Flat bands for electrons in rhombohedral graphene multilayers with a twin boundary Aitor Garcia-Ruiz12 Sergey Slizovskiy12 Vladimir I. Falko1 2 3 1National Graphene Institute University of Manchester Booth Street East Manchester M13 9PL UK

2025-05-06 0 0 2.62MB 9 页 10玖币

侵权投诉

Flat bands for electrons in rhombohedral graphene multilayers with a twin boundary

Aitor Garcia-Ruiz1,2, Sergey Slizovskiy1,2, Vladimir I. Fal’ko1, 2, 3

1National Graphene Institute, University of Manchester, Booth Street East, Manchester M13 9PL, UK

2Department of Physics and Astronomy, University of Manchester, Oxford Road,Manchester, M13 9PL, UK

3Henry Royce Institute for Advanced Materials, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

Topologically protected ﬂat surface bands make thin ﬁlms of rhombohedral graphite an appealing

platform for searching for strongly correlated states of 2D electrons. In this work, we study rhom-

bohedral graphite with a twin boundary stacking fault and analyse the semimetallic and topological

properties of low-energy bands localised at the surfaces and at the twinned interface. We derive an

eﬀective 4-band low energy model, where we implement the full set of Slonczewski-Weiss-McClure

(SWMcC) parameters, and ﬁnd the conditions for the bands to be localised at the twin boundary,

protected from the environment-induced disorder. This protection together with a high density of

states at the charge neutrality point, in some cases – due to a Lifshitz transition, makes this system

a promising candidate for hosting strongly-correlated eﬀects.

I. INTRODUCTION

In the recent years, multilayer graphenes were found

to host various correlated phases of matter driven by

electron-electron interactions: superconductivity [1–5],

ferromagnetism [6, 7], nematic state [8], and Mott in-

sulator [9–11]. The electron correlation eﬀects in these

systems are promoted by the characteristically ﬂat low-

energy bands [12–17]. Among all these systems, few-layer

rhombohedral (ABC) graphenes are the only ones which

can be grown using chemical vapour deposition [18] with-

out the need to assemble twistronic structures with a high

precision of crystallographic alignment. The low-energy

bands in ABC ﬁlms are set by topologically protected

surface states, hence, it is aﬀected by external environ-

ment. As a result, their dispersion depends both on the

number of layers in the ﬁlm, encapsulation and verti-

cal electric bias, so that the ABC graphenes may behave

both as compensated semimetals and gapful semiconduc-

tors [14].

A rhombohedral graphitic ﬁlm with one stacking fault

such as as twin boundary, Fig. 1, also host low-energy

ﬂat bands [19, 20]: four rather than two speciﬁc for ABC

graphene. The additional two bands come from the twin

boundary inside the ﬁlm, hence, they can be protected

from the environmental inﬂuences due to screening by the

surface states. Here, we study the low-energy spectra of

thin ﬁlms of twinned ABC graphenes with N=m+n+3

layers such as ’mABAn’ multilayer sketched in Fig. 1,

where the twin boundary appears as a Bernal (ABA) tri-

layer buried inside the ﬁlm with nand mrhombohedrally

stacked (ABC and CBA) layers above and underneath it.

In Fig. 1 we also present four low-energy bands in a 9-

layer ﬁlm (3ABA3) with a twin boundary at the middle

layer, which illustrates that such systems are semimet-

als and that - in some of these systems - there might

be at least one low energy band located at the twinned

interface. Moreover, we notice that a neutral (undoped)

3ABA3 multilayer has an additional feature: the electron

Fermi energy in it is close to the Lifshitz transition [21–

23], marked by the van Hove singularity in the density of

FIG. 1: Left: Sketch of multilayer rhombohedral graphene

(mABAn) with one twin boundary (ABA ’trilayer’). The

low-energy basis is highlighted in red/green for the relevant

twin boundary sites/surface layers. Middle: Low-energy band

structure of a 3ABA3 ﬁlm across energy window ±30 meV,

with the colour coding of bands according to the dominant lo-

cation of their wave function. Right: Density of states (DoS)

of electrons in a 3ABA3 multilayer with the Fermi level in an

undoped structure coinciding with the van Hove singularity.

states, Fig. 1.

