Magneto-optical properties of multilayer graphene

Mikito Koshino and Tsuneya AndoDepartment of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

�Received 19 November 2007; revised manuscript received 4 February 2008; published 11 March 2008�

The magneto-optical absorption properties of graphene multilayers are theoretically studied. It is shown thatthe spectrum can be decomposed into sub-components effectively identical to the monolayer or bilayergraphene, allowing us to understand the spectrum systematically as a function of the layer number. Odd-layered graphenes always exhibit absorption peaks which shifts in proportion to �B, with B being the magneticfield, due to the existence of an effective monolayer-like subband. We propose a possibility of observing themonolayer-like spectrum even in a mixture of multilayer graphene films with various layers numbers.

DOI: 10.1103/PhysRevB.77.115313 PACS number�s�: 78.20.�e, 81.05.Uw, 73.22.�f

I. INTRODUCTION

The unusual electronic property of the atomically thingraphene films has been of great interest. Recently, the opti-cal absorption spectra were measured in graphene-relatedsystems under magnetic fields.1–6 In this paper, we theoreti-cally study magneto-optical spectra of the graphenemultilayer.

The monolayer graphene is a zero-gap semiconductorwith the linear dispersion analogous to the zero-mass relativ-istic particle. In the presence of a magnetic field B, it givesan unusual sequence of the Landau levels with spacing pro-portional to �B in both of the electron and hole sides.7 Thetransport properties in such a unique band structure werestudied and found to be significantly different from the con-ventional system.8–13 Recent experimental realizations ofmonocrystalline graphene opened the way for direct probingof those unusual properties.14–16 The Landau-level structureof the single-atomic sheet of graphene was investigatedthrough the quantum Hall effect15,16 and the cyclotronresonance.2,3 The optical response in a graphene monolayerwas theoretically studied.11,17,18

The multilayer systems containing a few layers ofgraphene are also fabricated15,19 and have attracted much in-terest as well.20 There, the interlayer coupling drasticallychanges the structure around the band touching point.19–28

On the other hand, recent observations of the magnetoab-sorption spectra of thin epitaxial graphite1 show�B-dependent transition peaks just as in monolayergraphene.2,3 Similar evidences for the linear dispersion werealso found in thicker graphite systems.4,5,29,30 Quite recently,the cyclotron resonance was measured in a graphene bilayer.6

In theories, the electronic structure in magnetic fields hasbeen extensively studied for a three-dimensional �3D�graphite7,31–35 and for few-layered graphenes.21,24,37 The op-tical absorption was theoretically investigated for the bilayergraphene.36,37

Here, we systematically study the optical absorption prop-erties of the AB-stacked multilayer graphenes in magneticfields as a function of the layer number. We decompose theHamiltonian into subsystems effectively identical to a mono-layer or a bilayer graphene28 and express the spectrum as asummation over each of them. We present in Sec. II theHamiltonian decomposition and the Landau-level structure

of the multilayer graphene as well as the formulation of theoptical absorption. We show the numerical results in Sec. IIIand present our discussion in Sec. IV.

II. FORMULATION

We consider a multilayer graphene composed of N layersof a carbon hexagonal network, which are arranged in the AB�Bernal� stacking. The system can be described by a k ·pHamiltonian based on a 3D graphite model.39–41 The effec-tive models were derived for the monolayer graphene,7,42–44

the bilayer,21 and the trilayer and more.24,28 For simplicity,we include the nearest-neighbor intralayer coupling param-eter �0 and the interlayer coupling �1 between A and B atomslocated vertically with respect to the layer plane. The bandparameters were experimentally estimated in bulk graphite as�0�3.16 eV �Ref. 29� and �1�0.39 eV.45 The effects ofother band parameters neglected here will be discussed inSec. IV.

The low-energy spectrum is given by the states in thevicinity of K and K� points in the Brillouin zone. Let �Aj� and�Bj� be the Bloch functions at the K point, corresponding tothe A and B sublattices, respectively, of layer j. For conve-nience, we divide carbon atoms into two groups as

Group I: B1,A2,B3, . . . , �1�

Group II: A1,B2,A3, . . . �2�

The atoms of group I are arranged along vertical columnsnormal to the layer plane, while those in group II are aboveor below the center of hexagons in the neighboring layers.The lattice constant within a layer is given by a=0.246 nm,and the distance between adjacent layers by c0 /2=0.334 nm.

