A computational investigation of topological insulator Bi₂Se₃ film

Abstract

Topological insulators have a bulk band gap like an ordinary insulator and conducting states on their edge or surface which are formed by spin–orbit coupling and protected by time-reversal symmetry. We report theoretical analyses of the electronic properties of three-dimensional topological insulator Bi₂Se₃ film on different energies. We choose five different energies (–123, –75, 0, 180, 350 meV) around the Dirac cone (–113 meV). When energy is close to the Dirac cone, the properties of wave function match the topological insulator’s hallmark perfectly. When energy is far way from the Dirac cone, the hallmark of topological insulator is broken and the helical states disappear. The electronic properties of helical states are dug out from the calculation results. The spin-momentum locking of the helical states are confirmed. A 3-fold symmetry of the helical states in Brillouin zone is also revealed. The penetration depth of the helical states is two quintuple layers which can be identified from layer projection. The charge contribution on each quintuple layer depends on the energy, and has completely different behavior along K and M direction in Brillouin zone. From orbital projection, we can find that the maximum charge contribution of the helical states is p_z orbit and the charge contribution on p_yand p_x orbits have 2-fold symmetry.

Graphical abstract

Keywords

topological insulator / spin–orbit coupling / helical state

References

[1] C. L. Kane and E. J. Mele, Z₂ topological order and the quantum spin Hall effect, Phys. Rev. Lett., 2005, 95(14): 146802

[2] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Quantumu spin Hall effect and topological phase transition in HgTe quantum wells, Science, 2006, 314(5806): 1757

[3] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys., 2010, 82(4): 3045

[4] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys., 2011, 83(4): 1057

[5] X. L. Qi and S. C. Zhang, The quantum spin Hall effect and topological insulators, Phys. Today, 2010, 63(1): 33

[6] J. E. Moore, The birth of topological insulators, Nature, 2010, 464(7286): 194

[7] L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B, 2007, 76(4): 045302

[8] X. L. Qi, T. L. Hughes, and S. C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B, 2008, 78(19): 195424

[9] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of a large-gap topological-insulator class with a single Dirac cone on the surface, Nat. Phys., 2009, 5(6): 398

[10] H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Topological insulators in Bi₂Se₃, Bi₂Te₃ and Sb₂Te₃ with a single Dirac cone on the surface, Nat. Phys., 2009, 5(6): 438

[11] D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of time-reversal-protected single-Dirac-cone topologicalinsulator states in Bi₂Te₃ and Sb₂Te₃, Phys. Rev. Lett., 2009, 103(14): 146401

[12] K. He, Y. Zhang, K. He, C. Z. Chang, C. L. Song, L. L. Wang, X. Chen, J. F. Jia, Z. Fang, X. Dai, W. Y. Shan, S. Q. Shen, Q. Niu, X. L. Qi, S. C. Zhang, X. C. Ma, and Q. K. Xue, Crossover of the three-dimensional topological insulator Bi₂Se₃ to the two-dimensional limit, Nat. Phys., 2010, 6(8): 584

[13] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A tunable topological insulator in the spin helical Dirac transport regime, Nature, 2009, 460(7259): 1101

[14] Y. S. Hor, A. Richardella, P. Roushan, Y. Xia, J. G. Checkelsky, A. Yazdani, M. Z. Hasan, N. P. Ong, and R. J. Cava, p-type Bi₂Se₃ for topological insulator and low-temperature thermoelectric applications, Phys. Rev. B, 2009, 79(19): 195208

[15] S. R. Park, W. S. Jung, C. Kim, D. J. Song, C. Kim, S. Kimura, K. D. Lee, and N. Hur, Quasiparticle scattering and the protected nature of the topological states in a parent topological insulator Bi₂Se₃, Phys. Rev. B, 2010, 81: 041405(R)

[16] C. X. Liu, X. L. Qi, H. J. Zhang, X. Dai, Z. Fang, and S. C. Zhang, Model Hamiltonian for topological insulators, Phys. Rev. B, 2010, 82(4): 045122

[17] W. Zhang, R. Yu, S. J. Zhang, X. Dai, and Z. Fang, Firstprinciples studies of the three-dimensional strong topological insulators Bi₂Te₃, Bi₂Se₃ and Sb₂Te₃, New J. Phys., 2010, 12(6): 065013

[18] Jeongwoo Kim, Jinwoong Kim, and Seung-Hoon Jhi, Prediction of topological insulating behavior in crystalline Ge-Sb-Te, Phys. Rev. B, 2010, 82: 201312(R)

[19] O. V. Yazyev, J. E. Moore, and S. G. Louie, Spin polarization and transport of surface states in the topological insulators Bi₂Se₃ and Bi₂Te₃ from first principles, Phys. Rev. Lett., 2010, 105(26): 266806

[20] Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, STM imaging of electronic waves on the surface of Bi₂Te₃: Topologically protected surface states and hexagonal warping effects, Phys. Rev. Lett., 2010, 104(1): 016401

[21] Y. H. Zhao, Y. B. Hu, L. Liu, Y. Zhu, and H. Guo, Helical states of topological insulator Bi₂Se₃, Nano Lett., 2011, 11(5): 2088

[22] R. Yu, W. Zhang, H. J. Zhang, S. C. Zhang, X. Dai, and Z. Fang, Quantized anomalous Hall effect in magnetic topological insulators, Science, 2010, 329(5987): 61

[23] T. M. Schmidt, R. H. Miwa, and A. Fazzio, Spin texture and magnetic anisotropy of Co impurities in Bi2Se3 topological insulators, Phys. Rev. B, 2011, 84(24): 245418

[24] P. Blaha, K. Schwarz, P. Sorantin, and S. B. Trickey, Fullpotential, linearized augmented plane wave programs for crystalline systems, Comput. Phys. Commun., 1990, 59(2): 399

[25] Nanodcal is developed by NanoAcademic Technologies Inc. Nanodcal is an LCAO implementation of density functional theory within the Keldysh nonequilibrium Greens function formalism. It is a general purpose tool for ab initio modeling of electronic structure, equilibrium and non-equilibrium quantum transport.

[26] J. Taylor, H. Guo, and J. Wang, Ab initio modeling of quantum transport properties of molecular electronic devices, Phys. Rev. B, 2001, 63(24): 245407

[27] L. Kleinman and D. M. Bylander, Efficacious form for model pseudopotentials, Phys. Rev. Lett., 1982, 48(20): 1425

[28] G. Theurich and N. A. Hill, Self-consistent treatment of spinorbit coupling in solids using relativistic fully separable ab initio pseudopotentials, Phys. Rev. B, 2001, 64(7): 073106

[29] L. Fernández-Seivane, M. A. Oliveria, S. Sanvito, and J. Ferrer, On-site approximation for spin–orbit coupling in linear combination of atomic orbitals density functional methods, J. Phys.: Condens. Matter, 2006, 18(34): 7999

[30] J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B, 1992, 45(23): 13244

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