- Email: [email protected]
Gap discrete breathers in strained boron nitride
Physics Letters A 381 (2017) 3553–3557
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Gap discrete breathers in strained boron nitride Elham Barani a,∗ , Elena A. Korznikova b , Alexander P. Chetverikov c , Kun Zhou d,e , Sergey V. Dmitriev b,f a
Department of Chemistry, Faculty of Science, Ferdowsi University of Mashhad, Mashhad, Iran Institute for Metals Superplasticity Problems, Russian Academy of Sciences, 450001 Ufa, Russia c Saratov State University, 83 Astrakhanskaya St., 410012 Saratov, Russia d Environmental Process Modelling Centre, Nanyang Environment and Water Research Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore e School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore f National Research Tomsk State University, 36 Lenin Ave, 634050 Tomsk, Russia b
a r t i c l e
i n f o
Article history: Received 20 June 2017 Accepted 29 August 2017 Available online 1 September 2017 Communicated by R. Wu Keywords: Boron nitride Discrete breather Molecular dynamics Lattice dynamics Extended nonlinear vibrational mode
a b s t r a c t Linear and nonlinear dynamics of hexagonal boron nitride (h-BN) lattice is studied by means of molecular dynamics simulations with the use of the Tersoff interatomic potentials. It is found that sufficiently large homogeneous elastic strain along zigzag direction opens a wide gap in the phonon spectrum. Extended vibrational mode with boron and nitrogen sublattices vibrating in-plane as a whole in strained h-BN has frequency within the phonon gap. This fact suggests that a nonlinear spatially localized vibrational mode with frequencies in the phonon gap, called discrete breather (also often termed as intrinsic localized mode), can be excited. Properties of the gap discrete breathers in strained h-BN are contrasted with that for analogous vibrational mode found earlier in strained graphene. It is found that h-BN modeled with the Tersoff potentials does not support transverse discrete breathers. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Among nonlinear lattice excitations of particular interest are nonlinear spatially localized vibrational modes called either discrete breathers or intrinsic localized modes, first discovered in simple 1D nonlinear chains [1–3]. A transition from solitons to discrete breathers with the change of relative strength of interparticle and on-site interactions was described in [4]. Discrete breathers have been modeled in a number of crystals [5] including metals [6–10], covalent crystals [11,12], ionic crystals [13,14], and in low-dimensional materials such as graphene [15–23,25,24,26], graphane (fully hydrogeneted graphene) [27–31], and carbon nanotubes [32–35]. Undoubted success in various technological developments of graphene has generated a growing interest of researchers to other two-dimensional crystalline materials such as hexagonal boron nitride (h-BN), silicene, phosphorene, molybdenum dioxide, and others [36–48]. Very little is known about the existence and properties of discrete breathers in these new materials.
In the present study we consider h-BN, whose properties are often compared to that of graphene, because both these materials have hexagonal lattice. It is even possible to achieve pressureinduced commensurate stacking of graphene on h-BN [49]. In spite of similarities in their structure, many properties of h-BN are different from that of graphene: graphene is a semimetal with no gap in the electron spectrum, while the h-BN sheet is a wide-band-gap (about 6 eV) electrical insulator. Graphene and h-BN possess interesting thermal properties [50,51] and high strength [52–54]. Elastically strained graphene supports discrete breathers and breather clusters with frequencies within the phonon gap [23,26]. Unstrained graphene modeled with the AIREBO potential supports transverse discrete breathers with frequencies within the phonon spectrum but above the phonon spectrum of out-of-plane vibrations [15–17]. Such discrete breathers have a very long lifetime because the large-amplitude out-of-plane vibrations weakly interact with the in-plane phonons. It is tempting to check if h-BN can support similar nonlinear excitations and this is the subject of the present study. 2. Simulation setup
*
Corresponding author. E-mail address: [email protected] (E. Barani).
http://dx.doi.org/10.1016/j.physleta.2017.08.057 0375-9601/© 2017 Elsevier B.V. All rights reserved.
Atoms in h-BN form a two-dimensional hexagonal lattice with a primitive translational cell containing one B and one N atoms.
3554
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
Atom mass of boron is equal to mB = 10.8 atomic mass units, while for nitrogen it is mN = 14.0 atomic mass units. Let the x and the y axes of the Cartesian coordinate system are directed along the zigzag and armchair directions of the lattice. This √ cell is generated by the vectors a1 = (a, 0, 0) and a2 = (a/2, a 3/2, 0), where a = 2.497 Å is the equilibrium lattice parameter. For definiteness, boron atom is located at the origin, while nitrogen atom at the middle of the primitive translational cell. The computational cell of N × N primitive cells is considered with periodic boundary conditions. The large-scale atomic/molecular massively parallel simulator (LAMMPS) [55] is employed to conduct the molecular dynamics simulations with the use of the Tersoff-like interatomic potential [56,57]. Fourth-order scheme with the time step of 0.1 fs is used to integrate numerically the equations of atomic motion. Phonon density of states (DOS) is calculated as the Fourier transform of the autocorrelation functions of trajectories of atoms at a temperature of 10 K. The results are compared for different computational cell sizes, N = 10, 20, and 40. It is found that N = 20 is sufficient and this size is used also for simulation of the extended vibrational modes and discrete breathers. In order to induce a gap into phonon spectrum, we apply inplane tensile elastic strain εxx > 0 along zigzag direction. The equilibrium positions of atoms in uniformly strained lattice are found by minimizing the potential energy of the crystal. The initial conditions used to excite the extended vibrational modes and the discrete breathers are described below. Thermal vibrations are not introduced in the study of extended vibrational modes and discrete breathers. Frequency and amplitude of the extended modes and discrete breathers are found by averaging over the simulation run of 0.6 ps which covers more than a dosen of the oscillation periods. 3. Results Fig. 1. Phonon DOS for unstraind h-BN.
