Table Of ContentInfluence of Landau level mixing on the properties of
elementary excitations in graphene in strong magnetic field
Yu.E. Lozovik1,2∗ and A.A. Sokolik1
1Institute for Spectroscopy RAS, 142190, Troitsk, Moscow Region, Russia
2Moscow Institute of Physics and Technology, 141700, Moscow, Russia
MasslessDiracelectronsingraphenefillLandaulevelswithenergiesscaledassquarerootsoftheir
numbers. Coulomb interaction between electrons leads to mixing of different Landau levels. The
2 relativestrengthofthisinteractiondependsonlyondielectricsusceptibilityofsurroundingmedium
1 andcanbelargeinsuspendedgraphene. WeconsiderinfluenceofLandaulevelmixingontheprop-
0 ertiesofmagnetoexcitonsandmagnetoplasmons—elementaryelectron-holeexcitationsingraphene
2 in quantizing magnetic field. We show that, at small enough background dielectric screening, the
n mixing leads to very essential change of magnetoexciton and magnetoplasmon dispersion laws in
a comparison with the lowest Landau level approximation.
J
PACSnumbers: 73.22.Pr,71.35.Ji,73.43.Mp,71.70.Gm
3
1
INTRODUCTION by electrontransitionsfromone ofthe filled Landaulev-
]
l els to one of the empty levels [14]. Such transitions can
l
a be observed in cyclotron resonance or Raman scatter-
Two-dimensional systems in strong magnetic field
h
ing experiments as absorption peaks at ceratin energies.
- are studied intensively since the discovery of integer
s and fractional quantum Hall effects [1–3]. For a long With neglect of Coulomb interaction, energy of the ex-
e citedelectron-holepairisjustadistancebetweenLandau
m time, such systems were representedby gallium arsenide
levels of electronand hole. Coulombinteraction leads to
heterostructures with two-dimensional electron motion
. mixing of transitions between different pairs of Landau
t within each subband [4].
a levels, changing the resulting energies of elementary ex-
m New and very interesting realization of two-
citations.
dimensional electron system appeared when graphene,
-
d a monoatomic layer of carbon, was successfully isolated Characteristic energy of Coulomb interaction in mag-
n [5, 6]. The most spectacular property of graphene is the neticfieldise2/εlH,whereεisadielectricpermittivityof
o factthatitselectronsbehaveasmasslesschiralparticles, surrounding medium. The relative strength of Coulomb
c obeying Dirac equation. Intensive experimental and interactioncanbe estimated as ratio of its characteristic
[
theoretical studies of this material over several recent value to a characteristicdistance betweenLandaulevels.
2 years yielded a plethora of interesting results [7–9]. In For massive electrons in semiconductor quantum wells,
v particular, graphene demonstrates unusual half-integer this ratio is proportional to H−1/2, thus in asymptoti-
6 quantum Hall effect [6], which can be observed even at cally strong magnetic field the Coulomb interaction be-
7
room temperature [10]. comesaweakperturbation[15,16]. Inthiscase,thelow-
1
In external perpendicular magnetic field, the mo- est Landaulevel approximation,neglecting Landau level
1
. tion of electrons along cyclotron orbits acquires zero- mixing,isoftenused. ItwasshownthatBose-condensate
1
dimensional character and, as a result, electrons fill dis- of noninteracting magnetoexcitons in the lowest Landau
1
crete Landau levels [11]. In semiconductor quantum levelasanexactgroundstateinsemiconductorquantum
1
wells, Landau levels are equidistant and separation be- well in strong magnetic field [17].
1
: tween them is determined by the cyclotron frequency A different situation arises in graphene. The relative
v
ω =eH/mc. Ingraphene,duetomasslessnatureofelec-
i c strength of Coulomb interaction in this system can be
X trons, “ultra-relativistic” Landau levels appear, which expressed as r = e2/εv and does not depend on mag-
s F
r are non-equidistant and located symmetrically astride netic field [18]. The only parameter which can influence
a the Dirac point [12, 13]. Energies of these levels are it is the dielectric permittivity of surrounding medium
En = sign(n) 2nvF/lH, where n = 0, 1, 2,..., ε. At small enough ε, mixing between different Landau
| | ± ±
vF 106m/s is the Fermi velocity of electrons and levels can significantly change properties of elementary
≈ p
l = c/eH is magnetic length, or radius of the cy- excitations in graphene.
