Pis'ma v ZhETF, vol. 101, iss. 2, pp. 94-100 © 2015 January 25
Boundary conditions and surface state spectra in topological insulators
y. y. Enaldiev+, I. V. Zagorodnev+, V. A. Volkov+
+Kotelnikov Institute of Radio-engineering and Electronics of the RAS, 125009 Moscow, Russia * Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia Submitted 21 November 2014
We study surface states spectra in 2D topological insulators (TIs) based on HgTe/(Hg,Cd)Te quantum wells and 3D Bi2Se3-type compounds by constructing a class of feasible time-reversal invariant boundary conditions (BCs) for an effective kp-Hamiltonian and a tight-binding model of the topological insulators. The BCs contain some phenomenological parameters which implicitly depend on both bulk Hamiltonian parameters and crystal potential behavior near the crystal surface. Space symmetry reduces the number of the boundary parameters to four real parameters in the 2D case and three in the 3D case. We found that the boundary parameters may strongly affect not only an energy spectrum but even the very existence of these states inside the bulk gap near the Brillouin zone center. Nevertheless, we reveal in frames of the tight-binding model that when surface states do not exist in the bulk gap in the Brillouin zone center they cross the gap in other points of the Brillouin zone in agreement with the bulk-boundary correspondence.
DOI: 10.7868/S0370274X15020058
Topological insulator (TI) is an intriguing quantum state of matter which is insulating in its interior and possesses surface states (SSs) crossing the bulk energy gap [1-3]. The existence of the SSs is provided by a Zi topological invariant and the bulk-boundary correspondence. The Zi invariant is a characteristic of the bulk band structure of a crystal and it comprises no information about the crystal surface. The bulk-boundary correspondence asserts that the odd numbers of pairs of gapless SSs should arise [1-5] at the interface between two crystals with different values of the Zi invariant, for instance, between vacuum (with the Zi index equals zero) and the TI (with the Zi index equals one), but it does not characterize the dispersion of the SS spectra.
Topological SSs have been studied in a number of papers in tight-binding approximations [4, 6] and in envelope function approximation [2,7-12]. Nevertheless in any approach spectra of the SSs should depend on details of crystal-vacuum interface structure. To take it into account one usually chooses appropriate boundary conditions (BCs) for a wave function of topological SSs in a specific model. Most of authors use open BCs (wave function vanishes at the surface) which guarantee mass-less Dirac spectrum of SSs in TIs. However, open BCs are not highlighted by nature and the other BCs might be realized. There are few papers that address the problem of BCs for a wave function of SSs at the surface.
Within a 3D tight-binding model of TI [4] it was shown that a strong topological insulator cannot be
-'-'e-mail: volkov.v.a@gmail.com
transformed into a trivial insulator by means of varying BCs. But that tight-binding model is a toy model and does not describe any really existing 3D TIs. Therefore, it is important to study BC influence on SSs in frames of more realistic kp-Hamiltonian model though it is only valid in a vicinity of Brillouin zone (BZ) center. Refs. [10-12] analyzed dispersion dependence of SSs on the BCs for envelope functions (eigenfunctions of the kp-Hamiltonian). In this approach one should exploit a matrix form of BCs with some unknown parameters [13] that connect envelope functions and their derivatives on the interface. The boundary parameters are determined by the interface structure (on the atomic scale) as well as bulk band parameters of materials. As microscopic details of a real interface are unknown it is involved problem to calculate values of the boundary parameters. More reliable method to determine the parameters is to extract them from experiment (for instance, in Ref. [14] it was done for the most studied GaAs/AlGaAs interface).
Aim of our paper is to demonstrate that the boundary parameters play crucial role not only for the spectrum of topological SSs but also the very existence of SSs in bulk gap near Brillouin zone center, where envelope function approximation is valid. In addition, we show that this is in agreement with the bulk-boundary correspondence. To that end we consider a 2D tight-binding model of TI with general BCs at the edge. Tuning the boundary parameters in the model we can modify spectra of SSs over the full BZ, for instance, so that they cross the bulk gap at the edge of the BZ.
The paper is organized as follows. First, we derive general BCs for the envelope functions in HgTe/(Hg,Cd)Te quantum wells and in Bi2Se3-type 3D TI and analyze edge state (ES) spectra in 2D TI and SS spectra in 3D TI. Then we study an effect of general BCs within a tight-binding model of 2D TI that allows us to clarify behavior of the SS spectra over the full BZ.
