Gap structure of the Hofstadter system of interacting Dirac fermions in graphene

Physical Review Letters

Volumn 112, Issue 17, 2014, Pages

  Apalkov, Vadim M ^a,b   Chakraborty, Tapash ^a,b  

^a GEORGIA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF MANITOBA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

COULOMB INTERACTIONS; ENERGY GAP; SOLIDS; SPECTROSCOPY;

DIRAC FERMIONS; ENERGY SPECTRA; GAP STRUCTURES; HOFSTADTER ENERGY SPECTRUMS; INTERACTION EFFECT; LANDAU LEVELS; NONINTERACTING; PERIODIC POTENTIALS;

GRAPHENE;

EID: 84899741388     PISSN: 00319007     EISSN: 10797114     Source Type: Journal    
DOI: 10.1103/PhysRevLett.112.176401     Document Type: Article

References (40)

1. U. Rössler and M. Shurke, in Advances in Solid State Physics, edited by B. Kramer (Springer, Berlin, 2000), Vol. 40, pp. 35-50.
2. P.G. Harper, Proc. Phys. Soc. London 68, 874 (1955). PPSOAU 0370-1328 10.1088/0370-1298/68/10/304
3. D. Langbein, Phys. Rev. 180, 633 (1969). PHRVAO 0031-899X 10.1103/PhysRev.180.633
4. D. Hofstadter, Phys. Rev. B 14, 2239 (1976). PLRBAQ 0556-2805 10.1103/PhysRevB.14.2239
5. M.C. Geisler, J.H. Smet, V. Umansky, K. von Klitzing, B. Naundorf, R. Ketzmerick, and H. Schweizer, Phys. Rev. Lett. 92, 256801 (2004); PRLTAO 0031-9007 10.1103/PhysRevLett.92.256801
6. M.C. Geisler, J.H. Smet, V. Umansky, K. von Klitzing, B. Naundorf, R. Ketzmerick, and H. Schweizer Physica (Amsterdam) 25E, 227 (2004). PELNFM 1386-9477 10.1016/j.physe.2004.06.021
7. C. Albrecht, J.H. Smet, K. von Klitzing, D. Weiss, V. Umansky, and H. Schweizer, Phys. Rev. Lett. 86, 147 (2001); PRLTAO 0031-9007 10.1103/PhysRevLett.86.147
8. C. Albrecht, J.H. Smet, K. von Klitzing, D. Weiss, V. Umansky, and H. Schweizer Physica (Amsterdam) 20E, 143 (2003). PELNFM 1386-9477 10.1016/j.physe.2003.09.031
9. T. Schlösser, K. Ensslin, J.P. Kotthaus, and M. Holland, Europhys. Lett. 33, 683 (1996); EULEEJ 0295-5075 10.1209/epl/i1996-00399-6
10. T. Schlösser, K. Ensslin, J.P. Kotthaus, and M. Holland Semicond. Sci. Technol. 11, 1582 (1996). SSTEET 0268-1242 10.1088/0268-1242/11/11S/022
11. U. Kuhl and H.-J. Stöckmann, Phys. Rev. Lett. 80, 3232 (1998). PRLTAO 0031-9007 10.1103/PhysRevLett.80.3232
12. M. Aidelsburger, M. Atala, M. Lohse, J.T. Barreiro, B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013); PRLTAO 0031-9007 10.1103/PhysRevLett.111.185301
13. H. Miyake, G.A. Siviloglou, C.J. Kennedy, W.C. Burton, and W. Ketterle, Phys. Rev. Lett. 111, 185302 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111. 185302
14. D.S.L. Abergel and T. Chakraborty, Phys. Rev. Lett. 102, 056807 (2009). PRLTAO 0031-9007 10.1103/PhysRevLett.102.056807
15. Physics of Graphene, edited by H. Aoki and M.S. Dresselhaus (Springer, New York, 2014).
16. D.S.L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty, Adv. Phys. 59, 261 (2010). ADPHAH 0001-8732 10.1080/00018732.2010. 487978
17. C.R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K.L. Shepard, J. Hone, and P. Kim, Nature (London) 497, 598 (2013). NATUAS 0028-0836 10.1038/nature12186
18. B. Hunt, J.D. Sanchez-Yamagishi, A.F. Young, M. Yankowitz, B.J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R.C. Ashoori, Science 340, 1427 (2013). SCIEAS 0036-8075 10.1126/science.1237240
19. L.A. Pomomarenko, R.V. Gorbachev, G.L. Yu, D.C. Elias, R. Jalil, A.A. Patel, A. Mishchenko, A.S. Mayorov, C.R. Woods, J.R. Wallbank, M. Mucha-Kruczynski, B.A. Piot, M. Potemski, I.V. Grigorieva, K.S. Novoselov, F. Guinea, V.I. Falko and A.K. Geim, Nature (London) 497, 594 (2013). NATUAS 0028-0836 10.1038/nature12187
