Filling-Factor-Dependent Magnetophonon Resonance in Graphene
M. O. Goerbig,1 J.-N. Fuchs,1 K. Kechedzhi,2 and Vladimir I. Fal’ko2
Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS UMR 8502, F-91405 Orsay, France and
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
(Dated: October 23, 2018)
We describe a peculiar fine structure acquired by the in-plane optical phonon at the Γ-point
in graphene when it is brought into resonance with one of the inter-Landau-level transitions in
this material. The effect is most pronounced when this lattice mode (associated with the G-band
in graphene Raman spectrum) is in resonance with inter-Landau-level transitions 0 ⇒ +, 1 and
−, 1 ⇒ 0, at a magnetic field B0 ≃ 30T. It can be used to measure the strength of the electron-
phonon coupling directly, and its filling-factor dependence can be used experimentally to detect
circularly polarized lattice vibrations.
PACS numbers: 78.30.Na, 73.43.-f, 81.05.Uw
In metals and semiconductors the spectra of phonons
are renormalized by their interaction with electrons.
Some of the best known examples include the Kohn
anomaly [1] in the phonon dispersion, which originates
from the excitation/de-excitation of electrons across the
Fermi level upon the propagation of a phonon through
the bulk of a metal and a shift in the longitudinal opti-
cal phonon frequency in heavily doped polar semiconduc-
tors [2]. However, despite the transparency of theoretical
models the observation of such effects is often obscured
by the difficulty to change the electron density in a mate-
rial, whereas in semiconductor structures containing two-
dimensional (2D) electrons the density of which can be
varied, the influence of the latter on the phonon modes is
weak due to a negligibly small volume fraction occupied
by the electron gas. In this context, a unique opportunity
arises in graphene-based field-effect transistors [3], where
the density of carriers in an atomically thin film (mono-
layer [4, 5, 6] or a bilayer [7]) can be continuously varied
from 1013cm−2 p-type to 1013cm−2 n-type. Several Ra-
man experiments have already been reported [8, 9] where
the variation of carrier density in graphene changes the
optical phonon frequency, in agreement with theoretical
expectations [10, 11, 12].
When graphene is exposed to a quantizing magnetic
field, its electronic spectrum quenches into discrete Lan-
dau levels (LLs) [13]. Then, the optical phonon energy in
graphene may coincide with the energy of one of the inter-
LL transitions, a condition known as magnetophonon
resonance [14, 15]. Recently, Ando has suggested [16]
that in undoped graphene the magnetophonon resonance
enhances the effect of the electron-phonon coupling on
a spectrum of the in-plane optical phonons - the E2g
modes attributed to the G-band in the Raman spectra
in Refs. [8, 9, 17, 18, 19]. In this paper, we investigate
a rich structure of the anti-crossing experienced by such
lattice modes when a magnetic field makes their energy
equal to the energy of one of the valley-antisymmetric
interband magnetoexcitons [20]. Most saliently, the dif-
ference between circular polarization of various inter-LL
transitions [21, 22] makes the magnetophonon resonance
distinguishable for lattice vibrations of different circular
polarization, which makes the number of split lines in
the fine structure acquired by a phonon and the value of
splitting dependent on the electronic filling factor, ν.
The in-plane optical phonons in graphene [relative dis-
placement u = (ux, uy) of sublattices A and B] have
the energy ω ≈ 0.2eV at the Γ-point (in the center of
the Brillouin zone). These phonons and their coupling to
electrons can be described using the Hamiltonian [10, 11],
Hph =
ωb†µ,qbµ,q + g
2Mω(σxuy − σyux), (1)
u(r) =
2NucMω
bµ,q + b
eµ,qe
−iq·r,
where b
µ,q are annihilation (creation) operators of a
phonon with polarisation eµ,q, M is the mass of a car-
bon atom, and Nuc is the number of unit cells. Here
and below, we use units ~ ≡ 1. Also, we shall uti-
lize a double degeneracy of the E2g mode at the Γ-point
(at q = 0) and describe the in-plane optical phonon in
terms of a degenerate pair of circularly polarized modes,
u	 = (ux+iuy)/
2 and u� = u
	. The constant g in Eq.
(1) characterizes the electron-phonon coupling [23]. This
coupling has the form of the only invariant linear in u per-
mitted by the symmetry group of the honeycomb crystal.
It is constructed using Pauli matrices σ = (σx, σy) acting
in the space of sublattice components of the Bloch func-
tions, [φK+A, φK+B] and [φK−B, φK−A] which describe
electron states in the valleys K± (two opposite corners of
the hexagonal Brillouin zone) and obey the Hamiltonian,
in terms of the electron charge −e < 0 [24],
Hel = ξvσ · p, p =− i∇+ eA, ∂xAy − ∂yAx = B.
Here, ξ = ± distinguishes between K±, and momentum
p is calculated with respect to the center of the corre-
sponding valley. This Hamiltonian represents the dom-
inant term of the next-neighbor tight-binding model of
graphene [25, 26, 27], and the electron-phonon coupling
http://arxiv.org/abs/0704.0027v4
:   B  sublattice
:   A  sublattice
(a) (b)
−,(n+1)
+,(n+1)
FIG. 1: (a) Optical phonons are lattice vibrations with an out-
off-phase oscillation of the two sublattices. (b) Interband electron-
hole excitations coupling to phonon modes with different circular
polarization.
in Eq. (1) takes into account the change in the A − B
hopping elements due to the sublattice displacement [28].
