Abstract
The search for strongly inversion asymmetric topological insulators is an active research field because these materials possess distinct properties compared with the inversion symmetric ones. In particular, it is desirable to realize a large Rashba spinsplitting (RSS) in such materials, which combined with the topological surface states (TSS) could lead to useful spintronics applications. In this report, based on first principles calculations, we predict that the heterostructure of BiTeI/Bi_{2}Te_{3} is a strong topological insulator with a giant RSS. The coexistence of TSS and RSS in the current system is native and stable. More importantly, we find that both the invariants and the Rashba energy can be controlled by engineering the layer geometries of the heterostructure, and the Rashba energy can be made even larger than that of bulk BiTeI. Our work opens a new route for designing topological spintronics devices based on inversion asymmetric heterostructures.
Introduction
The study of the effects of spinorbit coupling (SOC) has been one of the central themes in physics research in the past decade. A significant discovery is that a strong SOC can convert a normal insulator into a topological insulating phase, in which metallic surface states appear in the bulk gap and they are protected by a nontrivial bulk topological order^{1,2}. Another important consequence of SOC is the Rashba effect in inversion asymmetric structures^{3,4}. It has been extensively studied in the context of spintronics, which aims to utilize the electron's spin degree of freedom for device applications. It is of great interest to search for a material hosting both robust TSS and large RSS which could be used as a playground to study the interplay between these two interesting effects and also as a base material for technological applications.
A nonzero RSS requires structural inversion symmetry breaking. It occurs naturally at sample surfaces or interfaces. For example, RSS has been observed at metallic surfaces^{5,6,7}, ultrathin metal films^{8,9}, and semiconductor heterostructures^{10}. In particular, ARPES experiments on the topological insulator Bi_{2}Se_{3} have shown that RSS exists at surfaces with artificially deposited adsorbates^{11,12,13,14}. Such RSS, however, are typically small and sensitive to the surface environment. On the other hand, a large RSS of the bulk bands is possible in polar compounds with heavy elements, such as the noncentrosymmetric layered semiconductor BiTeI which was recently reported to have the largest known RSS strength^{15}. However, BiTeI is a trivial insulator at ambient condition. It was proposed that under pressure BiTeI might be turned into a topological insulator^{16}. But pressure induced structural transition might also occur as suggested by experiments^{17,18}.
Motivated by the recent progress in the experimental fabrication of layered heterostructures^{19,20}, here we predict that the coexistence of robust TSS and giant RSS can be realized in the [(BiTeI)_{m}(Bi_{2}Te_{3})_{n}] heterostructure in its native state without any need of strain or pressure (m and n represent the number of layers of each constituent in a unit cell). Moreover, we find that both the topological order and the RSS energy can be controlled by varying the number of layers. In particular, the topological invariants of heterostructure with even (odd) m are given by [1;000] ([1;001]). And the RSS strength is tunable by varying the ratio between m and n. Especially the structures [(BiTeI)_{2}(Bi_{2}Te_{3})_{1}] and [(BiTeI)_{3}(Bi_{2}Te_{3})_{1}] have Rashba energies even larger than that of BiTeI. The TSS at different crystalline surfaces are analyzed in detail, which demonstrate characteristic features of a strongly inversion asymmetric topological insulator. Our findings thus identify a promising topological material for future spintronics and quantum computation applications.
Results
Basic crystal structure
The well known topological insulator material Bi_{2}Te_{3} has a layered structure with five atomic layers (TeBiTeBiTe) as a quintuple layer (QL) unit^{21}. BiTeI has a similar layered structure with three atomic layers (TeBiI) as a triple layer (TL) unit^{15}. The interactions between QLs (or TLs) are of weak vanderWaals (vdW) type, while the interactions within one QL (or TL) are of strong covalent or ionic type. Since inplane lattice constants of Bi_{2}Te_{3} (4.383 Å) and BiTeI (4.339 Å) are very close, their heterostructure [(BiTeI)_{m}(Bi_{2}Te_{3})_{n}], which consists of alternating m TLs of BiTeI and n QLs of Bi_{2}Te_{3}, is possible to be epitaxially grown along the crystallographic caxis^{22}. The simplest structure [(BiTeI)_{1}(Bi_{2}Te_{3})_{1}] (hereafter referred to as Bi_{3}Te_{4}I) is shown in Fig. 1(a). In the following, we shall choose it as a representative example for the discussion of general structural and electronic properties of this family of heterostructures.
