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We have investigated the quantum magnetic oscillations of graphene subjected to the spin-orbit interaction(SOI) in the presence of crossed uniform electric and magnetic fields and scattered from impurities at finite temperatures. Landau levels are shown to be modified in an unexpected fashion by the spin-orbit interaction, the electrostatic potential and magnetic confinement; this is strikingly different from the non-relativistic 2D electron gas. Furthermore, we derive the analytical expressions of the thermodynamic quantities subject to the SOI, such as density of states, thermodynamic potential, magnetization, and magnetic susceptibility etc. At finite temperatures, the magnetization and magnetic susceptibility can both be predicted to oscillate periodically as a function of reciprocal field 1/B and shown to be modulated through the SOI and the dimensionless parameter ( = E/ F B). As approaches unity, the values of magnetization and magnetic susceptibility finally move to infinity, indicating a transformation of closed orbits into open trajectories, thereby, leading to the vanishing of magnetic oscillations. And, the magnetic susceptibility depends largely on the external fields, suggesting that graphene should be a non-linear magnetic medium. Besides, the associative effect of impurity scattering and temperature may make the standard 2D electron gas be deemed as the consequence of the relativistic type spectrum of low energy electrons and holes in graphene. Also, we comment on a possibility of using magnetic oscillations for detecting a gap that may open in the spectrum of quasiparticle excitations due to the SOI.
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Keywords:
- spin-orbit interaction /
- magnetic oscillations /
- graphene /
- Landau level
[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[3] Zhang Y B, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201
[4] Sharapov S G, Gusynin V P, Beck H 2003 Phys. Rev. B 67 144509
[5] Gusynin V P, Sharapov S G 2005 Phys. Rev. Lett. 95 146801
[6] Gusynin V P, Sharapov S G 2005 Phys. Rev. B 71 125124
[7] Gusynin V P, Sharapov S G 2006 Phys. Rev. B 73 245411
[8] Lukose V, Shankar R, Baskaran G 2007 Phys. Rev. Lett. 98 116802
[9] Gu N, Rudner M, Young A, Kim P, Levitov L 2011 Phys. Rev. Lett. 106 066601
[10] Zhang S L, Ma N, Zhang E H 2010 J. Phys. Condens. Matter 22 115302
[11] Ma N, Zhang S L, Liu D Q, Zhang E H 2011 Phys. Lett. A 375 3624
[12] Reis M S, Soriano S 2013 Appl. Phys. Lett. 102 112903
[13] Reis M S 2013 Solid State Commun. 161 19
[14] Alisultanov Z Z 2014 JETP Letters 99 232
[15] Alisultanov Z Z 2014 Physica B 438 41
[16] Alisultanov Z 2014 Phys. Letters A 378 2329
[17] Ji Q S, Hao H Y, Zhang C X, Wang R 2015 Acta Phys. Sin. 64 087302 (in Chinese) [季青山, 郝鸿雁, 张存熙, 王瑞 2015 物理学报 64 087302]
[18] Dresselhaus G, Dressehaus M S 1965 Phys. Rev. 140 A401
[19] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[20] Min H, Hill J E, Sinitsyn N A, Sahu B R, Kleinman L, MacDonald A H 2006 Phys. Rev. B 74 165310
[21] Huertas-Hernando D, Guinea F, Brataas A 2006 Phys. Rev. B 74 155426
[22] Gmitra M, Konschuh S, Ertler C, Ambrosch-Draxl C, Fabian J 2009 Phys. Rev. B 80 235431
[23] Yao Y, Ye F, Qi X L, Zhang S C, Fang Z 2007 Phys. Rev. B 75 041401(R)
