Effects of local exchange field in different directions on spin transport of stanene

Zheng Jun; Ma Li; Xiang Yang; Li Chun-Lei; Yuan Rui-Yang; Chen Jing

doi:10.7498/aps.71.20220277

Abstract

Topological insulator is a new quantum state of matter in which spin-orbit coupling gives rise to topologically protected gapless edge or surface states. The nondissipation transport properties of the edge or surface state make the topological device a promising candidate for ultra-low-power consumption electronics. Stanene is a type of two-dimensional topological insulator consisting of Sn atoms arranged similarly to graphene and silicene in a hexagonal structure. In this paper, the effects of various combinations of local exchange fields on the spin transport of stanene nanoribbons are studied theoretically by using the non-equilibrium Green's function method. The results show that the spin-dependent conductance, edge states, and bulk bands of stanene are significantly dependent on the direction and strength of the exchange field in different regions. Under the joint action of the exchange fields in [I: $ \pm Y $, II: $ +Z $, III: $ \pm Y $] direction, the edge states form a band-gap under the influence of the Y-direction exchange field. The band-gap width is directly proportional to the exchange field strength M, and the conductance is zero in an energy range of $ -M<E<M $. When the exchange fields in the direction of $ +Z $ or $ -Z $ are applied, respectively, to the upper edge region and the lower edge region at the same time, the spin-up energy band and the spin-down energy band move to a high energy region in opposite directions, and strong spin splitting occurs in the edge state and bulk bands. Increasing the strength of the exchange field, the range of spin polarization of conductance spreads from the high energy region to the low energy region. When the directions of the exchange field are [I: $ \mp Z $, II: $ \pm Y $, III: $ \pm Z $], the edge states are spin degenerate, but the weak spin splitting occurs in the bulk bands. Under the condition of different exchange field strengths, the spin-dependent conductance maintains a conductance platform of $ G_\sigma=e^2/h $ in the same energy range of $ -\lambda_{\rm so} <E<\lambda_{\rm so} $.

Keywords:

Authors and contacts

1.
College of Physics Science and Technology, Bohai University, Jinzhou 121013, China
2.
College of Elementary Education, Capital Normal University, Beijing 100048, China
3.
Department of Physics, Capital Normal University, Beijing 100048, China

Corresponding author: Zheng Jun, zhengjun@bhu.edu.cn ; Li Chun-Lei, licl@cnu.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12174038, 11604021), the Revitalization Talents Program of Liaoning Province, China (Grant No. XLYC2007141), and the Science Technology Foundation from Education Commission of Beijing, China (Grant No. KM201810028022)

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References

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Cited By

图 1 (a) 交换场作用下的锡烯纳米带俯视图. 图中沿Y轴方向将锡烯等分为I, II, III 3个区域, 并分别对这3个区域施加[I: $ +Z $, II: $ +Y $, III: $ -Z $]和[I: $ -Z $, II: $ -Y $, III: $ +Z $]方向组合的交换场; (b) 无外场作用时锡烯的电子能带结构; 铁磁交换场按照(c) [I: $ \pm Y $, II: $ +Z $, III: $ \pm Y $]和(e) [I: $ \pm Y $, II: $ -Z $, III: $ \pm Y $]分布时自旋相关电导随费米能E的变化; (d)交换场强度为$ M=\lambda_{{\rm{so}}}/2 $, 方向为(d) [I: $ \pm Y $, II: $ +Z $, III: $ \pm Y $]及(f) [I: $ \pm Y $, II: $ -Z $, III: $ \pm Y $]时的电子能带结构, 图中红色圈线和蓝色实线分别对应自旋向上和自旋向下的电子

Figure 1. (a) Top view of a stanene nanoribbon with local exchange field, where stanene is equally divided into three regions, (i.e., I, II, and III) along the Y-axis, and exchange fields in the directions of [I: $ + Z $, II: $ + Y $, III: $ - Z $] and [I: $ - Z $, II: $ - Y $, III: $ + Z $] are applied to these three regions respectively. (b) Energy-band diagram of stanene without external field. Spin dependent conductance $ G_\sigma $ as a function of the Fermi energy E with the ferromagnetic exchange fields distributed according to (c) [I: $ \pm Y $, II: $ + Z $, III: $ \pm Y $] and (e) [I: $ \pm Y $, II: $ - Z $, III: $ \pm Y $]. Energy-band diagram of stanene with the strength of external field $ M=\lambda_{{\rm{so}}}/2 $, the exchange field directions are (d) [I: $ \pm Y $, II: $ +Z $, III: $ \pm Y $] and (f) [I: $ \pm Y $, II: $ +Z $, III: $ \pm Y $]. The red circle-lines and blue solid-lines correspond to spin-up and spin-down electrons, respectively

