Superconductivity from electronic interactions and spin-orbit enhancement in bilayer and trilayer graphene. Alejandro Jimeno-Pozo1H ector Sainz-Cruz1Tommaso Cea1 2Pierre A. Pantale on1and Francisco Guinea1 3 4
2025-05-02
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Superconductivity from electronic interactions and spin-orbit enhancement in bilayer
and trilayer graphene.
Alejandro Jimeno-Pozo,1, ∗H´ector Sainz-Cruz,1Tommaso Cea,1, 2 Pierre A. Pantale´on,1and Francisco Guinea1, 3, 4
1Imdea Nanoscience, F araday 9,28049 M adrid, Spain
2Department of P hysical and Chemical Sciences, University of L0Aquila, via V etoio, Coppito, 67100 L0Aquila, Italy
3Donostia International P hysics Center, P aseo Manuel de Lardizabal 4,20018 San Sebastian, Spain
4Ikerbasque F oundation, Maria de Haro 3,48013 Bilbao, Spain
(Dated: 6th February 2023)
We discuss a Kohn-Luttinger-like mechanism for superconductivity in Bernal bilayer graphene
and rhombohedral trilayer graphene. Working within the continuum model description, we find that
the screened long-range Coulomb interaction alone gives rise to superconductivity with critical tem-
peratures that agree with experiments. We observe that the order parameter changes sign between
valleys, which implies that both materials are valley-singlet, spin-triplet superconductors. Adding
Ising spin-orbit coupling leads to a significant enhancement in the critical temperature, also in line
with experiment, and the superconducting order parameter shows locking between the spin and
valley degrees of freedom.
Introduction. Recent experiments report cascades of
correlated phases in Bernal bilayer graphene (BBG) [1–
3]. One of them is spin-polarized superconductivity [1]
with a critical temperature of Tc≈26 mK for fillings
near the hole-doped van Hove singularity (vHs) when
an out-of-plane electric field together with an in-plane
magnetic field are applied on the system. Moreover, the
authors of Ref. [4] have assembled a heterostructure in
which a transition metal dichalcogenide (TMD) and BBG
enter in synergy, making superconductivity appear over a
broader range of electron filling and magnetic field, even
without the latter, and enabling a striking increment in
critical temperature to Tc≈260 mK, an effect that is
attributed to Ising spin-orbit coupling (SOC). The sta-
bility and structural simplicity of BBG are clear advant-
ages for experimental reproducibility, a major obstacle
in Moir´e materials [5], in which all samples differ due
to angle disorder [6,7] and strains [8]. A previous ex-
periment showed that rhombohedral trilayer graphene
(RTG) is a superconductor as well [9] and several the-
ories about it have been proposed [10–20]. A key piece
in the puzzle of graphene superconductors is that they
all display cascades of flavour-symmetry-breaking phase
transitions [1–3,21–23], perhaps hinting at a common
origin of superconductivity.
In these materials, and in BBG in particular, the prox-
imity of superconductivity to flavour-polarized metal-
lic phases and its Pauli limit violation point to an
unconventional spin-triplet pairing mediated by elec-
trons. In Ref. [24], a model is discussed in which short-
range momentum-independent interactions and proxim-
ity to symmetry broken phases induce spin-triplet f-wave
pairing. The authors of Ref. [25] propose a superconduct-
ivity from repulsion mechanism, in which the Coulomb
interaction dressed by soft quantum-critical modes drives
pairing. In their model, the magnetic field induces spin
∗alejandro.jimeno@imdea.org
imbalance, which makes the interaction acquire a de-
pendence on frequency or on soft-mode momenta, leading
to valley-singlet, spin-triplet s-wave or valley-triplet spin-
triplet p-wave pairings, respectively. In Refs. [26,27], a
pairing mechanism mediated by acoustic phonons is in-
vestigated, which is compatible with spin-singlet s-wave
and spin-triplet f-wave pairings. However, the authors
assume that the Coulomb interaction is detrimental for
superconductivity. In contrast, we find that the Coulomb
interaction alone enables superconductivity, see also [25].
