Abstract
The quantum spin Hall insulators predicted ten years ago and now experimentally observed are instrumental for a break through in nanoelectronics due to nondissipative spinpolarized electron transport through their edges. For this transport to persist at normal conditions, the insulators should possess a sufficiently large band gap in a stable topological phase. Here, we theoretically show that quantum spin Hall insulators can be realized in ultrathin films constructed from a trivial band insulator with strong spinorbit coupling. The thinnest film with an inverted gap large enough for practical applications is a centrosymmetric sextuple layer built out of two inversely stacked noncentrosymmetric BiTeI trilayers. This nontrivial sextuple layer turns out to be the structure element of an artificially designed strong threedimensional topological insulator Bi_{2}Te_{2}I_{2}. We reveal general principles of how a topological insulator can be composed from the structure elements of the BiTeX family (X = I, Br, Cl), which opens new perspectives towards engineering of topological phases.
Introduction
Twodimensional (2D) topological insulators (TIs)—a new electronic phase also referred to as a quantum spin Hall (QSH) insulator—are characterized by an absolute band gap induced by spinorbit coupling (SOC) and helical gapless edge states inside the gap^{1}. These states protected by timereversal symmetry provide perfectly conducting spinfiltered channels, meeting the demands of lowpower nanoelectronics and spintronics. The existence of such states as the fingerprint of a topologically nontrivial 2D insulator was first predicted in Refs 2,3. It was also suggested that the QSH effect can be observed in graphene, where SOC opens a gap at the two inequivalent Dirac points. This gap in graphene appears to be too small for practical use, so heavyelements based analogs of graphene must be sought. Actually, the 2D materials with lowbuckled honeycomblattice structures^{4,5,6,7,8,9,10}—silicene, germanene, and stanene—possess a significantly larger SOCinduced gap at the Dirac points (up to ~0.1 eV in stanene), and the spinpolarized edge channels could be detected at easily accessible temperatures. However, the QSH effect in such systems has not been experimentally observed so far.
Further effective enhancement of the SOC to make the gap larger can be realized by chemical functionalization of the above 2D materials^{11,12}. Such a functionalization substantially enlarges the gap, in fact “destroying” the Dirac cones, and it may lead to a SOCinduced band inversion at the time reversal invariant momentum (TRIM) (normally at k = 0) with an absolute gap of several hundred meV at this momentum. If the band inversion occurs, the resulting 2D system is a 2D TI that should support the QSH effect^{12} similar to the inverted HgTe quantum wells predicted to be QSH insulators in ref. 13. It is important that this prediction has found experimental confirmation: In the inverted HgTe/CdTe and InAs/GaSb quantum wells^{14,15,16} the QSH effect was observed despite the very small gaps in these quantum wells, less than 10 meV. It has spurred a rising tide of theoretical propositions of different 2D TIs with honeycomb or squarelattice structures and a large inverted gap enabling roomtemperature operating^{17}.
Ab initio approaches to electronic structure, especially those based on the density functional theory (DFT), have become a powerful tool to search for new materials with unique properties. At the same time, the effective models that proved indispensable in predicting the QSH effect in graphenelike systems and quantum wells are currently widely used to analyze the effect of strain, quantum confinement, and external fields in 2D TIs, i.e., to solve the problems that presently are not accessible with ab initio methods. Thus, to efficiently model the nanoelectronics and spintronics devices, the microscopic methodology must be bridged with the effective Hamiltonian approach based on symmetry considerations and on the k·p perturbation theory.
With a few exceptions, none of the theoretically proposed 2D materials has been hitherto fabricated^{17}. Thus, the intensive search of robust and easily fabricated materials remains to be actual. In particular, it was suggested that 2D TIs can be produced from a thin film of layered 3D (threedimensional) TIs of the Bi_{2}Se_{3} family, where the hybridization between the opposite surfaces of the film opens a gap at the Dirac point (DP). Depending on film thickness, the 2D system may “oscillate” between band insulator and QSH insulator as was predicted by the 4band effective k·p model (see refs 18,19). The thinnest known topologically nontrivial film consists of at least two structural elements—quintuple or septuple layers^{20}.