In a perpendicular magnetic field, Hel determines [13]
a spectrum of 4-fold (spin and valley) degenerate LLs,
εα=±n = α
2nvλ−1B in the valence band (ε
n>0), con-
duction band (ε+n>0), and at zero energy (ε0 = 0, ex-
actly at the Dirac point in the electron spectrum), in
terms of the magnetic length λB = 1/
eB. Such a
spectrum has been confirmed by recent quantum Hall
effect measurements [4, 5, 6]. In each of the two val-
leys, the LL basis is given by two-component states
1 + δn,0φn,m, iξα(1−δn,0)φn−1,m], where φn,m are
the LL wave functions described by the quantum num-
bers n and m, the latter being related to the guiding
center degree of freedom. Here, we neglect the Zeeman
effect, and simply take into account the two-fold spin de-
generacy.
Excitations of electrons between LLs can be described
in terms of magnetoexcitons (see Fig. 1). Those relevant
for the magnetophonon resonance are
	(n, ξ) =
1 + δn,0
+,n,m;ξc−,(n+1),m;ξ,
�(n, ξ) =
1 + δn,0
+,(n+1),m;ξ
c−,n,m;ξ, (2)
where the index A =	,� characterizes the angular mo-
mentum of the excitation and the operators c
α,n,m;ξ
annihilate (create) an electron in the state α, n,m in
the valley Kξ. The normalization factors N	n =
[(1 + δn,0)NB(ν̄−,(n+1) − ν̄+,n)]1/2 and N�n = [(1 +
δn,0)NB(ν̄−,n − ν̄+,(n+1))]1/2 are used to ensure the
bosonic commutation relations of the exciton operators,
[ψA(n, ξ), ψ
′, ξ′)] = δA,A′δξ,ξ′δn,n′ , where NB is the
total number of states per LL in a sample, including
the two-fold spin-degeneracy. These commutation rela-
tions are obtained within the mean-field approximation
with 〈c†α,n,m;ξcα′,n′,m′;ξ′〉 = δξ,ξ′δα,α′δn,n′δm,m′(δα,− +
δα,+ν̄α,n), where 0 ≤ ν̄α,n ≤ 1 is the partial filling fac-
tor of the n-th LL. Similarly to magneto-optical selec-
tion rules in graphene [20, 21, 22], α, n ⇒ α′, n ± 1, 	-
polarized phonons are coupled to electronic transitions
with −, (n + 1) ⇒ +, n, and �-polarized phonons to
−, n ⇒ +, (n + 1) magneto-excitons, at the same en-
ergy Ωn ≡
2(v/λB)(
n+ 1) (Fig. 1), which
follows directly from the composition of the LL in
graphene and the form of the electron-phonon coupling
in Eq. (1). In contrast to photons that couple to
the valley-symmetric mode ψA,s(n) = [ψA(n,K+) +
ψA(n,K−)]/
2, electron-phonon interaction in Eq.(1)
couples phonons to the valley-antisymmetric magnetoex-
citon ψA,as(n) = [ψA(n,K+)− ψA(n,K−)]/
In terms of magnetoexcitons we can, now, rewrite the
electron-phonon Hamiltonian in a bosonized form, as
τ=s,as
A,τ (n)ψA,τ (n) +
AbA (3)
gA(n)
ψA;as(n) + bAψ
A;as(n)
g	(n) = g
(1 + δn,0)γ
ν̄−,(n+1) − ν̄+,n,
g�(n) = g
(1 + δn,0)γ
ν̄−,n − ν̄+,(n+1),
where gA are the effective coupling constants, with γ =
3a2/2πλ2B and a = 1.4Å (distance between neighbor-
ing carbon atoms). In the Hamiltonian (3), we have omit-
ted electronic excitations with a higher angular momen-
tum which do not couple to the in-plane optical phonon
modes (e.g., n ⇒ n′, with n′ 6= n ± 1). The dressed
phonon operator corresponding to the Hamiltonian (3)
is obtained by solving Dyson’s equation. The pole of the
propagator gives the antisymmetric coupled mode fre-
quencies ω̃A,
ω̃2A − ω2 = 4ω
n=nF+1
ω̃2A − Ω2n
∆nF g
A(nF )
ω̃2A −∆2nF
, (4)
where nF stands for the number of the highest fully occu-
pied LL in the spectrum, and ∆n =
2(v/λB)(
n+ 1−√
n). In Eq. (4), the sum (extended up to the high-
energy cut-off N ∼ (λB/a)2 above which the electronic
dispersion is no longer linear) takes into account inter-
band magnetoexcitons, and the last term gives a small
correction due to an intraband magnetoexciton. In the
small-field limit and large doping (nF ≫ 1), solution of
Eq. (4) reproduces the zero-field result [10, 11] if one
replaces the sum by an integral,
n=0 →
dn, ap-
proximates
n+ 1 ≈ 2
n and ∆nF ≈ 0, and,
then, linearizes Eq. (4) by replacing ω̃A by ω in the de-
nominator,
ω̃ ≃ ω̃0 + λ
ω + 2
2nFv/λB
ω − 2
2nFv/λB
ω̃0 ≃ ω + 2
ω2 − Ω2n
where λ = (2/
3π)(g/t)2 ≃ 3.3× 10−3 is the same as in
Refs. [10, 16] (t = 2v/3a ∼ 3eV is the A−B hopping am-
plitude), and ω̃0 is the renormalized phonon frequency in
an undoped graphene sheet at B = 0. The only variation
arises at high fields, ω̃0 &
2v/λB, where for nF = 0 the
linearized Eq. (4) yields
ω̃ ≃ ω̃0 −
g2(0)
(ω̃0λB/
2v)2 − 1
The strongest effect of the phonon coupling to elec-
tron modes occurs when the frequency of the former
coincides with the frequency Ωn of one of the magne-
toexcitons ψA,as(n). In such a case, the sum on the
right-hand-side of the eigenvalue equation (4) is domi-
nated by the resonance term and may be approximated
by 2ωg2A(n)/ (ω̃A − Ωn). This results in a fine structure
of mixed phonon-magnetoexciton modes, ψA,as(n) cos θ+
bA sin θ with frequency ω̃
A and ψA,as(n) sin θ − bA cos θ
with frequency ω̃−A [where cot 2θ = (Ωn−ω̃0)/2gA], which
are determined for each polarisation (A =	,� ) sepa-
rately,
(n) = 1
(Ωn + ω̃0)∓
(Ωn − ω̃0)2 + g2A(n). (5)
A generic form of the phonon-magnetoexciton anti-
crossing and formation of coupled modes, ω±
(n) in un-
doped graphene (i.e., ν = 0) is illustrated in Fig. 2(a).
