Quantum Hall ferromagnetic phases in the Landau level N=0 of a graphene bilayer

1.1 MB·English

by  J. Lambert

Quantum Hall ferromagnetic phases in the Landau level N = 0 of a graphene bilayer
J. Lambert1 and R. Cˆot´e1
1D´epartement de physique, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, J1K 2R1, Canada
(Dated: January 15, 2013)
InaBernal-stackedgraphenebilayer,anelectronicstateinLandaulevelN =0isdescribedbyits
guiding-centerindexX(intheLandaugauge)andbyitsvalley,spin,andorbitalindicesξ=±K,σ=
±1, and n = 0,1. When Coulomb interaction is taken into account, the chiral two-dimensional
3 electron gas (C2DEG) in this system can support a variety of quantum Hall ferromagnetic (QHF)
1 ground states where the spins and/or valley pseudospins and/or orbital pseudospins collectively
0 align in space. In this work, we give a comprehensive account of thephase diagram of theC2DEG
2 at integer filling factors ν ∈ [−3,3] in Landau level N = 0 when an electrical potential difference
n ∆B between the two layers is varied. We consider states with or without layer, spin, or orbital
a coherence. For each phase, we discuss the behavior of the transport gap as a function of ∆B, the
spectrumofcollectiveexcitationsandtheopticalabsorptionduetoorbitalpseudospin-wavemodes.
J
Wealsostudytheeffectofanexternalin-planeelectricfieldonacoherentstatethathasbothvalley
4
and spin coherence and show that it is possible, in such a state, to control thespin polarization by
1
varyingthestrength of theexternal in-plane electric field.

]
l PACSnumbers: 73.21.-b,73.22.Gk,72.80.Vp
l
a
h
- I. INTRODUCTION neously and collectively aligned in space13.
s
e Inbilayergraphene,atop-bottomgatesvoltageimbal-
m Electrons in a Bernal-stacked graphene bilayer1,2 be- ance can be applied to create a potential difference ∆
B
. have as a chiral two-dimensional gas of massive Dirac (whichwecallthe”bias”hereafter)betweenthetwolay-
t
a fermions (C2DEG)3. The chiral nature of the electrons ers. In dual-gated bilayer graphene, ∆B and the total
m leadtotransportandopticalpropertiesthataredifferent densityofelectronsinthe bilayercanbe controlledinde-
- from those of conventional semiconductor 2DEG’S or of pendently. Thisallowsthephasediagramofthe C2DEG
d the 2DEG in monolayer graphene. In particular, in the to be studied as a function of ∆B, magnetic field and
n absence of Coulomb interaction, the Landau level (LL) temperature. Such study has been done by Weitz et
o
spectrum is given by E = ℏω∗ N(N +1), where al.12 inhigh-qualitybilayergraphenesuspendedbetween
[c the Landau level index NN = 0±, 1,c 2,... and the effec- a top gate electrode and the substrate. The measure-
tive cyclotron frequency ω∗ = e±B/±mp∗c where B is the mentsshowaseriesofphasetransitionsbetweendifferent
1 c
magneticfieldandm∗ theeffectivemassoftheelectrons. QHFstates as ∆B is increasedatgivenfilling factorand
v
EachLandaulevelisfour-timedegeneratewhencounting magnetic field. Special attention has been given to the
4
9 valley and spin degrees of freedom with the exception of fillingfactorν =0wheretheprecisenatureoftheground
0 Landau level N = 0 which is eight-time degenerate. In- state when B 0 near zero bias is still debated14. All
→
3 deed, an electronic state in Landau level N =0 is speci- experiments were done at relatively small magnetic field
1. fied by its guiding-center X (in the Landau gauge), spin B <10Twiththeexceptionoftheexperimentsreported
0 σ = 1, valley ξ = K and orbital n = 0,1 indices1. in Ref. 9 where B reached 35 T.
± ±
3 (In Landau level N =0, valley and layer degrees of free- Various aspects of the QHF states in bilayer graphene
1 dom are equivalent.) When the Coulomb interaction is (in particular the nature and the evolution of the
:
v negligible with respectto the disorderbroadeningatlow ground state of the C2DEG near charge neutrality as
i temperature and when the small Zeeman splitting is ne- B 0) have been studied theoretically by a number
X glected,theeightstatesinN =0aredegenerateandthe of →authors15–23. In the work of Gorbar et al.17 and
ar Hall conductivity has plateaus at σxy = 4Me2/h where Shizuya19, the phase diagram of the C2DEG as a func-
M = 1, 2,...4. tion of bias for ν = 0,1,2,3 is presented. Gorbar et al.
± ±
Inrecenttransportexperiments5–12,itwasshownthat haveconsideredtheeffectofbothstatic17 anddynamical
in sufficiently pure sample,when disorderis low orwhen screenings18 of the Coulomb interaction. The modified
the magnetic field is large enough, the Coulomb inter- gapequation captures the linear scaling of the transport
action completely lifts the degeneracy of the N = 0 gapswiththemagneticfieldwhichisseeninalltheexper-
octet and lead to the formation of seven new plateaus imentsatlowmagneticfield. Thetransportgapsarealso
in the Hall conductivity i.e. σ = νMe2/h, where stronglyreducedwithrespecttotheunscreenedcaseand
xy
ν [ 3,3]. These plateaus were attributed to the for- become more comparable to those observed experimen-
∈ −
mation of broken-symmetry many-body ground states. tally. The dynamical screening was found to reproduce
These states can alternatively be described as quantum the offset in the behavior of the gap with magnetic field
Hall ferromagnets (QHF’s) where the spin and/or val- in the spin-polarized QHF state near zero bias which is
ley pseudospins and/or orbital pseudospins are sponta- seen in the experiments10. Shizuya points out that the
2
filled levels N 1 cannot be considered as completely of an electric field in the plane of the layers can produce
≤−
inert. Instead,they leadtoacorrectionΛ ofthe energy anewstatewhereallthreecoherences(layer,orbitaland
n
oftheorbitallevelsn=0,1thatcanchangetheordering spin)arepresent. Insuchastate,itispossibletocontrol
ofthesestatesinawaythatdependontheiroccupation. the degree of spin polarization by changing the strength
A negative capacitance effect is also found that suppress of the external in-plane electric field.
