An isolated Dirac cone on the surface of ternary tetradymite-like topological insulators

We have extended the search for topological insulators to the ternary tetradymite-like compounds M2X2Y (M = Bi or Sb; X and Y = S, Se or Te), which are variations of the well-known binary compounds Bi2Se3 and Bi2Te3. Our first-principles computations suggest that five existing compounds are strong topological insulators with a single Dirac cone on the surface. In particular, stoichiometric Bi2Se2S, Sb2Te2Se and Sb2Te2S are predicted to have an isolated Dirac cone on their naturally cleaved surface. This finding paves the way for the realization of the topological transport regime.

2 materials fail in this regard due to their material-specific complications, such as interference from numerous surface states [16], shielding of the Dirac cone inside either the bulk-valence band [17,18] or the conduction bands [19], or the involvement of crystal distortion in fabricating the material [20,21].
Recently, it was shown experimentally that an isolated Dirac cone can be achieved by tuning the Fermi level with appropriate hole doping, but the non-stoichiometric crystal structure and the requirement for surface deposition generally make this approach unsuitable for most practical applications [11,18]. On the other hand, solid solutions in the tetradymitelike compounds provide a platform to engineer topological surface states. For example, the theoretical work of Zhang et al [22] indicates the presence of a topological phase transition in Sb 2 (Te 1−x Se x ) 3 for 0 x 1. Interestingly, since Se atoms preferentially occupy the central site in the crystal, at x = 1/3 the stoichiometric compound Sb 2 Te 2 Se can admit an ordered phase. Our recent work on the topological insulator Bi 2 Te 2 Se reveals that the linewidth of the topological surface states in angle-resolved photoemission spectroscopy is narrow, indicating that disorder effects are suppressed in these compounds [23]. Ren et al [24] have shown that to date Bi 2 Te 2 Se is the most suitable material for the study of the topological surface states because it has the highest bulk resistivity among all known topological insulators.
In this paper, we report first-principles computations on five existing tetradymite-like compounds-Bi 2 Te 2 Se, Bi 2 Te 2 S, Bi 2 Se 2 S, Sb 2 Te 2 Se and Sb 2 Te 2 S-to show that these materials host salient topological insulating features in their stoichiometric crystal structures. Our important findings include that the last three of the five aforementioned compounds harbor an isolated Dirac cone with Dirac points that lie fully within the gap, allowing these states to reach the long-sought territory of the topological spin-transport regime [6,15,25], where the in-plane carrier transport would have a purely quantum topological origin.
Topological insulators in two and three dimensions have been predicted theoretically and observed experimentally to display the quantum spin Hall effect [3,4,26] and strong topological behavior due to the combined effects of relativistic Dirac fermions and quantum entanglement [1,[3][4][5][6][7][8][9][10][11]. For systems with inversion symmetry, a band inversion occurs due to a finite value of the SOC. This results in a bulk-insulating gap that ensures the existence of metallic surface states within the gap [16]. In bismuth or antimony metals, numerous surface states interfere with each other and an odd number of Dirac cones can be achieved by making composite alloys such as Bi 1−x Sb x [10]. In other novel classes of topological insulators, the Dirac node appears either below the valence band maximum (VBM) as in Bi 2 Te 3 or Sb 2 Te 3 [18] or above the conduction band minimum as in TlSbTe 2 [19]. Topologically non-trivial materials such as the half-Heusler [20] and Li 2 AgSb [21] series are semi-metals, which require lattice distortion to be tuned into the topological insulating state. For these reasons, it is difficult to electrically gate these materials for the manipulation and control of charge carriers for realizing a device. Therefore, a viable route to search for the isolated Dirac node in topological insulating materials is to find a chemically stable stoichiometric composition in an appropriate class of materials.