Such an anticrossing and mode mixing is simlar to that
described by Ando [16]. It can manifest itself in Raman
spectroscopy: in a fine structure acquired by the G-line
(earlier attributed [8, 9, 17, 18, 19] to the in-plane op-
tical phonon at the Γ-point, E2g mode) at the magneto-
phonon resonance conditions. The effect is the strongest
for the resonance Ωn=0 ≈ ω̃0 between the phonon and
magnetoexciton based upon −, 1 ⇒ 0 and 0 ⇒ +, 1 tran-
sitions. When approaching the resonance (by sweeping a
magnetic field), the phonon line becomes accompanied by
a weak satellite moving towards it and increasing its in-
tensity. Exactly at the magnetophonon resonance, where
both the upper mode [ω̃+A(n)] and the lower mode [ω̃
A(n)]
consist of an equal-weight superposition of the phonon
and the resonant exciton, with cos θ = sin θ = 1/
the G-band in graphene would appear as two lines. For
Ωn=0 =
2v/λB ≈ 36
B[T] meV (see [16, 24]) and
ω̃0 ≃ 200 meV, this resonance occurs in an experimen-
tally accessible field range, B0 ≃ 30 T. For the filling
factor ν = 0, the central LL (n = 0) is always half-filled.
Then, coupling and, therefore, splitting of the �- and 	-
polarized modes coincide, g� = g	, thus, giving rise to a
pair of peaks at the energies ω̃± = ω̃0 ± g� sketched
in part I in Fig. 2(b). For the magnetic field value
B0 ≃ 30 T and g ≃ 0.28eV [12], we estimate this splitting
as 2gA ∼ 16meV (∼ 130cm−1), which largely exceeds the
G-band width observed in Refs. [8, 9, 17, 18, 19].
Doping of graphene changes the strength of the cou-
pling constants g� and g	, as shown in Fig. 2(c). This
5 10 15 20 25 30 35
Magnetic Field [T]
3010 20 40
ν = 0
0 < |ν| < 2
|ν| = 2
2g 2g
mode splitting
−6 −4 −2 0
4n−2 4n+2 4n+6
B=Bn>02γ
FIG. 2: (a) Coupled phonon and magneto-excitons as a function
of the magnetic field. Energies are in units of the bare phonon
energy ω. Dashed lines indicate the uncoupled valley-symmetric
modes, with gA = 0. (b) Mode splitting as a function of the filling
factor, as may be seen in Raman spectroscopy, with the resonance
condition Ωn=0 ≈ ω̃0, for ν = 0 in (I), 0 < |ν| < 2 (in II), and
ν = ±2 (in III). The absolute intensity of the modes is in arbitrary
units, but the height and the width reflect the expected relative
intensities. (c) Mode splitting for n = 0, as a function of the filling
factor ν. (d) Same as in (c) for n ≥ 1.
is because a higher (lower) occupancy of the n = 0 LL
reduces (enhances) the oscillator strength of the 	 polar-
ized transition due to the availability of filled and empty
states in the involved LLs, whereas the same change in
the electron density has the opposite effect on g�. As
a result, for an arbitrary filling factor −2 < ν < 2,
we predict that, in the vicinity of magnetophonon reso-
nance, the phonon mode (and, therefore, G-band in Ra-
man spectrum) should split into four lines [part II in Fig.
2(b)], with ω̃±� = ω̃±g� for �-polarized and ω̃±	 = ω̃±g	
for 	-polarized phonons. In the quantum Hall state at
filling factor ν = 2, the transition −, 1 ⇒ 0 becomes suc-
cessively blocked and no longer affects the frequency of a
	-polarized phonon, whereas the transition 0 ⇒ +, 1 ac-
quires the maximum strength, thus, increasing the cou-
pling parameter g�. This leads to the magnetophonon
resonance fine structure consisting of three peaks, with
an even larger splitting between side lines, as sketched in
part III in Fig. 2(b). Interestingly, this may enable one
to directly observe lattice modes with a definite circu-
lar polarization. A further increase of the electron filling
factor reduces the side-line splitting which should com-
pletely disappear at ν = 6, after the transition 0 ⇒ +, 1
becomes blocked by a complete filling of the +, 1 LL [Fig.
2(c)]. The same arguments hold for p-doped graphene,
though in this case the roles of �- and 	-polarized modes
are interchanged.
Magnetophonon resonances with other possible inter-
LL transitions n ⇒ n+ 1 occur at much lower magnetic
fields, Bn = B0/(
n+ 1)2. For example, a resonant
phonon coupling with the magnetoexciton ψA;as(1) is ex-
pected to occur at B1 ≈ 5T. Its description remains qual-
itatively similar, though for n > 0 the mode splitting is
less pronounced because of the B-field dependence of the
coupling constants in Eq. (3). One finds that g� = g	
for |ν| < 2(2n − 1). At ν = 2(2n − 1), filling of the n-
th LL starts changing, which reduces splitting of the 	-
polarized mode and gives rise to the four-peak structure.