rotation of the valley pseudospins. We also present a study the properties of the different
ground states in the phase diagram. For all filling fac-
The work that we present in this paper extends our
tors,weshowhowthetransportgapsevolvewithbias. In
previous study of the phase diagram of the C2DEG at
zero bias13,24–26 and complements the study of Gorbar most cases, this evolution is qualitatively similar to that
obtained with screening corrections17. We compute all
et al. and Shizuya. We give a comprehensive account
the intra-LL collective excitations in the various phases
of the phase diagram of the C2DEG in bilayer graphene
showing that all coherent states but the orbital state
at all integer filling factors ν [ 3,3] in Landau level
∈ − are characterizedby a linearly-dispersing gapless (in the
N = 0 as a function of an applied bias ∆ at a fixed
B
long-wavelength limit) Goldstone mode. This mode be-
magnetic field. Our analysis is based on an effective
two-bandmodel27whichdescribesthelow-energyphysics comes gapped after the transition to an adjacent inco-
herentstate. Intheorbitalphase,theorbital-pseudospin
near the valleys K . We explicitly take into account as
±
Goldstone mode dispersion is anisotropic and this mode
symmetry-breaking terms the Zeeman splitting ∆ and
Z
becomes unstable at a finite wave vector indicating a
the bias ∆ . In the effective two-band model, the bias
B transition to a charge-density-wave state25,26. We iden-
∆ lifts the degeneracy between the orbital levels n=0
B
and n = 1 by a small amount ξβ∆ where β = γ /ℏω∗ tify the number of spin-waves and orbital modes in each
B 1 c phase. These later modes are active in optical absorp-
with γ the interlayer hopping between carbon atoms
1
tion. The inter-LL and some intra-LL magnetoexcitons
that are part of a dimer. We also include in our model
have been computed recently20–23 and we comment on
the interlayernextnearest-neighborhoppingtermγ be-
4
the difference with our results and how the presence of
tween carbons atoms in the same sublattices. This term
the inter-LLmagnetoexcitonsinthe spectrum maycom-
causes a small asymmetry in the electronic band struc-
ture and was neglected in previous studies17,19. It com- plicate the detection of the intra-LL excitations. The
mainresultsofourpaperaresummarizedinFig. 5(phase
bines with the correctionξβ∆ to give an energy differ-
B
diagram), Fig. 8 (transport gaps) and Fig. 9,10 (collec-
ence 2βγ γ /γ +ξβ∆ between the n=1 and n=0
1 4 0 B
≈ tive mode dispersions).
orbital levels where γ is the intralayer hopping energy
0
This paper is organized in the following way. Section
between nearest-neighbors. This correction is thus finite
II introduces the two-band model of bilayer graphene
atzerobiasandbreakstheorbitaldegeneracy. Ourphase
with the resulting LL spectrum in finite magnetic field.
diagramisnotelectron-holesymmetricaroundν =0and
SectionIIIsummarizestheHartree-Fockandgeneralized
the sequenceofphasetransitionsis differentforeachfill-
random-phase approximations that we use to take into
ing factor.
account the Coulomb interaction and gives the formal-
In our analysis, we treat the electron interaction in ismforthecalculationoftheelectromagneticabsorption.
theHartree-Fockapproximation(HFA)andcomputethe Our numerical results for the phase diagram, transport
collective excitations and electromagnetic absorption of gaps, collective excitations and optical absorption are
the different phases of the C2DEG in the generalized presented in Sec. IV. In Sec. V, we show how the ap-
random-phaseapproximation(GRPA).Weincludeinour plication of an in-plane electric field allow to control the
studybothuniformandnon-uniformstatesandallowfor spin polarization in some phases. We conclude in Sec.
thepossibilityofanytypeofcoherent(orQHF)state. By VI.
coherent state, we mean a state where the average value
c†ξ,σ,n,Xcξ′,σ′,n′,X 6=0 for ξ 6=ξ′ and/or σ 6=σ′ and/or
Dn=n′ where c† E creates an electron in a state with II. TWO-BAND MODEL OF BILAYER
6 ξ,σ,n,X GRAPHENE
quantumnumbersξ,σ,n,X.Inourphasediagramwhich
is summarized on Fig. 5, the layer-coherentstates occur
A. Crystal structure and tight-binding
at very small bias because of the small interlayer dis-
Hamiltonian
tance d = 0.34 nm in bilayer graphene. As the bias is
increased, we find around a critical bias corresponding
to the regions where the Hall conductivity ceases to be The crystalstructure of a Bernal-stackedgraphenebi-
quantized in the experiments12 a state with both layer layer is shown in Fig. 1. Each graphene layer is a two-
and spin coherence. The orbitalcoherentstates occur at dimensional crystal with a honeycomb lattice structure.
a much larger bias corresponding to the situation where The honeycomb lattice can be described as a triangu-
leveln=1getslowerinenergythanleveln=0invalley lar Bravais lattice with a basis of two carbon atoms
K− (see Fig. 4). In-between these coherent states are An and Bn where n = 1,2 is the layer index. The
variousincoherentstates,some ofwhichhave beenstud- two basis vectors are given by a1 = a0 1/2, √3/2 and
−
ied before17. Interestingly, we find that the application a = a (1,0), where a = 2.46 ˚A= √3c is the lattice
2 0 0 (cid:0) (cid:1)
3
constantof the underlying triangularBravaislattice and hopping between carbon atoms that are part of a dimer
c=1.42˚Aistheseparationbetweentwoadjacentcarbon (i.e. A B ,thesesitesarecalledthehigh-energysites),
1 2
−
atoms. The distance between the two graphene layers is γ theinterlayerNNhoppingtermbetweencarbonatoms
3
d=3.4˚A.IntheBernalstackingarrangement,theupper of different sublattices (i.e. A B ) and γ the inter-
2 1 4
−
A sublattice is directly on top of the lower B sublattice layer next NN hopping term between carbons atoms in
while the upper B sublattice is above the center of a the same sublattice (i.e. A A and B B ). The
1 2 1 2
− −
hexagonal plaquette of the lower layer. energy δ represents the difference in the crystal field be-
tween sites A ,B and sites A ,B . In this work, we
1 2 2 1
neglect the trigonal warping term γ , a correct approxi-
3
mation at sufficiently high magnetic field27.
