Hund's Rules for the N=0 Landau Levels of Trilayer Graphene

0.2 MB·English

by  Fan Zhang

Hund’s Rules for the N = 0 Landau Levels of Trilayer Graphene
Fan Zhang1,2,∗ Dagim Tilahun2,3, and A. H. MacDonald2
1 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104
2 Department of Physics, University of Texas at Austin, Austin TX 78712
3 Department of Physics, Texas State University, San Marcos, TX 78666
(Dated: January 27, 2012)
The N = 0 Landau levels of ABC and ABA trilayer graphene both have approximate 12 fold
degeneracies that are lifted by interactions to produce strong quantum Hall effects (QHE) at all
integer filling factors between ν = −6 and ν = 6. We discuss similarities and differences between
the strong-magnetic-field weak-disorder physics of the two trilayer cases, and between trilayer and
2
bilayer cases. These differences can be used to identify the stacking order of high-quality trilayer
1
0 samples by studyingtheir quantumHall effects.
2
PACSnumbers: 73.43.-f,75.76.+j,73.21.-b,71.10.-w
n
a
J I. INTRODUCTION ABC Trilayer AB Bilayer ABA Trilayer

6
2 A1 B1 A1 B1
Recent experimental advances1 have realized a A1 B1
A2 B2 A2 B2
] new family of quasi two-dimensional electron systems
l A2 B2
e (2DESs)consistingoftwoormoregraphenelayers. Many A3 B3 A3 B3
- of the unusual properties of these systems are related to
r
t momentum-space textures in the low-energy quasiparti-
.s cle bands, and to the correspondingly enhanced N = 0 FIG. 1: (Color online) Schematic electronic structure of few-
t layer graphene. The sites coupled by thestrongest interlayer
a (zero energy) Landau level (LL) degeneracies that ap-
m pear when an external magnetic field is applied. As in bonds characterized by the γ1 hopping parameter are indi-
cated by shading and have little weight in N = 0 Landau
the case of semiconductor 2DESs, Coulomb interactions
- levels.
d break LL degeneracies when disorder is weak2,3 and fill-
n ing factors are close to integer values. This phenomenon
o
issometimesreferredtoasquantumHallferromagnetism
c to n = 0 and n = 1. Their quantized LL energies are
because its simplest4 manifestation is spontaneous spin-
[ polarizationin the conduction bands of zincblende semi- En(3) ∼B3/2 n(n−1)(n−2). This property causes LL
1 conductors at odd integer filling factors. In few-layer mixing effectps to vanish strongly at very high magnetic
v fields. The N = 0 LLs of ABA trilayers, on the other
graphene 2DESs, LL degeneracies are larger because of
7
the presence of two low-energy valleys (K and K′)5–7, hand, consist of n=0,1 orbitals on the layer symmetric
3
states A + A and B ,andonlyn=0orbitalsonthe
4 andforN =0alsobecauseofthemomentum-spacepseu- | 1i | 3i | 2i
layer antisymmetric states A A and B B .
5 dospin textures. 1 3 1 3
| i−| i | i−| i
01. canInlbeaildayteor gcraanpthedenea,ntthifeerardomdiatigonneatlisdmeg8–r1e1e,s aonfdfreveadlolemy TsithieonNof6=m0onLoLlayspere-cltikruemLLcsanwibtehvEien(w1)ed∼aBs 1a/2s√upnerapnod-
2 andorbitalorder12–15amongother16–18phenomena. The bilayer-likeLLswithEn(2) B n(n 1). Thisstructure
1 trilayer graphene19–22 system, which has become exper- leads to LL crossing22,25,3∼5 anpd for N− = 0 increases the
v: imentally realized23–32 only recently, presents an even quantitative importance of LL mixing6at stronger fields
i richer many-electronphysics landscape. Both ABC (chi- compared to the ABC case.
