Patterned bilayer graphene as a tunable strongly correlated system Z. E. Krixand O. P. Sushkov School of Physics University of New South Wales Sydney 2052 and

2025-04-26 0 0 936.02KB 13 页 10玖币

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Patterned bilayer graphene as a tunable, strongly correlated system

Z. E. Krix∗and O. P. Sushkov

School of Physics, University of New South Wales, Sydney 2052 and

Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,

University of New South Wales, Sydney 2052, Australia

(Dated: October 13, 2022)

Recent observations of superconductivity in Moire graphene1have lead to an intense interest in

that system, with subsequent studies revealing a more complex phase diagram including correlated

insulators and ferromagnetic phases. Here we propose an alternate system, electrostatically pat-

terned bilayer graphene (PBG), in which a supermodulation is induced via metallic gates rather

than the moire eﬀect. We show that, by varying either the gap or the modulation strength, bilayer

graphene can be tuned into the strongly correlated regime. Further calculations show that this is not

possible in monolayer graphene. We present a general technique for addressing Coulomb screening

of the periodic potential and demonstrate that this system is experimentally feasible.

I. INTRODUCTION

Superconductivity in twisted bilayer graphene1occurs

at a twist angle which turns the lowest lying energy states

into a ﬂat band2. More generally, strongly correlated

phases due to ﬂat band physics arise in a broad range

of materials. Observations of superconductivity1,3–6,

correlated insulators3–5,7–9, ferromagnetism10,11 and ne-

matic order8,12,13 have been reported across the family

of twisted graphene systems. This includes twisted bi-

layer graphene1, twisted trilayer graphene6, and twisted

double bilayer graphene13. Flat bands also arise in

twisted TMDCs14 and kagome systems which exhibit

superconductivity, ferromagnetism, and charge density

waves15–20.

Given the high level of interest in strongly corre-

lated phases arising from ﬂat band systems, particularly

twisted bilayer graphene, the present work proposes an

alternative graphene-based system which is fully tunable

and contains a well-deﬁned, isolated ﬂat band. The sys-

tem we consider is a graphene bilayer with no twist an-

gle and a patterned electrostatic gate a vertical distance,

z, from the bilayer. For brevity we refer to this sys-

tem as patterned bilayer graphene (PBG). The guiding

idea is to restructure the bare energy bands of bilayer

graphene via periodic electrostatic gating rather than

with a twist-induced Moire superlattice. Conceptually,

this is a continuation of our previous work on semicon-

ductor artiﬁcial crystals21, which are less eﬃcient than

PBG at generating a strong modulation. A major ad-

vantage of this approach is that it bypasses the issue of

twist-angle disorder22 (i.e. long-range spatial variation

of the twist angle). The Moire ﬂat band occurs at a pre-

cise value of twist angle (θ≈1.1°) and a modest amount

of twist-disorder (<

∼10 %) can destroy this band23. We

demonstrate that PBG has no equivalent ﬁne-tuning or

disorder problem.

The central advantage of our system is its controlla-

bility. A designed superlattice potential induced by pat-

terned electrostatic gating can have any desired lattice

symmetry (e.g. square, triangular, honeycomb24, Lieb,

or kagome) and lattice constants as small as 40 nm25. It

is also possible to tune both the strength of the super-

modulation and the particle density independently25,26.

In contrast, Moire graphene superlattices have a trian-

gular symmetry which is ﬁxed by the crystal structure

of graphene. The superlattice constant, a≈13 nm, is

also ﬁxed by the ﬂat band condition, θ≈1.1°and tun-

ing the superlattice strength is only possible by applying

hydrostatic pressure4. Some prior works have focused

on patterning monolayer graphene, either by etching

holes directly into the graphene sheet27 or by patterned

electrostatic gating25,28. Ref.27 demonstrates, theoreti-

cally, that patterning introduces an energy gap in the

graphene dispersion while Refs.25,28 measure magneto-

transport properties of a real device and show that the

result of patterning is essentially a correction to single

particle physics. There is not, however, the possibility

for generating an isolated ﬂat band or strongly correlated

phases in these monolayer graphene systems.

