Superconductivity induced by the inter-valley Coulomb scattering in a few layers of graphene Tommaso Cea

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Superconductivity induced by the inter-valley Coulomb scattering in a few layers of

graphene

Tommaso Cea

Department of Physical and Chemical Sciences, Universit´a degli Studi dell’Aquila, I-67100 L’Aquila, Italy

We study the inter-valley scattering induced by the Coulomb repulsion as a purely electronic

mechanism for the origin of superconductivity in few layers of graphene. The pairing is strongly

favored by the presence of van Hove singularities (VHS’s) in the density of states (DOS). We consider

three diﬀerent hetherostructures: twisted bilayer graphene (TBG), rhombohedral trilayer graphene

(RTG) and Bernal bilayer graphene (BBG). We obtain trends and estimates of the superconducting

(SC) critical temperature in agreement with the experimental ﬁndings, which might identify the

inter-valley Coulomb scattering as a universal pairing mechanism in few layers of graphene.

I. INTRODUCTION

The discovery of superconductivity in TBG[1–3] led

the scientiﬁc community to a renewed interest in the

study of the SC properties of graphene, that has been

further motivated by the more recent observations of SC

behavior in other heterostructures based on graphene:

twisted trilayer graphene (TTG)[4], untwisted RTG[5]

and BBG[6] in a perpendicular electric ﬁeld. In all these

systems the SC transition can be controlled by experi-

mentally tunable parameters, like eg: the relative twist

between the layers, the electronic density and the ap-

plied displacement ﬁeld. Even though the critical tem-

peratures, Tc, observed so far in these materials do not

exceed the scale of a few Kelvin, the large ratios between

Tcand the Fermi energy, up to ∼10%, suggests that a

strong pairing interaction is at play. On the other hand,

the complex phase diagrams reported in the literature

clearly highlight the strongly correlated behavior. The

recent observation of the Ref. [7], that the value of Tcin

BBG can be increased by one order of magnitude by a

substrate of WSe2, emphasizes the highly tunable nature

of the pairing, paving the way towards engineering new

techniques for controlling the magnitude of Tc. Further-

more, the violation of the Pauli’s limit reported in the

experiments [4–6] suggests that spin-triplet Cooper pairs

are favored in these systems.

On the theoretical side, it is universally accepted that

the band ﬂattening and the vicinity of the VHS’s to

the Fermi level enhance the role of the electronic in-

teractions in TBG, TTG, RTG and BBG, favoring the

formation of symmetry broken phases (see eg the Refs.

[8–13]). However, the debate on the mechanism at the

origin of the superconductivity in these systems is still

open. Many models have been studied so far, that ei-

ther consider the superconductivity driven by purely elec-

tronic interactions[9,12,14–32] or by more conventional

phononic mechanisms[33–43]. The combined eﬀects of

the screened Coulomb interaction, the electronic Umk-

lapp processes and the electron-phonon coupling have

been shown to favor the pairing in TBG[44–46] and in

TTG[47]. Furthermore, the Refs. [48–52] explored other

unconventional mechanisms, in which the pairing is me-

diated by soft electronic collective modes. Remarkably,

there is not yet a general agreement on whether the su-

perconductivity observed in TBG and in TTG has the

same origin as in the untwisted RTG and BBG.

In this article, we study the inter-valley scattering in-

duced by the Coulomb interaction as a purely electronic

mechanism for the origin of superconductivity in few lay-

ers of graphene. The resulting Cooper pairs are spin-

triplets with the two electrons in opposite valleys, K, K0,

featuring p- or f-wave symmetry. Because the large mo-

mentum transfer, ∆K≡K−K0, involved in the process

makes the interaction strength negligible, a high DOS is

necessary to boost the pairing. This condition is often

realized in few layers of graphene, where the electronic

bands can be ﬂattened by tuning a number of experi-

mental parameters, thus giving rise to VHS’s. At ﬁrst

approximation, we neglect the contribution of the intra-

valley Coulomb repulsion, which is long-ranged, since it is

drastically screened in the van Hove scenario. We show

quantitatively that this assumption is fully justiﬁed in

the SI[53]. Using eﬀective continuum models with real-

istic parameters, we characterize the SC transition in-

duced by the inter-valley Coulomb scattering in TBG,

RTG and BBG, upon varying the relative twist between

the layers and/or the electronic density and/or the dis-

placement ﬁeld. We obtain estimates and trends of Tcin

good agreement with the experimental results, empha-

sizing the strong enhancement of Tcby the presence of

VHS’s. Remarkably, our calculations account for the dif-

ferent orders of magnitude of the critical temperatures

observed in diﬀerent materials. Considering also the ex-

perimental evidence of spin-triplet superconductivity in

these systems, our study might identify the inter-valley

Coulomb scattering as a universal driving mechanism for

the superconductivity observed so far in few layers of

graphene. We also identify a non-trivial structure of the

SC order parameter (OP) in real space.

