Robustness of edge states in graphene quantum dots

Physical Review B - Condensed Matter and Materials Physics

Volumn 82, Issue 4, 2010, Pages

  Wimmer, M ^a   Akhmerov, A R ^a   Guinea, F ^b  

^a UNIVERSITEIT LEIDEN   (Netherlands)

^b INSTITUTO DE CIENCIA DE MATERIALES DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77956693420     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.82.045409     Document Type: Article

References (82)

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, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Proc. Natl. Acad. Sci. U.S.A. 102, 10451 (2005). 10.1073/pnas.0502848102
3. A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 (2007). 10.1038/nmat1849
4. P. Avouris, Z. Chen, and V. Perebeinos, Nat. Nanotechnol. 2, 605 (2007). 10.1038/nnano.2007.300
5. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 10.1103/RevModPhys.81.109
6. D. J. Klein, Chem. Phys. Lett. 217, 261 (1994). 10.1016/0009-2614(93) E1378-T
7. M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jpn. 65, 1920 (1996). 10.1143/JPSJ.65.1920
8. K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996). 10.1103/PhysRevB.54.17954
9. A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B 77, 085423 (2008). 10.1103/PhysRevB.77.085423
10. E. V. Castro, N. M. R. Peres, J. M. B. Lopes dos Santos, A. H. Castro Neto, and F. Guinea, Phys. Rev. Lett. 100, 026802 (2008). 10.1103/PhysRevLett. 100.026802
11. B. Sahu, H. Min, A. H. MacDonald, and S. K. Banerjee, Phys. Rev. B 78, 045404 (2008). 10.1103/PhysRevB.78.045404
12. E. V. Castro, N. M. R. Peres, and J. M. B. Lopes dos Santos, EPL 84, 17001 (2008). 10.1209/0295-5075/84/17001
13. Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, and H. Fukuyama, Appl. Surf. Sci. 241, 43 (2005). 10.1016/j.apsusc.2004.09.091
14. Y. Kobayashi, K.-I. Fukui, T. Enoki, K. Kusakabe, and Y. Kaburagi, Phys. Rev. B 71, 193406 (2005). 10.1103/PhysRevB.71.193406
15. Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, and H. Fukuyama, Phys. Rev. B 73, 085421 (2006). 10.1103/PhysRevB.73.085421
16. K. Nakada, M. Igami, and M. Fujita, J. Phys. Soc. Jpn. 67, 2388 (1998). 10.1143/JPSJ.67.2388
17. Y.-W. Son, M. L. Cohen, and S. G. Louie, Nature (London) 444, 347 (2006). 10.1038/nature05180
18. J. Fernández-Rossier and J. J. Palacios, Phys. Rev. Lett. 99, 177204 (2007). 10.1103/PhysRevLett.99.177204
19. M. Ezawa, Phys. Rev. B 76, 245415 (2007). 10.1103/PhysRevB.76.245415
20. O. V. Yazyev and M. I. Katsnelson, Phys. Rev. Lett. 100, 047209 (2008). 10.1103/PhysRevLett.100.047209
21. M. Wimmer, I. Adagideli, S. Berber, D. Tománek, and K. Richter, Phys. Rev. Lett. 100, 177207 (2008). 10.1103/PhysRevLett.100.177207
22. B. Wunsch, T. Stauber, and F. Guinea, Phys. Rev. B 77, 035316 (2008). 10.1103/PhysRevB.77.035316
23. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, Phys. Rev. Lett. 101, 036803 (2008). 10.1103/PhysRevLett.101.036803
24. I. Romanovsky, C. Yannouleas, and U. Landman, Phys. Rev. B 79, 075311 (2009). 10.1103/PhysRevB.79.075311
25. L. A. Ponomarenko, F. Schedin, M. I. Katsnelson, R. Yang, E. W. Hill, K. S. Novoselov, and A. K. Geim, Science 320, 356 (2008). 10.1126/science.1154663
26. J. Güttinger, C. Stampfer, S. Hellmüller, F. Molitor, T. Ihn, and K. Ensslin, Appl. Phys. Lett. 93, 212102 (2008). 10.1063/1.3036419
27. C. Stampfer, J. Güttinger, F. Molitor, D. Graf, T. Ihn, and K. Ensslin, Appl. Phys. Lett. 92, 012102 (2008). 10.1063/1.2827188