If we define the spinor
γ
0 d† = a† b† a† b† , (2)
A B k,σ 1,k,σ 1,k,σ 2,k,σ 2,k,σ
1 1 (cid:16) (cid:17)
γ
γ 3 γ where a† b† creates an electron on site A(B) in
1 4 i,k,σ i,k,σ
layer i= 1,2(cid:16)with(cid:17)wave vector k and spin σ = 1, then
±
we can write the second-quantized tight-binding Hamil-
tonian in the basis (A ,B ,A ,B ) as
1 1 2 2
A B
2 2
H0 = d† H0(k)d , (3)
k,σ σ k,σ
k,σ
X
FIG. 1: (Color online) Crystal structure of a Bernal-stacked with the matrix
graphene bilayer.
H0(k)= (4)
σ
The Brillouin zone of the reciprocal Bravais lattice is 1∆ +δ
2 B γ Λ(k) γ Λ∗(k) γ
sphooinwtns itnoFbieg. 2. Wechoosethetwononequivalentvalley  −21σ∆Z −10∆ − 4 − 1 
γ Λ∗(k) 2 B 0 γ Λ∗(k)
where ξ =± is thKeξva=ll(cid:18)ey2aiπ0n(cid:19)de(cid:18)x.ξ23,0(cid:19), (1)  −−γ0−4Λγ1(k) −−γ124σ0Λ∆(kZ) −−−γ02121Λσ∆∗∆B(Zk) −−−−21γ4∆210σΛB∆(+kZ)δ ,
 
k where ∆Z = gµBB with g = 2 is the Zeeman energy.
y We havealsoincluded in H0 anexternaltransverseelec-
σ
tricfieldthatcreatesanelectricalpotentialdifference(or
bias) ∆ between the layers. The function Λ(k) is de-
B
fined by
K K
- + 3
kkk Λ(k)= eik·δi, (5)
xxx
i=1
X
where the summation is over the vectors connecting a
site A to its three nearest-neighbors in the same plane
1
i.e. δ = a 1/2,1/2√3 ,δ = a 1/2,1/2√3 ,δ =
1 0 2 0 3
−
a 0, 1/√3 .
0 − (cid:0) (cid:1) (cid:0) (cid:1)
Using Λ(K +k) √3a (ξk +ik )/2 and setting
(cid:0) (cid:1)ξ ≈ − 0 x y
γ = δ = ∆ = ∆ = 0, we find for the the band
4 B Z
FIG. 2: Brillouin zone of the triangular Bravais lattice and structure near the points Kξ the bands
definition of thevalleys indices K±.
p2
E (p) = γ , (6)
The electronic dispersion is obtained from a tight- 1,± ± 1± 2m∗
binding model with the parameters1: γ0 the nearest- p2
neighbor (NN) hopping in each layer, γ1 the interlayer E2,±(p) = ±2m∗, (7)
4
wherethemomentumpismeasuredwithrespecttoℏK i√nϕ (x) where ϕ (x) with n = 0,1,2,... are the
ξ n−1 n
−
and the effective mass is defined by eigenfunctions of the one-dimensional harmonic oscilla-
tor.
m∗ = 2ℏ2γ1. (8) For all calculations done in this paper, we choose2 for
3γ2a2 the value of the parameters
0 0
The band structure consists of four bands. In the ab- γ0 = 3.1 eV, (16)
sence of bias, the two middle bands meet at the six val- γ = 0.39 eV, (17)
1
ley points. The two high-energy bands are separated by
γ = 0.12 eV, (18)
a gap γ from the two middle, low-energy bands. The 4
1
bands E (p) remain degenerate at p = 0 when γ ,γ δ = 0.0156 eV. (19)
2,± 3 4
and ∆ are finite if ∆ = 0. This degeneracy is lifted by
B We havecheckedthatthe banddispersionobtainedwith
a finite ∆ .
B this choice of signs for the hopping terms is consistent
For a neutral bilayer, the chemical potential is at the
withthatreportedintheliterature29. Withthemagnetic
energy E = 0. The low-energy excitations (E << γ ) of
1 field in Tesla, we have
thetight-bindingmodelcanbestudiedusinganeffective
two-band model27,28. This model gives for each valley β = 8.86 10−3B, (20)
×
H0 (p)= (9) ζ1 = 0.404B meV, (21)
ξ,σ
ξ∆2B +η−ξp−p+ 1 (p ip )2 while
 −21σ∆Z 2m∗ x− y , ℏω∗ = 3.46B meV, (22)
 2m1∗ (px+ipy)2 −ξ∆2−B21+σ∆ηξZp+p−  ∆Zc = 0.1158B meV, (23)
  e2
α = =11.25√B meV. (24)
where we used the basis (A ,B ) for K and (B ,A )
2 1 − 1 2 κℓ
for K and defined p =p ip and
+ ± x y
± In the calculation of α, we take κ 5 for the effective
≈
1 ∆ γ δ dielectricconstantatthepositionofthe graphenelayers.
B 4
η = ξ +2 + . (10)
ξ 2m∗ γ γ γ At B =10 T,
(cid:18) 1 0 1(cid:19)
In this model, the presence of a quantizing perpen-
ζ /α = 3.5911 10−2√B =0.114, (25)
dicular magnetic field is accounted for by making the 1
×
Peierls substitution p P = p+eA/c (with e > 0), ∆ /α = 1.0291 10−2√B =0.0325, (26)
where A=B=B→z. Defining the ladder operators Z ×
a = (P∇ ×iP )ℓ/√2ℏ and a† = (P +iP )ℓ/√2ℏ with ℏωc∗/α = 0.30756√B =0.973. (27)
x y x y
−
the magnetic length ℓ=b ℏc/eB, we get
B. Landau levels and eigenstates of the
ξ∆2B +ζ1,−apa† ζ a2 non-interacting Hamiltonian
1σ∆ 2
H0 = −2 Z , (11)
ξ,σ  ζ2 a† 2 −ξ∆2−B21+σ∆ζ1Z,+a†a  speWcthreunmγo4f H=ξ0,δσ i=s g∆ivBen=by∆Z = 0, the Landau level
 (cid:0) (cid:1) 
where E0 =sgn(N) N (N +1)ℏω∗, (28)
N | | | | c
γ γ
ζ = β 2 1 4 +δ , (12) whereN =0, 1, 2,...isptheLandaulevelindexandsgn
1 γ ± ±
(cid:18) 0 (cid:19) is the signum function. The corresponding eigenvectors
ζ1,± = ζ1 ξβ∆B, (13) of a given spin are
±
2
δγ γ
ζ2 = βγ1 1+2 4 + 4 , (14) 1 h|N|−1,X(r) (29)
γ0γ1 (cid:18)γ0(cid:19) ! √2(cid:18)−sgn(N)h|N|+1,X(r) (cid:19)
and for N = 0. We use the Landau gauge A = (0,Bx,0)
6
where the eigenstates are
ℏω∗
β = c. (15)
γ1 h (r)= 1 e−iXy/ℓ2ϕ (x X) (30)
n,X n
L −
y
The effective cyclotron frequency is ω∗ = eB/m∗c.
