Vibrational Spectroscopy 53 (2010) 163–172

Calculated dependence of vibrational band frequencies of single-walled anddouble-walled carbon nanotubes on diameter

Metin Aydin a,*, Daniel L. Akins b

a Department of Chemistry, Faculty of Art and Sciences, Ondokuz Mayıs University, Kurupelit, Samsun 55139, Turkeyb Center for Analysis of Structures and Interfaces (CASI), Department of Chemistry, The City College of New York, 138th Street at Convert Ave., New York, NY 10031, USA

A R T I C L E I N F O

Article history:

Received 3 September 2009

Received in revised form 2 February 2010

Accepted 5 February 2010

Available online 17 February 2010

Keywords:

SWNT

DWNT

DFT calculation

Nanotube

Normal Raman

IR

A B S T R A C T

We have used density functional theory (DFT) at the B3LYP/6-31G level to calculate Raman and IR

spectra of zigzag (n,0) single-walled carbon nanotubes (SWNTs) and (n,0) and (2n,0) double-walled

carbon nanotubes (DWNTs), for n ranging from 6 to 19 and 6 to 8, respectively. In the low frequency RBM

region, calculated Raman spectra of SWNTs indicate that there are three vibrational modes, with

symmetries A1g, E1g and E2g, whose frequencies depend strongly on nanotube diameter. The E2g mode is

not only diameter dependent, but also depends on whether the number of hexagons formed in the

circumference direction of the CNT is even or odd. Two IR spectral modes (of A2u and E1u symmetries) are

found in calculated IR spectra that show strong diameter dependence. Also, three Raman bands with E1g,

A1g and E2g symmetries found are to exist in the G-band region. For this latter case, computed spectra

indicate that while Raman bands with A1g symmetry essentially remain constant for even number of

hexagons formed in the circumference direction, (e.g., (0,2n)-type CNTs with band position

1526 � 0.5 cm�1), bands corresponding to odd number of hexagons, i.e., (0,2n + 1)-type CNTs, are diameter

dependent. The frequencies of the E1g and E2g modes (in the G-band region) are not only strongly diameter

dependent, but also converge towards one another with increasing tube diameter. This latter type of behavior

can lead to erroneous classification of nanotubes as metallic or semiconducting, since partially overlapping

bands in the G-band region might result in bands that appear to have diffuse shoulders, a characteristic of

metallic SWNTs. The RBMs for DWNTs are also strongly diameter dependent and are blue-shifted relative to

their corresponding RBMs in the spectra of SWNTs. The relative distance between RBMs vibrational modes in

the spectrum of a selected DWNT is larger than that for the corresponding SWNTs. The electron density for

small-sized DWNT, e.g., (6,0)&(12,0), indicates an intratube (inner-outer tube) s-bonding in the excited state.

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1. Introduction

Carbon nanotubes, since their recent discovery [1–3], havereceived enormous attention by researchers. Such attentionderives from anticipated numerous applications of this uniquematerials. In particular, the optical and electronic properties ofsingle-walled carbon nanotubes (SWNTs) and multi-walled carbonnanotubes (MWNTs) [4–8] are being intensively studied due totheir potential applications in a variety of technological uses,namely, nanotechnology, functional nanodevices [9,10], materialsscience, heat conduction [11], electronics [12], molecular memo-ries [13,14], optics [15,16], unique electrical properties, transistors,electrically excited single-molecule light sources [17], DNAfunctionalization [18], high-performance adsorbent electrodematerial for energy-storage device [19], and proteins [20]. Mostof the research has been invested in understanding their optical

* Corresponding author. Tel.: +90 3623121919/5522.

E-mail address: aydn123@netscape.net (M. Aydin).

0924-2031/$ – see front matter � 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.vibspec.2010.02.001

and structural properties as well as the development andadvancement of carbon nanotubes [21].

As is well known, a carbon nanotube can be visualized as beingformed by rolling up a well defined projected area within thehexagonal lattice of a graphene sheet in a seamless fashion suchthat all carbon–carbon (C–C) valences are satisfied, and thedirection in which the roll up is performed transforms into thecircumference of the tube. The projected area is in fact ahomomorphic representation of a particular carbon nanotube[13]. The roll-up vector is also termed the chiral vector and isdefined as n~a1 þm~a2, where ~a1 and ~a2 are the unit vectors of thehexagonal lattice, and n and m are the so-called chiral indices. Ingeneral, an infinite number of nanotube geometries are possible,with specific nanotubes characterized by chiral indices (n,m),which, in turn, define the chiral angle (u) and tube diameter (dt);the latter is also dependent on the C–C bond length of thehexagonal lattice. For n = m, the nanotube is said to have the‘‘armchair’’ conformation; for n 6¼ 0 and m = 0, the conformation iscalled ‘‘zigzag’’; while for n 6¼ 0 and m 6¼ 0 the conformation istermed ‘‘chiral.’’ Double-walled carbon nanotubes (DWNTs) are in

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172164

the border between single-walled carbon nanotubes (SWNTs) andmulti-walled carbon nanotubes (MWNTs).

