Influence of nonlinear atomic interaction on excitation of intrinsic localized modes in carbon nanotubes

Physica D 239 (2010) 407–413

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Physica D journal homepage: www.elsevier.com/locate/physd

Influence of nonlinear atomic interaction on excitation of intrinsic localized modes in carbon nanotubes Takahiro Shimada a,∗ , Daisuke Shirasaki a , Yusuke Kinoshita b , Yusuke Doi c , Akihiro Nakatani c , Takayuki Kitamura a a

Department of Mechanical Engineering and Science, Kyoto University, Yoshida-hommachi, Sakyo-ku, Kyoto 606-8501, Japan

b

Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

c

Department of Adaptive Machine Systems, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan

article

info

Article history: Received 9 June 2009 Received in revised form 26 November 2009 Accepted 5 January 2010 Available online 11 January 2010 Communicated by A. Mikhailov Keywords: Intrinsic localized mode Nonlinear interaction Atomic system Selective excitation Carbon nanotubes

abstract Excitation of intrinsic localized modes (ILMs) in carbon nanotubes (CNTs) with different chiral structures was investigated by molecular dynamics simulations. For CNTs with a chiral angle less than or equal to 30◦ , energy concentration that continued for more than 200 fs was found in a localized area, where a pair of neighboring atoms strongly oscillated with a higher vibrational frequency than the upper bound of phonon bands. This evidently indicates excitation of the ILM. On the other hand, the ILM was not excited in CNTs with a chiral angle greater than or equal to 41◦ . Analyzing the nonlinearity of the interaction between excited atoms in the ILM vibration mode, we elucidated that nonlinearity predominates the selective ILM excitation. Furthermore, stronger nonlinearity excites ILMs with both higher frequency and longer lifetime. © 2010 Elsevier B.V. All rights reserved.

1. Introduction An intrinsic localized mode (ILM) or discrete breather (DB) is a time-periodic and spatially-localized vibration that characteristically appears in a lattice system consisting of discrete elements with nonlinear interaction. Unlike the well-known phonons, only a few lattices in a local area where the ILM is excited strongly vibrate with large amplitude accompanying a concentration of kinetic energy. In addition, the ILM possesses a higher frequency than the upper bound of phonon bands due to nonlinearity. Since the discovery of ILMs [1,2], their fundamental properties have been intensively studied both theoretically [3–15] and experimentally [16–22]. Existence of the stationary ILM has been proved in the wide range of lattice systems [23]. In recent years, the ILM has been observed experimentally in many physical contexts, e.g., Josephson junction arrays [16,17], antiferromagnetic structures [18,19], optical lattices [20], and micromechanical cantilever arrays [21,22]. The detailed review can be found in Ref. [24]. Since the atomic system also satisfies the requirement for the excitation of ILMs, i.e., the discreteness of atomic arrangement

∗

Corresponding author. Tel.: +81 75 753 5192; fax: +81 75 753 5256. E-mail address: [email protected] (T. Shimada).

0167-2789/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2010.01.001

and the nonlinearity of interatomic interaction, the excitation of the ILM is expected in the atomic system. Although, for the sake of simplicity, most studies on the ILM have dealt with onedimensional lattice systems [3–6], a crystal consisting of regularly arranged discrete atoms with nonlinear interactions is one of the candidates in which the ILM may develop. One of the possible extensions is the consideration of higher-dimensional systems. In particular, the ILM has been observed in various two-dimensional systems [7–11]. Consideration of realistic interatomic potentials is also possible. Cuevas et al. numerically analyzed the effect of a vacancy on the ILM in a crystal using an anharmonic interatomic potential of a model material [12]. Marín et al. simulated a moving ILM in a two-dimensional lattice using the L–J potential of a mica-like model material [15]. In recent years, theoretical studies on ILMs are extended to proteins by Sanejouand et al. [25,26]. They discussed the crucial role of ILM in proteins as enzymatic activation. For studies on ILMs in the atomic system, Yamayose et al. [27] demonstrated strong evidence for the existence of ILMs in an actual two-dimensional atomic component of a graphite sheet by means of molecular dynamics simulations using the Brenner potential [28]. The atomic study was extended to a quasi-threedimensional (3D) system of a carbon nanotube (CNT), which is made by rolling up a graphite sheet in a specific direction. Unlike the graphite sheet, CNTs characteristically have a variety

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Fig. 1. Schematic illustration explaining the chiral vector, Cv , of carbon nanotubes simulated in this study.

of microscopic structures depending upon the rolling direction (chirality). Kinoshita et al. [29] conducted molecular dynamics simulations for two typical chiralities, the zigzag and armchair CNTs, and demonstrated a distinct difference in the excitation of ILMs. The ILM was excited in the zigzag CNTs but not in the armchair CNTs. Although understanding the mechanism of ILM excitation in CNTs is essential for the nature of ILMs as well as the mechanical behavior of CNTs [30], it has not yet been thoroughly elucidated. In addition, the chiral samples were too limited to profoundly discuss the selective mechanism, as the previous study [29] addressed only two examples, zigzag and armchair CNTs, which have no helicity. In this paper, we conduct molecular dynamics simulations of various CNTs with different chirality in order to reveal the selective mechanism of excitation or nonexcitation of the ILMs. This paper is organized as follows. In Section 2, we describe the details of the simulation procedure. In Section 3, we examine whether the ILM was excited or not for each CNT based on both the energy localization and the vibrational frequency. In addition, the characteristics of the ILMs, such as the vibrational frequency, lifetime, and the number of ILMs per unit area are presented. In Section 4, we discuss why the ILM was excited in some CNTs while no ILM could be observed in others, in terms of the nonlinearity of interatomic interaction, which plays a significant role in the excitation of ILMs. Finally, we conclude in Section 5.

It is well known that the ILM is excited from the modulational instability of the phonon mode that has the maximum angular frequency (zone boundary mode) [34]. In the previous studies of a graphite sheet and a zigzag CNT [27,29], the ILM was successfully generated by initially applying a displacement that corresponds to the zone boundary phonon mode and a momentum for a minute disturbance. Hence, we follow the same procedure in this study. First, the fully-relaxed atomic configuration was calculated using the fast inertial relaxation engine (FIRE) algorithm [35] until the force acting on the atoms was less than 1.0 × 10−8 nN. Then, we applied an atomic displacement with the same amplitude of 0.1 Å and momentum corresponding to 10 K for a minute disturbance. The orientation of initial displacement for each CNT (vibration angle, ψ ), which is defined as the angle from the axial z direction, is shown in Fig. 2(b) and listed in Table 1. Interaction among carbon atoms was evaluated by a bond order potential (BOP) proposed by Brenner [28], where the functional (2) form consists of the two-body term, φij , and three-body term, (3) φijk . The Hamiltonian of the atomic system can be given by

H =

N X X (pα )2 i

α

i

+

2Mi

+

N N 1 X X (2) φ (rij ) 2 i j6=i ij

N N N 1 X X X (3) φ (rij , rik , θijk ), 6 i j6=i k6=i,j ijk

(2)

where N denotes the number of atoms, i, j, and k are the indices of atoms, α is the x, y, or z coordinate, pαi is the kinetic momentum in the α direction of the ith atom, Mi is the mass of the ith atom, rij is the distance between the ith and jth atoms, and θijk is the bond angle between the i–j and i–k bonds. Brenner developed two different sets of potential parameters for carbon: One precisely reproduces the C–C bond length at equilibrium, but produces a relatively large error in the forces acting on atoms. The other replicates a reasonable bond length as well as atomic forces within an error of several percent. In this study, we employ the latter because it is preferable for a dynamic simulation. The molecular dynamics simulations were conducted under constant total energy. The numerical integration was carried out using Verlet’s algorithm [36]. The time step was selected to be 0.01 fs, which is sufficiently small to avoid non-negligible truncation error which would lead to difficulty in analyzing the precise vibration of the atoms.

2. Simulation procedure

3. Results

Fig. 1 is a schematic illustration of the hexagonal honeycomb lattice of a graphene sheet. The circumferential structure of CNTs is uniquely represented by the chiral vector [31–33],

Within the framework of the Brenner potential, the potential pot energy of the ith atom, Ei , can be given by

Cv = ma1 + na2 ≡ (m, n),

Ei

(1)

which connects two crystallographically equivalent sites on a graphene sheet. Here, a1 and a2 denote the primitive cell vectors of the graphene sheet, and the chiral angle, θ , is defined as the ascending angle from the primitive cell vector a1 . Fig. 2(a) shows the simulation models of (8, 0) zigzag, (6, 3), (3, 6), (7, 1) chiral, and (5, 5) armchair CNTs, whose chiral vectors are also shown in Fig. 1. The periodic boundary condition was applied in the axial z direction. The CNTs were carefully chosen to ensure their diameter, axial length and the number of atoms would be as similar as possible so that any finite size effect other than chirality should be excluded. The detailed structural parameters are listed in Table 1. Note that the (5, 5)A (θ = 30◦ ) and (5, 5)B (θ = 90◦ ) CNTs possess the same atomic structure, but have different orientations of atomic vibration, as described later.

pot

=

N N N 1 X (2) 1 X X (3) φij + φijk . 2 j6=i 6 j6=i k= 6 i,j

(3)

Since the kinetic energy of ith atom, Eikin , can be also calculated from its momentum, the total energy of ith atom, Ei , can be simply pot obtained by Ei = Ei + Eikin . Fig. 3 shows the change in the highest total energy of an atom, Emax = max(Ei ), in each CNT. In the simulations of (8, 0), (6, 3) and (5, 5)A CNTs (see Fig. 3(a)), Emax began to dramatically increase at 0.4 ps, and remained high (2.0–4.0 eV) from 0.5 ps to 1.1 ps. Such a significant increase in Emax was characteristically observed when the ILM was excited in a graphite sheet [27]. Afterward, Emax fluctuated around a lower energy state of 1.3–2.0 eV. In contrast, no remarkable increase in Emax was observed in the (3, 6), (7, 1), or (5, 5)B CNTs (see Fig. 3(b)), although Emax momentarily exceeded 2.0 eV several times from 0.5 ps to 1.5 ps. Thus, we separately

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409

(a) Simulation models of CNTs.

(b) Initial displacement (vibration direction). Fig. 2. (a) Simulation models of zigzag, chiral, and armchair CNTs (see Table 1 for structural parameters). (b) Initial condition of atomic displacement for each CNT. The ψ represents the angle of the displacement vector from the axial direction (vibration angle).

(a) (8, 0), (6, 3), (5, 5)A CNTs.

(b) (3, 6), (7, 1), (5, 5)B CNTs.

Fig. 3. Changes in the maximum energy of an atom, Emax , in the (a) (8, 0), (6, 3), (5, 5)A CNTs and in the (b) (3, 6), (7, 1), (5, 5)B CNTs during the simulations. Table 1 Structural parameters of the simulation models of zigzag, chiral and armchair CNTs. Chiral index (m, n)

Chiral angle θ (◦)

Diameter (Å)

Axial length (Å)

Number of atoms

Vibration angle ψ (◦)

(8, 0) Zigzag (6, 3) Chiral (5, 5)A Armchair (3, 6) Chiral (7, 1) Chiral (5, 5)B Armchair

0 19 30 41 67 90

6.439 6.389 6.964 6.389 6.082 6.964

35.04 34.77 35.27 34.77 33.09 35.27

256 252 280 252 228 280

0 19 30 41 67 90

examine the results of two groups in terms of Emax ; (i) (8, 0), (6, 3), and (5, 5)A CNTs, and, (ii) (3, 6), (7, 1), and (5, 5)B CNTs. Fig. 4 displays the change in the distribution of total energy of an atom, Ei , in (8, 0), (6, 3), and (5, 5)A CNTs during the high

energy state (0.5–1.1 ps), where the circumferential direction of a CNT is projected on a plane. In the (8, 0) CNT, there existed four pairs of carbon atoms that maintained considerably high energy for more than 300 fs, as indicated by the blue circles. Such energy

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(a) (8,0) CNT

0.6 ps

0.7 ps

0.8 ps

0.9 ps

0.8 ps

0.9 ps

1.0 ps

0.8 ps 0.9 ps Circumferential direction

1.0 ps

(b) (6,3) CNT

Axial direction

0.7 ps (c) (5,5)A CNT

0.7 ps

Fig. 5. Power spectrum distribution in the (8, 0), (6, 3), and (5, 5)A CNTs. The dominant (peak) frequency is indicated as ωpeak . For comparison, the highest phonon

phonon frequency of each CNT, ωmax

Fig. 4. Change in the distribution of total energy of an atom in (a) (8, 0), (b) (6, 3), and (c) (5, 5)A CNTs. The pair of carbon atoms that maintained a higher total energy for over 200 fs is emphasized by blue circles. The atom that has the highest total energy in each snapshot is encircled by a red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

localization with a fairly long lifetime is one characteristic of the ILM. Moreover, it was reported in the previous study [27] that the vibration mode of the ILM in a graphite sheet is such that two neighboring atoms strongly vibrate along their bonding direction. Such pairs of atoms with localized energy were observed in the (6, 3) and (5, 5)A CNTs, although the number of these pairs was less than in the (8, 0) CNT. Since the vibrational frequency of ILMs is higher than the maximum frequency of phonon mode [27], we examine the frequency of the localized vibration mode observed in the simulations. The phonon dispersion relation can be calculated by solving the eigenvalue problem of the dynamical matrix, D(k ) =

1

X l0

p

Mi Mj

l0 jβ

Kliα exp[ik · (Rl0 j − Rli )],

(4)

where subscripts l and l0 denote the numbers of unit cells, Rli is the coordinate of the ith atom in the lth cell, and k is the wavenumber l0 jβ

vector. The interatomic force constant, Kliα , is defined as the second derivative of the potential energy of the system, Φ , with respect to the atomic coordinate, l0 jβ

Kliα =

, is shown as a vertical dashed line.

1.3 eV

0.5 eV

∂ 2Φ . ∂ Rliα ∂ Rl0 jβ

(5)

Fig. 5 shows the power spectrum distribution of the atom having the maximum energy (red circles in Fig. 4) in (8, 0), (6, 3), and (5, 5)A CNTs. For comparison, the upper bound of phonon phonon bands in each CNT, ωmax , is shown as the dashed vertical line. The spectrum was calculated by the discrete Fourier transform of the atomic displacement–time relation during the high energy state with a sampling interval of 0.1 fs. The detailed methodology was described in a previous report [27]. We take displacement to be along the vibration direction, ψ , because of the dominant moving orientation of the atom in the trajectory of velocity vector. For the (8, 0) CNT, the dominant frequency, ωpeak = 0.3139 × phonon

1015 rad/s, exceeded the phonon frequency, ωmax = 0.3111 × 1015 rad/s, providing clear evidence for the excitation of the ILM. The stable vibration of the ILM in the (8, 0) CNT continued for 340 fs, which corresponds to 17 cycles. The excitation of the ILM was also confirmed in the (6, 3) and (5, 5)A CNTs because the dominant frequencies, ωpeak = 0.3132 × 1015 rad/s and 0.3122 × 1015 rad/s, respectively, are located above the upper bounds of the phonon phonon bands, ωmax = 0.3128 × 1015 rad/s and 0.3119 × 1015 rad/s, respectively. The lifetime of the ILM was about 200 fs (10 cycles) and 260 fs (13 cycles) for the (6, 3) and (5, 5)A CNTs, respectively, which are somewhat shorter than in the (8, 0) CNT. Fig. 6 shows the change in the distribution of total energy of an atom, Ei , in (3, 6), (7, 1), and (5, 5)B CNTs from 0.4–1.1 ps. No pair of carbon atoms that maintained a high energy state for a long time was found in any CNT. Moreover, the location of the highest total energy, indicated by the red circles, changed in each snapshot, whereas it remained at the same site in the (8, 0), (6, 3), and (5, 5)A CNTs. Although some atoms had a momentarily high total energy, the state did not last for even a period as short as 100 fs. This suggests that the ILM may not be excited.

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(a) (3,6) CNT

0.8 ps

0.9 ps

1.0 ps

1.1 ps

0.5 ps

0.6 ps

0.7 ps

0.8 ps 0.9 ps Circumferential direction

1.0 ps

(b) (7,1) CNT

Axial direction

0.4 ps (c) (5,5)B CNT

0.7 ps

Fig. 7. Power spectrum distribution in (3, 6), (7, 1), and (5, 5)B CNTs. The dominant (peak) frequency is indicated as ωpeak . For comparison, the highest phonon

phonon frequency of each CNT, ωmax

0.5 eV

, is shown as a vertical dashed line.

1.3 eV

Fig. 6. Change in distribution of total energy of an atom in (a) (3, 6), (b) (7, 1), and (c) (5, 5)B CNTs. The atom that has the highest total energy in each snapshot is encircled by a red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 plots the power spectrum distribution of the atom that exhibited the highest total energy during the simulations of (3, 6), (7, 1), and (5, 5)B CNTs. For comparison, the upper bound of the phonon phonon bands in each CNT, ωmax , is shown as a dashed vertical line. Here, the atoms with the highest energy at 1.0 ps, 0.6 ps, and 0.9 ps for (3, 6), (7, 1), and (5, 5)B CNTs, respectively, were selected for the analysis of power spectrum distribution (see the red circles in Fig. 6). The vibration direction, ψ , was taken as the axis of atomic displacement for the discrete Fourier transform because this was the orientation of the dominant motion. The dominant phonon frequency, ωpeak , was below ωmax . Although we evaluated the power spectrum distribution of other atoms possessing a higher phonon total energy, no ωpeak exceeded ωmax . This indicates that there was no excitation of the ILM in the (3, 6), (7, 1), or (5, 5)B CNTs. Fig. 8 plots the relationship between the frequency of ILM phonon normalized by the highest phonon frequency, ωpeak /ωmax , of ¯ each CNT and time-averaged total energy of ILM-excited atom, E. The result for a graphite sheet is taken from the previous work [27]. The result of energy-dependent analysis clearly shows a linear phonon relation between the frequency and energy, ωpeak /ωmax = 1+ ¯ where the coefficient c is calculated to be 0.0459 eV−1 from c E, the gradient. This points out the strong evidence that the localized vibration obtained in the (8, 0), (6, 3), (5, 5)A CNTs is surely the ILM. All results are summarized in Table 2. The result for a graphite sheet from the previous study [27] is shown for comparison. There exists a clear dependence of the ILM excitation on the structure of

Fig. 8. Relationship between the frequency of ILM normalized by the highest phonon phonon frequency, ωpeak /ωmax , of each CNT and time-averaged total energy of an ILM-excited atom.

the CNT: As the vibrational direction, ψ , increases, the number of ILMs, their lifetime, and the dominant frequency normalized by the phonon highest phonon frequency, ωpeak /ωmax , all decrease, and no ILM was excited between the (5, 5)A and (3, 6) CNTs. 4. Discussion Focusing on the structure of the CNTs and their corresponding vibrational mode, the carbon atoms formed an in-plane vibration along the axial direction in the (8, 0) CNT, while the atoms vibrated along the tangential line of the wall (out-of-plane) in the other CNTs, as depicted in Fig. 9. The C–C interaction in the (5, 5)B CNT is

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Table 2 Summary of molecular dynamics simulation results for ILM. The result for a graphite sheet [27] is also shown for comparison. Chiral index

Vibrational direction ψ (◦)

phonon ωpeak /ωmax

Excitation of ILM

Count of ILMs

Averaged cycles

(8, 0) (6, 3) (5, 5)A (3, 6) (7, 1) (5, 5)B Sheet [27]

0 19 30 41 67 90 –

1.0090 1.0013 1.0012 0.9811 0.9686 0.9574 1.0401

◦ ◦ ◦ × × × ◦

4 1 2 0 0 0 4

340 fs (17 cycles) 200 fs (10 cycles) 260 fs (13 cycles) – – – 500 fs (26 cycles)

Fig. 9. Schematic illustration of the vibrational mode in (8, 0) and (5, 5)B CNTs.

expected to be relatively weaker due to the misfit of the vibrational plane compared with that in the (8, 0) CNT. This suggests that the misfit of atomic vibration affects the excitation of ILMs through the nonlinearity of C–C interaction. The ILM has usually been studied in Fermi–Pasta–Ulam (FPU) [1] and discrete Klein–Gordon lattices, where the potential function is symmetric with respect to the lattice displacement. The potential energy in the FPU lattice is described by the quadratic function (harmonic term) and quartic function (anharmonic term) of lattice displacement. This research has revealed that the terms are essential for the excitation of the ILM [3]. In other words, the force acting on the lattice, which consists of first-order (linear) and third-order (nonlinear) functions of the displacement, plays a key role in the excitation. The approximate nonlinearity of C–C interaction in each CNT was evaluated by the following procedure: A finite displacement along the vibration direction was applied to a pair of neighboring carbon atoms, while the other atoms were fixed at the initial lattice sites considering the vibrational mode of the ILM. Then, we calculated the forces acting on the atoms using the Brenner potential by applying a small displacement stepwise from −0.10 Å to 0.10 Å. The third-order coefficient, a, indicating the nonlinearity of the ILM, was evaluated by fitting the force–displacement relation to a cubic function, f (r ) = ar 3 + br 2 + cr ,

(6)

where r and f (r ) denote the displacement of a carbon atom and the force acting on a atom, respectively. Fig. 10 shows the relationship between the magnitude of the third-order coefficient, a, and the misfit angle, ξ , which is defined as the cross-angle of vibrational planes of a carbon pair (see Fig. 9). The magnitude of a evidently differs among the CNTs, and is well correlated to the misfit angle. This indicates that the nonlinearity is weakened by the misfit of atomic vibrations. The (8, 0), (6, 3), and (5, 5)A CNTs, in which the excitation of the ILM was observed, possess a strong nonlinearity (more than a = 2.40 × 1024 N/m3 ), whereas the (3, 6), (7, 1), and (5, 5)B CNTs exhibited a weaker nonlinearity. This shows that the ILM is likely to be excited by the strong nonlinearity. Note that the second-order coefficient (the

Fig. 10. Relationship between the misfit angle, ξ , and the magnitude of the thirdorder coefficient, a, in Eq. (6) for each CNT.

Fig. 11. Relationship between the ILM vibrational frequency normalized by the phonon highest phonon frequency, ωpeak /ωmax , and the magnitude of the third-order phonon

coefficient, a, in Eq. (6). The magnitude of ωpeak /ωmax taken from Ref. [27].

for a graphite sheet was

third-order for potential energy), b, is quite small that the C–C interaction can be sufficiently described as the third- and firstorder terms. In addition, no remarkable difference cannot be found in the magnitude of second-order coefficient. Fig. 11 plots the relationship between the ILM vibrational frequency normalized by the upper bound of the phonon bands, phonon ωpeak /ωmax , and the magnitude of the third-order coefficient, a, in Eq. (6). The result for a graphite sheet [27] is also shown. phonon The frequency of the ILM, ωpeak /ωmax , apparently increased as the nonlinearity became large. This indicates that the vibrational frequency of the ILM was enhanced by the nonlinearity. Moreover, the excited ILM continued for a longer time as the nonlinearity was strengthened (see Table 2). Thus, the characteristic features of ILMs (frequency and lifetime) are governed by the nonlinearity of C–C interaction.