c
In Eq. (11), the ladder operators are defined such with X the guiding-pcenter index. All Landau levels
that a†ϕ (x) = i√n+1ϕ (x) and aϕ (x) = N =0 are four-folddegenerate including spin and valley
n n+1 n
6
5
degrees of freedom in addition to the guiding-center de-
generacyN =S/2πℓ2 whereS isthe areaofthe 2DEG.
ϕ
The Landau level N = 0 is an exception because there 0.1
N=+2
are two degenerate spinors with zero energy which are
0.08
given,in the basis (A ,B ) for K and(B ,A ) for K ,
2 1 − 1 2 +
by 0.06
0.04 N=+1
0 0
, . (31) 0.02
h (r) h (r)
(cid:18) 0,X (cid:19) (cid:18) 1,X (cid:19) V)
e 0 N=0,n=0(2bands)
( N=0,n=1(2bands)
It follows that N = 0 is eight-fold degenerate. In this E N=0,n=0(4bands)
-0.02 N=0,n=1(4bands)
paper, we restrict the Hilbert space to the Landau level N=-1
N = 0 and use the index n = 0,1 to refer to the two -0.04
”orbitals” ϕn=0(x) and ϕn=1(x). With finite values of -0.06 N=-2
γ ,δ,∆ or∆ ,thevalley,spin,andorbitaldegeneracies
4 Z B
are lifted and the noninteracting energies become -0.08
-0.1
1 1 0 0.05 0.1 0.15
E0 = ξ∆ σ∆ , (32) ∆ (eV)
ξ,σ,n=0 −2 B − 2 Z B
1 1
E0 = ξ∆ σ∆ +ξβ∆ +ζ . (33)
ξ,σ,n=1 −2 B − 2 Z B 1
FIG. 3: (Color online) Comparison between the complete
The corresponding eigenspinors are still given by Eq. four-band model (symbols) and the approximate two-band
model (lines) for the electronic dispersion in Landau levels
(31). Note that the structure of the sublattice spinors
in Eqs. (31) is such that states from different valleys N =−2,−1,0,1,2 and valley K−.
(which are localized on different layers)have no overlap.
ForN =0,thelayerindexisthusequivalenttothevalley
index.
K_,n=1
ζ −β∆
C. Limit of validity of the two-band model 1 B
K_,n=0
Fig. 3 showsa comparisonbetweenthe four-bandand
∆
two-bandmodels for the electronic dispersion in Landau
B
K ,n=1
levels N = 2, 1,0,1,2 and valley K using the val-
− +
− −
ues of the parameters given previously. The agreement
ζ +β∆
between the two models is excellent for N = 0 where
1 B
the difference in energy is of the order of 1%. For levels
N >0,thedifferenceinenergybetweenthetwomodels
| | K ,n=0
is much more important. Note that the two sub-Landau +
levels of N = 0 intersect level N = 1 at ∆ 0.15 eV.
B
≈
For the valley K (not shown in the figure) the crossing
+
occurs at a smaller bias ∆ 0.10 eV corresponding to
B ≈ FIG. 4: Ordering of thefour levels of a given spin in Landau
an electric field E 300 meV/nm between the layers.
⊥ ≈ level N =0 at finite bias ∆B.
In our calculation we must keep the bias smaller than
0.10 eV (i.e. ∆ /α . 2.8 for κ = 5) for our model to
B
≈
be valid.
III. INTERACTING CHIRAL
Fig. 4 shows the ordering of the four levels of a given TWO-DIMENSIONAL ELECTRON GAS
spin in N = 0 at finite bias. The correction ζ opens a
1
gapbetweenthetwoorbitalstatesn=0andn=1which
We now add the Coulomb interaction to the noninter-
isindependentofthe bias. Theeffectivetwo-bandmodel
acting Hamiltonian. Hereafter, we use the same basis
introducesacorrectionβ∆ tothisgapthathasdifferent
B (A ,B ) for both valleys and define the field operators
signs in the two valleys as indicated in the figure. When 2 1
Ψ (r) by
combined with ζ , the gap in valley K is positive at ξ,σ,n
1 +
all biases while the gap in valley K changes sign (level
−
0
n∆=>10g.e0t4s6beeVlow(i.ele.v∆el n/α==0)1.w3hfoernκβ=∆B5)>. ζ1 i.e. for Ψ−,σ,n(r)= hn,X(r) ⊗|σic−,σ,n,X, (34)
B B X (cid:18) (cid:19)
X
6
and can be written in terms of these operators by (we adopt
the convention that repeated indices are summed over)
h (r)
Ψ (r)= n,X σ c . (35)
+,σ,n 0 ⊗| i +,σ,n,X
XX (cid:18) (cid:19) HHF =NϕEa,nρan,,an (41)
The second-quantized noninteracting part of the Hamil- N X(aξ,bξ) (0) ρa,b ρb,a ,
tonian is given by − ϕ n1,n4,n3,n2 n1,n2 n3,n4
where (cid:10) (cid:11)
H0 = drΨ†ξ,σ,n(r)Hξ0,σΨξ,σ,n(r) (36) E =E0 +α ν ν d, (42)
σX,ξ,nZ a,n a,n aξ − 2 ℓ
(cid:18) (cid:19)
= E0 c† c . e
ξ,σ,n ξ,σ,n,X ξ,σ,n,X withEa0,n givenby Eqs. (32-33)e. The Fockinteractionis