3.1. Effect of elastic strain on phonon DOS The phonon density of states (DOS) for unstrained h-BN is shown in Fig. 1. Panels (a)–(c) give the x, y, and z-components of DOS, while in (d) the cumulative DOS is presented. It can be seen that maximal frequency of in-plane small-amplitude vibrations is about 47 THz, while for out-of-plane phonon modes it is at 26 THz. A narrow gap in phonon DOS (width of about 2 THz) can be seen around 35 THz. Similar results but for elastically stretched h-BN along zigzag direction with εxx = 0.3 can be seen in Fig. 2. Maximal frequencies for in-plane and out-of-plane phonons have been reduced down to 40 THz and 15 THz, respectively. More important for our discussion is the appearance of the wide gap in the phonon spectrum ranging from 23 to 36 THz. This makes it possible for in-plane DBs to exist with frequencies inside the gap, as it has been shown for strained graphene [26]. We have also calculated phonon DOS for εxx = 0.1 and 0.2. For 10% strain, due to breaking of the lattice symmetry, the gap exists only in the x-component of DOS and no gap in the cumulative phonon spectrum is found. For 20% strain the gap does exist in the frequency range from 26 to 34 THz. 3.2. Extended nonlinear vibrational modes Symmetry of h-BN lattice under homogeneous strain along zigzag direction allows one to excite at least two extended shortwavelength vibrational modes [58]. The first one (in-plane mode) is obtained when, say, boron sublattice is shifted as a whole along the armchair direction relative to the nitrogen sublattice by 2 A. Initial velocities of all atoms are equal to zero. The second one
(transverse mode) is obtained by the relative shift of, say, boron sublattice relative to the nitrogen sublattice by 2 A in the direction normal to the sheet. In both cases the two sublattices will oscillate out-of-phase along the corresponding direction with the amplitudes A B and A N satisfying A B + A N = A. The amplitudes A B and A N are not equal because B and N have different atomic masses and because stiffness of B-N-B valence angle is different from that for N-B-N valence angle. These extended vibrational modes are the lattice symmetry dictated exact solutions to the equations of atomic motion, no matter what kind of interatomic potential is used and for any amplitude. Typically such modes demonstrate the effect of modulational instability [59–68] with the critical exponent growing with the increase of the amplitude. If frequency of an extended vibrational mode at large amplitudes lies outside the phonon spectrum, one can attempt to excite a discrete breather by applying a bell-shaped localization function [15,69]. Frequency as a function of amplitude A B for the in-plane mode is shown in Fig. 3. It can be seen that frequency of this mode bifurcates from the upper edge of the gap in the phonon spectrum (shown by the horizontal dashed line) and decreases with increasing amplitude being in the phonon gap. The same picture was observed for strained graphene in [26]. We thus conclude that h-BN, similar to graphene, can support gap discrete breathers, and in the following it will be shown that this is true. As for the transverse extended mode, its frequency also decreases with amplitude (see Fig. 4), bifurcating from the upper edge of the z-component of phonon DOS (see Fig. 1). This result is for unstrained h-BN, but the same is observed for h-BN under elastic strain εxx = 0.2 and 0.3. Since the transverse mode has fre-
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
3555
Fig. 4. Frequency-amplitude dependence of transverse extended mode in unstrained h-BN. Horizontal dashed line shows the upper edge of the z-component of DOS (see Fig. 1).
Fig. 2. Phonon DOS for h-BN elastically stretched along zigzag direction with 0.3.
εxx =
Fig. 5. Time evolution of the displacements along the y axis of B atom (blue solid line) and N atom (red dashed line) in the core of discrete breather in h-BN under strain εxx = 0.3. The discrete breather was excited with initial shift of these two atoms δ B = 0.2 Å and δ N = −0.18 Å. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
modeled with the AIREBO potential and they were not found when Savin potential was used [15]. 3.3. Discrete breathers
Fig. 3. Frequency-amplitude dependence for in-plane extended mode in h-BN stretched with εxx = 0.3. Horizontal dashed line shows the upper edge of the gap in the phonon spectrum (see Fig. 2).
quency inside the z-component phonon DOS, there is no chance to excite transverse discrete breathers in unstrained or strained h-BN. This result can be compared to similar results for graphene. The transverse mode in graphene has been modeled with the Savin [70] and AIREBO [71] potentials in [15]. In the first case the result is qualitatively same with h-BN, while AIREBO potential gave a qualitatively different result, namely, the frequency of the transverse mode growing with amplitude. That is why, transverse discrete breathers have been successfully excited in graphene
Very simple initial conditions are used to excite discrete breathers in h-BN strained with εxx = 0.3. For the B and N neighboring atoms connected by the valence bond oriented along the armchair direction an initial shift along the bond, δB and δN , is applied with all other atoms being at their lattice positions and all atoms in the system having initial zero velocities. For chosen δB the value of δN is found by the try and error method to achieve constancy of the vibration amplitudes for the initially excited atoms. In Fig. 5 we show the displacements along the y axis for the two initially excited atoms as the functions of time for the case δ B = 0.2 Å and δ N = −0.18 Å. The blue solid line is for B atom and red dashed line for N atom. It can be seen that the atoms vibrate with nearly constant amplitudes, A B = 0.167 Å and A N = 0.122 Å, without noticeable attenuation. In Fig. 6 positions of atoms in the vicinity of the discrete breather core are shown at the time when the excited two atoms are at the largest distance (dark blue for B and dark red for N) and when they are at the smallest distance (light blue for B and light red for N). Small black dots show the equilibrium lattice positions of h-BN at εxx = 0.3 strain. It can be seen from Fig. 5 and Fig. 6 that the lighter B atom has larger vibration amplitude than the heavier N atom. Fig. 6 reveals that vibration amplitudes of the initially excited B and N atoms are much larger than the amplitudes of their nearest neighbors.
3556
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
point masses, mB and mN , connected by an elastic bond. The deviation from this ratio is explained by the nonlinearity of interatomic forces and by the difference in stiffness of B–N–B and N–B–N valence angles. 4. Conclusions
Fig. 6. Displacements of atoms in the vicinity of the discrete breather in h-BN under strain εxx = 0.3. Black small dots show the lattice positions. Dark (light) blue dots show B atoms at the time when separation between B and N atoms in the core of the discrete breather is maximal (minimal). Similarly, dark (light) red dots show N atoms at the time when separation between B and N atoms in the core of the discrete breather is maximal (minimal). The discrete breather is excited with initial shift of B and N atoms equal to δ B = 0.2 Å and δ N = −0.18 Å, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Frequency-amplitude dependence for discrete breather in h-BN under strain of εxx .