DownLoad: Full-Size Img PowerPoint

图 2 交换场强度$M=\lambda_{{\rm{so}}}/2$, 方向为(a) [I: $ +Z $, II: $ \pm Y $, III: $ +Z $]及(b) [$ -Z $, II: $ \pm Y $, III: $ +Z $]时电子能带结构; 交换场按照(c) [I: $ +Z $, II: $ \pm Y $, III: $ +Z $]和(d) [$ -Z $, II: $ \pm Y $, III: $ +Z $]分布时电导G随费米能E 的变化, 图中红色圈线、蓝色三角线和黑色点线分别对应自旋向上、自旋向下以及总的电导

Figure 2. Energy-band diagram of stanene with the strength of external field $ M=\lambda_{{\rm{so}}}/2 $, the exchange field directions (a) [I: $ +Z $, II: $ \pm Y $, III: $ +Z $] and (b) [$ -Z $, II: $ \pm Y $, III: $ +Z $]. Conductance G as a function of the Fermi energy E with the ferromagnetic exchange fields distributed according to (c) [I: $ +Z $, II: $ \pm Y $, III: $ +Z $] and (d) [$ -Z $, II: $ \pm Y $, III: $ +Z $]. The red circle-lines, blue triangle-lines and black dot-lines correspond to spin-up, spin-down, and total conductance, respectively

DownLoad: Full-Size Img PowerPoint

图 3 交换场方向为(a) [I: $ + Y $, II: $ + Z $, III: $ +Y $], (b) [I: $ + Z $, II: $ + Y $, III: $ +Z $], (c) [I: $ - Z $, II: $ + Y $, III: $ +Z $], 交换场强度参数M分别取0.025, 0.050, 0.075, 0.100 eV 时, 自旋向上电导$ G_\uparrow $随费米能E的变化

Figure 3. Conductance G as a function of the Fermi energy E with different values of exchange field parameter $ M=0.025 $, 0.050, 0.075, 0.100 eV for the exchange field directions are (a) [I: $ + Y $, II: $ + Z $, III: $ +Y $], (b) [I: $ + Z $, II: $ + Y $, III: $ +Z $], (c) [I: $ - Z $, II: $ + Y $, III: $ +Z $]

