Probing the electronic structure of bilayer graphene by Raman scattering
L.M.Malard,1J.Nilsson,2D.C.Elias,1J.C.Brant,1F.Plentz,1E.S.Alves,1A.H.Castro Neto,2and M.A.Pimenta1 1Departamento de Física,Universidade Federal de Minas Gerais,30123-970Belo Horizonte,Brazil 2Department of Physics,Boston University,590Commonwealth Avenue,Boston,Massachutts02215,USA
͑Received14September2007;published1November2007͒
The electronic structure of bilayer graphene is investigated from a resonant Raman study of the GЈband
using different lar excitation energies.The values of the parameters of the Slonczewski-Weiss-McClure
model for bilayer graphene are obtained from the analysis of the dispersive behavior of the Raman features,
and reveal the difference of the effective mass of electrons and holes.The splitting of the two TO phonon
branches in bilayer graphene is also obtained from the experimental data.Our results have implications for
bilayer graphene electronic devices.
DOI:10.1103/PhysRevB.76.201401PACS number͑s͒:73.21.Ϫb,63.20.Kr,78.30.Ϫj,81.05.Uw
Differently from monolayer graphene,where the electrons behave like massless Dirac fermions and exhibit a linear dis-persion near the Dirac point,the electrons in bilayer graphene are described by nonzero effective mass Dirac fer-mions with a parabolic electronic dispersion.1Furthermore, while the unbiad bilayer graphene is a zero-gap micon-ductor,a biad bilayer is a tunable gap miconductor by electricfield effect.2,3Hence the development of bilayer graphene-bad bulk devices depends on the detailed under-standing of its electronic properties.This work shows that, by performing Raman scattering experiments in bilayer graphene with many different lar excitation energies,we can probe its electronic structure and we can obtain experi-mental values for the Slonczewski-Weiss-McClure͑SWM͒parameters4,5for bilayer graphene.
Figure1shows the atomic structure of a bilayer graphene, in which we can distinguish the two nonequivalent atoms A and B in each plane giving ri to a unit cell with four atoms. Since this unit c
ell is the same for graphite in the Bernal stacking structure,we can describe the electronic spectrum of bilayer graphene in terms of the SWM model for graphite,4,5by determining the parameters␥0,␥1,␥3,and␥4, that are associated with overlap and transfer integrals calcu-
lated for nearest neighbor atoms.The pair of atoms associ-
ated with the parameters is indicated in the atomic struc-
ture of a bilayer graphene shown in Fig.1͑a͒.The
tennis是什么意思
parameters,that are fundamental for the electronic process
in the system,are only roughly known to this date.
The graphene samples ud is this experiment were ob-
tained by a micromechanical cleavage of graphite on the sur-
face of a Si sample with a300nm layer of SiO2on the top.1
The bilayerflakes were identified by the slight color change
from monolayer graphene in an optical microscope,followed
by a Raman spectroscopy characterization using the proce-
dure described by Ferrari et al.6For the Raman measure-
ments,we ud a Dilor XY triple monochromator in the
backscattering configuration.The spot size of the lar was ϳ1m using a100ϫobjective and the lar power was kept at1.2mW in order to avoid sample heating.Raman spectra were obtained for11different lar lines of Ar-Kr and dye lars in the range1.91–2.71eV.
阅读题及答案
Recently,Ferrari et al.6showed that Raman spectroscopy
can be ud to identify the number of layers in a graphene
sample and,in particular,to clearly distinguish a monolayer from a bilayer graphene sample.Figure2shows the Raman spectra of the monolayer͓Fig.2͑a͔͒and bilayer͓Fig.2͑b͔͒graphene samples,where the most prominent features are the G and GЈRaman bands.7
The GЈband of the monolayer graphene can befitted by just one Lorentzian with a full width at half maximum ͑FWHM͒of24cm−1.A better adjustment can be obtained with V oigt functions,which have fourfitting parameters. However,different ts of four V oigt parametersfit the GЈband equally well,preventing a preci physical interpreta-tion of the parameters.Therefore we decided to analyze the data using the Lorentzian functions.The GЈband for bilayer graphene wasfitted using four Lorentzian peaks,all of them having the same FWHM of24cm−1ud tofit the GЈband of monolayer graphene,in agreement with the previous Ra-man studies of graphene systems.6,8,9The relative amplitudes of the four Lorentzians depend on the lar energy;two of them increa and the other two decrea with increasing lar energy.Thefit was done by following the trend of the lar energy dependence of the relative intensities.
