Finite size effects on the helical edge states on the Lieb lattice

doi:10.1088/1674-1056/25/6/067204

Abstract

For a two-dimensional Lieb lattice, that is, a line-centered square lattice, the inclusion of the intrinsic spin--orbit (ISO) coupling opens a topologically nontrivial gap, and gives rise to the quantum spin Hall (QSH) effect characterized by two pairs of gapless helical edge states within the bulk gap. Generally, due to the finite size effect in QSH systems, the edge states on the two sides of a strip of finite width can couple together to open a gap in the spectrum. In this paper, we investigate the finite size effect of helical edge states on the Lieb lattice with ISO coupling under three different kinds of boundary conditions, i.e., the straight, bearded and asymmetry edges. The spectrum and wave function of edge modes are derived analytically for a tight-binding model on the Lieb lattice. For a strip Lieb lattice with two straight edges, the ISO coupling induces the Dirac-like bulk states to localize at the edges to become the helical edge states with the same Dirac-like spectrum. Moreover, it is found that in the case with two straight edges the gapless Dirac-like spectrum remains unchanged with decreasing the width of the strip Lieb lattice, and no gap is opened in the edge band. It is concluded that the finite size effect of QSH states is absent in the case with the straight edges. However, in the other two cases with the bearded and asymmetry edges, the energy gap induced by the finite size effect is still opened with decreasing the width of the strip. It is also proposed that the edge band dispersion can be controlled by applying an on-site potential energy on the outermost atoms.

Keywords: quantum spin Hall state finite size effect spin--orbit coupling Lieb lattice

Received: 17 January 2016
Revised: 23 February 2016
Accepted manuscript online:

PACS:	72.25.Dc	(Spin polarized transport in semiconductors)
	73.43.-f	(Quantum Hall effects)
	85.75.-d	(Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated magnetic fields)

Fund:

Project supported by the National Natural Science Foundation of China (Grant No. 11274102), the Program for New Century Excellent Talents in University of the Ministry of Education of China (Grant No. NCET-11-0960), and the Specialized Research Fund for the Doctoral Program of the Higher Education of China (Grant No. 20134208110001).

Corresponding Authors: Bin Zhou
E-mail: binzhou@hubu.edu.cn

Cite this article:

Rui Chen(陈锐), Bin Zhou(周斌) Finite size effects on the helical edge states on the Lieb lattice 2016 Chin. Phys. B 25 067204