The optimized lattice parameters and bond lengths without and with vdW corrections are listed in Table 1. As can be seen from the table, the interlayer bond lengths (ITe_{1} and Te_{3}Te_{4}) are much larger than the intralayer bond lengths in both cases. With vdW correction, the intralayer bond lengths are hardly changed, while the interlayer bond lengths are significantly reduced, confirming that the interactions between TL and its adjacent QLs are of weak and nonlocal vdW type. The Te_{3}Te_{4} and ITe_{1} bond lengths relaxed by vdWDF method are similar to the interlayer TeTe bond length in Bi_{2}Te_{3} (3.748 Å) and the ITe bond length in BiTeI (3.924 Å) respectively, which is reasonable. The dynamic stability of the structure is further investigated through the phonon spectrum calculations. The calculated phonon spectrum with SOC is shown in Fig. 1(c). The absence of imaginary frequencies demonstrates that our structure of Bi_{3}Te_{4}I is dynamically stable. In the following electronic calculations, the optimized structure with vdW corrections is adopted.
Giant Rashba spin splitting
The calculated band structure of Bi_{3}Te_{4}I without SOC is shown in Fig. 2(a). As we can see, Bi_{3}Te_{4}I is a direct band gap semiconductor with E_{g} ≈ 0.52 eV, much smaller than that of BiTeI (≈1.2 eV)^{23}. The direct band gap is located at Apoint of the Brillouin zone (labeled in Fig. 1(b)). The states near the band edges are dominated by Bi6p_{z} orbitals for the conduction band bottom, and by Te5p_{z} orbitals for the valence band top.
When SOC is turned on, as shown in Fig. 2(b), the energy bands are spinsplitted and at the same time the conduction band (local) minima are shifted away from the high symmetry points Γ and A, which is similar to the case of bulk BiTeI. This is the signature of a Rashba type spin splitting^{15}. The RSS is characterized by the Rashba energy E_{R}, the momentum offset k_{0} and the Rashba coupling parameter α_{R} = 2E_{R}/k_{0}. These parameters are illustrated in Fig. 2(c) for the conduction band bottom near Γ point. The RSS at Γ point is slightly anisotropic. Along ΓK direction, we have E_{R} ≈ 76 meV and α_{R} ≈ 2.54 eV·Å, while along the ΓM direction, the corresponding values are smaller, with E_{R} ≈ 74 meV and α_{R} ≈ 2.48 eV·Å. The RSS of conduction band near A point has a slightly smaller Rashba energy (E_{R} ≈ 56 meV), but the value of α_{R} (≈2.51 eV·Å) is close to that of the Γ point. The value of RSS in the heterostructure Bi_{3}Te_{4}I is much larger than the RSS of conventional semiconductor heterostructures (on the order of meV). This giant RSS can be viewed as being inherited from the BiTeI component. The bulk BiTeI has a RSS of 116 meV (k_{0} ≈ 0.05 Å^{−1}) in our calculation, which is in good agreement with earlier calculation (113 meV)^{23} and experimental ARPES data (100 meV)^{15}.
Topological insulating phase
In addition to RSS, the band order is also inverted after the introduction of SOC. In the presence of SOC, spin and orbital angular momentum are mixed and the p_{z} states are transformed into j = 1/2 states. As a consequence, the conduction band bottom, which is mainly occupied by the Bip_{z} orbital, is pushed downward. Meanwhile the valence band top, which is occupied by the Tep_{z} orbital, is pushed upward. Due to the large atomic SOC from Bi and Te, the energy gap around A point is inverted. To facilitate our understanding of this band inversion process, we gradually increase the strength of SOC (λ_{SO}) from zero to its full value, and monitor the evolution of the band structure. The local band gap at k_{z} = 0.5 plane (around A points) and at k_{z} = 0.0 plane (around Γ point) is plotted in Fig. 2(d). As λ_{SO} increases, the local gap around Γ points decreases from 0.85 eV to 0.12 eV without closing. In contrast, the local gap around A points decreases, closes, reopens, and finally reaches a value of 0.22 eV. This band inversion induced by SOC signals a topological phase transition in the current system.
To pin down the topological phase of the heterostructure more precisely, we calculate the topological invariants. Since the inversion symmetry is explicitly broken in Bi_{3}Te_{4}I, the parity criterion can not apply^{24}. Instead, we evaluate the invariants ν_{0}; (ν_{1}ν_{2}ν_{3}) in terms of the Berry gauge potential and Berry curvature by using our recently developed general method within the fullpotential linearized augmented plane wave framework^{25}. The calculated for Bi_{3}Te_{4}I is consistent with the previous band inversion analysis. At first sight, this topological insulating phase seems to be inherited from the Bi_{2}Te_{3} component. However, it should be noted that the invariants of Bi_{3}Te_{4}I is different from that of Bi_{2}Te_{3} . Instead it is more like bulk BiTeI under pressure^{16}, which has the same topological invariants. The difference in the index ν_{3} reflects the different band inversion point in the Brillouin zone. For Bi_{2}Te_{3}, band inversion occurs at Γ point, while for Bi_{3}Te_{4}I band inversion occurs at A point. The nonzero value of ν_{3} can have interesting physical consequences. For example, it has been shown that a system with ν_{3} = 1 may offer a topologically protected onedimensional conducting channel along the caxis^{26}.