ξX,σ,nXX defined by
For the second-quantized Coulomb interaction, dpℓ2 1
X(aξ,bξ) (q) = α K (p) (43)
1 n1,n2,n3,n4 2π pℓ n1,n2
V = dr dr′Ψ† (r) (37) Z
2n1X,...,n4Xσ,σ′Xξ,ξ′Z Z ξ,σ,n1 ×Kn3,n4(−p)eiq×pℓ2e−pd(1−δaξ,bξ)
×Ψ†ξ′,σ′,n2(r′)Vξ,ξ′(r−r′)Ψξ′,σ′,n3(r′)Ψξ,σ,n4(r), with the form factors
where the Coulomb potential K0,0(q) = e−q24ℓ2, (44)
e2 K1,1(q) = e−q24ℓ2 1 q2ℓ2 , (45)
Vξ,ξ′(r)= κ r r′+(1 δξ,ξ′)dz (38) (cid:18) − 2 (cid:19)
has the Fourier transfor|m− − | K1,0(q) = e−q24ℓ2 (qy +√i2qx)ℓ , (46)
b (cid:18) (cid:19)
Vξ,ξ′(r)= S1 2κπqe2eiq·(r−r′)e−qd(1−δξ,ξ′), (39) K0,1(q) = e−q24ℓ2 (−qy√+2iqx)ℓ . (47)
q (cid:18) (cid:19)
X
These form factors capture the character of the two dif-
where q is a two-dimensional vector in the plane of the ferent orbital states. In Eq. (42), ν = ν + 4 is the
bilayer. The terms that do not conservethe valley index number of filled levels in N = 0. We reserve the sym-
in Eq. (37) are very small and usually neglected30. bol ν [ 3,3] for the filling factor of the C2DEG. In
e
∈ −
deriving Eq. (41), we have taken into account a neutral-
izing positive background so that the only contribution
A. Hartree-Fock Hamiltonian
fromthe Hartreeandbackgroundtermsis thecapacitive
energy given by the term in parenthesis in Eq. (42). In
Inordertodescribethedifferentphasesoftheuniform this term, ν = ρaξ,σ;aξ,σ(0) is the total filling
aξ n,σ n,n
C2DEG, we define the operators factor in valley a . Detailed expressions for the Hartree
ξ
P (cid:10) (cid:11)
(see next section) and Fock interactions H and X are
ρan,,an′′ = N1 c†a,n,Xca′,n′,X, (40) given in Apependix A of Ref. 25.
ϕ The Hartree-Fock energy per electron is given by
X
X
E 1
where c†a,n,X(ca,n,X) creates(destroys) an electron in a NHF = νEa0,n ρan,,an (48)
state (a,n,X). The index a combines the spin and val- e
slepyininodricveasllaenydinwdeexuseofaaξ a=nd(aaσ,ato)r.efeTrhteostehteosfpeacviefirc- +41ν dℓα eνK+ −(cid:10)νK−(cid:11)2
ξ σ
age values ρan,,an′′ gives a complete description of an −21νXn(a1ξ,n,b4ξe,)n3,n2(cid:0)(e0) ρan,1eb,n2 (cid:1)ρbn,3a,n4 ,
uniform grnouDnd stEatoe. They are the order parameters where N is the number of e(cid:10)lectron(cid:11)s(cid:10)in the(cid:11)2DEG and
of that state. The diagonal elements ρa,a are the e
filling factors of levels (a,n), while the offn-d,niagonal ele- νK± are the efilling factors of the two valleys.
(cid:8)(cid:10) (cid:11)(cid:9) Atq=0,theonlynonzeromatrixelementsoftheFock
ments are the ”coherences”. For nonuniform states, it is
necessary to define the order parameters ρa,a′ (G) interactions are
n,n′ 3
where G is a reciprocal lattice vector anndD ρan,,an′′(GE)o X0ξ,,0ξ,0,0(0) = ∆C,X1ξ,,1ξ,1,1(0)= 4∆C, (49)
the Fouriertransformof ρan,,an′′(r) . We refeDrthe readeEr X0ξ,,0ξ,1,1(0) = X1ξ,,1ξ,0,0(0)= 12∆C, (50)
to Refs. 25,26 where theDformalisEm for this case is dis- 1
cussed in more details. The Hartree-Fock Hamiltonian X1ξ,,0ξ,0,1(0) = X0ξ,,1ξ,1,0(0)= 2∆C, (51)
7
and the corresponding interlayer terms which must be inthegeneralizedrandom-phaseapproximation(GRPA).
computed numerically. We have defined In this approximation, χa,b,c,d (q,τ) is the solution
n1,n2,n3,n4
of the equation
π
∆ = α (52)
C
2
r
χa,b,c,d (q,iΩ ) (60)
with α=e2/κℓ. n1,n2,n3,n4 n
= χ(0)a,b,c,d (q,iΩ )
n1,n2,n3,n4 n
1
+ χ(0)a,b,e,e (q,iΩ )
B. Calculation of the order parameters ℏ n1,n2,n5,n6 n
H(eξ,gξ) (q)χg,g,c,d (q,iΩ )
We define the single-particle Matsubara Green’s func- × n5,n6,n7,n8 n7,n8,n3,n4 n
1
tion χ(0)a,b,e,f (q,iΩ )
−ℏ n1,n2,n5,n6 n
Gan,1b,n2(X,τ)=− Tτca,n1,X(τ)c†b,n2,X(0) , (53) ×Xn(e5ξ,n,f8ξ,)n7,n6(q)χfn,7e,,nc8,d,n3,n4(q,iΩn),
D E
where Tτ is the imaginary time ordering operator, such where Ωn is a bosonic Matsubura frequency and the
that the order parameters are given by Hartree interaction
1 1
ρa,b = Gb,a X,τ =0− . (54) H(aξ,bξ) (q) = K (q)K ( q) (61)
n1,n2 Nϕ n2,n1 n1,n2,n3,n4 qℓ n1,n2 n3,n4 −
X
Theeq(cid:10)uation(cid:11)ofmotioXnforthe G(cid:0)reen’sfunc(cid:1)tioninthe e−qd(1−δaξ,bξ).
