# Effective continuous model for surface states and thin films of three-dimensional topological insulators 

Wen-Yu Shan, Hai-Zhou Lu, and Shun-Qing Shen ∗ 1 

$^{1}$Centre of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China ∗ 

Two-dimensional effective continuous models are derived for the surface states and thin films of the three-dimensional topological insulator (3DTI). Starting from an effective model for 3DTI based on the first principles calculation [Zhang et al , Nat. Phys. 5, 438 (2009)], we present solutions for both the surface states in a semi-infinite boundary condition and in thin film with finite thickness. The coupling between opposite topological surfaces and structure inversion asymmetry (SIA) give rise to gapped Dirac hyperbolas with Rashba-like splittings in energy spectrum. Besides, the SIA leads to asymmetric distributions of wavefunctions for the surface states along the film growth direction, making some branches in the energy spectra much harder than others to be probed by light. These features agree well with the recent angle-resolved photoemission spectra of Bi 2 Se 3 films grown on SiC substrate [Zhang et al, arXiv: 0911.3706]. More importantly, using the parameters fitted by experimental data, the result indicates that the thin film Bi 2 Se 3 lies in quantum spin Hall region based on the calculation of the Chern number and the Z 2 invariant. In addition, strong SIA always intends to destroy the quantum spin Hall state. 

Topological insulators (TIs), which are band insula- tors with topologically protected edge or surface states, have attracted increasing attention recently $^{1}$. A well- known paradigm of topological insulator is the quantum Hall effect, in which the cyclotron motion of electrons in a strong magnetic field gives rise to insulating bulk states but one-way conducting states propagating along edges of system $^{2}$. The idea was generalized to a graphene model with spin-orbit coupling, which exhibits the quan- tum spin Hall (QSH) state $^{3,4}$. Later, the realization of an existing QSH matter was predicted theoretically 5 and soon confirmed experimentally 6,7 in two-dimensional (2D) HgTe/CdTe quantum wells. Furthermore it was found that the QSH state can be induced even by the dis- orders or impurities $^{8–10}$. Meanwhile, the concept was also generalized for three-dimensional (3D) TIs, which are 3D band insulators surrounded by 2D conducting surface states with quantum spin texture $^{11–14}$. Bi$_{x}$ Sb$_{1}$$_{−}$$_{x}$, an al- loy with complex structure of surface states, was first con- firmed as a 3DTI $^{15,16}$. Soon after that it was verified by both experiments 17,18 and first-principles calculations 19 that stoichiometric crystals Bi$_{2}$X$_{3}$ (X=Se, Te) are TIs with well-defined single Dirac cone of surface states and extra large band gaps comparable with room tempera- ture. The Dirac fermions in the surface states of 3DTI obey the 2+1 Dirac equations and reveal a lot of uncon- ventional properties and possible applications, such as the topological magneto-electric effect 20 and Majorana fermions for fault-tolerant quantum computing $^{21–26}$. 

Thanks to the state-of-art semiconductor technologies, low-dimensional structures of Bi$_{2}$X$_{3}$ can be routinely fab- ricated into ultra-thin films 27,28 and nanoribbons $^{29}$. This stimulates several theoretical works on the thin films of 3DTIs $^{30–32}$. For further studies of the transport and op- 

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## I. INTRODUCTION 

tical properties of 3DTI films and their potential appli- cations in spintronics and quantum information, it is de- sirable to establish an effective continuous model for thin films of TIs. 

In this paper, we present an effective continuous model for the surface states and ultra-thin film of TIs. Start- ing with a 3D effective low-energy model based on the first principles calculations $^{19}$, we first present the solu- tions for the surface states and the corresponding spectra for a semi-infinite boundary condition of gapless Dirac Fermions and for the thin film of TIs. The finite size effect of spatial confinement in a thin film leads to a massive Dirac model which may exhibit the QSH ef- fect. Within the same theoretical framework, a structure inversion asymmetry (SIA) term is further introduced in this work to account for the influence of substrate, providing a description of the Rashba-like energy spec- tra observed in the angle resolved photoemission spec- tra (ARPES) in the recent experiment on Bi$_{2}$Se$_{3}$ films $^{28}$. We derived the parameter conditions for the formation of QSH effect in a thin film in the absence and presence of the SIA. By analyzing the fitting parameters with the help of the Chern number and the Z$_{2}$ invariant, we iden- tified the ultrathin films of Bi$_{2}$Se$_{3}$ in the QSH phase in the experiment. 

The paper is organized as follows. In Sec. II we intro- duce an anisotropic 3D Hamiltonian for 3DTI, which is a starting point of the present work. With this Hamilto- nian, we present detailed solutions to the thin film in two different boundary conditions. In Sec. III, effective con- tinuous models are established for the surface states and thin film of 3DTI. Within the framework of this effective continuous model, the structure inversion asymmetry is taken into account and an effective Hamiltonian for SIA is derived in Sec. IV. In Sec. V, we apply the model to newly fabricated thin film Bi$_{2}$Se$_{3}$ and demonstrate that 

