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HgTe/CdTe quantum well is a typical two dimensional topological material which supports the helical edge states and quantum spin Hall effect that is imposing in applying of spin electronic devices. The special plateau valued with
$0.25\;h/e^{2}$ of nonlocal resistance in H-shaped four terminal devices can be used as the fingerprint of quantum spin Hall effect. Based on the HgTe/CdTe quantum well, with the aid of nonequilibrium Green's function theory and multi-terminal Landauer-Büttiker formula, we calculate the nonlocal resistance and study the dephasing effect of spin topological states in the presence of exchange field and external magnetic field. It is found the dephasing processes play a role completely different from exchange field and external magnetic field. The latter destroy time reversal symmetry and change the width and relative position of topological gap, but do not influent the topological stability of helical edge states. In the contrary, dephasing processes don't change the width and relative position, however, they broke the topological stability. We consider two kinds of dephasing: normal dephasing and spin dephasing. In the first kind, the carriers lose only the phase memory while maintaining the spin memory. In the second kind, the carriers lose both phase and spin memories. Because of the spin locking properties, normal dephasing almost have no influence on the helical edge states. While the spin dephasing will induce spin flip backscattering and finally destroy helical edge states seriously.-
Keywords:
- quantum transport /
- quantum spin Hall /
- nonlocal resistance /
- dephasing
[1] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801Google Scholar
[2] Zhang F, Zhou J, Xiao D, Yao Y 2017 Phys. Rev. Lett. 119 266804Google Scholar
[3] Schindler F, Cook A M, Vergniory M G, Wang Z, Parkin S S, Bernevig B A, Neupert T 2018 Sci. Adv. 4 eaat0346Google Scholar
[4] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar
[5] Alexandradinata A, Wang Z, Bernevig B A 2016 Phys. Rev. X 6 021008
[6] König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766Google Scholar
[7] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757Google Scholar
[8] Yang W, Chang K, Zhang S C 2008 Phys. Rev. Lett. 100 056602Google Scholar
[9] Liu C C, Feng W, Yao Y 2011 Phys. Rev. Lett. 107 076802Google Scholar
[10] Xu Y, Yan B, Zhang H J, Wang J, Xu G, Tang P, Duan W, Zhang S C 2013 Phys. Rev. Lett. 111 136804Google Scholar
[11] Hsieh D, Qian D, Wray L, Xia Y, Hor Y S, Cava R J, Hasan M Z 2008 Nature 452 970Google Scholar
[12] Chen Y L, Analytis J G, Chu J H, Liu Z K, Mo S K, Qi X L, Zhang H J, Lu D H, Dai X, Fang Z, Zhang S C, Fisher I R, Hussain Z, Shen Z X 2009 Science 325 178Google Scholar
[13] Zhang L, Zhuang J, Xing Y, Li J, Wang J, Guo H 2014 Phys. Rev. B 89 245107Google Scholar
[14] Xing Y, Xu F, Cheung K T, Sun Q F, Wang J, Yao Y 2018 New J. Phys 20 043011Google Scholar
[15] Xing Y, Xu F, Sun Q F, Wang J, Yao Y G 2018 J. Phys. Condens. Mat. 30 435303Google Scholar
[16] Wada M, Murakami S, Freimuth F, Bihlmayer G 2011 Phys. Rev. B 83 121310Google Scholar
[17] Chang K, Lou W K 2011 Phys. Rev. Lett. 106 206802Google Scholar
[18] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[19] Xing Y, Sun Q F 2014 Phys. Rev. B 89 085309Google Scholar
[20] Hinz J, Buhmann H, Schäfer M, Hock V, Becker C, Molenkamp L 2006 Semicond. Sci. Tech. 21 501Google Scholar