The nontrivial value is tied to the existence of TSS^{27}. As another verification, we calculate the surface energy spectrum directly. Fig. 2(e) shows the local density of states (LDOS) of a sidecut surface (the ac plane of Bi_{3}Te_{4}I). A Dirac cone in the bulk energy gap can be clearly seen, which is located at point of the projected twodimensional Brillouin zone. This again confirms the topological insulating phase of Bi_{3}Te_{4}I.
Controllability of properties
Thus far, we have shown that the simplest heterostructure Bi_{3}Te_{4}I was a strong topological insulator with giant bulk RSS. Now we extend our investigation to the family of heterostructures [(BiTeI)_{m}(Bi_{2}Te_{3})_{n}] with different number of layers m and n, and see how the topological invariants and the RSS depend on the layer configuration. The calculated results are summarized in Table 2. We observe that the coexistence of strong topological insulating phase and giant bulk RSS is common to all these heterostructures. Quite interestingly, the ν_{3} index of has a dependence on the parity of m. Namely ν_{3} = 1 when m is odd and ν_{3} = 0 when m is even. For odd m, band inversion occurs at A point, while for even m, it occurs at Γ point. The location of the surface Dirac cone is associated with the band inversion point, hence can also be controlled by the parity of m. (Note that this is meaningful for side surfaces where the projections of A and Γ are distinguishable.) From the results, we see that the Rashba energy is also tunable to a great extent by varying m and n. E_{R} generally increases with larger m and smaller n. It should be noted that heterostructures [(BiTeI)_{2}(Bi_{2}Te_{3})_{1}] and [(BiTeI)_{3}(Bi_{2}Te_{3})_{1}] have extremely large E_{R} of 130 meV (α_{R} ≈ 5.23 eV·Å) and 139 meV (α_{R} ≈ 5.45 eV·Å) respectively, which are both larger than that of bulk BiTeI.
Surface states
A salient feature of an inversion asymmetric topological insulator is that different crystal surfaces have different TSS. In our heterostructure system, due to the giant RSS and the freedom to control the properties by layer configurations, more interesting physics are expected to appear on the sample surface. As two representative examples with different invariants, we shall discuss Bi_{3}Te_{4}I with and Bi_{4}Te_{5}I_{2} ([(BiTeI)_{2}(Bi_{2}Te_{3})_{1}]) with in the following.
Due to the lack of inversion symmetry, there are two different surfaces of Bi_{3}Te_{4}I (or Bi_{4}Te_{5}I_{2}) perpendicular to the caxis, namely the Iterminated surface and the Teterminated surface. They are constructed by cleaving the adjacent QL and TL (TL and TL for Bi_{4}Te_{5}I_{2}) through the weakly coupled TeI gap. As can be seen in Fig. 3(a,c), although with different ν_{3} index in , the two heterostructures have similar surface spectra. There are two different kinds of electronic states in the energy window around the bulk band gap. The states with dispersions identical for both Iterminated and Teterminated surfaces are the bulk RSS states whose wavefunctions spread over the bulk region, thus are insensitive to the surface environment. The other states with linear dispersions that cross the bulk gap are the TSS. Owing to the strong inversion asymmetry, TSS are very different at different atomterminated surfaces. For both heterostructures, the Dirac point of TSS stays in bulk band gap at Iterminated surface [Fig. 3(a, c)], while it is buried in bulk valence bands at Teterminated surface [Fig. 3(b, d)]. If Fermi level lies in the bulk gap below the Dirac point of Iterminated surface, the two surfaces of a sample would have different types of charge carriers, which can be used to construct a topological pn junction^{28}.
