Summary
The discovery of the integer quantum Hall effect in the early eighties of the last century, with highly precise quantization values for the Hall conductance in multiples of e2/h, has been the first fascinating manifestation of the topological state of matter driven by magnetic field and disorder, and related to the formation of non-dissipative current flow. Throughout the 2000’s, several new phenomena such as the spin Hall effect and the quantum spin Hall effect were confirmed experimentally for systems with strong spin-orbit coupling effects and in the absence of external magnetic field. More recently, the Zeeman spin Hall effect and the formation of valley Hall topological currents have been introduced for graphene-based systems, under time-reversal or inversion symmetry-breaking conditions, respectively. This review presents a comprehensive coverage of all these Hall effects in disordered graphene from the perspective of numerical simulations of quantum transport in two-dimensional bulk systems (by means of the Kubo formalism) and multiterminal nanostructures (by means of the Landauer-Büttiker scattering and non-equilibrium Green’s function approaches). In contrast to usual two-dimensional electron gases in semiconductor heterostructures, the presence of defects in graphene generates more complex electronic features such as electron-hole asymmetry, defect-induced resonances in the electron density of states or percolation effect between localized impurity states, which, together with extra degrees of freedom (sublattice pseudospin and valley isospin), bring a higher degree of complexity and enlarge the transport phase diagram.
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References
Hall E. H., Am. J. Math., 2 (1879) 287.
von Klitzing K., Dorda G. and Pepper M., Phys. Rev. Lett., 45 (1980) 494.
von Klitzing K., Rev. Mod. Phys., 58 (1986) 519.
Tsui D. C., Stormer H. L. and Gossard A. C., Phys. Rev. Lett., 48 (1982) 1559.
Laughlin R. B., Phys. Rev. Lett., 50 (1983) 1395.
Jain K. J., Composite Fermions (Cambridge University Press) 2007.
Dyakonov M. I. and Perel V. I., JETP Lett., 13 (1971) 467.
Dyakonov M. I. and Perel V. I., Phys. Lett. A, 35 (1971) 459.
Sinova J., Valenzuela S. O., Wunderlich J., Back C. H. and Jungwirth T., Rev. Mod. Phys., 87 (2015) 1213.
Vignale G., J. Supercond. Nov. Magn., 23 (2010) 3.
Sinitsyn N. A., J. Phys.: Condens. Matter, 20 (2008) 023201.
Nagaosa N., Sinova J., Onoda S., MacDonald A. H. and Ong N. P., Rev. Mod. Phys., 82 (2010) 1539.
Kato Y. K., Myers R. C., Gossard A. C. and Awschalom D. D., Science, 306 (2004) 1910.
Wunderlich J., Kaestner B., Sinova J. and Jungwirth T., Phys. Rev. Lett., 94 (2005) 047204.
Saitoh E., Ueda M., Miyajima H. and Tatara G., Appl. Phys. Lett., 88 (2006) 182509.
Valenzuela S. O. and Tinkham M., Nature, 442 (2006) 176.
Zhao H., Loren E. J., van Driel H. M. and Smirl A. L., Phys. Rev. Lett., 96 (2006) 246601.
Kane C. L. and Mele E. J., Phys. Rev. Lett., 95 (2005) 226801.
Kane C. L. and Mele E. J., Phys. Rev. Lett., 95 (2005) 146802.
Gmitra M., Konschuh S., Ertler C., Ambrosch-Draxl C. and Fabian J., Phys. Rev. B, 80 (2009) 235431.
König M., Wiedmann S., Brüne C, Roth A., Buhmann H., Molenkamp L. W., Qi X. L. and Zhang S. C., Science, 318 (2007) 766.
Knez I., Du R. R. and Sullivan G., Phys. Rev. Lett., 107 (2011) 136603.
Bernevig B. A., Hughes T. L. and Zhang S.-C., Science, 314 (2006) 1757.
Hasan M. Z. and Kane C. L., Rev. Mod. Phys., 82 (2010) 3045.
Qi X.-L. and Zhang S.-C., Rev. Mod. Phys., 83 (2011) 1057.
Weeks C., Hu J., Alicea J., Franz M. and Wu R., Phys. Rev. X, 1 (2011) 021001.
Jiang H., Qiao Z., Liu H., Shi J. and Niu Q., Phys. Rev. Lett., 109 (2012) 116803.
Autés G. and Yazyev O. V., Phys. Rev. B, 87 (2013) 241404.
Jin K.-H. and Jhi S.-H., Phys. Rev. B, 87 (2013) 075442.
Liu W., Peng X., Wei X., Yang H., Stocks G. M. and Zhong J., Phys. Rev. B, 87 (2013) 205315.
