Skyrme crystal in bilayer and multilayer graphene

0.38 MB·English

by  Yasuhisa Sakurai

Skyrme Crystalinbilayerand multilayergraphene
Yasuhisa Sakurai and Daijiro Yoshioka
Department of BasicScience, TheUniversityof Tokyo, 3-8-1Komaba, Tokyo 153-8902, Japan
(Dated:February1,2012)
The ground state of the two-dimensional electron systems in Bernal bilayer and ABC-stacked multilayer
graphenesinthepresenceofastrongmagneticfieldisinvestigatedwiththeHartree-Fockapproximation.Phase
diagramsofthesystemsareobtained,focusingonchargedensitywavestatesincludingstateswithvorticesof
valleypseudospins(calledaSkyrmecrystal).Thesingle-electronstatesinthesestackedgraphenesaregivenby
2
two-component wavefunctions. ThatofthefirstexcitedLandaulevelhasthesamecomponent asthelowest
1
Landauleveloftheordinarytwo-dimensionalelectrons. Becauseofthislocalizedwavefunctions,theSkyrme
0
crystalhaslowenergyinthisfirstexcitedleveluptofourlayersofgraphene,whentheinter-layerdistanceis
2
assumedtobeinfinitesimal.Atthesametime,bubblecrystalsaresuppressed,sothephasediagramisdifferent

n fromthatofasingle-layergraphene.
a
J
PACSnumbers:73.21.-b,73.22.Gk,73.22.Pr

1
3
I. INTRODUCTION paredwith thatofa separatedparticle-holeexcitationto find

] the condition where the Skyrme crystal has low energy. In
l
l Sec. IV, numericalresults are presented at the Landau level
a Over the past decades, the conventional two-dimensional
h electronsystem(2DES)insemiconductorheterostructuresin wheretheSkyrmecrystalisexpected. InSec. V,thevalidity
- a strong magnetic field has been studied. It is found that oftheresultsandarelationtoexperimentsarediscussed.
s
e electron-electroninteractionbringsvariousphases[e.g.,frac-
m tional quantum Hall effect (FQHE) states1,2 and charge den-
II. MODEL
. sitywave(CDW)states3].Similarly,graphene,4aflatsheetof
t
a carbonswithahoneycomblatticeexfoliatedfromagraphitein
m 2004,5isunderintensiveinvestigationasanew2DESwhose A. Single-particlestateinamagneticfield
- electron has a valley degree of freedom (K,K′) and a linear
d dispersion. Exhibitionof FQHE6,7 and CDW8,9 states in the Forasingle-layergraphene(SLG),thelow-energyeffective
n new2DEShasbeenexpected;FQHEwasrecentlyobservedin HamiltonianaroundtheK valleyisgivenby
o
asingle-layergraphene(SLG).10,11 Furthermore,few-stacked
c SLG =v (p σ + p σ ), (1)
[ graphenes have recently attracted attention due to their in- HK F x x y y
3 triguing properties: a band structure tunable by the number whichhasanelectronwithalineardispersion.4,14–19Here,vF
oflayersandtheirstackingsequence,andabandgapcontrol- isFermivelocity,andσ andσ arePaulimatricesactingon
v x y
lablebyaperpendicularelectricalfield.12,13 sublattice(A,B)space. TheHamiltonianforthe K valleyis
3 ′
56 graWpheefnoecsu,swohnichBheranvaelcbhiilraayleerleacntdroAnsBcCo-nsctaecnktreadtemdounltiolauyteerr HBeKSL,′Gus=ing−a(HmKSaLgGn)∗e.ticInlaaddpeerrpoepnedraictourlaar=ma(gπneticiπfie)lld/B√2=~
z x y B
4 layers in low-energy states. In this paper, the CDW ground (π= p eA),thesingle-particleHamiltonianis−writtenas
1. state of the 2DESin bilayer graphene(BLG) and multilayer −
1 graphene(M-LG, M ≥ 3) in a strongmagneticfield is stud- SLG = √2~vF 0 a , (2)
1 ied with the Hartree-Fock approximation. As candidates of HK lB a† 0!
1 the groundstate, the electron (hole)Wigner crystal, electron
v: (hole) bubble crystal, and Skyrme crystal are considered. A wherelB = √~/eBismagneticlength.
i Skyrmecrystalisthestatewithatopologicaltextureofvalley Theeigenenergiesare
X
pseudospins.Ithasthelowestenergyaroundafillingν=1in
ar SLG.9Ourmean-fieldanalysispredictsthatthevalleySkyrme EN =±vF√2e~BN, (N =0,1,...), (3)
crystalalso haslowenergyin BLG and M-LG (M = 2,3,4) whereapositive(negative)signistakenastheelectron(hole)
atthefirstexcitedLandaulevelwhentheinterlayerdistanced state. Thecorrespondingeigenstateshavetheform
isassumedtobeinfinitesimal.