If the basis is taken as �A1� , �B1�; �A2� , �B2� ; . . . ; �AN� , �BN�,the Hamiltonian for the multilayer graphene around the Kpoint becomes

H =�H0 V

V† H0 V†

V H0 V

� � �

� , �3�

with

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H0 = 0 v�−

v�+ 0, V = 0 0

�1 0 , �4�

where ��=�x� i�y, with �=−i�� +eA, the vector poten-tial A, and v as the band velocity of the monolayer graphene,which is related to the band parameter via v=�3a�0 /2�. Theeffective Hamiltonian for K� is obtained by exchanging �+and �−.

The Hamiltonian �Eq. �3�� can be decomposed intosmaller subsystems for the basis appropriately chosen.28

First, we define the orthonormal sets

��l�I�� = �l�1��B1� + �l�2��A2� + �l�3��B3� + ¯ ,

��l�II�� = �l�1��A1� + �l�2��B2� + �l�3��A3� + ¯ , �5�

where

�l�j� =� 2

N + 1sin j�l, �l =

�

2−

l�

2�N + 1�, �6�

with

l = − �N − 1�,− �N − 3�, . . . ,N − 1. �7�

Here, l is an odd integer when the layer number N is even,while l is even when N is odd. Therefore, l=0 is allowedonly for odd N.

Next, for m0, we take the basis

���m�II�� + ��−m

�II���/�2, ���m�I�� + ��−m

�I� ��/�2,

���m�I�� − ��−m

�I� ��/�2, ���m�II�� − ��−m

�II���/�2� . �8�

For m=0, we take the basis ��0�II�� , ��0

�I���. Then, the Hamil-tonian has no off-diagonal elements between different m’s.For m0, the sub-Hamiltonian within the basis of Eq. �8�becomes

Hm =�0 v�− 0 0

v�+ 0 m�1 0

0 m�1 0 v�−

0 0 v�+ 0� , �9�

with

m = 2 cos �m, �10�

which is equivalent to the Hamiltonian of the bilayergraphene, while the interlayer coupling �1 is multiplied bym. For m=0, we have

Hm=0 = 0 v�−

v�+ 0 , �11�

which is identical to the Hamiltonian of the monolayergraphene. These subsystems are labeled as

m = 0,2,4, . . . ,N − 1 �odd N� ,

m = 1,3,5, . . . ,N − 1 �even N� . �12�

The eigenstate of a finite-layered graphene can be re-garded as a part of a standing wave in 3D limit, which is a

superposition of opposite traveling waves with �kz. Thequantity � �=�m� in our representation corresponds to the 3Dwave number via �= �kz�c0 /2. Thus, the monolayer-type sub-band �=� /2 is related to a H point in the 3D Brillouin zone,while no states exactly correspond to kz=0 since � neverbecomes zero.

The Landau levels of the monolayer-type states are givenby

�sn = s�B�n , �13�

with n=0,1 , . . . and s=�, where s=+ and � represent theelectron and hole bands, respectively, and only s=+ is al-lowed for n=0.7 Here, �B is the magnetic energy defined by

�B = �2�v2eB . �14�

The Landau-level structure of the bilayer-type Hamil-tonian �Eq. �9�� was obtained previously46 and can be ana-lytically derived by noting that �� are associated with theascending or descending operators of the Landau levels24 ina similar way to that for 3D graphite.31,32 The eigenfunctioncan be written as

�c1 n−1,k,c2 n,k,c3 n,k,c4 n+1,k� , �15�

with n�−1 and amplitudes ci. Here, n,k�x ,y� is the wavefunction of the nth Landau level in a conventional two-dimensional system, given in the Landau gauge A= �0,Bx�by n,k= in�2nn!��l�−1/2eikye−z2/2Hn�z�, with z= �x+kl2� / l andHn being the Hermite polynomial. We define n,k�0 for n�0.

For n�1, the Hamiltonian matrix for the vector�c1 ,c2 ,c3 ,c4� then becomes

�0 �B

�n 0 0

�B�n 0 �1 0

0 �1 0 �B�n + 1

0 0 �B�n + 1 0

� , �16�

where the index of m is dropped. This immediately givesfour eigenvalues

�n,�,s =s

�2���1�2 + �2n + 1��B

2

+ ����1�4 + 2�2n + 1���1�2�B2 + �B

4�1/2, �17�

where �=� correspond to the higher and lower subbands inthe limit of zero magnetic field, respectively.21 In the follow-ing, we use the notation �=H ,L instead of �,� to avoid theconfusion with s=�. The eigenstates can be labeled by n, �,s, and k.