X
ral) and ABA stacked trilayers are expected to have in- OurdescriptionoftheN =0physicsofABCandABA
r teger Hall plateaus at ν = 4(n+3/2). The ν = 6 trilayer graphene starts from a model that includes only
a ± −
and ν = 6 plateaus are separated by a set of 12 approx- nearest neighbor intra-layer and inter-layer hopping and
imately degenerate N = 0 LLs, the trilayer duodectet, focuses on energies smaller than the inter-layer hopping
due to a combination of the 4-fold spin-valley degener- strength γ . By treating other hoppings perturbatively
1
acy common to all levels, and trilayer momentum space at low-energies,it is possible to derive effective Hamilto-
pseudospin textures19,20. However, as we shall explain, nians for N = 0 LLs that are valid at realistic magnetic
theroleofelectron-electroninteractionsisquitedifferent field strengths. In Section II of this article we discuss
inthetwocasesbecauseofdifferences8,26,33 inthespatial N = 0 quantum Hall ferromagnetism in ABC trilayer
and orbital contents of the N =0 LL wavefunctions. graphene, predicting a simple set of Hund’s rules that
ABC trilayers have N = 0 layer wavefunctions that apply in the absence of inversion-symmetrybreaking ex-
are localized on the A site in Fig. I for one valley ternal electric fields. We then address the phase tran-
1
and on the B site for the other. ABC trilayers are sitions that can be driven experimentally by applying
3
like AB bilayers in that they have inversion symmetry electric fields across the trilayers using top and bottom
protected34 degeneracy,butthey differ inthat they have gates. TheinteractionphysicsoftheN =0levelinABA
N = 0 LLs with n = 2 orbital character, in addition graphene,whichis discussedin SectionIII, is more com-
2
plex even in the absence of an external electric field. We whichmustbedeterminedself-consistentlyateachfilling
conclude in Section IV with a discussionof prospects for factor. When we include only interactions37 within the
interesting experimental discoveries in these systems. N = 0 manifold we find the quasiparticle Hamiltonian
with Hartree and exchange interaction contributions is
II. ABC-STACKED TRILAYER GRAPHENE hαns|HAHBFC|βn′s′i=(ELnLδss′ −EZMσszs′)δnn′δαβ
+EH(∆B−∆T)ταzβδss′δnn′
B3Trheeprloewse-netnaetrigoynmiso:delforABCtrilayers19,20 intheA1- −EF Xnαnβ2,n1n′∆βαnn12ss′δss′, (2)
nX1n2
v3 0 π†3
= 0 where we have used the labels n = 0,1,2 for orbitals,
HABC γ2 (cid:18)π3 0 (cid:19)
1 index s for spin, and the Greek indices α,β = B(K)
1 0 v2 π†π 0 or T(K′) for layer(valley). Here EF = e2/εℓB charac-
+ ud(cid:20)(cid:18)0 1(cid:19)− γ02 (cid:18) 0 ππ† (cid:19)(cid:21) . (1) terizes the overall strength of the exchange interactions,
− 1 − E = (2d/2ℓ )E captures the capacitive electrostatic
H B F
In Eq.(1) π = ~k+eA/c is the 2D kinetic momentum, energy associated with shifting charge between the top
π = τzπ +iπ where τz = 1 represents valley K and and bottom layers, and d = 0.335 nm is the separation
x y
K′, v is the monolayer Dira±c velocity, γ 0.5 eV19,27, between adjacent graphene layers. The layer projected
0 1
and2u is the site-energydifferencebetwee∼nA andB . charge density is
d 1 3
Thismodelisapproximatelyvalidfor~v /ℓ γ where
0 B 1
ℓB = 25.6nm/ B[T] is the magnetic lengt≪h, i.e. for ∆α = ∆ααnnss, (3)
fields much smpaller than 400 T. For ud = 0, ABC Xns
∼ H
has three zero energy eigenstates for each value of τ .
z and the density matrix is
The corresponding eigenstates, which play the key role
in ABC trilayer quantum Hall ferromagnetism have the ∆βn′s′ = c† c . (4)
form (0,φ ) for valley K and (φ ,0) for valley K′. Here αns h βn′s′ αnsi
n n
n=0,1,2is an orbitallabel and φn is the nth LL eigen- The dimensionless exchange integrals Xαβ reflect
state of an ordinary 2DEG. The ABC trilayer duodectet nn2,n1n′
the form factors associated with n = 0,1,2 orbitals. We
isthedirectproductofthisLLorbitaltriplet,arealspin
find that
doublet,andawhich-layer (orequivalentlywhich-valley)
doublet. d2k 2πe2ηαβ
Gaps can open within the duodectet, giving rise to Xnα,βn2,n1n′ =Z (2π)2Fnn2(−k)Fn1n′(k) εk E , (5)
F
additional integer quantum Hall effects, due to single-
particle terms that lift degeneracies or due to interac- where ηαβ becomes 1 for α = β and e−2kd for α = β.