setTuhpisaploapwe-reniserogrygaenffiezcetdivaesHfoalmloiwltos.niIannthinetnheexptrseescetniocne,owfae √12e−iK·rφNφ−1,X((rr)) (N ≥1),
FmoacgkneatpicprfioexlidmfaotrioSnLGis,aBpLpGlie,dantdoMth-eLsGy.stTehmeno,fthientHeraarctrtieneg- ΦSKL,NG,X(r)= e−iK·r ±0N,X  (N =0), (4)
eClDecWtrosntsataensdasreelfi-nctorondsuiscteedn,teinqculautdioinngsaSrkeydremrievcerdy.sItnalSse(cm.oIIrIe, fortheKvalley.IntheLandauφNg,aXu(gr)e,φ (r)isgivenby
N,X
properly,themeronandmeronpaircrystal).Asapreliminary
cciatlactuiolantiaotna,tνhe=en1efregryroomfaagsnkeytircmsitoante(aisnteivskayluramteiodna)npdaicroemx-- φN,X(r)= √π2N1N!lBL!1/2iNe−ilX2By−(x2−lX2B)2HN x−lBX!, (5)
2
where L is the length of the system, H (x) is the Hermite l 80 Å, so we consider the vanishing limit of interlayer
N B
polynomial, and X = k l2 is the guiding center coordinate. dist∼anceasanapproximatemodel.
y B
A macroscopic number of states with different X are degen- Apartfromaconstantkineticterm,theHamiltonianforin-
eratedatoneLandaulevel. Asingle-particledensityoneach teractingelectronsin M-LG(M 1)isgivenby
≥
sublattice2πl2φ φ becomesbroaderasindexNincreases.
B ∗N,X NX 1
Tthheepsrtoafitetstowfiothrmbrsokaydrmdeinosnisty(sieneduSceec.aIIbIu)b.bleCDWandlose Hint = 2Z d2rZ d2r′σ1,σX2,σ3,σ4ψˆσM1,†N(r)ψˆσM2,†N(r′) (9)
muBlteirlanyael-rsgtarcakpehdenbei2l0ay(Mer-gLrGap)hheanveel(oBwL-Gen)earngdyqAuBasCip-satratcickleeds ×V(|r−r′|)ψˆσM3,N(r′)ψˆσM4,N(r),
wlaiytehrsa.1d2i,1s3p,2e1r,2s2ioInnEtw∝o apdMja,cwenhterlaeyMers(,≥BL2G)iasntdheMn-uLmGbehravoef wherethefieldoperatorψˆσM,N(r)isrepresentedby
interlayer hoppings t between the sublattice β in the lower
ψˆσ (r)= ΦM-LG(r)cˆσ , (10)
layer and the sublatti⊥ce α in the upper layer. In a magnetic M,N σ,N,X N,X
XX
field, the effective single-particle Hamiltonian for BLG and
M-LG,whichactontheoutermostlayer(top,bottom)space, andCoulombinteractioniswrittenas
havethesameform
e2
V(r)= . (11)
0 aM ǫr
M-LG =~ω , (M 2), (6)
HK M (a†)M 0 ! ≥ Here, ǫ is the dielectricconstant, σ = representsvalley K
and K , respectively, cˆσ is an annihil±ation operator of the
where off-diagonal components represent M-times hopping ′ N,X
electronatvalleyσwithguidingcenterX.Avalleyscattering
viahigh-energystatesintheinnerlayers. Itseigenenergyis
termisrelativelysmallwhenthecutoffq Kisused,8,9,23
max
≪
since l a ( a = 1.42 Å is the lattice constant), so we
EN =±~ωM N(N−1)···(N−M+1), (7) canignBor≫eitasfarastheCDWwithmagneticlengthscaleis
p
concerned.Then,theHamiltonianinqspaceiswrittenas
where~ω = t (√2~v /t l )M BM/2. Atzeroenergy, M
M F B
Landaulevelsar⊥edegenera⊥ted. Th∝ecorrespondingeigenstate 1
= V(q)ρσ1( q)ρσ2(q), (12)
hastheform Hint 2L2 ξ1 − ξ2
Xq σX1,σ2ξX1,ξ2
φ (r)
ΦKM,-NL,GX(r)=√12e−ei−KiK·r·rφN±N,0Xφ−(NMr,)X,X(r) ((NN ≥< MM)),, (8) ρ=σξ(12q)XX e−iqxX−41q2l2BLNσ,ξ(cid:16)q2l2B/2(cid:17)cˆσN†,X+12qyl2BcˆσN,X−12qyl2B, (13)
forthe K valley. Noticethatinthefirstexcitedstaterealized
atN = M,theuppercomponentofthewavefunctionisgiven 2πe2
V(q)= , (14)
byφ0,X(r),whichistheground-statewavefunctionoftheor- ǫq
dinary2delectrons.