×
Hartree-Fock approximationis
The two-particle Green’s functions χ(0)a,b,c,d (q,iΩ )
n1,n2,n3,n4 n
(iℏω +µ E )Ga,b (iω ) (55) satisfy the set of equations
n − a,n1 n1,n2 n
+Ua,c Gc,b (iω )=ℏδ δ ,
n1,n3 n3,n2 n n1,n2 a,b
[iℏΩ (E E )]χ(0)a,b,c,d (q,Ω )(62)
where µ is the chemical potential, ωn a fermionic Mat- n− b,n2 − a,n1 n1,n2,n3,n4 n
subara frequency and = ℏ ρa,d δ δ ℏ ρc,b δ δ
n1,n4 b,c n2,n3 − n3,n2 a,d n1,n4
Una1,c,n3 =Xn(a4ξ,n,c3ξ,)n1,n2(0) ρcn,4a,n2 (56) +(cid:10)Uma,,en1χ(cid:11)(m0,)ne,2b,,nc3,d,n4(q,Ω(cid:10)n) (cid:11)
Ue,b χ(0)a,e,c,d (q,Ω ).
are the self-consistent Fock potential(cid:10)s. (cid:11) − n2,m n1,m,n3,n4 n
The self-consistent Eq. (55) can be put in a 8 8 ma- Eq. (60)canbe representedby a setofbubbles (Hartree
×
trix form by defining superindices and then solved nu- terms) and ladder (Fock terms) diagrams. The function
merically in an iterative way in order to get the order χ(0)a,b,c,d is the Hartree-Fock approximations for the
parameters. n1,n2,n3,n4
two-particle Green’s functions. It includes the Hartree-
The Hartree-Fock equation of motion for the Green’s
Fock self-energy corrections but not the vertex correc-
function leads to the sum rule (at T =0 K)
tions. Note that two-particle Green’s functions depend
ρa,b 2 = ρa,a =νa, (57) only on the order parameters ρan,,bm computed in the
n,m n,n n HFA. Eqs. (60,62)canbe solvednumericallyby defining
Xb,m(cid:12)(cid:10) (cid:11)(cid:12) (cid:10) (cid:11) superindices and then writing t(cid:10)hem i(cid:11)n a 64 64 matrix
(cid:12) (cid:12) ×
where νa is the filling factor of the (a,n) level. By defi- form. The collective excitations are then given by the
n
nition polesoftheretardedGreen’sfunctionsχ(nR1,)na2,b,n,c3,d,n4(q,ω)
which are obtained by the analytic continuation iΩ
ρa,b = ρb,a ∗. (58) n →
n,m m,n ω + iδ of the corresponding two-particle Green’s func-
tions. To derive the dispersion relations, we follow these
(cid:10) (cid:11) (cid:10) (cid:11)
poles as the wave vector q is varied.
C. Collective modes in the generalized
random-phase approximation
D. Pseudospin description
To study the collective excitations, we compute the
two-particle Green’s functions
We showed above that the coherent states of the
C2DEG can be described by the set of order parame-
χa,b,c,d (q,τ) (59) ters ρa,a′ . These states are also quantum Hall fer-
n1,n2,n3,n4 n,n′
= N T ρa,b (q,τ)ρc,d ( q,0) romangDnets(QEoHF’s) andcanalsobe describedby using a
− ϕ τ n1,n2 n3,n4 − pseudospin language where the two valley states ξ = +
+Nϕ ρ(cid:10)an,1b,n2(q) ρcn,3d,n4(−q) (cid:11) (ξ = )areassociatedwithvalley-pseudospinup(down)
−
(cid:10) (cid:11)(cid:10) (cid:11)
8
and the two orbital states n = 0 (n = 1) with orbital- F. Electromagnetic absorption
pseudospin up (down).
Inthislanguage,thetotalspin,valleypseudospin,and The total current operator in second quantization is
orbital pseudospin components of the electron gas are given by J= J with
a a
given by
P
S = 1 ℏ c† σ(i) c , (63) Ja = drΨ†a,n(r)ja(r)Ψa,m(r), (73)
i 2Nϕ ξ,α,n,X α,β ξ,β,n,X Xn,mZ
ξX,n,XXα,β D E
1 The current operator ja(r) is derived from the Hamil-
Li = 2N c†ξ,α,n,Xσξ(i,ξ)′cξ′,α,n,X , (64) tonian in Eq. (9) by making the Peierls substitution
ϕ αX,n,XXξ,ξ′D E p → P = p+eAext/c and then taking the derivative
1 with respect to the external vector potential A
Oi = 2N c†ξ,α,n,Xσn(i,)n′cξ,α,n′,X , (65) ext
ϕ ξX,α,XnX,n′D E j = c ∂Ha0 (74)
a,i − ∂Ae
where σ(i)′s are the Pauli matrices and the total filling i (cid:12)(cid:12)Aei→0
factor is where i=x,y. This gives (cid:12)(cid:12)
1
ν = Nϕ ξ,αX,n,XDc†ξ,α,n,Xcξ,α,n,XE. (66) J= ℏ1 a ∆0a(z×da)= dddt, (75)
X
Note that tehese 10 fields do not provide a complete de- with b
scription of a state. One must also consider the 54 other
combinationsofindices(the 64orderparametersarenot ∆0a =Ea0,1−Ea0,0 =ζ1+aξβ∆B (76)
allindependent,however).WewillusebothS ,L ,O and
i i i i.e. the bare gap in valley ξ and
the order parameters ρa,b to characterize the ground
n,m
states of the C2DEG. dd i
(cid:10) (cid:11) dt =−ℏ HH0F,d , (77)
where H0 = N E0 ρa,a i(cid:2)s the no(cid:3)ninteracting Hamil-
E. Induced dipoles HF ϕ a,n n,n
tonian.