Kou L., Yan B., Hu F., Wu S.-C., Wehling T. O., Felser C., Chen C. and Frauenheim T., Nano Lett., 13 (2013) 6251.
Hu J., Alicea J., Wu R. and Franz M., Phys. Rev. Lett., 109 (2012) 266801.
Li Y., Tang P., Chen P., Wu J., Gu B.-L., Fang Y., Zhang S. B. and Duan W., Phys. Rev. B, 87 (2013) 245127.
Lu H.-Z., Shan W.-Y., Yao W., Niu Q. and Shen S.-Q., Phys. Rev. B, 81 (2010) 115407.
Yao W., Yang S. A. and Niu Q., Phys. Rev. Lett., 102 (2009) 096801.
Xu X., Yao W., Xiao D. and Heinz T. F., Nat. Phys., 10 (2014) 343.
Xiao D., Liu G.-B., Feng W., Xu X. and Yao W., Phys. Rev. Lett., 108 (2012) 196802.
Feng W., Yao Y., Zhu W., Zhou J., Yao W. and Xiao D., Phys. Rev. B, 86 (2012) 165108.
Mak K. F., McGill K. L., Park J. and McEuen P. L., Science, 344 (2014) 1489.
Berry M. V., Proc. R. Soc. Lond. A, 392 (1984) 45.
Dolcetto G., Sassetti M. and Schmidt T. L., Riv. Nuovo Cimento, 39 (2016) 113.
Aoki H., Rep. Prog. Phys., 50 (1987) 655.
Thouless D. J., Kohmoto M., Nightingale M. P. and den Nijs M., Phys. Rev. Lett., 49 (1982) 405.
Kubo R., J. Phys. Soc. Jpn., 12 (1957) 570.
Kohmoto M., Ann. Phys., 160 (1985) 343.
Niu Q., Thouless D. J. and Wu Y.-S., Phys. Rev. B, 31 (1985) 3372.
Avron J. E., Osadchy D. and Seiler R., Phys. Today, 56 (2003) 38.
Castro Neto A. H., Guinea F., Peres N. M. R., Novoselov K. S. and Geim A. K., Rev. Mod. Phys., 81 (2009) 109.
Goerbig M. O., Rev. Mod. Phys., 83 (2011) 1193.
Das Sarma S., Adam S., Hwang E. H. and Rossi E., Rev. Mod. Phys., 83 (2011) 407.
Avouris P., Nano Lett., 10 (2010) 4285.
Giannazzo F. and Raineri V., Riv. Nuovo Cimento, 35 (2012) 267.
Wehling T. O., Black-Schaffer A. M. and Balatsky A. V., Adv. Phys., 63 (2014) 1.
Foa Torres L. E. F., Roche S. and Charlier J., Introduction to graphene-based nanomaterials: from electronic structure to quantum transport (Cambridge University Press) 2014.
Bychkov Y. A. and Rasbha E. I., JETP Lett., 39 (1984) 78.
Datta S. and Das B., Appl. Phys. Lett., 56 (1990) 665.
Sinova J., Culcer D., Niu Q., Sinitsyn N. A., Jungwirth T. and MacDonald A. H., Phys. Rev. Lett., 92 (2004) 126603.
Dyakonov M. I. and Perel V. I., Sov. Phys. Solid State, 13 (1972) 3023.
Elliott R. J., Phys. Rev., 96 (1954) 266.
Manchon A., Koo H. C., Nitta J., Frolov S. and Duine R., Nat. Mater., 14 (2015) 871.
Ertler C., Konschuh S., Gmitra M. and Fabian J., Phys. Rev. B, 80 (2009) 041405(R).
Huertas-Hernando D., Guinea F. and Brataas A., Phys. Rev. Lett., 103 (2009) 146801.
Zhou Y. and Wu M. W., Phys. Rev. B, 82 (2010) 085304.
Ochoa H. and Roldán R., Phys. Rev. B, 87 (2013) 245421.
Wang L. and Wu M. W., Phys. Rev. B, 89 (2014) 115302.
Gmitra M. and Fabian J., Phys. Rev. B, 92 (2015) 155403.
Tombros N., Jozsa C., Popinciuc M., Jonkman H. and van Wees B., Nature, 448 (2007) 571.
Roche S. and Valenzuela S. O., J. Phys. D: Appl. Phys., 47 (2014) 094011.
Žutić I., Fabian J. and Das Sarma S., Rev. Mod. Phys., 76 (2004) 323.
Fabian J., Matos-Abiague A., Ertler C., Stano P. and Zutic I., Acta Phys. Slovaca, 57 (2007) 565.
Ochoa H., Castro Neto A. H. and Guinea F., Phys. Rev. Lett., 108 (2012) 206808.