AtLandaulevel N < M, thevalleydegreeoffreedomco- where Ln(x) is the Laguerre polynomial, and Nσ,ξ is defined
incideswiththelayerdegreesoffreedom. Ifthelayerdegree by N+, = N , = N M and N+, = N , = N. When
of freedom in the M-LG model is regardedas the sublattice Landau↑-leveli−n↓dexNis−smallerthan↓thenum−b↑eroflayersM,
degreeoffreedom,Eq. (8)isformallyidenticaltothatofthe thevalleydegreeoffreedomcoincideswiththatofthelayers
SLGmodelwhenM =1. Inthefollowing,thecaseofM =1 (sublattices for SLG), and ρ+(r) = ρ−(r) = 0 and ρ+(r) and
(M =2)representsthemodelofSLG(BLG). ρ−(r)aretwicetheamounto↑fEq. (13↓). ↓
↑For the Hartree-Fock decoupling of , we assume the
int
H
followingorderparameters:
B. Hartree-FockHamiltonian
2πl2
Althoughtheelectronstateismainlyconsideredinthefol- ∆σN,σ′(Q)= L2B XX hcˆσN†,X+cˆσN′,X−ie−iQxX, (15)
lowing, theholestate canbe treatedinthe same way. Inthe
whereQisareciprocallatticevectoroftheCDWstate. Then
present case of strong magnetic fields, electronic spins are
theHartree-FockHamiltonianisgivenby24–27
completelypolarizedandthegaparoundtheneutralitypointis
sufficientlylargeduetoexchangeenhancement;thuswetake
onlyonespincomponentandignoretheeffectofLandau-level HHF = HMσ,,σN′(Q)e−iQxX∆σN,σ(−Q)cˆσN′,X†+cˆσN′,X
transitions. XQ XX Xσσ′ −
weTakheeniendtebrayctaionninbteetrwlaeyeenrtdhiestealneccetrodn(s∼in3d.3iff5eÅrenftolrayBeLrsGi)s. −XQ XX Xσσ′ XMσ,,σN′(Q)e−iQxX∆σN,σ′(−Q)cˆσN′,X†+cˆσN,X−,
WhenB=10T,disquitesmallcomparedtomagneticlength (16)
3
whereX = X Q l2/2. TheHartree-Fockpotentialconsists TheequationofmotionforGσ,σ′(Q,τ)isderivedas
ofadire±ctterm± y B N
~(iω µ)Gσ,σ′(Q,ω ) ~δ δ
Hσ,σ′(Q)= Hσξ,σ′ξ′(Q), (17) n− N n − Q,0 σ,σ′
M,N Xξ,ξ′ M,N =Xσ′′ XQ′ ΣσM,,σN′′(Q,Q′)GσN′′,σ′(Q′,ωn), (26)
Hσξ,σ′ξ′(Q) fromtheHeisenbergequation.Here,ωn istheMatsubarafre-
M,N quencyforfermions,andtheself-energyΣσ,σ′(Q,Q)isgiven
= 1 e2 1 L Q2l2/2 L Q2l2/2 e Q2l2/2, (18) by M,N ′
4ǫlB QlB Nσξ(cid:16) B (cid:17) Nσ′ξ′ (cid:16) B (cid:17) −
ΣσM,,σN′(Q,Q′)
andanexchangeterm
= HMσ′,′Nσ(|Q−Q′|)e−Q×Q′l2/2∆σN′′,σ′′(Q−Q′)δσ,σ′ (27)
Xσ,σ′(Q)= Xσξ,σ′ξ′(Q), (19) Xσ′′
M,N Xξ,ξ′ M,N −XMσ′,σN(|Q−Q′|)e−Q×Q′l2/2∆σN′,σ(Q−Q′).
Tosolvethisself-consistentequation,wediagonalizetheself-
XMσξ,N,σ′ξ′(Q)=14ǫel2B Z0∞x2dxJ0(QxlxB2)e−12x2 (20) energymatrix Σ+M,,+N(Q,Q′) Σ+M,,−N(Q,Q′) V+j(Q′)
×LNσξ 2 !LNσ′ξ′ 2 !. XQ′ Σ−M,,+N(Q,Q′) Σ−M,,−N(Q,Q′)! V−j(Q′)! (28)
V+(Q)
Here,J (x)isaBesselfunctionofthefirstkind. =γ j ,
The rneal-space density of electrons at valley σ and layer j V−j(Q)!
(sublattice)ξisgivenby
where (V+,V ) is the jth eigenvector with eigenvalue γ .
j −j j
Theorderparametersareobtainedfromtheeigenvectorsand
1 1
ρσξ(r)= 22πl2 ∆σN,σ(Q)LNσ,ξ(Q2l2B/2)eiQ·r−Q2l2/4. (21) eigenvalues
B XQ
ThefillingfactoratLandaulevelN isdefinedas ∆σNσ′(Q)= f(γk−µ)Vkσ′(Q)Vkσ∗(0), (29)
Xk
ν = Ne [0,2], (22) where f(x) is the Fermi-Dirac distribution function. The
N
N ∈ chemicalpotentialµisdeterminedfrom
φ
whereNφ =S/2πl2B(S isaareaofthesystem)isthedegener- ∆σNσ(0)= Vσj(0)Vσj∗(0)f(γj−µ)=νN. (30)
acyofLandauorbitalsandNeisthenumberofelectrons.The Xσ Xσ,j
factor2comesfromthevalleydegreeoffreedom.
