7/30/2019 65609

1/9

D band Raman intensity calculation in armchair edgedgraphene nanoribbons

Citation Barros, E. et al. D Band Raman Intensity Calculation inArmchair Edged Graphene Nanoribbons. Physical Review B83.24 (2011).2011 American Physical Society.

As Published http://dx.doi.org/10.1103/PhysRevB.83.245435

Publisher American Physical Society

Version Final published version

Accessed Tue Aug 20 14:27:42 EDT 2013

Citable Link http://hdl.handle.net/1721.1/65609

Terms of Use Article is made available in accordance with the publisher's policyand may be subject to US copyright law. Please refer to thepublisher's site for terms of use.

Detailed Terms

The MIT Faculty has made this article openly available. Please sharehow this access benefits you. Your story matters.

http://dx.doi.org/10.1103/PhysRevB.83.245435http://hdl.handle.net/1721.1/65609http://libraries.mit.edu/forms/dspace-oa-articles.htmlhttp://libraries.mit.edu/forms/dspace-oa-articles.htmlhttp://hdl.handle.net/1721.1/65609http://dx.doi.org/10.1103/PhysRevB.83.245435

7/30/2019 65609

2/9

PHYSICAL REVIEW B 83, 245435 (2011)

D band Raman intensity calculation in armchair edged graphene nanoribbons

E. B. Barros,1,2,* K. Sato,2 Ge. G. Samsonidze,3 A. G. Souza Filho,1 M. S. Dresselhaus,4 and R. Saito2

1Departamento de F sica, Universidade Federal do Cear a, Fortaleza, Cear a, 60455-760 Brazil2Department of Physics, Tohoku University, Sendai 980-8578, Japan

3Department of Physics, University of California at Berkeley, and Materials Sciences Division, Lawrence Berkeley National Laboratory,

Berkeley, California 94720, USA4Department of Electrical Engineering and Computer Science and Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts, 02139, USA

(Received 22 March 2011; revised manuscript received 3 May 2011; published 27 June 2011)

The D band Raman intensity is calculated for armchair edged graphene nanoribbons using an extended

tight-binding method in which the effect of interactions up to the seventh nearest neighbor is taken into account.

The possibility of a double resonance Raman process with multiple scattering events is considered by calculating

a T matrix through a direct diagonalization of the nanoribbon Hamiltonian. We show that long-range interactions

play an important role in the evaluation of both the D band intensity and that the main effectof multiple scattering

events on the calculated D band is an overall increase in intensity by a factor of 4. The D band intensity is shown

to be independent of the nanoribbon widths for widths larger than 17 nm, leading to the well-known linear

dependence of the ID /IG ratio on the inverse of the crystalline size. The D band intensity was shown to be nearly

independent of the laser excitation energy and to have a maximum value for incident and scattering photons

polarized along the direction of the edge.

DOI: 10.1103/PhysRevB.83.245435 PACS number(s): 78.67.Wj, 73.22.Pr

I. INTRODUCTION

The D band Raman spectra in both graphene and single-wall carbon nanotubes (SWNTs) has been widely used tocharacterize defective carbon-based samples. It has beenknown that the D band intensitydepends both on thecrystallinesize and on the laser excitation energy (Elaser) in such away that the D band intensity is inversely proportional tothe in-plane crystalline size1,2 (for d larger than 20 nm)and the D band to G band relative intensity (I

D/I

G) is

proportional to E4laser.2 Experimental evidence suggests that

the main dependence of the ID/IG ratio on the laser excitationoriginates mainly from an increase in the IG intensity withincreasing laser excitation energy as IG E4laser, while the Dband intensity ID would be mainly independent of the laserexcitation energy.3 A recent independent experimental workby Narula et al.4 investigating the Raman scattering matrixelements of the D and G bands observed that the D bandintensity increases nearly linearly with laser excitation forgraphene (Elaser < 2.6 eV). Also, different Elaser dependenceswere found for the D and G band intensities for monolayergraphene on Si/SiO2 and pencil graphite.

4 Furthermore, Sasakiet al. have studied the polarization dependence of the D band

intensity for nanoribbons, observing that the D band intensityhas a maximum when both the incident and the scatteredphotons are polarized along the direction of the armchairedge.5

The D band intensity is usually explained considering adouble resonance Raman scattering (DRRS) mechanism6,7 inwhich the defect acts as an elastic scatterer forthe photoexcitedelectron in the second-order scattering processes in k space. Inorder to estimate the role of the defect on the double resonanceprocess, it is necessary to calculate the elastic scatteringamplitude between two k states in the Brillouinzone. Formally,the scattering amplitude for a given scattering potential can beexpressed by the so-called T matrix, in which the modification

of the electronic states by scattering is taken into account inan infinite series expansion.8 Sato et al. calculated the D bandintensity for armchair edged graphene ribbons in which theyconsider the first-order term of the scattering matrix, and theyobtained a dependence for the D band integrated intensityon Elaser.

9 In the previous calculation, the electron-defectscattering matrix elements were obtained within a simpletight-binding method (STB) for the nearest-neighbor atoms inwhich the scattering at an armchair edge is taken into account

as missing transfer integrals of the nearest-neighbor atoms atthe edges.