To compute the electromagnetic absorption per unit
The coupling of the C2DEG with a uniform external
area, we define the two-particle current-current Green’s
electric field in the plane of the layers is given by
function
1
HE =−e drn(r)φ(r) (67) χJα,Jβ(τ)=−S hTτJα(τ)Jβ(0)i, (78)
Z
where E = φ(r). The total density is given by which gives
ext
−∇
n(r)= Ψ† (r)Ψ (r). (68)
ξ,σ,n ξ,σ,m eℓ 2 1
σ,ξ,n,m χ (iΩ ) = ∆0∆0 (79)
X Jα,Jβ n ℏ πℓ2 a b
FouriertransformingEq. (67)andusingtheformfactors (cid:18) (cid:19) Xa,b
dneefiounsedstianteE2q0,s2.5,(4H4-47g)i,vewsetchaendsiphoowlarthcaotupinlinagnhomoge- ×χaρα,a,,ρbβ,b(q=0,iΩn),
E
where α,β = x,y and x = y,y = x. Using Eqs. (71,72),
we have for example
H = d E , (69)
E ext
− · 1
χa,b,c,d = χa,b,c,d+χa,b,c,d+χa,b,c,d+χa,b,c,d
withthe totaldipole vectordefinedby d= ada where ρx,ρx 4 1,0,1,0 0,1,1,0 1,0,0,1 0,1,0,1
(cid:16) (cid:17)(80)
d = √2ℓeN ρa,a(0)x ρa,a(0P)y (70) and similarly for the other components. The absorption
a − ϕ x − y
can only involve these four combinations of orbital in-
is the dipole moment in v(cid:0)alley a with spin a .(cid:1)We have
ξ σ diceswhateverthe polarizationofthe electricfieldofthe
b b
defined here
electromagnetic wave. The retarded current-current re-
1 sponse function χ (ω) is obtained from the analytic
ρa = ρa,a+ρa,a , (71) Jα,Jβ
x 2 0,1 1,0 continuation iΩn ω+iδ and the electromagnetic ab-
→
ρa = 1(cid:0) ρa,a ρa,a(cid:1) . (72) sorption for an electromagnetic wave of amplitude E0
y 2i 0,1 − 1,0 linearly polarized in the direction α is given by
Itispossibletocontrolthe(cid:0)orientation(cid:1)oftheorbitalpseu- P (ω)= 1 χJα,Jα(ω) E2. (81)
dospins in the x y plane with anexternal electric field. α −ℏℑ ω 0
− (cid:20) (cid:21)
9
This formula is valid at finite frequency only since we withα=x,yand∆0 givenbyEq. (76). Inthenumerical
a
have neglected the diamagnetic contribution to the cur- calculation, we introduce a small Landau level width in
rent. order to get a finite value for the optical absorption.
Wecanfollowthesametypeofreasoningtoshowthat,
in a phase with no orbital coherence but with possibly
G. Absorption in the incoherent phases layer and/or spin coherence, the functions χa,b,c,d(ω) =
1,0,1,0
χa,b,c,d(ω) = 0 and the absorption depends again only
0,1,0,1
If there is no coherence in a phase, then
on χa,a,b,b(ω) and χa,a,b,b(ω). In this special case, the
0,1,1,0 1,0,0,1
ρan,,bm = ρan,,an δn,mδa,b. (82) equation of motion for χa0,,1b,,c1,,d0(ω) is
In this case, we(cid:10)can so(cid:11)lve(cid:10)analy(cid:11)tically for the absorption
χa,a,b,b(ω) = χ(0)a,a,b,b(ω) (90)
because this restriction leads, from Eq. (62) to 0,1,1,0 0,1,1,0
1
χ(n01),an,2b,,nc,3d,n4(q,Ωn) (83) −ℏ χ(00,1)a,1,,a0,e,f(ω)X1e,ξ1,,f0ξ,0χf0,,1e,,b1,,b0(ω)
e,f
= χ(0)a,b,b,a (q,Ω )δ δ δ δ . X
n1,n2,n2,n1 n a,d b,c n1,n4 n2,n3
and a similar expression with 0⇋1 for χa,b,c,d(ω).
Now, at q=0 the only nonzero Fock interactions in the 1,0,0,1
GRPAequationsaregiveninEqs. (49-51)whiletheonly
Hartreeinteractionsthatneedtobeconsideredarethose
IV. PHASE DIAGRAM OF THE C2DEG
of the form
Hξ,−ξ (0),Hξ,−ξ (0),Hξ,−ξ (0),Hξ,−ξ (0). (84) At zero bias, the QHF states follow a set of Hund’s
0,0,0,0 1,1,1,1 0,0,1,1 1,1,0,0
rules: the spin polarization is maximized first, then the
These interlayer Hartree interaction involves combina- layer polarization is maximized to the greatest extent
tionsoftheforme−qd/q thatgiveafinitecontributionat possibleandfinallytheorbitalpolarizationismaximized
q=0(andalsoadivergingcontributionthatiscancelled to the extent allowed by the first two rules13. In this
bytheotherterms). ItfollowsthatEq. (60)givesforthe section, we study the phase transitions that occur when
GRPA response functions χa1,,0a,,1c,,0c(ω) = χa0,,1a,,0c,,1c(ω) = 0 a finite bias (or transverse electric field) is turned on.