[21] Piatrusha S, Khrapai V, Kvon Z, Mikhailov N, Dvoretsky S, Tikhonov E 2017 Phys. Rev. B 96 245417Google Scholar
[22] Xue F, MacDonald A H 2018 Phys. Rev. Lett. 120 186802Google Scholar
[23] Dolcetto G, Sassetti M, Schmidt T L 2015 arXiv preprint: 1511.06141
[24] Sticlet D, Cayssol J 2014 Phys. Rev. B 90 201303Google Scholar
[25] Väyrynen J I, Pikulin D I, Alicea J 2018 Phys. Rev. Lett. 121 106601Google Scholar
[26] Zhou B, Lu H Z, Chu R L, Shen S Q, Niu Q 2008 Phys. Rev. Lett. 101 246807Google Scholar
[27] Dai X, Hughes T L, Qi X L, Fang Z, Zhang S C 2008 Phys. Rev. B 77 125319Google Scholar
[28] Ohyama Y, Tsuchiura H, Sakuma A 2011 Journal of Physics: Conference Series 266 012103Google Scholar
[29] Xing Y, Yang Z L, Sun Q F, Wang J 2014 Phys. Rev. B 90 075435Google Scholar
[30] Zhang L B, Cheng F, Zhai F, Chang K 2011 Phys. Rev. B 83 081402Google Scholar
[31] Xing Y, Zhang L, Wang J 2011 Phys. Rev. B 84 035110Google Scholar
[32] Ren Y, Zeng J, Wang K, Xu F, Qiao Z 2017 Phys. Rev. B 96 155445Google Scholar
[33] Wang K T, Xu F, Xing Y, Zhao H K 2018 Front. Phys. 13 1Google Scholar
[34] Abanin D A, Morozov S V, Ponomarenko L A, Gorbachev R V, Mayorov A S, Katsnelson M L, Watanabe K, Taniguchi T, Novoselov K S, Levitov L S, Geim A K 2011 Science 332 328Google Scholar
[35] Wang Z, Liu H, Jiang H, Xie X C 2016 Phys. Rev. B 94 035409Google Scholar
[36] Renard J, Studer M, Folk J A 2014 Phys. Rev. Lett. 112 116601Google Scholar
[37] Shimazaki Y, Yamamoto M, Borzenets I V, Watanabe K, Taniguchi T, Tarucha S 2015 Nat. Phys. 11 1032Google Scholar
[38] Sui M, Chen G, Ma L, Shan W Y, Tian D, Watanabe K, Taniguchi T, Jin X, Yao W, Xiao D, Zhang Y 2015 Nat. Phys. 11 1027Google Scholar
[39] Wang J, Lian B, Zhang H, Zhang S C 2013 Phys. Rev. Lett. 111 086803Google Scholar
[40] Brüne C, Roth A, Buhmann H, Hankiewicz E M, Molenkamp L W, Maciejko J, Qi X L, Zhang S C 2012 Nat. Phys. 8 485Google Scholar
[41] Krishtopenko S, Teppe F 2018 Phys. Rev. B 97 165408Google Scholar
[42] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802Google Scholar
[43] Ahn J, Yang B J 2017 Phys. Rev. Lett. 118 156401Google Scholar
[44] Sun Q F, Xie X C 2010 Phys. Rev. Lett. 104 066805Google Scholar
[45] Xing Y, Sun Q F, Wang J 2008 Phys. Rev. B 77 115346Google Scholar
[46] Jiang H, Cheng S, Sun Q F, Xie X C 2009 Phys. Rev. Lett. 103 036803Google Scholar
[47] Qi J, Liu H, Jiang H, Xie X C 2019 Front. Phys. 14 43403Google Scholar
[48] Golizadeh-Mojarad R, Datta S 2007 Phys. Rev. B 75 081301Google Scholar
[49] Zhang L, Yu Z, Xu F, Wang J 2018 Carbon 126 183Google Scholar
[50] Datta S 1995 Electronic Transport in Mesoscopic Systems (1st ed.) (New York: Cambridge university press) pp79–93
[51] Jiang H, Wang L, Sun Q F, Xie X C 2009 Phys. Rev. B 80 165316Google Scholar
[52] Sun Q F, Li Y X, Long W, Wang J 2011 Phys. Rev. B 83 115315Google Scholar
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图 1 (a) 四端口HgTe/CdTe量子阱模型, 其中1、4端口为电流输入端口, 2、3端口为电压测量端口. 图中蓝色和绿色区域分别由两个独立的门电压控制, 通过调节门电压, 两个区域可分别独立地在图(b)中所示的三个区间自由切换; (b) HgTe/CdTe量子阱二维系统的能带结构, 包含简并的螺旋边缘态和明显不对称的价带和导带. 体带隙分割出三个区域: 量子自旋霍尔区(QSH), n型自旋霍尔区(n-SH)和p型自旋霍尔区(n-SH)
Figure 1. (a) Four-terminal device based on HgTe/CdTe quantum well with two current terminals 1, 4 and two voltage terminals 2, 3; the blue region and green region can be independently tuned into the three region shown in panel (b); (b) band structure of 2 D HgTe/CdTe quantum well. Bulk energy gap divides three region: QSH region, n-SH region and p-SH region.
