The valley degree of freedom of an electron

Sun Jia-Tao; Meng Sheng

doi:10.7498/aps.64.187301

Abstract

Under the periodic potential of solid, the movement of an electron obeys the Bloch theorem. In addition to the charge and real spin degree of freedom, Bloch electrons in solids are endowed with valley degree of freedom representing the local energy extrema of the Bloch energy bands. Here we will review the intriguing electronic properties of valley degree of freedom of solid materials ranging from conventional bulk semiconductors to two-dimensional atomic crystals such as graphene, silicene, and transition metal dichalcogenides. The attention is paid to how to break the valley degeneracy via different ways including strain, electric field, optic field, etc. Conventional semiconductors usually have multiple valley degeneracy, which have to be lifted by quantum confinement or magnetic field. This can alleviate the valley degeneracy problem, but lead to simultaneously more complex many-body problems due to the remnant valley interaction in the bulk semiconductor. Two-dimensional materials provide a viable way to cope with the valley degeneracy problem. The inequivalent valley points in it are in analogy with real spin as long as the inversion symmetry is broken. In the presence of electric field, the nonvanishing Berry curvature drives the anomalous transverse velocity, leading to valley Hall effect. The valley degree of freedom can be coupled with other degree of freedom, such as real spin, layer, etc, resulting in rich physics uncovered to date. The effective utilization of valley degree of freedom as information carrier can make novel optoelectronic devices, and cultivate next generation electronics–valleytronics.

Keywords:

Authors and contacts

1.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
2.
Collaborative Innovation Center for Quantum Matter, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

Corresponding author: Sun Jia-Tao, jtsun@iphy.ac.cn;smeng@iphy.ac.cn ; Meng Sheng, jtsun@iphy.ac.cn;smeng@iphy.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61306114, 11222431), and the National Basic Research Program of China (Grant Nos. 2013CBA01600, 2012CB921403), and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB07030100).