At ν = 2(2n+1), where the +, n LL becomes completely
filled, splitting of the 	-polarized phonon vanishes, thus,
resulting in the three-peak fine structure [part III in Fig.
2(b)] that would persist up to ν = 2(2n + 3). This is
because the splitting of the �-polarized modes remains
constant up to the filling factor ν = 2(2n + 1), above
which population of the +, (n+ 1) LL starts to suppress
the value of g�, until the latter vanishes at ν = 2(2n+3)
[see Fig. 2(d)].
In conclusion, we have predicted a filling-factor depen-
dence of the fine structure acquired by the in-plane (E2g)
optical phonon in graphene when the latter is in reso-
nance with one of the inter-LL transitions in this ma-
terial. The effect is expected to be most pronounced
when the phonon is resonantly coupled to the 0 ⇒ +, 1
and −, 1 ⇒ 0 transitions, which requires a magnetic field
B0 ≃ 30T. The predicted mode splitting may be used
to measure directly the strength of the electron-phonon
coupling, and also to distinguish between circularly (left-
and right- hand) polarized lattice modes.
We thank D. Abergel, A. Ferrari, P. Lederer, and A.
Pinczuk for useful discussions. This work was suported
by Agence Nationale de la Recherche Grant ANR-06-
NANO-019-03 and EPSRC-Lancaster Portfolio Partner-
ship EP/C511743. We thank the MPI-PKS workshop
‘Dynamics and Relaxation in Complex Quantum and
Classical Systems and Nanostructures’ and the Kavli
Institute for Theoretical Physics, UCSB (NSF PHY99-
07949) for hospitality.
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Erratum
In the previous version (v3) of this Letter, we have
underestimated the numerical value of the mode split-
ting of the magnetophonon resonance [see paragraph af-
ter Eq. (5)] by a factor of 2 (the text above takes into
account the corrected parameters). This is a result of
two mistakes. First, there is a factor of
2, which finds
its origin in an erroneous normalization of the circular
polarized phonons. They should indeed be defined as
u	 = (ux + iuy)/
2 and u� = (ux − iuy)/
2 [and not
as u	 = ux + iuy and u� = ux − iuy as incorrectly
assumed on page 1, second column], such that the asso-
ciated phonon operators bA obey the usual commutation
relations [bA, b
] = δA,A′ , with A =	,�. This yields
a factor of
2 in the definition of the effective coupling
constants [Eq. (3)], which read in the corrected form
g	(n) = g
(1 + δn,0)γ
ν̄−,(n+1) − ν̄+,n ,
g�(n) = g
(1 + δn,0)γ
ν̄−,n − ν̄+,(n+1) .
As a consequence, the zero-field dimensionless coupling
constant λ [defined in the first column page 3 of our
Letter] is multiplied by a factor of 2 and becomes λ =
3π)(g/t)2.
Second, we also underestimated the numerical value of
the electron-phonon coupling constant g by a factor of√
2. Indeed, g defined in our work [see Eq. (1)] is related
to 〈g2Γ〉F ≃ 0.0405 eV2 computed by Piscanec et al. [2]
as g =
2〈g2Γ〉F ≃ 0.28 eV and not as g =
〈g2Γ〉F ≃ 0.2
eV as incorrectly assumed in our Letter. In addition,
there is a substantial uncertainty in the precise value of
the constant g. In a tight-binding model, the latter may
be related to the derivative of the hopping amplitude t
as a function of the carbon-carbon distance a as g =
(−dt/da)× 3/(2
Mω) [1]. Harrison’s phenomenological
law t ∝ 1/a2 then implies that g ≃ 0.26 eV. Experiments
in graphene [3] and [4] in zero magnetic field give for the
dimensionless coupling constant λ the values 4.4 × 10−3
and 5.3×10−3 respectively. This determines g in between
0.3 eV and 0.36 eV, where we take into account that the
value of t lies between 2.7 and 3 eV. In the end, we have
to take g in the range between 0.26 and 0.36 eV [instead
of g ≃ 0.2 eV] and therefore the dimensionless coupling
constant becomes λ ≃ (2.8 to 5.3)× 10−3 [instead of λ ≃
10−3].
As a result of the two factors of
2, the numerical
estimate for the mode splitting 2gA at ν = 0 and B ≃ 30
T [at the discussed resonance −, 1 ⇒ 0 and 0 ⇒ +, 1, see
second column of page 3] becomes 2gA ∼ 15 meV (∼ 120
cm−1), for g ≃ 0.26 eV and 2gA ∼ 20 meV (∼ 160 cm−1)
for g ≃ 0.36 eV [instead of 2gA ∼ 8 meV]. The effect
is therefore twice larger than initially predicted. The
conclusions of our work remain unaltered.
We would like to thank C. Faugeras and M. Potemski
for having drawn our attention on the underestimated
value of the mode splitting. See also their recent preprint
where they measure the magnetophonon resonance [5].
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Lett. 98, 166802 (2007).
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Sprinkle, C. Berger, W.A. de Heer and M. Potemski,
arXiv:0907.5498.
http://arxiv.org/abs/0907.5498

  We describe a peculiar fine structure acquired by the in-plane optical phonon
at the Gamma-point in graphene when it is brought into resonance with one of
the inter-Landau-level transitions in this material. The effect is most
pronounced when this lattice mode (associated with the G-band in graphene Raman
spectrum) is in resonance with inter-Landau-level transitions 0 -> (+,1) and
(-,1) -> 0, at a magnetic field B_0 ~ 30 T. It can be used to measure the
strength of the electron-phonon coupling directly, and its filling-factor
dependence can be used experimentally to detect circularly polarized lattice
modes.

