Ž .Applied Surface Science 130–132 1998 870–875

Phonon dispersion of nano-graphite ribbons

Masatsura Igami a,), Mitsutaka Fujita a, Seiji Mizuno b

a Institute of Materials Science, UniÕersity of Tsukuba, Tsukuba 305, Japanb Department of Applied Physics, Hokkaido UniÕersity, Sapporo 060, Japan

Received 17 September 1997; accepted 22 December 1997

Abstract

Based on the classical force constant models, we study numerically and analytically out-of-plane phonon-dispersionrelations for two types of nano-graphite ribbons confined by the armchair- and zigzag-shaped edges. We demonstrate for tworibbons how the edge-localized vibrational modes, i.e., edge phonon modes, behave depending on edge environment bymeans of altering the effective mass of the edge sites. Special emphasis is placed on the qualitative difference in the phononmodes which are localized near two types of edges. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 63.20.Pw; 63.22.qm; 68.45.Kg; 68.35.Ja

Keywords: Nano-graphite; Graphite; Edge-localized phonon mode; Force-constant model

1. Introduction

Graphite fragments of a nanometer scale, called‘nano-graphites’, have a different electronic structurefrom both bulk graphite and aromatic moleculesw x1–4 . In a graphite fragment, we see two basicshapes for edges, namely, armchair and zigzag edges.While studying the p electronic states of nano-gra-phite based on the graphite ribbon models which arebounded by a pair of parallel armchair or zigzagedges with a nanometer width, we found the p

electronic states of nano-graphite ribbons show criti-cal dependence on their edge shape and size. Nano-graphite could be considered as a new mesoscopicsystem.

) Corresponding author. Fax: q81-298-55-7440; e-mail:igami@comet.ims.tsukuba.ac.jp.

Recently, there have been some experimental tri-als for synthesizing nano-graphite, one of which isthe formation of monolayer graphite on substrateshaving step edges. Since a graphite sheet rips at the

w xstep edges 5 , it might be possible to produce anysize and shape of graphite fragments by controllingthe morphology of terrace on the substrate. It sug-gests that even graphite ribbons can be producible ona substrate if a vicinal surface is prepared where theterraces are separated by the parallel step edges.

In the meaning that an edge of nano-graphiteterminates a sheet of graphite, it is equivalent to atwo-dimensional surface of three-dimensional bulkcrystal, namely, an edge of nano-graphite is a kind ofsurface of graphite sheet. One interesting physics atthe surface is the emergence of surface modes bothin the electronic and lattice-vibrational states whichare localized near the surface. We may expect suchlocalized states near the edge site of the nano-gra-phite as well.

0169-4332r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0169-4332 98 00168-8

( )M. Igami et al.rApplied Surface Science 130–132 1998 870–875 871

In fact, one of the factors which characterizes thesingular p electronic state in nano-graphite is theformation of localized edge states at a specific edge.

The purpose of this paper is to study the behaviorof lattice-vibrational states localized near the edgesof nano-graphite, i.e., edge phonon modes. Applyingclassical force-constant models to nano-graphite rib-bons, we examine the edge phonons, especially tak-ing notice of the edge-shape dependence. Restrictingourselves within the out-of-plane modes for simplic-ity, we study the dispersion relations of edge phononsfor two graphite ribbons, i.e., armchair and zigzagribbons, and how the edge phonons are sensitive tothe edge environment, by both numerical and analyt-ical methods. We qualitatively demonstrate it byaltering the mass parameter of edge sites. Edgephonon modes are strongly affected by edge environ-ment; this means that once the relationship betweenedge phonon mode and its circumstance is under-stood, we can estimate edge environment from ex-perimental data.

2. Phonon in graphite ribbons

Measurement of phonon-dispersion relations formonolayer graphite on some substrates shows that nohybridization occurs between the in-plane and the

w xout-of-plane vibrational modes 6 . This may justifydecomposing the model Hamiltonian between them.Here we are concerned with the out-of-plane modes

Fig. 1. Skeleton of the armchair and zigzag ribbons with the siteindices in the unit cell and the definition of the ribbon width N inŽ . Ž .a and b , respectively.

of nano-graphite ribbons because obviously the po-tential term has simpler structure, which opens a pathfor analytical approach as discussed in Section 3 andgives a simple relation to the electronic state prob-lem based on the tight binding model.

In Fig. 1, we show the skeletons of two prototyperibbons, namely, armchair and zigzag ribbons indi-cating the definition of site indices in the unit celland the ribbon width N. To study the out-of-planephonon-dispersion relations of the graphite ribbons,we introduce two force-constant models A and B.Each Hamiltonian is given by

1st n.n.1 gA 22HH s m z q z yz 1Ž . Ž .˙Ý ÝA i i j i2 2i i , j

and

1st n.n.1 g B 22HH s m z q z qz qz y3 z ,˙ Ž .Ý ÝB i i j k l i22 2 ri i , j ,k , l

2Ž .