Here, we present a framework in which superconduct-
ivity in BBG emerges only from the long-range Coulomb
interaction, arguably the simplest explanation. Due to
screening by particle-hole pairs, this repulsive interac-
tion between electrons becomes attractive and leads to
pairing. We use a diagrammatic technique similar to the
Kohn-Luttinger approach [28], which we have already ap-
plied to twisted bilayer and trilayer graphene [29,30], as
well as to RTG [15], thus allowing for a direct comparison
of superconductivity in these systems and in BBG. We
find superconductivity with critical temperature compar-
able to the experimental one [1], near the vHs. Including
Ising SOC leads to significant increments in the critical
temperature, as seen experimentally [4]. We observe that
the order parameter (OP) changes sign within each val-
ley. Adding a short-range Hubbard repulsion, we con-
clude that the OP also changes sign between valleys,
showing that BBG is a spin-triplet superconductor. The
rest of the paper is organized as follows: first we describe
the continuum model and the Kohn-Luttinger-like frame-
work for superconductivity. Then we present and discuss
the results of the critical temperature, the superconduct-
ing OP and the positive effect of adding SOC.
The continuum model. Bernal bilayer graphene (BBG)
refers to two stacked graphene layers so that atoms be-
longing to sublattice A of layer 1 lie over the atoms in
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 lin-
arXiv:2210.02915v3 [cond-mat.mes-hall] 3 Feb 2023
2
- 0 . 0 4 - 0 . 0 2 0 . 0 0 0 . 0 2 0 . 0 4
- 0 . 0 6 5
- 0 . 0 6 0
- 0 . 0 5 5
- 0 . 0 5 0
- 0 . 0 4 5
- 0 . 0 4 0
0 1 0 2 0 3 0
- 0 . 0 4 - 0 . 0 2 0 . 0 0 0 . 0 2 0 . 0 4
- 0 . 0 6 5
- 0 . 0 6 0
- 0 . 0 5 5
- 0 . 0 5 0
- 0 . 0 4 5
- 0 . 0 4 0
0 2 0 4 0 6 0 8 0
E n e r g y ( e V )
kx ( Å - 1 )
a ) b )
c )
d ) D O S ( a . u . )
E n e r g y ( e V )
kx ( Å - 1 )D O S ( a . u . )
Figure 1. (a) Crystal lattice and hoppings of BBG. (b) Bril-
louin zone of BBG and RTG. We perform the calculation
of superconductivity considering electron states in hexagonal
grids around the Dirac points, with ultraviolet momentum
cutoff kΛ= 0.025KDfor BBG (and 0.035KDfor RTG),
with KD= 4π/3athe Dirac point modulus, which con-
tain the Fermi surface and where the continuum model is a
good approximation. Red and blue hexagons are plotted with
kΛ= 0.2KDfor clarity. (c, d) Continuum model valence band
and DOS of BBG and RTG with electric-field-induced gaps
of 98 and 74 meV, respectively [31]. Dashed red lines mark
the energies of the vHs.
ear dispersion [32,33]. This band touching makes the
charge susceptibility and other susceptibilities diverge
[34], which leads to symmetry-broken phases [35–43], as
seen in experiments [44–50]. A perpendicular electric field
opens a gap and the bands acquire a ‘Mexican hat’ profile
[51,52], as shown in Fig. 1(c,d), making the density of
states diverge logarithmically near the band edges, caus-
ing vHs and setting the stage for new correlated phenom-
ena [1–4,53–56].
In the continuum approximation the hamiltonian of
BBG can be expressed in the basis {ΨA1,ΨB1,ΨA2,ΨB2}
as [52],
Hξ=
V/2v0π†−v4π†v3π
v0π V/2+∆0γ1−v4π†
−v4π γ1−V/2+∆0v0π†
v3π†−v4π v0π−V/2
,(1)
where π=ξpx+ ipy,ξ=±1 is the valley index and
p=k−Kξis the momentum with respect to the Kξ
Dirac point, vi=√3aγi
2~are effective velocities, ais the
lattice constant and γithe different hoppings in the sys-
tem, shown in Fig. 1(a). Vcorresponds to an applied
electric field and ∆0is the energy difference between di-
mer (A1,B2) and non-dimer (A2,B1) sites. In Ref. [57]
we list the values [52,58], and describe the hamiltonian
of RTG.
Kohn-Luttinger-like superconductivity. We propose
that electronic interactions alone are enough to induce
superconductivity in BBG and RTG and that the pairing
glue is the screened long-range Coulomb potential Vscr .