tions that break symmetries. For example a Zeeman Here Fnn′(k) are the ordinary38 2DEG LL form fac6 tors
term splits spin levels by 2EZM =gµBB =0.116 B[T] which can be expressed in terms of associated Laguerre
meV, while ud splits both layers and orbitals: ×ELnL = polynomials that capture the spatial profile of the LL
τzu (1 n(~ω /γ )2/3). The notation used in this wavefunctions. The exchangeintegralsarenon-zeroonly
d 3 1
−equation−is motivated by viewing the ABC multilayer for n n′ = n n ; numerical values of the relevant
2 1
− −
graphene Hamiltonian as the J = 3 generalization of form factors for the case d=0 are listed in Table I.
Hamiltonians that apply to single-layer graphene for The self-consistent solutions of the N = 0 ABC
J = 1 and to bilayer graphene for J = 2. In each Hartree-Fock equations for balanced (u = 0) ABC tri-
d
case ~ω is proportional to the energy of the N = 1 layers are summarized in Fig.2 for a typical field of
J
LL. ~ω = (√2v ~/γ ℓ )Jγ is the cyclotron frequency B = 20 T. Gaps appear at all intermediate integer fill-
J 0 1 B 1
of quasiparticles in chirality J 2DES’s. The Zeeman en- ingfactors,andasexpectedarealwaysmuchlargerthan
ergyisinvariablysmallcomparedtotheinteractionscale E . The electronic ground state at duodectet filling
ZM
e2/ǫℓ ,whereasu canbelargeenoughtocompletelyre- factors from ν = 6 to 6 follows a Hund’s rule behav-
B d
−
organize the duodectet. ior progression in which spin polarization is maximized
The full electron-electron interaction Hamiltonian in first, then layer (valley) polarization to the greatest ex-
theduodectetsubspaceisreadilyavailablegiventhesim- tent possible, and finally LL orbital polarization to the
ple form factors of the n = 0,1,2 Landau levels. In extent allowed by the first two rules. In these calcula-
our discussion of N = 0 quantum Hall ferromagnetism tions, the quasiparticle energies of spins along the field
we will nevertheless use the Hartree-Fockapproximation direction(say spin ) are muchlowerthan those for spin
which is expected to be accurate36 at the integer filling states because the↑spin-splitting is exchangeenhanced.
↓
factors of interest, and more physically transparent. We Since d/ℓ is always small in trilayer graphene systems,
B
solve the Hartree-Fock equations in the duodectet sub- the exchange integrals that favor spin-polarization are
space allowing any symmetry, other than translational only slightly larger than those that favor layer polariza-
invariance,to be broken. The Hartree-FockHamiltonian tion. Because these differences and Zeeman energies are
is expressed in terms of the duodectet density-matrix both small, inter-valley exchange39 (which is neglected
3
TABLE I: Summary of thenumerical valuesof exchange integrals. The intralayer integrals are independent of magnetic fields
whiletheinterlayeronesdeclineslowlyasthefieldstrengthincreases. NumericalvaluesarereportedforB=20TandB=30
′
T. Wehavedefined indicesa=|n−n2|, b=min(n1,n) and c=min(n,n2). Exchangeintegrals are symmetricbetween b and
c. Forthe interlayerexchange integrals thevalues reported hereare for neighboring layer separation d=0.335 nm.