Due to the coexistence of two types of spinmomentum locking—TSS and RSS, the surface spin texture is expected to be more complicated in our system. In Fig. 3(e–g), we show the spin projections of S_{x}, S_{y}, and S_{z} for the Iterminated surface of Bi_{3}Te_{4}I at different constant energies. In Fig. 3(e1) at energy 250 meV, there are three contours: inner, middle, and outer ones. All of them have the largest spin distributions along k_{x} direction while suppressed values along k_{y} direction. The outer and inner ones are from the RSS states which have opposite spin polarizations, while the middle one is from the TSS. When shifting the energy level below the RSS crossing point [Fig. 3(e2)], the inner spin contour will change its polarization and then switches place with the middle TSS contour and merges with the outer one [Fig. 3(e3)], leaving only two spin contours. Below the conduction band edge, only the TSS contour remains and its spin distribution changes sign when the energy level crosses the Dirac point [Fig. 3(e4)]. The spin projection of S_{y} shown in Fig. 3(f) shows similar features. Combining results in Fig. 3(e) and 3(f), one can easily obtain the inplane spin chiralities of TSS and RSS states. For TSS, the upper and lower Dirac cones have the left and righthanded spin textures respectively. For RSS states, the innerbranch has the left (right) handed spin textures above (below) the RSS crossing point, while the outerbranch always has the righthanded spin texture. The S_{z} projection shown in Fig. 3(g) represents the outofplane spin texture. It can be observed that the outer RSS branch has the largest outofplane spin component than TSS and the inner branch of RSS. For the Teterminated surface, only the spin projections for the upper Dirac cone are shown [Fig. 3(h)–3(j)] because the lower Dirac cone is deeply buried in the bulk valence bands. The spin textures of RSS states are identical to those of Iterminated surface, hence are not shown here.
Apart from the spinmomentum locking discussed above, recent theoretical and experimental works found that in topological insulator Bi_{2}Se_{3} there was an additional locking between spin and orbital for TSS, leading to a spinorbital texture^{29,30,31}. Following the method of Zhang et al.^{30}, we calculate the inplane (p_{x} and p_{y}) and outofplane (p_{z}) orbital projections together with the orbitalselective spin textures for TSS of both Iterminated and Teterminated surfaces of Bi_{3}Te_{4}I. First let's examine the Iterminated surface. As shown in Fig. 4, for the inplane orbitals, the righthanded spin texture is coupled to tangential (radial) orbital texture on the upper (lower) Dirac cone. (tangential and radial inplane orbitals refer to linear combinations of p_{x} and p_{y} orbitals with their orbital axis along tangential or radial directions^{30}.) For the outofplane p_{z} orbital, it is coupled to the left(right) handed spin texture on the upper (lower) Dirac cone. As for the Teterminated surface, the orbitalselective spin textures are opposite to the Iterminated ones. We have therefore demonstrated that the interesting spinorbital texture of TSS, initially observed in Bi_{2}Se_{3}^{31}, also appears in our inversion asymmetric heterostructure systems.
Discussion
As we mentioned before, in several aspects, the BiTeI/Bi_{2}Te_{3} heterostructure retains the properties of the bulk BiTeI. However, the bulk BiTeI is a trivial insulator. It can only be converted to a topological insulator under pressure. In contrast, for the system which we study, no external constraint is required. It is the intercalation by layers of Bi_{2}Te_{3} that changes the topological order of the structure. To better understand this result, we start from the orbital analysis of Bi_{2}Te_{3} and BiTeI. Let's focus on the conduction band edge. The states there are mainly from Bi p orbitals. For a QL of Bi_{2}Te_{3}, the p orbitals from the two Bi atoms can be combined to form bonding and antibonding orbitals. When crystal field is taken into account, p_{z} orbital is split from p_{x,y} orbitals, with the bonding orbital having the lower energy^{21}, as shown in the left shadowed area of Fig. 5. Similar energy splitting due to crystal field also occurs for BiTeI^{23}.
When the layers of two materials are stacked together to form a heterostructure, due to the overlap between orbitals, the levels with close energies and the same symmetry character are coupled and repel each other. Using Bi_{3}Te_{4}I as an example, on the right hand side of Fig. 2(a), we show the Bi_{1}p_{z} (Bi in TLs) and (Bi in QLs) orbital projected character of the low energy conduction bands. At Γ point, the and Bi_{1}p_{z} orbital are mainly concentrated in the lowest and the second lowest conduction band respectively. As moving towards A point, the coupling between and Bi_{1}p_{z} orbitals is getting stronger, which pushes the lowest conduction band downward. Therefore the intercalated Bi_{2}Te_{3} not only provides the orbital which has a lower energy than the Bi_{1}p_{z} orbital, but also through the coupling between the two orbitals it helps to further reduce the band gap (from 1.2 eV in bulk BiTeI to 0.52 eV in Bi_{3}Te_{4}I). Hence it becomes easier to achieve the band gap inversion when SOC is turned on.