We employ a similar approach to the Kohn-Luttinger
theory [28], following an analogous method as in [15].
We use the Random Phase Approximation to take into
account the screened direct interaction to infinite order,
while neglecting the contribution from the exchange in-
teraction. The multiplicity of the direct diagrams equals
the number of flavours, Nf= 4, so the approximation
can be considered an expansion in powers of 1/Nf. Un-
der these assumptions the screened interaction is given
by,
Vscr(q) = VC(q)
1−Π(q)VC(q),(2)
where VC(q) = 2πe2
|q|tanh (d|q|) is the bare Coulomb po-
tential, with ethe electron charge, = 4 the dielectric
constant associated hBN encapsulation and d= 40 nm
the distance to the metallic gate. Π(q) corresponds to
the zero-frequency limit of the charge susceptibility,
Π(q) = Nf
ΩX
k,m,n
f(n,k)−f(m,k+q)
n,k−m,k+q|hΨm,k+q|Ψn,ki|2,
(3)
where m, n are band indexes, fis the Fermi-Dirac dis-
tribution and =E−µwith µbeing the Fermi en-
ergy, E are the eigenvalues, Ψ the eigenvectors of (1)
and Ω is the area of the system. Once we perform
this calculation within each valley, we include a short-
range, repulsive Hubbard U=3eV[59,60], which al-
lows electrons to exchange valley and fixes the sign of
the order parameter in each valley. We define the di-
mensionless anomalous expectation values in both val-
leys ˜
∆+,i,j (k) = Dc†
k,i,K+,↑c†
−k,j,K−,↓Eand ˜
∆−,i,j (k) =
Dc†
k,i,K−,↑c†
−k,j,K+,↓E. Therefore the gap equation is
3
- 0 . 0 5 5 - 0 . 0 5 4 - 0 . 0 5 3 - 0 . 0 5 2 - 0 . 0 5 1 - 0 . 0 5 0
0
5
1 0
1 5
-0.0465 -0.0460 -0.0455 -0.0450
0
2 0
4 0
6 0
- 0 . 0 3 0 . 0 0 0 . 0 3
- 0 . 0 3
0 . 0 0
0 . 0 3
- 0 . 0 3 0 . 0 0 0 . 0 3
- 0 . 0 3
0 . 0 0
0 . 0 3
Tc ( m K )
E n e r g y ( e V )
a )
Tc ( m K )
E n e r g y ( e V )
b )
ky ( Å - 1 )
kx ( Å - 1 )
Vscr ( e V ⋅Å2)
ky ( Å - 1 )
kx ( Å - 1 )
Vscr ( e V ⋅Å2)
Figure 2. Superconducting critical temperature due to only the screened long-range Coulomb interaction (blue), and including
also the short-range interaction (green), and DOS (a.u., dashed black) versus Fermi energy in (a) BBG with hole doping, near
the vHs (ne≈ −0.87 ·1012 cm−2), we observe a Tc≈10 mK, and (b) RTG with hole doping, near the vHs (ne≈ −0.6·1012
cm−2), Tc≈33 mK and it raises to Tc≈65 mK after including the short-range interaction. The top insets show the shape
of the Fermi surfaces, which experience a change in topology at the vHs due to Lifshitz transitions. Right insets: screened
Coulomb potential to kΛat the energy for which Tcis maximum.
