Tunable superconductivity and Moebius Fermi surface

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Tunable superconductivity and Möbius Fermi surfaces

in an inversion-symmetric twisted van der Waals heterostructure

Harley D. Scammell1and Mathias S. Scheurer2

1School of Physics, the University of New South Wales, Sydney, NSW, 2052, Australia

2Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria

We study theoretically a moiré superlattice geometry consisting of mirror-symmetric twisted tri-

layer graphene surrounded by identical transition metal dichalcogenide layers. We show that this

setup allows to switch on/oﬀ and control the spin-orbit splitting of the Fermi surfaces via application

of a perpendicular displacement ﬁeld D0, and explore two manifestations of this control: ﬁrst, we

compute the evolution of superconducting pairing with D0; this features a complex admixture of

singlet and triplet pairing and, depending on the pairing state in the parent trilayer system, phase

transitions between competing superconducting phases. Second, we reveal that, with application

of D0, the spin-orbit-induced spin textures exhibit vortices which lead to “Möbius fermi surfaces”

in the interior of the Brillouin zone: diabatic electron trajectories, which are predicted to domi-

nate quantum oscillation experiments, require encircling the Γpoint twice, making their Möbius

nature directly observable. We further show that the superconducting order parameter inherits the

unconventional, Möbius spin textures. Our ﬁndings suggest that this system provides a promising

experimental avenue for studying systematically the impact of spin-orbit coupling on the multitude

of topological and correlated phases in near-magic-angle twisted trilayer graphene.

Graphene moiré superlattices [1,2] in the small twist

angle regime, θ'1◦−2◦, have emerged in the past few

years as a versatile platform for realizing and probing a

variety of correlated quantum many-body phases [3–29].

While the intrinsic spin-orbit coupling (SOC) of graphene

is very small [30,31], increasing it is expected to enrich

the phenomenology of these systems even further: SOC

opens up completely new avenues for stabilizing topolog-

ical phases [32,33], is expected to aﬀect the energetic

balance of closely competing [34–39] instabilities as well

as their form [40], and has an enormous potential for

spintronics applications [41–43].

Bringing graphene in close proximity to a transition

metal dichalcogenide (TMD) layer, which involves heav-

ier atoms, is known to induce SOC [44–46]; the form of

the resultant SOC terms is well established for single-

layer [47–52], and non-twisted multi-layer graphene [53,

54]. Importantly, the choice of TMD and the twist angle

θTMD relative to the proximitzed graphene layer can be

l=1, θ/2

l=3, θ/2

l=2, –θ/2

θ +θ/2

TMD

θ +π+θ/2

TMD

FIG. 1: Moiré superlattice studied in this work, which

consists of three layers of graphene (l= 1,2,3) with

alternating twist angle ±θ/2symmetrically surrounded by

TMDs with relative twist angle θTMD '30◦. A displacement

ﬁeld D0breaks inversion symmetry and, thus, allows to tune

the spin-splitting of the bands.

used to tune the strength and nature of the induced SOC

[48–54]. Although some experiments, see, e.g., Refs. 18

and 25, clearly indicate that the proximitized TMD lay-

ers inﬂuence the correlated physics, in many cases it is

not established what role the proximitized TMD layer

plays for the observed phases. In general, understanding

the impact of SOC on the correlated physics of graphene

moiré systems is an open question.

To help elucidate the role of SOC, we here propose and

analyze an inversion-symmetric graphene moiré superlat-

tice setup, shown in Fig. 1, which has a key feature that

the SOC-induced spin splitting of the Fermi surfaces can

be tuned in situ, by applying a perpendicular displace-

ment ﬁeld D0. While the twisted graphene moiré sys-

tem with the fewest number of layers—twisted bilayer

graphene—exhibits spin-split bands already at D0=

0[33], the situation is diﬀerent for mirror-symmetric

twisted trilayer graphene (tTLG): it has inversion sym-

metry I, which persists when being surrounded symmet-

rically by TMD layers, see Fig. 1; together with time-

reversal symmetry, Iguarantees pseudospin-degenerate

bands, despite the presence of orbital SOC. Breaking it

via D06= 0 allows to tune eﬀective SOC terms lifting the

bands’ pseudospin degeneracy.