Although these approximations can give insight into theoverall properties of the D band Raman peak, the modificationof the electronic wave functions by the scattering wave wasnot considered by higher-order terms in the perturbationexpansion. Furthermore, in the previous calculation, weconsidered only first nearest-neighbor interactions for thescattering potential, which does not always represent the fullrange of the scattering potential.

In this paper, we evaluate both of these missing effects

by considering farther neighbors and by considering the

effect of multiple scattering processes on the total elastic

scatteringamplitude. The effect of considering other neighborscan be taken into account by using the extended tight-

binding method (ETB), where the tight-binding parameters

dependence on the interatomic distance was obtained from

density functional theory.10 Furthermore, the effect of multiple

scattering processes can be taken into account by considering a

T matrix.

In Sec. II, we briefly describe the T matrix formalism in

the scattering theory and then introduce the scattering potential

for an armchair edged graphene nanoribbon. In Sec. III, we

introduce the model used for the D band Raman intensity

calculation and in Sec. IV, we discuss the calculated results

for the scattering amplitude and for the D band intensity as a

245435-11098-0121/2011/83(24)/245435(8) 2011 American Physical Society

http://dx.doi.org/10.1103/PhysRevB.83.245435http://dx.doi.org/10.1103/PhysRevB.83.245435

7/30/2019 65609

3/9

E. B. BARROS et al. PHYSICAL REVIEW B 83, 245435 (2011)

function ofElaser, the nanoribbon width, and its dependence on

the incident and scattered light polarizations. Finally, in Sec.

V, a summary of the present calculation is given.

II. THEORETICAL BACKGROUND

Within scattering theory, an elastic scattering amplitude

between the initial (i) and final (f) states of the unperturbedsystem due to an arbitrary perturbation is given as

Sfi = f i 2 (Ef Ei)Tf i(E), (1)where the first Kronecker delta function (f i ) becomes unityfor identical initial (i) and final (f) states, and the seconddelta function corresponds to the conservation of energy. TheT matrix Tf i (E) can be calculated in terms of the T operator,which can be expressed in terms of a Greens function for theperturbed system G(E) = (E H)1 as

T(E) = V + V G(E)V , (2)where H

=H0

+V is the perturbed Hamiltonian (graphene

ribbon withedges), H0 is the unperturbed Hamiltonian (infinitegraphene sheet), and V is the scattering potential associatedwith the edge.

The T operator can also be expressed by an unperturbedGreens function G0(E) = (E H0)1 as the following seriesexpansion:

T(E) = V + V G0(E)V + V G0(E)V G0(E)V + . (3)Each of the terms in the series expansion can be interpretedas processes with a fixed number of scattering events. Thereare several difficulties related to using the method given bythe expansion of Eq. (3). The first difficulty is the fact that, formost scattering potentials, it is not possible to know a priori

how fast the series converges and thus, the series is usuallytruncated at a finite number of scattering processes.

Such difficulty is avoided if the Hamiltonian H for theperturbed systemcan be diagonalizedso that G(E) is expandedin terms of its eigenfunctions, and then the T matrix can beobtained explicitly. This formalism takes into considerationall the possible scattering events and does not depend on the

convergence of the series. However, there is a requirement thatthe eigenfunctions and eigenenergies of the perturbed systemmust be known in order to allow for the calculation of the Tmatrix.

In this paper, we will apply the T matrix formalism to cal-culate the scattering amplitude for armchair edges in graphene(the choice of the armchair edges is based on the fact that

purely zigzag edges will not contribute to the D band Ramanscattering).11 To calculate the scattering matrix elements T(E)in Eq. (2) for the case of armchair edged graphene ribbons,the supercell shown in Fig. 1(a) is considered. The ribbonwidth L is given by the number of 4-atom unit cells (shownby a rectangle in the figure) used to construct the supercell(Ny). Figure 1(b) shows the real-space structure of graphenewhere the 4-atom unit cell mentioned above is depicted by alight blue rectangle. The graphene reciprocal space structureis shown in Fig. 1(c), where the Brillouin zone correspondingto the 4-atom unit cell in Fig. 1(b) is also shown in a lightgray rectangle. In order to calculate the electronic propertiesof graphene, it is sufficient to consider only the wave vectors,

which are within the dark gray area shown in Fig. 1(c). Weare mainly interested in the T matrix for large values ofL forwhich we can disregard quantum confinement effects.

