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OPENTrigonalquasicrystallinestates
◦
in30rotateddoublemoiré
superlattices
,2&PilkyungMoon1,2,3*
Westudythelatticeconfigurationandelectronicstructureofadoublemoirésuperlattice,whichis
composedofagraphenelayerencapsulatedbytwootherlayersinawaysuchthatthetwohexagonal
moiré
between0and4suchconfigurationsdependingonthelatticemismatchbetweengrapheneandthe
,whichisdistinctfromtheconventional
2-,3-,4-wavemixingofmoirésuperlattices,thatbringstogetherandhybridizestwelvedegenerate
,theirwavefunctions
exhibittrigonalquasicrystallineorder,whichlacksinversionsymmetry,attheenergiesmuchcloserto
thechargeneutralitypointofgraphene.
Whentwoormoretwo-dimensionalatomiclayerswhichdonotshareacommonperiodicityareoverlaid,an
additionalperiodicityintheformofmoiré
systems—forexampletwistedbilayergraphene2–6,grapheneonhexagonalboronnitride(hBN)7–13,andtwisted
bilayertransitionmetaldichalcogenides14,15withsmalltwistanglesθ≈1◦—havebeeninvestigatedextensively.
ThesematerialshaveverylongmoirésuperlatticevectorsLiM(i=1,2),and,hence,exhibitmanyexoticproper-
tiessuchastheFermivelocityrenormalization2,3,miniDiracpointsformation11,13,Hofstadter’sbutterfly5,7–9,16,
theemergenceofsuperconductivity17,correlatedphases18,andorbitalmagneticmoment19.
Aspecialcaseoccurswhentwohexagonallatticesareoverlappedatθ=30◦(Fig. 1a).Inthisinstancethe
atomicarrangementismappedontoaquasicrystallinelattice,whichisorderedbutnotperiodic,witha12-fold
rotationalsymmetry20–,quasicrystallinetwistedbilayergrapheneexhibits
,italsohosts
uniqueelectronicstateswhichsatisfythe12-foldrotationalsymmetry22,27,
fromtheresonantinteractionbetweenthestatesatspecificwavevectorsviatherotationalsymmetryofthe
quasicrystalaswellasthetranslationalsymmetryoftheconstituentatomiclayers22,-
gonsinFig. 
wavevectorsoftheconstituentmonolayerstatesandtheinterlayerinteractionwhichformthequasicrystalline
ofsitesinacharacteristic12-foldrotationallysymmetricpattern(Fig. 1c).Thesestates,however,appearatthe
energies(about±)-
statesalsoariseinanybilayerstackedinaquasicrystallineconfigurationifallthedominantinterlayerinterac-
,eventransition
metaldichalcogenides(TMDC)orsquarelatticescanshowthequasicrystallinestates.
Recently,rapidprogresshasbeenmadeinstackingmorethantwoincommensurateatomiclayersandanum-
berofstudieshaveinvestigatedtheeffectsofmultiplemoirésuperlatticepotentialsontheelectronicstructure.
Themostnotableexampleamongthemisadoublemoirésystem,whichiscomposedofagraphenelayerencap-
sulatedbyhBNlayers(BN/G/BN)29,
moirésuperlatticepotentialwithasuperlatticeperiodLiM(i=1,2)thatcanbeaslongas14nm(Fig. 2a).Sucha
longperiod[whichresultsinshortsuperlatticereciprocallatticevectors,Fig. 2b]carvesthegrapheneelectronic
structuresintosuperlatticebandswithanenergyscalemuchsmallerthanthatofpristinegraphene(Fig. 2c)13.
Recently,LeconteandJungshowthatBN/G/BNatspecificconfigurationscanhosttwohexagonalmoirépat-
ternsoverlaidatatwistangleof30◦,andclaimedthatthesystemhostsquasicrystallineelectronicstructure31.
