Nonlinear vibrational modes in graphene: group-theoretical results

G. Chechin1, D. Ryabov1, S. Shcherbinin1
1Southern Federal University, Institute of physics, Stachki Ave., 194, 344090, Rostov-on-Don, Russia
Abstract
In-plane nonlinear delocalized vibrations in uniformly stretched single-layer graphene (space group P6mm) are considered with the aid of the group-theoretical methods. These methods were developed by authors earlier in the framework of the theory of the bushes of nonlinear normal modes (NNMs). Each bush represents a set of delocalized NNMs which is conserved in time, and the energy of initial excitation turns out to be trapped in the given bush. The number m of modes entering into the bush defines its dimension. One-dimensional bushes (m=1) represent individual nonlinear normal modes by Rosenberg which describe periodic dynamical regimes, while bushes of higher dimension (m>1) describe quasiperiodic motion with m basis frequencies in the Fourier spectrum. Each bush is characterized by a space group, which is a subgroup of the symmetry group of the system equilibrium state. There exist bushes of NNMs of different physical nature. In this paper, we restrict ourselves to study of vibrational bushes. We have found that only 4 symmetry-determined NNMs (one-dimensional bushes), as well as 14 two-dimensional, 1 three-dimensional and 6 four-dimensional vibrational bushes are possible in the single-layer graphene. These dynamical regimes are exact solutions to the motion equations of this two-dimensional crystal. Prospects of further research are discussed.
Accepted: 17 February 2016
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References
1.
S. Aubry. Physica D 103, 201 (1997).
2.
S. Flach, C. R. Willis. Phys. Rep. 295, 181 (1998).
3.
S. Aubry. Physica D 216, 1 (2006).
4.
S. Flach, A. Gorbach. Phys. Rep. 467, 1 (2008).
5.
V. P. Sakhnenko, G. M. Chechin. Dokl. Akad. Nauk 330, 308 (1993); V. P. Sakhnenko, G. M. Chechin. Phys. Dokl. 38, 219 (1993).
6.
V. P. Sakhnenko, G. M. Chechin. Dokl. Akad. Nauk 338, 42 (1994); V. P. Sakhnenko, G. M. Chechin. Phys. Dokl. 39, 625 (1994).
7.
G. M. Chechin, V. P. Sakhnenko. Physica D 117, 43 (1998).
8.
R. M. Rosenberg. J. Appl. Mech. 29, 7 (1962).
9.
G. M. Chechin, N. V. Novikova, A. A. Abramenko. Physica D 166, 208 (2002).
10.
G. M. Chechin, D. S. Ryabov, K. G. Zhukov. Physica D 203, 121 (2005).
11.
G. M. Chechin, V. P. Sakhnenko, H. T. Stokes, A. D. Smith, D. M. Hatch. Int. J. Non-Linear Mech. 35, 497 (2000).
12.
G. M. Chechin, A. V. Gnezdilov, M. Yu. Zekhtser. Int. J. Non-Linear Mech. 38, 1451 (2003).
13.
G. M. Chechin, K. G. Zhukov. Phys Rev E 73, 36216 (2006).
14.
G. M. Chechin, S. A. Scsherbinin. Commun. in Nonlinear Sci. and Numer. Simulat. 22, 244 (2015).
15.
W. Kohn. Rev. Mod. Phys. 71, 1253 (1999).
16.
G. M. Chechin, S. V. Dmitriev, I. P. Lobzenko, D. S. Ryabov. Phys. Rev. B 90, 045432 (2014).
17.
G. M. Chechin, I. P. Lobzenko. Letters on Materials 4, 226 (2014).
18.
I. P. Lobzenko, G. M. Chechin, G. S. Bezuglova, Y. A. Baimova, E. A. Korznikova, S. V. Dmitriev. Phys. Solid State 58, 616 (2016). (In Russian).
19.
G. M. Chechin, D. Ryabov, and S. Shcherbinin. Phys. Rev. E 92, 012907 (2015).
20.
A. Lyapunov, Ann. Fac. Sci., Toulouse 9, 203 (1907).
21.
A. F. Vakakis, L. I. Manevich, Yu. V. Mikhlin, V. N. Pilipchuk, A. A. Zevin. Normal modes and localization in nonlinear systems (New York: Wiley, 1996).
22.
T. Bountis, G. M. Chechin, V. P. Sakhnenko. Int. J. Bif. Chaos 21, 1539 (2011).
23.
G. M. Chechin, V. A. Koptsik. Comput. Math. Appl. 16, 521 (1988).
24.
V. P. Sakhnenko, G. M. Chechin. Comput. Math. Appl. 16, 453 (1988).
25.
G. M. Chechin, G. S. Bezuglova. J. Sound Vibr. 322, 490 (2009).
26.
G. S. Bezuglova, G. M. Chechin, P. P. Goncharov. Phys. Rev. E. 84, 036606 (2011).
27.
G. M. Chechin, D. S. Ryabov. Phys. Rev. E 69, 036202 – 1 (2004).
28.
G. M. Chechin. Comput. Math. Appl. 17 (1-3), 255 (1989).
29.
G. M. Chechin, T. I. Ivanova, V. P. Sakhnenko. Phys. Stat. Sol. (b) 152, 431 (1989).
30.
G. M. Chechin, E. A. Ipatova, V. P. Sakhnenko. Acta Cryst. A49, 824 (1993).
31.
L. D. Landau, E. M. Lifshitz. Course of Theoretical Physics, Volume 5, 3rd Edition (Butterworth-Heinemann, Oxford, 1980).
32.
O. V. Kovalev. Representations of the Crystallographic Space Groups: Irreducible Representations, Induced Representations and Corepresentations (Gordon and Breach, Amsterdam, 1993).
33.
P. L. Hagelstein, D. Letts, D. Cravens. J. Condensed Matter Nucl. Sci. 3, 59 (2010).
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