For n=0, the first component of the wave function �Eq.�15�� disappears, and we have only three levels,

�0,L = 0, �18�

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�0,H,s = s��12 + �B

2 . �19�

At n=−1 only the last component survives in Eq. �15�, sothat we have only a single level in the lower subband,�−1,L=0 �the level �−1,H does not exist�.

In small magnetic fields, the Landau levels for the lowersubband in the region ���1 are approximately given by�n,L,s�s��eB /m*��n�n+1�, with the effective mass m*

=�1 / �2v2�.21 Thus, the level spacing shrinks much faster inB→0 than in the monolayer ��B. The ratio of the first gapof the bilayer-type subband, �eB /m*, to that of the mono-layer, �B, is given by �B / ��1�.

Figure 1 shows the Landau levels of the bilayer-typeHamiltonian as a function of � in the magnetic field given by�B /�1=0.5. The bilayer levels become those of two indepen-

dent monolayers at �=� /2, where the effective interlayercoupling �1 vanishes. The levels become flat around �=0,where d /d� vanishes. In the bottom panel, we show the listof � for every layer number N. The top and bottom panelsshare the horizontal axis; the Landau levels in the specificpoint in the bottom panel are shown directly above.

The velocity operator for the sub-Hamiltonian Hm isgiven by vx=−�i /���x ,Hm�=�Hm /��x. There are no matrixelements connecting different m’s. For a bilayer-type sub-band, vx has a nonzero matrix element only between theLandau levels with n and n�1 for arbitrary combinations of�=H ,L and s=�. This is explicitly written as

�n�,��,s�;k��vx�n,�,s;k� = v�k,k���c1�*c2 + c3

�*c4��n,n�−1

+ �c2�*c1 + c4

�*c3��n,n�+1� , �20�

where ci and ci� are the eigenvectors of matrix �16�, corre-sponding to the Landau levels �n ,� ,s� and �n� ,�� ,s��, re-spectively. For the monolayer-type band, we have

�n�,s�;k��vx�n,s;k� = v�k,k��c1�*c2�n,n�−1 + c2

�*c1�n,n�+1� ,

�21�

where �c1 ,c2� is �0,1� for n=0 and �s ,1� /�2 for n�1 for a Kpoint.8

To estimate the optical absorption intensity, we calculatethe real part of the dynamical conductivity �xx���. The rela-tive transmission of the sheet to the vacuum, for the linearlypolarized light incident perpendicular to the plane, is relatedto this quantity via38

T = �1 +2�

c�xx����−2

� 1 −4�

cRe �xx��� . �22�

As will be shown below, the expansion is valid except inthick multilayer graphenes for which the absorption is sig-nificant. The dynamical conductivity can be written in ausual manner as

�xx��� =e2�

iS��,�

f���� − f������ − ��

����vx����2

�� − �� + �� + i�, �23�

where S is the area of the system, vx is the velocity operator,� is the positive infinitesimal, f��� is the Fermi distributionfunction, and ��� and �� describe the eigenstate and theeigenenergy of the system.

In the simplest approximation, we include the disordereffect by replacing � with the phenomenological constant� /��2� and taking the ideal eigenstates as � and �. Theconductivity can then be written as a summation over all thecontributions of the subsystems, which are independentlycalculated. Correspondingly, we compute the density ofstates per unit area as

D��� = −1

�SIm �

�

1

� − �� + i�, �24�

with the ideal eigenstates �.The dynamical conductivity at zero magnetic field was

calculated for the monolayer11,17 and the bilayer graphene.36

For the ideal monolayer at �F=0, the expression apart from

-1L, 0L

1L+2L+3L+4L+

4L−3L−2L−1L−

3H−2H−1H−0H−

0H+1H+2H+3H+

1+

0

2+3+4+

4−3−2−1−

0 π/4 π/2

0

5

10

15

20

Num

ber

ofLa

yers

2

-2

Energy(units of γ

1)

0

κ

0 π/4 π/2κ

π/3

π/3

∆B = 0.5 γ1

FIG. 1. �Top� Landau levels of the bilayer-type subband as afunction of � with =2 cos �. The magnetic field strength is takenas �B /�1=0.5. �Bottom� Lists of � in N-layered graphene. Emptyand filled circles represent even and odd N’s, respectively.

