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