Self-consistent equations are numerically calculated to yield
theorderparametersandtheHartree-Fockenergyperparticle
C. Green’sfunctionmethod forseveralCDWstatesintroducedinSec. III.
The order parameter sum rule at zero temperature,28 ex-
tendedtothecaseofvalleydegeneracy
To determine the order parameters self-consistently, the
Green’s function method is used.8,9,25,26 The single-particle
∆σσ′(Q)2 =∆σσ(0)=νσ,
MatsubaraGreen’sfunctionisdefinedby | N | N N (31)
XQ Xσ′
GσN,σ′(X,X′,τ)=−hTcσN,X(τ)cσN′,X†′(0)i. (23) is easily derivedfromEq. (29). Here, νσN is the contribution
from valley σ electrons to the partial filling factor ν . This
TheFouriertransformation N
relationisusedtocheckconvergenceoftheresults.
Gσ,σ′(Q,τ)
N
2πl2 (24)
= L2B XX,X e−(i/2)Qx(X+X′)δX,X′−Qyl2GσN,σ′(X,X′,τ), III. CHARGEDENSITYWAVESTATES
′
A. Valleyskyrmion
relatestotheorderparameters∆σ′σ(Q)by
N
Itisusefultomapthevalleydegreesoffreedomtoapseu-
∆σN′σ(Q)=GσN,σ′(Q,τ=0−). (25) dospin.9 In this language, the componentsof the pseudospin
4
vectordensity P(Q) = P (Q)xˆ+P (Q)yˆ+P (Q)zˆaredefined
x y z
TABLEI. Hartree-Fock quasiparticle and skyrmion (antiskyrmion)
by
particle-holeexcitationgaps(inunitofe2/ǫl √π/2)atLandaulevel
B
Px(Q)= ∆+N−(Q)+2∆−N+(Q), (32) ∆aNncSdoKfnov∆.r)3,2SDLLGE)G,St(ei∆ntrSPaHLa-GLcaGonndv(∆e∆n4SStiLKLoGGn)a,alnBdsLe∆Gm4i(c∆LoGPBn)HLd,Guacantndodrp∆setSBnrLKtuGac-)tL,uGtrrei-((L∆∆G5cPoHLn(v∆G.3Paa−HnnLGdd
S−K P−H S−K P−H
∆5 LG). For conventional and SLG 2DES, the energies of the two
S−K
∆+ (Q) ∆ +(Q) excitationswerecomparedbyYangetal.32 Thesituationwherethe
P (Q)= N− − −N , (33) skyrmionisfavoredisemphasizedbythick“>”.
y
2i
N ∆conv. ∆conv. ∆SLG ∆SLG ∆BLG ∆BLG
PH SK PH SK PH SK
P (Q)= ∆+N+(Q)−∆−N−(Q). (34) 0 1 > 1/2 1 > 1/2 1 > 1/2
z 2 1 0.75 < 0.875 0.6875 > 0.2188 0.75 < 0.875
2 0.6406 < 1.1328 0.5664 > 0.3301 0.5977 > 0.3770
In this paper, states with a topological pseudospin texture,
3 0.5742 < 1.3418 0.5029 > 0.4097 0.5029 < 0.5151
whichis called a skyrmion,29 are considered. A skyrmionis
4 0.5279 < 1.5522 0.4608 < 0.4754 0.4528 < 0.6181
akindofspintextureusuallyusedtodescribeamagneticor-
derandwasfirstintroducedinhadronphysics.30Forconven- 5 0.4927 < 1.6834 0.4298 < 0.5328 0.4187 < 0.7048
tionalandSLG2DES,astatewithalignedskyrmions,which N ∆3P−HLG ∆3S−KLG ∆4P−HLG ∆4S−KLG ∆5P−HLG ∆5S−KLG
iscalledaSkyrmecrystal,hasbeenshowntheoreticallytobe 0 1 > 1/2 1 > 1/2 1 > 1/2
thegroundstatearoundν=1.9,31 1 0.75 < 0.875 0.75 < 0.875 0.75 < 0.875
2 0.6406 < 1.1328 0.6406 < 1.1328 0.6406 < 1.1328
3 0.5498 > 0.4448 0.5742 < 1.3418 0.5742 < 1.3418
B. Preliminaryconsiderations
4 0.4660 < 0.5808 0.5285 > 0.4870 0.5279 < 1.5522
5 0.4223 < 0.6829 0.4406 < 0.6284 0.4963 < 0.5390
Theexcitationenergyofaskyrmioninaferromagneticuni-
form state at ν = 1 is evaluated to find the conditions on
N
whichtheSkyrmecrystalispreferred.32,33Whentheinterlayer
distanced 0,theenergyofaskyrmion(antiskyrmion)pair angularmomentum,andis concentrated. On the otherhand,
→
excitation∆ andtheenergyofa widelyseparatedparticle- when introduction of a hole is accompanied by pseudospin-
SK
holepairexcitation∆ aregivenby flipofotherelectrons,manystateswiththesameangularmo-
PH
mentum are connected by the Coulomb interaction, and the
∆SK = 41πZ ∞q3V(q) FM,N(q) 2e−q2l2B/2dq, (35) cnheacrtegdedqeunasnittuymofsttahteehcoolrerehsapsoandwsidtoeradsisktyrirbmuitoionn..WThhiesncothne-
0
(cid:2) (cid:3)
chargeisconfinedlikeawavefunctionφ givenbyEq. (5),
0,X
the formation of a skyrmion reduces the charge locality. At
1
∆PH = 2πZ0∞qV(q) FM,N(q) 2e−q2l2B/2dq, (36) hsiicgahllLyabnrdoaaud,lesvoetlhNe,bheonwefievtetor,faowrmavsekfyurnmcitoionnsφisNl,Xaciksiningt.riInn-
(cid:2) (cid:3)
M-layered graphene,the spinorwave functionhasthe local-
wheretheformfactor (q)hastheform
FM,N ized componentφ0,X at Landaulevel N = M, so it can drive
thesystemtoformskyrmions.
