Simulation of energy transport in crystal with NaCl structure assisted by discrete breathers

Discrete breather (DB) is a spatially localized vibrational mode of large amplitude with vibration frequency outside the phonon band of the crystal. DB frequency can leave phonon spectrum due to the anharmonicity of interatomic bonds owing to the fact that the frequency of a nonlinear oscillator is amplitude dependent. In the case of soft (hard) anharmonicity the nonlinear oscillator frequency decreases (increases) with amplitude. Crystals having a gap in the phonon spectrum can, in principle, support the so-called gap DBs, i.e., DBs with frequencies within the gap. The alkali halide NaI crystal possesses a wide gap in the phonon spectrum and the existence of gap DBs in this crystal has been shown by Kiselev and Sievers with the use of the molecular dynamics method. Later on, several experimental works have been undertaken to support the results of the numerical study and also the possibility of energy exchange between two closely positioned DBs was shown by atomistic simulations. In the present study the energy exchange between DBs in larger clusters is simulated by molecular dynamics. It is shown that the energy initially given to the DB cluster stays in the localized form for a long time (hundreds of DB oscillation periods) even though the energy can travel from one lattice site to another and even polarization of DBs can change. These results contribute to our better understanding of the mechanism of energy localization and transport in crystals.


Introduction
The phenomenon of energy localization in nonlinear lattices in the form of discrete breathers (DBs) was first demonstrated by Dolgov [1] and then in a very famous work by Sievers and Takeno [2]. These works have initiated extensive studies on DBs in model systems as well as in crystals [3 -7]. In fact, in real nonlinear lattices DBs are not single-frequency modes and they do not demonstrate exact time-periodicity. Chechin et al. have suggested the concept of quasi-breathers for such objects [8]. DB has a very long lifetime because its frequency lies outside the spectrum of small-amplitude running waves of the lattice. DB frequency, depending on the type of nonlinearity of chemical bonds, can decrease or increase with increase in its amplitude. In the former (latter) case we have soft (hard) type of anharmonicity. Note that soft-type anharmonicity DBs can exist only in crystals featuring gaps (in other words, forbidden zones) in their phonon spectra. Crystals with simple lattices, e.g., pure fcc and bcc metals do not have gaps in the phonon spectra, on the other hand, phonon spectra of crystals with more complex structure may have gaps. In particular, alkali halide crystals with large difference in the atomic weights of anions and cations, e.g., NaI or LiBr have a wide gap in the phonon spectrum. The first investigation of DBs in crystals based on molecular dynamics simulations has been done for NaI [9]. This work has stimulated experimental measurements of crystal spectra to support the existence of DBs [10 -12]. In [13] it has been argued that the concentration of large energy DBs at thermal equilibrium in NaI is insufficient for their reliable experimental observation. In [14] DBs in NaI have not been detected.
Molecular dynamics studies on DBs in alkali halide crystals with NaCl structure were continued in the works [15 -21]. In particular, in [19] it was shown that two closely placed DBs exchange by their energy quasiperiodically. Note that energy exchange between DBs in graphene has been discussed in the works [22 -24]. In the present study the effect of energy exchange between many DBs is investigated by molecular dynamics.

Simulation setup
We consider a model of crystal with NaCl structure having lattice parameter a. This structure consists of two fcc lattices shifted relative to each other by the vector (a / 2,0,0), at the points of one of them light atoms are located, and the points of the other one are occupied by heavy atoms, as shown in Fig. 1a. Each of the atoms has eight neighbors of the opposite type, located on the tops of the regular octahedron. Each cubic translational cell consists of four light and four heavy atoms. The interaction of atoms is described by pair potentials that take into account Coulomb interactions, Born-Mayor repulsion and dispersion interaction [15]. The atomic mass of the light (heavy) component was assumed to be 10 (100) g / mol. The computational cell with superimposed periodic boundary conditions includes 14 × 14 × 14 cubic periodic cells of the crystal. Each cell contains 8 atoms, so that the total number of atoms in the computational cell is equal to 21952.
The DB pair was excited by shifting the atoms 1 and 2 from their equilibrium positions along the vectors (-0.3, -0.3, 0) and (0.3, 0.3, 0), respectively. Vector components are given in angstrom. Other atoms initially are in their equilibrium positions. Initials velocity of all atoms is zero.

Simulation results
Firstly, for the sake of clarity, we reproduce the results reported in [15,19]. Phonon DOS of the crystal is depicted in Fig. 2a. One can see a wide gap in the phonon spectrum, which appears due to the big difference in the atomic weights of anions and cations. DB in the considered model crystal is highly localized on a single light atom.  [20] because the simple method does not work. In Fig. 2b [011] DB frequency is shown as the function of its amplitude. It lies within the phonon gap and decreases with increase in DB amplitude.
The dynamics of DB pair excited on the sites 1 and 2 [denoted in Fig. 1a] is presented in Fig. 3a and 3b, respectively. In this case DBs oscillate out-of-phase along [110] direction, as it can be seen from Fig. 3c, where the phase difference of oscillation of atoms 1 and 2 is shown. Note that time is normalized to the DB oscillation period Θ. Displacement component along z axis is nearly equal to zero for these two atoms.
In Fig. 4 we plot the same as in Fig. 3, but for the case when DBs 1 and 2 oscillate with the phase difference different from π. In this case quasi-periodic energy exchange between two DBs can be clearly seen. This DB pair was excited by applying initial shifts (-0.31, -0.31, 0) and (0.29, 0.29, 0), in angstrom, to the atoms 1 and 2, respectively.
Our next step is to analyze dynamics of a larger number of interacting DBs. In Fig. 5, 10  Displacements as the functions of time are shown in Fig. 6a -6j for the atoms labeled as (a) -(j) in Fig. 5, respectively. It can be seen that initially excited atoms (a), (b), (c), and (d) stop moving after 150t / Θ, but their energy is given to the other light atoms of the chain. Even the atom labeled as (j), which is six sites away from the atom (d), gets excited at about 500t / Θ.

Conclusions
From the results presented in Fig. 6 it follows that not only energy exchange between excited DBs in DB clusters is possible but excitation of DBs on the atoms which were not initially excited can be observed. Thus, a new mechanism of energy transport through the excitation of new DBs in expense of the energy of the existing DBs is revealed in the present study.
In the future studies it would be interesting to check if DBs can change macroscopic properties of crystals [X,Y] [25,26].    Fig. 3 but for the case when atoms 1 and 2 vibrate with the phase difference different from π, and it changes quasiperiodically from −π to π and this change is correlated with the energy exchange between DBs.