MAGNETO-OPTICAL PROPERTIES OF MULTILAYER GRAPHENE PHYSICAL REVIEW B 77, 115313 �2008�

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�=0 becomes a frequency-independent value11,17

Re �xx��� =gvgs

16

e2

�, �25�

with gv=2 being the valley �K ,K�� degeneracy and gs=2 thespin degeneracy. Note that the dynamical conductivity has asingularity at ��F ,��= �0,0�, which is removed if level-broadening effect is included properly.11 The expression forthe effective bilayer Hamiltonian �Eq. �9�� with �F=0 isgiven by36

Re �xx��� =gvgs

16

e2

��22�1

2

�2�2 ���� − �1�

+�� − 2�1

�� − �1���� − 2�1� +

�� + 2�1

�� + �1� ,

�26�

where ��x�=1 for x0 and ��x�=0 for x�0. The aboveshows that the typical value of the real part of the conduc-tivity for �� /�1�1 is �gvgs /16��e2 /�� per layer. By notingthat e2 /�c�1 /137, we see that the expansion in Eq. �22� isvalid roughly for N�20.

III. NUMERICAL RESULTS

Figure 2 shows the plots of Re�xx��� for the monolayerand bilayer graphenes in several magnetic fields. Here wetake � /�1=0.01, �F=0, and zero temperature. Dotted linespenetrating panels represent the transition energies betweenseveral specific Landau levels as a continuous function of�B. The peak positions of each panel correspond to the in-tersections of those and the bottom line of the panel.

In the monolayer, the peak position obviously shifts inproportion to �B �i.e., ��B�. In the limit of vanishing mag-netic field, the conductivity eventually becomes the valuegiven by Eq. �25�. The spectrum in the bilayer is rather com-plicated; starting from �=0, we first see the series of thetransition peaks within L bands from �=0, and then thosebetween L and H enter for ����1, and lastly those within Hbands for ���2�1. Every peak position behaves as a linearfunction of B ���B

2� in weak fields, but it switches over to�B dependence as the corresponding energy goes out of theparabolic band region. In small fields, the peaks are smearedout more easily in the bilayer than in the monolayer. Theconductivity converges to the zero-field curve with a steplikestructure at �=�1, which is expressed as Eq. �26� in the cleanlimit.

Figure 3 shows the plots of Re�xx��� from N=1 to 5 withtwo different magnetic fields with �B /�1=0.1 and 0.2. Weagain take � /�1=0.01, �F=0, and zero temperature. The re-sults are shown separately for each subband. In every oddlayers the monolayer-type subband gives the identical spec-trum. All other bilayer-types give different spectra dependingon . The quantized feature is more easily resolved in asubband with a smaller , because of its narrower level spac-ings. In the zero-field limit, every bilayer-type spectrum hasa step at ���1, where the excitation between L and Hbands starts.

It is intriguing to consider how the absorption spectrumlooks like when the sample is a mixture of thin graphenefilms with various layer numbers. One might think that thediscreteness of � is easily smeared out and we just get the 3Dlimit, but it is not always the case as we will show in thefollowing. We here calculate the dynamical conductivity av-eraged over the samples N=1,2 ,3 , . . . ,20. We show plots ofRe�xx��� for different magnetic fields with �B /�1=0.1, 0.2,

0 1 2

1 2

0

1

2

0

1

2

Re

σ xx(u

nits

ofg vg

se2 /

h)

hω / γ1

(0L, 1L±)

(1L , 2L±)±

(2L , 3L±)±

(-1L, 0H±)

(0L, 1H±)(0H , 1L±)±

(1L , 2H±)±

(1H , 2L±)±

(0H , 1H±)±

∆B / γ1 = 0.1(B ~ 1.1T)

0.2 (4.5T)

0.3 (10T)

0.4 (18T)

0.5 (28T)

0

N = 1 (monolayer)

N = 2 (bilayer)

hω / γ1

(0, 1±) (1 , 2±)±

(2 , 3±)±

Re

σ xx(u

nits

ofg vg

se2 /

h)∆B / γ1 = 0.1

(B ~ 1.1T)

0.2 (4.5T)

0.3 (10T)

0.4 (18T)

0.5 (28T)

FIG. 2. Real part of the dynamical conductivity of the mono-layer �top� and bilayer �bottom� graphenes plotted against the fre-quency � calculated for different magnetic fields �specified by �B�and � /�1=0.01. Dashed curves indicate the transition energies be-tween several Landau levels in the ideal limit.

