated for b)
and e) zero and c) and f) $20$ meV perpendicular displacement field. At the
side of the low energy bands is the corresponding density of states. The model
and parameters for the figures are given in Ref. [16]

Both RTG and BBG become superconductors at very low electronic densities
$n_{e}\sim 10^{12}\text{ cm}^{-2}$ when an electric field is applied
perpendicular to the graphene system. From the theoretical point of view is
universally accepted that the band flattening resulting from the displacement
field and the consequent van Hove singularities enhance the role of
interactions. The study of the electronic properties is usually performed with
low energy continuum models with the exception of Ref. [106] which studied the
superconductivity using a tight binding model. The non-interacting tight
binding band structure [16], low energy regions and density of states of RTG
and BBG is shown in Fig. 5. In BBG there are two vertically aligned graphene
layers so that atoms belonging to the sublattice $A$ of layer $1$ lie over the
atoms in the sublattice $B$ of layer 2. Similarly to monolayer graphene, BBG
is a semi-metal in which the low energy bands touch at the Dirac points, but
with parabolic instead of linear dispersion [107; 15]. This band touching
makes the charge susceptibility and other susceptibilities diverge [17], which
leads to symmetry-broken phases [18; 19; 20; 21; 108; 109; 110]. As a
perpendicular electric field is applied, inversion symmetry is broken, and the
Dirac points connecting the bands become gapped. As a result, the local band
dispersion can be nearly quenched. As shown in Fig. 5(b) and (c), the low
energy bilayer bands acquire a ‘Mexican hat’ profile [111].

Rhombohedral trilayer graphene is a semi-metal with an approximate cubic band
degeneracy at the zone corners [7; 37; 30; 27; 28; 24; 112; 34; 35; 56; 26]
where optimal parameters have been reported in the literature [24; 27; 63;
113]. Close to the Dirac points, the cubic degeneracy resulting from
diagonalizing the Hamiltonian splits into three Dirac cones, creating a
trigonally-warped Fermi surface, shown in Fig. 5(e). A perpendicular electric
field breaks inversion symmetry, and these Dirac points acquire a finite mass,
Fig. 5(f). As a result, the local band dispersion is flattened, generating a
van Hove singularity that favors the emergence of correlated electronic
phases.

[ caption = Overview of theories of superconductivity in bilayer and trilayer
graphene systems, according to their proposed pairing mechanism, estimated
critical temperature $T_{c}$ and symmetry of the superconducting order
parameter (OP)., label = tab:mergedTable ]—c—p5cm—p2.5cm—p4cm—p2.5cm— System
Pairing mechanism $T_{c}$ (mK) OP symmetry OP sign change

BBG Phonons [114; 115] 1000 - 2000 (no $e$-$e$ interactions), 20-500 (with
$e$-$e$ interactions) Spin-triplet [116], valley singlet ($f-$wave), intra-
valley: singly degenerate, $A$. Most likely, no

Short range, momentum independent interactions, proximity to broken symmetry
phases [117] —Spin-triplet ($f-$wave). —

Quantum critical modes [118] 35 Spin-triplet, valley-singlet $s-$wave or spin-
triplet, valley-triplet ($f-$wave). —

Screened long-range Coulomb [119] 10 Spin-triplet, valley-singlet ($f-$ wave),
intra-valley: singly degenerate, $A$, or doubly degenerate, $E$. Yes

Short-range Coulomb [120] 20 Spin-triplet, valley-singlet ($f-$wave). Most
likely, no

BBG-$\text{WSe}_{2}$ Phonons + substrate tunneling [121] — Mixture of spin-
singlet, valley-triplet ($s-$wave) and spin-triplet, valley-singlet ($f-$wave)
—

Screened long-range Coulomb [119] 12-40 Ising superconductor, intra-valley:
singly, $A$, or doubly degenerate, $E$. Yes

RTG Phonons [122] 500 - 1600 Spin-singlet, valley-triplet (unpolarized normal
state), spin-triplet, valley-singlet (polarized normal state) Most likely, no

Short range, momentum independent interactions, proximity to broken symmetry
phases [123] —Spin-triplet ($f-$wave). —

Flavour fluctuations [124] — Spin-singlet, valley-triplet, singly degenerate,
$A$ Most likely, no

Flavour fluctuations [125] — Dependent on details of interactions Dependent on
details of interactions

Intervalley fluctuations [126] — Spin-singlet, valley-triplet —

Screened long-range Coulomb [127; 119] 65-140 Spin-triplet, valley-singlet
($f-$ wave), intra-valley: singly degenerate, $A$, or doubly degenerate, $E$.
Yes