H
clotron orbit (here and below we assume ~=1).
Coulomb interaction leads to appearance of two types
p
Inthecaseofintegerfilling,whenseveralLandaulevels
of elementary excitations from the filled Landau levels.
arecompletelyfilledbyelectronsandallhigherlevelsare
From summation of “ladder” diagrams we get magne-
empty, elementary excitations in the system are caused
toexcitons, which can be imagined as bound states of
electron and hole in magnetic field [14, 16, 19]. Prop-
erties of magnetoexcitons in graphene were considered
in several works, mainly in the lowest Landau level ap-
∗ [email protected] proximation [20–24]. At ε 3, Landau level mixing was
≈
2
shown to be weak in the works [20, 25]. I. MAGNETOEXCITONS
Note that influence of Landau level mixing on proper-
Electrons in graphene populate vicinities of two
tiesofaninsulatinggroundstateofneutralgraphenewas
nonequivalent Dirac points in the Brillouin zone, or two
consideredin[26]bymeansoftight-bindingHartree-Fock
valleys K and K′. We do not consider intervalley scat-
approximation. It was shown that Landau level mixing
tering and neglect valley splitting, thus it is sufficient to
favors formation of insulating charge-density wave state
consider electrons in only one valley and treat existence
instead of ferromagnetic and spin-density wave states in
of the other valley as additional twofold degeneracy of
suspendedgraphene,i.e. atweakenoughbackgrounddi-
electron states.
electric screening.
We consider magnetoexciton as an electron-hole pair,
and we will denote all electron and hole variables by the
From the experimental point of view, the most inter-
indices 1 and 2 respectively. In the valley K, Hamilto-
esting are magnetoexcitons with zero total momentum,
nian of free electrons in graphene in the basis A ,B
whichareonlyabletocouplewithelectromagneticradia- 1 2
{ }
of sublattices takes a form [7]:
tionduetoverysmallphotonmomentum. Forusualnon-
relativisticelectrons,magnetoexcitonenergyatzeromo-
mentum is protected against corrections due to electron H1(0) =vF√2 p0 p10− , (1)
1+
interactionsbytheKohntheorem[27]. However,forelec- (cid:18) (cid:19)
trons with linear dispersion in graphene the Kohn theo- where p = (p ip )/√2 are the cyclic components
1± 1x 1y
rem is not applicable [21, 24, 28–32]. Thus, observable of electron momen±tum and v 106m/s is the Fermi
F
energiesofexcitonicspectrallinescanbeseriouslyrenor- velocity of electrons. ≈
malized relatively to the bare values, calculated without For external magnetic field H, parallel to the z axis,
taking into account Coulomb interaction. we take the symmetrical gauge, when A(r) = 1[H
2 ×
r]. Introducing the magnetic field as substitution of the
The other type of excitations can be derived using
momentum p p +(e/c)A(r ) in (1) (we treat the
1 1 1
therandomphaseapproximation,correspondingtosum- →
electron charge as e), we get the Hamiltonian of the
mation of “bubble” diagrams. These excitations, called −
form:
magnetoplasmons, are analogue of plasmons and have
beenstudiedbothintwo-dimensionalelectrongas[14,33] H = vF√2 0 a1 . (2)
andgraphene[18,20,21,24,34–39](bothwithandwith- 1 l a+ 0
H (cid:18) 1 (cid:19)
out taking into account Landau level mixing).
Here the operators a = l p ir /2l and a+ =
1 H 1− − 1− H 1
In the present article, we consider magnetoexcitons l p + ir /2l (where r = (x iy )/√2) obey
H 1+ 1+ H 1± 1 1
and magnetoplasmons with taking into account Landau bosonic commutation relation [a ,a+]=±1.