图 3 纵向非局域电阻
$R_{23, 14}$ 和部分透射系数$T_{14}$ 和$T_{41}$ 随电流输入区(图1绿色区)在位能$E_1$ 的变化. 左右两栏分别对应铁磁交换场$M=0$ 和$M=5\;{\rm{meV}}$ 的情况, 探测区(图1蓝色区)在位能$E_2=0$ . 为便于比较, 顶栏给出了相应能量区间的能带图. 能带图中, 蓝色和红色分别表示自旋上和自旋下的能带Figure 3. Nonlocal resistors
$R_{23, 14}$ and transmission coefficient$T_{14}$ and$T_{41}$ vs energy$ E_1 $ with energy of detector region$E_2=0$ . left and right panels are for$ M= 0$ and$ M=5 \;{\rm{meV}} $ , respectively. Top panels are band structure of infinite ribbon, the blue lines and red lines correspond to spin up and spin down, respectively.图 4 (a)不同退相干条件下
$\varGamma_{\rm {dc}}$ 随费米能的变化; (b)和(c): 在普通退相干(图(b))和自旋退相干(图(c))作用下, 纵向非局域电阻$R_{23, 14}$ 随入射端能量$E_1$ 的变化, 不同曲线对应不同的退相干强度. 图(b)和图(c)共享图例Figure 4. Panel (a)
$\varGamma_{\rm {dc}}$ vs energy$E_1$ . Panel (b) and (c): longitudinal non-local resistance$R_{23, 14}$ vs energy$E_1$ with different$\varGamma_d$ for normal dephasing [panel (b)] and spin dephasing [panel (c)].图 5 磁场从弱到强作用下无穷长条带在零磁交换场(上栏)和非零磁交换场(下栏)条件下的能带演化
Figure 5. Band structures of infinite ribbon with various magnetic field B for exchange field
$M=0$ (top panels) and$5\rm {\;meV}$ (bottom panels). Red lines denotes the chiral edge states. The red lines and blue lines are correspond to the states of spin up and spin down.图 7 普通退相干和磁场作用下, 纵向非局域电阻
$R_{23, 14}$ 随入射端能量$E_1$ 的变化. 左栏: 零磁交换场; 右栏: 非零磁交换场. 上下两栏共享图例Figure 7.
$R_{23, 14}$ vs energy$E_1$ for different normal dephasing strength$\varGamma_{\rm d}$ with (right panels) or without (left panels) exchange field. The gray region signs the energy gap in zero magnetic field. -
[1] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801Google Scholar
[2] Zhang F, Zhou J, Xiao D, Yao Y 2017 Phys. Rev. Lett. 119 266804Google Scholar
[3] Schindler F, Cook A M, Vergniory M G, Wang Z, Parkin S S, Bernevig B A, Neupert T 2018 Sci. Adv. 4 eaat0346Google Scholar
[4] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar
[5] Alexandradinata A, Wang Z, Bernevig B A 2016 Phys. Rev. X 6 021008
[6] König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766Google Scholar
[7] Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757Google Scholar
[8] Yang W, Chang K, Zhang S C 2008 Phys. Rev. Lett. 100 056602Google Scholar
[9] Liu C C, Feng W, Yao Y 2011 Phys. Rev. Lett. 107 076802Google Scholar
[10] Xu Y, Yan B, Zhang H J, Wang J, Xu G, Tang P, Duan W, Zhang S C 2013 Phys. Rev. Lett. 111 136804Google Scholar
[11] Hsieh D, Qian D, Wray L, Xia Y, Hor Y S, Cava R J, Hasan M Z 2008 Nature 452 970Google Scholar
[12] Chen Y L, Analytis J G, Chu J H, Liu Z K, Mo S K, Qi X L, Zhang H J, Lu D H, Dai X, Fang Z, Zhang S C, Fisher I R, Hussain Z, Shen Z X 2009 Science 325 178Google Scholar
[13] Zhang L, Zhuang J, Xing Y, Li J, Wang J, Guo H 2014 Phys. Rev. B 89 245107Google Scholar
[14] Xing Y, Xu F, Cheung K T, Sun Q F, Wang J, Yao Y 2018 New J. Phys 20 043011Google Scholar
[15] Xing Y, Xu F, Sun Q F, Wang J, Yao Y G 2018 J. Phys. Condens. Mat. 30 435303Google Scholar
[16] Wada M, Murakami S, Freimuth F, Bihlmayer G 2011 Phys. Rev. B 83 121310Google Scholar
[17] Chang K, Lou W K 2011 Phys. Rev. Lett. 106 206802Google Scholar