A. Calculation of the scattering amplitude

The electronic structure of both the graphene ribbon andthe infinite graphene layer can be calculated using an extendedtight-binding method, where several nearest-neighbor inter-actions are taken into account. For both an infinite graphenesheet and a graphene ribbon, the wave functions can be writtenby considering a unit cell consisting of 2N carbon atoms as

b(

k,r )

=l Cbl (

k)

1

U u exp(ik Rul)(r Rul),(4)

where Cbl (k) is the coefficient of the Bloch function, and

(r Rul ) is the atomic orbital at the lth atom of the uthsupercell. Here, l and u go over 2N atomic sites in a unit celland over all U unit cells in the lattice. The infinite graphene

-1/2)a

Armchari Edge

Ribbon

widt

h

L=

(Ny-

A B

by

K

a1

a2

b2

Real Space Reciprocal Space)c()b(

)a(

ax

ay

bx

K

b1

'

FIG. 1. (Color online) (a) Schematics of a graphene ribbon unit cell for Ny = 5, corresponding to L 1.1 nm. The filled symbols representthe atoms in the ribbon unit cell, while the open symbols represent adjacent unit cells in the x direction. The box shows a 4-atom unit cell,

which is used to generate the graphene ribbon. (b) and (c) show, respectively, the real- and reciprocal-space structures of graphene, highlighting

the 4-atom unit cell and its corresponding Brilluoin zone.

245435-2

7/30/2019 65609

4/9

D BAND RAMAN INTENSITY CALCULATION IN . . . PHYSICAL REVIEW B 83, 245435 (2011)

plane is obtained by making U and 2N. TheHamiltonian matrix for b is written in terms of tight-bindingwave functions as

b(k,r )|H0| b(k,r ) =

ll

Cbl (k)Cbl (k)H0ll (k), (5)

where H0ll

(k) is given by

H0ll(k) =

u

exp[ik ( Rul Rul )]H0uull , (6)

where

H0uull = (r Rul )|H0|(r Rul) (7)are the atomic matrix elements. The values for the atomic ma-trix elements and for the overlap parameter (S0uull ), defined as

S0uull = (r Rul )|(r Rul), (8)were obtained from the density functional theory (DFT)calculations of Porezag et al.10 A cutoff radius of

10a0 0.53 nm (where a0 is the Bohr radius) is usedfor the calculation of H0uull and S

0uull parameters in order

to save computation time, which corresponds to consideringinteractions of up to the seventh nearest neighbors. This choiceof parameters has been shown to reproduce well the electronicproperties of graphene and single-wall carbon nanotubes.12

For the graphene ribbon, we set to zero the values ofH0uull and S

0uull between atoms in the edge and missing

atomic positions outside of the edge. Then, we can definethe single scattering amplitude between two eigenfunctions

[ b (k,r)|V| b(k,r ), where V denotes the potential changeat the edges relative to a bulk atomic site] as

b (k,r)|V|

b

(k,r ) =dl

Cbd (k)C

bl (k)Vdl (k,k) (9)

and

Vdl (k,k)=

u

exp(ik Rulik Rud)H0uuld(kk + Q),

(10)

where d represents a missing atom position outside the edge.

The vector Q in Eq. (10) represents the reciprocal latticevectors for the supercell, and the delta function in Eq. (10)represents the conservation of momentum.

Since both the perturbed and unperturbed Hamiltonians

are defined for the same unit cell, the wave functions of thegraphene ribbon n(k,r ) will also be written in terms of thesame atomic orbitals as those of the unperturbed graphenestructure, but having different tight-binding coefficients Pnl (k)such that

n(k,r ) =

l

Pnl (k)1U

u

exp(ik Rul)(r Rul), (11)

which can be obtained by diagonalizing the Hamiltonianfor the graphene ribbon. Since the electronic states in thegraphene ribbon are confined by the edges, the coefficentsPnl (k) are defined only for k values along the direction of

the edges, while when writing the Cbl (k) coefficients for the

infinite graphene sheet, we note that the vector k is defined ina two-dimensional reciprocal space.

The complete T matrix b (k,r)|T(E)| b(k,r ) can beobtained from Eq. (2) by expanding G(E) in terms of theeigenfunctions of the graphene ribbon |n(k,r ), which resultsin the following equation:

Tf i (E) = Vf i + f nniE En + i , (12)

where Tf i is the total scattering amplitude between the initial(i) and final (f) pristine graphene states

Tf i (E) = bf(

kf,r)|T(E)| bi (ki ,r), (13)

Vf i is the value of the first Born approximation term

Vf i = bf (kf,r)|V| bi (ki ,r), (14)and

ni = n(k,r )|V| bi (ki ,r) (15)is the matrix element for the scattering between the ithstate to an nth eigenstate [n(k,r )] of the graphene ribbon.Similarly,

f n = bf (kf,r)|V|n(k,r ) (16)is the matrix element for the scattering from this sameeigenstate n of the graphene ribbon to a final state f.The broadening factor appears in Eq. (12) in order toremove the singularity in the denominator of this equationand is related to the time scale of the electron-defectinteraction.