1 1
level mixing and show how the properties of these ex- Using this relation, by means of successive action of
citations change in comparison with the lowest Landau the raising operator a+ we can construct Landau levels
1
level approximation. For magnetoexcitons, we take into for electron [18] with energies
account the mixing of asymptotically large number of
v
Landau levels and find the limiting values of cyclotron EnL =sn 2|n|lF (3)
resonance energies. H
p
and wave functions
For simplicity and in order to stress the role of vir-
tual transitions between different pairs of electron and ψ (r)= √2 δn0−1 snφ|n|−1,k(r) . (4)
nk φ (r)
hole Landau levels (i.e. the role of two-particle pro- (cid:16) (cid:17) (cid:18) |n|k (cid:19)
cesses),herewedonottakeintoaccountrenormalization Here k =0,1,2,... is the index of guiding center, which
ofsingle-particleenergiesviaexchangewiththefilledlev- enumerateselectronstatesonthen-thLandaulevel(n=
els. This issue have been studied in several theoretical ,...,+ ), having macroscopically large degeneracy
works [20, 21, 24, 30, 40]. Correction of Landau level −N∞=S/2π∞l2 ,equaltoanumberofmagneticfluxquanta
φ H
energies can be treated as renormalization of the Fermi penetrating the system of the area S. Eigenfunctions
velocity,dependentontheultravioletcutoffforanumber φ (r)ofatwo-dimensionalharmonicoscillatorhavethe
nk
ofthefilledLandaulevelstakenintoaccountinexchange explicit form:
processes.
The rest of this article is organized as follows. In Sec- φnk(r)= i|n−k| min(n,k)! e−r2/4l2H
tion 2 we present a formalism for description of magne- √2πlHsmax(n,k)!
toexcitons in graphene, which is applied in Section 3 to x+is y |n−k| r2
study influence ofCoulombinteractionandLandaulevel n−k L|n−k| , (5)
× √2l min(n,k) 2l2
mixing on their properties. In Section 4 we study mag- (cid:18) H (cid:19) (cid:18) H(cid:19)
netoplasmons in graphene in the random phase approxi- s =sign(n) and Lα(x) are associated Laguerre polyno-
n n
mation and in Section 5 we formulate the conclusions. mials.
3
Consider now the hole states. A hole wave function b = l p ir /2l , b+ = l p + ir /2l , b =
is a complex conjugated electron wave function, and the l1p Hir−/−2l ,−b+ H= l1 p +Hir+/2l +contHain o2nly
H + − + H 2 H − − H
hole charge is +e. Thus, we can obtain Hamiltonian of therelativeelectron-holecoordinateandmomentumand
the hole in magnetic field fromthe electron Hamiltonian obey commutation relations [b ,b+]=1, [b ,b+]=1 (all
1 1 2 2
(2) by complex conjugation and reversal of the sign of other commutators vanish).
the vectorpotential A(r ). In the representationof sub- Thus, the Hamiltonian (7) of electron-hole pair in its
2
lattices A ,B it is center-of-mass reference frame takes the form
2 2
{ }
v √2 0 a 0 b2 b1 0
where the operaHto2rs=a FlH= l(cid:18)pa+2 02ir(cid:19),/2l and a+(6=) H0′ = vFlH√2−b0b+1+2 −b00+ 00b+ −b0b12 . (10)
clHomp2m−u+taitrio2n−/r2ellHatioconm[2amu,ate+H]w=2it+h1−.a1E,n2a+er+1giaeHsndofotbheey2htohlee Afour-componentwavefunctio1nof−ele2ctron-holerelative
2 2 motionΦ ,beinganeigenfunctionof(10),canbecon-
Landau levels are the same as these of electron Landau n1n2
structed by successive action of the raising operators b+
levels (3), but have an opposite sign. 1
and b+ (see also [20, 21]):
Hamiltonian of electron-hole pair without taking into 2
a(6c)c,oaunndt cLaanndbaeurelpevreeslemntiexdinignitshejucsotmtbhienesudmbaosfis(o2f)ealencd- Φn1n2(r)= √2 δn1,0+δn2,0−2
tron and hole sublattices {A1A2,A1B2,B1A2,B1B2} as sn1sn2φ(cid:16)|n1|−(cid:17)1,|n2|−1(r)
s φ (r)
0 a2 a1 0 n1 |n1|−1,|n2| . (11)
H0 =H1+H2 = vFlH√2 aa+2+1 00 00 aa12 . (7) × sn2φφ||nn11|||,n|n22|(|−r)1(r)
0 a+1 a+2 0 The bare energy of magnetoexciton in this state is a dif-
ferencebetweenenergies(3)ofelectronandholeLandau
