LayerMaterialsintheFlatland:TwistedGeometryandtheStrainEffects

ShiangFang,StephenCarr,MiguelA.Cazalilla,EfthimiosKaxirasHarvardUniversity

Supported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319, and by ARO MURI Award No. W911NF-14-0247.

May 19, 2017, Mathematical Modeling of 2D Materials Institute for Mathematics and its Applications, Minneapolis

Two-DimensionalPhysics

A.K.Geimetal.,Nature499,419(2013)

VanderWaalsheterostructures

• Superconductivity• CDW• Topologicalphases• Magnetism• Spintronics• Valleytronics• Optoelectronics• Straintronics

2DLayeredMaterials

Semimetal

Graphene hBN TMDC

Insulator Semiconductor

a=2.46A a=2.50A a=3.18A(MoS2)

• Rela^vis^clinearDiracdispersionatKvalleys

• Inversionsymmetry• Mechanicalstrength

• MX2,M=Mo/W,X=S/Se• Brokeninversionsymmetry

• Directbandgap1-2eVatKvalleys

• Spin-orbitcoupling

• Brokeninversionsymmetry

• Usedtoencapsulategraphene

• Stability• Largebandgap

WannierTransformation

DFTWannier90:AToolforObtainingMaximally-LocalisedWannierFunc^ons,A.A.Mostofi,J.R.Yates,Y.-S.Lee,I.Souza,D.VanderbiltandN.Marzari,Comput.Phys.Commun.178,685(2008)

EF

DFT/TBH

EF

Blochwave/energyandhamiltonian

TMDCInterlayerCouplingEmpiricalinterlayerpoten:al:

6

2 1 43

5

61

23 4 5

Empiricalfunc^onalform: Vpp,b(r) = ⌫b exp(�(r/Rb)�b)

perspec^veviewofTMDCinterface:

layer1

layer2

Ref:ShiangFangetal.,Phys.Rev.B92,205108(2015).

GrapheneInterlayercoupling

Ref:ShiangFang,EghimiosKaxiras,Phys.Rev.B93,235153(2016)

GrapheneWannierpz

TwistronicswithBilayerGraphene

StephenCarretal,Phys.Rev.B95,075420(2017)

Densityofstatesversustwistangle

VanHovesingularitiesinthedensityofstates

RenormalizationofFermivelocity

ExperimentatJarillo-Herrerogroup

Si

Graphene

Si

Si

Si

h-BN

Polymer

Si

……

Pick up graphene and bottom h-BN

Bottom gate

.

!

sampleprepara^on “tear-and-stack”enablessubdegreecontrolofthetwistangle!

FullyhBNencapsulateddual-gatetwistedbilayergraphenedevice(cross-sec^on)

BG TG 4

3

2

1

6

5

5µm

"↓$%&' =50~200µV

~Vxx

Vxy

A

Ref:YCaoetal,Phys.Rev.Lek.117,116804(2016).

ExperimentatJarillo-HerrerogroupB=0

temperaturedependence

TransportinTwistedBilayerGraphene

Ref: Y Cao et al, Phys. Rev. Lett. 117, 116804 (2016).

Super-lalceinducedinsula^ngstateswithsmalltwistangleforbilayergraphene

Insula^nggapfromsinglepar^clebandstructure

ElectronicStructurefor(19,18)TwBLG

Constantenergycontour

HallPlateausandLandaulevels

Energycontour

• 2inequivalentKpoints• 2setsoforbitalseach• 2spinstates• 8degeneracies

K

jump by 8!

HallPlateausandLandaulevels

Energycontour

• 1Gammapoint• 2setsoforbitals• 2spinstates• 4degeneracies

jump by 4!

StrainedandRippledLayers

Ubiquitousripplesingraphene!Effects:• Fermivelocity• Workfunc^on• Pseudogaugefield• Scakeringandmobility• Topologicaldefects

MelinaK.Bleesetal.,Nature524,204(2015)

Graphenekirigamistretchablegraphenetransistors

CurvedSpaceandEmergentGeometry

Ubiquitous ripples in graphene!

Effec^veLow-Energytheory:Diracequa^onincurvedspacewithEmergentgeometry,metricandconnec^ons

��µ, �⌫

= 2gµ⌫

i�µ(@µ � �µ) = 0

Curvedspace-^meingeneralrela^vity(NASA)

Ref:AlbertoCor^jo,MariaA.H.Vozmediano,Europhys.Lek.77:47002(2007)BoYang,Phys.Rev.B91,241403(R)(2015)

Strain-inducedPseudoMagneticFieldinGraphene

N.Levyetal.,Science329,544(2010)

Themagne^cfieldiseffec^velygreaterthan300T.

Reductionism:MicroscopicModels

Deformedgrapheneunitcell

compress

t~r

⇡ t0~r

+ ↵~r

(uxx

+ uyy

) + �~r

[!~r

y

(uxx

� uyy

) + 2!~r

x

uxy

]

Tight-bindingmodelforstrainedgraphene/hBN

Manuscriptinprepara^on.beyondcentralforcetwo-centerapproxima^on

TMDCHamiltonianwithUniformStrain

Manuscriptinprepara^on.

Transi^onMetalDichalcogenides(TMDC):MX2(M=Mo,W,X=S,Se).

TMDCcrystalstructure

Xisallowedtorelax:generalizedCauchy-Bornrule

Orbitalsaregroupedandclassified(x,y,orz-like)

Hamiltonianfor1st/3rdneighborcoupling

DFT/TBHComparison

Manuscriptinprepara^on.

Strain-dependent gap

shig~-100meV/%

Low-EnergyHamiltonians

Symmetryirreduciblerepresenta^on

uxx

+ uyy

1dim.

2dim.

Otherobjects

(kx

, ky

) (�̂x

, �̂y

)

Ref:JuanL.Manesetal.,Phys.Rev.B88,155405(2013)

(uxx

� uyy

,�2uxy

)

SymmetryinvariantinkpHamiltonian

C3virreduciblerepresenta^on

StrainPhysics/Applications

Ref:RodrickKuateDefoetal,Phys.Rev.B94,155310(2016)

• Bandstructureengineering

• Interplaybetweenspin/valley/orbital

• Pseudomagne^cfield(300Tingraphene)

• Dynamicalperturba^on(phonons);Floquet

physics

• Probe/controlknobformany-bodycorrelated

statesandquantumphasetransi^on

(anisotropywithcompositefermions)

Summary

Supported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319, and by ARO MURI Award No. W911NF-14-0247.

• Wanniertransforma^onisusedtoderivetheabini^o^ghtbindingHamiltoniansforlayers(PRB92,205108(2015)).

• Forbilayergraphene,theelectronicproper^esdependsensi^velyonthetwistangle(PRB93,235153(2016);PRB95,075420(2017)).

• Insula^ngstatescanbeinducedatsmalltwistangle(PRL117,116804(2016)).• Straincanbeusedtoengineerdesireproper^es(PRB94,155310(2016))• Thecodewillbeavailableonline:hkps://sites.google.com/view/shiangfang

Acknowledgements

Grants: • STC CIQM, NSF Grant No. DMR-1231319• ARO MURI Award No. W911NF-14-0247

EghimiosKaxiras BertrandHalperin

StephenCarr

PhilipKim

YuanCaoJasonLuo

PabloJarillo-Herrero

AppliedMath• MitchellLuskin• PaulCazeaux• DanielMassak

VallaFatemi