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Cited By

[1]	Tikhonenko F V, Horsell D W, Gorbachev R V, Savchenko A K 2008 Phys. Rev. Lett. 100 056802
[2]	Wu G Y, Lue N Y, Chen Y C 2013 Phys. Rev. B 88 125422
[3]	Zhu Z, Collaudin A, Fauqué B, Kang W, Behnia K 2012 Nat. Phys. 8 89
[4]	Gunlycke D, White C T 2011 Phys. Rev. Lett. 106 136806
[5]	Liu Y, Song J, Li Y, Liu Y, Sun Q F 2013 Phys. Rev. B 87 195445
[6]	Guinea F, Katsnelson M I, Geim A K 2009 Nat. Phys. 6 30
[7]	Jiang Y J, Low T, Chang K, Katsnelson M I, Guinea F 2013 Phys. Rev. Lett. 110 046601
[8]	Prada E, San-Jose P, Schomerus H 2009 Phys. Rev. B 80 245414
[9]	Low T, Jiang Y J, Katsnelson M I, Guinea F 2012 Nano Lett. 12 850
[10]	Wu Z, Zhai F, Peeters F M, Xu H Q, Chang K 2011 Phys. Rev. Lett. 106 176802
[11]	Ezawa M 2014 Phys. Rev. B 89 195413
[12]	Xiao D, Liu G, Feng W, Xu X, Yao W 2012 Phys. Rev. Lett. 108 196802
[13]	Mak K F, McGill K L, Park J, McEuen P L 2014 Science 344 1489
[14]	Gong Z, Liu G B, Yu H, Xiao D, Cui X, Xu X, Yao W 2013 Nat. Commun. 4 2053
[15]	Zhang L F, Niu Q 2014 arXiv:1502.02573
[16]	Loss D, DiVincenzo D P 1998 Phys. Rev. A 57 120
[17]	Friesen M, Rugheimer P, Savage D E, Lagally M G, van der Weide D W, Joynt R, Eriksson M A 2003 Phys. Rev. B 67 121301
[18]	Ando T, Fowler A B, Stern F 1982 Rev. Mod. Phys. 54 437
[19]	McWhorter A L 1957 Semiconductor Surface Physics (Philadelphia: University of Pennsylvania Press) p55
[20]	Ramos L E, Teles L K, Scolfaro L M R, Castineira J L P, Rosa A L, Leite J R 2001 Phys. Rev. B 63 165210
[21]	Dziekan T, Zahn P, Meded V, Mirbt S 2007 Phys. Rev. B 75 195213
[22]	Yu D C, Zhang Y, Liu F 2008 Phys. Rev. B 78 245204
[23]	Grosso G, Parravicini G P, Piermarocchi C 1996 Phys. Rev. B 54 16393
[24]	Koiller B, Hu X D, Das Sarma S 2001 Phys. Rev. Lett. 88 027903
[25]	Gunawan O, Shkolnikov Y P, Vakili K, Gokmen T, de Poortere E P, Shayegan M 2006 Phys. Rev. Lett. 97 186404
[26]	Fowler A B, Fang F F, Howard W E, Stiles P J 1966 Phys. Rev. Lett. 16 901
[27]	Khrapai V S, Shashkin A A, Dolgopolov V T 2003 Phys. Rev. B 67 113305
[28]	Shkolnikov Y P, de Poortere E P, Tutuc E, Shayegan M 2002 Phys. Rev. Lett. 89 226805
[29]	Takashina K, Ono Y, Fujiwara A, Takahashi Y, Hirayama Y 2006 Phys. Rev. Lett. 96 236801
[30]	Nebel C E 2013 Nat. Mater. 12 690
[31]	Löfas H, Grigoriev A, Isberg J, Ahuja R 2011 AIP Adv. 1 032139
[32]	Isberg J, Gabrysch M, Hammersberg J, Majdi S, Kovi K K, Twitchen D J 2013 Nat. Mater. 12 760
[33]	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
[34]	Novoselov K S, Jiang D, Schedin F, Booth T J, Khotkevich V V, Morozov S V, Geim A K 2005 PNAS 102 10451
[35]	Castro NetoA H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[36]	Fujita M, Wakabayashi K, Nakada K, Kusakabe K 1996 J. Phys. Soc. Jpn. 65 1920
[37]	Son Y W, Cohen M L, Louie S G 2006 Nature 444 347
