Pis'ma v ZhETF, vol. 97, iss. 5, pp. 297-303
© 2013 March 10
Interface induced states at the boundary between a 3D topological
insulator and a normal insulator
V. N. Men'shov^ V. V. Tugushev, E. V. Chulkov+ National Research Centre "Kurchatov Institute", 123182 Moscow, Russia * Prokhorov General Physics Institute of RAS, 119991 Moscow, Russia + Tomsk State University, 634050 Tomsk, Russia
+ Departamento de Fisica de Materiales, Facultad de Ciencias Quimicas, UPV/EHU and Centro de Fisica de Materiales CFM-MPC, Centro Mixto CSIC-UPV/EHU, 20080 San Sebastian, Basque Country, Spain
Submitted 29 January 2013
We show that, when a three-dimensional (3D) narrow-gap semiconductor with inverted band gap ("topological insulator", TI) is attached to a 3D wide-gap semiconductor with non-inverted band gap ("normal insulator", NI), two types of bound electron states having different spatial distributions and spin textures arise at the TI/NI interface. Namely, the gapless ("topological") bound state can be accompanied by the emergence of the gapped ("ordinary") bound state. We describe these states in the framework of the envelope function method using a variational approach for the energy functional; their existence hinges on the ambivalent character of the constraint for the envelope functions that correspond to the "open" or "natural" boundary conditions at the interface. The properties of the ordinary state strongly depend on the effective interface potential, while the topological state is insensitive to the interface potential variation.
DOI: 10.7868/S0370274X13050044
Introduction. It was predicted [1,2] that certain 3D semiconductors with strong spin-orbit coupling (SOC) and an inverted order of energy bands near some high-symmetry points in the Brillouin zone would have gapped electron states in the bulk, but gapless and spin-momentum locked surface states with the two-dimensional (2D) Dirac-like spectrum inside the bulk gap. An existence of these states, the properties of which are topologically protected by time-reversal symmetry, is the very fingerprint that distinguishes such the semiconductors, called the 3D topological insulators (TIs), from their topologically trivial counterparts, below called normal insulators (NIs).
Angle-resolved photo-emission spectroscopy experiments on a number of materials such as, for example, Bi2Se3 and Bi2Te3 or TlBiSe2 have proven the existence of the mid-gap surface helical states with nearly linear low-energy spectrum [3-6]. When the 3D TI has a free surface, i.e. an interface with vacuum, it shows the aforesaid remarkable features of the topologically robust surface states, which can host a great deal of quantum transport phenomena [7-10]. However, for practical applications, one often requires multiple interfaces or channels rather than a single surface. It was pre-
1)e-mail: vnmenshov@mail.ru
dieted that the topological electron states have also to exist at the TI/NI interface [11]. In Ref. [11], using density functional theory to design superlattiee structures based on Bi2Se3, it was shown that an interface state with an ideal Dirac cone is caused by alternating NI and TI. Recently Berntsen et al. have directly observed the Dirac states at the Bi2Se3/Si(111) buried interface [12]. Another photoemission study [13] revealed an existence of interface topological states in the layered bulk crystal (PbSe)5(Bi2Se3)3m, which forms a natural multilayer heterostructure composed of TI and NI. The evidence of a large shift of the Dirac point towards the conduction band edge relative to the vacuum interface, due to the In2Se3 capping layer on epitaxial Bi2Se3 thin film, was reported in [14] demonstrating the possibility of controlling the Dirac cone in TI-based systems. The magneto-transport signatures of the opening of a magnetic gap in the spectrum of interface states due to broken time-reversal symmetry in hybrid GdN/Bi2Se3 heterostructure were detected in Ref. [15].
So the topological states can occur not only on the free surface of a 3D TI but also at the TI/NI interface. On the other hand, it is well known [16,17] that ordinary electron states (bound or resonance) may also arise at the interface formed by two different semiconductors, even without a special requirement to their band in-
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version. Many of the proposed applications require an appropriate interface formed by a TI with another material, thus it is important to explore if and to what extent the electronic properties of a TI are modified by the TI/NI interface potential. In this letter, we show that two different types of in-gap bound states, topological and ordinary ones, exist at the TI/NI interface. Recently, we discussed a similar problem of a coexistence of the topological and ordinary ("convenient") electron states induced by a ferromagnetic 5 layer embedded into a 3D TI [18]. Here, within the k • p approximation, we investigate the energy spectra and the envelope functions for these topologically different states on the TI side of the TI/NI interface. We concentrate only on the key aspects of the phenomenon omitting the cumbersome details of analytical calculations.