[18] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802
[19] Xing Y, Sun Q F 2014 Phys. Rev. B 89 085309Google Scholar
[20] Hinz J, Buhmann H, Schäfer M, Hock V, Becker C, Molenkamp L 2006 Semicond. Sci. Tech. 21 501Google Scholar
[21] Piatrusha S, Khrapai V, Kvon Z, Mikhailov N, Dvoretsky S, Tikhonov E 2017 Phys. Rev. B 96 245417Google Scholar
[22] Xue F, MacDonald A H 2018 Phys. Rev. Lett. 120 186802Google Scholar
[23] Dolcetto G, Sassetti M, Schmidt T L 2015 arXiv preprint: 1511.06141
[24] Sticlet D, Cayssol J 2014 Phys. Rev. B 90 201303Google Scholar
[25] Väyrynen J I, Pikulin D I, Alicea J 2018 Phys. Rev. Lett. 121 106601Google Scholar
[26] Zhou B, Lu H Z, Chu R L, Shen S Q, Niu Q 2008 Phys. Rev. Lett. 101 246807Google Scholar
[27] Dai X, Hughes T L, Qi X L, Fang Z, Zhang S C 2008 Phys. Rev. B 77 125319Google Scholar
[28] Ohyama Y, Tsuchiura H, Sakuma A 2011 Journal of Physics: Conference Series 266 012103Google Scholar
[29] Xing Y, Yang Z L, Sun Q F, Wang J 2014 Phys. Rev. B 90 075435Google Scholar
[30] Zhang L B, Cheng F, Zhai F, Chang K 2011 Phys. Rev. B 83 081402Google Scholar
[31] Xing Y, Zhang L, Wang J 2011 Phys. Rev. B 84 035110Google Scholar
[32] Ren Y, Zeng J, Wang K, Xu F, Qiao Z 2017 Phys. Rev. B 96 155445Google Scholar
[33] Wang K T, Xu F, Xing Y, Zhao H K 2018 Front. Phys. 13 1Google Scholar
[34] Abanin D A, Morozov S V, Ponomarenko L A, Gorbachev R V, Mayorov A S, Katsnelson M L, Watanabe K, Taniguchi T, Novoselov K S, Levitov L S, Geim A K 2011 Science 332 328Google Scholar
[35] Wang Z, Liu H, Jiang H, Xie X C 2016 Phys. Rev. B 94 035409Google Scholar
[36] Renard J, Studer M, Folk J A 2014 Phys. Rev. Lett. 112 116601Google Scholar
[37] Shimazaki Y, Yamamoto M, Borzenets I V, Watanabe K, Taniguchi T, Tarucha S 2015 Nat. Phys. 11 1032Google Scholar
[38] Sui M, Chen G, Ma L, Shan W Y, Tian D, Watanabe K, Taniguchi T, Jin X, Yao W, Xiao D, Zhang Y 2015 Nat. Phys. 11 1027Google Scholar
[39] Wang J, Lian B, Zhang H, Zhang S C 2013 Phys. Rev. Lett. 111 086803Google Scholar
[40] Brüne C, Roth A, Buhmann H, Hankiewicz E M, Molenkamp L W, Maciejko J, Qi X L, Zhang S C 2012 Nat. Phys. 8 485Google Scholar
[41] Krishtopenko S, Teppe F 2018 Phys. Rev. B 97 165408Google Scholar
[42] Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802Google Scholar
[43] Ahn J, Yang B J 2017 Phys. Rev. Lett. 118 156401Google Scholar
[44] Sun Q F, Xie X C 2010 Phys. Rev. Lett. 104 066805Google Scholar
[45] Xing Y, Sun Q F, Wang J 2008 Phys. Rev. B 77 115346Google Scholar
[46] Jiang H, Cheng S, Sun Q F, Xie X C 2009 Phys. Rev. Lett. 103 036803Google Scholar
[47] Qi J, Liu H, Jiang H, Xie X C 2019 Front. Phys. 14 43403Google Scholar
[48] Golizadeh-Mojarad R, Datta S 2007 Phys. Rev. B 75 081301Google Scholar
[49] Zhang L, Yu Z, Xu F, Wang J 2018 Carbon 126 183Google Scholar
[50] Datta S 1995 Electronic Transport in Mesoscopic Systems (1st ed.) (New York: Cambridge university press) pp79–93
[51] Jiang H, Wang L, Sun Q F, Xie X C 2009 Phys. Rev. B 80 165316Google Scholar
[52] Sun Q F, Li Y X, Long W, Wang J 2011 Phys. Rev. B 83 115315Google Scholar
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