Theoretical calculations have indicated that, depending on thenanotube symmetry and diameter, carbon nanotubes (CNTs) can beeither metallic or semiconducting [22–26]. Optical spectral methodshave proven very useful in providing experimental information thatcan help characterize real CNT samples. In particular, Raman and IRspectroscopy have made it possible to quickly deduce variousproperties of CNTs. Most notably, radial breathing mode (RBM)frequencies of SWNTs have been empirically connected to tubediameters, and shapes of Raman bands that correspond tovibrational modes with transverse motions along the axis of aCNT (i.e., G-bands) have enabled assessment of whether samples ofCNTs contain metallic and/or semiconducting constituents.

In this paper, density functional theory (DFT) with the splitvalence type basis set was performed to calculate the spectroscopicand geometric properties of SWNTs and DWNTs in order to examinethe changes when the system changes from SWNT to DWNT at thesame level of the theory, B3LYP/6-31G. We report on quantumchemical calculations that suggest that some care must be exercisedin applying some widely accepted interpretations of the relation-ships of band positions and shapes to physical and electronicproperties of CNTs. We use density functional theory to calculateRaman and IR spectra of zigzag SWNTs with structural indices (n,0),with n ranging from 6 to 19. For (n,0)&(2n,0)-DWNT, n = 6–8, weused the same method to calculate the geometric spectroscopicproperties as well as the electron density in molecular orbitals of theCNTs, in order to examine the changes in these properties when asystem changes from a SWNT to a DWNT. The choice of this rangewas arbitrary, but does result in sufficient change in nanotubevibrational frequencies that can be resolved by most Raman and IRinstruments [27–29]; hence, the prospect of comparison toexperimental studies. Also, although we focus on zigzag nanotubes,the general finding for our calculations are interpreted as suggestingthat caution must be exercised when inferring electronic andcompositional characters for all types of carbon nanotube samples.

Our calculations indicate, in particular, that there are threeRaman vibrational modes of frequencies in the low frequencyregion that are strongly diameter dependent for zigzag carbonnanotubes. The calculated relationship between band position andSWNT diameter is found to be in agreement with the empiricalinverse expression relating wave number position of the RBMmode to SWNT diameter (i.e., n ¼ A=dt þ B, where A is a constantthat has a value in the range 220–250 cm�1 and dt is a diameter ofthe (n,m)-SWNT in nanometer units; B is effectively a correctionfactor that is associated with the nanotube’s environment) as wellas other theoretical predictions [22,30]. Also, we have calculated IRspectra for same zigzag systems as used for the Ramaninvestigations and have found that two vibrational modes in thelow frequency region exhibit size dependences that can be fit toequations similar to those found to apply to the RBM band.Furthermore, we have found that calculated, allowed opticaltransitions, using time-dependent DFT (at the same level as for theRaman and IR calculations), show a good correlation with theresults of tight-binding (TB) calculations. The Raman bands withE1g, E2g and A1g (RBMs) symmetries for the DWNTs are not onlydiameter dependent, but also red shifted when compared withtheir spectral position in the spectra of corresponding isolatedSWNTs. Furthermore, the electron density of (0,6)&(0,12) exhib-ited intertube CC bonding interactions in the excited state.

2. Computational methods

The ground state geometry of the zigzag single-walled carbonnanotubes (n,0), with n = 6–19, and double-walled carbon nano-tubes (n,0)&(2n,0), with n = 6–8, were optimized without symme-

try restriction on the initial structures. Both structure optimizationand vibrational analysis calculations were implemented by usingDFT with functionals, specifically, B3LYP, in which the exchangefunctional is of Becke’s three parameter type, including gradientcorrection, and the correlation correction involves the gradient-corrected functional of Lee, Yang and Parr. The basis set of splitvalence type 6-31G, as contained in the Gaussian 03 softwarepackage [31], was used. The vibrational mode descriptions weremade on the basis of calculated nuclear displacements using visualinspection of the animated normal modes (using GaussView03)[31], to assess which bond and angle motions dominate the modedynamics for the nanotube. The DFT method was chosen because itis computationally less demanding than other approaches asregards inclusion of electron correlation. Moreover, in addition toits excellent accuracy and favorable computation expense ratio, theB3LYP calculation of Raman frequencies has shown its efficacy innumerous earlier studies performed in this laboratory and by otherresearchers, often proving itself the most reliable and preferablemethod for many molecular species of intermediate size, includinganions and cations [32]. In our calculations we used one unit cell foreach SWNT, and hydrogen atoms have been placed at the endpoints of the unit cells.