Itis known[41]thatforelectron-holepairinmagnetic
levels:
field there exists a conserving two-dimensional vector of
magnetic momentum, equal in our gauge to E(0) =EL EL . (12)
n1n2 n1 − n2
e
P=p +p [H (r r )] (8) Here we label the state of relative motion by num-
1 2 1 2
− 2c × −
bers of Landau levels n and n of electron and hole re-
1 2
and playing the role of a center-of-mass momentum. spectively. The whole wave function of magnetoexciton
The magnetic momentum is a generator of simultane- (9) is additionally labeled by the magnetic momentum
ous translation in space and gauge transformation, pre- P. In the case of integer filling, when all Landau lev-
serving invariance of Hamiltonian of chargedparticles in els up to ν-th one are completely filled by electrons and
magnetic field [42]. all upper levels are empty, magnetoexciton states with
The magnetic momentum commutes with both n > ν, n ν are possible. For simplicity, we neglect
1 2
≤
the noninteracting Hamiltonian (7) and electron-hole Zeeman and valley splittings of electron states, leading
Coulomb interaction V(r r ). Therefore, we can find to appearance of additional spin-flip and intervalley ex-
1 2
−
a wave function of magnetoexciton as an eigenfunction citations [20, 21, 24].
of (8):
ΨPn1n2(r1,r2)= 21π exp iR P+ [ez2l×2 r] II. INFLUENCE OF COULOMB INTERACTION
(cid:26) (cid:18) H (cid:19)(cid:27)
×Φn1n2(r−r0). (9) NowwetakeintoaccounttheCoulombinteractionbe-
tweenelectronandholeV(r)= e2/εr,screenedbysur-
Here R=(r1+r2)/2, r=r1−r2, ez is a unit vector in rounding dielectric medium wit−h permittivity ε. Upon
the direction ofthe z axis. The wavefunction of relative
switching into the electron-hole center-of-mass reference
motion of electron and hole Φ (r r ) is shifted on
the vector r =l2 [e P]. Thni1sns2hift−can0 be attributed frame, it is transformed as V′(r) = V(r+r0). To ob-
0 H z × tain magnetoexciton energies with taking into account
to electric field, appearingin the moving referenceframe
Coulomb interaction, we should find eigenvalues of the
of magnetoexciton and pulling apart electron and hole.
full Hamiltonian of relative motion H′ = H′ + V′ in
Transformation (9) from Ψ to Φ can be considered 0
the basis of the bare magnetoexcitonic states (11). As
as a unitary transformation Φ = UΨ, corresponding a
discussed in the Introduction, a relative strength of the
to a switching from the laboratory reference frame to
Coulomb interaction is described by the dimensionless
the magnetoexciton rest frame. Accordingly, we should
parameter
transform operators as A SAS+. Transforming the
operators in (7), we get: →Sa S+ = b , Sa+S+ = b+, e2 2.2
Sa2S+ = −b2, Sb+2S+ = −1b+2. He1re th1e operato1rs rs = εvF ≈ ε . (13)
4
2.8 2.8 2.8
E -1 fi 1
n1n2 -1 fi 1
v /l 2.4 -1 fi 1 0 fi 4
F H 0 fi 4 0 fi 3
2.4 0 fi 3
0 fi 3 1.8
2 0 fi 2
0 fi 2
2
1.6 0 fi 1
0 fi 2
0.8
1.2 0 fi 1
1.6
0 fi 1
0.8
n =0 n =0 -0.2 n =0
1.2 r =0.5 0.4 rs =1 rs =2
s
a b c
0.8 0 -1.2
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
Pl Pl Pl
H H H
2 2 2
E
n1n2 0fi 2 0fi 2 0fi 2
v /l
F H1.6 1 fi 5 1.6 1 fi 5 1.5 1 fi 5
1 1 fi 4
1.2 1 fi 4
1.2 1 fi 4 1 fi 3
0.5
0.8 1 fi 3
1 fi 2
1 fi 3 0
0.8
1 fi 2 0.4 1 fi 2 -0.5
n =1 n =1 n =1
0.4 rs =0.5 0 rs =1 -1 rs =2
d e f
0 -0.4 -1.5
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
Pl Pl Pl
H H H
FIG. 1. Magnetoexciton dispersions En1n2(P), calculated in the first order in Coulomb interaction (dotted lines) and with
takingintoaccountmixingbetween16low-lyingmagnetoexcitonstates(solidlines). Thedispersionsarecalculatedatdifferent
filling factors ν and different r : (a) ν =0, r =0.5, (b) ν =0, r =1, (c) ν =0, r =2, (d) ν =1, r =0.5, (e) ν =1, r =1,
s s s s s s
(f) ν = 1, r = 2. Dispersions of 5 lowest-lying magnetoexciton states n → n , indicated near the corresponding curves, are
s 2 1
shown.