Screened long-range Coulomb [128] 0 - 2000, depending on dielectric constant
Two pockets in each valley, spin singlet, $d-$ wave (doubly degenerate), or
spin triplet, $p-$wave (doubly degenerate), depending on inter-pocket
couplings Most likely, yes

Screened long-range Coulomb [129] Depends on coupling constant, $\lambda$.
$T_{c}$ covers a broad range. Depends on details of intervalley coupling. Yes

Few momentum independent interactions [130] — Spin triplet, valley-singlet or
spin-singlet, valley-triplet, depending on sign of inter-valley Hund coupling
—

Short-range Coulomb [131] — Spin-singlet, $d-$wave (at half filling only) —

Short-range Coulomb [120] 20-140 Spin-triplet, valley-singlet ($f-$wave). Most
likely, no

Short-range Coulomb [132] — d + $\mathrm{i}$d. —

RTG-$\text{WSe}_{2}$ Screened long-range Coulomb [119] 65-70 Ising
superconductor, intra-valley: singly degenerate,$A$, or doubly degenerate,
$E$. Yes

###  III.2 Superconducting mechanisms

It is worth first to describe some general features of the superconducting
state. The symmetries of the hamiltonian constrain the possible
superconducting order parameters. The existence of two valleys implies that
the order parameter can be even or odd with respect to valley exchange. If we
neglect spin-orbit coupling, the Pauli principle implies that a valley
symmetric order parameter is a spin-singlet, while a valley antisymmetric
order parameter is a spin-triplet. The lowest symmetry that the bilayer and
trilayer in a perpendicular field has around the $K$ and $K^{\prime}$ points
is a ${\cal C}_{3}$ rotation. This symmetry allows for two representations, a
non degenerate one, $A$, and a doubly degenerate one, $E$. Spin-orbit coupling
mixes valley and spin exchange. The simplest case where the spin-orbit
coupling only affects the out of plane spin direction, $s_{z}$, leads to Ising
superconductivity, where the spin and valley degrees of freedom are locked
[133]. Finally, depending on the sign and relative strength of intra- and
inter-valley interactions, the order parameter can show changes of sign and
nodes at the Fermi surfaces, an example is shown in Fig. 6.

Using this analysis of the superconducting state, a classification of
different theoretical models is given in Table LABEL:tab:mergedTable for both
Bernal bilayer and $ABC$ trilayer. Three general pairing mechanisms have been
considered: i) electron-phonon coupling [114; 115; 121; 122], ii) electronic
fluctuations associated to the proximity of broken symmetry phases [117; 118;
123; 124; 125; 126], and iii) direct repulsive electron-electron interactions,
either the long range Coulomb potential [119; 127; 128; 130; 129] or short
range interactions [120; 131; 132]. These pairing mechanisms have been studied
using a variety of formalisms: i) standard BCS calculations [114; 115; 121;
122], ii) diagrammatic methods inspired by the Kohn-Luttinger [134] formalism
[126; 127; 119; 120], the renormalization group [128; 130], phenomenological
electronic susceptibilities describing nearby broken symmetry phases [117;
118; 124; 126], or numerical Monte Carlo analyses [131]. Different
approximations require the knowledge of different number of parameters.

The changes of sign of the order parameter at the Fermi surfaces determine the
robustness of the superconducting phase against elastic scattering. When the
order parameters has different signs in the two valleys, short range
scattering is pair breaking. If the order parameter changes sign within the
pockets in each valley, long range scattering is also pair breaking. It is
worth mentioning that long range repulsive interactions can lead to a
superconducting state because the screened interaction is less repulsive at
$\boldsymbol{q}=0$ than at finite values of $\boldsymbol{q}$, so that an order
parameter with sign changes is allowed [135].

Finally, we compare the results of the theoretical proposals collected in
Table LABEL:tab:mergedTable with the experimental findings in Refs. [3; 5; 4].
In BBG, which has a measured critical temperature of $T_{c}\approx 26$ mK, the
theories in which pairing is due to purely electronic interactions [117; 118;
119; 120] yield critical temperatures and superconducting dome widths of the
same order of magnitude as the experiments, while in the theory where pairing
is mediated by phonons [114; 115] the predicted critical temperatures and the
width of the superconducting dome are larger than in experiments. Identical
conclusions apply to RTG. When a monolayer of WSe2 is placed on top of BBG,
the critical temperature goes up by an order of magnitude, reaching
$T_{c}\approx 260$ mK. In this setup, Ref. [121] shows that virtual tunnelling
to the substrate can favour the attraction between two holes and boost the
phonon-driven superconducting $T_{c}$, although an estimate of the enhancement
is not given. Ref. [119] proposes a Kohn-Luttinger-like mechanism for
superconductivity and shows that, due to reduced screening of the Coulomb
interaction at intermediate momenta, Ising spin-orbit coupling enhances the
superconducing $T_{c}$ by a factor of 4 in BBG. The associated order parameter
shows locking between the spin and valley degrees of freedom.