This band gap reduction induced by intercalation also helps to explain the observed giant RSS. As been pointed out in the study of bulk BiTeI^{23}, one of the several factors contributing to a large RSS is that the band gap should be small. Approximately, the RSS strength is inversely proportional to the gap size. The reduced gap size of the heterostructure helps to realize a giant RSS. This point is reflected from the results in Table 2. The two structures ([(BiTeI)_{2}(Bi_{2}Te_{3})_{1}] and [(BiTeI)_{3}(Bi_{2}Te_{3})_{1}]) with RSS energy greater than that of bulk BiTeI indeed have very small band gaps.
To conclude, we have investigated the crystal structure and electronic properties of the heterostructures [(BiTeI)_{m}(Bi_{2}Te_{3})_{n}] by first principles calculations. We discover that these heterostructures host both strong topological insulating phase and giant RSS in their native states. Moreover, both the invariants that characterize the topological properties and the Rashba energy that characterizes strength of RSS can be controlled by varying the layer numbers m and n. The surface electronic structure and spin texture present unique features of the coexistence of TSS and RSS, and of strongly inversion asymmetric topological insulators^{32}. Our result indicates a rich physics in this class of heterostructures which provide an ideal platform for future development of spintronics and quantum computation devices, for example designing the topological pn junctions, creating majorana fermions via combining with superconductors^{33}.
Methods
Structure optimizations
The lattice parameters and the atomic positions of the heterostructures are optimized using the projector augmented wave method^{34} implemented in VASP code^{35,36}. PerdewBurkeErnzerhof parametrization of the generalized gradient approximation (GGAPBE) is used for the exchange correlation potential^{37}. Since the interaction between TLs and QLs is of the vdWtype similar to bulk BiTeI and Bi_{2}Te_{3}, the vdW corrections by the approach of Dion et al.^{38} (vdWDF) as implemented in VASP is employed to further relax the outofplane lattice parameter c and atomic positions^{39}. The plane wave energy cutoff is set to 300 eV, and the Brillouin zone is sampled by a 16 × 16 × 4 mesh. Both the crystal's shape and the atomic positions are relaxed until force on each ion becomes less than 0.01 eV·Å^{−1}. The phonon spectrum is calculated using the PHONOPY code^{40} through the DFPT approach^{41}.
Bulk electronic structure calculations
The electronic groundstate calculations are performed using fullpotential linearized augmented planewave (FPLAPW) method^{42}, implemented in the package WIEN2K^{43}. The convergence criteria are more stringent than those of structural optimizations, with K_{max}R_{MT} = 8.0 and a 20 × 20 × 4 kmesh, where R_{MT} represents the smallest muffintin radius (2.5 Bohr for all atoms) and K_{max} is the maximum size of reciprocallattice vectors. Spinorbit coupling is included by a secondvariational procedure^{42}.
Surface electronic structure calculations
Surface electronic structures are calculated using the combination of maximally localized wannier function (MLWF)^{44,45} and surface Green's function methods^{46,47}. The construction of MLWF using package wannier90^{48} is a nonselfconsistent process on a 8 × 8 × 3 kmesh with formerly converged selfconsistent charge potential. The surfaces with different atomterminations are considered as semiinfinite systems, from which the iterative method^{46,47} is used to obtain the surface Green's function. The LDOS is extracted from the imaginary part of the surface Green's function, which is used to analyze surface electronic structures. With the advantage of addressing an individual surface, the surface Green's function methods is more suitable for the analysis of surfacedependent TSS.
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Acknowledgements
This work was supported by the MOST Project of China (Grants Nos. 2014CB920903, 2011CBA00100), SUTDSRGEPD2013062, NSFC (Grant Nos. 11174337, 11225418, 11135001 and 11374033), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grants No. 20121101110046 and 20131101120052).
Author information
Affiliations
Institute of Physics, Chinese Academy of Sciences and Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China
 JinJian Zhou
School of Physics, Beijing Institute of Technology, Beijing 100081, China
 Wanxiang Feng
 & Yugui Yao
Department of Physics, Beijing Normal University, Beijing 100875, China
 Ying Zhang
Engineering Product Development, Singapore University of Technology and Design, Singapore 138682, Singapore
 Shengyuan A. Yang
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Contributions
Y.Y. conceived the idea and designed the research. J.J.Z. performed calculations. J.J.Z., W.F., Y.Z., S.A.Y. and Y.Y. contributed to the analysis and interpretation of the results and prepared the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Shengyuan A. Yang or Yugui Yao.
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This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/
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Further reading

1.
Scientific Reports (2015)

2.
Robustness of Rashba and Dirac Fermions against Strong Disorder
Scientific Reports (2015)
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