1 L q2l2B +L q2l2B (N M)
(q)=2(cid:20) N(cid:18) 2 (cid:19) N−M(cid:18) 2 (cid:19)(cid:21) ≥ . (37)
FM,N LN q22l2B (N < M) C. Crystalstructure
(cid:18) (cid:19)
Theratioofthetwoenergiesofthepairexcitationsisthesame
When Zeemanenergyfor valleypseudospinsdoesnotex-
asthatof single-particleexcitations, fromparticle-holesym-
ist,askyrmionsplitsintotwomerons(half-skyrmions). Four
metry.Forconventional,SLG,BLG,andM-LG(M =3,4,5)
texturesofameronarepossiblefromthetwodirectionatcen-
2DESs, these energies at Landau level N 5 are presented
≤ terandthetwovorticities. Chargedensitywavestateswhere
inTableI.Itshowsthatskyrmionexcitationisfavored(1)at
electrons,holes,ormeronsformatriangularorsquarelattice
N = 0 in all systemsassingle-particlewave functionsbeing
identicalto conventionalone, (2) at N = 1,2,3in SLG, and structure are considered. The order parameters ∆σN,σ′(Q) are
definedatpoints
(3) at N = M in M-LG (M = 2,3,4). Thus we can expect
thattheSkyrmecrystalbecomesthegroundstateinthesesit-
1 √3
uationsaroundνN =1. Q= j+ k Q0, kQ0 , foratriangularlattice, (38)
The reason why a skyrmion can be a low-energy excita-  2 ! 2 
tionisthefollowing.34 ThegroundstateatνN = 1isapseu-  
dospinferromagneticliquidstate. Whenaholeisintroduced Q=(jQ ,kQ ), forasquarelattice, (39)
0 0
in this state without flipping pseudospin of other electrons,
thechargedensityoftheholeisgivenbyaneigenstateofthe where jandkareintegers. Thefollowingstatesareassumed:
5
10 0.5 skyrmions per unit cell. Thus Q0 is determined from
0.4 theconditionthattheCDWhasn(= 2)skyrmionsina
unitcell:ν =2πl2n/s.Thezcomponentofpseudospin
0.3 h B
5 densityinrealspaceandthevorticityalternatefromone
0.2
site tothe next(Figs. 1 and2). The XY orientationof
0.1
B pseudospinshas U(1) symmetry (Fig. 2). The density
y/l 0 0 distributioninrealspaceisbipartiteinlayers.
-0.1
-0.2 4. Meronpaircrystal(MPC):atriangularlatticewithfour
-5
-0.3 meronsperunitcell.Themeronsarenotequallyspaced
and bound into pairs. The Q is determined from the
-0.4 0
samewayasMC.TheenergyofMPCissimilartoMC,
-10 -0.5
and has slightly lower energy in the low quasiparticle
-10 -5 0 5 10
densityregime(closetoν =1)ingeneral.
N
x/l
B
FIG.1.(Coloronline)P(r)formeroncrystalatν =0.86inSLG.
z 1 IV. RESULTS
10
Weusethepartialfillingfactorν atLandaulevelN;thus
N
the total filling factor is given by ν = 4N 2+ν for any-
N
−
layered graphene. Assume that Zeeman splitting is suffi-
5
ciently large, so the phase diagram for ν [0,2] is identi-
N
∈
calto thatforν [2,4]. Furthermore,the Hamiltonianhas
N
y/lB 0 electron-holesym∈metryaroundνN = 1,sothephasediagram
forν > 1iscaughtbyalternatingparticlesforν < 1toan-
N N
tiparticles. NumericalcalculationforMCandMPCaredone
-5 at 1 νN 0.06, since too many wave vectors are needed
| − | ≥
to get well-convergedsolutions at ν 1. In the following
N
results, the energieswith an accuracy≃of 10 6 are presented.