III. THE D BAND RAMAN INTENSITY CALCULATION

After calculating the elastic scattering matrix elements, it ispossible to calculate the D band Raman intensity. According tothe work of Martin and Falicov,13 the differential cross sectionfor the Raman scattering process is given by

d

d=

20 V

2E2e

4 2h4c4

K ,ea (Ea,Ee)2, (17)where Ea and Ee are the energies of the incident (absorbed)

and scattered (emitted) photons with and polarizations,respectively. is the solid angle for the collection of thescattered light and 0 is the vacuum permittivity, c is the speedof light, and V is the quantization volume for the irradiatedfield. According to the model for a double resonance Ramanprocess,6,7 thevalue of thekernel K ,

ea (Ea,Ee) canbe obtained

as

K ,

ea (Ea,Ee)

2

=

i

f

Ma (ki )Mce-p(ki ,kf)Mcelas(kf,ki )M

e (ki )

(Eki + i)

Ecki Eckf hph + i

Eckf E

cki + i

2

, (18)

245435-3

7/30/2019 65609

5/9

E. B. BARROS et al. PHYSICAL REVIEW B 83, 245435 (2011)

where

Eki = Ea

Ecki Evki

(19)

and

hph = Ea Ee. (20)For the matrix elements Mc

e-p, which correspond to the

electron-phonon matrix elements for electrons in the con-duction (c) band, we have used the previously calculatedexpression by Jiang et al.14 The factor in Eq. (18) wasconsidered to be equal to 0.06 eV for all the processes,which corresponds to the assumption that the time scale of theRaman process is mainly governed by the lifetime of electronsdue to the electron-phonon interaction.15 This assumption isonly valid for a low density of elastic scattering centers forwhich the mean distance between the scattering centers iswell below the mean-free path for electron-phonon scatteringprocesses. By using the uncertainty relation for 0.06 eV andthe Fermi velocity of graphene 106 m/s, we roughly estimatethe value of this low density of elastic scattering centers as

n = 1011 cm2 and by assuming the effective cross sectionfor elastic scattering in the two-dimensional system to be ofthe order of = 1.0 nm2. This value is consistent with theexperiment by Lucchese et al.16 in which the D band spectralwidth changes when the scattering densities increase and reachvalues of around 1012 cm2. Formally, the value of due tothe electron-phonon scattering should also depend on the laserexcitation energy and on the density of defects. However, forsimplicity, a constant value for is considered in this paper.Further studies of how electron-phononrelated resonancebroadening effects change the laser excitation energy andcrystalline size dependence of the D band intensity shouldbe the subject of future work. For the calculations shown in

this paper, we chose a particular process for which the defectscattering takes place between initial and final states in theconduction band. Other processes involving hole scatteringare also possible, and similar results are obtained (not shownhere) by considering hole scattering processes. The matrixelements Ma and M

e correspond to the light absorption

and light emission matrix elements, respectively.17 In sucha Raman process, it is expected that a stimulated absorption ofthe photons takes place while the emission should be usuallyspontaneous, such that the matrix elements for the absorptionand emission are, respectively,18

Ma (k)

=i

eh

mEaI

0c P

D(k) (21)

and

M

e (k) = ieh

mEe

Ee

2 0P D(k), (22)

where P and P

are the polarization vectors, I is the intensity

of the incident beam, and the dipolar vector D(k) is given,within the dipolar approximation [exp(ik a) 1], by17

D(k) = c(k)| | v(k). (23)To evaluate the importance of considering the long-range

scattering potential and the effect of multiple scattering

processes on the D band intensity calculation, the electron-defect matrix element (Melas) in Eq. (18) was calculated inthree different ways:

(1) STB: The simple tight binding is used for the scatteringpotential, for which only the first nearest-neighbor interactionis taken into account. Also, only single scattering processes(first Born approximation) are considered.

(2) ETB-s: The long-range scattering potential is taken intoaccount using the extended tight-binding model. However,only single scattering processes (s) are considered.

(3) ETB-m: Both the long-range interactions and multiplescattering processes (m) are taken into account.

To calculate the Raman scattering cross section, a fullcalculation was performed for initial and final states in a256 256 mesh of points centered at the K and K points,respectively, and spanning a 10/6

3a 10/63a area

of the Brillouin zone. For the STB-based calculation, thedefectscattering matrix element was obtained directly from theexpression derived by Sato et al.9 For the ETB-s and ETB-mmodels, the defect matrix element for some given k states was

obtained by a linear interpolation of the calculated T matrixin each case. The same set of initial and final points were usedfor all the three models.

IV. RESULTS

A. Full T matrix calculations

In order to calculate the T matrix [Eq. (12)], we adopt afinite value for to account for the uncertainty of the energyduring the scattering process. To verify that our choice of is physically sound, we note that it is expected that smallvariations of should not change the calculated value for theT matrix significantly. Figure 2 shows the calculated values

for the Tkk matrix elements as a function of the broadeningfactor for a graphene ribbon with Ny = 16 ( 0.38 nm).The value for the T matrix shown in Fig. 2 was calculatedfor k = 0.86/a and k = 0.86/a (a = 3acc = 2.460 A)electronic states in either the valence band (shown as bluesquares) or the conduction band (shown as red circles).

0.0 0.1 0.2 0.3 0.4

0.15

0.30

0.45

0.60

valence

conduction

Tkk'(e

V)

(eV)

FIG. 2. (Color online) Dependence of the scattering amplitude

Tkk for k = 0.86/a and k = 0.86/a on the broadening factor. Squares (circles) are for scattering within the valence (conduction)

band. In both cases, the T matrix converges for > 0.4 eV.