given by,
˜
∆+,i,j (k) = −KBT
ΩX
q,ω X
i0,j0
Vscr (k−q)Gi,i0
K+(q,iω)Gj,j0
K−(−q,−iω)˜
∆+,i0,j0(q)
+UGi,i0
K−(q,iω)Gj,j0
K+(−q,−iω)˜
∆−,j0,i0(−q),
(4)
where indexes i, j denote layer and sublattice degrees of
freedom, ωis the Matsubara frequency and Gi,j
Kξ(q,iω)
are electronic Green’s functions. Eq. (4) can be under-
stood in terms of the convolution of the gap with a ker-
nel,
˜
∆(k) = ˜
∆+(k)
˜
∆−(k)=X
k0Γ+(k,k0)˜
U(k,k0)
˜
U†(k,k0) Γ−(k,k0)˜
∆(k0),
(5)
so that the solution of this gap equation is achieved when
the maximum eigenvalue of the hermitian kernel reaches
a value of 1, indicating the Tcat which the scattering
amplitude for pairs of carriers of opposite momenta and
energy within the Fermi surface has a pole. The corres-
ponding eigenvector is the superconducting order para-
meter. The long-range Coulomb kernel is,
Γξ
m,n(k,k0) = −Vscr (k−k0)
ΩSξ,ξ
m,n(k,k0)Rξ,ξ
m,m(k,k)Rξ,ξ
n,n(k0,k0),
(6)
while the short-range kernel is,
˜
Um,n(k,k0) = −U
ΩS+,−
m,n (k,k0)R+,−
m,m(k,k)R−,+
n,n (k0,k0),
(7)
with
Rξ,ξ0
m,n(k,k0) = rf(−ξ
m,k)−f(ξ0
n,k0)/ξ
m,k+ξ0
n,k0
and Sξ,ξ0
m,n(k,k0) = DΨξ
m,kΨξ0
n,k0E
2. Once Eq. (5) is
solved, gaps with dimensions of energy can be obtained
as, ∆+(k) = Vscr(k)˜
∆+(k) + U˜
∆−(k) and ∆−(k) =
Vscr(k)˜
∆−(k) + U˜
∆+(k).
Results. In Figure 2we present the superconduct-
ing critical temperature of hole-doped BBG and RTG
as a function of Fermi energy. We obtain a Tcof 10 mK
for BBG and 65 mK for RTG, which match well with
those observed experimentally [1,9]. In both materials
superconductivity survives only in narrow energy inter-
vals around the vHs. It is worth noting that in RTG,
including the short-range interaction doubles Tc, but it
does not increase Tcin BBG. To obtain Tc, we use a k-
point density of up to O(107)˚
A2for BBG and O(106)˚
A2
for RTG. Since states close to the Fermi surface give
the principal contribution to Eq. (5), we cut off phase
space by considering only states with |n,k|630 meV.
These continuum model results are in good agreement
with our previous tight-binding calculation for RTG [15].
The insets in Fig. 2show the screened potential in mo-
mentum space, which is always repulsive, i.e. positive,
and its value increases with increasing momentum. This
dependence favors non s-wave superconductivity [61].
Moreover, inspired by the authors of Ref. [4], we study
the effect of stacking a TMD on top of BBG. We as-
sume that the main effect is an induced Ising spin-orbit
coupling (SOC) in BBG, which breaks the equivalence
between Cooper pairs |K+,↑;K−,↓i and |K+,↓;K−,↑i,
so the order parameter cannot be defined as a spin sing-
let or a spin triplet. The Ising hamiltonian which we add
to (1) is just HI,ξ,s =sξλII, where s is the spin index.
This interaction does not reshape the band structure, it
just promotes one type of Cooper pair, by lowering the
energy of two spin-valley flavours and raising that of the
other two, splitting the original vHs into two. As shown
in Figure 3, Ising SOC has a very positive effect on su-
4
0 1 2 3 4 5
3 5
4 0
4 5
5 0
5 5
6 0
6 5
7 0
012345
0
1 0
2 0
3 0
4 0
λI ( m e V )
Tc ( m K )
λI ( m e V )
a ) b )
Tc ( m K )
Figure 3. Ising spin-orbit coupling enhancement of critical
temperature in hole-doped (a) BBG, for which Tcaugments
by a factor of 4 and saturates at λI= 4 meV. (b) RTG, for
which Tcdue to long-range interactions (blue) augments by
a factor of 2 and saturates at λI= 1 meV. In contrast, Ising
SOC is detrimental to short-range interactions, so Tcdoes not
increase much when they are included (green).
perconductivity in both materials: it increases Tcby a
factor of 4 in BBG, to 40 mK, congruent with the ex-
periment, in which a factor of 10 increment was seen [4]
and in RTG it brings forth a comparable increment of
Tcdue to long-range interactions only. The two vHs lead
to two superconducting domes as a function of Fermi en-
ergy, as observed in Ref. [4], see [57]. It is worth noting
that the spin-orbit enhancements of Tcsaturate once the
SOC fully polarizes the non-superconducting state into a
half-metal [57].