−
-10 Wigner and bubble crystals are calculated only in triangular
-10 -5 0 5 10
symmetry;asquarelatticegenerallyhashigherenergythana
x/l
B triangularoneinalowquasiparticledensityregime.
It is difficult to get MC solutions close to ν = 1.0,
N
FIG.2. (Coloronline)TheXYorientationofpseudospinsformeron so we extrapolate the order parameters of the MC solutions
crystal at ν = 0.86 in SLG. One meron crystal has two types of
1 in ν 0.94. To execute the extrapolation, the quantity
N
merons. M,N(Q≤)∆σ,σ′(Q)isfitted byaquadraticcurve. Theenergies
F
oftheextrapolatedMCstatesarerepresentedinthefollowing
1. Electron Wigner crystal (eWC) and n-electron bubble figuresasadottedline. Itisnotedthatuncertaintyremainsin
crystal (eBCn): a triangular or square lattice with one theextrapolationespeciallyin the4-LGcase. Itcomesfrom
ornelectronsperunitcell. Thefundamentallengthin therelativelylowvalidityoffittingtheorderparameterswhich
q space Q0 is determined from the condition that the haveaninflectionpointnearνN =0.92.
CDWhasnelectronsinaunitcell:ν=2πl2n/s.Here,ν Theenergiesofthevalley-concentratedholeWignercrystal
B
isthefillingfactorofelectronsandsistheareaofaunit states which are quite accurately approximated by Gaussian
cell. Thelow-energystateispseudospinferromagnetic formorderparameters
becauseofthePauliprinciple.
∆σ,σ(0)=ν ,∆σ,σ(Q,0)=(ν 1)e Q2l2/4,
N N −
2. Hole Wigner crystal (hWC) and n-hole bubble crystal − (40)
∆σ,σ¯(Q)=∆σ¯,σ(Q)=∆σ¯,σ¯(Q)=0,
(hBCn): a triangular or square lattice with one or n
holesperunitcell. Thefundamentallengthin q space
are represented as “GhWC” in the following figures. In the
Q isdeterminedfromtheconditionthattheCDWhas
0 vanishinginterlayerdistancelimit,theGhWCstatewithP =
nholesinaunitcell: ν = 2πl2n/s. Here,ν isthefill- z
h B h (ν+ ν )/2 = ν /2 is degenerated to the hWC solutions
ingfactorofholesand sistheareaofa unitcell. The N − −N ± N
with P <ν /2(Figs. 4,6,8).
low-energy state is pseudospin ferromagnetic because | z| N
In the followingwe show phase diagramsobtainedby the
ofthePauliprinciple.
presentHFapproximationinthewholerangeofν . Itshould
N
3. Meroncrystal(MC):a squarelattice withfourmerons be remarked that the true phase diagram should contain re-
of charge e/2 (ν < 1) or e/2 (ν > 1) per unit gionsofincompressibleliquidstates thatcannotbeobtained
−
cell, equally spaced. A meron pair is equivalent to bytheHFapproximation.Thusthephasediagramsarepartly
one skyrmion, so MC can be seen as a state with two incorrect.However,HFcalculationgivesqualitativelycorrect
6
resultswhenthefractionalquantumHallstatesdonotappear. )
This is established from comparisonswith the results by the ε l B -0.1 eWC
exactdiagonalizationmethod35–37orthedensitymatrixrenor- 2e / -0.15 hWC
msibailliiztaytioofnmgerorounpcmryetshtaolds.3n8eIanrνthNis=pa1p,ewrhweerefothceuslioqnuitdhestpaotess- ctron ( -0.2 LiqMuiCd
arenotexpected,butonlythecharge-orderedstatescompete. e
Therefore,thefollowingdiscussionastotherealizationofthe r el -0.25
e
meroncrystalisreliable. p -0.3
y
g
er -0.35
n
e
A. Single-layergraphene F -0.4
H
0 0.2 0.4 0.6 0.8 1
TheHartree-Fock(HF)phasediagramofthe2DESinSLG ν
2
hasbeenobtainedandcomparedwiththatoftheconventional
one.8ItisshownthatSkyrmecrystals(MCandMPC)become
the groundstate aroundν = 1 at Landaulevels N = 0 and FIG.3. (Coloronline)Ground-stateenergyperparticle(inunitsof
N
1.9ConsideringtheexcitationenergiesforSLG(TableI),itis e2/ǫlB)atLandaulevelN =2inBLG(d/lB=0).