Besides the studies on the thin films of 3D TIs, recently it was heuristically suggested that a 3D TI can be constructed artificially via stacking 2D bilayers that are topologically trivial^{21}. It encourages our search for 2D TIs built out of trivial bandinsulator constituents. These constituents should have strong spinorbit coupling (SOC), and prospective candidates are bismuth tellurohalides BiTeX with X = I, Br, and Cl, among which the polar semiconductor BiTeI demonstrates the strongest spinorbit coupling providing the biggest known Rashba spinsplitting of bulk and surface states^{22,23}. The structure element of BiTeI is a trilayer (TL) with the IBiTe stacking. A single TL that possesses the Rashba spinsplit band structure^{24} can be grown epitaxially on a suitable substrate or be easily exfoliated from the bulk BiTeI, where the adjacent TLs couple through a weak vanderWaals (vdW) interaction. The samples of BiTeI always contain a large number of randomly distributed bulk stacking faults, which leads to a mixture of terminations at the surface, as experimentally observed in refs 25, 26, 27, 28, 29, 30. This implies that adjacent TLs may have different sequence order along the hexagonal z axis.
Here, based on DFT calculations we demonstrate that a centrosymmetric sextuple layer (SL) constructed from two BiTeI TLs with facing Telayer sides and a typical vdW spacing is a 2D TI with the gap of 70 meV at . The vdW interaction between these TLs is crucial to realize such a QSH insulator phase: The SL becomes topologically trivial with increasing the vdW spacing by 5% only. We consider the nontrivial SL as a structure element, a repetition of which along the z axis results in thin films that are found to “oscillate” between trivial and nontrivial phases with the number of SLs. The corresponding bulk system composed of SLs turns out to be a strong 3D TI (hereafter referred to as Bi_{2}Te_{2}I_{2}). It is energetically unfavourable by only 0.5 meV compared with the noncentrosymmetric BiTeI. This makes it plausible to suppose that crystals of BiTeI grown by the Bridgman method already contain the desired SLs, and that the alternative stacking can be experimentally observed and controllably manufactured. To describe the lowenergy properties of Bi_{2}Te_{2}I_{2} and its films, we derive fourband k·p Hamiltonians from the ab initio wave functions. They are similar to the Hamiltonians constructed for Bi_{2}Se_{3} family 3D TIs and their thin films^{18,31,32}. For a more accurate description of the SL, we derive an eightband Hamiltonian that involves Rashbasplit valence and conduction bands of the standalone TLs. We thus demonstrate that due to the bondingantibonding splitting the inversion occurs between one of the Terelated valence bands and one of the conduction bands formed by Bi orbitals. The proposed materials illustrate the effectiveness of the new way to design 2D TIs from trivial band insulators with giantRashba spit bands for roomtemperature operating.
Results
Figure 1(a–c) show the band structure of 1 and 5 SLs and the bulk crystal of Bi_{2}Te_{2}I_{2} obtained with the extended linearized augmented plane wave (ELAPW) method^{33} within the local density approximation (LDA) for the exchangecorrelation functional and with the use of the full potential scheme of ref. 34. (Details on the equilibrium bulk atomic structure, the bulktruncated slab geometry of the related thin films, and the calculations performed can be found in Supplementary Note 1.)
The 1SL film is constructed from two BiTeI trilayers with facing Telayer sides, Fig. 1(d). It is noteworthy that the band structure of this film with the gap of 56 meV (70 meV in the relaxed geometry, see Supplementary Fig. 1) differs substantially from that of its constituents (cf. Supplementary Fig. 2): there is no trace of Rashbatype split bands. The band structure of the 5SL film exhibits a gapless Dirac state residing in the band gap of 151 meV, see Fig. 1(b) and Supplementary Fig. 3. This is a signature of the topological character of the respective bulk band structure (Fig. 1(c)), which has an inverted gap of 234 meV at Γ and the fundamental gap of 169 meV in the Γ–A line close to the A point. As seen in Fig. 2(a), the Dirac surface state almost completely resides within the outer SL. Moreover, this state is localized stronger than the Dirac state of TIs like Bi_{2}Te_{3}, since 70% of its weight falls in the outermost half of the SL, i.e., in the surface BiTeI TL (see also Fig. 1(e)).