3. Results and discussion

3.1. Structure results

Calculated averaged C–C bond distances for the (n,0)-SWNTs,n = 6–19, were found to be, respectively, 1.430, 1.429, 1.425, 1.426,1.423, 1.424, 1.422, 1.423, 1.422, 1.423, 1.421, 1.421, 1.421, and1.409 A; the average C–C bond distance over the 14 nanotubes is1.423 A. The calculated diameters for the (n,0)-SWNTs, n = 6–19,were, respectively, 0.482, 0.549, 0.636, 0.702, 0.791, 0.857, 0.947,1.015, 1.104, 1.180, 1.259, 1.321, 1.417, and 1.476 nm. The plot ofthe averaged CC bond distance versus the calculated tube diameteris found to deviate substantially from linear when the above valuesare plotted, indicating that the CC bond distance does not simplydecrease with increasing tube diameter. Also, we have observedthat plots for bond distance versus tube diameter for (0,2n)- and(0,2n + 1)-type SWNTs are well separated, reflecting measurabledifferences between zigzag tubes with even and odd indices; thiscircumstance is also observed in the Raman spectra of (0,2n)- and(0,2n + 1)-type SWNTs. The calculated full natural bond orbitalanalysis (NBO) indicates that three of the four valence electrons ofthe carbon atoms in SWNTs are sp2 hybridized in the one-dimensional (1D) network, with �34% s and �66% pxy character,and the fourth electron is�100% pz in character. As expected, eachcarbon atom contributes three electrons to the sigma bonds withinthe surface of the CNT and has one electron left in the pz orbitalsthat is delocalized over the entire surface.

Fig. 1A provides the calculated electron density in some of theupper occupied molecular orbitals and lower unoccupied molecu-lar orbitals. It is to be noted that the HOMO and LUMO are purenonbonding p-orbitals (resulting from pz atomic orbitals). The plotof the calculated global energy per hexagon for the (n,0)-SWNTsreferenced to (6,0)-SWNT energies suggested that the curvatureenergy (or folding energy) of the SWNT rapidly decreases andstabilizes with increasing size of the isolated SWNT; the solid curvein Fig. 1S of SI is a fit to the calculated energies using a functionalform that depends inversely on nanotube diameter: fit parametersare shown in Eq. (1)

DEn;6 ðeVÞ ¼ En;0 � E6;0

¼ �1:55� 0:16 ðeV nmÞdt ðnmÞ þ 0:44 ðeV nm2Þ

d2t ðnmÞ

(1)

Fig. 1. Calculated electron densities in the lowest HOMO and LUMO states: (A) for

the (0,6)- and (0,12)-SWNTs and (B) for the (0,6)&(0,12)-DWNT.

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172 165

In Eq. (1), En,m is the energy (in eV) of the (n,0)-type isolatedSWNT and dt its diameter in nm unit. Clearly, larger diameterSWNTs can be more easily formed than the smaller diameter onesin gas phase. For example, while energy differences for the (7,0)-through (12,0)-SWNT, relative to the (6,0)-SWNT, rapidlyapproaches a limiting value, one notes that from the (11,0)- upto the (16,0)-SWNT a limiting value is nearly reached and changesin energy approach zero. Of course, this is to be expected since thesmaller the diameter, the more strained are the sp2 hybridizedsigma bonds; or stated another way, the smaller the diameter themore altered from planarity must be the sp2 hybridized orbitals. Itshould be noted that the formation of the nanotube in gas phaseand in any environment might be different. The size of thenanotube in an environment also depends on cavity size of theenvironment owing to the electrostatic interactions between thetube and its neighboring.

3.2. DWNT

The DFT technique, at same level of the theory, was performedto calculate the Raman and IR spectra for (n,0)&(2n,0)-DWNTs,(6,0)&(12,0), (7,0)&(14,0) and (8,0)&(16,0), as well as their inner-and outer-shell diameters and electron densities in gas phase. Thekey conclusions of these calculations on DWNTs are summarizedbelow. The diameter dependence of the curvature energies of theDWNTs reference to the global energies of their correspondinginner- and outer-SWNTs is well fitted by a Lennard-Jones potentialexpression:

DE ðeVÞ ¼ E½ð2n;0Þ&ðn;0Þ� � Eð2n;0Þ � Eðn;0Þ

¼ 3:6760:702 nm

Dt ðnmÞ

� �6

� 0:702 nm

Dt ðnmÞ

� �12( )

;

which may be interpreted as a van der Waals-type intertubeinteractions for DWNTs, where

1

Dt¼ 1

dtðInner ShellÞ �1

dtðOuter ShellÞ :