[38]	Son Y W, Cohen M L, Louie S G 2006 Phys. Rev. Lett. 97 216803
[39]	Rycerz A, Tworzydlo J, Beenakker C W J 2007 Nat. Phys. 3 172
[40]	Akhmerov A R, Bardarson J H, Rycerz A, Beenakker C W J 2008 Phys. Rev. B 77 205416
[41]	Zhang Z Z, Chang K, Chan K S 2008 App. Phys. Lett. 93 062106
[42]	Gunlycke D, Vasudevan S, White C T 2013 Nano Lett. 13 259
[43]	Jiang L, L X, Zheng Y 2011 Phys. Lett. A 376 136
[44]	L X L, Liu Z, Yao H B, Jiang L W, Gao W Z, Zheng Y Z 2012 Phys. Rev. B 86 045410
[45]	L X, Jiang L, Zheng Y 2013 Phys. Lett. A 377 2687
[46]	Chen J H, Autes G, Alem N, Gargiulo F, Gautam A, Linck M, Kisielowski C, Yazyev O V, Louie S G, Zettl A 2014 Phys. Rev. B 89 121407
[47]	Martin I, Blanter Y M, Morpurgo A F 2008 Phys. Rev. Lett. 100 036804
[48]	Garcia-Pomar J L, Cortijo A, Nieto-Vesperinas M 2008 Phys. Rev. Lett. 100 236801
[49]	Wang Z, Liu F 2010 ACS Nano 4 2459
[50]	Pereira Jr J M, Peeters F M, Costa Filho R N, Farias G A J 2009 Phys. Condens. Matter 21 045301
[51]	Giovannetti G, Khomyakov P A, Brocks G, Kelly P J 2007 Phys. Rev. B 76 073103
[52]	Shinde P P, Kumar V 2011 Phys. Rev. B 84 125401
[53]	Zhou S Y, Gweon G H, Fedorov A V, First P N, de Heer W A, Lee D H, Guinea F, Castro Neto A H, Lanzara A 2007 Nat. Mater. 6 770
[54]	Brar V W, Zhang Y, Yayon Y, Ohta T, McChesney J L, Bostwick A, Rotenberg E, Horn K, Crommie M F 2007 Appl. Phys. Lett. 91 122102
[55]	Naumov I I, Bratkovsky A M 2011 Phys. Rev. B 84 245444
[56]	Pereira V M, Castro Neto A H, Peres N M R 2009 Phys. Rev. B 80 045401
[57]	Ni Z H, Yu T, Lu Y H, Wang Y Y, Feng Y P, Shen Z X 2008 ACS Nano 2 2301
[58]	Ni Z H, Yu T, Lu Y H, Wang Y Y, Feng Y P, Shen Z X 2009 ACS Nano 3 483
[59]	Choi S M, Jhi S H, Son Y W 2010 Phys. Rev. B 81 081407
[60]	Zhai F, Zhao X, Chang K, Xu H Q 2010 Phys. Rev. B 82 115442
[61]	Zhai F, Chang K 2012 Phys. Rev. B 85 155415
[62]	Zhai F, Ma Y, Chang K 2011 New J. Phys. 13 083029
[63]	Song Y, Zhai F, Guo Y 2013 Appl. Phys. Lett. 103 183111
[64]	Jiang Y, Low T, Chang K, Katsnelson M I, Guinea F 2013 Phys. Rev. Lett. 110 046601
[65]	Brouwer PW 1998 Phys. Rev. B 58 10135
[66]	Wang J, Chan K W, Lin Z 2014 Appl. Phys. Lett. 104 013105
[67]	Abergel D S L, Chakraborty T 2009 Appl. Phys. Lett. 95 062107
[68]	Golub L E, Tarasenko S A, Entin M V, Magarill L I 2011 Phys. Rev. B 84 195408
[69]	Linnik T L 2014 Phys. Rev. B 90 075406
[70]	Oka T, Aoki H 2009 Phys. Rev. B 79 081406
[71]	Xiao D, Yao Y, Fang Z, Niu Q 2006 Phys. Rev. Lett. 97 026603
[72]	Xiao D, Chang M, Niu Q 2010 Rev. Mod. Phys. 82 1959
[73]	Chang M, Niu Q 1996 Phys. Rev. B 53 7010
[74]	Xiao D, Yao W, Niu Q 2007 Phys. Rev. Lett. 99 236809
[75]	Yao Y, Ye F, Qi X L, Zhang S C, Fang Z Phys. Rev. B 75 041401
[76]	Yao W, Xiao D, Niu Q 2008 Phys. Rev. B 77 235406
[77]	Cao T, Wang G, Han W, Ye H, Zhu C, Shi J, Niu Q, Tan P, Wang E, Liu B, Feng J 2012 Nat. Commun. 3 887