The spin texture of the Dirac state is illustrated in Fig. 2(b) and (c), which show spinresolved constant energy contours for the lower and upper cones of the Dirac surface state. Apart from the inplane polarization – clockwise above the DP and counterclockwise below it, both contours also have an outofplane spin component, which is an intrinsic feature of the hexagonal surface. However, in this case S_{z} is extremely small and varies in the range of ≈±0.01.
As has been shown for the Dirac state in Bi_{2}Se_{3} both experimentally and theoretically^{35,36,37,38,39}, the spin textures of p_{x}, p_{y}, and p_{z} orbitals are remarkably different, which leads to the dependence of the spin polarization of photoelectrons on the polarization of light. The spin texture provided by p_{z} orbitals has clockwise (counterclockwise) chirality for the upper (lower) cone, while the projections of the total spin on p_{x} and p_{y} orbitals are not chiral, and their spins are opposite to each other. Similar spinorbital texture we find in the Bi_{2}Te_{2}I_{2}, see Fig. 2(d–f) for the upper Dirac cone (for the lower Dirac cone, the coupling of spin and orbital textures is opposite, not shown). As can be seen, the spin orientations for p_{x} and p_{y} projections are antiparallel at each k_{} point, whereas the spin orientation of p_{z} projection coincides with the total spin.
It is noteworthy that similar spinorbital texture has been observed for the spinpolarized Rashba state in BiTeI^{40}. For the Teterminated surface, it was found that the outer Rashba branch demonstrates the same spin orientations for p_{x}, p_{y}, and p_{z} projections as those for the upper Dirac cone in Bi_{2}Te_{2}I_{2}, and the inner Rashba branch has opposite texture, i.e., the same as in the lower cone. Because the surface of the Bi_{2}Te_{2}I_{2} slab has iodine termination and its spintexture is reversed due to the opposite orientation of the z axis, the spinorbit texture of the upper (lower) Dirac cone in Bi_{2}Te_{2}I_{2} is the same as the texture of the inner (outer) Rashba branch in BiTeI.
To construct a simple effective k·p model for the centrosymmetric Bi_{2}Te_{2}I_{2} we derive a model Hamiltonian of a desired dimension and accurate up to the second order in k from the LDA spinor wave functions Ψ_{n↑(↓)} of the doubly degenerate bands E_{n} found at k = 0 (see Supplementary Note 2 and ref. 41 for details). The subscripts ↑ or ↓ in Ψ_{n↑(↓)} refer to the zcomponent of the total angular momentum J = L + S in the atomic sphere that has the largest weight in the nth band, see Fig. 1(e). The Hamiltonian is constructed in terms of the matrix elements^{42} of the velocity operator , where n and m run over the relativistic bands (from semicore levels up to highlying unoccupied bands). Here, σ is the vector of the Pauli matrices, and V(r) is the crystal potential.
For the bulk Bi_{2}Te_{2}I_{2}, in the basis of the two valence bands and two conduction bands , our ab initio fourband Hamiltonian reads:
where , , , and the direct matrix product of the Pauli matrices τ and σ is implied (the explicit matrix form of is presented in Supplementary Note 3). Note that this Hamiltonian is the same (to within a unitary transformation) as that constructed for Bi_{2}Se_{3} in ref. 31 within the theory of invariants. The matrices τ and σ in Eq. (1) have different meaning: τ operates in the valenceconduction band space, while σ refers to the total angular momentum J.