When ε 1, r 1 and we can treat Coulomb inter- order in Coulomb interaction (14) have been performed
s
≫ ≪
action as a weak perturbation and calculate magnetoex- in several works [20–24]. However, such calculations are
citon energy in the first order in the interaction as: well-justified only at small enough r , i.e. at large ε.
s
When ε 1 (this is achievable in experiments with sus-
E(1) (P)=E(0) + Φ V′ Φ . (14) ∼
n1n2 n1n2 h n1n2| | n1n2i pended graphene [43–46]), the role of virtual electron
transitions between different Landau levels can be sig-
Due to spinor nature of electron wave functions in
nificant.
graphene, the correction (14) to the bare magnetoexci-
ton energy (12) is a sum of four terms, each of them
having a form of correctionto magnetoexciton energy in To take into account Landau level mixing, we should
usual two-dimensional electron gas [20–22]. Dependence perform diagonalization of full Hamiltonian of Coulomb
ofmagnetoexcitonenergyonmagneticmomentumPcan interacting electrons in some basis of magnetoexcitonic
be attributed to Coulomb interaction between electron states ΨPn1n2, where electron Landau levels n1 > ν are
and hole, separated by the average distance r P. unoccupied and hole Landau levels n ν are occupied.
0 2
∝ ≤
Calculationsofmagnetoexcitondispersionsinthe first To obtain eigenvalues of the Hamiltonian, we need to
5
0.95 0.52 -0.2
E(N) a n =0 b n =0 c n =0
10 r =0.5 r =1 r =2
v /l s s -0.4 s
F H 0.48
0.94
-0.6
0.44
-0.8
0.93
0.4
-1
0.36 -1.2
0.92
-1.4
0.32
0.91 -1.6
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
N N N
0.266 -0.05 -0.6
E(N)
21 d n =1 e n =1 f n =1
vF/lH r =0.5 r =1 r =2
0.264 s s s
-0.06 -0.7
0.262
-0.07 -0.8
0.26
-0.08 -0.9
0.258
-0.09 -1
0.256
0.254 -0.1 -1.1
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
N N N
FIG. 2. Magnetoexciton energies at rest En(N1n)2(P = 0), calculated with taking into account N electron and N hole Landau
levels,withstepwiseincreasingN (crosses). Thefitstotheseenergieswithinverse-square-rootfunction(solidlines)andlimiting
values of En(N1n)2(P = 0) at N → ∞ (dotted lines) are also shown. The results are presented for different filling factors ν and
different r : (a) ν =0, r =0.5, (b) ν =0, r =1, (c) ν =0, r =2, (d) ν =1, r =0.5, (e) ν =1, r =1, (f) ν =1, r =2.
s s s s s s s
solve the equation: takingintoaccountthe mixingbetween16lowest-energy
states. The results are shown for Landau level fillings
det δn′1n1δn′2n2(En(01)n2 −E) ν = 0 and ν =1, and for different values of rs. Close to
P = 0, magnetoexciton can be described as a compos-
(cid:13)
+hΨ(cid:13)(cid:13)Pn′1n′2|V|ΨPn′1n′2i =0. (15) ite particle with parabolic dispersion, characterized by
some effective mass M = [d2E (P)/dP2]−1 .