Filling-Factor-Dependent Magnetophonon Resonance in Graphene
M. O. Goerbig,1 J.-N. Fuchs,1 K. Kechedzhi,2 and Vladimir I. Fal’ko2
Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS UMR 8502, F-91405 Orsay, France and
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
(Dated: October 23, 2018)
We describe a peculiar fine structure acquired by the in-plane optical phonon at the Γ-point
in graphene when it is brought into resonance with one of the inter-Landau-level transitions in
this material. The effect is most pronounced when this lattice mode (associated with the G-band
in graphene Raman spectrum) is in resonance with inter-Landau-level transitions 0 ⇒ +, 1 and
−, 1 ⇒ 0, at a magnetic field B0 ≃ 30T. It can be used to measure the strength of the electron-
phonon coupling directly, and its filling-factor dependence can be used experimentally to detect
circularly polarized lattice vibrations.
PACS numbers: 78.30.Na, 73.43.-f, 81.05.Uw
In metals and semiconductors the spectra of phonons
are renormalized by their interaction with electrons.
Some of the best known examples include the Kohn
anomaly [1] in the phonon dispersion, which originates
from the excitation/de-excitation of electrons across the
Fermi level upon the propagation of a phonon through
the bulk of a metal and a shift in the longitudinal opti-
cal phonon frequency in heavily doped polar semiconduc-
tors [2]. However, despite the transparency of theoretical
models the observation of such effects is often obscured
by the difficulty to change the electron density in a mate-
rial, whereas in semiconductor structures containing two-
dimensional (2D) electrons the density of which can be
varied, the influence of the latter on the phonon modes is
weak due to a negligibly small volume fraction occupied
by the electron gas. In this context, a unique opportunity
arises in graphene-based field-effect transistors [3], where
the density of carriers in an atomically thin film (mono-
layer [4, 5, 6] or a bilayer [7]) can be continuously varied
from 1013cm−2 p-type to 1013cm−2 n-type. Several Ra-
man experiments have already been reported [8, 9] where
the variation of carrier density in graphene changes the
optical phonon frequency, in agreement with theoretical
expectations [10, 11, 12].
When graphene is exposed to a quantizing magnetic
field, its electronic spectrum quenches into discrete Lan-
dau levels (LLs) [13]. Then, the optical phonon energy in
graphene may coincide with the energy of one of the inter-
LL transitions, a condition known as magnetophonon
resonance [14, 15]. Recently, Ando has suggested [16]
that in undoped graphene the magnetophonon resonance
enhances the effect of the electron-phonon coupling on
a spectrum of the in-plane optical phonons - the E2g
modes attributed to the G-band in the Raman spectra
in Refs. [8, 9, 17, 18, 19]. In this paper, we investigate
a rich structure of the anti-crossing experienced by such
lattice modes when a magnetic field makes their energy
equal to the energy of one of the valley-antisymmetric
interband magnetoexcitons [20]. Most saliently, the dif-
ference between circular polarization of various inter-LL
transitions [21, 22] makes the magnetophonon resonance
distinguishable for lattice vibrations of different circular
polarization, which makes the number of split lines in
the fine structure acquired by a phonon and the value of
splitting dependent on the electronic filling factor, ν.
The in-plane optical phonons in graphene [relative dis-
placement u = (ux, uy) of sublattices A and B] have
the energy ω ≈ 0.2eV at the Γ-point (in the center of
the Brillouin zone). These phonons and their coupling to
electrons can be described using the Hamiltonian [10, 11],
Hph =
ωb†µ,qbµ,q + g
2Mω(σxuy − σyux), (1)
u(r) =
2NucMω
bµ,q + b
eµ,qe
−iq·r,
where b
µ,q are annihilation (creation) operators of a
phonon with polarisation eµ,q, M is the mass of a car-
bon atom, and Nuc is the number of unit cells. Here
and below, we use units ~ ≡ 1. Also, we shall uti-
lize a double degeneracy of the E2g mode at the Γ-point
(at q = 0) and describe the in-plane optical phonon in
terms of a degenerate pair of circularly polarized modes,
u	 = (ux+iuy)/
2 and u� = u
	. The constant g in Eq.
(1) characterizes the electron-phonon coupling [23]. This
coupling has the form of the only invariant linear in u per-
mitted by the symmetry group of the honeycomb crystal.
It is constructed using Pauli matrices σ = (σx, σy) acting
in the space of sublattice components of the Bloch func-
tions, [φK+A, φK+B] and [φK−B, φK−A] which describe
electron states in the valleys K± (two opposite corners of
the hexagonal Brillouin zone) and obey the Hamiltonian,
in terms of the electron charge −e < 0 [24],
Hel = ξvσ · p, p =− i∇+ eA, ∂xAy − ∂yAx = B.
Here, ξ = ± distinguishes between K±, and momentum
p is calculated with respect to the center of the corre-
sponding valley. This Hamiltonian represents the dom-
inant term of the next-neighbor tight-binding model of
graphene [25, 26, 27], and the electron-phonon coupling
http://arxiv.org/abs/0704.0027v4
:   B  sublattice
:   A  sublattice
(a) (b)
−,(n+1)
+,(n+1)
FIG. 1: (a) Optical phonons are lattice vibrations with an out-
off-phase oscillation of the two sublattices. (b) Interband electron-
hole excitations coupling to phonon modes with different circular
polarization.
in Eq. (1) takes into account the change in the A − B
hopping elements due to the sublattice displacement [28].