The parameters in Eq. (1) obtained within the LDA are: C_{0} = 0.03 eV, C_{z} = 0.13 a.u., C_{} = 4.19 a.u., M_{0} = −0.12 eV, M_{z} = 1.35 a.u., M_{} = 5.88 a.u., V_{} = 0.52 a.u., and V_{z} = 0.13 a.u. (we use Rydberg atomic units: ). Since the basis functions explicitly refer to the valence and conduction bands rather than to atomic orbitals, the parameter M_{0} that defines the band gap at k = 0 is negative and does not change sign upon moving from the topologically nontrivial insulator to the trivial one. The eigenvalues E(k) of the Hamiltonian (1) with the above parameters are shown in Fig. 1(c) by red lines, nicely reproducing the LDA curves over a quite large kregion and providing an absolute gap in the k·p spectrum. Moreover, these parameters reflect the band inversion and meet the condition of the existence of topological surface states (see, e.g., ref. 43) in accord with the ℤ_{2} topological invariant ν_{3D} = 1 obtained from the parities of the bulk LDA wave functions at the TRIM points^{44}. Actually, the diagonal dispersion term M_{z()} is positive, and it is larger than the electronhole asymmetry: C_{z()} < M_{z()}.
For the Bi_{2}Te_{2}I_{2} thin films, we derive the Hamiltonian in the basis as
where , , and τ refers now to the two decoupled sets of massive Dirac fermions. The Hamiltonian (2) is similar to the one obtained for 3D TI thin films within the effective continuous model based on the substitution in the Hamiltonian of ref. 31 and on the imposition of the open boundary conditions (see, e.g., refs 18,43). The crucial difference is that in our ab initio approach within the same formalism for 3D and 2D systems we obtain the Hamiltonian and its parameters from the original spinor wave functions. We do not a priori impose the form of the Hamiltonian based on symmetry arguments and do not resort to the fitting of ab initio band dispersion curves or to a solution of 1D Schrödinger equations derived by using the above substitution with special boundary conditions.
All the considered Bi_{2}Te_{2}I_{2} films are characterized by the velocity V_{} = 0.45 ± 0.01 a.u. and the electronhole asymmetry C_{} = 4.15 ± 0.10 a.u., which are weakly sensitive to the number of SLs, where the ± ranges indicate the variations of V_{} and C_{} in moving from 1 to 5 SLs. On the contrary, as seen in Fig. 1(g) and (h) the parameters M_{0} and M_{} depend strongly on the film thickness, approaching monotonically zero.
In order to explicitly indicate whether a given film is a QSH insulator, in Fig. 1(g) we also plot the gap parameter with ν_{2D} being the ℤ_{2} invariant obtained from the parities of the wave functions at the TRIM points of the 2D Brillouin zone. This parameter is negative for a topologically nontrivial film and positive for a trivial one. As follows from the figure, Δ “oscillates” with the period of 2 SLs within the examined thickness interval. (The parity of and is (+) and (−), respectively, for Δ < 0, and it is (−) and (+) for Δ > 0). As in 3D TI films^{45}, the thickness dependence of Δ may be sensitive to the quasiparticle approximation employed, and it may change if manybody corrections beyond DFT are introduced. However, even the simplest quasiparticle method, the GW approximation for the selfenergy, is methodologically challenging and computationally too demanding to study a large series of complex systems. Thus, DFT remains the method of choice, and its good performance for a wide range of TIs justifies the use of the KohnSham band structure as a reasonable starting point.
The diagonalization of the Hamiltonian (2) then leads to E(k) shown by red lines in Fig. 1(a) and (b). The absence of the absolute gap in the resulting k·p spectrum is the general feature of all the films studied. It is caused by the rather big electronhole asymmetry C_{} compared with the diagonal dispersion parameter M_{}, Fig. 1(h). It should be noted that the conclusion on whether the edge states exist in a TI film is often made based on the signs and relative values of the parameters M_{0}, M_{}, and C_{}. On the contrary, we find that the asymmetry C_{} is larger than M_{} everywhere, breaking one of the conditions for the film to be a QSH insulator, see, e.g., refs 18,46. Focusing on the behaviour of the diagonal dispersion M_{} (as, e.g., in the topology analysis of ref. 15), we note that it is positive for all the thicknesses, Fig. 1(h). Along with the negative M_{0}, this should signify an inverted band gap for the respective films. However, it does not correlate with the oscillating Δ, Fig. 1(g).