Kochan D., Gmitra M. and Fabian J., Phys. Rev. Lett., 112 (2014) 116602.
Soriano D., Tuan D. V., Dubois S. M.-M., Gmitra M., Cummings A. W., Kochan D., Ortmann F., Charlier J.-C., Fabian J. and Roche S., 2D Mater., 2 (2015) 022002.
Gmitra M., Kochan D. and Fabian J., Phys. Rev. Lett., 110 (2013) 246602.
Van Tuan D., Ortmann F., Soriano D., Valenzuela S. and Roche S., Nat. Phys., 10 (2014) 857.
Van Tuan D., Ortmann F., Cummings A., Soriano D. and Roche S., Sci. Rep., 6 (2016) 21046.
Cummings A. W. and Roche S., Phys. Rev. Lett., 116 (2016) 086602.
Rashba E. I., Phys. Rev. B, 79 (2009) 161409.
Guimarães M. H. D., Zomer P. J., Ingla-Aynés J., Brant J. C., Tombros N. and van Wees B. J., Phys. Rev. Lett., 113 (2014) 086602.
Dery H., Wu H., Ciftcioglu B., Huang M., Yang S., Kawakami R., Shi J., Krivorotov I., Zutic I. and Sham L., IEEE Trans. Electron Dev., 59 (2012) 259262.
Kawakami R. K., 2D Mater., 2 (2015) 034001.
Roche S., Åkerman J., Beschoten B., Charlier J.-C., Chshiev M., Dash S. P., Dlubak B., Fabian J., Fert A., Guimarães M., Guinea F., Grigorieva I., Schönenberger C, Seneor P., Stampfer C, Valenzuela S. O., Waintal X. and van Wees B., 2D Mater., 2 (2015) 030202.
Van Tuan D. and Roche S., Phys. Rev. Lett., 116 (2016) 106601.
McClure J. W., Phys. Rev., 104 (1956) 666.
Novoselov K. S., Geim A. K., Morozov S. V., Jiang D., Katsnelson M. I., Grigorieva I. V., Dubonos S. V. and Firsov A. A., Nature, 438 (2005) 197.
Zhang Y., Tan Y.-W., Stormer H. L. and Kim P., Nature, 438 (2005) 201.
Novoselov K. S., Jiang Z., Zhang Y., Morozov S. V., Stormer H. L., Zeitler U., Maan J. C., Boebinger G. S., Kim P. and Geim A. K., Science, 315 (2007) 1379.
Poumirol J.-M., Cresti A., Roche S., Escoffier W., Goiran M., Wang X., Li X., Dai H. and Raquet B., Phys. Rev. B, 82 (2010) 041413.
Aoki H. and Ando T., Solid State Commun., 38 (1981) 1079.
Peters E. C., Giesbers A. J. M., Burghard M. and Kern K., Appl. Phys. Lett.104 (2014) 203109.
Guillemette J., Sabri S. S., Wu B., Bennaceur K., Gaskell P. E., Savard M., Lévesque P. L., Mahvash F., Guermoune A., Siaj M., Martel R., Szkopek T. and Gervais G., Phys. Rev. Lett., 110 (2013) 176801.
Leconte N., Ortmann F., Cresti A., Charlier J.-C. and Roche S., 2D Mater., 1 (2014) 021001.
Leconte N., Ortmann F., Cresti A. and Roche S., Phys. Rev. B, 93 (2016) 115404.
García J. H., Covaci L. and Rappoport T. G., Phys. Rev. Lett., 114 (2015) 116602.
Garcia J. H. and Rappoport T. G., 2D Mater., 3 (2016) 024007.
Ortmann F. and Roche S., Phys. Rev. Lett., 110 (2013) 086602.
Ortmann F., Leconte N. and Roche S., Phys. Rev. B, 91 (2015) 165117.
Roche S., Phys. Rev. B, 59 (1999) 2284.
Ribeiro-Palau R., Lafont F., Brun-Picard J., Kazazis D., Michon A., Cheynis F., Couturaud O., Consejo C., Jouault B., Poirier W. and Schopfer F., Nat. Nanotechnol., 10 (2015) 1.
Lafont F., Ribeiro-Palau R., Han Z., Cresti A., Delvallée A., Cummings A. W., Roche S., Bouchiat V., Ducourtieux S., Schopfer F. and Poirier W., Phys. Rev. B, 90 (2014) 115422.
Van Tuan D., Kotakoski J., Louvet T., Ortmann F., Meyer J. C. and Roche S., Nano Lett., 13 (2013) 1730.
Nam Y., Sun J., Lindvall N., Jae Yang S., Kireev D., Rae Park C., Woo Park Y. and Yurgens A., Appl. Phys. Lett., 103 (2013) 233110.