Intralayer Interlayer Interlayer Intralayer Interlayer Interlayer Intralayer Interlayer Interlayer
a b c a b c a b c
AnyB B=20T B=30T AnyB B=20T B=30T AnyB B=20T B=30T
0 0 0 1.2533 1.1444 1.1219 1 0 0 0.6267 0.5215 0.5008 2 0 0 0.4700 0.3676 0.3481
0 0 1 0.6267 0.6229 0.6211 1 0 1 0.2216 0.2177 0.2160 2 0 1 0.1357 0.1319 0.1302
0 0 2 0.4700 0.4689 0.4684 1 0 2 0.1357 0.1348 0.1343 2 0 2 0.0720 0.0711 0.0708
0 1 0 0.6267 0.6229 0.6211 1 1 0 0.2216 0.2177 0.2160 2 1 0 0.1357 0.1319 0.1302
0 1 1 0.9400 0.8365 0.8165 1 1 1 0.5483 0.4470 0.4280 2 1 1 0.4308 0.3315 0.3133
0 1 2 0.5483 0.5434 0.5412 1 1 2 0.2398 0.2348 0.2325 2 1 2 0.1592 0.1542 0.1520
0 2 0 0.4700 0.4689 0.4684 1 2 0 0.1357 0.1348 0.1343 2 2 0 0.0720 0.0711 0.0708
0 2 1 0.5483 0.5434 0.5412 1 2 1 0.2398 0.2348 0.2325 2 2 1 0.1592 0.1542 0.1520
0 2 2 0.8029 0.7029 0.6844 1 2 2 0.4994 0.4010 0.3832 2 2 2 0.4027 0.3058 0.2887
in the continuum model we employ here) can potentialy F0
H
E
alterthecompetitionbetweenspinandlayerpolarization y
g
compared to the results reported in Fig.2. We return to er−0.5
n
this point in the discussion section. For balanced ABC E
trilayers, layer-order appears as spontaneous inter-layer ock −1
F
phase coherence40–42 rather than layer polarization, and −
ee−1.5
to Hartree-Fock quasiparticles that have symmetric (S) artr
or anti-symmetric (AS) layer states. The three LL or- H
−2
bitals are filled in the order of increasing n because ex- −6 −4 −2 0 2 4 6
change integrals decrease with n as indicated in Table I. ns 6
o
It follows that the first six filled LLs are S,0 , S,1 ati 5
| ↑i | ↑i z
and S,2 followedbytheAScounterparts;thenextsix ari 4
filled| LLs↑aire the spin states in the same order. Here Pol 3
S and AS are orthogon↓alcoherent bilayer states, chosen pin 2
s
as symmetric (S) and antisymmetric (AS) states as a o
d 1
u
matter of convenience. e
Ps 0
These Hund’s rules imply that the interaction driven
−6 −4 −2 0 2 4 6
integer quantumHall states are spin andpseudospin po- Filling Factor ν
larizedat all the intermediate integer filling factors from
ν = 5 to 5, as summarized in Fig.2. All 11 states are FIG.2: (Coloronline)Upperpanel: fillingfactordependence
−
spin polarized ferromagnets with a maximum spin po- oftheHartree-FockenergiesofoccupiedandunoccupiedLLs
lariza↑tion at ν = 0. The LL orbital dependence of the for a balanced ABC trilayer at 20 T. Energies are in unitsof
microscopicHamiltonianleadstoorbitalpolarizationex- (π/2)1/2e2/εℓB. Unoccupied LLs are sextets (blue), triplets
(green), or singlets (cyan); occupied LLs are doublets (ma-
cept at filling factors that are multiples of 3. In the ab-
genta) or singlets (red); LLs at ν = ±6 are split only by the
sence of an interlayer potential which breaks inversion
smallZeemansplitting. Theseenergiesincludeonlyexchange
symmetry, the layer pseudospin symmetry is broken by
interactions within the N = 0 level. Particle-hole symmetry
states with spontaneousinterlayercoherencewhichopen
within the N = 0 space is recovered if exchange interactions
a gap at no cost in Hartree energy. The Hartree en- with remote Landau levels are included; these interactions
ergyand the difference between intralayerand interlayer do not influence the energy spacings between Landau levels.
exchange are both the same order d/ℓ e2/ℓ , but Lower panel: filling factor dependenceof thepolarizations of
B B
∼ ·
it turns out that Hartree energy slightly dominates over upspin(blue)andlayer(red),andpopulationdifferencesbe-
exchangedifferencefavoringspontaneouscoherencestate tweenn=2andn=0(green) andn=1(cyan)LLorbitals.
over competing layer polarized states.
We have so far neglected the strong dependence of
duodectet quantum Hall ferromagnets on an interlayer first-orderphasetransitionfromthespinpolarizedferro-
electric field. A very small potential difference between magneticstatetoalayer(valley)polarizedstatewhenthe
the top and bottom layers is sufficient to destabilize potential difference between the top and bottom layers
spontaneous-coherence states in favor of layer polarized becomes dominant over the exchange induced spin split-
states. Quantum phase transitions vs. u are expected ting. For B =20 T (ε=1) we find that the layer polar-
d
at all filling factors. For ν = 0, for example, there is a izedstatehasthelowestenergyifthepotentialdifference
4
between the top and bottom layers exceeds 0.2 eV. The of magnetic field. The single-particle splitting of N = 0
gap between the highest occupied LL T,2, and the can therefore lead to quantum Hall effects at ν = 2,
| ↑i −
lowest unoccupied LL B,0, is 0.46 eV for u =0 and ν =2 and ν =4.