[78]	Cahangirov S, Topsakal M, Akturk E, Sahin H, Ciraci S 2009 Phys. Rev. Lett. 102 236804
[79]	Ezawa M 2013 Phys. Rev. B 87 155415
[80]	Pan H, Li Z, Liu C C, Zhu G, Qiao Z, Yao Y 2014 Phys. Rev. Lett. 112 106802
[81]	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, Geim A K 2014 Science 346 448
[82]	Sui M, Chen G, Ma L, Shan W, Tian D, Watanabe K, Taniguchi T, Jin X, Yao W, Xiao D, Zhang Y 2014 arXiv: 1501.04685
[83]	Wang Q H, Kalantar-Zadeh K, Kis A, Coleman J N, Strano M S 2012 Nat. Nano 7 699
[84]	Mak K F, Lee C, Hone J, Shan J, Heinz T F 2010 Phys. Rev. Lett. 105 136805
[85]	Scholes G D, Rumbles G 2006 Nat. Mater. 5 683
[86]	Law M, Goldberger J, Yang P D 2004 Annu. Rev. Mater. Res. 34 83
[87]	Qiu D Y, da Jornada F H, Louie S G 2013 Phys. Rev. Lett. 111 216805
[88]	Ye Z, Cao T, O’Brien K, Zhu H, Yin X, Wang Y, Louie S G, Zhang X 2014 Nature 513 214
[89]	Shi H, Pan H, Zhang Y, Yakobson B I 2013 Phys. Rev. B 87 155304
[90]	Ramasubramaniam A 2012 Phys. Rev. B 86 115409
[91]	Shishkin M, Kresse G 2006 Phys. Rev. B 74 035101
[92]	Shishkin M, Kresse G 2007 Phys. Rev. B 75 235102
[93]	Zeng H, Dai J, Yao W, Xiao D, Cui X 2012 Nat. Nano 7 490
[94]	Mak K F, He K, Shan J, Heinz T F 2012 Nat. Nano 7 494
[95]	Zhu Z Y, Cheng Y C, Schwingenschlogl U 2011 Phys. Rev. B 84 153402
[96]	Xu X, Yao W, Xiao D, Heinz T F 2014 Nat. Phys. 10 343
[97]	MacNeill D, Heikes C, Mak K F, Anderson Z, Kormányos A, Zólyomi V, Park J, Ralph D C 2015 Phys. Rev. Lett. 114 037401
[98]	Aivazian G, Gong Z, Jones A M, Chu R L, Yan J, Mandrus D G, Zhang C, Cobden D, Yao W, Xu X 2015 Nat. Phys. 11 148
[99]	Srivastava A, Sidler M, Allain A V, Lembke D S, Kis A, Imamoğlu A 2015 Nat. Phys. 11 141
[100]	Urbaszek B, Marie X 2015 Nat. Phys. 11 94
[101]	Yu H, Wu Y, Liu G, Xu X, Yao W 2014 Phys. Rev. Lett. 113 156603
[102]	Zhang Y J, Oka T, Suzuki R, Ye J T 2014 Science 344 725
[103]	Ross J S, Klement P, Jones A M, Ghimire N J, Yan J, Mandrus D G, Taniguchi T, Watanabe K, Kitamura K, Yao W, Cobden D H, Xu X 2014 Nat. Nano 9 268
[104]	Wu S, Ross J S, Liu G, Aivazian G, Jones A, Fei Z, Zhu W, Xiao D, Yao W, Cobden D, Xu X 2013 Nat. Phys. 9 149
[105]	Jones A M, Yu H, Ross J S, Klement P, Ghimire N J, Yan J, Mandrus D G, Yao W, Xu X 2014 Nat. Phys. 10 130
[106]	Koshino M, Ando T 2010 Phys. Rev. B 81 195431
[107]	Koshino M 2011 Phys. Rev. B 84 125427
[108]	Tahir M, Manchon A, Schwingenschlogl U 2014 Phys. Rev. B 90 125438
[109]	Dyrdal A, Barnas J 2012 J. Phys.: Condens. Matter 24 275302
[110]	Bergman D L, Oganesyan V 2010 Phys. Rev. Lett. 104 066601
[111]	Xiao D, Yao Y, Fang Z, Niu Q 2006 Phys. Rev. Lett. 97 026603
[112]	Konabe S, Yamamoto T 2014 Phys. Rev. B 90 075430

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Vol. 64, Issue 18 - 2015-09-20

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