With this motivation, we consider in this paper five existing stoichiometric tetradymite-like ternary compounds in the structure M 2 X 2 Y , namely Bi 2 Te 2 Se [27], Bi 2 Te 2 S [28], Bi 2 Se 2 S [29], Sb 2 Te 2 Se [30] and Sb 2 Te 2 S [31]. The tetradymite structure is with a rhombohedral unit cell of the space group R3m. The commonly invoked hexagonal cell shown in figure 1(a) consists of three quintuple layers. The stacking order of two consecutive units may be represented as unit. The natural surface termination is between the two X -atom layers. The known topological insulators Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 have the same crystal structure with X = Y . Figure 1(b) shows a schematic illustration of the bulk and surface band structures of Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 in which the surface Dirac point lies below the VBM. In our earlier experimental work, we lowered the Fermi level (E F ) of Bi 2 Se 3 into the bulk bandgap by substituting trace amounts of Ca 2+ for Bi 3+ , with Ca acting as a hole donor to the bulk states. In order to lift the Dirac point above the VBM, we deposited NO 2 on the surface. An isolated Dirac cone is then obtained, as illustrated in figure 1(c). In this study, we predict new topological insulators Bi 2 Te 2 Se, Bi 2 Te 2 S, Bi 2 Se 2 S, Sb 2 Te 2 Se and Sb 2 Te 2 S, among which the latter three have a Dirac point located in the bulk gap above the VBM, as shown in figure 1(c). Therefore, these three compounds are well suited as materials with an isolated Dirac cone without requiring tuning via surface deposition. With proper electron/hole doping control or gating, one can position the E F at the Dirac point and the material can be in the transport regime for potential device applications. We did not attempt to tune the electronic structure of the studied compounds using virtual crystal [32,33] or other first-principles approaches [34,35].
The first-principles band structures were computed on the basis of experimental lattice data [27][28][29][30][31] within the framework of the density functional theory (DFT) using the generalized gradient approximation (GGA). Our results, shown in figure 2, predict that all these materials naturally host a topological insulating phase with band gaps in the range of 224-297 meV. A band inversion occurs at the -point, as highlighted by gray-shaded areas. As illustrated in figure 2(f), the conduction band (red solid line) and the valence band (blue dashed line) cross each other near the -point, so that at this point the occupied state possesses the character of the conduction band and the unoccupied state possesses that of the valence band. Since the crystal has inversion symmetry, we have applied band parity analysis [5] to evaluate the value of the topological invariant Z 2 . All five compounds are thus found to be strong topological insulators with Z 2 = −1. This has come about due to band inversion at the -point through which the dispersion. All five compounds exhibit a single Dirac cone inside the bulk band gap. For Bi 2 Te 2 Se and Bi 2 Te 2 S, the Dirac point lies below the VBM. This is the same as the case of the stoichiometric binary compound Bi 2 Te 3 in which bulk scattering channels at the energy of the Dirac point prevent interesting quantum phenomena in the topological transport regime being realized. The most interesting results are found in Bi 2 Se 2 S, Sb 2 Te 2 Se and Sb 2 Te 2 S where an isolated Dirac cone is obtained. Here, the Dirac point lies above the VBM and below the conduction bands. When E F is tuned to the Dirac point, the density of states approaches zero and there are no other scattering channels. Therefore, Bi 2 Se 2 S, Sb 2 Te 2 Se and Sb 2 Te 2 S would be stoichiometric compounds that are well suited for investigating the topological transport regime.
In conclusion, we predict that the stoichiometric tetradymite-like compounds Bi 2 Te 2 Se, Bi 2 Te 2 S, Bi 2 Se 2 S, Sb 2 Te 2 Se and Sb 2 Te 2 S are strong topological insulators with a large enough band gap for room temperature applications. The isolated Dirac cone on the surface of Bi 2 Se 2 S, Sb 2 Te 2 Se and Sb 2 Te 2 S will be particularly important for the study of topological transport properties on a bulk material without the need for surface deposition [11]. Experiments with magnetic scattering techniques should be able to uniquely access the helical spin texture of the isolated surface states that are essential for device applications. The stoichiometric crystals of the investigated compounds are very easy to produce and manipulate for the manufacture of devices for microchips [36], spintronics [14] and information [12,13] technologies. Furthermore, the full exposure of the topological transport regime for dissipationless spin current with the unique advantage of tunable surface states is suitable for the study of novel topological phenomena such as the intrinsic quantum spin Hall effect [3,4,26], the universal topological magneto-electric effect [37] and anomalous half-integer quantization of Hall conductance [6,38] and quantum electrodynamical phenomena such as the image magnetic monopole induced by an electric charge [39] or Majorana fermions induced by the proximity effects from a superconductor [12].
Methods. First-principles band calculations were performed with the linear augmented plane wave method using the WIEN2K package [40] within the framework of DFT. GGA was used to describe the exchange-correlation potential [41]. SOC was included as a second variational step using the basis of scalar relativistic eigenfunctions. The surface was simulated by placing a slab of six quintuple layers for Bi 2 Te 2 Se and Bi 2 Te 2 S. For Bi 2 Se 2 S, Sb 2 Te 2 Se and Sb 2 Te 2 S, we used 12 quintuple layers to ensure convergence with the number of layers.