In a perpendicular magnetic field, Hel determines [13]
a spectrum of 4-fold (spin and valley) degenerate LLs,
εα=±n = α
2nvλ−1B in the valence band (ε
n>0), con-
duction band (ε+n>0), and at zero energy (ε0 = 0, ex-
actly at the Dirac point in the electron spectrum), in
terms of the magnetic length λB = 1/
eB. Such a
spectrum has been confirmed by recent quantum Hall
effect measurements [4, 5, 6]. In each of the two val-
leys, the LL basis is given by two-component states
1 + δn,0φn,m, iξα(1−δn,0)φn−1,m], where φn,m are
the LL wave functions described by the quantum num-
bers n and m, the latter being related to the guiding
center degree of freedom. Here, we neglect the Zeeman
effect, and simply take into account the two-fold spin de-
generacy.
Excitations of electrons between LLs can be described
in terms of magnetoexcitons (see Fig. 1). Those relevant
for the magnetophonon resonance are
	(n, ξ) =
1 + δn,0
+,n,m;ξc−,(n+1),m;ξ,
�(n, ξ) =
1 + δn,0
+,(n+1),m;ξ
c−,n,m;ξ, (2)
where the index A =	,� characterizes the angular mo-
mentum of the excitation and the operators c
α,n,m;ξ
annihilate (create) an electron in the state α, n,m in
the valley Kξ. The normalization factors N	n =
[(1 + δn,0)NB(ν̄−,(n+1) − ν̄+,n)]1/2 and N�n = [(1 +
δn,0)NB(ν̄−,n − ν̄+,(n+1))]1/2 are used to ensure the
bosonic commutation relations of the exciton operators,
[ψA(n, ξ), ψ
′, ξ′)] = δA,A′δξ,ξ′δn,n′ , where NB is the
total number of states per LL in a sample, including
the two-fold spin-degeneracy. These commutation rela-
tions are obtained within the mean-field approximation
with 〈c†α,n,m;ξcα′,n′,m′;ξ′〉 = δξ,ξ′δα,α′δn,n′δm,m′(δα,− +
δα,+ν̄α,n), where 0 ≤ ν̄α,n ≤ 1 is the partial filling fac-
tor of the n-th LL. Similarly to magneto-optical selec-
tion rules in graphene [20, 21, 22], α, n ⇒ α′, n ± 1, 	-
polarized phonons are coupled to electronic transitions
with −, (n + 1) ⇒ +, n, and �-polarized phonons to
−, n ⇒ +, (n + 1) magneto-excitons, at the same en-
ergy Ωn ≡
2(v/λB)(
n+ 1) (Fig. 1), which
follows directly from the composition of the LL in
graphene and the form of the electron-phonon coupling
in Eq. (1). In contrast to photons that couple to
the valley-symmetric mode ψA,s(n) = [ψA(n,K+) +
ψA(n,K−)]/
2, electron-phonon interaction in Eq.(1)
couples phonons to the valley-antisymmetric magnetoex-
citon ψA,as(n) = [ψA(n,K+)− ψA(n,K−)]/
In terms of magnetoexcitons we can, now, rewrite the
electron-phonon Hamiltonian in a bosonized form, as
τ=s,as
A,τ (n)ψA,τ (n) +
AbA (3)
gA(n)
ψA;as(n) + bAψ
A;as(n)
g	(n) = g
(1 + δn,0)γ
ν̄−,(n+1) − ν̄+,n,
g�(n) = g
(1 + δn,0)γ
ν̄−,n − ν̄+,(n+1),
where gA are the effective coupling constants, with γ =
3a2/2πλ2B and a = 1.4Å (distance between neighbor-
ing carbon atoms). In the Hamiltonian (3), we have omit-
ted electronic excitations with a higher angular momen-
tum which do not couple to the in-plane optical phonon
modes (e.g., n ⇒ n′, with n′ 6= n ± 1). The dressed
phonon operator corresponding to the Hamiltonian (3)
is obtained by solving Dyson’s equation. The pole of the
propagator gives the antisymmetric coupled mode fre-
quencies ω̃A,
ω̃2A − ω2 = 4ω
n=nF+1
ω̃2A − Ω2n
∆nF g
A(nF )
ω̃2A −∆2nF
, (4)
where nF stands for the number of the highest fully occu-
pied LL in the spectrum, and ∆n =
2(v/λB)(
n+ 1−√
n). In Eq. (4), the sum (extended up to the high-
energy cut-off N ∼ (λB/a)2 above which the electronic
dispersion is no longer linear) takes into account inter-
band magnetoexcitons, and the last term gives a small
correction due to an intraband magnetoexciton. In the
small-field limit and large doping (nF ≫ 1), solution of
Eq. (4) reproduces the zero-field result [10, 11] if one
replaces the sum by an integral,
n=0 →
dn, ap-
proximates
n+ 1 ≈ 2
n and ∆nF ≈ 0, and,
then, linearizes Eq. (4) by replacing ω̃A by ω in the de-
nominator,
ω̃ ≃ ω̃0 + λ
ω + 2
2nFv/λB
ω − 2
2nFv/λB
ω̃0 ≃ ω + 2
ω2 − Ω2n
where λ = (2/
3π)(g/t)2 ≃ 3.3× 10−3 is the same as in
Refs. [10, 16] (t = 2v/3a ∼ 3eV is the A−B hopping am-
plitude), and ω̃0 is the renormalized phonon frequency in
an undoped graphene sheet at B = 0. The only variation
arises at high fields, ω̃0 &
2v/λB, where for nF = 0 the
linearized Eq. (4) yields
ω̃ ≃ ω̃0 −
g2(0)
(ω̃0λB/
2v)2 − 1
The strongest effect of the phonon coupling to elec-
tron modes occurs when the frequency of the former
coincides with the frequency Ωn of one of the magne-
toexcitons ψA,as(n). In such a case, the sum on the
right-hand-side of the eigenvalue equation (4) is domi-
nated by the resonance term and may be approximated
by 2ωg2A(n)/ (ω̃A − Ωn). This results in a fine structure
of mixed phonon-magnetoexciton modes, ψA,as(n) cos θ+
bA sin θ with frequency ω̃
A and ψA,as(n) sin θ − bA cos θ
with frequency ω̃−A [where cot 2θ = (Ωn−ω̃0)/2gA], which
are determined for each polarisation (A =	,� ) sepa-
rately,
(n) = 1
(Ωn + ω̃0)∓
(Ωn − ω̃0)2 + g2A(n). (5)
A generic form of the phonon-magnetoexciton anti-
crossing and formation of coupled modes, ω±
(n) in un-
doped graphene (i.e., ν = 0) is illustrated in Fig. 2(a).