Model Hamiltonian and effective energy functional within the k • p method. To approximately describe the band electron states of a bulk semiconductor, |nk) (k is a wave vector, n is a band index), in the region of the Brillouin zone around the point of band extrema k0, the k • p method is reputed to be adequate enough [19]. Under a perturbation smooth enough on the atomic scale, this method makes it possible to qualitatively predict evolution of the electron state wave function ^„(r) in terms of a product of a slowly varying envelope function f„(r) and the Bloch functions |nk0) = exp(ik0r)u„ko (r) of the unperturbed crystal at the point k0: ^„(r) = f„(r)|nk0), u„k(r) is the Bloch periodic function. The envelope function approach with generalized boundary conditions may also be applied to the description of localized and resonant interface states in the semiconductor junctions of different types, using Hermiticity and symmetry requirements (see, for example, [20]). It is evident that some information about the interface states is lost in this continual approach that is compensated for by its relative simplicity allowing a transparent description of the envelope functions and spectra.
We consider the TI/NI layered system with both the TI and NI layers treated as semi-infinite slabs joined at the interface, the boundary between the constituent materials is assumed to be perfectly flat. The system displays translational symmetry in the (x,y) interface plane located at z = 0. Let us denote by ©(r) and $(r) the envelope functions for the TI half-space (z > 0) and the NI half-space (z < 0), respectively. An important point to note is that, in general case, the Bloch amplitudes of quantum states in the TI and NI bulks correspond to different irreducible representations of the different space groups of the crystal symmetry. So the envelope functions ©(r) and $(r) (as well as their space
derivatives) being defined on the different bases (u and w, respectively) do not keep continuity at the TI/NI interface.
The low energy and long wavelength bulk electronic states of the prototypical TI BizSe3 are described by the four band k • p Hamiltonian with strong SOC [21, 22]:
Ht (k) = £0(k)I
4x4
l(k)rz ® a0 +
+ A\\ (kxrx + kyTy) ® ax + Azkzrx ® az, (1)
where S(k) = S—B\\ k+k"2) — Bzk2 I4x4 is an unit matrix, a0,xy ,z and T0,xy ,z denote the Pauli matrices in the spin and orbital space, respectively. The Hamiltonian (1) is written in the basis u = {|+,t), I —,t), |+,I), I —,I )} of the four states at the r point with k0 = 0. The superscripts ± denote the even and odd parity states and the arrows It indicate the spin projections onto the z quantization axis. In the BizSe3-type materials, these four states originate from the bonding combinations of Bi P 1z-orbitals and anti-bonding combinations of Se P2z-orbitals. The parameters S, B||, Bz and A||, Az are connected with matrix elements of momentum [22]. An important feature is that the orbitals |+, t (I)) and —, t (I)) at the r point have the opposite parities, so that the off-diagonal terms are linear in k^- — kx ±tky and kz. The simple model (1) captures the remarkable features of the band structure, especially, under the condition S, Bz, Bi > 0, the inverted order of the energy terms |+, t (I)) and —, t (I)) around k0 =0 as compared with large k, which correctly characterizes the topologically non-trivial nature of the system due to strong SOC. In what follows, without loss of generality we assume e0(k) = 0 and three-dimensional isotropy Bz = B| = B and Az = A| = A, which does not affect the topology. The dispersion of the bulk bands is given by uj{k) = ±y/E2(k) + A2k2, k = |k|.
The low energy and long wavelength bulk electronic states in NI are modeled by the four band Hamiltonian without SOC:
HN(k) = £0(k)l4x4 + A(k)Tz ® G0
(2)
written in the basis w = ^c, t), ^, t), ^, I), ^, I)} which is constructed from the spin-degenerate states of the conduction band (c) and valence band (v) of the direct gap NI at the r point. For the sake of convenience we use a simple two-band effective mass approximation so that A(k) = A + Nkz, A, N > 0 and £0(k) = E0; the electron and hole effective masses are implied to be the same, mc,v = (2N)_1. We regard NI as a wide-gap semiconductor in the sense that Ec,v | > S, where Ec = E0 + A and Ev = E0 — A are the edges of the corresponding bands.
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We write the full electron energy of the TI/NI hete-rocontact in the following form:
J dr0+ (r)HT(—iV)0(r) +
z>0
+ J dr$+ (r)Hn(—iV)$(r)+ftj, (3)
z<0
Qr = J dr[0+(r)V(r)$(r)+$+ (r)V +(r)0(r)j. (4)
Here, the operators HT( —iV) and HN(-iV) determined in Eqs. (1), (2) act in the space of the spinor functions 0(r) = [0i(r), 02(r), 03(r), 04(r)]T and
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