[1]	Lieb E H 1989 Phys. Rev. Lett. 62 1201
[2]	Tasaki H 1992 Phys. Rev. Lett. 69 1608
[3]	Kusakabe K and Aoki H 1994 Phys. Rev. Lett. 72 144
[4]	Shen S Q, Qiu Z M and Tian G S 1994 Phys. Rev. Lett. 72 1280
[5]	Noda K, Inaba K and Yamashita M 2014 Phys. Rev. A 90 043624
[6]	Noda K, Inaba K and Yamashita M 2015 Phys. Rev. A 91 063610
[7]	Niţă M, Ostahie B and Aldea A 2013 Phys. Rev. B 87 125428
[8]	Leykam D, Bahat-Treidel O and Desyatnikov A S 2012 Phys. Rev. A 86 031805
[9]	Iglovikov V I, Hébert F, Grémaud B, Batrouni G G and Scalettar R T 2014 Phys. Rev. B 90 094506
[10]	Wang H, Yu S L and Li J X 2014 Phys. Lett. A 378 3360
[11]	Cao X, Chen K and He D 2015 J. Phys.: Condens. Matter 27 166003
[12]	Gouveia J D and Dias R G 2015 J. Magn. Magn. Mater. 382 312
[13]	Jaworowski B, Manolescu A and Potasz P 2015 Phys. Rev. B 92 245119
[14]	Palumbo G and Meichanetzidis K 2015 Phys. Rev. B 92 235106
[15]	Emery V J 1987 Phys. Rev. Lett. 58 2794
[16]	Scalettar R T, Scalapino D J, Sugar R L and White S R 1991 Phys. Rev. B 44 770
[17]	Shen R, Shao L B, Wang B and Xing D Y 2010 Phys. Rev. B 81 041410
[18]	Apaja V, Hyrkäs M and Manninen M 2010 Phys. Rev. A 82 041402
[19]	Goldman N, Urban D F and Bercioux D 2011 Phys. Rev. A 83 063601
[20]	Vicencio R A and Mejía-Cortés C 2014 J. Opt. 16 015706
[21]	Guzmán-Silva D, Mejía-Cortés C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Segev M, Szameit A and Vicencio R A 2014 New J. Phys. 16 063061
[22]	Vicencio R A, Cantillano C, Morales-Inostroza L, Real B, Mejía-Cortés C, Weimann S, Szameit A and Molina M I 2015 Phys. Rev. Lett. 114 245503
[23]	Mukherjee S, Andersson E and Thomson R R 2015 Phys. Rev. Lett. 114 245504
[24]	Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[25]	Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[26]	Weeks C and Franz M 2010 Phys. Rev. B 82 085310
[27]	Zhao A and Shen S Q 2012 Phys. Rev. B 85 085209
[28]	Beugeling W, Everts J C and Smith C M 2012 Phys. Rev. B 86 195129
[29]	Weeks C and Franz M 2012 Phys. Rev. B 85 041104(R)
[30]	Tsai W F, Fang C, Yao H and Hu J 2015 New J. Phys. 17 055016
[31]	Zhou B, Lu H Z, Chu R L, Shen S Q and Niu Q 2008 Phys. Rev. Lett. 101 246807
[32]	Linder J, Yokoyama T and Sudbø A 2009 Phys. Rev. B 80 205401
[33]	Liu C X, Zhang H J, Yan B H, Qi X L, Frauenheim T, Dai X, Fang Z and Zhang S C 2010 Phys. Rev. B 81 041307
[34]	Lu H Z, Shan W Y, Yao W, Niu Q and Shen S Q 2010 Phys. Rev. B 81 115407
[35]	Shan W Y, Lu H Z and Shen S Q 2010 New J. Phys. 12 043048
[36]	Brüne C, Roth A, Novik E G, König M, Buhmann H, Hankiewicz E M, Hanke W, Sinova J and Molenkamp L W 2010 Nat. Phys. 6 448
[37]	Zhang Y, He K, Chang C Z, Song C L, Wang L L, Chen X, Jia J F, Fang Z, Dai X, Shan W Y, Shen S Q, Niu Q, Qi X L, Zhang S C, Ma X C and Xue Q K 2010 Nat. Phys. 6 584
[38]	Mao S J, Kuramoto Y, Imura K and Yamakage A 2010 J. Phys. Soc. Jpn. 79 124709
[39]	Mao S J and Kuramoto Y 2011 Phys. Rev. B. 83 085114
[40]	Imura K, Yamakage A, Mao S, Hotta A and Kuramoto Y 2010 Phys. Rev. B. 82 085118
[41]	Wada M, Murakami S, Freimuth F and Bihlmayer G 2011 Phys. Rev. B. 83 121310
[42]	Michetti P, Penteado P H, Egues J C and Recher P 2012 Semicond. Sci. Technol. 27 124007
[43]	Cheng Z, Chen R and Zhou B 2015 Chin. Phys. B 24 67304
[44]	Creutz M and Horváth I 1994 Phys. Rev. D 50 2297
[45]	Creutz M 2001 Rev. Mod. Phys. 73 119
[46]	König M, Buhmann H, Molenkamp L W, Hughes T, Liu C X, Qi X L and Zhang S C 2008 J. Phys. Soc. Jpn. 77 031007
[47]	Yao W, Yang S A and Niu Q 2009 Phys. Rev. Lett. 102 096801