d
| ↓i
is reduced to 0.28 eV at the phase transition. We find The self-consistentHartree-FockHamiltonianthatde-
that the critical screened electric field for this first order scribes the broken symmetry ABA trilayer N = 0
phasetransitionis 15mV/(nmT) forε=1,compara- duodectet is
∼ ·
ble to the experimental value of the corresponding tran-
sition field in bilayers.43–46 The linear dependence of the hinis|HAHBFA|jnjs′i=(EL′Lδss′ −EZMσszs′)δninjδij
critical field on B follows from the fact that all relevant EH
microscopicenergysales(thebiassupportedenergiesE + 2 ∆B2(2δB2,i−1)δss′δninjδij
H
eaxncdhEanLgLe,st,haenddifftehreenZceeebmeatwneeenneirngtyraElayer)aanrdeianltlerlilnaeyaerr −EF Xnijin2,n1nj∆jinn21ss′δss′, (7)
ZM nX1n2
functions of B.
In high mobility and low disorder suspended sam- where the LL index ni range depends on the atomic
ples, we anticipate that the ν = 6,0 states and the orbital i, i.e., ni = 0 only for J = 1 branch while
bias supported ν = 3 states, pe±rsist down to nearly ni = 0,1 for J = 2 branch. The exchange integrals
zeromagneticfieldexh±ibitingspontaneousquantumHall Xij are still defined by Eq.(5) but with a more gen-
effects.8,26 ABCtrilayersarehighlysusceptibletobroken eral definition ηij = mcmVm/V0, where V0,1,2 respec-
symmetries that are accompanied by large momentum tively denote intralaPyer, nearest interlayer, and next-
space Berrycurvatures anddifferent types of topological nearest interlayer Coulomb interactions, and cm is ob-
order.8 Remote weaker hopping processes19 in ABC tri- tained from the interaction matrix element decomposi-
layersthatwehaveignoredsofarcouldinfluencethefield tion hij|V|iji = mcmVm. Otherwise the notation in
to which the quantum Hallstates survive. The mostim- Eq.(7)isthesamePasintheABCcase. Sinceitisalready
portantoftheseappearslikelytobethetrigonalwarping quite complicated, we focus on the balanced case47,48
energyscale 7meV.19 Thestudyofweakfieldquantum with ud = 0. ELL is therefore absent and replaced by
Halleffects i∼nABCtrilayersshouldshedlightonthe na- EL′L which is given by the ~π =0 eigenvalues of Eq.(6).
ture of the B =0 ground state.8 We find that the ABA duodectet intermediate filling
factor states follow the Hunds rule for maximum spin-
polarization, but are otherwise strongly influenced by
III. ABA-STACKED TRILAYER GRAPHENE the single-particle gaps. The J = 2 LLs, which have
lower single-particle energies are filled first. When the
magnetic field is smaller than B = 17 T (ε = 1 here-
Unlike ABC trilayerquasiparticleswith pure chirality, c1
after), the first six filled LLs are all majority spin levels:
thebandstructureofanABAtrilayerconsistsofamass-
A +A with n = 0 and 1, then B with n = 0
less monolayer (J = 1) and a massive bilayer (J = 2) | 1 3 ↑i | 2 ↑i
and 1, and lastly A A and B B with
subbands. Unbiased ABA trilayers have mirror symme- | 1 − 3 ↑i | 1 − 3 ↑i
n = 0. The next six filled LLs are the spin states
trywithrespecttothe middle layerandtheirlowenergy ↓
physics is governed by22 in the same order. Within a given spin the LL filling
patternfollowsthe ordersuggestedbythe single-particle
0 π† γ2 0 physics. The Hartree-Fock theory predictions for field
HABA = (cid:20)v0(cid:18)π 0 (cid:19)+(cid:18)−02 δ′ γ5 (cid:19)(cid:21) strength B = 10 Tesla are summarized in Fig.3(a). At
− 2 J=1 ν = 5andν = 4 A +A LLsareoccupiedfirst. In
1 3
− − | ↑i
v2 0 π†2 γ2 0 addition to being favored by the single-particle energies,
− 0 + 2 , (6)
⊕ (cid:20)√2γ (cid:18)π2 0 (cid:19) (cid:18) 0 0(cid:19)(cid:21) thischoicegainsexchangeenergyandminimizesHartree
1 J=2
energy. At ν = 3 the B 0, LL is then occupied
2
− | ↑i
wheretheJ =1subbandsarelayerantisymmetricstates in agreement with the single-particle Hamiltonian, but
A A and B B while the J = 2 subbands also loweringthe interactionenergy. Notice the the gaps
1 3 1 3
| i−| i | i−| i
are layer symmetric states A + A and B . Note between occupied and empty states, which are impor-