Let us now analyze the behaviour of the diagonal dispersion parameter M_{} together with the topological invariant under a continuously varying geometry. We choose the 1SL film–the thinnest film, for which the k·p prediction of the band inversion does not contradict the actual topological property–and gradually expand the vanderWaals spacing d_{vdW}. The evolution of the band structure with increasing d_{vdW} is shown in Fig. 3. According to the gap parameter Δ, see Fig. 4(a), a topological phase transition occurs at d_{vdW} that is just around the mentioned 5% larger than its bulk value, and the 1SL film becomes topologically trivial. Further expansion leads to a larger band gap at , which is not inverted anymore. It is noteworthy that such a behaviour of Δ as a function of d_{vdW} with the topological phase transition around 5% is stable with respect both to the choice of the approximation to the DFT exchangecorrelation functional (LDA, GGA, dispersion corrected GGA) and to the SL geometry (bulk truncated or relaxed). In the limit of very large d_{vdW}, when the BiTeI trilayers composing the 1SL film are too far from each other, the band structure is identical to that of a freestanding BiTeI trilayer (see Supplementary Fig. 2). Similarly, artificial reduction of the spinorbit interaction strength λ relative to its actual value λ_{0} in the equilibrium SL leads to a decrease in the gap, which closes at λ/λ_{0} = 0.95. A further decrease in λ causes a widening of the already uninverted gap of the trivial phase. In general, the dependence of the relative gapwidth on the spinorbit interaction strength is almost linear and can be approximated as .
The 1SL parameters of the 4band k·p Hamiltonian (2) strongly depend on d_{vdW} (the respective eigenvalues E(k) of this Hamiltonian are shown by red lines in Fig. 3(a–c)). With the d_{vdW} expansion (given in percents of the bulk value ) up to 50%, the velocity V_{} decreases monotonically from 0.470 a.u. to 0.342 a.u., and the electronhole asymmetry C_{} becomes smaller as well, Fig. 4(c). At , C_{} is already smaller than M_{}, ensuring an absolute gap in the 4band k·p spectrum, see Fig. 3(c). With further increasing it even becomes negative, but it remains C_{} < M_{}. A stepwise behaviour of the parameter M_{} that changes sign at the small indicates that M_{} keeps following the actual ν_{2D} and, thus, predicts a gap without inversion. With increasing d_{vdW} this parameter again goes through zero around , telling us that the band gap becomes inverted again, and at ~35% with the given C_{} and M_{0} meets the conditions of the existence of the edge states^{18,46}. However, as seen in Fig. 4(a), the 1SL film is too far from a topological phase transition at such . With this example we illustrate the strong limitations of the predictive capabilities of the effective continuous model.
Let us now analyze the formation of the SL band structure with the inverted band gap. Starting from wellseparated layers, Fig. 3(f), and going back to the bulk value of the vanderWaals spacing, Fig. 3(a), we retrace the valence bands (ν_{1} with the energy and ν_{2} with ) with the predominant contribution coming from the p_{z} orbitals of Te and the conduction bands (c_{1} with and c_{2} with ) mainly formed by Bi p_{z} orbitals, see Supplementary Fig. 2. We derive an 8band Hamiltonian which is presented in Supplementary Note 3. Its eigenvalues are shown in Fig. 3 by blue lines, and the corresponding parameters as a function of the d_{vdW} expansion are depicted in Fig. 4(b) and (d), see also Supplementary Fig. 4.