A comparison the diameters of the inner- and outer-shells ofthe DWNTs with their corresponding SWNTs diameters showsthat the averaged inner-shell diameters decrease (��0.08 A) andthe averaged outer-shells diameters increased as much as 0.25 A.These changes also found in the averaged C–C bond distances:about �0.014, 0.004 and 0.009 A in inner-shells and 0.044, 0.028and 0.023 A in outer-shells for (6,0)&(12,0)-, (7,0)&(14,0)- and(8,0)&(16,0)-DWNTs, respectively, referenced to their corre-sponding averaged C–C bond distances for the SWNTs. Thesepredictions explicitly indicate the existence of intertube interac-tions in DWNT systems, which may be expressed by a van derWaals-type interaction; not like chemical bonding interactions inthe ground state. Furthermore, Fig. 1B provides the calculatedelectron density of (0,6)&(0,12)-DWNT, showing that the first fourhighest occupied molecular orbitals (from HOMO to HOMO-3with the A1u, A2g and 2E1g symmetries, respectively) belong to theouter-shell, and the next highest occupied molecular orbitals fromHOMO-4 to HOMO-24 include both inner- and outer-shells of(0,6)&(0,12)-DWNT. The lowest unoccupied molecular orbitalLUMO (E1u), lying about 0.780 eV above the HOMO (A1u), belongsto the outer-shell, while the next one (B2u) belongs to the inner-shell and lies 0.849 eV above the HOMO (A1u). The calculatedelectron density also indicates that an intratube (inner and outertube) interaction may possibly take place in the excited state: theLUMO+7 with A2u symmetry and 2.494 eV above the HOMO (A1u),LUMO+8 (E1u; 2.557 eV), LUMO+10 (E1g; 2.563 eV) and LUMO+15(E1g; 3.637 eV). The intratube CC s-bonding interaction in theexcited state may lead to an intertube charge transfer, which can

be observed by a significant change in the tangential modes (TMs)of Raman spectra when the tube is excited to its intratube chargetransfer state. The TM may provide information not only about themetallic or semiconducting character of nanotubes, but also onthe inner-outer tube (intratube) charge transfer. In addition, veryrecently, resonant Raman measurements [33], photoemissionmeasurements [4] and theoretical calculations [5] have providedevidence for charge transfer between the inner- and outer-shellsof DWNTs.

3.3. Raman spectra

We calculated Raman spectra for the zigzag (n,0)-SWNTs with n

ranging from 6 to 19; as shown in Fig. 2. Table 1 providesvibrational mode assignments and frequencies, while Fig. 3Aprovides diagrams of the atomic motions associated with thevibrational frequencies for the (11,0)-SWNT, used as a representa-tive case. All assignments to motions of atoms or groups of atomsin Table 1 have been accomplished through use of vibrationvisualization software (specifically, GaussView03). The calculatedRaman spectra indicate that there are two additional Raman bandsbesides the RBM in the low frequency region. The frequencies ofthese latter bands are also found to depend on SWNT diameter, asshown in Fig. 2A, and Fig. 2B indicates the calculated vibrationalfrequencies as a function diameters of SWNTs. As seen in Fig. 2B,the RBM (with A1g symmetry) and two other Raman bands (withE1g and E2g symmetries) have frequencies that inversely depend ona nanotube’s diameter. A linear fit to the calculate RBM frequencydependence on nanotube diameter is provided; a linear equation,vRBM(A1g) = 12.04 cm�1 + 221.4 cm�1 nm/dt (nm), is in excellentagreement with the empirically determined expression [34];indeed, a popular one has the value 12.5 for the offset constant and223.5 for the constant shift parameter that appear on the right-hand side of Eq. (2). Even though a linear equation reproduces theRBMs within about �3 cm�1 error ranges for the large size SWNTs, itoverestimates RBMs for (0,n < 7)-SWNTs that have diameters smallerthan 0.55 nm such as about 14 cm�1 for (0,6)-SWNT. In actual fact,this is not so surprising, because the C–C–C bond strain rapidlyincreases with decreasing CNTs diameters (as seen in Fig. 1S of SI), theplot of curvature energy per hexagon of the isolated SWNTs.

Fig. 2. Calculated Raman spectra: (A) increased resolution in the low frequency region, showing diameter dependence of the calculated Raman band frequencies for the (n,0)-

SWNTs, n = 0–19; (B) the plots of the frequencies of vibrational modes of symmetries A1g, E1g and E2g versus 1/dt in the high energy region; (C) for (2n + 1,0)-SWNTs; (D) the

plots of the frequencies of vibrational modes of symmetries A1g, E1g and E2g versus 1/dt; (E) for (2n,0)-SWNTs; and (F) the plots of the frequencies of vibrational modes of

symmetries A1g, E1g and E2g versus 1/dt, where �, * and + stand for A1g, E1g and E2g, respectively.

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172166

Therefore, a curve fit may be obtained using a quadratic equation asgiven in Eq. (2), which reproduces the RBMs within a �2 cm�1 errorrange when, comparing with the calculated Raman spectra of theSWNTs from (0,6) to (0,19) using the DFT technique.

vRBMðA1gÞ ¼269:67 cm�1 nm

dt ðnmÞ � 20:24 cm�1 nm2

½dt ðnmÞ�2

� 14:12 cm�1 (2)

The analytical expressions for the two other accompanyingcalculated low frequency bands (vBD of E1g symmetry and vED of

E2g symmetry) as functions of the inverse of the SWNT diameterare given by the following linear equations:

vBDðE1gÞ ¼ 51:9 cm�1 þ 136:8 cm�1 nm

dt ðnmÞ

vEDðE2gÞ ¼ �43:8 cm�1 þ 80:9 cm�1 nm

dt ðnmÞ

However, the best fit parameters carried out to second order ininverse diameter parameter are shown in Eqs. (3a) and (3b) and 4,respectively. It is to be noted that both the E1g and E2g bands havelower frequencies than the RBM, with the E2g band being of lowest

Table 1DFT-calculated Raman vibrational frequencies (in cm�1) and assignments for (n,0)-CNT at the B3LYP/6-31G level.