Cummings A. W., Cresti A. and Roche S., Phys. Rev. B, 90 (2014) 161401.
Sheng D. N., Sheng L. and Weng Z. Y., Phys. Rev. B, 73 (2006) 233406.
Bolotin K. I., Sikes K. J., Hone J., Stormer H. L. and Kim P., Phys. Rev. Lett.101 (2008) 096802.
Gattenlöhner S., Hannes W.-R., Ostrovsky P. M., Gornyi I. V., Mirlin A. D. and Titov M., Phys. Rev. Lett., 112 (2014) 026802.
Evers F. and Mirlin A. D., Rev. Mod. Phys., 80 (2008) 1355.
Bergvall A., Carlsson J. M. and Löfwander T., Phys. Rev. B, 91 (2015) 245425.
Löfwander T., San-Jose P. and Prada E., Phys. Rev. B, 87 (2013) 205429.
Yager T., Lartsev A., Mahashabde S., Charpentier S., Davidovikj D., Danilov A., Yakimova R., Panchal V., Kazakova O., Tzalenchuk A., Lara-Avila S. and Kubatkin S., Nano Lett., 13 (2013) 4217.
Nam Y., Sun J., Lindvall N., Jae Yang S., Rae Park C., Woo Park Y. and Yurgens A., Appl. Phys. Lett., 104 (2014) 021902.
Pereira A. L. C. and Schulz P. A., Phys. Rev. B, 78 (2008) 125402.
Qiao Z., Yang S. A., Feng W., Tse W.-K., Ding J., Yao Y., Wang J. and Niu Q., Phys. Rev. B, 82 (2010) 161414.
Coraux J., Marty L., Bendiab N. and Bouchiat V., Acc. Chem. Res., 46 (2013) 2193.
Jia Z., Yan B., Niu J., Han Q., Zhu R., Yu D. and Wu X., Phys. Rev. B, 91 (2015) 085411.
Sutter E., Albrecht P., Wang B., Bocquet M.-L., Wu L., Zhu Y. and Sutter P., Surf. Sci., 605 (2011) 1676.
McCreary K. M., Pi K. and Kawakami R. K., Appl. Phys. Lett., 98 (2011) 192101.
Qiao Z., Tse W.-K., Jiang H., Yao Y. and Niu Q., Phys. Rev. Lett., 107 (2011) 256801.
Cresti A., Van Tuan D., Soriano D., Cummings A. W. and Roche S., Phys. Rev. Lett., 113 (2014) 246603.
Pi K., McCreary K. M., Bao W., Han W., Chiang Y. F., Li Y., Tsai S.-W., Lau C. N. and Kawakami R. K., Phys. Rev. B, 80 (2009) 075406.
Santos J. E., Peres N. M. R., Lopes dos Santos J. A. M. B. and Castro Neto A. H., Phys. Rev. B, 84 (2011) 085430.
McCreary K. M., Pi K., Swartz A. G., Han W., Bao W., Lau C. N., Guinea F., Katsnelson M. I. and Kawakami R. K., Phys. Rev. B, 81 (2010) 115453.
Alemani M., Barfuss A., Geng B., Girit C., Reisenauer P., Crommie M. F., Wang F., Zettl A. and Hellman F., Phys. Rev. B, 86 (2012) 075433.
Eelbo T., Waśniowska M., Thakur P., Gyamfi M., Sachs B., Wehling T. O., Forti S., Starke U., Tieg C., Lichtenstein A. I. and Wiesendanger R., Phys. Rev. Lett., 110 (2013) 136804.
Katsnelson M. I., Guinea F. and Geim A. K., Phys. Rev. B, 79 (2009) 195426.
Yuan S., Roldán R., De Raedt H. and Katsnelson M. I., Phys. Rev. B, 84 (2011) 195418.
Yuan S., Roldán R., Katsnelson M. I. and Guinea F., Phys. Rev. B, 90 (2014) 041402.
Maekawa S., Valenzuela S. O., Saitoh E. and Kimura T., Spin Current (Oxford University Press, Oxford) 2011.
Balakrishnan J., Koon G. K. W., Avsar A., Ho Y., Lee J. H., Jaiswal M., Baeck S.-J., Ahn J.-H., Ferreira A., Cazalilla M. A., Castro Neto A. H. and Özyilmaz B., Nat. Commun., 5 (2014) 4748.
Ferreira A., Rappoport T. G., Cazalilla M. A. and Castro Neto A. H., Phys. Rev. Lett., 112 (2014) 066601.
Hankiewicz E. M., Li J., Jungwirth T., Niu Q., Shen S.-Q. and Sinova J., Phys. Rev. B, 72 (2005) 155305.