1NewYorkUniversityShanghai,ArtsandSciences,Shanghai200122,-ECNUInstituteofPhysics
atNYUShanghai,Shanghai200062,,NewYorkUniversity,NewYork10003,
USA.*email:pilkyung.******@
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(a)(b)a2*(c)
~a2*
3
702
251
910~0
a1*y[nm]
K
43−1
~
K−2
yky118a1*
61−3
xkx−3−12−1023
x[nm]
Figure 1.(a)Latticestructuresofquasicrystallinetwistedbilayergraphene21,22,
representtheunitcellsofeachlayer.(b)ThewavevectorsofthetwelvemonolayerstatesCn(n=0,1,...,11)
,andtheredandbluearrows[a∗
∗i
anda˜i(i=1,2)]
symmetry,thesetwelvestatesarealldegenerateinenergy,andthedashedlinesshowthatthesetwelvestates
Ŵpoint.(c)
tothesquaredwaveamplitude,andredandbluecirclesrepresentthestatesintheupperandthelowerlayers,
respectively.
(a)(b)(c)
G2
K
LMkya*1
kxY
X
0XYKXY
XE[eV]YX
LM1M1K
Y−
GM
3
Y
a2X
BMa*1−
Aa1G1
Figure 2.(a)LatticestructureofgrapheneonhBN8,
ofgrapheneandhBN,respectively,
LM(i=1,2)representtheunitcellsandsuperlatticevectorsofthemoirésuperlattice,,
i
thelatticeconstantofhBNisdrawn15%largerthanthatofgraphenetoenhancethevisibilityofthemoiré
pattern(%).Insetshowsthelatticeconfigurationofgraphene;theblackand
whitecirclesrepresenttheAandBsublattices,aiandτXshowtheprimitivelatticevectorsandthecoordinates
ofthesublattices,respectively.(b)SuperlatticeBrillouinzone(bluehexagon)neartheDiracpoint(theregion
surroundedbybluelinesintheinset)
iM
theBrillouinzonecornerswhereminiDiracpointappear,andφshowstherelativeorientationofG1tothe
reciprocallatticevectora∗,wheretheblack
1′
andwhitecirclesrepresentthethreeequivalentDiracpoints,KandK,respectively.(c)Thebanddispersionof
thefirsttwobandsintheconductionandvalencebandsofgrapheneonhBNwithθ=0◦,whichshowtheband
openingattheprimaryandtheminiDiracpoints.
However,theinteractionmechanismresponsibleforsuchuniqueelectronicstatesinBN/G/BN,aswellasthe
actualelectronicbandstructure,andwhetherthewavefunctionsactuallysatisfythesymmetryofthequasicrystal
havenotyetbeeninvestigated.
Here,weinvestigatetheconditionswherethetwohexagonalmoirépatternsindoublemoirésuperlatticeare
arrangedinadodecagonal(12-fold)
thatbringtogetherandhybridizetwelvedegenerateBlochstatesofmonolayergrapheneandshowthatsuch
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statesofquasicrystallinetwistedbilayergraphenewheretheactualatomiclatticesarearrangedinadodecagonal
configuration22,27,28,theresonantstatesofBN/G/BNappearattheenergiesmuchclosertothechargeneutrality
,theirwavefunctionsshowthequasicrystallineorderwithathree-foldrotational
symmetryratherthanfullysatisfyingthe12-foldrotationalsymmetryofthedoublemoirépattern.
Methods
Hexagonalmoirésuperlatticesstackedat30◦.Weconsideratrilayersystemcomposedofgraphene
-dimensionalhoneycomblatticewhoseunitcellcomprises
oftwo(AandB),whilehBNhasnitrogenatomon
,a˜≈,isslightlylargerthanthatof
graphene,a≈,andweuseaconstantinterlayerdistanceofd=
,wedonotconsiderthelatticerelaxationbetweengrapheneandhBN34,35,sincetheeffectsofsuch
,oureffectivetheorythatrespects
thelatticesymmetryisabletoproperlydescribeboththegapattheprimaryDiracpointandtheasymmetricgap
openingatthetwoinequivalentminiDiracpoints13,aswellastheorbitalmagnetismofthestructure19.
