Large-amplitude in-plane atomic vibrations in strained graphene monolayer: bushes of nonlinear normal modes

Abstract

Conventional normal modes are independent of each other in the framework of harmonic approximation. When small anharmonic terms in the Hamiltonian are taken into account, these dynamical objects are only approximate and one can speak about their interactions. In our previous works, the theory of bushes of nonlinear normal modes has been developed for dynamical systems with discrete symmetry groups. With the aid of this theory it is possible to find some exact solutions beyond the harmonic approximation. Each bush is an invariant manifold corresponding to a collection of m nonlinear normal modes, where m is the bush dimension. During the evolution of a given bush, this collection conserves, but amplitudes of the modes, entering the bush, change in time. Previously, we proved with the aid of group-theoretical methods, that in graphene under uniform strain (space symmetry group P6mm) there can exist 4 bushes with m = 1, 14 bushes with m = 2, 1 bush with m = 3, 6 bushes with m = 4, and etc. In this paper, we study some low-dimensional bushes in graphene using ab initio calculations based on the density functional theory. The amplitude-frequency dependencies of one-dimensional bushes are found. The excitation transfer between nonlinear vibrational modes of different symmetry that belong to the same bush is investigated.

Delocalized Nonlinear Vibrational Modes in Graphene: Second Harmonic Generation and Negative Pressure

“…There exist four one‐component and twelve two‐component DNVM, all of them are shown in Figure . The four one‐component DNVM are labeled with Roman numerals from I to IV.…”

“…It can be seen that in addition to the main oscillation component a small component appears and that the vibration is not periodic. From the theoretical consideration it follows that excitation of mode X inevitably excites the additional DNVM III. We then combine the mode X with DNVM III and find that for the mode amplitudes A X = 0.3 A X = 0.3Å and A III = 0.065 Å periodic motion is realised, see Figure b, and this periodic mode is the two‐component DNVM IIIa.…”

“…All the studied DNVM are excited by applying initial displacements to the atoms according to the patterns described in ref. and shown in Figure . Initial velocities of atoms are equal to zero.…”

“…Delocalized nonlinear vibrational modes supported by hexagonal lattice of graphene . Yellow circles show the unperturbed hexagonal lattice points.…”

“…Recently, with the help of the group theory methods, delocalized nonlinear vibrational modes (DNVM), originally called “bushes of nonlinear normal modes,” have been derived for the hexagonal lattice (graphene also has hexagonal lattice). This approach takes into account only symmetry of crystal lattice, that is why DNVM are exact solutions to the dynamical equations of atomic motion for any amplitude and for any type of interatomic potentials.…”