We can constrain our basis to N2 (cid:13)(cid:13)terms, involving N AtlargeP,theCoulomnb1inn2teractionnw1ne2akensandthe|Pd=is0-
(cid:13)
Landaulevelsforelectron(n =ν+1,...,ν+N)andN persions tend to the energies of one-particle excitations
1
Landaulevelsforahole(n =ν,...,ν N+1). Sincethe (12). However,the dispersion can be rather complicated
2
−
HamiltoniancommuteswithmagneticmomentumP,the structurewithseveralminimaandmaximaatintermedi-
procedure of diagonalization can be performed indepen- ate momenta P l−1.
∼ H
dently at different values of P, resulting in dispersions Weseethatthemixingatsmallr hasaweakeffecton
s
E(N) (P) of magnetoexcitons, affected by a mixing be- the dispersions (solid and dotted lines are very close in
n1n2
tween N electron and N hole Landau levels. Fig. 1(a,d)). However, at r 1 the mixing changes the
s
∼
We present in Fig. 1 dispersion relations for 5 low- dispersions significantly. We can observe avoided cross-
est magnetoexciton states, calculated with and without ings between dispersions of different magnetoexcitons,
6
and even reversal of a sign of magnetoexciton effective using magnetoexciton wave functions (11) and energies
masses (see Fig. 1(b,c,e,f)). Also we see that the high (12) (see also [18, 32, 34–38]):
levels are more strongly mixed than the low-lying ones.
f f
Scoimncillaursiorensuthltastwtehreempirxeisnegntiesdwienak[2.0] for rs = 0.73 with Π(q,ω)=gnX1n2 ω−nE2n(−01)n2n+1 iδFn1n2(q), (18)
Aswesee,atlarger themixingofseveralLandaulev-
s
elsalreadystronglychangesmagnetoexcitondispersions. F (q)=Φ+ (ql2 )
Important question arises here: how many Landau lev- n1n2 n1n2 H
1 0 0 1
elsshouldwetakeintoaccounttoachieveconvergencyof
0 0 0 0
results? Toanswerthisquestion,weperformdiagonaliza- Φ (ql2 ), (19)
tion of the type (15), increasing step-by-step a quantity ×0 0 0 0 n1n2 H
1 0 0 1
N of electron and hole Landau levels. For simplicity, we
perform these calculations at P = 0 only. Energies of
where g = 4 is the degeneracy factor and f is the oc-
n
magnetoexcitons at rest, renormalized by electron inter-
cupation number for the n-th Landau level, i.e. f = 1
n
actions due to breakdown of the Kohn theorem, are the
at n ν and f = 0 at n > ν (we neglect temperature
n
most suitable to be observed in optical experiments. ≤
effects since typical separation between Landau levels in
The results of such calculations of En(N1n)2(P = 0) as graphene in quantizing magnetic field is of the order of
functions of N are shown in Fig. 2 by cross points. We room temperature [10]). The matrix between magne-
found semi-analytically that eigenvalues of the Hamilto- toexcitonic wave functions in (19) ensures that electron
nian under consideration should approach a dependence and hole belong to the same sublattice, that is needed
for Coulomb interaction in exchange channel treated as
C
E(N) E(∞) + n1n2 (16) annihilation of electron and hole in one point of space
n1n2 ≈ n1n2 √N and subsequent creation of electron-hole pair in another
point.
at large N. We fitted the numerical results by this de-
Unlike electron gas without magnetic field, having a
pendence and thus were able to find the limiting values
single plasmon branch, Eqs. (17)–(19) give an infinite
En(∞1n)2 of magnetoexciton energies with infinite number number of solutions ω = Ωn1n2(q), each of them can
of Landau levels taken into account. be attributed to specific inter-Landau level transition
We see in Fig. 2 that the differences between mag- n n , affected by Coulomb interaction [18, 37, 38].
2 1
netoexciton energies calculated in the first order in Not→e that at q 0, when Coulomb interaction V(q) be-
Coulombinteraction(thecrossesatN =1)andtheener- comes weak, di→spersion of each magnetoplasmon branch
gies calculated with taking into account mixing between Ω (q) tends to the corresponding single-particle exci-
n1n2
allLandaulevels(dottedlines)areverysmallatrs =0.5 tation energy E(0) .