The Raman spectra of both the monolayer and bilayer graphene have been measured with many different lar en-ergies in the visible range.Figure3shows the lar energy dependence of the GЈ-band frequency for the monolayer sample͓Fig.3͑a͔͒and for each one of the four peaks that compri the GЈband for bilayer graphene͓Fig.3͑b͔͒.
The origin of the GЈband in all graphitic materials is due to an intervalley double-resonance͑DR͒Raman process10,
11
FIG.1.͑Color online͒͑a͒Atomic structure of bilayer graphene. The A atoms of the two layers are over each other,whereas the B atoms of the two layers are displaced with respect to each other.The SWM constants␥0,␥1,␥3,and␥4label the corresponding pair of atoms associated with the hopping process.͑b͒First Brillouin zone of monolayer graphene,showing the high symmetry points⌫, K,KЈ,and M.
PHYSICAL REVIEW B76,201401͑R͒͑2007͒
involving electronic states near two nonequivalent K points ͑K and K Ј͒in the first Brillouin zone of graphene ͓e Fig.1͑b ͔͒,and two phonons of the iTO branch.7,12As a result of the angular dependence of the electron-phonon matrix elements 13and the existence of interference effects in the Raman cross ction,12the main contribution for the G Јband comes from the particular DR process that occ
urs along the ⌫-K -M -K Ј-⌫direction,in which the wave vectors k and k Јof the two intermediate electronic states in the conduction band ͑measured from the K and K Јpoints,respectively ͒are along the K ⌫and K Ј⌫directions,respectively.6Therefore the wave vector q of the phonons involved in this specific process is along the KM direction and is related to the elec-tron wave vectors by the condition q =k +k Ј.7
In the ca of the bilayer graphene,where the two graphene layers are stacked in a Bernal configuration,7the -electrons dispersion in the valence and in the conduction
band splits into two parabolic branches near the K point,14as shown in Fig.4.In this figure,the upper and lower branches of the valence band are labeled as 1and 2,respectively.The lower and upper branches of the conduction band are
called 1*and 2*
,respectively.Along the high symmetry ⌫-K -M -K Ј-⌫direction,the branches belong to different irreducible reprentations of the P 63/mmc space group and
therefore only the 1 1*and 2 2*
optical transitions between the valence and conduction bands are allowed.Therefore there are four possible intervalley DR process involving electrons along the ⌫-K -M -K Ј-⌫direction that lead to the obrvation of the four peaks of the G Јband of bilayer graphene.6
The four DR process are reprented in Fig.4.In pro-cess P 11͓Fig.4͑a ͔͒,an electron with wave vector k 1is reso-nantly excited from the valence band 1to the conduction band 1*by absorbing a photon with energy E L .This electron is then resonantly scattered to a state with wave vector k 1Јby
emitting a phonon with momentum q 11and energy E p 11
.Fi-nally,the electron is scattered back to state k 1by emitting a cond phonon,and it recombines with a hole producing a scattered photon with an energy E S =E L −2E p 11.The phonon wave vector q 11,measured from the K point and along the KM direction,is given by q 11=k 1+k 1Ј.归属感英文
Figure 4͑b ͒shows the DR process P 22,which involves an electron that is optically excited between the 2and 2
*branches.The energy of the associated phonon is E p 22
and its wave vector is given by q 22=k 2+k 2Ј.Figures 4͑c ͒and 4͑d ͒show process P 12and P 21that i
nvolve electrons with wave vectors k 1,k 2Јand k 2,k 1Ј,respectively,which belong to different electron branches.The wave vectors of the phonons associated with process P 12and P 21are given by q 12=k 1+k 2Јand q 21=k 2+k 1Ј,respectively.
The two iTO phonon branches of a bilayer graphene along the ⌫KM line belong to the irreducible reprentations T 1and T 3.The scattering of an electron in the conduction band between states around K and K Јhas to satisfy the electron-phonon lection rule,so that the allowed
transi-
confidant
FIG.2.͑a ͒Raman spectrum of the monolayer graphene and ͑b ͒Raman spectrum of the bilayer graphene performed with the 2.41eV lar
line.