II. THE MODEL: EFFECTIVE ATTRACTION

FROM THE INTER-VALLEY SCATTERING

Our theoretical description of the pairing interaction

starts from considering the Coulomb repulsion between

the pzelectrons within the minimal lattice model for a

arXiv:2210.11873v5 [cond-mat.str-el] 30 Jan 2023

multilayer of graphene:

Hint =1

RR0X

ijσσ0

c†

iσ(R)c†

jσ0(R0)Vij

C(R−R0)cjσ0(R0)ciσ(R),(1)

where Rare the coordinates of the Bravais lattice, i, j

are the labels of the sub-lattice/layer, ciσ (R) is the the

quantum operator for the annihilation of one electron

with spin σin the pzorbital localized at the position

R+δi,δibeing the internal coordinate in the unit cell,

and:

Vij

C(R−R0) = e2

4π R−R0+δi−δj

(2)

is the Coulomb potential, where eis the electron charge

and is the dielectric constant of the environment, =0

in the vacuum. Next, we consider the continuum limit of

the lattice model, by expanding the operators cas:

ciσ(R)≡A1/2

chψK

iσ(R)eiK·R+ψK0

iσ (R)eiK0·Ri,(3)

where Ac=√3a2/2 is the area of the unit cell of

graphene, a= 2.46˚

A being the lattice constant, K, K0are

the non equivalent corners of the BZ and ψK

iσ(r), ψK0

iσ (r)

are fermionic operators, which vary smoothly with the

continuum position, r, and represent the valley projec-

tions of ciσ(R). Replacing the Eq. (3) into the Eq.

(1), among all the terms one ﬁnds the following valley-

exchange interaction:

Hexc =A2

RR0X

ijσσ0

ψK,†

iσ (R)ψK0,†

jσ0(R0)ψK

jσ0(R0)ψK0

iσ (R)Vij

C(R−R0)e−i∆K·(R−R0),(4)

which describes the inter-valley scattering processes. As

we show in detail in the supplementary information

(SI)[53], the Eq. (4) can be safely approximated by the

continuum Hamiltonian:

Hexc '

−JX

iσσ0Zd2rψK,†

iσ (r)ψK0,†

iσ0(r)ψK0

iσ (r)ψK

iσ0(r),(5)

where: J≡e2

2|∆K|is the Fourier transform of the

Coulomb potential in 2D, evaluated at ∆K. It’s worth

noting that the interaction described by the Eq. (5) is

purely local, not only in the space coordinates, but also

in the sub-lattice and layer indices. Because J > 0,

Hexc provides an eﬀective attraction in the spin-triplet

channel (see the SI[53]), favoring the Cooper pairing

with electrons in opposite valleys. This kind of interac-

tion belongs to the universality class identiﬁed by Crepel

and Fu[54,55], who have demonstrated the relevance of

the valley-exchange interaction in inducing the pairing

in narrow band systems. As we already mentioned, we

stress that the spin triplet superconductivity is a general

claim of the experimental works. On the other hand, the

valley-exchange from the Coulomb interaction has been

shown to favor spin-triplet superconductivity in various

materials (see for example the Refs. [29,56,57]). The SC

OP, ∆i(r), is purely local and the value of Tccan be ob-

tained within the BCS theory as the largest temperature

for which it exists a nonzero solution of the linearized

gap equation:

∆i(r) = J

βX

jZdr0

+∞

l=−∞ ×(6)

× GK

ij (r,r0;iωl)GK0

ij (r,r0;−iωl) ∆j(r0),

where β= (KBT)−1is the inverse of the temperature,

ωl=π(2l+ 1)/(~β) are fermionic Matsubara frequencies

and GK,K0are the Green’s function for the K, K0val-

leys, respectively, computed in the normal phase. The

Eq. (6) is written in real space in order to be as general

as possible, holding also for non-translationally invariant

systems, as is the case of the TBG that we will consider

below.

It’s worth noting that: i) Jis generally small. For

example, if we consider /0= 4, which mimics the

screening by a substrate of hBN, then J'13.25eV˚

A2,

consistent with the estimates of the Hubbard interaction

strength in graphene[58–62]. Such a small value requires

a large DOS at the Fermi energy, NF, for making the

dimensionless SC coupling, λ=NFJ, sizeable. While

the DOS is suppressed in the monolayer graphene close

to charge neutrality, multilayer stacks of graphene oﬀer a

way to increase the value of NF, and hence to strengthen

λ, upon tuning a number of experimental parameters,

like eg the relative twist between the layers, the electronic

density, the displacement ﬁeld etc.; ii) we are not consid-

ering the eﬀects of the intra-valley scattering induced by

the Coulomb interaction at small momenta, which are

repulsive. In a van Hove scenario, these terms are sup-

pressed by the strong internal screening. As we show in

the SI[53] for the case of the RTG, the strength of the

screened intra-valley Coulomb repulsion is orders of mag-

nitude smaller than J, which fully justiﬁes its omission;

iii) we are not considering the internal screening of J,

which is supposed to be negligible as it is induced by the

particle-hole excitations with the particle and the hole

in opposite valleys. These kind of processes are indeed

suppressed despite of a large DOS. This assumption is

justiﬁed quantitatively in the SI[53], where we show that

the screening essentially does not aﬀect the value of Jas

compared to its bare value.