245435-4

7/30/2019 65609

6/9

7/30/2019 65609

7/9

E. B. BARROS et al. PHYSICAL REVIEW B 83, 245435 (2011)

1260 1330 1400 1470

1.50

1.752.00

2.252.50

2.753.00

3.253.50

3.75

DB

andIntensity(arb.units)

Raman Shift (cm-1)

Laser Energy

(a)

1.5 2.4 3.30

40

80

2 3 41E-3

0.01

0.1

1

(b)

STB

ETB-s

D

bandintensity

LASER ENERGY (eV)

ETB-mI

D/I

Gratio

(c)

STB

ETB-s

ETB-m

FIG. 4. (Color online) (a) Laser excitation energy dependence of

the D band Raman spectra for the ETB-m model. Dependence of

(b) the D band integrated intensity and (c) ID /IG ratio on the laser

excitation energy (log-log scale). The lines are a guide to the eyes

showing the expected E4laser dependence.

It can be seen that the D band intensity remains almostconstant within this energy range. A more detailed analysis

shows that, for the ETB-m and ETB-s models, there is a weaklinear dependence that, however, is very small compared to theoverall intensity and would thus be very difficult to observeexperimentally. For a better comparison with experiments,we also show in Fig. 4(c) the expected value for the ID/IGintensity ratio (log-log scale) dependence on the excitationlaser energy. Since the calculation of the G band intensitydependence on the laser excitation energy is still not wellestablished in the literature, we considered the IG to be givenby IG E4laser, and adjusted the values so that the calculatedID/IG ratio for the ETB-m model fits the expected dependencefor a 50-nm crystalline sized nanographite.3 The solid linesshow the expected ID/IG = AE4laser dependence, indicatingthat, for the ETB-s and ETB-m models, the calculated resultsare in good agreement with the experimental results.3,4 It isimportant to note that, although this E4laser behavior is wellknown for nonresonant Raman scattering, the G band Ramanprocess involves resonant electronic states and thus, the Elaserdependence of the G band intensity should deviate from thisbehavior. This point has been discussed in the literature byBasko,21 where he concludes that, for very low excitationenergies (Elaser 3 eV), the G band should follow a E2laser.This different G band intensity behavior would change theID/IG ratio evaluation performed in this work. Also, recentexperimental work indicates that both the D and G bandintensities behave differently for graphene and graphite as afunction of the laserexcitation energy.4 Further experiments on

graphene nanoribbon edges and G band intensity calculationsfor graphene ribbons are needed in order to improve ourunderstanding of this effect. It is important to comment on thefact that, in our calculations, we have assumed spontaneousphoton emission, which is a reasonable assumption for thelow laser power regime used in Ref. 3. For higher laserpowers, it is possible that a stimulated emission process may

take place,22 which should have an altogether different laserpower dependence. Also, we would like to point out that,when comparing the ETB-s and ETB-m models, there is littlechange in the energy dependence of the D band intensityand of the ID /IG ratio calculation, indicating that, for largeribbons, the main effect of considering multiple scatteringprocesses is an overall increase in the D band intensity bya factor of 4. In the case of the STB model, the D bandintensity does not show a weak linear dependence on Elaser,being mostly constant throughout this energy range. However,this difference between the STB and the ETB models haslittle effect on the ID /IG ratio dependence on Elaser, whichis dominated by the fact that IG increases rapidly with the

excitation laser energy. Another important point is that theelectron-phonon interaction is expected to increase for higherlaser excitation energies, and this should cause a change inthe electron-phonon lifetime and consequently a change in thevalue of, thereby adding an extra effect to the dependence ofboth the D band intensity and the ID /IG ratio on Elaser. Thiseffect has not been taken into account in the present calculationand should be the focus of further studies.

We have also investigated the dependence of the D bandintensity on the polarization of the incident and the scatteredlight with respect to the orientation of the edge. The calculatedresults can be well described by the equation

I = I0[cos2

a cos

2

e + 0.14 sin(a + e)+ 0.11 sin2(a)sin2(e)], (24)

0.0

0.5

1.0

90

180

2700.0

0.5

1.0

e

a Armchair

Edge

(a)

0.0

0.1

0.2

90

180

2700.0

0.1

0.2

e

Armchair

Edge

a

(b)

FIG. 5. (Color online) Dependence of the calculated D band

intensity on the scattered light polarization (e) for (a) a = 0and (b) a = 90. The polarization scheme is shown for each case.The green vertical arrow in (a) (horizontal arrow in (b)) represents

the direction ofa relative to the graphene nanoribbon edge.