FIG.3.͑a ͒Lar energy dependence of the G Ј-band energy for a monolayer graphene.͑b ͒Lar energy dependence of the posi-tions of the four peaks that compri the G Јband of bilayer graphene.The four Raman peaks originate from the P 11,P 12,P 21,and P 22DR process illustrated in Fig.4.The black squares are the experimental data and the full lines are the fitted curves ͑e dis-cussion in the
text ͒
.
FIG.4.Schematic view of the electron dispersion of bilayer graphene near the K and K Јpoints showing both 1and 2bands.The four DR process are indicated:͑a ͒process P 11,͑b ͒process P 22,͑c ͒process P 12,and ͑d ͒process P 21.The wave vectors of the electrons ͑k 1,k 2,k 1Ј,and k 2Ј͒involved in each of the four DR process are also indicated.
MALARD et al.PHYSICAL REVIEW B 76,201401͑R ͒͑2007͒
tions are1* 1*or2* 2*for the T1phonon and2* 1* for the T3phonon.Since process P11and P22involve elec-trons states around K and KЈwhich belong to the same elec-tronic branch,the associated phonons belong to the T1pho-non branch.On the other hand,phonons involved in process P12and P21belong to the T3branch.
From Fig.4one can e that the phonon associated with the P11process has the largest wave vector͑q11͒.Since the energy of the iTO phonon increas with increasing q,the highest frequency component of the GЈband of a bilayer
graphene is related to the P11process.On the other hand,P22 process is associated with the smallest phonon wave vector q22,and gives ri to the lowest frequency component of the GЈband of a bilayer graphene.The two intermediate peaks of the GЈband are associated with process P12and P21.
In order to analyze the experimental results shown in Fig. 3͑b͒,we mustfind a relation between the electronic and the phonon dispersions of a bilayer graphene,according to the DR Raman process.The electronic dispersion of the bilayer graphene can be described in terms of the standard SWM model for graphite͓,Eq.͑2.1͒in Ref.4͔and using the full tight-binding dispersion introduced
originally by Wallace.15Along the K-⌫direction this amounts to replacing by␥0͓2cos͑2/3−kaͱ3/2͒+1͔in McClure’s expressions. Here k is measured from the K point and a=1.42Åis the in plane nearest neighbor carbon distance.Since there is no k z dependence we may t⌫=1and␥2=␥5=0͑␥2and␥5cor-respond to third layer interactions in graphite͒.We have veri-fied that the parameter⌬does not make any noticeable dif-ference in our results and will be ignored in our analysis. With the simplifications,the bands in the bilayer are pa-rametrized by the four parameters␥0,␥1,␥3,and␥4.A more detailed account of this approach to the band structure cal-culation of the graphene bilayer can be found in Ref.16.
In fact,along the high symmetry K-⌫direction,the 4ϫ4matrix factorizes and the dispersion of the four bands are given by
E
2
=͑−␥1−v3−+͒/2,͑1a͒
E
1
=͑␥1+v3−−͒/2,͑1b͒
E
1
*=͑−␥1−v3++͒/2,͑1c͒
E
2
*=͑␥1+v3+−͒/2,͑1d͒where
±=ͱ͑␥1−v3͒2+4͑1±v4͒22,͑2͒and v jϵ␥j/␥0.
In order to obtain the dependence of the phonon energy E p ij on the lar energy E L,let us consider a generic process P ij,where i,j=1or2,which describes the four process shown in Fig.4.In thefirst step of this process͑electron-hole creation͒,the incident photon is in resonance with the elec-tr
onic states in the valence and conduction bands at the k i point.Thus the lar energy E L can be written as
E L=E
i
*͑k i͒−Ei͑k i͒.͑3͒Equation͑3͒allows us to determine the momentum k i of the electron excited in the process.
The electron is then resonantly scattered from the vicinity of the K point to the vicinity of the KЈpoint by emitting a ͑iTO͒phonon with energy E p ij given by
E p ij͑k i+k jЈ͒=E
i
*͑k i͒−Ej*͑k jЈ͒.͑4͒Assuming that we know the iTO phonon dispersion near the K point͓A+B͑k i+k jЈ͔͒as well as the bands involved͓Eqs.͑1͒and͑2͔͒Eq.͑4͒uniquely determines the momentum of the scattered electron k jЈ.We then compute the phonon en-ergy E p ij,that is directly related to the Raman shift for this speci
fic P ij process,obtained with a given lar energy E L. Finally,we perform a least-squaresfit to determine the pa-rameters␥0,␥1,␥3,and␥4of the model͓Eqs.͑1͒and͑2͔͒that give the bestfit of the dispersion of the four GЈpeaks in bilayer graphene shown in Fig.3.The four solid lines in Fig. 3͑b͒reprent the bestfitting of the experimental E p ij vs E L data.
