Splitting of the Raman 2D band of graphene subjected to strain

Physical Review B - Condensed Matter and Materials Physics

Volumn 82, Issue 20, 2010, Pages

  Mohr, M ^a   Maultzsch, J ^a   Thomsen, C ^a  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 78649742712     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.82.201409     Document Type: Article

References (32)

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 10.1126/science.1102896
2. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature (London) 438, 197 (2005). 10.1038/nature04233
3. Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature (London) 438, 201 (2005). 10.1038/nature04235
4. K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, Science 315, 1379 (2007). 10.1126/science.1137201
5. A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, and A. K. Geim, Phys. Rev. Lett. 97, 187401 (2006). 10.1103/PhysRevLett.97.187401
6. M. M. Lucchese, F. Stavale, E. H. M. Ferreira, C. Vilani, M. V. O. Moutinho, R. B. Capaz, C. A. Achete, and A. Jorio, Carbon 48, 1592 (2010). 10.1016/j.carbon.2009.12.057
7. Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, ACS Nano 2, 2301 (2008). 10.1021/nn800459e
8. T. M. G. Mohiuddin, A. Lombardo, R. R. Nair, A. Bonetti, G. Savini, R. Jalil, N. Bonini, D. M. Basko, C. Galiotis, N. Marzari, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, Phys. Rev. B 79, 205433 (2009). 10.1103/PhysRevB.79. 205433
9. M. Huang, H. Yan, C. Chen, D. Song, T. F. Heinz, and J. Hone, Proc. Natl. Acad. Sci. U.S.A. 106, 7304 (2009). 10.1073/pnas.0811754106
10. J. Yan, Y. Zhang, P. Kim, and A. Pinczuk, Phys. Rev. Lett. 98, 166802 (2007). 10.1103/PhysRevLett.98.166802
11. G. Tsoukleri, J. Parthenios, K. Papagelis, R. Jalil, A. C. Ferrari, A. K. Geim, K. S. Novoselov, and C. Galiotis, Small 5, 2397 (2009). 10.1002/smll.200900802
12. M. Huang, H. Yan, T. F. Heinz, and J. Hone, Nano Lett. 10, 4074 (2010). 10.1021/nl102123c
13. O. Frank (private communication).
14. C. Thomsen and S. Reich, Phys. Rev. Lett. 85, 5214 (2000). 10.1103/PhysRevLett.85.5214
15. J. Maultzsch, S. Reich, and C. Thomsen, Phys. Rev. B 70, 155403 (2004). 10.1103/PhysRevB.70.155403
16. M. Cardona, in Light Scattering in Solids II, Topics in Applied Physics Vol. 50, edited by, M. Cardona, and, G. Güntherodt, (Springer, Berlin, 1982), p. 19.
17. P. R. Wallace, Phys. Rev. 71, 622 (1947). 10.1103/PhysRev.71.622
18. A. Grüneis, J. Serrano, A. Bosak, M. Lazzeri, S. L. Molodtsov, L. Wirtz, C. Attaccalite, M. Krisch, A. Rubio, F. Mauri, and T. Pichler, Phys. Rev. B 80, 085423 (2009). 10.1103/PhysRevB.80.085423
19. Wolfram Research, Inc., Mathematica Edition: Version 6.0 (Wolfram Research, Champaign, Illinois, 2007).
20. A. Grüneis, R. Saito, G. G. Samsonidze, T. Kimura, M. A. Pimenta, A. Jorio, A. G. Souza Filho, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 67, 165402 (2003). 10.1103/PhysRevB.67.165402
21. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, J. Phys.: Condens. Matter 21, 395502 (2009). 10.1088/0953-8984/21/39/395502
22. Calculations were performed with the code QUANTUM-ESPRESSO (Ref.). We used a plane-wave basis set, RRKJ pseudopotentials (Ref.) and the generalized gradient approximation in the Perdew, Burke, and Ernzerhof parameterization for the exchange-correlation functional (Ref.). A Methfessel-Paxton broadening with a width of 0.02 Ry was used (Ref.). The valence electrons were expanded in a plane-wave basis with an energy cutoff of 60 Ry. A 42 × 42 × 1 sampling grid was used for the integration over the Brillouin zone. The dynamical matrices were calculated on a 12 × 12 × 1 q grid using the implemented linear-response theory. Force constants were obtained via a Fourier transformation and interpolated to obtain phonons at arbitrary points in the Brillouin zone.
23. -1 ε R, where R is the rotational matrix. After applying a finite amount of strain we relax the coordinates of the basis atoms until forces are below 0.001 Ry/a.u. and minimize the total energy with respect to the Poisson's ratio ν. For small strain values we obtain a Poisson's ratio ν = 0.164, in excellent agreement with experimental tension measurements on pyrolytic graphite that yield a value of ν = 0.163 (Ref.).
24. M. Mohr, K. Papagelis, J. Maultzsch, and C. Thomsen, Phys. Rev. B 80, 205410 (2009). 10.1103/PhysRevB.80.205410
25. C. Thomsen, S. Reich, and P. Ordejón, Phys. Rev. B 65, 073403 (2002). 10.1103/PhysRevB.65.073403
26. J. Kürti, V. Zolyomi, A. Grüneis, and H. Kuzmany, Phys. Rev. B 65, 165433 (2002). 10.1103/PhysRevB.65.165433
27. R. Narula and S. Reich, Phys. Rev. B 78, 165422 (2008). 10.1103/PhysRevB.78.165422
28. P. Giannozzi, http://www.quantum-espresso.org
29. A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos, Phys. Rev. B 41, 1227 (1990). 10.1103/PhysRevB.41.1227
30. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 10.1103/PhysRevLett.77.3865
31. M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989). 10.1103/PhysRevB.40.3616
32. O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and T. Weng, J. Appl. Phys. 41, 3373 (1970). 10.1063/1.1659428