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Stability of in-plane delocalized vibrational modes in triangular Morse lattice

Delocalized nonlinear vibrational modes (DNVMs) play a very important role in the study of the dynamics of a nonlinear lattice in solid state physics. Such modes are exact solutions to the equations of motion of atoms dictated by the lattice space symmetry. If the amplitude of DNVM is above the threshold value, it is modulationally unstable. In the present study, we consider the stability of DNVMs in a two-dimensional (2D) triangular lattice with atoms interacting via the Morse potential. Critical exponents are calculated numerically as functions of the DNVM amplitude. Extrapolation to the zero value of the critical exponent gives an estimation of the DNVM amplitude, below which it is stable.


Introduction
Spatially localized nonlinear excitations, called discrete breathers (DBs), are actively studied in various crystals and model nonlinear lattices by means of molecular dynamics  and ab initio simulations [22,23]. Existing attempts to experimentally identify DBs in crystals were summarized in the review [2]. It has recently been demonstrated that delocalized nonlinear vibrational modes (DNVMs), synonymously called bushes of nonlinear normal modes [24], have a close relationship to DB. Particularly, it has been shown that DBs can be obtained by superimposing a bell-shaped function upon a DNVM having a frequency outside the phonon spectrum [4][5][6]. DBs can also emerge spontaneously from a DNVM having a frequency outside the phonon spectrum as a result of modulational instability [18][19][20][21]. From this it is clear that the stability of DNVM is a very important issue.
DNVMs are the exact solutions to the nonlinear equations governing atomic motion, and this is true for any interatomic potential used in simulations. This is so because DNVMs are found considering only the lattice space symmetry using the group theory [24]. For a two-dimensional (2D) triangular lattice, Chechin and Ryabov have recently derived eight DNVMs with in-plane atomic motion [25]. Here we study the stability of all eight DNVMs by estimating critical exponents as functions of the initial atomic displacements that define the amplitude of DNVM. Extrapolating the results to the zero value of the critical exponent, we find the maximal initial displacement for a stable DNVM.

Simulation details
A two-dimensional (2D) close-packed triangular lattice is considered. The interatomic interactions are described by the classical Morse potential where r is the distance between the considered pair of atoms, A, β, r m are the potential parameters. The function V(r) has a minimum at r=r m . A is the bond breaking energy (depth of the potential) and β defines the bond stiffness. With the proper choice of units of energy, distance, and time we set A=1, r m =1, and the atom mass m=1. We take β=5 and the cut-off radius equal to 5r m . Then the equilibrium interatomic distance is ρ=0.9881329. The computational cell includes 24×24 atoms. Periodic boundary conditions are applied.
In figure 1 eight DNVMs with in-plane displacements of atoms are shown [22]. All arrows have equal length D 0 . The initial conditions are set by setting the initial displacements according to these patterns with zero initial velocities of atoms. The choice of D 0 specifies the DNVM amplitude. To facilitate the development of the instability, random in-plane initial displacements are added to all atoms with a maximal deviation from the equilibrium position equal to 10 -9 .

Numerical results and discussion
In figure 2 we plot, as an example, the numerical results obtained for DNVM 2. Note that in this DNVM all moving atoms vibrate in the horizontal (x) direction. As a measure of the instability of this mode, one can analyse the appearance of displacements in the vertical (y) and out-of-plane (z) directions. It was found that the z-component of the displacements remained nearly zero but the ycomponent firstly remained at the level of the initial perturbation, and then started to grow exponentially with time, max|∆y|~e αt , with the critical exponent α. This is illustrated in figure 2

Conclusions
Using molecular dynamics, the modulational instability of eight DNVMs with in-plane atomic vibrations for a triangular Morse lattice was studied. It is found that the critical value of the DNVM amplitude, above which it becomes unstable, is of the order of 1% of the interatomic distance ρ. In the future studies, it would be important to analyze the effect of the computational cell size on the development of modulational instability. The stability of transverse DNVMs, as well as twodimensional DNVMs should also be analyzed.

Acknowledgements
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