We have also tried tofit,unsuccessfully,our experimental data for the four DR process taking only␥0,␥1,and␥3. Therefore in order to get a goodfitting,the parameter␥4, which is associated with the splitting of the two GЈinterme-diate peaks,has to be included.For the dispersion of the iTO phonon branches near the K point,we could notfit satisfac-torily the dispersion data in Fig.3͑b͒considering the same phonon dispersion for the four P ij process.The bestfit was obtained when we considered different dispersions for the two iTO phonon branches of bilayer graphene.Table I shows the parameters obtained for each phonon branch,which ex-hibit a linear dispersion near the K point.Notice that the difference between the two phonon branches corresponds to a splitting of about3cm−1,which is in clo agreement with that reported previously.6,9
Table II shows the␥values obtained experimentally.The parameter␥0,associated with the in-plane nearest neighbor TABLE I.Values obtained for the phonon dispersion of the two iTO phonon branche
s of bilayer graphene near the K point,where E p ii=A ii+B ii q corresponds to the P11and P22process and E p ij =A ij+B ij q to the P12and P21process.
A ii
B ii A ij B ij
153.7͑meV͒38.5͑meVÅ͒154.0͑meV͒38.8͑meVÅ͒1238͑cm−1͒310͑cm−1Å͒1241͑cm−1͒313͑cm−1Å͒
TABLE II.Experimental SWM parameters͑in eV͒for the band structure of bilayer graphene.The parameters for graphite are taken from Refs.18and19.
急急如律令怎么翻译␥0␥1␥2␥3␥4␥5␥6=⌬Bilayer graphene 2.90.30n/a0.100.12n/a n/a Graphite 3.160.39−0.020.3150.0440.0380.008
PROBING THE ELECTRONIC STRUCTURE OF BILAYER…PHYSICAL REVIEW B76,201401͑R͒͑2007͒
hopping energy,is ten times larger than␥1,which is associ-ated with atoms from different layers along the vertical di-
rection͑e Fig.1͒.The values are in good agreement with
the previous angle resolved photoemission spectroscopy ͑ARPES͒measurements in bilayer graphene.17The resolu-tion of our experiment allows,however,the measurement of
weaker hopping parameters͑␥3and␥4͒,that are beyond the current resolution of ARPES.The value of␥1is about three times larger than␥3and␥4,both associated with the inter-layer hopping not along the vertical direction͑e Fig.1͒.
The corresponding parameters found experimentally for
graphite are also shown in Table II.18,19The parameters␥0 and␥1for bilayer graphene are slightly smaller than tho for graphite.This difference is more accentuated for the pa-rameter␥3.The results are in good agreement with the weak interlayer coupling parameters extracted from thefirst-principles calculations on the three dimensional graphite.20,21 On the other hand,our value of␥4for bilayer graphene is significantly higher than the value for graphite measured by Mendez et al.in a magnetoreflection experiment.22Notice that this parameter is especially important since it is related to the difference of electron and hole effective mass in the valence and conduction bands.19
blissfulFinally,we must also comment that there are also other
环球职业教育在线网symmetry allowed weak terms which are not prent in the
standard SWM model.In particular,the next-nearest neigh-
bor hopping process within each layer,denoted by tЈ,23was already considered in the minal paper by Wallace15and later on in the model reported by Johnson and Dreslhaus.24 Similarly to the␥4parameter,tЈalso breaks the electron hole symmetry that is prent in the simplest ca considering only␥0,␥1,and␥3.However,with the introduction of this new term in the least-squarefitting process,we are faced with the problem of having too manyfitting parameters,pre-venting an accurate determination of all parameters from the data.However,the important point is that the data shown in Fig.3cannot be explained without considering the asymme-try between electrons and holes.
In summary,from the resonant Raman study of the GЈband of bilayer graphene using veral lar excitation ener-gies,we have been able to probe the dispersion of electrons and phonons of this material near the Dirac point.From the fitting of the experimental data using the SWM model,we have obtained experimental values for the interlayer hopping parameters for bilayer graphene,and the results reveal the asymmetry between the electronic dispersion curves in the valence and condu
ction bands.Finally,in a future work we will study DR process involving holes,many-body effects,25as well as the biad bilayer graphene.