alsopossiblefortheSkyrmecrystalphasetooccurat N = 2
and3, butthis hasnotbeenfoundyetin a mean-fieldcalcu- -0.345
GhWC
lShaiLtgiGohne,8r.,9fiAlwllitenhgoeuxregeghciumttheeed.saadmdeitiHonFaclaclchuelcaktsioannhdadinvbeesetnigdaotendetfhoer 2εn (e / l )B -0-.03.5355 extrapolateLdhiq WMMuiCCCd
AtLandaulevelN =0,thephasediagramisthefollowing: o
νeW0A∈Ct[L0fo.a5rn4νd,0a0u∈.6l[3e0v].,e1al0nN,d0=M.501P],C,thhfWeorpCνh0afos∈er[dν00i.a6∈g3r[,a00m..59i02s,]t0.h9.e54fo],llMowCinfogr: y per electr -0-.03.6356
g
νeWC[f0o.r73ν1,0∈.8[40].,1a0n,d0.5M0P],ChWfoCrνforν[10∈.8[40,.05.09,20]..793T]h,MerCanfgoer ener -0.37
1 ∈ 1 ∈ F d / l = 0.0
ofaskyrmionic(MCorMPC)phaseisnarrowerthanthatof H B
theN =0case. -0.375
0.84 0.88 0.92 0.96 1
At Landau level N = 2, the phase diagram is the follow- ν
2
ing: eWC for ν [0.10,0.27], eBC2 for ν [0.27,0.50],
2 2
∈ ∈
hBC2 for ν [0.50,0.73], and hWC for ν [0.73,0.94].
2 2
∈ ∈ FIG.4. (Coloronline)Ground-stateenergyperparticle(inunitsof
Itischaracteristicthat2-electron(hole)bubblecrystalsexist
e2/ǫl )aroundfillingν =1atLandaulevelN =2inBLG(d/l =
around ν = 0.50. The skyrmionic ground state is not seen B 2 B
2 0).
inthe rangeν 0.94. Theextrapolatinganalysis, however,
2
≤
suggests thatthe hWC and MC state are almost degenerated
inν [0.94,1.0].
2
At∈Landau level N = 3, the phase diagram is the follow- forν2 [0.50,0.94]. Thebubblestatedoesnotappearunlike
∈
the phase diagramat Landaulevel N = 2 for SLG. Figure 4
ing: eWC for ν [0.10,0.20], eBC2 for ν [0.20,0.30],
3 3
∈ ∈ showstheenergiesin thearea close to ν = 1. The extrapo-
eBC3forν [0.30,0.50],hBC3forν [0.50,0.70],hBC2 2
3 3
∈ ∈ latedenergyoftheMCsolutionshavelowervaluethanhWC
for ν [0.70,0.80], and hWC for ν [0.80,0.94]. The
3 3
∈ ∈ forν [0.94,1.0].
skyrmionicstatedoesnotappearinν 0.94. 2
∈
≤
AlthoughtheMCandMPCsolutionsarenotfoundinν
N
≤
0.94forSLGat N = 2and3,theSkyrmecrystalisexpected
tohavelowerenergyintheimmediatevicinityofν =1from
N
theanalysisinSec. IIIB. C. LandaulevelN =3intrilayergraphene
Figure5showstheenergiesperelectronforseveralcrystal
B. LandaulevelN =2inbilayergraphene structuresat Landaulevel N = 3 in tri-LG. Itshowsthe fol-
lowingsequenceofgroundstates: eWCforν [0.10,0.46],
3
∈
The HF calculation for degenerated zero-energy Landau eBC2forν [0.46,0.50],hBC2forν [0.50,0.54],hWC
3 3
∈ ∈
levelsN =0,1inBLGsuggeststhattheSkyrmecrystalstates for ν [0.54,0.94]. Although the bubble states are found
3
ofrealspinororbitalpseudospinoccur.39 around∈ν =0.50,itsrangeisnarrowerthanthatoftheN =3
3
Inwhatfollows,theresultsforthefirstexcitedLandaulevel case in SLG. The skyrmionicgroundstate is not seen in the
N =2inBLGarepresented. Figure3showstheenergiesper rangeν 0.94. Figure6showstheenergiesnearν =1. The
3
≤
electronforseveralcrystalstructures. Itshowsthefollowing extrapolatedstatesofthe MCsolutionshaveanenergyclose
sequence of groundstates: eWC for ν [0.10,0.50],hWC tothatofhWCsinthevicinityofν =1.
2 3
∈
7
) )
ε l B -0.1 eWC ε l B -0.1 eWC
2e / -0.15 eBC2 2e / -0.15 eBC2
( hBC2 ( hBC2
on hWC on hWC
ctr -0.2 MC ctr -0.2 MC
e Liquid e Liquid
el el
er -0.25 er -0.25
p p
y y
rg -0.3 rg -0.3
e e
n n
e e
F -0.35 F -0.35
H H
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ν ν
3 4
FIG.5. (Coloronline)Ground-stateenergyperparticle(inunitsof FIG.7. (Coloronline)Ground-stateenergyperparticle(inunitsof
e2/ǫl )atLandaulevelN =3intri-LG(d/l =0). e2/ǫl )atLandaulevelN =4intetra-LG(d/l =0).