1 3 2
| i | i | i
that the effective interlayer hopping energy is enhanced tant for transport, are greatly enhanced relative to the
by a factor of √2 compared to the AB bilayer case. single-particlegapsatallintermediatefillingfactors. The
Again,the mirrorsymmetry leadsto spontaneouscoher- smallest Fermi level gaps occur at filling factors ν = 5
−
ence states. The next-nearest-neighbor tunneling pro- and ν = 3, for which occupied and empty states are
−
cesses play a more essential role than in the ABC tri- separated by a gap between n = 0 and n = 1 orbitals
layer case. Here γ = 20 meV, γ = 40 meV and with the same layer structure. These two filling factors
2 5
δ′ = 50 meV22,25 lead to−band gaps for the J = 1 and should therefore have the weakest quantum Hall effects
J =2 chiralbranches separately,but no directgap over- inthe weak-disordersamples to whichthe presentcalcu-
all. When Zeeman splittings and interactions ignored, lations apply.
the duodectet has four-fold degenerate J = 2 eigenener- Although the LL filling order is universal for ν > 3,
−
gies at ǫ 10,0meV and two-fold degenerate J = 1 there is some field dependence for ν = 5, 4 and 3
≃ − − − −
eigenenergiesatǫ 10,30meV. Theseenergiesareequal as indicated in Fig.3. When the field strength exceeds
≃
to the B = 0 band energy extrema and are independent B = 17 T a LL reordering occurs at ν = 4. We find
c1
−
5
(a) B = 10 T
0
F
H
E
y −0.1 IV. DISCUSSION
g
er
n
E−0.2
F Single-layer, bilayer, and ABC and ABA trilayer
H
−0.3 graphene are a family of related but distinct two-
dimensional electron systems. Single-layer and bilayer
−6 −4 −2 0 2 4 6
graphenehavefourandeightfoldLandauleveldegenera-
(b) B = 15 T (c) B = 25 T (d) B = 35 T
cies that are lifted by interactions in the quantum Hall
0 regime when disorder is sufficiently weak, giving rise to
F
H a rich variety of broken symmetry states. In this pa-
E
gy −0.2 per we have exploredthe way in which interactions alter
ner the quantum Hall effect in both ABC and ABA trilayer
E−0.4
F graphene in the range of filling factors between ν = 6
H −
and ν = 6, over which their twelve-fold N = 0 Landau
−0.6
−5 −4 −3 −5 −4 −3 −5 −4 −3 levelis filled. As in the single-layerand bilayercasesour
Filling Factor ν calculations suggest that quantum Hall effects will even-
tually occur at all intermediate filling factors as sample
FIG. 3: (Color online) (a) Filling factor dependence of the qualityimprovestolessenthe roleofdisorder. Thereare
Hartree-Fock(HF)energies(eV)ofoccupiedandunoccupied
anumberofcleardifferencesbetweentheABCandABA
LLsforabalancedABAtrilayerat10T.UnoccupiedLLsare
cases which could provide a practical alternative to Ra-
quartets(green),doublets(blue),orsinglets(cyan);occupied
manscatteringforidentifyingthestackingordersofhigh
LLs are doublets (magenta) or singlets (red). (b)-(d) Field
quality trilayer samples: i) Because the N = 0 layer or-
dependenceoftheHFenergies(eV)ofthelowest fourLLsat
bitalsarelocalizedeitherinthetoporbottomlayerinthe
ν = −5,−4 and −3. Results for magnetic field B equal to
15 T are shown in (b), for 25 T in (c) and for 35 T in (d). ABCcase,whiletheyareoftenspreadacrosstwoormore
Magentaandreddenotethen=0and1orbitalsrespectively layersintheABAcase,interactioneffectsarestrongerfor
of LL |A1+A3 ↑i; green and blue denote the n = 0 and 1 ABCtrilayersandmorelikelytoinduceintermediatefill-
orbitals respectively of LL |B2 ↑i. The Zeeman splitting is ingfactorquantumHalleffects. ii)Perpendicularelectric
small enough to ignore in this figure and we assumed that fields have relativelysimple consequences in ABC trilay-
ε=1. ers, opening up gaps between layersthat will strengthen
thequantumHalleffectatν = 3andν =0,buthavea
±
muchmorecomplexbehaviorwithmanyquantumphase
that the pair of LLs B2 are then filled before the transitions in the ABC case. For ABC trilayers there is
| ↑i
A1+A3 pair. This transition occurs when Coulomb a single sharp phase transition from a spin-polarized (at