Nitta J., Akazaki T., Takayanagi H. and Enoki T., Phys. Rev. Lett., 78 (1997) 1335.
Hirsch J. E., Phys. Rev. Lett., 83 (1999) 1834.
Zhang S., Phys. Rev. Lett., 85 (2000) 393.
Hoffmann A., IEEE Trans. Magn., 49 (2013) 5172.
Ehlert M., Song C., Ciorga M., Hupfauer T., Shiogai J., Utz M., Schuh D., Bougeard D. and Weiss D., Phys. Status Solidi B, 251 (2014) 1725.
Ando K., Takahashi S., Ieda J., Kajiwara Y., Nakayama H., Yoshino T., Harii K., Fujikawa Y., Matsuo M., Maekawa S. and Saitoh E., J. Appl. Phys., 109 (2011) 103913.
Jungwirth T., Wunderlich J. and Olejník K., Nat. Mater., 11 (2012) 382.
Nikolić B. K., Zârbo L. P. and Souma S., Phys. Rev. B, 73 (2006) 075303.
Shi J., Zhang P., Xiao D. and Niu Q., Phys. Rev. Lett., 96 (2006) 076604.
Sugimoto N., Onoda S., Murakami S. and Nagaosa N., Phys. Rev. B, 73 (2006) 113305.
Wang L., Wesselink R. J. H., Liu Y., Yuan Z., Xia K. and Kelly P. J., Phys. Rev. Lett., 116 (2016) 196602.
Mellnik A. R., Lee J. S., Richardella A., Grab J. L., Mintun P. J., Fischer M. H., Vaezi A., Manchon A., Kim E.-A., Samarth N. and Ralph D. C., Nature, 511 (2014) 449.
Balakrishnan J., Gavin K. W., Jaiswal M., Neto A. H. C. and Özyilmaz B., Nat. Phys., 9 (2013) 284.
Avsar A., Lee J. H., Koon G. K. W. and Özyilmaz B., 2D Mater., 2 (2015) 044009.
Asmar M. M. and Ulloa S. E., Phys. Rev. B, 91 (2015) 165407.
Wang Y., Cai X., Reutt-Robey J. and Fuhrer M. S., Phys. Rev. B, 92 (2015) 161411.
Kaverzin A. A. and van Wees B. J., Phys. Rev. B, 91 (2015) 165412.
Mott N. F., Proc. R. Soc. Lond. A, 124 (1929) 425.
Nikolić B. K., Zârbo L. P. and Welack S., Phys. Rev. B, 72 (2005) 075335.
Adagideli I. and Bauer G. E. W., Phys. Rev. Lett., 95 (2005) 256602.
Nikolić B. K., Zârbo L. P. and Souma S., Phys. Rev. B, 72 (2005) 075361.
Nikolić B. K. and Dragomirova R. L., Semicond. Sci. Technol., 24 (2009) 064006.
Sheng L., Sheng D. N. and Ting C. S., Phys. Rev. Lett., 94 (2005) 016602.
Nikolić B. K. and Souma S., Phys. Rev. B, 71 (2005) 195328.
Nikolić B. K., Souma S., Zârbo L. P. and Sinova J., Phys. Rev. Lett., 95 (2005) 046601.
Schiró M. and Fabrizio M., Phys. Rev. B, 79 (2009) 153302.
Gradhand M., Fedorov D. V., Zahn P. and Mertig I., Phys. Rev. Lett., 104 (2010) 186403.
Gorbachev R. V., Song J. C. W., Yu G. L., Kretinin A. V., Withers F., Cao Y., Mishchenko A., Grigorieva I. V., Novoselov K. S., Levitov L. S. and Geim A. K., Science, 346 (2014) 448.
Song J. C. W., Samutpraphoot P. and Levitov L. S., Proc. Natl. Acad. Sci. U.S.A., 112 (2015) 10879.
Ju L., Shi Z., Nair N., Lv Y., Jin C., Velasco Jr. J., Ojeda-Aristizabal C., Bechtel H. A., Martin M. C., Zettl A., Analytis J. and Wang F., Nature, 520 (2015) 650.
Kirczenow G., Phys. Rev. B, 92 (2015) 125425.
Ando T., J. Phys. Soc. Jpn., 84 (2015) 114705.
Abanin D. A., Shytov A. V., Levitov L. S. and Halperin B. I., Phys. Rev. B, 79 (2009) 035304.
Chen C.-L., Chang C.-R. and Nikolić B. K., Phys. Rev. B, 85 (2012) 155414.
Streda P., J. Phys. C: Solid State Phys., 15 (1982) L1299.
Sinitsyn N. A., Hill J. E., Min H., Sinova J. and MacDonald A. H., Phys. Rev. Lett., 97 (2006) 106804.