(19,0) (18,0) (17,0) (16,0) (15,0) (14,0) (13,0) (12,0) (11,0) (10,0) (9,0) (8,0) (7,0) (6,0) Assignment

E1g 1586.9

1507.7

1508.7 1510.8 1525.6 1536.0 1542.1 1550.8 1558.3 1564.6 1571.2 1575.6 1576.5 1574.0 1550.2 Skeletal deformation due to the

asymmetric stretching of the

C–C–C bonds: nas(C–C–C)

E2g 1525.5 1537.3 1537.6 1539.3 1533.1 1541.8 1533.6 1544.2 1536.8 1543.7 1534.3 1533.8 1515.6 1498.3 Skeletal deformation due to the

stretching of the C–C bonds along

the tube axis: n(C–C); asymmetric

stretching of the C–C–C bonds:

nas(C–C–C). Note that wagging of

CH bonds also occurs

A1g 1524.3 1527.9 1542.7 1527.5 1521.2 1527.6 1518.2 1527.7 1512.9 1528.2 1503.4 1527.8 1486.2 1526.6 Skeletal deformation due to the

stretching of the C–C bonds,

n(C–C–C) and n(C–C) along

the tube axis

E2g 1399.1 1402.7 1408.3

weak

1410.5 1411.7 1417.5 1417.9 1422.9 1421.6 1423.8 1421.3 1415.9 1411.7 1375.8 Skeletal deformation due to

asymmetric and symmetric

stretching of the C–C–C bonds:

n(C–C–C) and ns(C–C–C), and

C–C bond stretching along the

nanotube axis, n(C–C)

E2g 472.2 463.9 450.9 464.9

457.8

477.1 471.0

462.6

442.2

479.7 464.2 483.3 467.1

461.2

489.1 478

454.4

500

463

461.2

402.0 Bending deformation of the C–C–C

bonds in the radial direction. Also

contribution from the wagging of

the CH bonds along tube

circumference

A1g 158.9 167.6 177.0 188.2 200.3 214.3 230.3 248.7 270.3 295.8 326.8 362.9 410.2 457.1 Radial breathing mode of CNT

E1g 132.9 147.4 148.5 163.9 163.9 183.6 184.9 208.0 209.9 238.5 242.2 278.3 284.4 332.4 Bending deformation of the

CNT due to expansion of the

CNT along diagonal axis, with the

motion of two end groups being

in opposite directions

E2g 15.4 17.1 18.9 21.9 24.7 28.3 32.9 38.4 46.1 54.7 68.4 83.7 110.8 145.5 Stretching of the CNT along its

diameter, resulting in an

elliptical shape

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172 167

frequency (see Fig. 2B). These two latter bands are labeled as BD forbonding deformation and ED for elliptical deformation, whichderives from the predominate motions that define vibrationalmode motions, as ascertained with the vibration visualizationsoftware mentioned earlier (see Table 1). Furthermore, thecalculated E2g band for (0,2n)-SWNTs is not only diameterdependent, but is also dependent on whether the number ofhexagons formed in the circumference direction of (0,2n)-type or(0,2n + 1)-type SWNTs are even or odd, respectively. As shown inFig. 2B, the E2g band for (0,2n)- and (0,2n + 1)-type SWNTs is wellseparated with decreasing tube diameter, but they again merge atlarge tube diameters. Therefore, in order to obtain a more precisefitting equation for this Raman band (of symmetry E2g), weobtained two fitting equations as given in Eq. (3a) for (0,2n)-typeSWNTs and Eq. (3b) for (0,2n + 1)-type SWNTs.

vBDðE1g; ð0;2nÞÞ ¼ 227:82 cm�1 nm

dt ðnmÞ � 33:60 cm�1 nm2

½dt ðnmÞ�2

þ 4:15 cm�1 (3a)

vBDðE1g; ð0;2nþ 1ÞÞ ¼ 224:23 cm�1 nm

dt ðnmÞ � 37:34 cm�1 nm2

½dt ðnmÞ�2

� 0:43 cm�1 (3b)

vEDðE2g; ð0;nÞÞ ¼ 1:13 cm�1 nm

dt ðnmÞ þ 33:23 cm�1 nm2

½dtðnmÞ�2

� 0:31 cm�1 (4)

Moreover, as seen in Fig. 2B, for large sized SWNTs, thevRBM(A1g) and vBD(E1g) mode frequencies converge. The calculated

frequency separation between the RBM and BD is found to be 0, 5, 9and 14 cm�1, when n has the values 27, 24, 22 and 20, respectively.Thus, one can anticipate the (27,0)-SWNT would have unresolvableRBM and BD bands for the experimental spectra. We can anticipatethat the acquisition of Raman spectra for experimental samplesconsisting of large diameter SWNT with the purpose of character-izing the sample in terms of electronic properties and purity maybe complicated by the existence of this BD band, which, in general,can lead to apparent broadening of bands as well as the presence ofadditional bands that may lead to the erroneous conclusion thatmore than one type of SWNT is present in the sample. Of course,this issue is not expected to be of great significance since thesynthesis routes that are presently in vogue do not lead tonanotubes with diameter as large as that corresponding to the(27,0) index.