MIKITO KOSHINO AND TSUNEYA ANDO PHYSICAL REVIEW B 77, 115313 �2008�

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and 0.3 in Fig. 4 and a gray-scale plot of Re�xx�� ,�B� inFig. 5.

Surprisingly, we still see the series of peaks in the mono-layer graphene ����B. This comes from the monolayer-type subbands, which appear in every odd layered graphene.The visibility of the monolayer-type signal depends on the

ratio of the number of monolayer-type subbands to the total;in the present case, this is 10 to 110. We expect the signal ofthe monolayer to gradually become invisible as the maxi-mum layer number Nmax becomes larger because the totalsubband number increases as �Nmax

2 while the number of themonolayer type increases as �Nmax.

We have another set of dominant peaks, which can beidentified as a bilayer type with �=0 �=2�. Although thereis no subband that exactly takes this value, many subbandsaround ��0 have almost the same peak positions as theLandau level is flat against � there and gives similar spectra.Every peak shifts upward with respect to the original posi-tion of �=0 since the Landau-level spacing is generallywider for larger �. Unlike the monolayer-type signal, thiswould survive even in the 3D limit since the finite region in� �not a point� can contribute to this spectrum. When de-creasing the magnetic field, however, the bilayer-type peaksare immediately blurred due to rapid B-linear dependence,while the monolayer peaks survive even in relatively smallermagnetic fields. In the zero-field limit, we are left with abump at ��=2�1, which comes from the H-L transition stepof the bilayer-type subbands with ��0.

Just in the same way as the monolayer-type subband ��=� /2� appears in every two layers, the bilayer-type subbandwith �=� /3 enters in every three layers �N=2,5 ,8 , . . . � andthat with �=� /4 in every four layers �N=3,7 ,11, . . . �. Wecan see the H-L transition peaks of those �’s in Fig. 5, whileL-L peaks are hidden by other dominant contributions.

κ = 2π/5(λ = 0.62)

κ = π/2(λ = 0)

κ = π/5(λ = 1.62)

κ = π/3(λ = 1)

hω /γ1

hω (units of γ1)

N=1

κ = π/2(λ = 0)

κ = π/4(λ = 1.41)

κ = π/2(λ = 0)

κ = π/3(λ = 1)

κ = π/6(λ = 1.73)

N=2

N=3

N=5

N=4

∆B / γ1 = 0.1 (B ~ 1.1T) ∆B / γ1 = 0.2 (B ~ 4.5T)R

eσ xx

(uni

tsof

g vgse

2 /h)

hω /γ1

hω /γ1 hω /γ1

hω /γ1 hω /γ1

hω /γ1 hω /γ1

hω /γ1 hω /γ10 1 2

0 1 2

0 1 2

0 1 2

0 1 2

0 1 20

0.2

0.4

0.6

0.8

1

0 1 20

0.2

0.4

0.6

0.80 1 2

0

0.2

0.4

0.6

0.8

0 1 20

0.2

0.4

0 1 20

0.2

0.4

FIG. 3. Real part of the dynamical conductivity of N=1,2 , . . . ,5 layered graphenes as functions of the frequency, calcu-lated for two different magnetic fields �B /�1=0.1 and 0.2. We take� /�1=0.01.

1 20

0.6

hω / γ1

1 20 hω / γ1

1 20 hω / γ1

0.8

0.7

0.5

0.6

0.8

0.7

0.5

0.6

0.8

0.7

0.5

0.9

0.9

0.9

0.010.02

Γ / γ1Re

σ xx

(uni

tsof

g vgse

2 /h)

∆B / γ1 = 0.2 (B ~ 4.5T)

∆B / γ1 = 0.3 (B ~ 10T)

∆B / γ1 = 0.1 (B ~ 1.1T)

FIG. 4. Plots of Re �xx��� averaged over the layer numbers ofN=1–20 for magnetic fields with �B /�1=0.1, 0.2, and 0.3. Verticalsolid and dashed lines represent the ideal transition energies for themonolayer-type ��=� /2� and bilayer-type ��=0� subbands,respectively.