The presented-below analysis of band structure of

twinned multilayers of ABC graphene is based on the

hybrid k·p- tight binding theory which accounts for

the full set of Sloczweski-Weiss-McClure (SWMcC) pa-

rameters for graphite [24–26], in section II. Taking all

SWMcC parameters into account appear to be impor-

tant, as (similarly to what has been found in monolithic

ABC ﬁlms [14]) the next-neighbour/layer hoppings and

coordination-dependent on-carbon potentials lift an ar-

tiﬁcial degeneracy of band edges predicted by the mini-

mal model accounting for only closest neighbour hopping

[20, 27]. In Sec. III, we develop and test an eﬀective 4-

band model for rhombohedral structures with one twin

boundary, which improves the low-energy Hamiltonian

derived in Ref. [20], and use it to study the Berry cur-

vature and the magnetic moment of the bands, Sec. IV.

arXiv:2210.07610v3 [cond-mat.mes-hall] 25 Nov 2022

This eﬀective Hamiltonian could provide an analytical

tool for further studies of correlation eﬀects.

II. SWMCC MODEL FOR MULTILAYERS

WITH VARIOUS STACKINGS

In the basis of sublattice amplitudes for electron states

in a mABAn N-layer ﬁlms (N=n+m+ 3), Ψ†=

ψA1, ψB1,··· , ψAN, ψBN†, the Hamiltonian, which

will be used to describe the subbands in it, is written as







gV W ··· 0··· 0 0 0

V†Hb

gV··· 0··· 0 0 0

W†V†Hb

g··· 0··· 0 0 0

.....

0 0 0 ··· HABA ··· 0 0 0

.··· .

.....

0 0 0 ··· 0··· Hb

gV†W†

0 0 0 ··· 0··· V Hb

gV†

0 0 0 ··· 0··· W V Hs







HABA =



gV˜

V†Hb

g−σz∆0V†

W†V Hb



,(1)

g=Hg+∆00

0 0+ ∆sˆ

12, Hb

g=Hg+ ∆0ˆ

12,

Hg=v0π∗

πξ0, V =−v4πξγ1

−v3π∗

ξ−v4πξ,

W=0 0

γ2/2 0˜

W=γ5/2 0

0γ2/2.

Here, ˆ

12is a 2 ×2 unit matrix, πξ≡ξpx+ipy, with

p= (px, py) being the valley momentum measured from

~Kξ=~ξ4π

3a(1,0). Hgand Hb/s

gare Hamiltonians of

free-standing graphene and graphene inside/ at the sur-

face of the structure, respectively. Matrices Vand Wde-

scribe the nearest and next-nearest layer couplings, and

they are assumed to be independent of the distance to

the surface layers. Below we use the following values of

parameters implemented in Eq.(1): v= 1.02 ·106m/s,

v3= 0.102·106m/s, v4= 0.022·106m/s, γ1= 390 meV,

∆0= 25 meV, γ2=−17 meV, γ5= 38 meV [12, 28]. In

addition, we account for energy shift, ∆s, of the surface

orbitals which captures the inﬂuence of the encapsulation

and other environmental conditions.

In the Hamiltonian for the band energies around the

centre of K±valleys, parameter γ1sets the largest en-

ergy scale. As a consequence, in rhombohedral graphite

with a twin boundary, we observe a clear spectral sep-

aration of the bands in the dispersion, where a set of

m+n+ 1 conduction (valence) bands are split by ±γ1

from the two isolated pairs of conduction and valence

bands with dispersions illustrated in Fig. 2 for several

exemplary multilayers. The bulk (split) band edges ap-

pear at [11, 14, 29] ±πγ1

max(m,n)≈ ± 1 eV

max(m,n)near the

Fermi level at |p| ≈ pc≡γ1/v ≈0.058 ˚

A−1~, as in Fig.3.