28. J. Güttinger, C. Stampfer, F. Libisch, T. Frey, J. Burgdörfer, T. Ihn, and K. Ensslin, Phys. Rev. Lett. 103, 046810 (2009). 10.1103/PhysRevLett.103.046810
29. S. Schnez, F. Molitor, C. Stampfer, J. Guettinger, I. Shorubalko, T. Ihn, and K. Ensslin, Appl. Phys. Lett. 94, 012107 (2009). 10.1063/1.3064128
30. J. Güttinger, T. Frey, C. Stampfer, T. Ihn, and K. Ensslin, arXiv:1002.3771 (unpublished).
31. N. Shima and H. Aoki, Phys. Rev. Lett. 71, 4389 (1993). 10.1103/PhysRevLett.71.4389
32. T. G. Pedersen, C. Flindt, J. Pedersen, N. A. Mortensen, A.-P. Jauho, and K. Pedersen, Phys. Rev. Lett. 100, 136804 (2008). 10.1103/PhysRevLett.100. 136804
33. M. Vanević, V. M. Stojanović, and M. Kindermann, Phys. Rev. B 80, 045410 (2009). 10.1103/PhysRevB.80.045410
34. J. A. Fürst, J. G. Pedersen, C. Flindt, N. A. Mortensen, M. Brandbyge, T. G. Pedersen, and A.-P. Jauho, New J. Phys. 11, 095020 (2009). 10.1088/1367-2630/11/9/095020
35. T. Shen, Y. Q. Wu, M. A. Capano, L. P. Rokhinson, L. W. Engel, and P. D. Ye, Appl. Phys. Lett. 93, 122102 (2008). 10.1063/1.2988725
36. J. Eroms and D. Weiss, New J. Phys. 11, 095021 (2009). 10.1088/1367-2630/11/9/095021
37. J. Bai, X. Zhong, S. Jiang, Y. Huang, and X. Duan, Nat. Nanotechnol. 5, 190 (2010). 10.1038/nnano.2010.8
38. R. Balog, B. Jørgensen, L. Nilsson, M. Andersen, E. Rienks, M. Bianchi, M. Fanetti, E. Lægsgaard, A. Baraldi, S. Lizzit, Z. Sljivancanin, F. Besenbacher, B. Hammer, T. G. Pedersen, P. Hofmann, and L. Hornekær, Nature Mater. 9, 315 (2010). 10.1038/nmat2710
39. L. G. Cançado, M. A. Pimenta, B. R. A. Neves, M. S. S. Dantas, and A. Jorio, Phys. Rev. Lett. 93, 247401 (2004). 10.1103/PhysRevLett.93.247401
40. A. L. Vázquez de Parga, F. Calleja, B. Borca, M. C. G. Passeggi, Jr., J. J. Hinarejos, F. Guinea, and R. Miranda, Phys. Rev. Lett. 100, 056807 (2008). 10.1103/PhysRevLett.100.056807
41. X. Jia, M. Hofmann, V. Meunier, B. G. Sumpter, J. Campos-Delgado, J.-M. Romo-Herrera, H. Son, Y.-P. Hsieh, A. Reina, J. Kong, M. Terrones, and M. S. Dresselhaus, Science 323, 1701 (2009). 10.1126/science.1166862
42. C. Ö. Girit, J. C. Meyer, R. Erni, M. D. Rossell, C. Kisielowski, L. Yang, C.-H. C.-H. Park, M. F. Crommie, M. L. Cohen, S. G. Louie, and A. Zettl, Science 323, 1705 (2009). 10.1126/science.1166999
43. Z. Liu, K. Suenaga, P. J. F. Harris, and S. Iijima, Phys. Rev. Lett. 102, 015501 (2009). 10.1103/PhysRevLett.102.015501
44. C. Casiraghi, A. Hartschuh, H. Qian, S. Piscanec, C. Georgi, A. Fasoli, K. S. Novoselov, D. M. Basko, and A. C. Ferrari, Nano Lett. 9, 1433 (2009). 10.1021/nl8032697
45. Y. J. Xu and J. Q. Li, Chem. Phys. Lett. 400, 406 (2004). 10.1016/j.cplett.2004.11.010
46. D. E. Jiang, B. G. Sumpter, and S. Dai, J. Phys. Chem. B 110, 23628 (2006). 10.1021/jp065980+
47. F. Cervantes-Sodi, G. Csányi, S. Piscanec, and A. C. Ferrari, Phys. Rev. B 77, 165427 (2008). 10.1103/PhysRevB.77.165427
48. B. Huang, F. Liu, J. Wu, B.-L. Gu, and W. Duan, Phys. Rev. B 77, 153411 (2008). 10.1103/PhysRevB.77.153411
49. Close to a boundary, the edge states occupy a single sublattice only. In order to avoid the oscillatory pattern on the lattice scale that inevitably arises when plotting both sublattices simultaneously, for every unit cell we only plot the atom with the largest occupation probability.
50. S. Schnez, K. Ensslin, M. Sigrist, and T. Ihn, Phys. Rev. B 78, 195427 (2008). 10.1103/PhysRevB.78.195427
51. E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989). 10.1103/PhysRevLett.62.1201
52. M. Ezawa, Physica E 42, 703 (2010). 10.1016/j.physe.2009.11.046