As regards other general conclusions that can be drawn fromour calculations for the SWNTs, we note that the lowest frequencyvED(E2g) mode may not be observable for large diameter nanotubedue to Rayleigh scattering; however, our calculation suggests that(6,0) and (7,0) zigzag SWNTs, with computed vED’s of 145.5 and110.8 cm�1, should be resolvable from Rayleigh scattering. Also,we have found that calculated Raman bands in the mid-frequencyregion exit nearly size-independent peak positions. While, asindicated in Table 1 or Fig. 2C and E, in the high frequency regionthere are three Raman bands of symmetries E1g/E2g/A1g that lieclose to one another. Raman bands of A1g symmetry essentiallyremain constant for (2n,0)-type CNTs (with band position1526 � 0.5 cm�1, see Fig. 2E), but that for (2n + 1,0)-type CNTs arediameter dependent (Fig. 2C), A1g (1486–1525 cm�1 or1526þ 3:55=dt ðnmÞ � 14:86=d2

t ðnmÞ). We further observe that the

Fig. 3. Calculated molecular motions for some vibrational bands of the (11,0)-SWNT and (6,0)&(12,0)-DWNT. The nuclear motions of the other SWNTs studied are provided in

Table 1.

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172168

E1g (�1547 � 25 cm�1) and E2g (�1532 � 12 cm�1) Raman modesfirst approach one another in frequency then separate as onecalculates these frequencies for increase diameter of the SWNT.These shifts in the peak positions may result from the nanotubecurvature effect. As mentioned by Asthana et al. [8], the curvatureenergy (as given by Eq. (1)) of the nanotube brings about dissimilarforce constants along the nanotube axis and the circumferencedirection. Therefore, the nanotube geometry causes a forceconstant reduction along the tube axis compared to that in thecircumferential direction. As a result, the curvature effect might playcrucial role in the shift of the peak positions of the G-band as well asthe RBM band, as mentioned earlier. Furthermore, we note that theseRaman bands, since they are overlapping, may pose a problem inassessing, based on the shape of bands in this transverse vibrations

region, whether SWNT samples are metallic or semiconducting. Inaddition, the calculated Raman band positions for bands around1330 cm�1, the disordered graphite region, are found to be slightlysize dependent, exhibiting a small blue shift with increasingdiameter of the SWNTs.

3.4. Raman spectra for DWNTs

Fig. 4 provides the calculated Raman spectra of (0,6)&(0,12)-and (0,7)&(0,14)-DWNTs in the low energy region. The calculationsshow that the frequencies of the RBMs and tangential modes (TMs)of DWNT significantly differ from those calculated for SWNT. Thecalculated Raman spectra of these DWNTs exhibited two RBMmodes resulting from the radial motion of the inner- and outer-

Fig. 4. Calculated Raman spectra of the (6,0)&(12,0)- and (7,0)&(14,0)-DWNTs: (A) low frequency region and (B) high frequency region. Also shown are the Raman band

frequencies and their corresponding calculated Raman spectra of SWNTs for comparison purposes. The spectra for the (0,8) and (0,16) are not provided because of

computation difficulties.

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172 169

shells, in-phase and out-of-phase, as seen in Fig. 3B, and both ofthese RBM modes are strongly diameter dependent. A large gapbetween RBMs of DWNT decreases with increasing diameter of theinner- and outer-shells. Comparing these calculated RBMs withtheir corresponding bands in the SWNTs spectra, we note that theRBMs at 457 cm�1 in the Raman spectrum of (6,0)-SWNT and at249 cm�1 in the (12,0)-SWNT spectrum are, respectively, blue-shifted to 525 and 276 cm�1 in the Raman spectrum of(6,0)&(12,0)-DWNT. Additionally, the RBMs for (7,0)-SWNT(410 cm�1) and (14,0)-SWNT (214 cm�1) are, respectively, upwardshifted to 450 and 237 cm�1 in the spectrum of (7,0)&(14,0)-DWNT. The relative distances between RBMs in the spectrum of(n,0)&(2n,0)-DWNTs are larger than the distances betweencorresponding RBMs in Raman spectra of (n,0)- and (2n,0)-SWNTs.

