Zero m phonons in MoS2 nanotubes

Edib Dobardžić, Borivoje Dakić, Milan Damnjanović, and Ivanka Milošević*Faculty of Physics, University of Belgrade, P.O. Box 368, 11001 Belgrade, Serbia and Montenegro

sReceived 28 January 2005; published 30 March 2005d

Phonon dispersions of single-wall MoS2 nanotubes are calculated within a full symmetry implementedvalence force-field model. For transversal, twisting, and breathing modes symmetry assignation, chirality, anddiameter dependence of their frequencies and displacements are discussed. Tubular structure is found to becharacterized by two Raman active modes: by the in-phase breathing modesin full analogy to carbon nano-tubesd with frequency approaching Brillouin scattering domainsas diameter approaches nmd, and by thehigh-energy breathing mode with sulfur shells breathing in phase, but out of phase relative to the molybdenumatoms. Likewise, the longitudinal rigid-shell mode, where sulfur shells vibrate out of phase whilst molybdenumatoms barely move, is predicted to be a fingerprint of the cylindrical configuration in infrared spectra. It is alsofound that twisting rigid-layer modes characterize chirality of the tubes. Finally, the large diameter limit isdiscussed and related to the measured Raman and infrared spectra of the layered structure.

DOI: 10.1103/PhysRevB.71.121405 PACS numberssd: 61.46.1w, 61.50.2f, 63.22.1m

Transition-metal disulfide nanotubes1,2 had been synthe-sized soon after the discovery of the carbon ones. However,besides a number of experimental reportsson opticalabsorption4 and Raman spectra,3 field emission,6 and shearand Young’s moduli5 measurementsd there are only a fewtheoretical works selectronic band calculations7,8 andsymmetry-based study9d and an apparent lack of the latticedynamics calculations. This is somewhat surprising, espe-cially in view of the fact that Raman spectroscopy proved tobe one of the most efficient methods in carbon nanotubesample characterization.10

Here we studyG point phonons with vanishing angularmomentum quantum numberm, Raman and infraredsIRdactive ones, in particular. We predict some of them to befingerprintssin Raman and IR absorption spectrad of the tu-bular structure.

A single-wall MoS2 tube can be imagined as a singleMoS2 layer with trigonal prismatic coordination11 rolled upinto a cylinder: sulfur shells are symmetrically arranged withrespect to the molybdenum one. The chiral vectorsn1,n2d isdefined within the molybdenum plane. Symmetry groups ofthe chiralsn1,n2d, n1.n2, zigzagsn,0d, and armchairsn,ndnanotubes are9

LC = Tqr sadCn, s1ad

LZ = Ls2ndnmc= T2n1 sÎ3a0dCnv, s1bd

LA = Ls2ndn/m= T2n1 sa0dCnh. s1cd

Here, n is the greatest common divisor ofn1 and n2, q=2sn1

2+n1n2+n22/nRd sR=3 if n1−n2 is divisible by 3n,

while R=1 otherwised is the order of the principle axis ofrotationssi.e., of the nanotube axisd, anda=Îs3q/2nRda0 isthe unit-cell lengthswherea0 is the hexagonal lattice param-eterd; finally, the helicity parameter is r =fn1+2n2

−sn2/ndwqR /n1Rg smodq/nd, with w giving the number ofcoprimes withn1/n among the integers 2, . . . ,n1/n−1. Thecorresponding isogonal groups are

PC = Cq, PZ = C2nv, PA = C2nh. s2d

Hence, the order of the principle rotation axis of the isogonalgroups is always even.

Unlike the carbon nanotubes, MoS2 zigzag and armchairtubes differ in symmetry: mirror planes are verticalscontain-ing the tube axisd in the former while horizontalsperpendicu-lar onto the tube axisd in the latter sFig. 1d. Therefore, inaddition to linear and angular quasimomenta quantum num-bers sk and md there are parities related to the verticalsPv

=A/Bd and horizontalsPh= ± d mirror planes.Although horizontal twofold axissU axisd, which is a

symmetry of single-wall carbon nanotubes,12 is not a sym-metry of MoS2 tubes, the effect of systematic singularities13

in phonon density of states for all opticalG point modes withm=0 is present in the MoS2 tubes as well. Namely, the vi-brational Hamiltonian, being real, possesses time-reversalsymmetry whichsreversing both linear and angular quasimo-mentad doubles the band degeneracy yielding the singulari-ties, just as theU-axis transformation does in the case ofcarbon nanotubes. Consequently, whether Raman or IR ac-tive, such modes are expected to be pronounced in the spec-tra.

Symmetry classification of the normal modes of

FIG. 1. Vertical mirror plane in the zigzags15,0d sleftd andhorizontal mirror plane in the armchairs10,10d tube srightd. Sulfuratoms are in light gray while the molybdenum atoms are in darkgray.