where z is the out-of-plane displacement at ith site,iŽ .and g asA, B and r are the force and latticea

constants of bulk graphite, respectively. As regardsthe effective mass parameter m at the ith site, wei

put as m for the three-coordinated sites and mbulk edge

for the two-coordinated sites considering the envi-ronmental difference. We will show the edge phonon

Ž .behavior depending on m 'm rm for twoedge bulk

kinds of graphite ribbons.Applying both models to a graphite sheet, a con-

spicuous difference arises in low frequency and longwave length modes. Model A gives a linear disper-sion curve, while model B a parabolic one. Thedifference originates from the structure of potentialterms, i.e., Hooke’s law is considered only betweenthe nearest neighbour sites in model A, while up tothe 2nd-nearest neighbour sites in model B. Since theparabolic dispersion curve is observed in monolayergraphite on some substrates, model B turns out to bemore realistic in the point of describing the out-of-

w xplane angle-bending mode of a graphite sheet 7 .We should mention that the dispersion relation ofmodel B is exactly in proportion to the square of thatof model A for a graphite sheet and ribbons withms1.

( )M. Igami et al.rApplied Surface Science 130–132 1998 870–875872

For systematic understanding of the phonon-dis-persion relations in the graphite ribbons, it is worthpointing out the projected dispersion relations of two

Ž .dimensional 2D graphite onto armchair and zigzagŽ . Ž .axis in Fig. 2 a and b , respectively. The wave

vector k is normalized by the primitive translationvector of each graphite ribbon, and the frequenciesv in the models A and B are scaled by v0

Ž .s g rm and g rm r2 r, respectively( (A bulk B bulk

throughout this paper. The upper and lower shadedareas correspond to the optical and acoustic phononmodes in the graphite band, respectively. In thearmchair ribbons, these bands are degenerated at asingle point ks0, and only unique inner-gap regionexists between these bands. On the other hand, thisdegenerated point is mapped to ks2pr3 in thezigzag ribbons leading the presence of two sorts of

< < Ž .inner-gap regions for k -2pr3 X and 2pr3-< < Ž .k Fp Y . Here we emphasize that the essentialfeature of inner-gap structure is almost the same forboth models.

We exhibit the phonon-dispersion relations for thearmchair ribbon when Ns13 and ms1 based on

Ž . Ž .the models A and B in Fig. 3 a and b , respec-tively. The most noticeable point between them isthe appearance of two modes within the gap region.These modes correspond to a vibration localized nearthe edge site, i.e., edge phonon modes, whose dis-placement decays exponentially from edge sites tothe inner region. The edge phonon modes are indi-

Fig. 3. Calculated phonon-dispersion relations for an armchairŽ . Ž .ribbon of Ns13 when ms1 by the models A a and B b , and

Ž . Ž .when c ms0.4 and d ms5.0 by the model A. The arrowsshow the edge phonon modes.

Ž . Ž .Fig. 2. The projected out-of-plane phonon-dispersion relations of 2D graphite onto armchair a and zigzag axis b by the models A and B.

( )M. Igami et al.rApplied Surface Science 130–132 1998 870–875 873

cated by arrow symbols in Fig. 3. In addition, wefind two modes corresponding the Rayleigh wavessituated below the bulk bands for acoustic phonons.The structure of the bulk bands can be well ex-plained by using the projected dispersion relation.

On the other hand, we can see no edge phononmode in both models for the zigzag ribbons whenms1, but this is not the case when m/1. Thefrequency of the edge phonon modes varies depend-ing on m. What is important is that throughout ourcalculation using models A and B, the edge phononsbehave qualitatively much the same when we alter m

as far as we are concerned with the edge phononmodes which appear in and around the gap. Thismay permit to adopt model A for discussing thoseedge phonons in graphite ribbons. This is of greatbenefit for the analytical treatment and the compari-son with the electronic state on the tight bindingmodel because of the structural simplicity. In Section3, based on model A we will see how the edgephonon modes are affected by the environment ofedge sites through the mass parameter m.

3. Edge phonon

We examine the m dependence of the edge phononmodes both numerically and analytically. The calcu-lated phonon-dispersion relations for the armchairribbon with Ns13 when ms0.4 and ms5.0 are

Ž . Ž .shown in Fig. 3 c and d , respectively. Let us recallthat this ribbon has the edge phonon modes within

Ž Ž ..the gap for ms1 Fig. 3 a . We can not see,however, any edge phonon mode within the gap forboth cases of ms0.4 and 5.0, but four branches of

Ž .edge phonon emerge in and above below the opti-Ž .cal acoustic phonon branches as seen in Fig.

Ž . Ž .3 c , d . This may be attributed to the effectiveŽ .stiffening softening of edge phonons when the edge

Ž .mass becomes lighter heavier . The important pointto note is that one of them in each case hybrids withbulk phonon modes forming the mixing states.

The global behavior of the edge phonons depend-ing on m can be shown analytically by considering asemi-infinite graphite sheet with an armchair edge.Assuming that the displacements are followed by therelation z sT 2 z , where z is the out-jq2AŽB. jAŽB. jAŽB.