With the help of molecular dynamics based on the Tersoff-like interatomic potentials the following main results are obtained in the study of linear and nonlinear dynamics of h-BN lattice. A wide gap in the phonon DOS can be induced by applying sufficiently large (20% and more) homogeneous tensile elastic strain in the zigzag direction (the x direction in our simulations). The two extended vibrational modes are studied in unstrained and strained h-BN: the in-plane mode with boron and nitrogen sublattices oscillating as a whole out-of-phase along the armchair direction and the transverse mode when they oscillate along the normal to the sheet. Both modes demonstrate soft-type nonlinearity as frequency decreases with the amplitude. Frequencies of the in-plane mode in strained h-BN lie in the gap of phonon DOS, while frequencies of the transverse mode are always within the phonon DOS for out-of-plane oscillations. From this it was concluded that discrete breathers with in-plane oscillations can exist in strained h-BN, but discrete breathers with out-of-plane oscillations cannot. Discrete breathers with in-plane oscillations are excited in strained h-BN by initial shift of two neighboring B and N atoms, one along and one opposite to the y-direction. Frequency of the discrete breather lies within the phonon gap and decreases with increasing amplitude. The lighter boron atom has larger vibration amplitude than the heavier nitrogen atom. This is the main difference of the in-plane discrete breather in strained h-BN in comparison to the analogous discrete breather in strained graphene [26]. Attempts to excite transverse discrete breathers in unstrained h-BN have failed, which was expected from the fact that the frequency of the extended transverse mode is within the phonon DOS for out-of-plane oscillations. Recall that in graphene modeled with the AIREBO potential transverse discrete breathers do exist [15–17] and the frequency of the extended transverse mode is above the phonon DOS for out-of-plane oscillations [15]. We believe that the search for discrete breathers in various lowdimensional materials should be continued. Acknowledgements
Fig. 8. Ratio of N atom to B atom amplitudes as a function of B atom amplitude for the discrete breather in h-BN under strain of εxx = 0.3. The horizontal dashed line shows the estimation of the amplitude ratio for a one-degree-of-freedom oscillator, A N / A B = mB /mN = 0.771.
In Fig. 7 frequency of the discrete breather is shown as a function of the vibration amplitude of B atom. Frequency decreases with amplitude and thus, the discrete breather demonstrates soft type nonlinearity. Discrete breather frequency lies within the gap of the phonon spectrum, same with the frequency of the in-plane extended mode shown in Fig. 3. Ratio of the amplitudes A N / A B is plotted in Fig. 8 as a function of A B . The ratio is growing with increasing A B and, as mentioned above, it is always less than 1. It is possible to give a simple estimation of the amplitude ratio as follows, A N / A B = mB /mN = 0.771 (shown in Fig. 8 by the horizontal dashed line). This estimation comes from consideration of one degree of freedom harmonic oscillator in the form of two
The work of E.A.K. (discussion of the numerical results) is supported by the Russian Foundation for Basic Research, grant no. 17-02-00984. A.P.Ch., who contributed to the design of the research, would like to thank financial support from the Russian Science Foundation, grant no. 16-12-10175. S.V.D. has contributed to writing and he thanks the financial support from the Russian Science Foundation, grant no. 14-13-00982. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
A.S. Dolgov, Sov. Phys., Solid State 28 (1986) 907. A.J. Sievers, S. Takeno, Phys. Rev. Lett. 61 (1988) 970. J.B. Page, Phys. Rev. B 41 (1990) 7835. M.G. Velarde, A.P. Chetverikov, W. Ebeling, S.V. Dmitriev, V.D. Lakhno, Eur. Phys. J. B 89 (2016) 233. S.V. Dmitriev, E.A. Korznikova, J.A. Baimova, M.G. Velarde, Phys. Usp. 59 (2016) 446. S.V. Dmitriev, Nonlinear Theory Appl., IEICE 8 (2017) 85. M. Haas, V. Hizhnyakov, A. Shelkan, M. Klopov, A.J. Sievers, Phys. Rev. B 84 (2011) 144303. D.A. Terentyev, A.V. Dubinko, V.I. Dubinko, S.V. Dmitriev, E.E. Zhurkin, M.V. Sorokin, Model. Simul. Mater. Sci. 23 (2015) 085007. R. Murzaev, R. Babicheva, K. Zhou, E.A. Korznikova, S.Y. Fomin, V. Dubinko, S.V. Dmitriev, Eur. Phys. J. B 89 (2016) 168.
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
[10] R.T. Murzaev, A.A. Kistanov, V.I. Dubinko, D.A. Terentyev, S.V. Dmitriev, Comput. Mater. Sci. 98 (2015) 88. [11] N.K. Voulgarakis, G. Hadjisavvas, P.C. Kelires, G.P. Tsironis, Phys. Rev. B 69 (2004) 113201. [12] R.T. Murzaev, D.V. Bachurin, E.A. Korznikova, S.V. Dmitriev, Phys. Lett. A 381 (2017) 1003–1008. [13] S.A. Kiselev, A.J. Sievers, Phys. Rev. B 55 (1997) 5755. [14] L.Z. Khadeeva, S.V. Dmitriev, Phys. Rev. B 81 (2010) 214306. [15] E. Barani, I.P. Lobzenko, E.A. Korznikova, E.G. Soboleva, S.V. Dmitriev, K. Zhou, A.M. Marjaneh, Eur. Phys. J. B 90 (2017) 38. [16] V. Hizhnyakov, M. Klopov, A. Shelkan, Phys. Lett. A 380 (2016) 1075. [17] V. Hizhnyakov, A. Shelkan, M. Haas, M. Klopov, Lett. Mater. 