Such an anticrossing and mode mixing is simlar to that
described by Ando [16]. It can manifest itself in Raman
spectroscopy: in a fine structure acquired by the G-line
(earlier attributed [8, 9, 17, 18, 19] to the in-plane op-
tical phonon at the Γ-point, E2g mode) at the magneto-
phonon resonance conditions. The effect is the strongest
for the resonance Ωn=0 ≈ ω̃0 between the phonon and
magnetoexciton based upon −, 1 ⇒ 0 and 0 ⇒ +, 1 tran-
sitions. When approaching the resonance (by sweeping a
magnetic field), the phonon line becomes accompanied by
a weak satellite moving towards it and increasing its in-
tensity. Exactly at the magnetophonon resonance, where
both the upper mode [ω̃+A(n)] and the lower mode [ω̃
A(n)]
consist of an equal-weight superposition of the phonon
and the resonant exciton, with cos θ = sin θ = 1/
the G-band in graphene would appear as two lines. For
Ωn=0 =
2v/λB ≈ 36
B[T] meV (see [16, 24]) and
ω̃0 ≃ 200 meV, this resonance occurs in an experimen-
tally accessible field range, B0 ≃ 30 T. For the filling
factor ν = 0, the central LL (n = 0) is always half-filled.
Then, coupling and, therefore, splitting of the �- and 	-
polarized modes coincide, g� = g	, thus, giving rise to a
pair of peaks at the energies ω̃± = ω̃0 ± g� sketched
in part I in Fig. 2(b). For the magnetic field value
B0 ≃ 30 T and g ≃ 0.28eV [12], we estimate this splitting
as 2gA ∼ 16meV (∼ 130cm−1), which largely exceeds the
G-band width observed in Refs. [8, 9, 17, 18, 19].
Doping of graphene changes the strength of the cou-
pling constants g� and g	, as shown in Fig. 2(c). This
5 10 15 20 25 30 35
Magnetic Field [T]
3010 20 40
ν = 0
0 < |ν| < 2
|ν| = 2
2g 2g
mode splitting
−6 −4 −2 0
4n−2 4n+2 4n+6
B=Bn>02γ
FIG. 2: (a) Coupled phonon and magneto-excitons as a function
of the magnetic field. Energies are in units of the bare phonon
energy ω. Dashed lines indicate the uncoupled valley-symmetric
modes, with gA = 0. (b) Mode splitting as a function of the filling
factor, as may be seen in Raman spectroscopy, with the resonance
condition Ωn=0 ≈ ω̃0, for ν = 0 in (I), 0 < |ν| < 2 (in II), and
ν = ±2 (in III). The absolute intensity of the modes is in arbitrary
units, but the height and the width reflect the expected relative
intensities. (c) Mode splitting for n = 0, as a function of the filling
factor ν. (d) Same as in (c) for n ≥ 1.
is because a higher (lower) occupancy of the n = 0 LL
reduces (enhances) the oscillator strength of the 	 polar-
ized transition due to the availability of filled and empty
states in the involved LLs, whereas the same change in
the electron density has the opposite effect on g�. As
a result, for an arbitrary filling factor −2 < ν < 2,
we predict that, in the vicinity of magnetophonon reso-
nance, the phonon mode (and, therefore, G-band in Ra-
man spectrum) should split into four lines [part II in Fig.
2(b)], with ω̃±� = ω̃±g� for �-polarized and ω̃±	 = ω̃±g	
for 	-polarized phonons. In the quantum Hall state at
filling factor ν = 2, the transition −, 1 ⇒ 0 becomes suc-
cessively blocked and no longer affects the frequency of a
	-polarized phonon, whereas the transition 0 ⇒ +, 1 ac-
quires the maximum strength, thus, increasing the cou-
pling parameter g�. This leads to the magnetophonon
resonance fine structure consisting of three peaks, with
an even larger splitting between side lines, as sketched in
part III in Fig. 2(b). Interestingly, this may enable one
to directly observe lattice modes with a definite circu-
lar polarization. A further increase of the electron filling
factor reduces the side-line splitting which should com-
pletely disappear at ν = 6, after the transition 0 ⇒ +, 1
becomes blocked by a complete filling of the +, 1 LL [Fig.
2(c)]. The same arguments hold for p-doped graphene,
though in this case the roles of �- and 	-polarized modes
are interchanged.
Magnetophonon resonances with other possible inter-
LL transitions n ⇒ n+ 1 occur at much lower magnetic
fields, Bn = B0/(
n+ 1)2. For example, a resonant
phonon coupling with the magnetoexciton ψA;as(1) is ex-
pected to occur at B1 ≈ 5T. Its description remains qual-
itatively similar, though for n > 0 the mode splitting is
less pronounced because of the B-field dependence of the
coupling constants in Eq. (3). One finds that g� = g	
for |ν| < 2(2n − 1). At ν = 2(2n − 1), filling of the n-
th LL starts changing, which reduces splitting of the 	-
polarized mode and gives rise to the four-peak structure.