B B B B
-0.31 -0.29
GhWC GhWC
2εn (e / l )B -0-0.3.3125 extrapolateLdhiq WMMuiCCCd 2εn (e / l )B -0.-209.35 extrapolateLdhiq WMMuiCCCd
o o -0.305
ctr -0.325 ctr
ele ele -0.31
er -0.33 er
p p -0.315
y y
g -0.335 g
er er -0.32
n n
e e
F -0.34 d / l = 0.0 F -0.325 d / l = 0.0
H B H B
-0.345 -0.33
0.84 0.88 0.92 0.96 1 0.84 0.88 0.92 0.96 1
ν ν
3 4
FIG.6. (Coloronline)Ground-stateenergyperparticle(inunitsof FIG.8. (Coloronline)Ground-stateenergyperparticle(inunitsof
e2/ǫl )aroundfillingν =1atLandaulevelN =3intri-LG(d/l = e2/ǫl ) around filling ν = 1 at Landau level N = 4 in tetra-LG
B 3 B B 4
0). (d/l =0).
B
V. DISCUSSION
TheHartree-Fock(HF)calculationsuggeststhatthemeron
D. LandaulevelN =4fortetralayergraphene crystal (MC) phase appears at Landau level N = 2 in BLG,
and hole Wigner crystal (hWC) and MC states are degener-
atedaroundν = 1atLandaulevelN = 2inSLG, N = 3in
N
tri-LGinthecaseofvanishinginterlayerdistance. Ourcalcu-
Figure7showstheenergiesperelectronforvariouscrystal lationstronglysuggeststhatmeronpaircrystals(MPCs)will
structuresatLandaulevelN =4fortetra-LG.Itshowsthefol- havelowerenergyintheimmediatevicinityofνN = 1forits
lowingsequenceofgroundstates: eWCforν [0.10,0.32], triangularsymmetry.
4
∈
eBC2 for ν [0.32,0.50], hBC2 for ν [0.50,0.68],and Skyrme crystals (MC and MPC) have a charge distribu-
4 4
∈ ∈
hWCforν [0.68,0.94].Thebubblestatesarefoundinthe tionandcollectivemodedifferentfromthatofhWC,sothese
4
∈
broadrangeν [0.32,0.68].Theskyrmionicgroundstateis states can be distinguished by transport properties40,41 and a
4
notseeninthe∈rangeν 0.94. Figure8showstheenergies microwave absorption spectrum.9,43 Furthermore, the CDWs
4
≤
nearν =1. TheextrapolatedstatesoftheMCsolutionshave exist in outer layers, so the local density of states (LDOS)
4
higherenergythanthatofhWCsinthevicinityofν =1. As can be measured in a spectroscopic manner.42 In this paper,
4
previously mentioned, however, the extrapolation method is the unidirectional stripe phase is not considered. The stripe
nolongervalidintetra-LG.AccordingtotheanalysisinSec. phase,however,alsowillappeararoundν =0.5inBLGand
N
IIIB,theSkyrmecrystalisexpectedtohavethelowestenergy M-LG, as SLG8 and conventional 2DES if fractional quan-
inthevicinityofν =1. tumHallstatesarenotrealized.Suchastate,ifexists,willbe
4
8
identifiedbyanisotropicconduction.41 It has been pointed out that anisotropy in the pseudospin
arisesintheorderofa/l .23,45–48 Inthemultilayergraphene,
B
finitelayerseparationalsobringsanisotropy. Thisanisotropy
Although we ignored the effects of disorder, a finite val- is quite small, since a < d l , but may have some effect
B
≪
leyZeemanenergy,andLandau-leveltransitions,itisunclear when the energies of two phases are quite close. We have
howtheseaffectdegeneracyofhWCandMCaroundνN = 1. doneacalculationtakingintoaccountonlytheeffectofd/lB
Inparticular,Landau-leveltransitionsinBLGand M-LGare as a preliminary investigation, and found that the finite d is
larger than that of SLG under a magnetic field B 10 T. slightlyunfavorableforthe Skyrmecrystal. However,toob-
∼
The gap near a charge neutrality pointis ∆ = √2~v /l tainadefiniteconclusionfortheeffectoffinited,weneedto
1 F B
∼
380√B/T K for SLG, ∆ = √2~ω 45 (B/T) K for takeintoaccounttheeffectofaalso. Suchcalculationisleft
2 c
BLG, ∆ = √6~ω 6.6 (B/T)3/∼2 K f×or tri-LG, and forfutureinvestigation.
3 3
∆ = √24~ω 1.1 ∼(B/T×)2 K for tetra-LG. The typical
4 4
Coulomb energy∼is E ×= e2/ǫl 100√B/T K. For SLG,
C B
theratioE /∆ = 0.39isindepend∼entoffield B. Inthiscase ACKNOWLEDGMENTS
C 1
itisshownthattheLandau-levelmixingdoesnotchangethe
CDW phase diagram (except the skyrmionic crystal).44 For Y.S. thanksR. Coˆte´ forhelpinghimtofind self-consistent
BLG and M-LG, high magnetic fields are neededto achieve solutionsto MC states. The numericalcalculation was done
acomparableratio: B 70TforBLG,60Tfortri-LG,and bySR11000atInformationTechnologyCenter,Universityof
∼
50Tfortetra-LG. Tokyo.
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