Ado I. A., Dmitriev I. A., Ostrovsky P. M. and Titov M., EPL, 111 (2015) 37004.
Milletarì M. and Ferreira A., Phys. Rev. B, 94 (2016) 201402(R).
Milletarì M. and Ferreira A., Phys. Rev. B, 94 (2016) 134202.
Van Tuan D., Marmolejo-Tejada J. M., Waintal X., Nikolić B. K., Valenzuela S. O. and Roche S., Phys. Rev. Lett., 117 (2016) 176602.
Irmer S., Frank T., Putz S., Gmitra M., Kochan D. and Fabian J., Phys. Rev. B, 91 (2015) 115141.
Zollner K., Frank T., Irmer S., Gmitra M., Kochan D. and Fabian J., Phys. Rev. B, 93 (2016) 045423.
Gmitra M., Kochan D., Högl P. and Fabian J., Phys. Rev. B, 93 (2016) 155104.
Kochan D., Irmer S. and Fabian J., arXiv:1610.08794 (2016).
Chang P.-H., Bahramy M. S., Nagaosa N. and Nikolić B. K., Nano Lett., 14 (2014) 3779.
Liu Z., Zhu M. and Zheng Y., Phys. Rev. B, 92 (2015) 245438.
Shevtsov O., Carmier P., Groth C., Waintal X. and Carpentier D., Phys. Rev. B, 85 (2012) 245441.
Sheng L., Sheng D. N., Ting C. S. and Haldane F. D. M., Phys. Rev. Lett., 95 (2005) 136602.
Dyrdał A., Dugaev V. K. and Barnaś J., Phys. Rev. B, 80 (2009) 155444.
Ferreira A., private communication.
Groth C. W., Wimmer M., Akhmerov A. R. and Waintal X., New J. Phys., 16 (2014) 063065.
Cresti A., Nemec N., Biel B., Niebler G., Triozon F., Cuniberti G. and Roche S., Nano Res., 1 (2008) 361.
Tworzydło J., Trauzettel B., Titov M., Rycerz A. and Beenakker C. W. J., Phys. Rev. Lett., 96 (2006) 246802.
Cresti A., Grosso G. and Pastori Parravicini G., Phys. Rev. B, 76 (2007) 205433.
Mihajlović G., Pearson J. E., Garcia M. A., Bader S. D. and Hoffmann A., Phys. Rev. Lett., 103 (2009) 166601.
Titov M., EPL, 79 (2007) 17004.
Abanin D. A., Morozov S. V., Ponomarenko L. A., Gorbachev R. V., Mayorov A. S., Katsnelson M. I., Watanabe K., Taniguchi T., Novoselov K. S., Levitov L. S. and Geim A. K., Science, 332 (2011) 328.
Renard J., Studer M. and Folk J. A., Phys. Rev. Lett., 112 (2014) 116601.
Wei P., Lee S., Lemaitre F., Pinel L., Cutaia D., Cha W., Katmis F., Zhu Y., Heiman D., Hone J., Moodera J. S. and Chen C.-T., Nat. Mater., 15 (2016) 711.
Abanin D. A., Gorbachev R. V., Novoselov K. S., Geim A. K. and Levitov L. S., Phys. Rev. Lett., 107 (2011) 096601.
McEuen P. L., Szafer A., Richter C. A., Alphenaar B. W., Jain J. K., Stone A. D., Wheeler R. G. and Sacks R. N., Phys. Rev. Lett., 64 (1990) 2062.
Haug R. J., Semicond. Sci. Technol., 8 (1993) 131.
Roth A., Brüne C., Buhmann H., Molenkamp L. W., Maciejko J., Qi X.-L. and Zhang S.-C., Science, 325 (2009) 294.
Li G., Luican-Mayer A., Abanin D., Levitov L. and Andrei E. Y., Nat. Commun., 4 (2013) 1744.
Winkler R., Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, Berlin) 2003.
Castro Neto A. H. and Guinea F., Phys. Rev. Lett., 103 (2009) 026804.
Brüne C., Roth A., Novik E. G., König M., Buhmann H., Hankiewicz E. M., Hanke W., Sinova J. and Molenkamp L. W., Nat. Phys., 6 (2010) 448.
Adam S., Brouwer P. W. and Das Sarma S., Phys. Rev. B, 79 (2009) 201404.
Kłos J. W. and Zozoulenko I. V., Phys. Rev. B, 82 (2010) 081414.
Stefanucci G. and van Leeuwen R., Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction (Cambridge University Press, Cambridge) 2013.
Golizadeh-Mojarad R. and Datta S., Phys. Rev. B, 75 (2007) 081301.