Most theories either predict, or are compatible with spin-triplet pairing, in
agreement with the Pauli limit violations seen in experiments. Furthermore,
some proposals based on the screened long-range Coulomb interaction [127; 119]
predict the form of the superconducting order parameter, which shows
intravalley nodes and sign changes, see Fig. 6. As mentioned above, these sign
changes make superconductivity fragile against long-range disorder, and may
explain why it has eluded discovery until now.

Figure 6: (a) Superconducting order parameter (OP) of BBG in valley $K^{+}$,
(b) in valley $K^{-}$, near the hole-doped van Hove singularity, with
$T_{c}\approx 10$ mK. Within a single valley, the OP peaks along the edges of
the Fermi surface and changes sign between the inner and outer edges. (c) OP
of RTG in valley $K^{+}$, (d) in valley $K^{-}$, near the hole-doped van Hove
singularity and with $T_{c}\approx 65$ mK, showing intensity stripes along the
edges of the Fermi surface. $C_{3}$-symmetry is broken because the intravalley
symmetry is $E$, degenerate. In both materials, the OP changes sign between
valleys, which means that BBG and RTG are valley-singlet, spin-triplet
superconductors. Data replotted from Ref. [119]

###  III.3 Other theoretical results.

The analyses described in the previous subsection give also information on the
nature of the normal state, especially those ones which assume that
superconductivity emerges from a broken symmetry phase. In addition, the role
of the electron-electron interaction in the normal state of RTG has bee
studied in [136], where the effects of strong local interactions are described
using Dynamical Mean Field Theory. Non superconducting broken symmetry phases
in RTG are analyzed phenomenologically in [137]. The information that the
optical conductivity gives on correlated phases in RTG has been studied in
[138]. Superconductivity has also been predicted to exist in rhombohedral
tetralayer graphene [115; 139].

##  IV Conclusions

The crystalline nature of bilayer and trilayer graphene is a crucial advantage
with respect to twisted systems, which suffer from twist angle disorder [140]
and strains [141]. These issues place strict demands on the fabrication
process and hinder experimental reproducibility [142]. In contrast, bilayer
and trilayer graphene are mechanically stable, easy to fabricate, and have
very low disorder and strain. Given the complexity of the phase diagrams to
elucidate, bypassing the issue of disorder may prove decisive in the pursuit
of a unified theory of superconductivity and strongly correlated phases in
graphene based superconductors.

The observation of superconductivity and other non trivial phases in non-
twisted graphene bilayers and trilayers suggests that other graphene based
compounds, suitably modified, can also support correlated electronic phases.
Non-twisted stacks, like the ones discussed here, probably can be tuned to
show flat bands. Note that, on the other hand, not all twisted graphene stacks
show superconductivity [143; 144].

Superconductivity has not been found yet in twisted transition metal
dichalcogenides (TMD’s), or in TMD heterobilayers, although there is ample
evidence of other correlated phases [145].

Finally, it is worth considering whether the mechanism which triggers
superconductivity is the same in twisted and non-twisted carbon compounds.

##  V Acknowledgments.

We are thankful to Andrey V. Chubukov, José A. Silva-Guillen, Elena Bascones
and Gerardo Naumis for illuminating discussions. We acknowledge support from
the Severo Ochoa programme for centres of excellence in R&D (Grant No.
SEV-2016-0686, Ministerio de Ciencia e Innovación, Spain); from the European
Commission, within the Graphene Flagship, Core 3, grant number 881603 and from
grants NMAT2D (Comunidad de Madrid, Spain), SprQuMat (Ministerio de Ciencia e
Innovación, Spain) and (MAD2D-CM)-MRR MATERIALES AVANZADOS-IMDEA-NC. V.T.P.
acknowledges support from the Department of Energy under grant DE-
FG02-84ER45118, the NSF Graduate Research Fellowships Program, and the P.D.
Soros Fellowship for New Americans.

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