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vibration9 at k=0, which we give by Eq.s3d, is based on theisogonal point groups, Eq.s2d. Thus, at theG point there are9q modes for the chiral tubessout of which nine are sym-metricd and 18n modesssix symmetricd for the achiral ones,

DCGdyn = 9 o

m=1−q/2

q/2

Am, s3ad

DZGdyn = 6sA0 + And + 3sB0 + Bnd + 9o

m=1

n−1

Em, s3bd

DAGdyn = 3 o

m=−n+1

n

sAm− + 2Am

+ d. s3cd

In the chiral tubes, two acoustic modes are symmetric: thelongitudinalsLA d and the twistingsTWAd one. On the otherside, in the achiral tubes only one of them is symmetric asTWA changes sign upon the vertical mirror reflectionsB0symmetryd in the case of the zigzag tubes, while in the arm-chair ones, LA is odd with respect to the horizontal mirrorreflectionsA0

−, i.e., A2u symmetryd.Out of seven remaining symmetric optical phonons in the

chiral tubes, four are rigid-layer modes and the other threeare breathing modes. Acoustic, LA, and TWA modes arerigid-layer displacements with shells vibrating in phaseslon-gitudinally and circumferentiallyd with all the atoms being, ateach instant, equally displaced from the equilibrium. If lon-gitudinal rigid-shell vibrations are combined in such a waythat sulfur shells move out of phase while molybdenum shellremains fixed one gets symmetric optical LOslongitudinalout of phased mode. On the other hand, if sulfur shells movein phase but altogether out of phase relative to the molybde-num shell, one gets symmetric optical LIslongitudinal inphased mode. Analogously, optical symmetric rigid-shelltwisting modes, TWOstwisting out of phased and TWIstwisting in phased, are obtained when sulfur shells rotate outof phase while molybdenum atoms are in equilibrium posi-tions, and when sulfur shells rotate in phase but altogetherout of phase with respect to the rotations of the molybdenumshell sFig. 2d.

Thus, so far we have determined six symmetric displace-ments, three of them being longitudinal and three twisting.Consequently, the remaining three symmetric modes of vi-bration, due to the orthogonality, are radial. When all shellsbreath in phase one gets radial all-breathing modesBAd. Theother two symmetric breathing modes, BOsbreathing out ofphased and BI sbreathing in phased, can be described follow-ing the pattern of LO and LI modes on the one side or TWOand TWI on the other. Only now, instead of the longitudinaland rotational rigid-shell modes the radial displacements areto be combined.

In a case of the achiral tubes, the same argument can beapplied to their six symmetric modes. Hence, in the armchairtubes twisting and breathing modes are symmetric while thelongitudinal ones change sign upon horizontal mirror reflec-tion. As in the zigzag tubes there is no horizontal mirror

symmetry; longitudinal modes are symmetric, as well as thebreathing ones. However, with respect to the vertical mirrorreflection the twisting modes are odd.

Likewise in the carbon nanotubes,14 symmetry allowsbreathing modes to have nonradial componentssin theachiral tube the component perpendicular to the mirror planeis forbidden thoughd. However, the nonradial component isless effective here due to the larger atomic masses.

We perform valence force-field calculations18,20 of thephonon dispersions in the single-wall MoS2 nanotubes bymeans of the full symmetry implementedPOLSYM code.15

Such an approach enables us to study lattice dynamics of thetubes with the diameters up to 50 nm. As MoS2 nanotube isa three-orbit system, the whole tube can be constructed fromthree atoms9 by application of the symmetry operations ofthe nanotube group, Eq.s1d. Hence, the dimension of theeigenproblem to be solved is ninesin a case of the chiraltubesd or at most 18sachiral tubesd.16

Configuration of the tubes with the diameters less than4 nm is optimizedsi.e., unless the corrections fall below 1%dusing a density-functional tight-binding method.17 The relax-ation yields an increase of the hexagonal lattice parameterfor ,0.25 Å, and a slight lengthening of Mo-S bond lengthssin average, 0.01 Å for the inner and 0.04 Å for the outersulfur shelld, as in Ref. 7.

The force constantssrelated to the bond lengths and bondangles, see Fig. 3d are obtained from Ref. 18. In order to geta better description of the tubular lattice dynamics, wechange slightly the force parameters for bulk 2HuMoS2.The corrections are proportional to the deviation of the bondlengths and bond angles from the corresponding bulk values.

In Fig. 4, phonon dispersions in the tubes17,17d are pre-sented. Especially indicated are zerom branches, as well asBA and BI modes which are predicted to be unique featuresin the single-wall MoS2 nanotube Raman spectra. These

FIG. 2. Longitudinalsupper panelsd and twisting modesslowerpanelsd of the tubes12,8d: LO, LI and TWO, TWI from the left tothe right. Sulfur atoms are in light gray while the molybdenumatoms are in dark gray.