MAGNETO-OPTICAL PROPERTIES OF MULTILAYER GRAPHENE PHYSICAL REVIEW B 77, 115313 �2008�

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A similar analysis is available for the density of states�DOS�. In Fig. 6, the top panel shows DOS averaged overN=1,2 ,3 , . . . ,20 as a function of the Fermi energy. The bot-tom panel shows the corresponding plot for the local densityof states �LDOS� on the top layer, defined by the number ofstates per unit energy width and per unit area on the layer.We also present in Fig. 7 the two-dimensional plots of DOSand LDOS on the �� ,�B� plane, where the gray scale showsthe relative value from the zero magnetic field.

In DOS, we observe several peaks coming from themonolayer-type subband similar to the optical absorptionspectra. The peaks from the bilayer-type subband with �=0become prominent in the high-field region �B /�1�0.2. InLDOS, interestingly, the peaks of the monolayer-type sub-band are much more pronounced, while those of �=0 arestrongly suppressed. This can be understood by the wavefunction defined by Eq. �8�. If we look at a state in thesubband m in an N-layered graphene, the wave amplitude onthe top layer �j=1� always acquires the factor �m�1�=�2 / �N+1�sin �m. Obviously, this takes maximum in themonolayer type ��=� /2� and zero at �=0, and thus themonolayer-type state contributes the most to the surfaceLDOS. The bilayer-type signals of the subbands �=� /3 and� /4 are also visible in LDOS, while they are hidden by �=0 in DOS.

IV. DISCUSSION

Recently, the optical absorption spectrum was measuredin the epitaxial thin graphite films and the monolayerlikesignal was observed, while the detailed profile of the systemremains unclear. Similar �B-dependent features were alsoobserved in the samples containing a large number ofgraphene layers ��100� grown on a SiC substrate4 and in athin graphite sample of thickness �100 nm exfoliated from ahighly oriented pyrolytic graphite.5 Those results are non-trivial because if the system is a real three-dimensional bulkgraphite, the spectrum would be contributed mainly from thestates around kz=0 ��=0 in our discussion� where the Lan-dau levels are flat with respect to kz. One possible scenariofor this is that the system can be regarded as a compound ofmultilayer fragments with various small layer numbers, andthe monolayerlike spectra of all the odd layers are observed.It should also be mentioned that the local density of states onthe surface of the graphite was observed in theexperiment.47,48 Our calculation predicts that the pronouncedmonolayer-type spectrum would be observable in amultilayer graphene.

While we adopted a simplified effective-mass model inwhich only �0 and �1 are included, we briefly mention herethe effects of other hopping parameters. The parameter �3neglected here couples group II atoms on neighboring layers.This is responsible for the trigonal warping of the band dis-persion but gives only a slight shift in the Landau-level en-ergies except for the low-energy region �10 meV�.33–35

Therefore, it would hardly affect the peak positions in theabsorption spectra while it might modify the amplitudesthrough the matrix element changes. The parameter �4couples group I and II atoms sitting on the neighboring lay-ers, such as Aj↔Aj+1 or Bj↔Bj+1. This parameter introduces

0

0.2

0.4

0

0.2

0.4

0.6 0.7 0.8

Re σxx

(units of gvgse2/h)

κ = π/4(L, H)

κ = π/3(L, H)

κ = 0(L, H)κ = π/2

κ = 0(L, L)

∆ B/γ

1∆ B

/γ1

0 1 2 hω / γ1

0 1 2 hω / γ1

0

12

5

10

20

B(T

)

0

12

5

10

20

B(T

)

FIG. 5. �Top� Density plot of Re�xx�� ,�B� averaged over thelayer numbers of N=1–20. � is set to 0.01�1. �Bottom� Corre-sponding plot for the transition energies between the several Landaulevels.

0

5

10

0

0.5

1

∆B / γ1 = 0.3 (B ~ 10T)

DO

S(u

nits

ofg vg

sγ1/

2πh2 v

2 )LD

OS

(uni

tsof

g vgsγ

1/2π

h2 v2 )

0 0.2 0.4 0.6 0.8 ε / γ1

0 0.2 0.4 0.6 0.8 ε / γ1

1L+2L+3L+4L+

1+ 2+ 3+ 4+

5L+

(κ = 0)

(κ = π/2) 0.01

0.02

Γ / γ1

FIG. 6. �Top� Density of states averaged over the layer numbersof N=1–20 at �B /�1=0.3. �Bottom� Corresponding plot for thelocal density of states on the top layer. Vertical solid and dashedlines indicate the ideal Landau level energies for the monolayer-type ��=� /2� and bilayer-type ��=0� subbands, respectively.