Our results are derived from band structure calcu-

lations in a continuum, bilayer graphene model with

imposed superlattice potential. We ﬁnd that bilayer

graphene can be driven into the Mott regime by ap-

plication of a suﬀciently strong band gap and potential

modulation. This occurs because a ﬂat band develops

in the lowest-energy band of the PBG dispersion. By

detuning either the band gap or potential modulation

the system can be tuned out of the Mott regime while

keeping the total electron density ﬁxed. Within the ﬂat

band it is possible to mimic the dispersion of many dif-

ferent two-dimensional lattices including square, triangu-

lar, kagome, and Lieb, by varying the symmetry of the

patterned gate. We show, using an analogous calcula-

tion, that it is not possible to generate a ﬂat band in

monolayer graphene. Lastly, we study electron-electron

screening in bilayer graphene. Current techniques are not

able to address a system with both an unbounded dis-

persion and a strong potential modulation; we develop a

general technique to address Coulomb screening in this

limit. The technique we develop is general and could

also be applied to, for example, the problem of impurity

screening in bilayer graphene. We show that screening of

the periodic potential is strong but can be overcome by

arXiv:2210.05827v1 [cond-mat.str-el] 11 Oct 2022

FIG. 1: Sketch of the two Brillouin zones. The larger is that

of the underlying bilayer graphene system and the smaller is

that of the artiﬁcial crystal. Since we use an expansion about

the Kpoints of the bilayer graphene BZ, the artiﬁcial BZ is

centered at a Kpoint. The inset shows the artiﬁcial crystal

in real space.

Bilayer Graphene BZ

Artiﬁcial BZ

K' K

experimentally realistic gate voltages. Our results show

that patterned bilayer graphene is an experimentally vi-

able way to engineer an isolated ﬂat with almost complete

control over the underlying eﬀective Hubbard model.

II. THEORETICAL TECHNIQUES

Our starting point is a plain bilayer graphene sheet

with external, perpendicular electric ﬁeld, E, inducing an

energy gap, ∆. The relationship between ﬁeld and gap is

roughly |E| ∝ ∆, where the constant of proportionality

is such that a displacement ﬁeld of 1 V nm−1leads to a

gap ∆ = 100 meV.29 We ﬁnd that the value of the gap

is important but does not need to be ﬁnely tuned, we

discuss conditions on ∆ below. The low-energy eﬀective

Hamiltonian for a single valley of bilayer graphene is30

HBLG =





∆/2vp−0γ

vp+∆/2 0 0

0 0 −∆/2vp−

γ0vp+−∆/2





(1)

Where v≈1×106m s−1is the Fermi velocity of

monolayer graphene, γ≈0.38 eV is the coupling be-

tween graphene layers, and the operator p±is deﬁned

by p±=px±ipy. These values are taken from Ref.30.

The Hamiltonian is composed of 2 ×2 blocks. Each

diagonal block is the Hamiltonian of a single graphene

layer, and each layer has a diﬀerent energy shift ±∆/2

depending on its position in the external ﬁeld. The oﬀ-

diagonal blocks, which couple the two layers, arise from

the simplest kind of interlayer hopping, between two car-

bon atoms which are vertically aligned: γis the matrix

element for this hopping. One can also include terms in

the Hamiltonian which describe longer-range interlayer

hopping (these are denoted by γ3and γ4in Ref.30). As

discussed in Ref.30 they contribute to trigonal warping

and particle-antiparticle asymmetry of the band disper-

sion. These additional terms are secondary to the ma-

jor terms, vand γ, and, for the sake of physical trans-

parency, we neglect them here. In the limit |ε|  γ,

the two low-energy bands which arise from Eqn. 1 are

roughly quadratic: ε(p) = ±p(p2/2m∗)2+ ∆2/4, where

the eﬀective mass is m∗=γ/2v2≈0.03.

Over the top of this Hamiltonian we wish to introduce

a spatially modulated electrostatic potential, U(r), due

to the patterned gating. Suppose, ﬁrst, that the periodic

potential deﬁned at the gate is given by

Ugate(r) = WX

eiG·rUG

Where the vectors, G, are the reciprocal lattice vec-

tors of the artiﬁcial superlattice and Wis a parameter

taking dimensions of energy which controls the strength

of the superlattice. In our calculation, the dimensionless

parameters, UG, deﬁne a muﬃn-tin shaped, periodic po-

tential with square-lattice symmetry (see the inset to Fig.

1). In this case Wis equal to the total variation in poten-

tial energy from minimum to maximum, the “height” of

the muﬃn-tin. For concreteness we study W > 0, which

corresponds to an array of anti-dots. The opposite limit

W < 0 is very similar and we discuss this below. The po-

tential, U(r), at the plane of the bilayer graphene sheet

is then

U(r) = WX

eiG·re−GzUG(2)

Where zis the vertical distance between the pat-

terned gate and the bilayer. This exponential suppression

of the higher harmonics follows directly from Poisson’s

equation. For a square lattice, the ﬁrst harmonic has

G=g= 2π/a, where ais the superlattice period. We

choose parameters a= 80 nm and z= 10 nm which are

reasonable from an experimental standpoint. The supres-

sion of the second harmonic relative to the ﬁrst is then

e−gz ∼0.5 meaning that higher harmonics should not

be neglected. For a smaller lattice constant, a= 40 nm,

e−gz ∼0.2 and so higher harmonics are slightly less sig-

niﬁcant. Given this periodic potential the total Hamilto-

nian becomes

HP BG =HBLG +U(r)