| ↑i
interaction physics overwhelms the single-particle next- highfield) orantiferromagnetic(atlow field) ν =0state
nearest layer tunneling effect. More localized layer or- to a layer polarized state with a critical field strength
bitals are preferred when interactions dominate because that is linear in B. iii) In ABA samples on substrates,
intralayerexchange energy gains dominate interlayerex- the single-particle splitting of N = 0 LLs might lead
change and Hartree energy costs. When the field is fur- to quantum Hall effects at ν = 2, ν = 2 and ν = 4
−
ther strengthened to Bc2 = 33 T LL B20 is filled with gaps 10 20 meV before the interaction driven
| ↑i ∼ −
before A1+A30 , at ν = 5. The two critical fields quantum Hall states appear at larger fields. iv) There
are rou|ghly determ↑iined by the−condition EH/2 ∼ ∆EL′L arephasetransitionsinABA trilayersatν =−5,−4,−3
which coincides approximately with the self-consistent between low-field states in which single-particle physics
numerical results for Bc1 and Bc2. determinestheorderinwhichLL’sareoccupied,tohigh-
We have focused here on interaction physics in the field states in which more localized orbitals with larger
N =0LL’sofABCandABAtrilayergraphene. Interest- exchange energies are occupied first. v) In high quality
ingeffectsareneverthelessparticularlylikelyforlargerN suspended ABC trilayers, ν = 0, 3, 6 states persist
± ±
intheABAcase. UnlikeABCtrilayerswhereonlyJ =3 down to zero magnetic field and smoothly connect with
bandsarepresentatlowenergies,both J =1 and2 sub- the electron-electron interaction driven broken symme-
bands appear at all energiesin ABA trilayers,leading to trystates8. We expectthatthesedifferenceswillbecome
LLcrossing22,25,35 betweenthetwochiralbrancheswhen apparent in future experiments.
ω /ω = n (n 1)/2n is satisfied. We anticipate InthispaperwehavereportedonHartree-Fockground
1 2 2 2 1
−
that Coulopmb physics will open gaps at the LL crossing states and quasiparticle energies as a function of filling
points,whosecharacteristicswoulddependontheorbital factor predicted by calculations which were restricted
indices of the crossing LLs. Because of the LL crossing only by the assumption of translational invariance but
effectspresentevenwithoutinteractions,the sequenceof otherwise allowedany symmetries of the Hamiltonian to
theplateausinABAtrilayershighlydependonthemag- be broken. We find that in the strong-magnetic-field
netic field strength. weak-disorder case the LL filling order tends to follow
6
a pattern in which states of a particular spin are filled couple to long-wavelengthelectromagnetic probes which
first,andwithinaspinstateswithdifferentvalleycontent wouldprovideinformationaboutthe many-bodyground
next. For ABAtrilayers,however,single-particlephysics state that is complementary to the transport activation
which places a bilayer-like group of states lower in en- gaps most related to the present study.
ergy than a monolayer-like group of states, can play an
important role. The rule which favors spin-polarization
overvalley(orequivalentlylayer)polarizationisweakbe-
cause the layers in this system are quite closely spaced,
Acknowledgments
anditis possiblethatthis rulewillbe overturnedbylat-
tice effects - as it appears to be in bilayers39. When M
of the 12 N = 0 Landau levels are occupied, there will F.Z. gratefully acknowledges helpful discussion with
be a set of M (12 M) collective modes whose ener- C.N. Lau, I. Zaliznyak, J. Velasco Jr, W. Bao and K.
× −
gies approach differences between quasiparticle energies Zou. ThisworkhasbeensupportedbyWelchFoundation
in the limit of long-wavelength. Many of these collec- grantTBF1473,NRI-SWAN,DOEgrantDivisionofMa-
tivemodes,whichcanbestudied49 usingtime-dependent terials Sciences and Engineering DE-FG03-02ER45958,
versions of the calculations reported in this paper, will and ARO W911NF-09-1-0527.
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