(Fig. 2(a,b)), moderate at r = 1 (Fig. 2(b,e)) and very n1n2
s At r 1, we can suppose that magnetoplasmon en-
s
large at r = 2 (Fig. 2(c,f)). Since convergency of the ≪
s ergyΩ (q)doesnotdiffersignificantlyfromthesingle-
inverse-square-root function is very slow, even the mix- n1n2
particle energy E(0) . In this case a dominant contribu-
ing of rather large (of the order of tens) number of Lan- n1n2
tion to the sum in (18) comes from the term with the
dau levels is not sufficient to obtain reliable results for
givenn andn . Neglectingallotherterms,wecanwrite
magnetoexciton energies, as clearly seen in the Fig. 2. 1 2
(18) as
Note that the mixing increases magnetoexciton bind-
ing energies, similarly to results on magnetoexcitons in F (q)
Π(q,ω) g n1n2 , (20)
semiconductor quantum wells [47, 48]. ≈ ω E(0) +iδ
− n1n2
and from (17) we obtain an approximation to plasmon
III. MAGNETOPLASMONS dispersion in the first order in the Coulomb interaction:
Ω (q) E(0) +gV(q)F (q). (21)
Magnetoplasmonsarecollectiveexcitationsofelectron n1n2 ≈ n1n2 n1n2
gas in magnetic field, occurring as poles of density-to- Magnetoplasmons in graphene were considered with-
density response function. In the random phase approx- out taking into account Landau level mixing in a man-
imation, dispersion of magnetoplasmon is determined as ner of Eq. (21) in the works [20, 39]. Other authors
a root of the equation [21, 24, 34] took into account several Landau levels,
and the others [35–38] performed full summation in the
1 V(q)Π(q,ω)=0, (17) frameworkoftherandomphaseapproximation(17)–(19)
−
to calculate magnetoplasmon dispersions.
where V(q) = 2πe2/εq is the two-dimensional Fourier Here we state the question: how many Landau lev-
transform of Coulomb interaction and Π(q,ω) is a po- els one should take into account to calculate magneto-
larization operator (or polarizability). Polarization op- plasmon spectrum with sufficient accuracy? To answer
erator for graphene in magnetic field can be expressed it, we performed calculations with successive taking into
7
3 3 3
W a n =0 b n =0 c n =0
n1n2 r =0.5 r =1 r =2
v /l s s s
F H 0 fi 3
2.6 0 fi 3 2.6 0 fi 3 2.6
2.2 2.2 2.2 0 fi 2
0 fi 2 0 fi 2
1.8 1.8 1.8
0 fi 1 0 fi 1 0 fi 1
1.4 1.4 1.4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
ql ql ql
H H H
2.5 2.5 2.5
W d n =1 e n =1 f n =1
n1n2 r =0.5 r =1 r =2
s s s
v /l
F H
2 2 2
1.5 1 fi 4 1.5 1 fi 4 1.5 1 fi 4
1 fi 3 1 fi 3 1 fi 3
1 1 1
1 fi 2 1 fi 2 1 fi 2
0.5 0.5 0.5
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
ql ql ql
H H H
FIG. 3. Magnetoplasmon energies Ωn1n2, calculated in the lowest Landau level approximation (solid lines), with taking into
account mixing between 2 (short dash lines) and 3 (long dash lines) Landau levels of electron and hole, and with taking into
account mixing between all Landau levels (dotted lines). The results are presented for different filling factors ν and different
r : (a) ν =0,r =0.5, (b)ν =0, r =1, (c)ν =0, r =2, (d)ν =1, r =0.5, (e) ν =1, r =1, (f)ν =1, r =2. Dispersions
s s s s s s s
of 3 lowest-lying magnetoplasmon modes n →n , indicated nearthe corresponding curves, are shown.
2 1
account increasing number of Landau levels at different high-lying magnetoplasmon modes. It is also seen, that
ν and r . In Fig. 3, dispersions of magnetoplasmons in the mixing considerably changes the dispersions even at
s
graphene calculated numerically are shown. Results ob- moderate r (see, e.g., Fig. 3(d) at r = 0.5). Note that
s s
tained without taking into accountLandaulevel mixing, the mixing usually decrease magnetoplasmon energies
with taking into account a mixing of two or three lowest and does not affect the long-wavelength linear asymp-
Landau levels and with taking into account all Landau totics of their dispersions.
levels are plotted with different line styles.