[24] Varykhalov A, Sanchez-Barriga J, Shikin A M, Biswas C, Vescovo E, Rybkin A, Marchenko D, Rader O 2008 Phys. Rev. Lett. 101 157601
[25] Castro Neto A H, Guinea F 2009 Phys. Rev. Lett. 103 026804
[26] Dresselhaus G 1955 Phys. Rev. 100 580
[27] Yang Y E, Xiao Y, Yan X H, Dai C J 2015 Chin. Phys. B 24 117204
[28] Cahangirov S, Topsakal M, Aktrk E, Sahin H, Ciraci S 2009 Phys. Rev. Lett. 102 236804
[29] Fang Y M, Hang Z Q, Hsu C H, Li X D, Xu Y X, Zhou Y H, Wu Z S, Chuang F C, Zhu Z Z 2015 Scientific Reports 5 14196
[30] Landau L D, Diamagnetismus D M 1930 Z. Phys. 64 629
[31] Landau L D, Lifshitz E M 1971 Relativistic Quantum Theory (New York: Pergamon Press) p100
[32] Zutic I, Fabian J, Sarma S D 2004 Rev. Mod. Phys. 76 323
[33] Dresselhaus G, Dressehaus M S 1965 Phys. Rev. 140 A401
[34] Meng L, Wang Y L, Zhang L Z, Du S X, Gao H J 2015 Chin. Phys. B 24 086803
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[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197
[3] Zhang Y B, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201
[4] Sharapov S G, Gusynin V P, Beck H 2003 Phys. Rev. B 67 144509
[5] Gusynin V P, Sharapov S G 2005 Phys. Rev. Lett. 95 146801
[6] Gusynin V P, Sharapov S G 2005 Phys. Rev. B 71 125124
[7] Gusynin V P, Sharapov S G 2006 Phys. Rev. B 73 245411
[8] Lukose V, Shankar R, Baskaran G 2007 Phys. Rev. Lett. 98 116802
[9] Gu N, Rudner M, Young A, Kim P, Levitov L 2011 Phys. Rev. Lett. 106 066601
[10] Zhang S L, Ma N, Zhang E H 2010 J. Phys. Condens. Matter 22 115302
[11] Ma N, Zhang S L, Liu D Q, Zhang E H 2011 Phys. Lett. A 375 3624
[12] Reis M S, Soriano S 2013 Appl. Phys. Lett. 102 112903
[13] Reis M S 2013 Solid State Commun. 161 19
[14] Alisultanov Z Z 2014 JETP Letters 99 232
[15] Alisultanov Z Z 2014 Physica B 438 41
[16] Alisultanov Z 2014 Phys. Letters A 378 2329
[17] Ji Q S, Hao H Y, Zhang C X, Wang R 2015 Acta Phys. Sin. 64 087302 (in Chinese) [季青山, 郝鸿雁, 张存熙, 王瑞 2015 物理学报 64 087302]
[18] Dresselhaus G, Dressehaus M S 1965 Phys. Rev. 140 A401
[19] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802
[20] Min H, Hill J E, Sinitsyn N A, Sahu B R, Kleinman L, MacDonald A H 2006 Phys. Rev. B 74 165310
[21] Huertas-Hernando D, Guinea F, Brataas A 2006 Phys. Rev. B 74 155426
[22] Gmitra M, Konschuh S, Ertler C, Ambrosch-Draxl C, Fabian J 2009 Phys. Rev. B 80 235431
[23] Yao Y, Ye F, Qi X L, Zhang S C, Fang Z 2007 Phys. Rev. B 75 041401(R)
[24] Varykhalov A, Sanchez-Barriga J, Shikin A M, Biswas C, Vescovo E, Rybkin A, Marchenko D, Rader O 2008 Phys. Rev. Lett. 101 157601
[25] Castro Neto A H, Guinea F 2009 Phys. Rev. Lett. 103 026804
[26] Dresselhaus G 1955 Phys. Rev. 100 580
[27] Yang Y E, Xiao Y, Yan X H, Dai C J 2015 Chin. Phys. B 24 117204
[28] Cahangirov S, Topsakal M, Aktrk E, Sahin H, Ciraci S 2009 Phys. Rev. Lett. 102 236804
[29] Fang Y M, Hang Z Q, Hsu C H, Li X D, Xu Y X, Zhou Y H, Wu Z S, Chuang F C, Zhu Z Z 2015 Scientific Reports 5 14196
[30] Landau L D, Diamagnetismus D M 1930 Z. Phys. 64 629
[31] Landau L D, Lifshitz E M 1971 Relativistic Quantum Theory (New York: Pergamon Press) p100
[32] Zutic I, Fabian J, Sarma S D 2004 Rev. Mod. Phys. 76 323
[33] Dresselhaus G, Dressehaus M S 1965 Phys. Rev. 140 A401
[34] Meng L, Wang Y L, Zhang L Z, Du S X, Gao H J 2015 Chin. Phys. B 24 086803
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