Tentative equations for the RBMs for different sizes of DWNTs arethe following:

vinnerðRBMÞ ¼ 138 cm�1 nm

dt ðnmÞ þ 39 cm�1 nm2

ðdt ðnmÞÞ2þ 62 cm�1;

and

vouterðRBMÞ ¼ 240 cm�1 nm

dt ðnmÞ � 30 cm�1 nm2

ðdt ðnmÞÞ2þ 45 cm�1;

where dt stand for the shell diameter. In the high frequency region,Fig. 4 provides the calculated Raman modes (with E1g/A1g/E2g

symmetries). When we compare these tangential bands with theirband position in the corresponding SWNT spectra, it can be seen

Table 2DFT-calculated vertical singlet–singlet transitions (in eV) of the (n,0)-SWNTs, n = 7–11, at the B3LYP/6-31G level.

SWNT (7,0) (8,0) (9,0) (10,0) (11,0)

TBa DFT fb TB DFT f TB DFT f TB DFT f TB DFT f

1 0.91 1.04 0.77 0.02 0.56 1.10 0.64 0.01 0.36

2 1.25 1.46 0.80 1.09 0.51

3 1.25 1.46 0.80 1.09 0.52

4 1.42 1.45 0.06 2.44 c 0.93 0.04 1.77 0.98 0.58 0.02

5 2.52 2.40 2.51 0.36 2.32 2.04 1.80

6 2.53 2.51 0.36 2.32 2.04 1.80

7 2.93 2.53 2.50 2.20 2.14 0.50 1.78 1.97 0.50

8 2.93 2.00 2.50 2.14 0.50 1.97 0.50

9 3.02 2.78 2.64 2.23 0.12 2.06 0.12

10 3.02 2.78 2.64 2.53 2.06

11 3.08 2.77 2.72 2.75 2.14

12 3.08 2.98 0.13 2.72 2.75 2.14

dt (nm) 0.56 0.64 0.71 0.79 0.87

Oscillator strengths and the results of the tight-binding calculation for each nanotube are provided for comparison.a eV units.b f stands for oscillator strength.c The TB approximation does not yield any transition around 0.93 eV (the lowest occurs at ca. 3 eV for the (9,0) nanotube).

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172170

that they are downward shifted (relative to SWNTs). Theanimations of the normal modes showed that the strong Ramanpeaks are mostly result from the nuclear motions of the outer-shell.

3.5. IR spectra

Fig. 5 provides calculated IR spectra for the (n,0)-SWNTs, wheren ranges from 6 to 19. As evidenced in Fig. 5, there are two IRvibrational modes of A2u and E1u symmetries, whose frequenciesare strongly diameter dependent. Least squares fits to thecomputed frequencies as functions of diameter for the fourteenzigzag SWNT are shown in Fig. 5B. The fits to the data, for the twobands (of A2u and E1u symmetries) are provided in Eqs. (5) and (6),respectively. The principal motions that these bands correspond tohave been determined, by the visualization software mentionedearlier, for the A2u band to be the wagging of the SWNT along itsradial direction, where the motion of the two end groups move inopposite directions; while for the E1u band, the principal motioninvolves wagging of the SWNT along its circumference direction.Our short hand notation for these two principal vibrations are thesubscripts shown on the frequencies, where RW and CW,

Fig. 5. (A) Calculated IR spectra of the (n,0)-SWNTs, with n varying from 0 to 19; (B) the p

function of nanotube diameter for vibrational modes of symmetries A2u and E1u.

respectively, specify radial and circumference wagging.

vRWðA2uÞ ¼ 10:4 cm�1 þ 192:3 cm�1 nm

dt ðnmÞ (5)

vCWðE1uÞ ¼ 35:2 cm�1 þ 283:7 cm�1 nm

dt ðnmÞ (6)

It is to be noted (for completeness sake), that in the highfrequency vibrational region of Fig. 5, as revealed upon closeinspection, there are three IR vibrational mode of frequencies�1520 cm�1 (E1u symmetry), �1365 cm�1 (E1u) and �1332 cm�1

(A2u) whose positions show weak dependence on diameter;additionally, in the mid-frequency region, there is one IR-modeof frequency �780 cm�1 (A2u) that is also weakly diameterdependent. Hence, the strong diameter dependence of the twolow frequency bands may be useful for determining structuralindices of SWNT samples. A more detailed description of theassignments for bands in the IR spectra for the various SWNTs iscontained in Table 2S of SI.

For DWNTs (Fig. 2S of SI), the calculated IR bands with A2u

symmetries at 387 and 216 cm�1 for (0,6)- and (0,12)-SWNTs are

lots of the dependence of calculated IR frequencies, in the low frequency region, as a

M. Aydin, D.L. Akins / Vibrational Spectroscopy 53 (2010) 163–172 171

blue-shifted to 395 and 253 cm�1 in IR spectrum of (0,6)- and(0,12)-DWNTs, respectively. The IR bands with A2u symmetries at358 and 187 cm�1 in spectra of (0,7)- and (0,14)-SWNTscorrespond to the bands at 411 and 247 cm�1 in spectrum of(0,7)&(0,14)-DWNT. Furthermore, the IR bands with E1u symme-tries at 580 and 339 cm�1 for (0,6)- and (0,12)-SWNTs are alsoblue-shifted to 617 and 350 cm�1 in the spectrum of (0,6)&(0,12)-DWNT, respectively. The bands (with E1u symmetries) at 541 and295 cm�1 in spectra of (0,7)- and (0,14)-SWNTs correspond to thebands at 567 and 300 cm�1 in the spectrum of (0,7)&(0,14)-DWNT,respectively.