6 (2016) 61. [18] I. Evazzade, I.P. Lobzenko, E.A. Korznikova, I.A. Ovid’Ko, M.R. Roknabadi, S.V. Dmitriev, Phys. Rev. B 95 (2017) 035423. [19] I.P. Lobzenko, G.M. Chechin, G.S. Bezuglova, Y.A. Baimova, E.A. Korznikova, S.V. Dmitriev, Phys. Solid State 58 (2017) 633. [20] J.A. Baimova, E.A. Korznikova, I.P. Lobzenko, S.V. Dmitriev, Rev. Adv. Mater. Sci. 42 (2015) 68. [21] E.A. Korznikova, J.A. Baimova, S.V. Dmitriev, Europhys. Lett. 102 (2013) 60004. [22] E.A. Korznikova, A.V. Savin, Y.A. Baimova, S.V. Dmitriev, R.R. Mulyukov, JETP Lett. 96 (2012) 222. [23] J.A. Baimova, S.V. Dmitriev, K. Zhou, Europhys. Lett. 100 (2012) 36005. [24] S.V. Dmitriev, L.Z. Khadeeva, Yu.S. Kivshar, Nonlinear Theory Appl., IEICE 3 (2012) 77. [25] Y. Doi, A. Nakatani, Proc. Eng. 10 (2011) 3393. [26] L.Z. Khadeeva, S.V. Dmitriev, Y.S. Kivshar, JETP Lett. 94 (2011) 539. [27] J.A. Baimova, R.T. Murzaev, I.P. Lobzenko, S.V. Dmitriev, K. Zhou, J. Exp. Theor. Phys. 122 (2016) 869. [28] J.A. Baimova, S.V. Dmitriev, Russ. Phys. J. 58 (2015) 785. [29] G.M. Chechin, S.V. Dmitriev, I.P. Lobzenko, D.S. Ryabov, Phys. Rev. B 90 (2014) 045432. [30] G.M. Chechin, I.P. Lobzenko, Lett. Mater. 4 (2014) 226. [31] B. Liu, J.A. Baimova, S.V. Dmitriev, X. Wang, H. Zhu, K. Zhou, J. Phys. D, Appl. Phys. 46 (2013) 305302. [32] Y. Doi, A. Nakatani, Lett. Mater. 6 (2016) 49. [33] T. Shimada, D. Shirasaki, Y. Kinoshita, Y. Doi, A. Nakatani, T. Kitamura, Physica D 239 (2010) 407. [34] T. Shimada, D. Shirasaki, T. Kitamura, Phys. Rev. B 81 (2010) 035401. [35] Y. Kinoshita, Y. Yamayose, Y. Doi, A. Nakatani, T. Kitamura, Physica D 77 (2008) 024307. [36] Y. Kubota, K. Watanabe, O. Tsuda, T. Taniguchi, Science 317 (2007) 932. [37] J.S. Ponraj, et al., Nanotechnology 27 (46) (2016) 462001. [38] J. Zhao, et al., Prog. Mater. Sci. 83 (2016) 24.
[39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71]
3557
Q. Tang, Z. Zhou, Prog. Mater. Sci. 58 (2013) 1244. M. Xu, T. Liang, M. Shi, H. Chen, Chem. Rev. 113 (2013) 3766. P. Miro, M. Audiffred, T. Heine, Chem. Soc. Rev. 43 (2014) 6537. S. Balendhran, S. Walia, H. Nili, S. Sriram, M. Bhaskaran, Small 11 (2015) 640. M. Yagmurcukardes, F.M. Peeters, R.T. Senger, H. Sahin, Appl. Phys. Rev. 3 (2016) 041302. I. Shtepliuk, V. Khranovskyy, R. Yakimova, Semicond. Sci. Technol. 31 (2016) 113004. L.C. Yan Voon, J. Zhu, U. Schwingenschlogl, Appl. Phys. Rev. 3 (2016) 040802. S. Chowdhury, D. Jana, Rep. Prog. Phys. 79 (2016) 126501. B. Liu, J.A. Baimova, C.D. Reddy, A.W.-K. Law, S.V. Dmitriev, H. Wu, K. Zhou, ACS Appl. Mater. Interfaces 6 (2014) 18180. A.A. Kistanov, Y. Cai, K. Zhou, S.V. Dmitriev, Y.-W. Zhang, J. Phys. Chem. C 120 (2016) 6876. M. Yankowitz, K. Watanabe, T. Taniguchi, P. San-Jose, B.J. Leroy, Nat. Commun. 7 (2016) 13168. C.-C. Chen, Z. Li, L. Shi, S.B. Cronin, Appl. Phys. Lett. 104 (2014) 081908. N. Sakhavand, R. Shahsavari, ACS Appl. Mater. Interfaces 7 (2015) 18312. A. Shekhawat, R.O. Ritchie, Nat. Commun. 7 (2016) 10546. J.A. Baimova, S.V. Dmitriev, K. Zhou, A.V. Savin, Phys. Rev. B 86 (2012) 035427. T. Han, F. Scarpa, N.L. Allan, Thin Solid Films 632 (2017) 35. S. Plimpton, J. Comput. Phys. 117 (1995) 1. J. Tersoff, Phys. Rev. B 39 (1989) 5566. C. Sevik, A. Kinaci, J.B. Haskins, T. Cagin, Phys. Rev. B 84 (2011) 085409. G. Chechin, D. Ryabov, S. Shcherbinin, Lett. Mater. 6 (2016) 9. V.M. Burlakov, S.A. Kiselev, V.I. Rupasov, Phys. Lett. A 147 (1990) 130. V.M. Burlakov, S.A. Kiselev, V.I. Rupasov, JETP Lett. 51 (1990) 544. T. Cretegny, T. Dauxois, S. Ruffo, A. Torcini, Physica D 121 (1998) 109. T. Dauxois, R. Khomeriki, F. Piazza, S. Ruffo, Chaos 15 (2005) 015110. Yu.A. Kosevich, G. Corso, Physica D 170 (2002) 1. Yu.A. Kosevich, S. Lepri, Phys. Rev. B 61 (2000) 299. K. Ikeda, Y. Doi, B.F. Feng, T. Kawahara, Physica D 225 (2007) 184. L.Z. Khadeeva, S.V. Dmitriev, Phys. Rev. B 81 (2010) 214306. L. Kavitha, E. Parasuraman, D. Gopi, A. Prabhu, R.A. Vicencio, J. Magn. Magn. Mater. 401 (2016) 394. E.A. Korznikova, D.V. Bachurin, S.Y. Fomin, A.P. Chetverikov, S.V. Dmitriev, Eur. Phys. J. B 90 (2017) 23. E.A. Korznikova, S.Y. Fomin, E.G. Soboleva, S.V. Dmitriev, JETP Lett. 103 (2016) 277. A.V. Savin, Yu.S. Kivshar, B. Hu, Phys. Rev. B 82 (2010) 195422. S.J. Stuart, A.B. Tutein, J.A. Harrison, J. Chem. Phys. 112 (2000) 6472.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Gap discrete breathers in strained boron nitride Elham Barani a,∗ , Elena A. Korznikova b , Alexander P. Chetverikov c , Kun Zhou d,e , Sergey V. Dmitriev b,f a
Department of Chemistry, Faculty of Science, Ferdowsi University of Mashhad, Mashhad, Iran Institute for Metals Superplasticity Problems, Russian Academy of Sciences, 450001 Ufa, Russia c Saratov State University, 83 Astrakhanskaya St., 410012 Saratov, Russia d Environmental Process Modelling Centre, Nanyang Environment and Water Research Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore e School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore f National Research Tomsk State University, 36 Lenin Ave, 634050 Tomsk, Russia b
a r t i c l e
i n f o
Article history: Received 20 June 2017 Accepted 29 August 2017 Available online 1 September 2017 Communicated by R. Wu Keywords: Boron nitride Discrete breather Molecular dynamics Lattice dynamics Extended nonlinear vibrational mode