MIKITO KOSHINO AND TSUNEYA ANDO PHYSICAL REVIEW B 77, 115313 �2008�

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a small electron-hole asymmetry in the band structure butdoes not change the qualitative feature of the low-energyspectrum.41

We also neglected the vertical hopping between thesecond-nearest-neighboring layers for group II and I atoms,

which are parametrized by �2 and �5, respectively. Includingthose parameters mainly shifts the zero energy �the bandtouching point� upward or downward, depending on eachsubsystem.25–27 The tight-binding model26,27 and the densityfunctional theory25 estimate the shift �E at the order of10 meV. In the 3D limit, this corresponds to the band dis-persion along the kz direction.7,31–35 The zero-energy shiftleads to the electron or hole doping and gives a change of theabsorption spectrum in the region ���2�E.

The effective-mass model is no longer valid when theenergy is as high as the intralayer coupling �0�3 eV. Thelattice effect appears as trigonal warping in the band disper-sion in higher energies,49–51 while this should be distin-guished from the trigonal warping discussed above, which isdue to the extra band parameter within the effective-massmodel. The frequency region covered in our calculation,���2.5�1�1 eV, roughly corresponds to the energy region����0.5 eV. The deviation in eigenenergy is estimated at 5%at �=0.5 eV and can be treated perturbationally,49 although itgrows as the energy increases out of this region. This aniso-tropy constitutes a major part of the chirality dependence ofoptical spectra in carbon nanotubes, enabling the assignmentof the structure of individual nanotubes.52

Lastly, while our model is based on the bulk 3D graphite,it should be noted that the band parameters in few-layeredgraphenes are not exactly the same as those for the bulkgraphite, but generally vary depending on the layernumber.19 There is a theoretical attempt to obtain accurateelectronic structures for few-layered graphenes using thedensity functional theory with a local densityapproximation.25 A calculation beyond a local density ap-proximation was also proposed, which properly treats nonlo-cal van der Waals interaction coupling graphene layers in thedensity functional framework.53 The study of optical absorp-tion in a refined band model is left for a future work.

In conclusion, we present a systematic study of the opticalabsorption properties and the density of states in themultilayer graphenes as a function of layer numbers. Thespectrum can be understood through the decomposition intosubcomponents, each of which is equivalent to the mono-layer graphene or the bilayer graphene with a single param-eter �. We propose that the monolayerlike spectra is possiblyobserved in the mixture of the multilayered graphene, con-tributed by the effective monolayer subbands existing in ev-ery odd-layered graphene.

ACKNOWLEDGMENTS

The authors acknowledge helpful interactions with E. A.Henriksen, Z. Jiang, K. F. Mak, P. Kim, and T. F. Heinz. Thiswork has been supported in part by the 21st Century COEProgram at Tokyo Tech “Nanometer-Scale Quantum Phys-ics” and by Grants-in-Aid for Scientific Research and Prior-ity Area “Carbon Nanotube Nano-Electronics” from the Min-istry of Education, Culture, Sports, Science and Technology,Japan.

0

0.2

0.4

DOS difference

(units of gvg

sγ

1/2πh2v2)

-1 0 1

LDOS difference

(units of gvg

sγ

1/2πh2v2)

0 0.2 0.4 0.6 0.8 ε / γ1

∆B

/γ 1

0

12

5

10

20

B(T

)

0

0.2

0.4

∆B

/γ 1

0

12

5

10

20

B(T

)

-0.1 0 0.1

0

0.2

0.4

∆B

/γ 1

0

12

5

10

20

B(T

)

κ = π/2

κ = π/3

κ = 0

0 0.2 0.4 0.6 0.8 ε / γ1

0 0.2 0.4 0.6 0.8 ε / γ1

FIG. 7. �Top� Density of states �measured from the zero-fieldvalue� averaged over the layer numbers of N=1–20, plotted in the�� ,�B� plane. � is set to 0.01�1. �Middle� A similar plot for thelocal density of states on the top layer. �Bottom� Ideal Landau levelenergies corresponding to several dominant peaks in the above twopanels.

MAGNETO-OPTICAL PROPERTIES OF MULTILAYER GRAPHENE PHYSICAL REVIEW B 77, 115313 �2008�

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