In this Letter, we study the resulting tunability of the

band structure and ﬁnd that (for θTMD '30◦) the spin-

orbit vector gkdetermining the spin-polarizations of the

bands exhibits vortices in the interior of the moiré Bril-

louin zone. These imply that there is a ﬁlling fraction—

found to be around half ﬁlling of the conduction or va-

lence band—with Fermi surfaces that cross each other

an odd number of times. We show that this feature can

be observed in quantum oscillations where the dominant

frequency corresponds to the sum of the inner and outer

arXiv:2210.03125v1 [cond-mat.mes-hall] 6 Oct 2022

spin-orbit-split Fermi sheets. To demonstrate the impact

of the tunable SOC on the correlated physics of tTLG, we

compute the superconducting order parameter as a func-

tion of D0assuming diﬀerent dominant pairing states in

the parent tTLG system. We see that D0not only allows

to drive superconducting phase transitions and tune the

order parameter, it also has the potential of being used

as a tool to probe the nature of pairing in tTLG [19–26],

which is currently under debate.

Non-interacting bandstructure.—To model the non-

interacting bandstructure of the system in Fig. 1we use

an appropriate generalization of the continuum-model

description of twisted bilayer graphene [55–60]. The as-

sociated Hamiltonian is diagonal in the graphene layers’

valley degree of freedom, η=±, and consists of four

terms, hη=h(g)

η+h(t)

η+h(SOC)

η+h(D)

η[61]. Here h(g)

η(∇),

∇= (∂x, ∂y), and h(t)

η(r),r= (x, y), capture, respec-

tively, the Dirac cones, rotated by (−1)lθ/2and with op-

posite chirality for η=±, of the three layers, l= 1,2,3,

and the tunneling between them, respectively. The lat-

ter is spatially modulated on the emergent moiré scale

and reconstructs the graphene cones into moiré bands.

As a result of the mirror symmetry [58], σh, the band-

structure of h(g)

η+h(t)

ηis that of twisted bilayer graphene

(σh-even) and single-layer graphene (σh-odd bands) in

valley η. Surrounding tTLG with TMD layers as shown

in Fig. 1induces SOC [47–52] in the outer two layers as

described by

h(SOC)

η=X

l=1,3

(−1)(l−1)

2Pl[λIszη+λR(ηρxsy−ρysx)] ,

where Plprojects onto the lth graphene layer and ρj

(sj), j=x, y, z, are Pauli matrices in sublattice (spin)

space. The relative minus sign between the l= 1,3

terms in h(SOC)

ηis dictated by the inversion symmetry,

I, of the heterostructure. Note that h(SOC)

ηnot only

breaks spin-rotation symmetry, but also σh, which can be

seen in the bandstructure shown Fig. 2(a): the graphene

Dirac cone of the σh-odd sector, located around the K0

point for the valley shown, hybridizes with the σh-even,

twisted-bilayer graphene sector. Despite the presence of

SOC, all bands are still doubly degenerate, which follows

from the momentum-space-local, anti-unitary symmetry

IΘs, with Θsbeing spin-1/2time-reversal, that obeys

(IΘs)2=−.

Applying a displacement ﬁeld D0—as is routinely done

in experiments on tTLG [19–26] and captured by the

last term, h(D)

η, in the Hamiltonian—breaks Iand IΘs.

As such, the pseudospin-degeneracy of the bands at

D0= 0 will be removed when D06= 0, see Fig. 2(b),

providing direct experimental control over the inversion-

antisymmetric SOC terms, gk6= 0, in the eﬀective Hamil-

tonian, heﬀ

k,η =s0ξη·k+ηgη·k·s, for the bands of tTLG

near the Fermi level. To further discuss the form of gk,

FIG. 2: Band structure, including Fermi surfaces and spin

texture in valley η= +. Here twist angle θ= 1.75◦,

µ=µ∗'16 meV, and λR= 20 meV. The cut through the

band structure in (a) at D0= 0 shows spin-degenerate

bands. The degeneracy is removed when D06= 0 as can be

seen in (b), where D0= 20 meV. Using the same parameters

as in (b), the direction of the two non-zero components of gk

in (c) reveals three vortices (+) and three anti-vortices (−).