a b s t r a c t Linear and nonlinear dynamics of hexagonal boron nitride (h-BN) lattice is studied by means of molecular dynamics simulations with the use of the Tersoff interatomic potentials. It is found that sufficiently large homogeneous elastic strain along zigzag direction opens a wide gap in the phonon spectrum. Extended vibrational mode with boron and nitrogen sublattices vibrating in-plane as a whole in strained h-BN has frequency within the phonon gap. This fact suggests that a nonlinear spatially localized vibrational mode with frequencies in the phonon gap, called discrete breather (also often termed as intrinsic localized mode), can be excited. Properties of the gap discrete breathers in strained h-BN are contrasted with that for analogous vibrational mode found earlier in strained graphene. It is found that h-BN modeled with the Tersoff potentials does not support transverse discrete breathers. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Among nonlinear lattice excitations of particular interest are nonlinear spatially localized vibrational modes called either discrete breathers or intrinsic localized modes, first discovered in simple 1D nonlinear chains [1–3]. A transition from solitons to discrete breathers with the change of relative strength of interparticle and on-site interactions was described in [4]. Discrete breathers have been modeled in a number of crystals [5] including metals [6–10], covalent crystals [11,12], ionic crystals [13,14], and in low-dimensional materials such as graphene [15–23,25,24,26], graphane (fully hydrogeneted graphene) [27–31], and carbon nanotubes [32–35]. Undoubted success in various technological developments of graphene has generated a growing interest of researchers to other two-dimensional crystalline materials such as hexagonal boron nitride (h-BN), silicene, phosphorene, molybdenum dioxide, and others [36–48]. Very little is known about the existence and properties of discrete breathers in these new materials.
In the present study we consider h-BN, whose properties are often compared to that of graphene, because both these materials have hexagonal lattice. It is even possible to achieve pressureinduced commensurate stacking of graphene on h-BN [49]. In spite of similarities in their structure, many properties of h-BN are different from that of graphene: graphene is a semimetal with no gap in the electron spectrum, while the h-BN sheet is a wide-band-gap (about 6 eV) electrical insulator. Graphene and h-BN possess interesting thermal properties [50,51] and high strength [52–54]. Elastically strained graphene supports discrete breathers and breather clusters with frequencies within the phonon gap [23,26]. Unstrained graphene modeled with the AIREBO potential supports transverse discrete breathers with frequencies within the phonon spectrum but above the phonon spectrum of out-of-plane vibrations [15–17]. Such discrete breathers have a very long lifetime because the large-amplitude out-of-plane vibrations weakly interact with the in-plane phonons. It is tempting to check if h-BN can support similar nonlinear excitations and this is the subject of the present study. 2. Simulation setup
*
Corresponding author. E-mail address: [email protected] (E. Barani).
http://dx.doi.org/10.1016/j.physleta.2017.08.057 0375-9601/© 2017 Elsevier B.V. All rights reserved.
Atoms in h-BN form a two-dimensional hexagonal lattice with a primitive translational cell containing one B and one N atoms.
3554
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
Atom mass of boron is equal to mB = 10.8 atomic mass units, while for nitrogen it is mN = 14.0 atomic mass units. Let the x and the y axes of the Cartesian coordinate system are directed along the zigzag and armchair directions of the lattice. This √ cell is generated by the vectors a1 = (a, 0, 0) and a2 = (a/2, a 3/2, 0), where a = 2.497 Å is the equilibrium lattice parameter. For definiteness, boron atom is located at the origin, while nitrogen atom at the middle of the primitive translational cell. The computational cell of N × N primitive cells is considered with periodic boundary conditions. The large-scale atomic/molecular massively parallel simulator (LAMMPS) [55] is employed to conduct the molecular dynamics simulations with the use of the Tersoff-like interatomic potential [56,57]. Fourth-order scheme with the time step of 0.1 fs is used to integrate numerically the equations of atomic motion. Phonon density of states (DOS) is calculated as the Fourier transform of the autocorrelation functions of trajectories of atoms at a temperature of 10 K. The results are compared for different computational cell sizes, N = 10, 20, and 40. It is found that N = 20 is sufficient and this size is used also for simulation of the extended vibrational modes and discrete breathers. In order to induce a gap into phonon spectrum, we apply inplane tensile elastic strain εxx > 0 along zigzag direction. The equilibrium positions of atoms in uniformly strained lattice are found by minimizing the potential energy of the crystal. The initial conditions used to excite the extended vibrational modes and the discrete breathers are described below. Thermal vibrations are not introduced in the study of extended vibrational modes and discrete breathers. Frequency and amplitude of the extended modes and discrete breathers are found by averaging over the simulation run of 0.6 ps which covers more than a dosen of the oscillation periods. 3. Results Fig. 1. Phonon DOS for unstraind h-BN.
