Nonlinear vibrational modes in graphene: group-theoretical results

G. Chechin, D. Ryabov, S. Shcherbinin show affiliations and emails
Accepted  17 February 2016
Citation: G. Chechin, D. Ryabov, S. Shcherbinin. Nonlinear vibrational modes in graphene: group-theoretical results. Lett. Mater., 2016, 6(1) 9-15


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.

References (33)

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).

Cited by (25)

E. Barani, Elena A. Korznikova, Alexander P. Chetverikov, K. Zhou, Sergey V. Dmitriev. Physics Letters A. 381(41), 3553 (2017). Crossref
Elena A. Korznikova, Dmitry V. Bachurin, Sergey Yu. Fomin, Alexander P. Chetverikov, Sergey V. Dmitriev. Eur. Phys. J. B. 90(2) (2017). Crossref
E. Barani, Ivan P. Lobzenko, Elena A. Korznikova, Elvira G. Soboleva, Sergey V. Dmitriev, K. Zhou, A. Marjaneh. Eur. Phys. J. B. 90(3) (2017). Crossref
G.M. Chechin, D.A. Sizintsev, O.A. Usoltsev. Computational Materials Science. 138, 353 (2017). Crossref
Sergey V. Dmitriev, Julia A. Baimova, Elena A. Korznikova, Alexander P. Chetverikov. Understanding Complex Systems: Nonlinear Systems, Vol. 2, Chapter 7, p.175 (2018). Crossref
Ivan P. Lobzenko. J. Micromech. Mol. Phys. 04(02), 1950002 (2019). Crossref
G. Chechin, D. Ryabov, S. Shcherbinin. J. Micromech. Mol. Phys. 03(01n02), 1850002 (2018). Crossref
М. Старостенков, П. Захаров, Н. Медведев. Известия АлтГУ. , 49 (2019). Crossref
A. Semenov, R. Murzaev, Y. Bebikhov, A. Kudreyko, S. Dmitriev. Lett. Mater. 10(2), 185 (2020). Crossref
O.V. Bachurina, A.A. Kudreyko. Computational Materials Science. 182, 109737 (2020). Crossref
S. A. Shcherbinin, M. N. Semenova, A. S. Semenov, E. A. Korznikova, G. M. Chechin, S. V. Dmitriev. Phys. Solid State. 61(11), 2139 (2019). Crossref
Dina U. Abdullina, Maria N. Semenova, Aleksander S. Semenov, Elena A. Korznikova, Sergey V. Dmitriev. Eur. Phys. J. B. 92(11) (2019). Crossref
G. Chechin, D. Ryabov. Lett. Mater. 10(4), 523 (2020). Crossref
Denis S. Ryabov, George M. Chechin, A. Upadhyaya, Elena A. Korznikova, Vladimir I. Dubinko, Sergey V. Dmitriev. Nonlinear Dyn. 102(4), 2793 (2020). Crossref
Y. Watanabe, S. Izumi. J. Phys. Soc. Jpn. 90(1), 014003 (2021). Crossref
O. V. Bachurina, A. A. Kudreyko. Eur. Phys. J. B. 94(11) (2021). Crossref
I. Sunagatova, A. Subkhangulova, M. Semenova, D. Borisov, A. Semenov, S. Dmitriev. IOP Conf. Ser.: Mater. Sci. Eng. 1008(1), 012073 (2020). Crossref
O. V. Bachurina, R. T. Murzaev, A. A. Kudreyko, S. V. Dmitriev, D. V. Bachurin. Eur. Phys. J. B. 95(7) (2022). Crossref
R. T. Murzaev, A. S. Semenov, A. I. Potekaev, M. D. Starostenkov, P. V. Zakharov, V. V. Kulagina, S. V. Dmitriev. Russ Phys J. 64(2), 293 (2021). Crossref
S. A. Shcherbinin, A. M. Kazakov, Yu. V. Bebikhov, A. A. Kudreyko, S. V. Dmitriev. Phys. Rev. E. 109(1) (2024). Crossref
O. Bachurina, R. Murzaev, S. Shcherbinin, A. Kudreyko, S. Dmitriev, D. Bachurin. Modelling Simul. Mater. Sci. Eng. 31(7), 075009 (2023). Crossref
S. V. Dmitriev, E. A. Korznikova, D. I. Bokij, K. Zhou. Physica Status Solidi (b). 253(7), 1310 (2016). Crossref
I.V. Kosarev, S.A. Shcherbinin, A.A. Kistanov, R.I. Babicheva, E.A. Korznikova, S.V. Dmitriev. Computational Materials Science. 231, 112597 (2024). Crossref
S.A. Shcherbinin, Yu.V. Bebikhov, D.U. Abdullina, A.A. Kudreyko, S.V. Dmitriev. Communications in Nonlinear Science and Numerical Simulation. , 108033 (2024). Crossref
I.D. Kolesnikov, S.A. Shcherbinin, Yu.V. Bebikhov, E.A. Korznikova, I.A. Shepelev, A.A. Kudreyko, S.V. Dmitriev. Chaos, Solitons & Fractals. 178, 114339 (2024). Crossref

Similar papers