When µ=µ∗, we obtain vanishing Fermi-surface splitting,

see (d), at their Cs

3z-related k-space locations, leading to a

Möbius winding of the spin texture (arrows).

let us focus on the limit θTMD = 30◦, where λIvanishes

by symmetry [48–51] and (gk)z= 0 as a consequence of

2zΘs(Cs

nz is the n-fold rotation symmetry along z). As

can be seen in Fig. 2(c), the remaining two components

of gkexhibit vortices at three, Cs

3z-related, generic posi-

tions k=k∗

j,j= 1,2,3, in the Brillouin zone, compen-

sating those with opposite chirality at Γ,Kand K0. Since

these vortices—and, thus, the associated zeros of |gk|and

type-II Dirac cones in the band structure—cannot be adi-

abatically removed, we are guaranteed to ﬁnd a chemical

potential µ=µ∗where the Fermi surface of ξkcrosses all

three k∗

j. For µ=µ∗(µ'µ∗), the spin-orbit splitting

of the Fermi surfaces of the system has to (almost) van-

ish at these three points, see Fig. 2(d). Having an odd

number of points on the Fermi surface where the spin

splitting (almost) vanishes also leads to exotic spin tex-

tures: following the spin polarizations of the Bloch states

[also shown in Fig. 2(d)] diabatically along the Fermi sur-

face, one ends up on the other Fermi sheet after one full

revolution, akin to an object traversing a Möbius strip.

For this reason and due to the related, but not identical

concept of “Möbius fermions” [62], which occur as edge

states crossing the zone boundary in systems with non-

symmorphic symmetries, we refer to the Fermi surfaces

at µ=µ∗as “Möbius Fermi surfaces”. Note that Möbius

Fermi surfaces are only possible since a single valley nei-

ther has two-fold out-of-plane rotation nor time-reversal

symmetry, which would necessarily lead to an even num-

ber of crossing points.

Quantum oscillations.—As our ﬁrst example of an

observable phenomenon associated with these Möbius

Fermi surfaces, we discuss quantum oscillations, which

are routinely observed in resistivity in small-angle

graphene moiré systems [3,63,64]. In the semi-classical

picture, the conduction electrons will undergo periodic

orbits on quantized constant-energy contours in momen-

tum space when a perpendicular magnetic ﬁeld B6= 0 is

applied. The oscillations of physical quantities are asso-

ciated with these contours crossing the Fermi level and

the frequency Fof the oscillations as a function of 1/B

are proportional to the momentum-space area enclosed

by the respective Fermi-surface contours [65,66].

Let us assume that µis close to but not identical to

µ∗, leading to a ﬁnite but small minimal splitting δ=

|gk∗

j|. In the adiabatic limit at small magnetic ﬁelds [B

δ2/(evgvF)], the electrons simply follow the outer and

inner Fermi surface, indicated as trajectory αand βin

Fig. 3(b). Meanwhile, ﬁnite Bwill lead to a non-zero

transition probability [61]

ρ=e−παB, αB=1

δ2

vgvF

eB "1 + gµBB

2δ2#,(1)

between the Fermi sheets at each of the three crossing

points k=k∗

jdue to Landau-Zener tunneling [67,68].

Here, vFis the Fermi velocity at k=k∗

jand vg=

|∂kkgk∗

j|, with kkdenoting the momentum along the

Fermi surface. We can see in Fig. 3(d) that ρreaches

a value close to 1already at moderately small magnetic

ﬁelds, B&10 mT, for the values of vg,vF,δextracted

from the continuum model at the indicated parameters.