3.1. Effect of elastic strain on phonon DOS The phonon density of states (DOS) for unstrained h-BN is shown in Fig. 1. Panels (a)–(c) give the x, y, and z-components of DOS, while in (d) the cumulative DOS is presented. It can be seen that maximal frequency of in-plane small-amplitude vibrations is about 47 THz, while for out-of-plane phonon modes it is at 26 THz. A narrow gap in phonon DOS (width of about 2 THz) can be seen around 35 THz. Similar results but for elastically stretched h-BN along zigzag direction with εxx = 0.3 can be seen in Fig. 2. Maximal frequencies for in-plane and out-of-plane phonons have been reduced down to 40 THz and 15 THz, respectively. More important for our discussion is the appearance of the wide gap in the phonon spectrum ranging from 23 to 36 THz. This makes it possible for in-plane DBs to exist with frequencies inside the gap, as it has been shown for strained graphene [26]. We have also calculated phonon DOS for εxx = 0.1 and 0.2. For 10% strain, due to breaking of the lattice symmetry, the gap exists only in the x-component of DOS and no gap in the cumulative phonon spectrum is found. For 20% strain the gap does exist in the frequency range from 26 to 34 THz. 3.2. Extended nonlinear vibrational modes Symmetry of h-BN lattice under homogeneous strain along zigzag direction allows one to excite at least two extended shortwavelength vibrational modes [58]. The first one (in-plane mode) is obtained when, say, boron sublattice is shifted as a whole along the armchair direction relative to the nitrogen sublattice by 2 A. Initial velocities of all atoms are equal to zero. The second one
(transverse mode) is obtained by the relative shift of, say, boron sublattice relative to the nitrogen sublattice by 2 A in the direction normal to the sheet. In both cases the two sublattices will oscillate out-of-phase along the corresponding direction with the amplitudes A B and A N satisfying A B + A N = A. The amplitudes A B and A N are not equal because B and N have different atomic masses and because stiffness of B-N-B valence angle is different from that for N-B-N valence angle. These extended vibrational modes are the lattice symmetry dictated exact solutions to the equations of atomic motion, no matter what kind of interatomic potential is used and for any amplitude. Typically such modes demonstrate the effect of modulational instability [59–68] with the critical exponent growing with the increase of the amplitude. If frequency of an extended vibrational mode at large amplitudes lies outside the phonon spectrum, one can attempt to excite a discrete breather by applying a bell-shaped localization function [15,69]. Frequency as a function of amplitude A B for the in-plane mode is shown in Fig. 3. It can be seen that frequency of this mode bifurcates from the upper edge of the gap in the phonon spectrum (shown by the horizontal dashed line) and decreases with increasing amplitude being in the phonon gap. The same picture was observed for strained graphene in [26]. We thus conclude that h-BN, similar to graphene, can support gap discrete breathers, and in the following it will be shown that this is true. As for the transverse extended mode, its frequency also decreases with amplitude (see Fig. 4), bifurcating from the upper edge of the z-component of phonon DOS (see Fig. 1). This result is for unstrained h-BN, but the same is observed for h-BN under elastic strain εxx = 0.2 and 0.3. Since the transverse mode has fre-
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
3555
Fig. 4. Frequency-amplitude dependence of transverse extended mode in unstrained h-BN. Horizontal dashed line shows the upper edge of the z-component of DOS (see Fig. 1).
Fig. 2. Phonon DOS for h-BN elastically stretched along zigzag direction with 0.3.
εxx =
Fig. 5. Time evolution of the displacements along the y axis of B atom (blue solid line) and N atom (red dashed line) in the core of discrete breather in h-BN under strain εxx = 0.3. The discrete breather was excited with initial shift of these two atoms δ B = 0.2 Å and δ N = −0.18 Å. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
modeled with the AIREBO potential and they were not found when Savin potential was used [15]. 3.3. Discrete breathers
Fig. 3. Frequency-amplitude dependence for in-plane extended mode in h-BN stretched with εxx = 0.3. Horizontal dashed line shows the upper edge of the gap in the phonon spectrum (see Fig. 2).
quency inside the z-component phonon DOS, there is no chance to excite transverse discrete breathers in unstrained or strained h-BN. This result can be compared to similar results for graphene. The transverse mode in graphene has been modeled with the Savin [70] and AIREBO [71] potentials in [15]. In the first case the result is qualitatively same with h-BN, while AIREBO potential gave a qualitatively different result, namely, the frequency of the transverse mode growing with amplitude. That is why, transverse discrete breathers have been successfully excited in graphene
Very simple initial conditions are used to excite discrete breathers in h-BN strained with εxx = 0.3. For the B and N neighboring atoms connected by the valence bond oriented along the armchair direction an initial shift along the bond, δB and δN , is applied with all other atoms being at their lattice positions and all atoms in the system having initial zero velocities. For chosen δB the value of δN is found by the try and error method to achieve constancy of the vibration amplitudes for the initially excited atoms. In Fig. 5 we show the displacements along the y axis for the two initially excited atoms as the functions of time for the case δ B = 0.2 Å and δ N = −0.18 Å. The blue solid line is for B atom and red dashed line for N atom. It can be seen that the atoms vibrate with nearly constant amplitudes, A B = 0.167 Å and A N = 0.122 Å, without noticeable attenuation. In Fig. 6 positions of atoms in the vicinity of the discrete breather core are shown at the time when the excited two atoms are at the largest distance (dark blue for B and dark red for N) and when they are at the smallest distance (light blue for B and light red for N). Small black dots show the equilibrium lattice positions of h-BN at εxx = 0.3 strain. It can be seen from Fig. 5 and Fig. 6 that the lighter B atom has larger vibration amplitude than the heavier N atom. Fig. 6 reveals that vibration amplitudes of the initially excited B and N atoms are much larger than the amplitudes of their nearest neighbors.
3556
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
point masses, mB and mN , connected by an elastic bond. The deviation from this ratio is explained by the nonlinearity of interatomic forces and by the difference in stiffness of B–N–B and N–B–N valence angles. 4. Conclusions
Fig. 6. Displacements of atoms in the vicinity of the discrete breather in h-BN under strain εxx = 0.3. Black small dots show the lattice positions. Dark (light) blue dots show B atoms at the time when separation between B and N atoms in the core of the discrete breather is maximal (minimal). Similarly, dark (light) red dots show N atoms at the time when separation between B and N atoms in the core of the discrete breather is maximal (minimal). The discrete breather is excited with initial shift of B and N atoms equal to δ B = 0.2 Å and δ N = −0.18 Å, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Frequency-amplitude dependence for discrete breather in h-BN under strain of εxx .
With the help of molecular dynamics based on the Tersoff-like interatomic potentials the following main results are obtained in the study of linear and nonlinear dynamics of h-BN lattice. A wide gap in the phonon DOS can be induced by applying sufficiently large (20% and more) homogeneous tensile elastic strain in the zigzag direction (the x direction in our simulations). The two extended vibrational modes are studied in unstrained and strained h-BN: the in-plane mode with boron and nitrogen sublattices oscillating as a whole out-of-phase along the armchair direction and the transverse mode when they oscillate along the normal to the sheet. Both modes demonstrate soft-type nonlinearity as frequency decreases with the amplitude. Frequencies of the in-plane mode in strained h-BN lie in the gap of phonon DOS, while frequencies of the transverse mode are always within the phonon DOS for out-of-plane oscillations. From this it was concluded that discrete breathers with in-plane oscillations can exist in strained h-BN, but discrete breathers with out-of-plane oscillations cannot. Discrete breathers with in-plane oscillations are excited in strained h-BN by initial shift of two neighboring B and N atoms, one along and one opposite to the y-direction. Frequency of the discrete breather lies within the phonon gap and decreases with increasing amplitude. The lighter boron atom has larger vibration amplitude than the heavier nitrogen atom. This is the main difference of the in-plane discrete breather in strained h-BN in comparison to the analogous discrete breather in strained graphene [26]. Attempts to excite transverse discrete breathers in unstrained h-BN have failed, which was expected from the fact that the frequency of the extended transverse mode is within the phonon DOS for out-of-plane oscillations. Recall that in graphene modeled with the AIREBO potential transverse discrete breathers do exist [15–17] and the frequency of the extended transverse mode is above the phonon DOS for out-of-plane oscillations [15]. We believe that the search for discrete breathers in various lowdimensional materials should be continued. Acknowledgements
Fig. 8. Ratio of N atom to B atom amplitudes as a function of B atom amplitude for the discrete breather in h-BN under strain of εxx = 0.3. The horizontal dashed line shows the estimation of the amplitude ratio for a one-degree-of-freedom oscillator, A N / A B = mB /mN = 0.771.
In Fig. 7 frequency of the discrete breather is shown as a function of the vibration amplitude of B atom. Frequency decreases with amplitude and thus, the discrete breather demonstrates soft type nonlinearity. Discrete breather frequency lies within the gap of the phonon spectrum, same with the frequency of the in-plane extended mode shown in Fig. 3. Ratio of the amplitudes A N / A B is plotted in Fig. 8 as a function of A B . The ratio is growing with increasing A B and, as mentioned above, it is always less than 1. It is possible to give a simple estimation of the amplitude ratio as follows, A N / A B = mB /mN = 0.771 (shown in Fig. 8 by the horizontal dashed line). This estimation comes from consideration of one degree of freedom harmonic oscillator in the form of two
The work of E.A.K. (discussion of the numerical results) is supported by the Russian Foundation for Basic Research, grant no. 17-02-00984. A.P.Ch., who contributed to the design of the research, would like to thank financial support from the Russian Science Foundation, grant no. 16-12-10175. S.V.D. has contributed to writing and he thanks the financial support from the Russian Science Foundation, grant no. 14-13-00982. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
A.S. Dolgov, Sov. Phys., Solid State 28 (1986) 907. A.J. Sievers, S. Takeno, Phys. Rev. Lett. 61 (1988) 970. J.B. Page, Phys. Rev. B 41 (1990) 7835. M.G. Velarde, A.P. Chetverikov, W. Ebeling, S.V. Dmitriev, V.D. Lakhno, Eur. Phys. J. B 89 (2016) 233. S.V. Dmitriev, E.A. Korznikova, J.A. Baimova, M.G. Velarde, Phys. Usp. 59 (2016) 446. S.V. Dmitriev, Nonlinear Theory Appl., IEICE 8 (2017) 85. M. Haas, V. Hizhnyakov, A. Shelkan, M. Klopov, A.J. Sievers, Phys. Rev. B 84 (2011) 144303. D.A. Terentyev, A.V. Dubinko, V.I. Dubinko, S.V. Dmitriev, E.E. Zhurkin, M.V. Sorokin, Model. Simul. Mater. Sci. 23 (2015) 085007. R. Murzaev, R. Babicheva, K. Zhou, E.A. Korznikova, S.Y. Fomin, V. Dubinko, S.V. Dmitriev, Eur. Phys. J. B 89 (2016) 168.
E. Barani et al. / Physics Letters A 381 (2017) 3553–3557
[10] R.T. Murzaev, A.A. Kistanov, V.I. Dubinko, D.A. Terentyev, S.V. Dmitriev, Comput. Mater. Sci. 98 (2015) 88. [11] N.K. Voulgarakis, G. Hadjisavvas, P.C. Kelires, G.P. Tsironis, Phys. Rev. B 69 (2004) 113201. [12] R.T. Murzaev, D.V. Bachurin, E.A. Korznikova, S.V. Dmitriev, Phys. Lett. A 381 (2017) 1003–1008. [13] S.A. Kiselev, A.J. Sievers, Phys. Rev. B 55 (1997) 5755. [14] L.Z. Khadeeva, S.V. Dmitriev, Phys. Rev. B 81 (2010) 214306. [15] E. Barani, I.P. Lobzenko, E.A. Korznikova, E.G. Soboleva, S.V. Dmitriev, K. Zhou, A.M. Marjaneh, Eur. Phys. J. B 90 (2017) 38. [16] V. Hizhnyakov, M. Klopov, A. Shelkan, Phys. Lett. A 380 (2016) 1075. [17] V. Hizhnyakov, A. Shelkan, M. Haas, M. Klopov, Lett. Mater. 6 (2016) 61. [18] I. Evazzade, I.P. Lobzenko, E.A. Korznikova, I.A. Ovid’Ko, M.R. Roknabadi, S.V. Dmitriev, Phys. Rev. B 95 (2017) 035423. [19] I.P. Lobzenko, G.M. Chechin, G.S. Bezuglova, Y.A. Baimova, E.A. Korznikova, S.V. Dmitriev, Phys. Solid State 58 (2017) 633. [20] J.A. Baimova, E.A. Korznikova, I.P. Lobzenko, S.V. Dmitriev, Rev. Adv. Mater. Sci. 42 (2015) 68. [21] E.A. Korznikova, J.A. Baimova, S.V. Dmitriev, Europhys. Lett. 102 (2013) 60004. [22] E.A. Korznikova, A.V. Savin, Y.A. Baimova, S.V. Dmitriev, R.R. Mulyukov, JETP Lett. 96 (2012) 222. [23] J.A. Baimova, S.V. Dmitriev, K. Zhou, Europhys. Lett. 100 (2012) 36005. [24] S.V. Dmitriev, L.Z. Khadeeva, Yu.S. Kivshar, Nonlinear Theory Appl., IEICE 3 (2012) 77. [25] Y. Doi, A. Nakatani, Proc. Eng. 10 (2011) 3393. [26] L.Z. Khadeeva, S.V. Dmitriev, Y.S. Kivshar, JETP Lett. 94 (2011) 539. [27] J.A. Baimova, R.T. Murzaev, I.P. Lobzenko, S.V. Dmitriev, K. Zhou, J. Exp. Theor. Phys. 122 (2016) 869. [28] J.A. Baimova, S.V. Dmitriev, Russ. Phys. J. 58 (2015) 785. [29] G.M. Chechin, S.V. Dmitriev, I.P. Lobzenko, D.S. Ryabov, Phys. Rev. B 90 (2014) 045432. [30] G.M. Chechin, I.P. Lobzenko, Lett. Mater. 4 (2014) 226. [31] B. Liu, J.A. Baimova, S.V. Dmitriev, X. Wang, H. Zhu, K. Zhou, J. Phys. D, Appl. Phys. 46 (2013) 305302. [32] Y. Doi, A. Nakatani, Lett. Mater. 6 (2016) 49. [33] T. Shimada, D. Shirasaki, Y. Kinoshita, Y. Doi, A. Nakatani, T. Kitamura, Physica D 239 (2010) 407. [34] T. Shimada, D. Shirasaki, T. Kitamura, Phys. Rev. B 81 (2010) 035401. [35] Y. Kinoshita, Y. Yamayose, Y. Doi, A. Nakatani, T. Kitamura, Physica D 77 (2008) 024307. [36] Y. Kubota, K. Watanabe, O. Tsuda, T. Taniguchi, Science 317 (2007) 932. [37] J.S. Ponraj, et al., Nanotechnology 27 (46) (2016) 462001. [38] J. Zhao, et al., Prog. Mater. Sci. 83 (2016) 24.
[39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71]
3557
Q. Tang, Z. Zhou, Prog. Mater. Sci. 58 (2013) 1244. M. Xu, T. Liang, M. Shi, H. Chen, Chem. Rev. 113 (2013) 3766. P. Miro, M. Audiffred, T. Heine, Chem. Soc. Rev. 43 (2014) 6537. S. Balendhran, S. Walia, H. Nili, S. Sriram, M. Bhaskaran, Small 11 (2015) 640. M. Yagmurcukardes, F.M. Peeters, R.T. Senger, H. Sahin, Appl. Phys. Rev. 3 (2016) 041302. I. Shtepliuk, V. Khranovskyy, R. Yakimova, Semicond. Sci. Technol. 31 (2016) 113004. L.C. Yan Voon, J. Zhu, U. Schwingenschlogl, Appl. Phys. Rev. 3 (2016) 040802. S. Chowdhury, D. Jana, Rep. Prog. Phys. 79 (2016) 126501. B. Liu, J.A. Baimova, C.D. Reddy, A.W.-K. Law, S.V. Dmitriev, H. Wu, K. Zhou, ACS Appl. Mater. Interfaces 6 (2014) 18180. A.A. Kistanov, Y. Cai, K. Zhou, S.V. Dmitriev, Y.-W. Zhang, J. Phys. Chem. C 120 (2016) 6876. M. Yankowitz, K. Watanabe, T. Taniguchi, P. San-Jose, B.J. Leroy, Nat. Commun. 7 (2016) 13168. C.-C. Chen, Z. Li, L. Shi, S.B. Cronin, Appl. Phys. Lett. 104 (2014) 081908. N. Sakhavand, R. Shahsavari, ACS Appl. Mater. Interfaces 7 (2015) 18312. A. Shekhawat, R.O. Ritchie, Nat. Commun. 7 (2016) 10546. J.A. Baimova, S.V. Dmitriev, K. Zhou, A.V. Savin, Phys. Rev. B 86 (2012) 035427. T. Han, F. Scarpa, N.L. Allan, Thin Solid Films 632 (2017) 35. S. Plimpton, J. Comput. Phys. 117 (1995) 1. J. Tersoff, Phys. Rev. B 39 (1989) 5566. C. Sevik, A. Kinaci, J.B. Haskins, T. Cagin, Phys. Rev. B 84 (2011) 085409. G. Chechin, D. Ryabov, S. Shcherbinin, Lett. Mater. 6 (2016) 9. V.M. Burlakov, S.A. Kiselev, V.I. Rupasov, Phys. Lett. A 147 (1990) 130. V.M. Burlakov, S.A. Kiselev, V.I. Rupasov, JETP Lett. 51 (1990) 544. T. Cretegny, T. Dauxois, S. Ruffo, A. Torcini, Physica D 121 (1998) 109. T. Dauxois, R. Khomeriki, F. Piazza, S. Ruffo, Chaos 15 (2005) 015110. Yu.A. Kosevich, G. Corso, Physica D 170 (2002) 1. Yu.A. Kosevich, S. Lepri, Phys. Rev. B 61 (2000) 299. K. Ikeda, Y. Doi, B.F. Feng, T. Kawahara, Physica D 225 (2007) 184. L.Z. Khadeeva, S.V. Dmitriev, Phys. Rev. B 81 (2010) 214306. L. Kavitha, E. Parasuraman, D. Gopi, A. Prabhu, R.A. Vicencio, J. Magn. Magn. Mater. 401 (2016) 394. E.A. Korznikova, D.V. Bachurin, S.Y. Fomin, A.P. Chetverikov, S.V. Dmitriev, Eur. Phys. J. B 90 (2017) 23. E.A. Korznikova, S.Y. Fomin, E.G. Soboleva, S.V. Dmitriev, JETP Lett. 103 (2016) 277. A.V. Savin, Yu.S. Kivshar, B. Hu, Phys. Rev. B 82 (2010) 195422. S.J. Stuart, A.B. Tutein, J.A. Harrison, J. Chem. Phys. 112 (2000) 6472.