As seen in Fig. 4(b), at in the larged_{vdW} limit there are two doubly degenerate energy levels, and . Upon decreasing d_{vdW}, the TLs start to interact primarily by their Telayer sides to cause the bondingantibonding splitting of the two degenerate levels: The Terelated energies as a function of d_{vdW} disperse stronger than those of Bi. Near the bulk value , the splitting is large enough to invert the order of the and levels, ensuring the topological phase transition. Thus, the stacking procedure that leads to the 3D TI is based on SL building blocks principally different from the Rashba bilayers used in ref. 21. It is essential that in our case the two Rashba constituents of the block (the standalone TLs) bring not only the Rashbasplit conduction band but also the valence band, see Supplementary Note 3. Then the gap in the SL (which may be inverted or not) is quite naturally the gap between the valence and conduction bands, in contrast to the scenario of ref. 21, where the band gap in the bilayer block is achieved by a dispersive “finite quantum tunneling” between the two Rashba constituents – the 2D electron gases of the adjacent layers.
Figure 4(d) shows the behaviour of the inverse effective masses of the chosen bands over the d_{vdW} interval considered. We find that the conductionband inverse masses, which are equal in the larged_{vdW} limit, , change smoothly with decreasing d_{vdW}: At the parameter becomes twice as large, while falls below zero. On the contrary, the valenceband inverse masses ( in the larged_{vdW} limit) “diverge” because the band ν_{1} moves down and “goes through” the Iorbital dominated bands, and ν_{2} moves up and hybridizes with Te p_{x,y} bands, see Fig. 3 and Supplementary Fig. 2. Finally, at the parameter reaches its large d_{vdW} limit, while becomes negative. Thus, in the topologically nontrivial 1 SL we have and , where C_{} and M_{} are the 1SL parameters of the Hamiltonian (2). At that, the interband coupling of the bands ν_{2} and c_{1} is equal to V_{} of the 4band k·p description. This reveals a close relation between the 4band and 8band Hamiltonians. However, already with 8 bands there is an absolute gap (see Fig. 3(a)), which is reasonably accurate and quite suitable for the theoretical research on linear response, Hall conductance, and motion of Dirac fermions in external fields.
Additional Information
How to cite this article: Nechaev, I. A. et al. Quantum spin Hall insulators in centrosymmetric thin films composed from topologically trivial BiTeI trilayers. Sci. Rep. 7, 43666; doi: 10.1038/srep43666 (2017).
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Acknowledgements
This work was supported by the Spanish Ministry of Economy and Competitiveness MINECO (Project Nos FIS201348286C21P, FIS201348286C22P, and FIS201676617P), the Basque Country Government, Departamento de Educación, Universidades e Investigación (Grant No. IT75613) and Saint Petersburg State University (Grant No. 15.61.202.2015).
Author information
Affiliations
Centro de Física de Materiales CFM  MPC and Centro Mixto CSICUPV/EHU, 20018 San Sebastián/Donostia, Spain
 I. A. Nechaev
 & E. V. Chulkov
Tomsk State University, 634050, Tomsk, Russia
 I. A. Nechaev
 , S. V. Eremeev
 & E. V. Chulkov
Saint Petersburg State University, 198504, Saint Petersburg, Russia
 I. A. Nechaev
 , S. V. Eremeev
 & E. V. Chulkov
Institute of Strength Physics and Materials Science, 634055, Tomsk, Russia
 S. V. Eremeev
Donostia International Physics Center, 20018 San Sebastián/Donostia, Spain
 S. V. Eremeev
 , E. E. Krasovskii
 , P. M. Echenique
 & E. V. Chulkov
Departamento de Física de Materiales UPV/EHU, Facultad de Ciencias Químicas, UPV/EHU, Apdo. 1072, 20080 San Sebastián/Donostia, Spain
 E. E. Krasovskii
 , P. M. Echenique
 & E. V. Chulkov
IKERBASQUE, Basque Foundation for Science, 48013, Bilbao, Spain
 E. E. Krasovskii
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Contributions
I.A.N. and S.V.E. conceived the idea and designed the research within the projects coordinated by P.M.E. and E.V.C., S.V.E. performed atomic structure optimization and GGA band structure calculations. I.A.N. and E.E.K. derived the smallsize k·p Hamiltonians from ab initio wave functions and performed the corresponding LDAbased calculations. I.A.N., S.V.E., and E.E.K. wrote the manuscript. All the authors discussed the results and commented on the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to I. A. Nechaev.
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