Ž .of-plane displacement at the jth site in the A B

w Ž .xsublattice site see Fig. 1 a and T is the dampingfactor, we can derive the physical solution in whichthe displacement decays exponentially. We show them dependence of the edge phonon frequencies atks0 in Fig. 4. When m)1, one mixing modeappears inside the acoustic band region. On the otherhand when m-1, two edge phonon modes areformed, one of whose frequencies stands at the topof the optical band and another starts from themixing state and goes above the band as m de-creases. We find that these analytical solutions forthe semi-infinite graphite sheet essentially expressthe edge phonons of the ribbons with finite width ifthey are compared with the numerical calculation forthe ribbons, although the interference effect fromboth edges must be considered for thinner ribbons.

The behavior of edge phonons in zigzag ribbonsis qualitatively different from that in armchair rib-bons. We can also derive the analytical solutions fora semi-infinite graphite sheet with a zigzag edge aswell as the case of an armchair edge assumingz sT for the displacements. The disper-jq1AŽB. z jAŽB.sion curves for edge phonons calculated from theanalytical expression are plotted in Fig. 5.

Staring from very small m, the edge phonon modegets down to touch the top of the optical band as m

increases. Beyond ms0.5, the edge phonon entersin the gap region Y up to ms1. After m becomes

Fig. 4. Analytical solution for the relation between the frequencyof edge phonon modes v in a semi-infinite graphite sheet with anarmchair edge and mass parameter m at ks0. Broken lines showexponentially increasing solution.

( )M. Igami et al.rApplied Surface Science 130–132 1998 870–875874

Fig. 5. Analytical solution for m dependence of the edge phonon frequencies for semi-infinite graphite sheet with a zigzag edge when m-1Ž . Ž .a and m)1 b .

larger than 1, the edge phonon mode is formed in thegap region X and below the acoustic band as theRayleigh mode. At ksp the edge phonon is com-pletely localized at the edge site where the frequency

'is determined as vs 2rm . The localization lengthof the edge phonon in the inner-gap regions elon-

'gates towards infinity approaching vs 3 and ks2pr3. It should be noted that when ms2r3 the

'dispersion curve becomes perfectly flat at vs 3which is exactly equivalent to the edge state whichwe found in the p electronic system. Meanwhile, inthe limit of m™` we have the perfect flat band at

'vs 3 as well, though in the gap region X. Thissolution corresponds to Klein’s edge state in the

w xelectronic system 8 . In both edge states the ampli-tude has a value only at one of the sublattice siteswhich includes the edge sites.

4. Conclusion

In the present paper we studied the edge phononmodes in out-of-plane vibrations of nano-graphiteribbons treating the simple force constant model bothnumerically and analytically. The behavior of theedge phonons are found to be qualitatively differentdepending on the edge shape. Furthermore, weshowed how the edge phonon modes are sensitively

affected by the edge environment by means ofchanging the effective mass of edge sites. Whenms1, the edge phonon modes are formed in theinner gap for armchair ribbons, but not for zigzagribbons. In the armchair ribbons, as m decreasesŽ . Ž .increases the edge phonon mode lifts falls toward

Ž . Ž .the bottom top of the optical acoustic bands.When the deviation is large, the edge phonons enterthe bands forming the mixing state. While in thezigzag ribbons, such mixing bands never occur. Re-flecting that there are two different inner-gap regions< < < <k -2pr3 and 2pr3- k Fp , the edge phononmode stays in the former when m)1 and in thelatter when m-1. These results suggest that themeasurement of edge phonons in nano-graphite pro-vides helpful information for the environment of theedge sites.

Acknowledgements

The authors are grateful to C. Oshima, K. Kusak-abe, K. Wakabayashi, and K. Nakada for helpfuldiscussions. This work has been supported by a

ŽGrant-in-Aid for Scientific Research No. 09874066,.M.F. and a Grant-in-Aid for Scientific Research on

Priority Areas ‘Carbon Alloys’ from the Ministry ofEducation, Science and Culture, Japan.

( )M. Igami et al.rApplied Surface Science 130–132 1998 870–875 875

References

w x1 M. Fujita, M. Yoshida, K. Nakada, Fullerene Sci. Technol. 4Ž .1996 565.

w x2 M. Fujita, K. Wakabayashi, K. Nakada, K. Kusakabe, J. Phys.Ž .Soc. Jpn. 65 1996 1920.

w x Ž .3 M. Fujita, M. Igami, K. Nakada, J. Phys. Soc. Jpn. 66 19971864.

w x4 K. Nakada, M. Fujita, G. Dresselhaus, S. Dresselhaus, Phys.Ž .Rev. B 54 1996 17954.

w x5 A. Nagashima, H. Itoh, T. Ichinokawa, C. Oshima, S. Otani,Ž .Phys. Rev. B 50 1994 4756.

w x6 T. Aizawa, R. Souda, S. Otani, Y. Ishizawa, C. Oshima, Phys.Ž .Rev. Lett. 64 1990 768.

w x7 T. Aizawa, R. Souda, S. Otani, Y. Ishizawa, C. Oshima, Phys.Ž .Rev. B 42 1990 11469.

w x Ž .8 D.J. Klein, Chem. Phys. Lett. 217 1994 261.