For m, n .50 the separation of bulk states from inter-

face and surface states exceeds the low energy dispersion,

c/(2me)≈40 meV. In this case, the bulk remains in-

sulating, and the eﬀective theory for low-energy bands,

presented below, applies in full.

In the formal ‘bulk limit’, m, n 50, where the bulk

modes cross the Fermi level, the surface and the twin

boundary states remain the same for |p|< pc, with

similar parabolic dispersions, Esurface 1,2 =p2

2me+ ∆s,

Eψa= ∆0−γ5

2+p2

2me,EψB,m+2 =p2

2m0

e(see Fig.3), where

ψa= (ψAm+3 −ψAm+1 )/√2, me=4vv4

γ1+2v2∆0

γ2

1−1,

and m0

e=4vv4

γ1+v2(∆0+γ5/2)

γ2

1−1.However, close to

momentum pc, the interfacial states blend into the bulk

spectrum. Also, Fermi level may shift due to a charge

redistribution between the bulk and the interface states.

We do not consider this case in detail since in the re-

cent experimental studies it is hard to ﬁnd rhombohe-

dral crystals with >50 ABC layers and without stacking

faults [30].

III. EFFECTIVE 4-BAND MODEL FOR

TWINNED RHOMBOHEDRAL FILMS

To study the low-energy dispersion, we employ degen-

erate perturbation theory [31], which allows us to con-

struct an eﬀective 4×4 Hamiltonian out of the low-energy

basis, highlighted in red in Fig. 1. This basis consists

of three sublattice amplitudes of the non-dimer orbitals,

ψB1, ψAm+2 and ψBN, and the antisymmetric combina-

tion of sublattice amplitude of orbitals dimerised with the

layer at the twin boundary, ψa= (ψAm+3 −ψAm+1 )/√2.

In turn, the high-energy basis encompasses the rest of

pzorbitals the dimer sites and a symmetric orbital ψs=

(ψAm+3 +ψAm+1 )/√2. The matrix elements of the low-

energy eﬀective Hamiltonian can be determined from the

degenerate perturbation theory [31] around the valley

~Kξpoint,

hψi|Heﬀ |ψji=hψi|HgnQ[−H−1

⊥,X]QHgon−1|ψji,(2)

where H⊥,X(X = R or T) is the high energy Hamiltonian

acting on the high-energy basis of B-A dimer bonds inside

the two rhombohedral stacks,

H−1

⊥,R≈−∆0/γ2

11/γ1

1/γ1−∆0/γ2

1,(3)

or between the high-energy basis around the twin-

boundary, ψBm+2 , ψs,

H−1

⊥,T≈−(2∆0+γ5)/(4γ2

1) 1/(√2γ1)

1/(√2γ1)−∆0/γ2

1,(4)

in the high-energy basis adjacent to the twin-boundary.

In Eq. (2), Qis a projector onto the subspace spanned

by the low-energy basis.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

FlatbandsforelectronsinrhombohedralgraphenemultilayerswithatwinboundaryAitorGarcia-Ruiz1;2,SergeySlizovskiy1;2,VladimirI.Fal'ko1,2,31NationalGrapheneInstitute,UniversityofManchester,BoothStreetEast,ManchesterM139PL,UK2DepartmentofPhysicsandAstronomy,UniversityofManchester,OxfordRoad,Manchester,M139P...

展开>> 收起<<

Flat bands for electrons in rhombohedral graphene multilayers with a twin boundary Aitor Garcia-Ruiz12 Sergey Slizovskiy12 Vladimir I. Falko1 2 3 1National Graphene Institute University of Manchester Booth Street East Manchester M13 9PL UK.pdf

共9页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Flat bands for electrons in rhombohedral graphene multilayers with a twin boundary Aitor Garcia-Ruiz12 Sergey Slizovskiy12 Vladimir I. Falko1 2 3 1National Graphene Institute University of Manchester Booth Street East Manchester M13 9PL UK

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: