I02(Invited)

Nonlinear Energy Localization Modes in Crystals: Discrete BreatherYusuke Doi
Department of Adaptive Machine Systems, Graduate School of Engineering, Osaka University
2-1 Yamadaoka, Suita, Osaka 567-0046, Japan
doi@ams.eng.osaka-u.ac.jp

ABSTRACT
Nonlinear vibration modes called discrete breathers (DBs) in crystals are investigated by molecular dynamicsimulations. Numerically precise solutions of the DB in two dimensional graphene are calculated and its properties areinvestigated. It is found that the calculated DB reflects nonlinearity of interactions between atoms. Moreover linear
stability analysis of the DB under uniaxial strain is performed. Complex behavior of growth rate has observed
depending on the strain as well as the period of the DB. Displacement of the some unstable modes localized aroundthe DB. This result shows the possibility that the DB can be a starting point of change of the structure of crystals.
Key words: Energy Localization, Discrete Breather, Molecular Dynamics

1. INTRODUCTION
Vibrational property is one of the most fundamental dynamical properties of crystal. Phonon dispersion relationreflects not only the structure of crystals but also macroscopic properties of solid such as elastic constants. It should beemphasized that theories of vibration are based on the lattice dynamics of theoretical lattices that is investigated inmathematical physics. From these points of view, it can be said that the material science and the mathematical physics
can be related closely. When we consider problems of vibration, small amplitude limit is often considered. In thiscase, we deal with the linear equations of motion. Linearization makes the problem much easier without losing some
important characteristics. However if we interested in problems in that the linear approximation is invalid, nonlinear
dynamics should be considered. It has been known that nonlinearity of the system reproduce richer dynamics such as
instability, bifurcation and pattern formation. Recently, energy localization called discrete breathers (DB) or intrinsic
localized modes (ILM) [1] has been extensively studied in physics and mathematics [2]. DB is characterized as(i)vibration modes that have frequency out of the linear phonon band, (ii)large amplitude vibrations with small
numbers of atoms, and (iii)vibration modes that exist wade class of the system, that is, the DB can be excited in any
dimension of the system, without requiring any integrability of the system. Therefore it is expected that the DB
appears in various physical phenomena. Indeed, it has been reported that DB is observed in various physical systems
such as micromechanical cantilever array [3], macroscopic mechanical structures [4], and waveguide array [5]. Crystal
structure is one of the most desirable physical systems that the DB is excited in atomic scales. In this study, we show
some numerical results that appearance of DB in two dimensional atomic structures by molecular dynamics
simulations.

2. MODELS
We show the some results of two dimensional dynamics of carbon atoms in graphene. Figure 1 shows the models of
our simulation. Carbon atoms form two-dimensional sheet with honeycomb structure. Total number of atoms in thesystem is 180. Periodic boundary conditions are considered in x-direction, and fixed boundary condition is consideredin y-direction, respectively. We investigate the dynamics of atoms by molecular dynamics method. Interaction
between carbon atoms is described by the Brenner potential [6]. Equilibrium length between neighboring atoms is0.145nm. Note that we perform simulation without any heat bath. We use the velocity Verlet method for numericalintegration of the equations of motion. Time step for the numerical integration is 0.01fs.

Fig1. Model
3. RESULTS AND DISCUSSION
At first, we search precise numerical solutions of DBs in graphene. Numerical method is described in [7] in detail.
Figure 2 shows snapshots of the DB that is excited in graphene. Two neighboring atoms in center vibrate with large
energy. However surrounding atoms are almost rest. In small amplitude vibration, it is expected that energy ofvibration spread out to the surrounding atoms. In our case, however, the vibration mode is confined to narrow region
and energy does not spread out to the surrounding atoms. This fact is also confirmed by Fig.3. Figure 3 shows thetemporal evolution of displacement of two atoms at which DB is excited. As long as the period of the numericalsimulation, atoms vibrate stably. Figure 4 shows the relation between the amplitude and frequency of DB. It is foundthat the frequency of DB depends on its amplitude. This is quite different from those in the linear phonon mode,
which does not depend on the amplitude of the vibration. Therefore the observed DB is highly affected by
nonlinearity of the interaction between atoms.

t=0ps

t=3.1ps

t=6.1ps

t=12.6ps

t=15.8ps

t=19psFig.2 Snapshot of vibration of DB in graphene

t=9.5ps

Fig.3 Temporal evolution of displacement(a)Fig.4 Relation between frequency and amplitude(b)

Fig.5 Growth ratio of unstable mode of DB: (a)period of DB T = 19ps, (b) T = 19.5ps.

(a)

(b)

Fig.6 Displacement of unstable perturbation mode with maximum growth ratio: (a)T=19ps, (b)T=19.5ps.
Stability of the DB is an important feature for the study of dynamics. We investigate linear stability of the numericalprecise solution of DB by solving the eigenvalue problem of Floquet matrix of DB. Floquet matrix can be constructedby calculating temporal evolution of the variational equations around the DB solution. This can be also performed by
the molecular dynamic simulations. We show the result of stability analysis under applying uniaxial strain tographene. Figure 5 shows the growth rate of unstable perturbation mode around the DB. Complex change of thegrowth rate against the strength of strain is observed. Some unstable modes appear and disappear as the strainchanges. Behavior of the growth rate also depends on the period of DB. We find two types of typical behavior. In one
case, the maximum growth rate appears in the case of the positive strain shown in Fig. 5(a). Another case, themaximum growth rate appears in the case of negative strain shown in Fig. 5(b). Instability of DB means that the DB
with perturbation or external force collapse and emit energy to the surrounding atoms. Stability analysis gives theinformation of displacement of the unstable perturbation modes. Figure 6 shows the displacement of the unstable
perturbation modes with the maximum growth rate of the above-mentioned two typical cases. Red colored atoms
indicate the position of DB. In both cases, it is found that unstable motion is excited around the atoms at which the DBis excited. In the right figure of Fig.6(a), shear motion of atoms is observed, which is perpendicular to the DB’svibration and localized on the DB’s position. In the right figure of Fig.6 (b), unstable displacement is also localized on
the position of DB. There results suggest that if the DB is excited in the crystals, their instability process leads torearrangement of atoms because the DB has large energy than the phonon modes. Recently, Shimada et al has
reported very interesting result of the molecular dynamic simulation, in which atomic structure of carbon nanotube
changes due to excitation of DBs [8]. Our results suggest that the starting point of change of crystal structure due toDB.

4. CONCLUSION
We investigate the DB excited in graphene as nonlinear energy vibration modes in crystals. We successfully obtain
precise numerical solutions of DB in graphene. Using these solutions, the linear stability of DB in graphene is also
investigated. Some unstable perturbation modes with the maximum growth rate are also localized around the DBs.
Therefore displacement pattern of the unstable perturbation modes can become a trigger of change of structure ofatom using energy of the DB.

ACKOWLEADGMENT
This work was supported by JSPS KAKENHI Grant Numbers 22760066 and 24560074.

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