245435-6

7/30/2019 65609

8/9

D BAND RAMAN INTENSITY CALCULATION IN . . . PHYSICAL REVIEW B 83, 245435 (2011)

0 32 64 960

2

4

6

0.0 8.0 15.9 23.9

0.00 0.03 0.060

2

4

6

8

ETB-s

L(nm)

Db

andIntensity

Ny

(a)

ETB-m

(b)

ID/I

G

1/Ny

ETB-s

ETB-m

FIG. 6. (Color online) Dependence of (a) the D band intensity

on the nanoribbon width Ny and of (b) the ID /IG ratio on the

inverse nanoribbon width (1/Ny ) for the ETB-s (squares) and ETB-m

(triangle) models for Elaser = 2.0 eV. The solid lines are a guide tothe eye showing the expected 1/Ny dependence of the ID /IG ratio.

where I0 is the maximum intensity and a and e arethe angles between the direction parallel to the edge andthe absorbed and emitted light polarization, respectively. Itcan be seen that the leading term shows a cos2 a cos

2 edependence, as predicted by Sasaki et al.5. In Figs. 5(a) and5(b), we show the intensity dependence on the polarization ofthe scattered light (e in polar coordinates for a = 0 anda = 90, respectively). It can be seen that, for a = 90, theD band intensity is almost independent of e and about 10times weaker than the intensity for a = e = 0. This resultis in good agreement with the experimental results of Conget al.23 who obtained a ratio of 0.2 between the maximumvalue (a = 0 and e unpolarized) and the minimum value(a

=90 and e unpolarized) of the D band intensity.23

Finally, in Fig. 6(a), we show the D band intensitydependence on the nanoribbon width (Elaser = 2.0 eV). Forcomparison, we show the results obtained using the ETB-s(black square) and ETB-m (red triangle) models. For theETB-m model, the D band intensity seems to be independentof the ribbon width for Ny > 64. For such large ribbonwidths, the two edges act as independent scattering centersand thus, the scattering amplitude is independent of the width.In this regime, the D band intensity is localized in thenanoribbon edges and its localization is dominated by thephase-coherent length.24 For thinner nanoribbons, multiplescattering processes involving the opposite edges becomeincreasingly important, causing a decrease in the D band

intensity, as explained above. In the case of the ETB- s model,the D band intensity is independent of the nanoribbon widthsince it disregards the multiple scattering processes. In order

to allow for a comparison with experiments, we also show inFig. 6(b) the expected ID /IG ratio by taking into considerationthat the D band intensity is proportional to the ribbon length,while the G band intensity is proportional to the total ribbonarea, thus leading to an ID/IG ratio that is proportional to1/Ny . Once more, we scaled the ID/IG ratio in order to fitthe experimental results from Cancado et al.3 In Fig. 6(b),

the solid lines are a guide to the eye, showing the expected1/Ny behavior. A deviation of the ID/IG ratio from the 1/Nydependence for smaller nanoribbons can be clearly seen for theETB-m model as a result of the fact that, for nanoribbons withwidths L < 17 nm, multiple scattering processes involvingboth edges start to play an important role in the scatteringamplitude. For this reason, the scattering rate ceases to beindependent of the nanoribbon size, causing a departure fromthe 1/Ny behavior.

V. CONCLUSION

In this paper, we have studied the calculation of the D band

intensity for edge scattering in graphene nanoribbons usingthree different models for the elastic scattering amplitude.The effect of considering the number of nearest neighbors inthe tight-binding model and of multiple scattering processeswas analyzed. The D band intensity was shown to be weaklydependent on the laser excitation energy, indicating that theE4laser dependence observed experimentally for the ID/IG ratiois mainly due to changes in the G band intensity, which is inagreement with the latest experiments on nanographite.3 Also,we have shown that, for nanoribbons with a width larger than17 nm, the D band intensity is independent of the nanoribbonwidth, which leads to the conclusion that for these ribbonsthe D band intensity is localized at either of the edges. Forsmaller nanoribbons, we have shown that multiple scatteringprocesses involving the two opposite edges cause a decrease inthe D band intensity. This effect indicates that the ID /IG ratioshould start decreasing even if the crystalline sizes are largerthan the phonon mean-free path. The dependence of the Dband intensity on the polarization of the incident and scatteredlight was also investigated.

ACKNOWLEDGMENTS

R.S. acknowledges support from MEXT GrantNo. 20241023. E.B.B. acknowledges support from CNPq andCAPES. M.S.D. acknowledges support from NSF-DMR GrantNo.10-04147. A.G.S.F. acknowledges support from CNPq

grants 577489/2008-9, 307317/2010-2, Rede Nacionalde Pesquisa em Nanotubos de Carbono, and INCTNanoBioSimes.

*ebarros@fisica.ufc.br1F. Tuinstra and J. L. Koenig, J. Chem. Phys. 53, 1126

(1970).2L. Cancado, K. Takai, T. Enoki, M. Endo, Y. Kim, H. Mizusaki,

A. Jorio, L. Coelho, R. Magalhaes-Paniago, and M. Pimenta, Appl.

Phys. Lett. 88, 163106 (2006).

3L. G. Cancado, A. Jorio, and M. A. Pimenta, Phys. Rev. B 76,

064304 (2007).4R. Narula, R. Panknin, and S. Reich, Phys. Rev. B 82, 045418

(2010).5K. I. Sasaki, K. Wakabayashi, andT. Enoki, Phys. Rev. B 82, 205407

(2010).