[1]	Control of topological phase transitions in Dirac semimetal films by exchange fields Fei Yang(杨菲), Hai-Long Wang(王海龙), Hui Pan(潘晖). Chin. Phys. B, 2017, 26(1): 017102.
[2]	Finite size effects on the quantum spin Hall state in HgTe quantum wells under two different types of boundary conditions Cheng Zhi (成志), Chen Rui (陈锐), Zhou Bin (周斌). Chin. Phys. B, 2015, 24(6): 067304.
[3]	Finite size effects on helical edge states in HgTe quantum wells with the spin–orbit coupling due to bulk-and structure-inversion asymmetries Cheng Zhi (成志), Zhou Bin (周斌). Chin. Phys. B, 2014, 23(3): 037304.
[4]	Finite size effect on Raman frequency of phonons in nano-semiconductors Xia Lei (夏磊), Zhong Kai (钟凯), Song Yang (宋阳), Lu Xin (陆欣), Xu Lü-Shun (许旅顺), Yan Yan (阎研), Li Hong-Dong (李红东), Yuan Fang-Li (袁方利), Jiang Jian-Zhong (蒋建中), Yu Da-Peng (俞大鹏), Zhang Shu-Lin (张树霖). Chin. Phys. B, 2012, 21(9): 097801.
[5]	Interfacial spin Hall current in a Josephson junction with Rashba spin–orbit coupling Yang Zhi-Hong(杨志红), Yang Yong-Hong(杨永宏), and Wang Jun(汪军) . Chin. Phys. B, 2012, 21(5): 057402.
[6]	Theoretical investigations of the local distortion and electron paramagnetic resonance parameter for CdCl₂:V²⁺ and CsMgX₃:V²⁺ (X=Cl, Br) systems Li Cheng-Gang(李成刚), Kuang Xiao-Yu(邝小渝), Duan Mei-Ling(段美玲), Zhang Cai-Xia(张彩霞), and Chai Rui-Peng(柴瑞鹏). Chin. Phys. B, 2010, 19(6): 067103.
[7]	Influence of spin－orbit coupling induced by in-plane external electric field on the intrinsic spin-Hall effect in a Rashba two-dimensional electron gas Yan Yu-Zhen(颜玉珍) and Hu Liang-Bin(胡梁宾). Chin. Phys. B, 2010, 19(4): 047203.
[8]	Relativistic density functional investigation of UX₆ (X = F, Cl, Br and I) Zhang Yun-Guang(张云光) and Li Yu-De(李育德). Chin. Phys. B, 2010, 19(3): 033302.
[9]	The influence of the Dresselhaus spin--orbit coupling on the tunnelling magnetoresistance in ferromagnet/ insulator /semiconductor/ insulator /ferromagnet tunnel junctions Wang Xiao-Hua(王晓华), An Xing-Tao(安兴涛), and Liu Jian-Jun(刘建军). Chin. Phys. B, 2009, 18(2): 749-756.
[10]	Spin-dependent electron transport of a waveguide with Rashba spin--orbit coupling in an electromagnetic field Xiao Xian-Bo(肖贤波),Li Xiao-Mao(李小毛), and Chen Yu-Guang(陈宇光) . Chin. Phys. B, 2009, 18(12): 5462-5467.
[11]	Polarized spin transport in mesoscopic quantum rings with electron--phonon and Rashba spin--orbit coupling Liu Ping(刘平) and Xiong Shi-Jie(熊诗杰). Chin. Phys. B, 2009, 18(12): 5414-5419.
[12]	Spin--orbit ab initio curves of ⁸⁰Se₂⁺ ion and theassignment of photoelectron spectra of ⁸⁰Se₂ molecule Yan Bing(闫冰), Pan Shou-Fu(潘守甫), and Guo Qing-Qun(郭庆群). Chin. Phys. B, 2008, 17(9): 3318-3321.
[13]	Electron resonance-transmission through a driven quantum well with spin--orbit coupling Zhang Cun-Xi(张存喜), Wang Rui(王瑞), Nie Yi-Hang(聂一行), and Liang Jiu-Qing(梁九卿). Chin. Phys. B, 2008, 17(7): 2662-2669.
[14]	Field-assisted electron transport through a symmetric double-well structure with spin--orbit coupling and the Fano-resonance induced spin filtering Zhang Cun-Xi(张存喜), Nie Yi-Hang(聂一行), and Liang Jiu-Qing(梁九卿). Chin. Phys. B, 2008, 17(7): 2670-2677.
[15]	Analytical potential energy function for the electronic states ²П_1/2 and ²П_3/2 of O₂^x(x=+1, --1) Lü Bing(吕兵), Linghu Rong-Feng(令狐荣锋), Zhou Xun(周勋), Yang Xiang-Dong(杨向东), Zhu Zheng-He(朱正和), and Cheng Xin-Lu(程新路) . Chin. Phys. B, 2008, 17(5): 1738-1742.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Finite size effects on the helical edge states on the Lieb lattice

Cite this article:

Online attention