and
χ(0)a,a,a,a(ω)δ
χa,a,c,c(ω)= 0,1,1,0 a,c , (85) A. Types of phases
0,1,1,0 1+ 1X(aξ,aξ)(0)χ(0)a,a,a,a(ω)
ℏ 1,1,0,0 0,1,1,0
h i Fig. 5 shows our numerical result for the phase dia-
i.e. valley and spin must be conserved in an optically gramoftheC2DEGasafunctionofanappliedtransverse
active electronic transition. Since electric field E = ∆ /ed for B = 10 T and κ = 5 and
B
χ(0)a,a,a,a(ω)= ρa0,,0a − ρa1,,1a (86) feoigrhatllnionnt-eignetrerfialclitninggfalecvteolrssbνy∈ho[−riz3o,n3]ta.lWlieneisndaincdatneutmhe-
0,1,1,0 ω+iδ ∆0+Ua,a Ua,a /ℏ
−(cid:10) a(cid:11) (cid:10)0,0 −(cid:11) 1,1 berthemaccordingtotheschemeindicatedinthetopin-
set. Note thatthe lines areonlyoffsetverticallyforclar-
(and a similar expression w(cid:0)ith 0 ⇋ 1 for χ(10,0)a,0(cid:1),,a1,a,a(ω)), ity. Their position does not reflect the true ordering of
we have easily the energy levels which changes with bias. We name the
phasesI ,L ,O andSL accordingtothetypeofcoher-
ρa,a ρa,a ν ν ν ν
χa,a,a,a(ω)= 0,0 − 1,1 (87) ence that is present: incoherent, layer-coherent, orbital-
0,1,1,0 ω+iδ−(cid:10) ∆0a(cid:11)+ ∆(cid:10)4C ρ(cid:11)a1,,1a /ℏ coherentorspin-layercoherentrespectively. When there
is more than one incoherent phase at a given filling fac-
and (cid:2) (cid:10) (cid:11)(cid:3) tor, we use the notation I∗ for the second phase, I∗∗ for
ν ν
ρa,a ρa,a the third phase and so on. The critical electric field for
χa,a,a,a(ω)= 1,1 − 0,0 . (88) thetransitionbetweentwophasesisindicatedbyE(1) to
1,0,0,1 ω+iδ+(cid:10) ∆0a(cid:11)+ (cid:10)∆4C ρ(cid:11)a1,,1a /ℏ E(18) and is in units of mV/nm. A circle on an energy
The functions χa,a,a,a(ω)(cid:2)and χa,a,a(cid:10),a(ω(cid:11))(cid:3)are the re- level represents a fully filled level while an ellipse that
0,1,1,0 1,0,0,1 connects two levels indicates a coherent superposition of
sponse to the two circular polarizations of light. The
thesetwostates. WelistinFig. 5somepropertiesofeach
absorption in an incoherent phase is finally given by
phase: spin polarizationS , number of Goldstonemodes
z
E2 eℓ 2 (G), number of collective modes gapped at the Zeeman
Pα(ω) = 4ℓ20ω ℏ ρa0,,0a − ρa1,,1a (89) energy(Z)andnumberofpeaksintheopticalabsorption
(cid:18) (cid:19) a spectrum(A). The transitionbetweena coherentandan
X(cid:0)(cid:10) (cid:11) (cid:10) (cid:11)(cid:1)
∆0 2δ ℏω ∆0 ∆C ρa,a incoherentphaseiscontinuouswhileatransitionbetween
× a − a− 4 1,1 two incoherent phases is discontinuous.
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:10) (cid:11)
10
L :S=1/2(1G,2Z,2A) I :S=1/2(0G,1Z,1A)
-3 z -3 z K,+,1 K,+,1 K,-,1 K,-,1
+ - + -
7 3 8 4
E(1)=0.11 5 1 6 2
L-2:Sz=1(1G,2Z,2A) I-2:Sz=1(0G,1Z,0A) K+,+,0 K-,+,0 K+,-,0 K-,-,0 I-2*:Sz=0(0G,0Z,1A)
E(3)=459
E(2)=0.35
L-1:Sz=3/2(1G,2Z,2A) I-1:Sz=3/2(0G,2Z,1A) S(L1G-1:,S2zZ∈,2[A3/)2,1/2] I-1*:Sz=1/2(0G,1Z,1A)
E(4)=0.23 E(5)=11.94 E(6)=12.15
I0,Sz=2(0G,2Z,0A) S(1LG0:,S2zZ∈,2[2A,0)] I0*:Sz=0(0G,0Z,0A)
E(7)=12.88 O:S=1/2
E(8)=13.62 (11G,1zZ,0A) I**:S=1/2 O
L1:Sz=3/2(1G,2Z,2A) I1:Sz=3/2(0G,2Z,1A) SL1:Sz∈[3/2,1/2] I1*:Sz=1/2(0G,1Z,1A) +nonuniformstates (01G,z1Z,1A) (
(1G,2Z,2A)
E(9)=0.11 E(10)=14.44 E(12)=135 E(13)=500
L2:Sz=1(1G,2Z,2A) I2:Sz=1(0G,1Z,0A) E(11)=14.85 I2*:Sz=0(0G,0Z,1A)
E(15)=1097
O:S=1/2
E(14)=0.35 (13G,1zZ,0A) I*:S=1/2
L3:Sz=1/2(1G,2Z,2A) I3:Sz=1/2(0G,1Z,1A) +nonuniformstates (30Gz,1Z,1A)
E(16)=0.23 E(17)=135 E(18)=500
E(mV/nm)
FIG. 5: (Color online) Phase diagram of the C2DEG in Landau level N = 0 at B = 10 T and for κ = 5 as a function of
the transverse electric field beween the layers for integer filling factors ν ∈ [−3,3] (from top to bottom). A filled (red) circle
representsafilledstatewhileafilled(blue)ellipseindicatesacoherentsuperpositionoftwolevels. Thenumberingofthelevels
is indicated in the inset at the top right of the figure. The E(i)′s indicate the critical perpendicular electric field (in mV/nm)
required for the transition between two phases. Also indicated for each phase are the spin polarization Sz, the number of
Goldstone mode(G), of modesgapped at theZeeman energy (Z) andthenumberof peaksin theoptical absorption spectrum
(A). The mention ”+ nonuniform states” for the O and O phases signals that this portion of the phase diagram is further
1 3
subdivided intouniform and nonuniform states as indicated in Eq. (109).
1. Incoherent phases Iν cent incoherent phases are readily compared to extract
the critical biases. We find, with ∆(i) =edE(i),
B
The I phases have no coherence of any kind and so
ν
L ,S ,O = 0 (the parallel component is in the plane
k k k
of the bilayer). Each level is either full or empty and
so L ,S and O vary from one phase to another. The
z z z
state corresponding to a specific diagram is easily read
∆ ζ + 3∆
from Fig. 5. For I2, we have for example I−2 → I−∗2 :∆(B3) = Z − 1β 8 C, (92)
Ψ = c† c† 0 , (91) ∆ +ζ + 5∆
| I2i 7,X 5,X| i I I∗ :∆(15) = Z 1 8 C. (93)
YX +2 → +2 B β
andthe orderparameters ρ = ρ =1.We include
5,5 7,7
h i h i
the phases I∗ ,I∗∗ and I∗ in the phase diagram only to
±2 1 3
make it more complete. Indeed, the bias can’t produce
any more transition after these states. But, the critical
biasneededtoreachthesestatesiswelloutsidethelimits If we ignore the SL phases at ν = 0, 1, we find that
ν
±
of validity of our two-band model. the transition between the two incoherent phases occurs
Using Eq. (48), the Hartree-Fock energy of two adja- in the middle of the SL phase (see Sec. IV.A.4 below)
ν