53. A. R. Akhmerov, J. H. Bardarson, A. Rycerz, and C. W. J. Beenakker, Phys. Rev. B 77, 205416 (2008). 10.1103/PhysRevB.77.205416
54. N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 73, 125411 (2006). 10.1103/PhysRevB.73.125411
55. K. Sasaki, S. Murakami, and R. Saito, Appl. Phys. Lett. 88, 113110 (2006). 10.1063/1.2181274
56. K.-i. Sasaki, Y. Shimomura, Y. Takane, and K. Wakabayashi, Phys. Rev. Lett. 102, 146806 (2009). 10.1103/PhysRevLett.102.146806
57. The dot has the shape of a deformed disk, with a radius R (θ) depending on the angle of direction θ. The numerical calculations presented in the text use R (θ) = R + 0.2 R sin (θ) + 0.05 R sin (2 θ) - 0.025 R sin (3 θ) + 0.02 R sin (4 θ) - 0.01 R sin (5 θ).
58. R. Peierls, Z. Phys. 80, 763 (1933). 10.1007/BF01342591
59. E. R. Mucciolo, A. H. Castro Neto, and C. H. Lewenkopf, Phys. Rev. B 79, 075407 (2009). 10.1103/PhysRevB.79.075407
60. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999).
61. N. E. Gibbs, W. G. Poole,Jr., and P. K. Stockmeyer, SIAM J. Numer. Anal. 13, 236 (1976). 10.1137/0713023
62. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1998).
63. For the solution of the sparse linear system arising in the shift-and-invert problem, we apply the MUMPS package: P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L'Excellent, SIAM J. Matrix Anal. Appl. 23, 15 (2001). 10.1137/S0895479899358194
64. Ken-ichi Sasaki, Kentaro Sato, Riichiro Saito, Jie Jiang, Seiichiro Onari, and Yukio Tanaka, Phys. Rev. B 75, 235430 (2007). 10.1103/PhysRevB.75. 235430
65. D. A. Bahamon, A. L. C. Pereira, and P. A. Schulz, Phys. Rev. B 79, 125414 (2009). 10.1103/PhysRevB.79.125414
66. S. C. Kim, P. S. Park, and S.-R. Eric Yang, Phys. Rev. B 81, 085432 (2010). 10.1103/PhysRevB.81.085432
67. R. J. Bell and P. Dean, Discuss. Faraday Soc. 50, 55 (1970). 10.1039/df9705000055
68. R. J. Bell, Rep. Prog. Phys. 35, 1315 (1972). 10.1088/0034-4885/35/3/306
69. F. Libisch, S. Rotter, J. Güttinger, C. Stampfer, and J. Burgdörfer, Phys. Rev. B 81, 245411 (2010). 10.1103/PhysRevB.81.245411
70. J. Wurm, A. Rycerz, I. Adagideli, M. Wimmer, K. Richter, and H. U. Baranger, Phys. Rev. Lett. 102, 056806 (2009). 10.1103/PhysRevLett.102.056806
71. F. Libisch, C. Stampfer, and J. Burgdörfer, Phys. Rev. B 79, 115423 (2009). 10.1103/PhysRevB.79.115423
72. M. L. Mehta, Random Matrices (Elsevier/Academic Press, Amsterdam, 2004).
73. The result does not change qualitatively if the threshold participation ratio is changed. In particular, the classification into the different types of level statistics does not depend on this threshold.
74. L. Brey and H. A. Fertig, Phys. Rev. B 73, 235411 (2006). 10.1103/PhysRevB.73.235411
75. In Ref., the chiral symmetry was broken by a mass term.
76. K. Harigaya, Chem. Phys. Lett. 340, 123 (2001). 10.1016/S0009-2614(01) 00430-4
77. K. Harigaya and T. Enoki, Chem. Phys. Lett. 351, 128 (2002). 10.1016/S0009-2614(01)01338-0
78. L. Brey, H. A. Fertig, and S. Das Sarma, Phys. Rev. Lett. 99, 116802 (2007). 10.1103/PhysRevLett.99.116802
79. J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A. Yacoby, Nat. Phys. 4, 144 (2008). 10.1038/nphys781
80. E. H. Lee, K. Balasubramanian, R. T. Weitz, M. Burghard, and K. Kern, Nat. Nanotechnol. 3, 486 (2008). 10.1038/nnano.2008.172
81. S. Heydrich, M. Hirmer, C. Preis, T. Korn, J. Eroms, D. Weiss, and C. Schüller, arXiv:1006.2067 (unpublished).
82. H. D. Raedt and M. I. Katsnelson, JETP Lett. 88, 607 (2009). 10.1134/S0021364008210133