As a secondary focus, we have calculated the vertical transitionsof the SWNTs, for the transition (7,0) to (11,0), using time-dependent DFT methods at the TD-B3LYP/6-31G level. Table 2provides the calculated electronic transitions below 4 eV. Thecalculated optically allowed vertical transitions are found tocorrelate well with the calculated values using the tight-binding(TB) method [13].

4. Conclusion and remarks

We have used DFT at the B3LYP/6-31G level to calculate Ramanand IR spectra of 14 zigzag SWNTs, with chiral indices ranging from(6,0) to (19,0), and DWNTs. The calculated Raman and IR spectra, inthe low frequency region, contain three vibrations with A1g, E1g

and E2g symmetries in the Raman case, and two vibrations with A2u

and E1u symmetries in the IR case, all of which depend significantlyon nanotube diameter. We have least squares fitted the diameterdependent vibrational frequencies with a functional form in whichthe frequencies are related inversely to second order of diameterfor the Raman bands and first order of the diameter for the IRbands. For the A1g vibration, also referred to as the RBM band, theresultant expression (see text) is essentially identical to theempirical expression found when real experimental samples areused, but produce significant error for (0, n < 7)-SWNTs. Therefore,the best fitting may be given by a quadratic equation (Eq. (2)) Thebands with E1g and E2g symmetries were also fitted (to highconfidence) to an identical type of expression found for the RBMband, having an inverse relationship to SWNT diameter, but theconstants are different since the two bands occur at frequenciesthat are different from that of the RBM band and from each other.Interestingly, high confidence least squares fits, with the sameinverse dependency on tube diameter, were acquired for two lowfrequency bands (of symmetries A2u and E1u) in IR spectra for thesame zigzag nanotubes for which Raman calculations were made.We also provided the calculated Raman and IR intensities, in caseof nonresonance, in order to compare the relative peak intensitiesin a spectrum of a desired CNT. In nonresonance case, thecalculated peak intensities indicated that, in some of the spectrum,the peak (with E2g symmetry) at the side of RBM is stronger thanthe intensity of the RBM. The identification and measurement ofsuch diameter dependent, low frequency bands in IR spectra,especially when such measurements are coupled with Ramanmeasurements, may provide higher resolution in the determina-tion of the chiral indices and characterization of carbon nanotubesamples. For (0,n)&(0,2n)-DWNTs, n = 6 and 7, the RBMs arestrongly diameter dependent and they are blue-shifted referenceto their corresponding position in the spectrum of the SWNTs. Thecalculations exhibited an intertube CC bonding interaction for thesmall-sized DWNTs as well as a slight change in their geometricparameters.

We have deduced, based on calculated spectra, that one shouldbe cautious in making the immediate assessment that nanotubesof different diameters are present when more than one lowfrequency band exists in the RBM region in Raman measurements,since our calculations indicate the existence of two other diameter

dependent low frequency bands for single diameter zigzag tubes.One might anticipate that calculations would indicate the presenceof low frequency bands in addition to the RBM band even forarmchair and chiral SWNTs.

A key, second point to be made based on our Raman spectracalculations is the implications of the existence of more than oneband in the high frequency region of 1500–1600 cm�1. Throughour calculations, we have found three Raman bands with E1g/A1g/E2g symmetries exist in the aforementioned region. The frequencyof the A1g symmetry mode for (0,n)-type SWNTs remains diameterindependent, whereas, for (0,2n + 1)-type SWNTs, it is consider-ably diameter dependent. Moreover, the frequencies of the E1g andE2g modes are expected to converge towards one another withincreasing tube diameter. This latter type of behavior couldpotentially lead to erroneous classification of large diameternanotubes as metallic or semiconducting, since partially over-lapping bands in the G-band region might result in bands withdiffuse shoulder, a characteristic of metallic SWNTs. However, thisissue is unlikely to be important because it only arises for verylarge diameter SWNT, not diameters that are normally synthe-sized. Additionally, we have deduced that in the case ofnonresonance, for DWNTs, the Raman bands (with E1g/A1g/E2g

symmetries) at high frequency are downward shifted relative toSWNTs and the peaks with high intensity mostly result from thenuclear motions of the outer-shells.

Acknowledgments

We would like to thank the NSF and DoD-ARO for support of thiswork, in part, through the following awards: (1) NSF-IGERTprogram under grant DGE-9972892, (2) NSF-MRSEC programunder grant DMR-0213574, (3) NSF-CREST program for supportunder Cooperative Agreement HRD-0833180, and (4) DoD-AROunder Cooperative Agreement DAAD19-01-1-0759. We also wouldlike to thank Dr. Philippe Mercier for his valuable suggestions.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in

the online version, at doi:10.1016/j.vibspec.2010.02.001.

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