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5. Conclusion Molecular dynamics simulations with the Brenner potential were carried out in order to investigate the mechanism of selective excitation of the intrinsic localized modes (ILMs) in various carbon nanotubes (CNTs) with different chirality. A pair of neighboring carbon atoms that maintained a higher energy state for a fairly long time was observed in (8, 0) zigzag, (6, 3) chiral, and (5, 5)A armchair CNTs with vibration angles from the axial direction of 0◦ , 19◦ , and 30◦ , respectively. Some of the neighboring carbon atoms formed a localized vibration along their bonding direction, the frequency of which exceeded the upper bound of the phonon bands. This evidently indicates excitation of the ILM at the pair of atoms. In contrast, such energy localization was not found in (3, 6), (7, 1) chiral, or (5, 5)B CNTs with vibration angles of 41◦ , 67◦ , and 90◦ , respectively. The atomic vibration, which did not last long, exhibited a lower frequency than the maximum frequency of the phonon modes. This suggests that there was no excitation of ILMs. The approximate nonlinearity of C–C interaction, which was evaluated by the third-order coefficient of the force–displacement relation along the vibrational mode of the ILM, differs among the CNTs because of the misfit of the vibration axis in the two neighboring atoms. The former ((8, 0), (6, 3), and (5, 5)A CNTs) have a stronger nonlinearity than the third-order coefficient of a = 2.40 × 1024 N/m3 . On the other hand, the ILM cannot be excited in the latter ((3, 6), (7, 1), and (5, 5)B CNTs) because of their weaker nonlinearity than in the former. Thus, the selective excitation of ILMs is governed by the nonlinearity of the C–C interaction. Furthermore, nonlinearity dominates the characteristics of ILMs; frequency and lifetime of the ILM increase as nonlinearity becomes stronger. Acknowledgement This work was supported in part by a Grant-in-Aid for Challenging Exploratory Research (Grant No. 21656031) of the Japan Society of the Promotion of Science. References [1] A.J. Sievers, S. Takeno, Intrinsic localized modes in anharmonic crystals, Phys. Rev. Lett. 61 (1988) 970–973. [2] S. Takeno, K. Kisoda, A.J. Sievers, Intrinsic localized vibrational-modes in anharmonic crsytals — Stationary modes, Prog. Theor. Phys. Suppl. 94 (1988) 242–269. [3] S.R. Bickham, S.A. Kiselev, A.J. Sievers, Stationary and moving intrisic localized modes in one-dimensional monatomic lattices with qubic and quartic anharmonicity, Phys. Rev. B 47 (1993) 14206–14211. [4] S.A. Kiselev, S.R. Bickham, A.J. Sievers, Anharmonic gap mode in a onedimensional diatomic lattice with nearest-neighbor Born–Mayer–Coulomb potentials and its interaction with a mass-defect impurity, Phys. Rev. B 50 (1994) 9135–9152. [5] J.L. Marín, J.C. Eilbeck, F.M. Russell, Localized moving breathers in a 2D hexagonal lattice, Phys. Lett. A 248 (1998) 225–229. [6] Y. Doi, Energy exchange in collisions of intrinsic localized modes, Phys. Rev. E 68 (2003) 066608. [7] S. Flach, K. Kladko, C.R. Willis, Localized excitions in 2-dimensional hamiltonian lattices, Phys. Rev. E 50 (1994) 2293–2303.

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