Our model also yields the superconducting order para-
meters (OPs), shown in Figure 4. The OPs have structure
within each valley: in BBG they display hotspots and
sign changes along the edges of the Fermi surface. The
sign is opposite for the inner and outer Fermi edges. In
RTG, the maximum eigenvalue of the kernel is degener-
ate, leading to C3-symmetry breaking. Such degeneracy
is not typical of conventional superconductors and hints
at an exotic type of superconductivity. We observe that
the OPs in both materials change sign between valleys,
forming valley-singlets. Since the overall electron wave-
function must be antisymmetric, this implies that the
pairs are spin-triplets. The order parameter that we find
in the presence of spin-orbit coupling is similar to the
spin-valley locking model in [62]. Our results are consist-
ent with the phenomenological model discussed in [63].
Discussion. BBG and RTG are 2D superconductors
with great structural stability and low disorder. These are
decisive advantages to understand the physics at play, as
they enable very reproducible experiments. There is also
compelling evidence that both support spin-triplet su-
perconducting phases and hence the pairing is most likely
unconventional and mediated by electrons. Here, we show
that a Kohn-Luttinger-like mechanism for superconduct-
ivity suffices to give rise to superconductivity in BBG and
RTG, with critical temperatures in good agreement with
experiments on both materials [1,4,9]. The screened
long-range Coulomb interaction is the sole responsible
for pairing. At fillings near the vHs, electron interactions
- 0 . 0 2 0 . 0 0 0 . 0 2
- 0 . 0 2
0 . 0 0
0 . 0 2
- 0 . 0 2 0 . 0 0 0 . 0 2
- 0 . 0 2
0 . 0 0
0 . 0 2
- 0 . 0 2 0 . 0 0 0 . 0 2
- 0 . 0 2
0 . 0 0
0 . 0 2
- 0 . 0 2 0 . 0 0 0 . 0 2
- 0 . 0 2
0 . 0 0
0 . 0 2
ky ( Å - 1 )
kx ( Å - 1 )
b )
c ) d )
ky ( Å - 1 )
kx ( Å - 1 )
a )
ky ( Å - 1 )
kx ( Å - 1 )
ky ( Å - 1 )
kx ( Å - 1 )
Figure 4. (a) Superconducting order parameter (OP) of BBG
in valley K+, (b) in valley K−, with hole doping and Tc≈10
mK. Within a single valley, the OP has hotspots 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−, with hole doping and Tc≈65 mK, showing intensity
stripes along the edges of the Fermi surface. In both materials,
the OP changes sign between valleys, which means that BBG
and RTG are valley-singlet, spin-triplet superconductors.
strongly screen the Coulomb potential, which glues pairs
of carriers, one in each valley, and causes superconduct-
ivity. Of course, other excitations may dress the Coulomb
interaction and contribute to superconductivity in BBG
[24–26] and RTG [10–14,16–20].
Furthermore, our results support the proposal in Ref.
[4] that Ising SOC is what causes the enhancement of
Tcin BBG when the TMD WSe2is stacked on top of
it. Based on the results presented here, we predict that
the same idea will boost superconductivity in RTG by a
comparable amount as in BBG. Since the four times in-
crement we obtain in BBG underestimates the ten times
increment observed in the experiment, it is likely that
our result for RTG similarly underestimates the SOC
enhancement that superconductivity will experience in
this material. Hence, we conjecture that RTG with WSe2
stacked on top will reach Tc∼1 K, putting it on par with
magic-angle twisted bilayer graphene and the other twis-
ted stacks [64–70].
The OPs reveal two key aspects of superconductiv-
ity. First, the OPs show sign changes within each val-
ley, which are characteristic of weak coupling supercon-
ductivity, in which electrons interact and form pairs with
high angular momentum, such as those with p-wave or f-
wave symmetry. Another consequence of this intravalley
structure is that long-range disorder will suppress super-
conductivity in these materials, unlike in twisted stacks.
Second, the sign of the OPs changes between valleys,
hence the pairs are spin-triplets.
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Superconductivityfromelectronicinteractionsandspin-orbitenhancementinbilayerandtrilayergraphene.AlejandroJimeno-Pozo,1,HectorSainz-Cruz,1TommasoCea,1,2PierreA.Pantaleon,1andFranciscoGuinea1,3,41ImdeaNanoscience;Faraday9;28049Madrid;Spain2DepartmentofPhysicalandChemicalSciences;UniversityofL0Aquil...
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