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modes are also diameter,D, and chirality,u, sensitive,

vBA =664

D−

1633

D2 +1215 cos 3u

D2 +1194 sin 3u

D2 , s4ad

vBI = 485.80 −245

D2 −102 cos 3u

D2 −144 sin 3u

D2 . s4bd

Taking the infinite diameter limit of Eq.s4d one gets thefrequencies of the corresponding modes in the layered struc-ture: symmetric Raman-active BA and BI modes become,respectively, acoustic transversalsTAd and optical transversalrigid-layer sTId modes withA2u sA0

−d symmetrysthus, Ramaninactive and still IR actived. Hence, according to Eq.s4bd theTI mode has a frequency of,486 cm−1. Such a mode hasnot been observed in Raman-scattering measurements on a

bulk 2H-MoS2 crystal, but has been observed in IR absorp-tion measurements,19 which confirms our prediction. Themeasured frequency was,481 cm−1. It should be noted thatfrequencies of the rigid-layer modes in the bulk are expectedto be essentially the same as the frequencies of the corre-sponding single-layer modes due to the extremely weak in-terlayer coupling.

Like the radial breathing mode in the single-wall carbonnanotubes, the frequency of the BA mode shows inverse di-ameter dependence. However, it falls quite low:,50 cm−1

for the tubes withD<1 nm. Therefore, the maximal, BImode is more convenient for the tubular structure identifica-tion by means of Raman spectroscopy.

On the other hand, the longitudinal optical mode of thelayered structure withE1g sE1

−d symmetrysthus Raman, butnot IR actived in the tubular geometry splits into the LO andTWO modes which remain Raman active. However, the LOmode becomes IR active, independently of the wrappingangle. The diameter and chirality dependence of the splittingis given by Eq. s5d. The measured19 value of the bulks287 cm−1d and the calculated20 frequency of the single layers280 cm−1d is in very good agreement with the largeD limits,284 cm−1d of Eq. s5d,

vLO = 283.75 +147

D2 +109 cos 3u

D2 +59 sin 3u

D2 , s5ad

vTWO = 283.75 +256

D2 +147 cos 3u

D2 +212 sin 3u

D2 . s5bd

Similarly, the IR and Raman-active longitudinal rigid-layer LI mode of the layered structure withE1u symmetrysplits into two modes in the nanotube: TWI and LI modes,both being Raman active while only the latter remains IRactive. Their frequencies are given by Eq.s6d. The agreementwith the measured,sby Wieting and Verble19d and calculated,sby Sandovalet al.20d values384 cm−1d with the leading terms385 cm−1d in Eq. s6d is excellent.

vTWI = 385.07 −274

D2 −171 cos 3u

D2 −100 sin 3u

D2 , s6ad

vLI = 385.07 +256

D2 +264 cos 3u

D2 +203 sin 3u

D2 . s6bd

Concerning the symmetric rigid-layer mode of the TOtype, which was observedsin the Raman-scattering measure-mentsd at 409 cm−1 and theoretically predicted20 to be at407 cm−1, it becomes, in a tube, a breathing mode of the BOtype, remaining symmetricsfor any chiralityd, thus Ramanactive. As for the frequency, our calculations confirm theresults of Ref. 20 and yield the following dependence on thetubes’ structural parameters:

vBO = 406.84 +976

D4 −213 cos 3u

D4 +334 sin 3u

D4 . s7d

While the longitudinal and breathing modes distinguishbetween the cylindrical and the planar geometry, the twistingmodes can help to identify the chirality. Namely, in the chiral

FIG. 3. Structure of 2H-MoS2 with trigonal prism coordination.Mo-S bond lengthssr andr8d, S-Mo-S bond anglessc andud, andMo-S-Mo bond anglesfd are indicated. Sulfur atoms are in whitewhile the molybdenum atoms are in black.

FIG. 4. Phonon dispersions in the tubes17,17d. Zerom branchesare in bold. The arrows point to the modes BAs21 cm−1d and BIs485 cm−1d, which are predicted to distinguish between the layeredand the tubular structure in Raman measurements.

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and armchair tubes the TWO mode pertains to the symmetricpart of the Raman tensor while in the zigzag ones it contrib-utes only to the antisymmetric part and is therefore expectedto be suppressed.21 Further, to distinguish between armchairsand achiral, generallyd and chiral tubes, IR measurementscan help as the TWI mode is only, in the chiral case, IRactive.

Finally, it should be noted that the symmetry analysis pre-sented in this work is valid, not only for the MoS2, but forthe all transition-metal dichalcogenide nanotubes. To get the

full experimental verification of our results, IR and Ramanmeasurements on the single nanotube are required. However,the peak observed at 495 cm−1 in the MoS2 nanoparticle Ra-man spectraswith no counterpart in the bulk spectrad3 can beidentified as the BI mode.

This work was supported by the Serbian Ministry of Sci-ence sProject No. MNTR-1924d, Deutsche Forschungsge-meinschaft sProject No. DFG 436 YUG 113/2/0-1d; andSlovenian-Serbian Project No. BI-CS/04-05-037.

*Electronic address: ivag@afrodita.rcub.bg.ac.yusURL:http://www.ff.bg.ac.yu/nanoscienced

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