DownLoad: Full-Size Img PowerPoint

[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 Google Scholar
[2]	Geim A K, Novoselov K S 2007 Nat. Mater. 6 183 Google Scholar
[3]	Chang H, Wang H, Song K K, Zhong M, Shi L B, Qian P 2021 J. Phys.: Condens. Matter 34 013003
[4]	Cahangirov S, Topsakal M, Akturk E, Sahin H, Ciraci S 2009 Phys. Rev. Lett. 102 236804 Google Scholar
[5]	Sahin H, Cahangirov S, Topsakal M, Bekaroglu E, Ciraci S 2009 Phys. Rev. B 80 155453 Google Scholar
[6]	Liu C C, Feng W X, Yao Y G 2011 Phys. Rev. Lett. 107 076802 Google Scholar
[7]	Liu C C, Jiang H, Yao Y G 2011 Phys. Rev. B 84 195430 Google Scholar
[8]	Xu Y, Yan B H, Zhang H J, Wang J, Xu G, Tang P Z, Duan W H, Zhang S C 2013 Phys. Rev. Lett. 111 136804 Google Scholar
[9]	Ni Z, Liu Q, Tang K, Zheng J, Zhou J, Qin R, Gao Z, Yu D, Lu J 2012 Nano Lett. 12 113 Google Scholar
[10]	Ezawa M 2013 Appl. Phys. Lett. 102 172103 Google Scholar
[11]	Kaneko S, Tsuchiya H, Kamakura Y, Mori N, Ogawa M 2014 Appl. Phys. Express 7 035102 Google Scholar
[12]	Ni Z Y, Zhong H X, Jiang X H, Quhe R G, Luo G F, Wang Y Y, Ye M, Yang J B, Shi J J, Lu J 2014 Nanoscale 6 7609 Google Scholar
[13]	Zhai X C, Jin G J 2016 J. Phys.: Condens. Matter 28 355002 Google Scholar
[14]	Katayama Y, Yamauchi R, Yasutake Y, Fukatsu S, Ueno K 2019 Appl. Phys. Lett. 115 122101 Google Scholar
[15]	Zheng J, Xiang Y, Li C L, Yuan R Y, Chi F, Guo Y 2020 Phys. Rev. Appl. 14 034027 Google Scholar
[16]	Zheng J, Xiang Y, Li C L, Yuan R Y, Chi F, Guo Y 2021 Phys. Rev. Appl. 16 024046 Google Scholar
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[20]	Xu C Z, Chan Y H, Chen P, Wang X X, Flototto D, Hlevyack J A, Bian G, Mo S K, Chou M Y, Chiang T C 2018 Phys. Rev. B 97 035122 Google Scholar
[21]	Yuhara J, Fujii Y, Nishino K, Isobe N, Nakatake M, Xian L, Rubio A, Le-Lay G 2018 2D Mater. 5 025002
[22]	Deng J L, Xia B Y, Ma X C, Chen H Q, Shan H, Zhai X F, Li B, Zhao A D, Xu Y, Duan W H, Zhang S C, Wang B, Hou J G 2018 Nat. Mater. 17 1081 Google Scholar
[23]	Liu Y, Gao N, Zhuang J, Liu C, Wang J, Hao W, Dou S X, Zhao J, Du Y 2019 J. Phys. Chem. Lett. 10 1558 Google Scholar
[24]	Pang W, Nishino K, Ogikubo T, Araidai M, Nakatake M, Le Lay G, Yuhara J 2020 Appl. Surf. Sci. 517 146224 Google Scholar
[25]	Li J, Lei T, Wang J, Wu R, Qian H, Ibrahim K 2020 Appl. Phys. Lett. 116 101601 Google Scholar
[26]	Dhungana D S, Grazianetti C, Martella C, Achilli S, Fratesi G, Molle A 2021 Adv. Funct. Mater. 31 2102797 Google Scholar
[27]	Ezawa M 2015 J. Phys. Soc. Jpn. 84 121003 Google Scholar
[28]	Wang D, Chen L, Wang X, Cui G, Zhang P 2015 Phys. Chem. Chem. Phys. 17 26979 Google Scholar
[29]	Kuang Y D, Lindsay L, Shi S Q, Zheng G P 2016 Nanoscale 8 3760 Google Scholar
[30]	van den Broek B, Houssa M, Iordanidou K, Pourtois G, Afanas’ev V V, Stesmans A 2016 2D Mater. 3 015001
[31]	Nakamura Y, Zhao T, Xi J, Shi W, Wang D, Shuai Z 2017 Adv. Electron. Mater. 3 1700143 Google Scholar
[32]	Shen L, Lan M, Zhang X, Xiang G, 2017 RSC Adv. 7 9840 Google Scholar
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[36]	M. Ezawa 2012 Phys. Rev. B 86 161407 Google Scholar
[37]	Salazar C, Muniz R A, Sipe J E 2017 Phys. Rev. Mater. 1 054006 Google Scholar
[38]	Rachel S, Ezawa M 2014 Phys. Rev. B 89 195303 Google Scholar
[39]	Lado J L, Fernandez-Rossier J 2014 Phys. Rev. Lett. 113 027203 Google Scholar
[40]	Li S S, Zhang C W 2016 Mater. Chem. Phys. 173 246 Google Scholar
[41]	W. Xiong, C. Xia, T. Wang, Y. Peng, Y. Jia 2016 J. Phys. Chem. C 120 10622 Google Scholar
[42]	Krompiewski S, Cuniberti G 2017 Phys. Rev. B 96 155447 Google Scholar
[43]	Zhou H, Cai Y, Zhang G, Zhang Y W 2016 Phys. Rev. B 94 045423 Google Scholar
[44]	Peng B, Zhang H, Shao H, Xu Y, Ni G, Zhang R, Zhu H 2016 Phys. Rev. B 94 245420 Google Scholar
[45]	Peng X F, Zhou X, Jiang X T, Gao R B, Tan S H, Chen K Q 2017 J. Appl. Phys. 122 054302 Google Scholar
[46]	Noshin M, Khan A I, Subrina S 2018 Nanotechnology 29 185706 Google Scholar
[47]	郑军, 李春雷, 王小明, 袁瑞旸 2021 物理学报 70 147301 Google Scholar Zheng J, Li C L, Wang X M, Yuan R Y 2021 Acta Phys. Sin. 70 147301 Google Scholar
[48]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802 Google Scholar
[49]	Ezawa M 2012 Phys. Rev. Lett. 109 055502 Google Scholar
[50]	Zheng J, Chi F, Guo Y 2018 Appl. Phys. Lett. 113 112404 Google Scholar
[51]	Zheng J, Chi F, Guo Y 2018 Phys. Rev. Appl. 9 024012 Google Scholar
[52]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015 Google Scholar
[53]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[54]	Williams J R, Carlo L D, Marcus C M 2007 Science 317 638 Google Scholar
[55]	Pastawski H M 1991 Phys. Rev. B 44 6329 Google Scholar
[56]	Datta S 1992 Phys. Rev. B 45 1347
[57]	Lee D H, Joannopoulos J D 1981 Phys. Rev. B 23 4997 Google Scholar
[58]	Sancho M P L, Sancho J M L, Rubio J 1984 J. Phys. F: Met. Phys. 14 1205 Google Scholar
[59]	Sancho M P L, Sancho J M L, Sancho J M L, Rubio J 1985 J. Phys. F: Met. Phys. 15 851 Google Scholar

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