This work was supported by Rede Nacional de Pesquisa em Nanotubos de Carbono–MCT,Brasil and FAPEMIG. L.M.M.,J.C.B,and D.C.E.acknowledge the support from the Brazilian Agency CNPq.We would like to thank Millie Dreslhaus and Ado Jorio for critical reading of the manu-script and for helpful discussions.
1A.K.Geim and K.S.Novolov,Nat.Mater.6,183͑2007͒.
2E.V.Castro,K.S.Novolov,S.V.Morozov,N.M.R.Peres,J.
M.B.Lopes dos Santos,J.Nilsson,F.Guinea,A.K.Geim,and
A.H.Castro Neto,arXiv:cond-mat/0611342.
3J.B.Oostinga,H.B.Heersche,X.Liu,A.F.Morpurgo,and L.
M.K.Vandersypen,arXiv:0707.2487.
滑雪装4J.W.McClure,Phys.Rev.108,612͑1957͒.
5J.C.Slonczewski and P.R.Weiss,Phys.Rev.109,272͑1958͒. 6A.C.Ferrari,J.C.Meyer,V.Scardaci,C.Casiraghi,M.Lazzeri,
F.Mauri,S.Piscanec,D.Jiang,K.S.Novolov,S.Roth,and
A.K.Geim,Phys.Rev.Lett.97,187401͑2006͒.
7M.A.Pimenta,G.Dreslhaus,M.S.Dreslhaus,L.G.Can-cado,A.Jorio,and R.Saito,Phys.Chem.Chem.Phys.9,1276͑2007͒.
8A.Gupta,G.Chen,P.Joshi,S.Tadigadapa,and P.C.Eklund, Nano Lett.6,2667͑2006͒.
9D.Graf,F.Molitor,K.Ensslin,C.Stampfer,A.Jungen,C.Hie-rold,and L.Wirtz,Nano Lett.7,238͑2007͒.
10C.Thomn and S.Reich,Phys.Rev.Lett.85,5214͑2000͒.
11R.Saito,A.Jorio,A.G.Souza Filho,G.Dreslhaus,M.S.
Dreslhaus,and M.A.Pimenta,Phys.Rev.Lett.88,027401͑2001͒.
12J.Maultzsch,S.Reich,and C.Thomn,Phys.Rev.B70, 155403͑2004͒.13G.Ge.Samsonidze,Ph.D.th
cultureesis,Massachutts Institute of Tech-nology,2007.
14E.McCann and V.I.Falko,Phys.Rev.Lett.96,086805͑2006͒. 15P.R.Wallace,Phys.Rev.71,622͑1947͒.
16B.Partoens and F.M.Peeters,Phys.Rev.B74,075404͑2006͒. 17T.Ohta,A.Bostwick,J.L.McChesney,T.Seyller,K.Horn,and Eli Rotenberg,Phys.Rev.Lett.98,206802͑2007͒.
18N.B.Brandt S.M.Chudinov,and Ya.G.Ponomarev,Semimetals 1:Graphite and its Compounds͑North-Holland,Amsterdam, 1988͒.
19M.S.Dreslhaus,G.Dreslhaus,K.Sugihara,I.L.Spain,and
H.A.Goldberg,Carbon Fibers and Filaments͑Springer-Verlag,
Berlin,Heidelberg,1988͒,Chap.7.
20R.C.Tatar and S.Rabii,Phys.Rev.B25,4126͑1982͒.
21J.-C.Charlier,J.-P.Michenaud,and X.Gonze,Phys.Rev.B46, 4531͑1992͒.
22E.Mendez,A.Misu,and M.S.Dreslhaus,Phys.Rev.B21, 827͑1980͒.
四级英语23N.M.R.Peres,F.Guinea,and A.H.Castro Neto,Ann.Phys.
͑N.Y.͒321,1559͑2006͒.
24L.G.Johnson and G.Dreslhaus,Phys.Rev.B7,2275͑1973͒. 25S.V.Kusminskiy,J.Nilsson,D.K.Campbell,and A.H.Castro Neto,arXiv:0706.2359.
MALARD et al.PHYSICAL REVIEW B76,201401͑R͒͑2007͒