The matrix structure of U(r) is trivial, it is a 4 ×4

identity matrix. Strictly speaking there should also be a

spatially varying correction to the gap, ∆, due to each

graphene layer being a slightly diﬀerent distance from

the patterned gate. This correction, however, produces

a small eﬀect relative to the major contribution, U(r),

and can be neglected. Note that the applied, constant

ﬁeld (which induces ∆) will give a similar, diagonal con-

tribution to the Hamiltonian. Since that contribution is

spatially invariant, it simply shifts all energy levels by the

same amount, and can be left out of the Hamiltonian.

Since the artiﬁcial superlattice period, a, is larger than

the period of the graphene lattice and since we are focus-

ing on a single valley, our approach amounts to deﬁning a

smaller, artiﬁcial Brillouin zone (either square or hexag-

onal) around one of the vertices of the original bilayer

graphene Brillouin zone. This is sketched in ﬁgure 1.

The energy levels of the Hamiltonian, HP BG, can be

obtained exactly by numerical diagonalisation. We must

ﬁrst compute the matrix elements of HP BG in a par-

ticular basis. For simplicity, we choose basis vectors,

FIG. 2: Evolution of dispersion from W= 0 meV to W=

30 meV. In each panel a= 80 nm, ∆ = 15 meV, R= 15 nm

and z= 10 nm. At W= 0 meV we have the bare BLG

dispersion band-folded into the artiﬁcial Brillouin zone. By

W= 30 meV a single, ﬂat band (width ≈0.5 meV) has sep-

arated from the rest of the hole bands. Points around the

artiﬁcial Brillouin zone are deﬁned in the inset.

W = 0 meV W = 15 meV

W = 30 meV

M¡

Artificial

|k+G, ii, deﬁned by

|k+G, ii=1

√Aei(k+G)·r|ii

(|ii)j=δij , i, j = 1,··· ,4

Here |iiis a 4-tuple, Gis a reciprocal lattice vector of

the superlattice and kis the quasi-momentum, which sits

somewhere within the artiﬁcial Brillouin zone. The nor-

malisation factor, A, is just the total area of the sample.

For example, at i= 1, we have

|1i=











The matrix elements of HP BG are then

hk+G, i|HP BG|k+G0, ji=hi|HBLG(k+G)|jiδG,G0

+δi,j W UG−G0e−|G−G0|z

In HBLG(k) the operator, p±, is replaced by the com-

plex number, k±=kx±iky. The matrix elements of the

potential are given by,

UG= 2πR2

Acell

J1(|G|R)

|G|R

Where Ris the radius of a single anti-dot, taken to

be R= 15 nm in our calculations, and Acell is the unit

cell area of the artiﬁcial crystal; J1is a Bessel function

of the ﬁrst kind. The only diﬀerences between square an

triangular lattices are the choice of reciprocal lattice vec-

tors, G, and the size of Acell in the above equation. To

diagonalise this Hamiltonian numerically we must make

the set of basis vectors ﬁnite by choosing a maximal G

vector. In practice, we simply increase the size of the ba-

sis until the energy levels or eigenvectors converge. The

result of this procedure is a set of energy levels, εn(k),

and eigenvectors, ψn,k(r), each being a function of the

quasi-momentum, k, of the artiﬁcial superlattice. The

band index can take values n=±1,±2,··· with the

charge neutrality point occurring between n=−1 and

n= +1. Each band, n, is degenerate across valleys and

spins and contains a number of particles, n0= 4/Acell,

corresponding to complete ﬁlling of one artiﬁcial Bril-

louin zone (accounting for spin and valley degeneracy).

At a= 80 nm we have n0= 6.25 ×1010 cm−2, while at

a= 40 nm we have n0= 2.5×1011 cm−2.

The parameter Whas been deﬁned at the patterned

gate. This allows us to fairly compare the strength of a

square lattice with that of a kagome lattice (for example).

Still, it is useful to have a general idea of the amplitude of

the potential at the 2DEG. Looking at the ﬁrst harmonic

only, this quantity is

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Patternedbilayergrapheneasatunable,stronglycorrelatedsystemZ.E.KrixandO.P.SushkovSchoolofPhysics,UniversityofNewSouthWales,Sydney2052andAustralianResearchCouncilCentreofExcellenceinLow-EnergyElectronicsTechnologies,UniversityofNewSouthWales,Sydney2052,Australia(Dated:October13,2022)Recentobservatio...

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