245435-7

http://dx.doi.org/10.1063/1.1674108http://dx.doi.org/10.1063/1.1674108http://dx.doi.org/10.1063/1.1674108http://dx.doi.org/10.1063/1.1674108http://dx.doi.org/10.1063/1.2196057http://dx.doi.org/10.1063/1.2196057http://dx.doi.org/10.1063/1.2196057http://dx.doi.org/10.1063/1.2196057http://dx.doi.org/10.1103/PhysRevB.76.064304http://dx.doi.org/10.1103/PhysRevB.76.064304http://dx.doi.org/10.1103/PhysRevB.76.064304http://dx.doi.org/10.1103/PhysRevB.76.064304http://dx.doi.org/10.1103/PhysRevB.82.045418http://dx.doi.org/10.1103/PhysRevB.82.045418http://dx.doi.org/10.1103/PhysRevB.82.045418http://dx.doi.org/10.1103/PhysRevB.82.045418http://dx.doi.org/10.1103/PhysRevB.82.205407http://dx.doi.org/10.1103/PhysRevB.82.205407http://dx.doi.org/10.1103/PhysRevB.82.205407http://dx.doi.org/10.1103/PhysRevB.82.205407http://dx.doi.org/10.1103/PhysRevB.82.205407http://dx.doi.org/10.1103/PhysRevB.82.205407http://dx.doi.org/10.1103/PhysRevB.82.045418http://dx.doi.org/10.1103/PhysRevB.82.045418http://dx.doi.org/10.1103/PhysRevB.76.064304http://dx.doi.org/10.1103/PhysRevB.76.064304http://dx.doi.org/10.1063/1.2196057http://dx.doi.org/10.1063/1.2196057http://dx.doi.org/10.1063/1.1674108http://dx.doi.org/10.1063/1.1674108

7/30/2019 65609

9/9

E. B. BARROS et al. PHYSICAL REVIEW B 83, 245435 (2011)

6C. Thomsen and S. Reich, Phys. Rev. Lett. 85, 5214 (2000).7R. Saito, A. Jorio, A. G. Souza Filho, G. Dresselhaus, M. S.

Dresselhaus, and M. A. Pimenta, Phys. Rev. Lett. 88, 027401

(2001).8J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrela-

tivistic Collisions (Dover, New York, 2000).9K. Sato, R. Saito,Y. Oyama,J. Jiang, L. G. Cancado,M. A. Pimenta,

A. Jorio, G. G. Samsonidze, G. Dresselhaus, and M. S. Dresselhaus,Chem. Phys. Lett. 427, 117 (2006).

10D. Porezag, T. Frauenheim, T. Kohler, G. Seifert, and R. Kaschner,

Phys. Rev. B 51, 12947 (1995).11L. G. Cancado, M. A. Pimenta, R. A. Neves, G. Medeiros-Ribeiro,

T. Enoki, Y. Kobayashi, K. Takai, K. Fukui, M. S. Dresselhaus,

R. Saito, and A. Jorio, Phys. Rev. Lett. 93, 047403 (2004).12G. G. Samsonidze, R. Saito, N. Kobayashi, A. Gruneis, J. Jiang,

A. Jorio, S. G. Chou, G. Dresselhaus, and M. S. Dresselhaus, Appl.

Phys. Lett. 85, 5703 (2004).13M. Cardona, Light Scattering in Solids (Springer, Berlin, 1982).14J. Jiang, R. Saito, A. Gruneis, S. G. Chou, G. G. Samsonidze,

A. Jorio, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 71,

205420 (2005).

15M. S. Dresselhaus, G. Dresselhaus, R. Saito, and A. Jorio, Phys.

Rep. 409, 47 (2005).16M. M Lucchese, F. Stavale, E. H Martins Ferreira, C. Vilani, M. V.

O. Moutinho, R. B. Capaz, C. A Achete, and A. Jorio, Carbon 48,

1592 (2010).17A. Gruneis, R. Saito, G. G. Samsonidze, T. Kimura, M. A. Pimenta,

A. Jorio, A. G. Souza Filho, G. Dresselhaus, and M. S. Dresselhaus,

Phys. Rev. B 67, 165402 (2003).18Y. Oyama,R. Saito,K. Sato, J. Jiang,G. G. Samsonidze, A. Gruneis,

Y. Miyauchi, S. Maruyama, A. Jorio, G. Dresselhaus et al., Carbon

44, 873 (2006).19A. Jorio, M. M. Lucchese,F. Stavale, and C. A. Achete, Phys. Status

Solidi B 246, 2689 (2009).20A. C. Ferrari and J. Robertson, Phys. Rev. B 61, 14095 (2000).21D. M. Basko, New J. Phys. 11, 095011 (2009).22B. P. Zhang, K. Shimazaki, T. Shiokawa, M. Suzuki, K. Ishibashi,

and R. Saito, Appl. Phys. Lett. 88, 241101 (2006).23C. Cong, T. Yu, and H. Wang, ACS Nano 4, 3175

(2010).24R. Beams, L. G. Cancado, and N. Lukas, Nano Letters 11, 1177

(2011).