At very large magnetic ﬁelds, B&10 T, the second term

in Eq. (1), which describes the Zeeman-ﬁeld-induced ef-

fective increase of the splitting between the inner and

outer Fermi surfaces, starts to dominate and ρdecreases

again. Using the frequently applied semi-classical ap-

proach [68–72] and taking into account that interval-

ley scattering is typically negligible in clean moiré sam-

ples [63] due to the large momentum-space separation,

we computed the Fourier spectrum of quantum oscilla-

tions shown in Fig. 3(a). As can be seen most clearly

in Fig. 3(c), where the intensities at the frequencies Fµ

associated with the fundamental trajectories are shown,

there is a large magnetic ﬁeld range, 0.1−2T, where

the Möbius trajectory dominates; more precisely, for

this ﬁeld range, which also signiﬁcantly overlaps with the

regime where quantum oscillations are most clearly visi-

(a)

Fα

Fβ

Fγ

Fδ

Fε

(b)

(c)

(d)

I(Fμ)

FIG. 3: Magnetic ﬁeld dependence of the quantum

oscillation frequency spectrum (a), where we also indicate

the frequencies Fµcorresponding to the ﬁve orbits

µ=α, β, γ, δ depicted in (b). The intensities at these ﬁve

frequencies and the tunnel probability are plotted in (c) and

(d) as a function of magnetic ﬁeld. In the red region, the

Möbius trajectory dominates. The eﬀective parameters

entering Eq. (1) have been extracted from the continuum

model for λR= 20 meV, D0= 30 meV, and µ= 16.5meV.

ble in experiment [63], the most prominent fundamental

frequency corresponds to the sum of the areas enclosed

by the inner and outer Fermi surface—a hallmark signa-

ture of its Möbius nature, requiring two revolutions to

be closed. In fact, is visible and dominant over its con-

stituent trajectories α,βin most of the experimentally

accessible ﬁeld range. Importantly, the experimental con-

trol over µallows to adjust δwhich, in turn, determines

the magnetic ﬁeld value 2δ/gµBwhere is most promi-

nent; this should facilitate its successful experimental de-

tection. We reiterate that for a Möbius trajectory to

dominate quantum oscillations, an odd number of cross-

ing points is required. Setting aside quasi-crystals [70]

and the case without any rotation symmetry, the single-

valley rotation symmetry Cs

3zof the system is the only

possibility to obtain an odd number of crossing points.

Superconductivity.—We now examine the inﬂuence of

SOC and D0on the superconducting state of tTLG,

which we refer to as the parent state. The parent su-

perconducting state comprises Cooper pairs formed from

a partially-ﬁlled, spin-degenerate band. To be speciﬁc,

we consider partial ﬁlling of the upper moiré bands, c.f.

those of Fig. 2(a,b), where also superconductivity is ob-

served experimentally [19,25,26,73]. Moreover, we ex-

clusively consider intervalley pairing since it is expected

[74–77] to be dominant over intravalley pairing due to

time-reversal symmetry Θs.

To study the evolution of the superconducting state

under combined SOC and D0we appeal to the linearized

gap equation [61],

dµ,k1=−X

k2,µ0,ν

Γµµ0,k1,k2Wµ0ν,k2dν,k2,(2)

Wµν,k=X

ni,sj

tanh ε+,n1,k

2T+ tanh ε+,n2,k

2T

2(ε+,n1,k+ε+,n2,k)×

(sµ)s2,s3C+,n1,s1,kC∗

+,n1,s2,kC+,n2,s3,kC∗

+,n2,s4,k(sν)s4,s1.

The intervalley superconducting order parameter dµ,ken-

codes the momentum and spin structure, where µ= 0

and µ= 1,2,3refer to the spin-singlet and triplet

components, respectively. With perturbed Hamiltonian

hη=h(g)

η+h(t)

η+h(SOC)

η+h(D)

η, the factors C∗

η,n,s,k≡

ψ†

η,n,kψ0

η,s,kaccount for the projection of the perturbed

eigenstates hηψη,n,k=εη,n,kψη,n,kin band nonto the

unperturbed eigenstates (h(g)

η+h(t)

η)ψ0

η,s,k=ε0

η,s,kψ0

η,s,k

of spin s, which comprise the parent superconducting

state. We specialize to η= + in (2); the η=−order pa-

rameter follows from the fermion anticommutation rela-

tion. Finally, the interaction vertex Γµ,ν,k1,k2assumes an

Anderson-Morel-type momentum structure, and encodes

the spin-symmetry of the parent state [61]; we consider

three cases: (i) SO(4), (ii) triplet-favored SO(3), and (iii)

singlet-favored SO(3), with all states being invariant un-

der Cs

3z[76].