Therefore we conclude here that convergency of mag-
As we see, even taking into account the mixing be- netoplasmondispersions in rather fast upon increasing a
tween 2 Landau levels changes the dispersions consider- numberofLandaulevelstakenintoaccount. Severallow-
ably (see the differences between solid and short dash est Landau levels are sufficient to obtain rather accurate
lines in Fig. 3). However, the calculations with mixing results. On the other hand, calculations in the lowest
between 3 Landau levels (long dash lines) are already Landau level approximation,i.e. without taking into ac-
close to the exact results (dotted lines), except for the count the mixing, can give inaccurate results, especially
8
in a region of intermediate momenta q l−1. graphene were calculated in the random phase approx-
∼ H
imation with taking into account different numbers of
Landau levels. We showed that even few Landau lev-
els for electronandhole aresufficient do obtainaccurate
IV. CONCLUSIONS
results, however the lowest Landau level approximation
(i.e. calculations without taking into account the mix-
ing) provide inaccurate results, especially for intermedi-
We studied influence of Landau level mixing in
ate momenta and high-lying magnetoplasmon modes.
graphene in quantizing magnetic field on properties of
In out study, we completely disregarded renormaliza-
elementary excitations — magnetoexcitons and magne-
tionofsingle-particleenergiesduetoexchangewithfilled
toplasmons — in this system. Virtual transitions be-
Landaulevelsinthe valence bandofgraphene,sincethis
tweenLandaulevels,causedbyCoulombinteraction,can
questionwasconsideredelsewhere[20, 21, 24, 30, 40]. In
change dispersions of the excitations in comparisonwith
our article, we focused on the role of Coulomb interac-
the lowest Landau level approximation.
tion in the electron-hole channel only. In this context,
Strength of Coulomb interaction and thus a degree of
animportantresultofourworkisthatbreakdownofthe
Landau level mixing can be characterized by dimension-
Kohntheorem in graphene leads to strong correctionsof
lessparameterr ,dependentinthecaseofgrapheneonly
s magnetoexciton energies not only due to exchange self-
on dielectric permittivity of surrounding medium. By
energies, but also due to virtual transitions caused by
embedding graphene in different environments, one can
Coulomb interaction between electron and hole.
change r from small values to r 2 [49].
s s ≈ We considered magnetoexcitons in the ladder approx-
We calculated dispersions of magnetoexcitons in
imation and magnetoplasmons in the random phase ap-
graphene and showed that the mixing even between few
proximations without taking into account vertex correc-
Landau levels can change significantly the dispersion
tions andscreening. Estimating the roleofthese factors,
curves at r > 1. However, at small r the role of the
s s especially in the strong-interacting regime at large r , is
s
mixing is negligible, in agreement with the other works
a difficult task and will be postponed for future studies.
[20, 25]. Then the question about convergency of such
Theresultsobtainedinthisworkshouldberelevantfor
calculations upon increasing a number of involved Lan-
magneto-opticalspectroscopyofgraphene[28,29,31,50–
dau levels have been raised.
52] and for the problem of Bose-condensation of mag-
We performed calculations of magnetoexciton ener- netoexcitons [53–55]. Excitonic lines in optical absorp-
gies at rest with taking into account stepwise increasing tion or Raman spectra of graphene can give experimen-
number of Landau levels and found their inverse-square- talinformationaboutenergiesofelementaryexcitations.
root asymptotics. By evaluating limiting values of these Magnetoexcitons andmagnetoplasmons canbe observed
asymptotics,wecalculatedmagnetoexcitonenergieswith alsoasconstituentsofvarioushybridmodes—polaritons
infinite number of Landau levels taken into account. We [56],trions [57],Bernsteinmodes [58]ormagnetophonon
demonstrated that influence of remote Landau levels of resonances [59].
magnetoexciton energies is strong, especially at large rs. The work was supported by grants of Russian Foun-
Also it was found that calculations with taking into ac- dation for Basic Research and by the grant of the Pres-
count even several Landau levels provide results, rather ident of Russian Federation for Young Scientists MK-
far from exact ones. 5288.2011.2. One of the authors (A.A.S.) also acknowl-
Also dispersion relations of magnetoplasmons in edges support from the Dynasty Foundation.
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