Adagideli I., Bauer G. E. W. and Halperin B. I., Phys. Rev. Lett., 97 (2006) 256601.
Yazyev O. V. and Katsnelson M. I., Phys. Rev. Lett., 100 (2008) 047209.
Kunstmann J., Özdoğan C., Quandt A. and Fehske H., Phys. Rev. B, 83 (2011) 045414.
Frederiksen T., Paulsson M., Brandbyge M. and Jauho A.-P., Phys. Rev. B, 75 (2007) 205413.
Mahfouzi F. and Nikolić B. K., Phys. Rev. B, 90 (2014) 045115.
Cresti A. and Pastori Parravicini G., Phys. Rev. B, 78 (2008) 115313.
Pryadko L. P. and Auerbach A., Phys. Rev. Lett., 82 (1999) 1253.
Sun Q.-f. and Xie X. C., Phys. Rev. Lett., 104 (2010) 066805.
Richter C. A., Wheeler R. G. and Sacks R. N., Surf. Sci., 305 (1994) 145.
Cheng J., Yao W., Xu X., Steel D. G., Bracker A. S., Gammon D. and Sham L. J., Phys. Rev. B, 77 (2008) 115315.
Liu R.-B., Yao W. and Sham L., Adv. Phys., 59 (2010) 703.
Lee J., Mak K. F. and Shan J., Nat. Nanotechnol., 11 (2016) 421.
Rycerz A., Tworzydło J. and Beenakker C. W. J., Nat. Phys., 3 (2007) 172.
Xiao D., Yao W. and Niu Q., Phys. Rev. Lett., 99 (2007) 1.
Zhang Z. Z., Chang K. and Chan K. S., Appl. Phys. Lett., 93 (2008) 062106.
Wakabayashi K. and Aoki T., Int. J. Mod. Phys. B, 16 (2002) 4897.
Cresti A., Grosso G. and Pastori Parravicini G., Phys. Rev. B, 77 (2008) 233402.
Akhmerov A. R., Bardarson J. H., Rycerz A. and Beenakker C. W. J., Phys. Rev. B, 77 (2008) 10.
Nakabayashi J., Yamamoto D. and Kurihara S., Phys. Rev. Lett., 102 (2009) 2.
Gunlycke D. and White C. T., Phys. Rev. Lett., 106 (2011) 1.
Chen J.-H., Autés G., Alem N., Gargiulo F., Gautam A., Linck M., Kisielowski C., Yazyev O. V., Louie S. G. and Zettl A., Phys. Rev. B, 89 (2014) 121407.
Fujita T., Jalil M. B. A. and Tan S. G., Appl. Phys. Lett., 97 (2010) 15.
Zhai F., Zhao X., Chang K. and Xu H. Q., Phys. Rev. B, 82 (2010) 1.
Low T. and Guinea F., Nano Lett., 10 (2010) 3551.
Zhai F., Ma Y. and Zhang Y.-T., J. Phys.: Condens. Matter, 23 (2011) 385302.
Song Y., Zhai F. and Guo Y., Appl. Phys. Lett., 103 (2013) 183111.
Myoung N. and Ihm G., Curr. Appl. Phys., 14 (2014) 1455.
Garcia-Pomar J. L., Cortijo A. and Nieto-Vesperinas M., Phys. Rev. Lett., 100 (2008) 236801.
Pereira Jr. J. M., Peeters F. M., Costa Filho R. N. and Farias G. A., J. Phys.: Condens. Matter, 21 (2008) 045301.
Hsieh S. H. and Chu C. S., Appl. Phys. Lett., 108 (2016) 033113.
Abergel D. S. L. and Chakraborty T., Appl. Phys. Lett., 95 (2009) 062107.
da Costa D. R., Chaves A., Sena S. H. R., Farias G. A. and Peeters F. M., Phys. Rev. B, 92 (2015) 045417.
Péterfalvi C. G., Oroszlány L., Lambert C. J. and Cserti J., New J. Phys., 14 (2012) 063028.
Pratley L. and Zülicke U., Appl. Phys. Lett., 104 (2014) 082401.
Xiao D., Chang M. C. and Niu Q., Rev. Mod. Phys., 82 (2010) 1959.
Lensky Y. D., Song J. C. W., Samutpraphoot P. and Levitov L. S., Phys. Rev. Lett., 114 (2015) 256601.
Zawadzki W., Am. J. Phys., 73 (2005) 756.
Jackson J. D., Classical Electrodynamics (Wiley, New York) 1998.
Fisher G. P., Am. J. Phys., 39 (1971) 1528.
Engel H.-A., Halperin B. I. and Rashba E. I., Phys. Rev. Lett., 95 (2005) 166605.