20. R. Decker, Y. Wang, V.W. Brar, W. Regan, H.-Z. Tsai, Q. Wu, W. Gannett, A. Zettl, and M.F. Crommie, Nano Lett. 11, 2291 (2011). NALEFD 1530-6984 10.1021/nl2005115
21. J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B.J. LeRoy, Nat. Mater. 10, 282 (2011). NMAACR 1476-1122 10.1038/nmat2968
22. A superlattice with a shorter period requires a larger magnetic field to fill up the unit cell. The larger magnetic field values make for larger Landau level separation in general. In graphene, the Landau level separation is larger for low filling fractions than in conventional semiconductor systems due to the linear dispersion relation. Further, the superlattice modulation potential between graphene and the hexagonal boron nitride is (Equation presented), which is quite substantial. All of these attributes make the butterfly spectrum more robust in graphene; P. Kim (private communication).
23. J.-W. Rhim and K. Park, Phys. Rev. B 86, 235411 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.86.235411
24. N. Nemec and G. Cuniberti, Phys. Rev. B 75, 201404 (2007); PRBMDO 1098-0121 10.1103/PhysRevB.75.201404
25. R. Bistritzer and A.H. MacDonald, Phys. Rev. B 84, 035440 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.84.035440
26. V. Gudmundsson and R.R. Gerhardts, Surf. Sci. 361, 505 (1996); SUSCAS 0039-6028 10.1016/0039-6028(96)00456-6
27. V. Gudmundsson and R.R. Gerhardts Phys. Rev. B 52, 16744 (1995); PRBMDO 0163-1829 10.1103/PhysRevB.52.16744
28. V. Gudmundsson and R.R. Gerhardts Phys. Rev. B 54, R5223 (1996). PRBMDO 0163-1829 10.1103/PhysRevB.54.R5223
29. A. Kol and N. Read, Phys. Rev. B 48, 8890 (1993). PRBMDO 0163-1829 10.1103/PhysRevB.48.8890
30. H. Doh and S.-H. Suck Salk, Phys. Rev. B 57, 1312 (1998). PRBMDO 0163-1829 10.1103/PhysRevB.57.1312
31. T. Chakraborty and V.M. Apalkov, Solid State Commun. 175, 123 (2013). SSCOA4 0038-1098 10.1016/j.ssc.2013.04.002
32. V.M. Apalkov and T. Chakraborty, Phys. Rev. Lett. 97, 126801 (2006). PRLTAO 0031-9007 10.1103/PhysRevLett.97.126801
33. V.M. Apalkov and T. Chakraborty, Phys. Rev. Lett. 105, 036801 (2010); PRLTAO 0031-9007 10.1103/PhysRevLett.105.036801
34. V.M. Apalkov and T. Chakraborty, Phys. Rev. Lett. 107, 186803 (2011). PRLTAO 0031-9007 10.1103/PhysRevLett.107.186803
35. The reason for (Equation presented) between the gaps is that the Coulomb interaction strength between the electrons in the (Equation presented) and (Equation presented) is less by (Equation presented) than that between the electrons in the (Equation presented) and (Equation presented). The interaction potential was evaluated for the wave vector (Equation presented).
36. See, for example, J.P. Eisenstein, H.L. Stormer, V. Narayanamurti, A.Y. Cho, A.C. Gossard, and C.W. Tu, Phys. Rev. Lett. 55, 875 (1985); PRLTAO 0031-9007 10.1103/PhysRevLett.55.875
37. I. Meinel, D. Grundler, S. Gargstädt-Franke, C. Heyn, and D. Heitmann, Appl. Phys. Lett. 70, 3305 (1997); APPLAB 0003-6951 10.1063/1.119145
38. I. Meinel, T. Hengstmann, D. Grundler, D. Heitmann, W. Wegscheider, and M. Bichler, Phys. Rev. Lett. 82, 819 (1999); PRLTAO 0031-9007 10.1103/PhysRevLett.82.819
39. I. Meinel, D. Grundler, D. Heitmann, A. Manolescu, V. Gudmundsson, W. Wegscheider, and M. Bichler, Phys. Rev. B 64, 121306 (2001); PRBMDO 0163-1829 10.1103/PhysRevB.64.121306
40. M. Zhu, A. Usher, A.J. Matthews, A. Potts, M. Elliott, W.G. Herrenden-Harker, D.A. Ritchie, and M.Y. Simmons, Phys. Rev. B 67, 155329 (2003). PRBMDO 0163-1829 10.1103/PhysRevB.67.155329