Having in mind θTMD '30◦, we focus on λR6= 0 and

λI= 0 here. When D0is turned on, the point symme-

try is reduced to Cs

6, generated by six-fold out-of-plane

rotations. It lacks Iand spin-rotation symmetry such

that singlet and triplet can mix. More precisely, the

parent triplet and singlet phases we consider separate

into three distinct superconducting phases when D06= 0,

transforming under the irreducible representations (IRs)

A,B, and E2of Cs

6. Here the Astate is predominantly

spin-singlet, with admixed in-plane triplet; the Bstate is

pure out-of-plane triplet, since Cs

2zprohibits any admix-

ture of singlet or in-plane triplet components; and the

E2state’s two components are predominantly in-plane

triplet, with admixed singlet. At ﬁxed λR6= 0 we see

in Fig. 4(a) that switching on the displacement ﬁeld D0

generates a splitting of these three distinct IRs. Hence,

physically, the application of D0allows to change the

nature of the superconducting state.

Most strikingly, for the triplet favored parent state, D0

drives a phase transition from E2pairing to an Astate,

as signalled by the crossing of the respective eigenval-

ues in Fig. 4(a)(ii). This can be more clearly seen in

Fig. 4(b)(ii), where we show the changes of the transi-

tion temperatures Tccorresponding to the eigenvalues of

these two states in the vicinity of the transition; both

decrease with D0but Tcdecreases faster for the predom-

inantly triplet state. As expected, the critical value of

D0of this transition decreases with increasing λR, see

Fig. 4(b)(i).

FIG. 4: Properties of the superconducting order. (a)

Superconducting eigenvalues vs D0[normalized to unity at

D0= 0] and at ﬁxed λR= 30 meV. The distinct IRs

(A, B, E2)of Cs

6are indicated. The three panels consider

diﬀerent spin-symmetries of the parent state: (ai) SO(4),

(aii) triplet-favored SO(3) and (aiii) singlet-favored SO(3).

(bi) Phase diagram: colored points mark the dominant IR,

with in-plane triplet E2(singlet A) state in blue (red). (bii)

Tcvs D0at λR= 5 meV. (c) Order parameter of the leading

Astate, decomposed into components dµ,k; the µ= 3

component is zero due to Cs

2z(not shown). Here θ= 1.75◦

and {λR, D0}={20,20}meV [as per Fig. 2(b,c,d)]. The d0,k

component transforms trivially under Cs

3z, while the

admixed in-plane triplet d= (dx, dy)follows a Möbius

winding, with dkkgk.

Finally, we turn to the spin and spatial structure of

the superconducting state, encoded in dµ,k. Figure 4(c)

presents the leading state, A. Crucially, we see that the

admixed in-plane triplet vector dk= (d1,k, d2,k)behaves

as dkkgkand has a sign change between the Fermi sur-

faces; the superconducting order parameter dkthereby

inherits the Möbius winding around the Fermi surface.

Discussion and conclusion.—We showed that the tun-

able SOC of tTLG with TMD layers on both sides allows

to stabilize Möbius-like Fermi surfaces, which result from

vortices in the spin texture and are thus of a topological

origin. We subsequently proposed quantum oscillations

as a smoking gun probe. Turning to superconductiv-

ity, we showed that the parent superconducting state of

tTLG is readily manipulated in our setup. As only a

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摘要：

TunablesuperconductivityandMöbiusFermisurfacesinaninversion-symmetrictwistedvanderWaalsheterostructureHarleyD.Scammell1andMathiasS.Scheurer21SchoolofPhysics,theUniversityofNewSouthWales,Sydney,NSW,2052,Australia2InstituteforTheoreticalPhysics,UniversityofInnsbruck,InnsbruckA-6020,AustriaWestudytheor...

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