III. RESULTS

The case of the TBG. A relative small twist, θ,

between the two layers of a bilayer graphene gener-

ates a moir´e superlattice with periodicity: Lm'a/θ,

much larger than the lattice constant of the monolayer

graphene. The inter-layer hopping varies smoothly at

the scale of the moir´e, breaking the translational invari-

ance within each moir´e unit cell and strongly hybridizing

the pzorbitals of the constitutive graphene sheets. The

superconductivity has been observed at the ”magic” an-

gle, θ= 1.05◦[1–3], where the free electron spectrum fea-

tures two weakly dispersing narrow bands at the charge

neutrality point (CNP), generating strong VHS’s in the

DOS[63,64].

Using the continuum model of the TBG[63–66], we

solve the linearized gap equation (6) as detailed in the

SI[53]. The Fig. 1shows Tcas a function of the ﬁlling

per moir´e unit cell, ν, obtained for three values of the

twist angle, as coded in the caption, and for /0= 4.

We ﬁnd values of Tcof the order of 1K, in good agree-

ment with the experimental ﬁndings. Tcis the largest

for θ= 1.05◦, where the bandwidth at the CNP is min-

imum. The band structure and the DOS corresponding

to the two central bands of the TBG at θ= 1.05◦are

shown in the Fig. 2for: ν=−1,0,1. The continuous

and the dashed lines refer to the Kand K0valleys, re-

spectively, while the horizontal lines identify the Fermi

energies. Note that the reshaping of the bands with the

ﬁlling is induced by the Hartree corrections[8,10]. Com-

paring the Figs. 1and 2makes it clear that the value of

Tcincreases with NF, which explains why we obtain the

largest Tcat ν= 0, where the bandwidth is minimum.

It’s worth noting that the TBG is actually a ﬂavor po-

larized insulator at ν= 0. As we are not considering

ﬂavor polarized phases, the superconductivity that we

obtain at ν= 0 is not realistic. Nonetheless, it serves as

an example to grab the relevance of the VHS’s in induc-

ing the pairing. As we show in the SI[53], the order of

magnitude of Tcand its ν-dependence do not change for

realistic values of the external screening: /0∼4−6.

The OP, ∆i(r), that we obtain as solution of the Eq.

-2-1 0 1 2

0.5

1.5

2.5

(

)

1.05°

1.12°

1.16°

Figure 1. Tcas a function of the ﬁlling, obtained for: /0= 4

and θ= 1.05◦,1.12◦,1.16◦.

K2M

M K1

-20

-10

energy (meV)

(a)

0 5 10

DOS (meV-

)

(b)

-1

Figure 2. Band structure (a) and DOS (b) corresponding to

the two central bands of the TBG, obtained for: θ= 1.05◦

and ν=−1,0,1. The continuous and the dashed lines refer

to the Kand K0valleys, respectively, while the horizontal

lines identify the Fermi energies. The DOS is expressed in

units of meV−1A−1

m, where Am=√3L2

m/2 is the area of the

moir´e unit cell.

(6) at T=Tc, is almost uniform in the sub-lattices and

layers, has a constant phase and varies in the moir´e unit

cell, reaching its maxima in the regions with local AA

stacking. Maps of ∆i(r) are shown in the SI[53].

The case of the RTG and BBG. Both RTG and

BBG become superconductors at very low electronic den-

sities: ne∼1012cm−2, when an electric ﬁeld is applied

perpendicular to the graphene’s ﬂakes[5,6]. The experi-

mental Tc’s are ∼10−1K for the RTG and ∼10−2K for

the BBG. From the electronic point of view, a perpen-

dicular electric ﬁeld breaks the inversion symmetry and

gaps out the Dirac points in both RTG and BBG. The

nearly ﬂat dispersion close to the gap’s edge gives rise

to a pronounced VHS. For values of neclose to those

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Superconductivityinducedbytheinter-valleyCoulombscatteringinafewlayersofgrapheneTommasoCeaDepartmentofPhysicalandChemicalSciences,UniversitadegliStudidell'Aquila,I-67100L'Aquila,ItalyWestudytheinter-valleyscatteringinducedbytheCoulombrepulsionasapurelyelectronicmechanismfortheoriginofsuperconductiv...

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Superconductivity induced by the inter-valley Coulomb scattering in a few layers of graphene Tommaso Cea

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