Wedefinetheatomicstructureofthedoublemoirésuperlatticesbystartingfromanonrotatedarrangement,
wherethehexagoncenterofthethreelayerssharethesamein-planeposition√(x,y)=(0,0),andtheA-Bbonds
=a(1,0)anda2=a(1/2,3/2)(a=)astheprimitivelattice
vectorsofgraphene,andτA=−τ1andτB=τ1[τ1=−(1/3)(a1−2a2)]asthecoordinatesoftheAandBsub-
((l)l=t)andthebottom(l=b)hBNlayersbecome
a˜i=(lM)ai(i=(l)1,2),whereM(l)=(1+(l)ε)Irepresentstheisotropicexpansionbythefactor(l)(l)(l)1+ε=˜a/a≈,
andτN=−τ1±dezandτB=τ1±dez[τ1=−(1/3)(a˜1−2a˜2)],wheretheupperandlowersignsare
forthetopandbottomlayers,respectively,representthecoordinatesofthenitrogenandboronatomsintheunit
cell(NotethatwedefinedthesublatticecoordinatesτX(X=A,B)andτX˜(l)(X˜=N,B,l=t,b)differentlyfrom
thoseinourpreviouswork13,tomakethepointswiththehighestrotationalsymmetry,.,thehexagonalcenter,
,bothdefinitionskeeptheinteractionmatrices(Eq. 9)thesame.).Wedefine
thereciprocallatticevectorsa∗anda˜∗forgrapheneandhBN,respectively,soastosatisfyai·a∗=˜ai·˜a∗=2πδij.
iij(t)j(b)
WethenrotatethetopandbottomhBNlayerswithrespecttographenebyarbitraryanglesθandθaround
theorigin,,weuse“BN/G/BN”
thelattice,0≤θ(l)≤30◦(l∈t,b)spansalltheindependentconfigurations.
Figure 3ashowsthemoiréinterferencepatternswhicharisefromthelatticemismatchbetweenthetophBN
andgraphene(leftside),andalsothatfromthebottomhBNandgraphene(rightside),respectively,andFig. M,(l)3b
,(l)Liandthereciprocallatticeperiod
Gi(i=1,2)ofeachmoirésuperlatticeare
LM,(l)=cR(φ(l))a,
ii
M,(l)−1(l)∗(1)
Gi=cR(φ)ai,
(l)(l)(l)
respectively,wherec=(1+ε)/ε2+2(1+ε)(1−cosθ(l)),φ=arctan[−sinθ/(1+ε−cosθ)],and
R(φ)isarotationbyφ13,|LM|and|GM|againstθinFig. 3cinredandbluelines,respectively.
ii
Now,wewillfindtheconfigurationwheretheunitcellsofthetwohexagonalmoirésuperlatticeshavethe
samesizeandareoverlaidwitharelativetwistangleof30◦.Insuchaconfiguration,theoverlaidtwohex-
agonalsuperlatticesaremappedontoa12-foldrotationallysymmetricquasicrystallinelatticewithoutany
translationalsymmetry,. (1),theformerandthelatterconditionsgive
|θ(t)|=|θ(b)|andφ(t)−φ(b)≡30◦(mod60◦),whichcanbesimultaneouslysatisfiedbyθ(t)=−θ(b)and
φ(t)=−φ(b)≡15◦(mod30◦).Figure ,green,bluelinescor-
respondtoε>0,ε=0,ε<0,respectively,
moirésuperlatticesformadodecagonalquasicrystallineconfigurationatθwherethelineandthedashedhori-
,graphene
layer,.,ε=0,thenthetwohexagonalmoirépatternscannothavearelativetwistangleof30◦.Ontheother
hands,thesystemswithε<0,0<ε<,andε≥,four,andtwoθwhichsatisfythe
,.,ε≈,thefouranglesareθ1=◦,θ2=◦,
θ3=◦,θ4=◦,andthecorresponding|GM|,,,|a∗|.Notethatθ4
MiMi
givesverylong|Gi|,andaccordinglyveryshort|Li|,whichcompeteswiththelengthscaleofmonolayergra-
(t)=−θiandθ(b)=θi(i=1,2,3,4),wegetφ(t)=−φiandφ(b)=φi,wh