Nikolić B. K. and Zârbo L. P., EPL, 77 (2007) 47004.
Boettger J. C. and Trickey S. B., Phys. Rev. B, 75 (2007) 121402.
Boykin T. B., Luisier M., Klimeck G., Jiang X., Kharche N., Zhou Y. and Nayak S. K., J. Appl. Phys., 109 (2011) 104304.
Konschuh S., Gmitra M. and Fabian J., Phys. Rev. B, 82 (2010) 245412.
Huertas-Hernando D., Guinea F. and Brataas A., Phys. Rev. B, 74 (2006) 155426.
Jeong J.-S., Shin J. and Lee H.-W., Phys. Rev. B, 84 (2011) 195457.
Qiao Z., Ren W., Chen H., Bellaiche L., Zhang Z., MacDonald A. H. and Niu Q., Phys. Rev. Lett., 112 (2014) 116404.
Han W., Kawakami R. K., Gmitra M. and Fabian J., Nat. Nanotechnol., 9 (2014) 794.
Mahfouzi F., Nikolić B. K., Chen S.-H. and Chang C.-R., Phys. Rev. B, 82 (2010) 195440.
Xu Y., Gan Z. and Zhang S.-C., Phys. Rev. Lett., 112 (2014) 226801.
Yang H. X., Hallal A., Terrade D., Waintal X., Roche S. and Chshiev M., Phys. Rev. Lett., 110 (2013) 046603.
O’Farrell E. C. T., Tan J. Y., Yeo Y., Koon G. K. W., Ozyilmaz B., Watanabe K. and Taniguchi T., Phys. Rev. Lett., 117 (2016) 076603.
Marchenko D., Varykhalov A., Scholz M. R., Bihlmayer G., Rashba E. I., Rybkin A., Shikin A. M. and Rader O., Nat. Commun., 3 (2012) 1232.
Wang Z., Ki D. K., Chen H., Berger H., MacDonald A. H. and Morpurgo A. F., Nat. Commun., 6 (2015) 8339.
Avsar A., Tan J. Y., Taychatanapat T., Balakrishnan J., Koon G. K. W., Yeo Y., Lahiri J., Carvalho A., Rodin A. S., O’Farrell E. C. T., Eda G., Castro Neto A. H. and Özyilmaz B., Nat. Commun., 5 (2014) 4875.
Büttiker M., Phys. Rev. Lett., 57 (1986) 1761.
Datta S., Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge) 1995.
Scheid M., Bercioux D. and Richter K., New J. Phys., 9 (2007) 401.
Pareek T. P., Phys. Rev. Lett., 92 (2004) 076601.
Meir Y. and Wingreen N. S., Phys. Rev. Lett., 68 (1992) 2512.
Kazymyrenko K. and Waintal X., Phys. Rev. B, 77 (2008) 115119.
Thygesen K. S. and Rubio A., Phys. Rev. B, 77 (2008) 115333.
Profumo R. E. V., Groth C., Messio L., Parcollet O. and Waintal X., Phys. Rev. B, 91 (2015) 245154.
Areshkin D. A. and Nikolić B. K., Phys. Rev. B, 81 (2010) 155450.
Sanvito S., Chapter 7 electron transport theory for large systems, in Computational Nanoscience (The Royal Society of Chemistry) 2011, pp. 179–224.
Golub H. and van Van Loan C. F., Matrix Computations (Johns Hopkins University Press) 1996.
Kilgour M. and Segal D., J. Chem. Phys., 144 (2016) 124107.
Büttiker M., Phys. Rev. B, 32 (1985) 1846.
Metalidis G. and Bruno P., Phys. Rev. B, 73 (2006) 113308.
Cresti A., Farchioni R., Grosso G. and Pastori Parravicini G., Phys. Rev. B, 68 (2003) 075306.
Cresti A., Grosso G. and Pastori Parravicini G., Eur. Phys. J. B, 53 (2006) 537.
Mahan G. D., Many-Particle Physics, 2nd edition (Plenum, New York, N.Y.) 1993.
Bastin A., Lewiner C., Betbeder-Matibet O. and Nozieres P., J. Phys. Chem. Solids, 32 (1971) 1811.
Silver R. and Röder H., Int. J. Mod. Phys. C, 05 (1994) 735.
Weisse A., Wellein G., Alvermann A. and Fehske H., Rev. Mod. Phys., 78 (2006) 275.
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Cresti, A., Nikolić, B.K., García, J.H. et al. Charge, spin and valley Hall effects in disordered graphene. Riv. Nuovo Cim. 39, 587–667 (2016). https://doi.org/10.1393/ncr/i2016-10130-6
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DOI: https://doi.org/10.1393/ncr/i2016-10130-6