At ν = 2(2n+1), where the +, n LL becomes completely
filled, splitting of the 	-polarized phonon vanishes, thus,
resulting in the three-peak fine structure [part III in Fig.
2(b)] that would persist up to ν = 2(2n + 3). This is
because the splitting of the �-polarized modes remains
constant up to the filling factor ν = 2(2n + 1), above
which population of the +, (n+ 1) LL starts to suppress
the value of g�, until the latter vanishes at ν = 2(2n+3)
[see Fig. 2(d)].
In conclusion, we have predicted a filling-factor depen-
dence of the fine structure acquired by the in-plane (E2g)
optical phonon in graphene when the latter is in reso-
nance with one of the inter-LL transitions in this ma-
terial. The effect is expected to be most pronounced
when the phonon is resonantly coupled to the 0 ⇒ +, 1
and −, 1 ⇒ 0 transitions, which requires a magnetic field
B0 ≃ 30T. The predicted mode splitting may be used
to measure directly the strength of the electron-phonon
coupling, and also to distinguish between circularly (left-
and right- hand) polarized lattice modes.
We thank D. Abergel, A. Ferrari, P. Lederer, and A.
Pinczuk for useful discussions. This work was suported
by Agence Nationale de la Recherche Grant ANR-06-
NANO-019-03 and EPSRC-Lancaster Portfolio Partner-
ship EP/C511743. We thank the MPI-PKS workshop
‘Dynamics and Relaxation in Complex Quantum and
Classical Systems and Nanostructures’ and the Kavli
Institute for Theoretical Physics, UCSB (NSF PHY99-
07949) for hospitality.
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Erratum
In the previous version (v3) of this Letter, we have
underestimated the numerical value of the mode split-
ting of the magnetophonon resonance [see paragraph af-
ter Eq. (5)] by a factor of 2 (the text above takes into
account the corrected parameters). This is a result of
two mistakes. First, there is a factor of
2, which finds
its origin in an erroneous normalization of the circular
polarized phonons. They should indeed be defined as
u	 = (ux + iuy)/
2 and u� = (ux − iuy)/
2 [and not
as u	 = ux + iuy and u� = ux − iuy as incorrectly
assumed on page 1, second column], such that the asso-
ciated phonon operators bA obey the usual commutation
relations [bA, b
] = δA,A′ , with A =	,�. This yields
a factor of
2 in the definition of the effective coupling
constants [Eq. (3)], which read in the corrected form
g	(n) = g
(1 + δn,0)γ
ν̄−,(n+1) − ν̄+,n ,
g�(n) = g
(1 + δn,0)γ
ν̄−,n − ν̄+,(n+1) .
As a consequence, the zero-field dimensionless coupling
constant λ [defined in the first column page 3 of our
Letter] is multiplied by a factor of 2 and becomes λ =
3π)(g/t)2.
Second, we also underestimated the numerical value of
the electron-phonon coupling constant g by a factor of√
2. Indeed, g defined in our work [see Eq. (1)] is related
to 〈g2Γ〉F ≃ 0.0405 eV2 computed by Piscanec et al. [2]
as g =
2〈g2Γ〉F ≃ 0.28 eV and not as g =
〈g2Γ〉F ≃ 0.2
eV as incorrectly assumed in our Letter. In addition,
there is a substantial uncertainty in the precise value of
the constant g. In a tight-binding model, the latter may
be related to the derivative of the hopping amplitude t
as a function of the carbon-carbon distance a as g =
(−dt/da)× 3/(2
Mω) [1]. Harrison’s phenomenological
law t ∝ 1/a2 then implies that g ≃ 0.26 eV. Experiments
in graphene [3] and [4] in zero magnetic field give for the
dimensionless coupling constant λ the values 4.4 × 10−3
and 5.3×10−3 respectively. This determines g in between
0.3 eV and 0.36 eV, where we take into account that the
value of t lies between 2.7 and 3 eV. In the end, we have
to take g in the range between 0.26 and 0.36 eV [instead
of g ≃ 0.2 eV] and therefore the dimensionless coupling
constant becomes λ ≃ (2.8 to 5.3)× 10−3 [instead of λ ≃
10−3].
As a result of the two factors of
2, the numerical
estimate for the mode splitting 2gA at ν = 0 and B ≃ 30
T [at the discussed resonance −, 1 ⇒ 0 and 0 ⇒ +, 1, see
second column of page 3] becomes 2gA ∼ 15 meV (∼ 120
cm−1), for g ≃ 0.26 eV and 2gA ∼ 20 meV (∼ 160 cm−1)
for g ≃ 0.36 eV [instead of 2gA ∼ 8 meV]. The effect
is therefore twice larger than initially predicted. The
conclusions of our work remain unaltered.
We would like to thank C. Faugeras and M. Potemski
for having drawn our attention on the underestimated
value of the mode splitting. See also their recent preprint
where they measure the magnetophonon resonance [5].
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K. Geim, A. C. Ferrari, and F. Mauri, Nature Materials
6, 198 (2007).
[4] J. Yan, Y. Zhang, P. Kim, and A. Pinczuk, Phys. Rev.
Lett. 98, 166802 (2007).
[5] C. Faugeras, M. Amado, P. Kossacki, M. Orlita, M.
Sprinkle, C. Berger, W.A. de Heer and M. Potemski,
arXiv:0907.5498.
http://arxiv.org/abs/0907.5498