245435-8

http://dx.doi.org/10.1103/PhysRevLett.85.5214http://dx.doi.org/10.1103/PhysRevLett.85.5214http://dx.doi.org/10.1103/PhysRevLett.85.5214http://dx.doi.org/10.1103/PhysRevLett.88.027401http://dx.doi.org/10.1103/PhysRevLett.88.027401http://dx.doi.org/10.1103/PhysRevLett.88.027401http://dx.doi.org/10.1103/PhysRevLett.88.027401http://dx.doi.org/10.1016/j.cplett.2006.05.107http://dx.doi.org/10.1016/j.cplett.2006.05.107http://dx.doi.org/10.1016/j.cplett.2006.05.107http://dx.doi.org/10.1103/PhysRevB.51.12947http://dx.doi.org/10.1103/PhysRevB.51.12947http://dx.doi.org/10.1103/PhysRevB.51.12947http://dx.doi.org/10.1103/PhysRevLett.93.047403http://dx.doi.org/10.1103/PhysRevLett.93.047403http://dx.doi.org/10.1103/PhysRevLett.93.047403http://dx.doi.org/10.1063/1.1829160http://dx.doi.org/10.1063/1.1829160http://dx.doi.org/10.1063/1.1829160http://dx.doi.org/10.1063/1.1829160http://dx.doi.org/10.1103/PhysRevB.71.205420http://dx.doi.org/10.1103/PhysRevB.71.205420http://dx.doi.org/10.1103/PhysRevB.71.205420http://dx.doi.org/10.1103/PhysRevB.71.205420http://dx.doi.org/10.1016/j.physrep.2004.10.006http://dx.doi.org/10.1016/j.physrep.2004.10.006http://dx.doi.org/10.1016/j.physrep.2004.10.006http://dx.doi.org/10.1016/j.physrep.2004.10.006http://dx.doi.org/10.1016/j.carbon.2009.12.057http://dx.doi.org/10.1016/j.carbon.2009.12.057http://dx.doi.org/10.1016/j.carbon.2009.12.057http://dx.doi.org/10.1016/j.carbon.2009.12.057http://dx.doi.org/10.1103/PhysRevB.67.165402http://dx.doi.org/10.1103/PhysRevB.67.165402http://dx.doi.org/10.1103/PhysRevB.67.165402http://dx.doi.org/10.1016/j.carbon.2005.10.024http://dx.doi.org/10.1016/j.carbon.2005.10.024http://dx.doi.org/10.1016/j.carbon.2005.10.024http://dx.doi.org/10.1002/pssb.200982314http://dx.doi.org/10.1002/pssb.200982314http://dx.doi.org/10.1002/pssb.200982314http://dx.doi.org/10.1002/pssb.200982314http://dx.doi.org/10.1103/PhysRevB.61.14095http://dx.doi.org/10.1103/PhysRevB.61.14095http://dx.doi.org/10.1103/PhysRevB.61.14095http://dx.doi.org/10.1088/1367-2630/11/9/095011http://dx.doi.org/10.1088/1367-2630/11/9/095011http://dx.doi.org/10.1088/1367-2630/11/9/095011http://dx.doi.org/10.1063/1.2211054http://dx.doi.org/10.1063/1.2211054http://dx.doi.org/10.1063/1.2211054http://dx.doi.org/10.1021/nn100705nhttp://dx.doi.org/10.1021/nn100705nhttp://dx.doi.org/10.1021/nn100705nhttp://dx.doi.org/10.1021/nn100705nhttp://dx.doi.org/10.1021/nl104134ahttp://dx.doi.org/10.1021/nl104134ahttp://dx.doi.org/10.1021/nl104134ahttp://dx.doi.org/10.1021/nl104134ahttp://dx.doi.org/10.1021/nl104134ahttp://dx.doi.org/10.1021/nl104134ahttp://dx.doi.org/10.1021/nn100705nhttp://dx.doi.org/10.1021/nn100705nhttp://dx.doi.org/10.1063/1.2211054http://dx.doi.org/10.1088/1367-2630/11/9/095011http://dx.doi.org/10.1103/PhysRevB.61.14095http://dx.doi.org/10.1002/pssb.200982314http://dx.doi.org/10.1002/pssb.200982314http://dx.doi.org/10.1016/j.carbon.2005.10.024http://dx.doi.org/10.1016/j.carbon.2005.10.024http://dx.doi.org/10.1103/PhysRevB.67.165402http://dx.doi.org/10.1016/j.carbon.2009.12.057http://dx.doi.org/10.1016/j.carbon.2009.12.057http://dx.doi.org/10.1016/j.physrep.2004.10.006http://dx.doi.org/10.1016/j.physrep.2004.10.006http://dx.doi.org/10.1103/PhysRevB.71.205420http://dx.doi.org/10.1103/PhysRevB.71.205420http://dx.doi.org/10.1063/1.1829160http://dx.doi.org/10.1063/1.1829160http://dx.doi.org/10.1103/PhysRevLett.93.047403http://dx.doi.org/10.1103/PhysRevB.51.12947http://dx.doi.org/10.1016/j.cplett.2006.05.107http://dx.doi.org/10.1103/PhysRevLett.88.027401http://dx.doi.org/10.1103/PhysRevLett.88.027401http://dx.doi.org/10.1103/PhysRevLett.85.5214