2D topological insulators. The phase of 2D TI was obtained in HgTe/(Hg,Cd)Te quantum wells with a certain thickness and composition [15]. The electronic spectrum in the quantum wells near the critical thickness is described by the effective 2D Hamiltonian [2, 16]
#2D = 00® [m{k)Tz - dk2r0 + vkyTy] +vkxaz(g)Tx. (1)
kx+ky,
Here, ftk = h(kx, ky) is the 2D momentum, k2 v/h is the effective speed of light (v > 0), ao,x,y,z and To,x,y,z are the Pauli matrices in the standard representation acting in spin and orbital subspaces, and <g> is the symbol for the direct product. The parameters 6, d < 0 are responsible for the dispersion of the mass rri(k) = mo — bk2, leading to the modification of the Dirac spectrum E = ±\Jm2 + v2k2, and are expected to have significant importance for the appearance of the TI (m0 < 0 in TI phase) [2].
To use the Hamiltonian (1) in a restricted area, it should be supplemented by the BCs at the edge of the system. To do this, we use general physical requirements that significantly restrict the form of the BCs. First of all, since the Hamiltonian (1) is of the second order in the momenta, we assume that the BCs are a linear combination of the wave function and its first derivative
(FcW + GVO |s = o,
(2)
where n = (cos a, sin a) is the outer normal to the edge, and G is an arbitrary 4x4 matrix. We introduce a matrix F as follows
F = —<TO ® T0 v
-o~o <8> tz
(3)
to make G matrix dimensionless in BCs (2). Second, we use the Hermiticity of the Hamiltonian in a restricted area. Therefore, we perform partial integration of (ip | H2 -DI for arbitrary wave functions f, rtp, and equating the surface term to zero, we find the restriction
G+a0 <g> tz - a0 <g> tzG - iaz <g> (r • n) = 0. (4)
It could be shown that Eq. (4) implies the absence of current normal to the edge. Next, we take into consideration the time-reversal symmetry with respect to the operator
T = iay® t0K, (5)
where K is the operator of complex conjugation. Applying commutation of the operator T with G and Eq. (4) we obtain the matrix G of the most general form:
{ 9i 92+m
i(eta+93)-92 94
0 ige-95 \ -><96-95 0
0 95+ig6^
95+igpj 0
91 92-Í93
-i{e~ta+g3)-g2 94 /
(6)
where gyg- are dimensionless real phenomenological boundary parameters, which depend both on the behavior of the crystal potential near the edge and the bulk Hamiltonian parameters. Their values should be determined by microscopic calculations or by experiments. The open BCs correspond to gi,g4 —> oo. The "natural" BCs [10] correspond to g2 = icosa + (sina)/2, #3 = — (cosa)/2, and the other parameters equal zero.
Consider two important particular cases. The first case occurs when boundary potential does not mix the spin of electrons. Since the upper (lower) two components of the wave function correspond to spin-up (down), this case results in gz = ge, = 0. The second case is when the edge, assumed clS X = 0, possesses spatial inversion, e.g. y —> —y, which is described by the operator
I y — (Tx eg) Tz'iy,
(7)
where iy is the coordinate inversion y —> —y. The commutation of the operator Iy with G gives 32 = 35 = 0 in (6) with a = 0.
To analyze the ES spectra, we solve the Schroedinger equation H2d^ = Ei^ on the half-plane x > 0 with the BCs (2) and (6). The wave vector along the edge, is a good quantum number. The bulk solutions are located in the energy region |E — dk21 >
> y'fmo — bk2)2 + (vk||)2, while the ES solutions are situated outside the region. The wave function of the ESs contains two exponents, exp(—n\x) and exp(—«2^)-The energy spectrum of the ESs is shown in Fig. 1. The spectrum of the ESs corresponding to the open BCs is shown by solid curves. In this case, the ESs have a strictly linear dispersion and their decay lengths in the bulk gap are estimated as l/n\ ~ \/b2 — d2/v « 1 nm (which is comparable to the atomic spacing) and l/«2 ~ 50 nm with slight dependence on (the bulk parameters was taken from [2] for mo = —0.01 eV). However, if the parameter g\ significantly decreases the ESs pushed out of the band gap, see the dashed curves. To better understand this extraordinary behavior of the ESs, we may note that usually the parameters b and d, describing the quadratic in k terms of the Hamiltonian (1), are small enough we therefore consider a limit 0.
-0.02-
-0.015 -0.010 -0.005
0.005 0.010 0.015
Fig. 1. The electron spectrum E(k||) of semi-infinite HgTe/(Hg,Cd)Te quantum well. The shaded region corresponds to the continuous spectrum (bulk solutions). The edge states are described by solid curves for open boundary conditions (g\ = g± = oo, <72 = gs = 0), and dashed curves for gi = — 2, g± = oo (all the other boundary parameters equal zero for both curves). Thus, the edge states vanish from the gap of the TI for some values of the boundary parameters. Nevertheless there are edge states in the bulk gap but they cross the bulk gap at large momenta that are beyond the scope of
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