Quantum spin Hall (QSH) insulators are a peculiar phase of matter exhibiting excellent quantum transport properties with potential applications in lower-power-consuming electronic devices. Currently, among all predicted or synthesized QSH insulators, square and hexagonal atomic rings are the dominant structural motifs, and QSH insulators composed of pentagonal rings have not yet been reported. Here, based on first-principles calculations, we predict a family of large-gap QSH insulators in SnX2 (X=S, Se, or Te) two-dimensional (2D) crystals by the direct calculation of Z2 topological invariants and edge states. Remarkably, in contrast to all known QSH insulators, the QSH insulators predicted here are composed entirely of pentagonal rings. Moreover, these systems can produce sizeable nontrivial gaps ranging from 121 to 224 meV, which is sufficiently large for practical applications at room temperature. Additionally, we propose a quantum well by sandwiching an SnTe2 2D crystal between two BiOBiS2 sheets and reveal that the considered 2D crystal remains topologically nontrivial with a sizeable gap. This finding demonstrates the robustness of its band topology against the effect of the substrate and provides a viable method for further experimental studies.
Topological insulators (TIs) have sparked extensive research activities in recent years because of their rich physics and promising applications in quantum devices and spintronics.1, 2 The resultant Dirac surface states in three-dimensional (3D) TIs as well as the helical edge states in two-dimensional (2D) TIs are spin locked because of protection by time-reversal symmetry; thus, they are robust against perturbations. Particularly in 2D TIs, also known as quantum spin Hall (QSH) insulators,3, 4 all the low-energy scatterings of the edge states caused by the nonmagnetic defects are completely forbidden because the edge electrons can only propagate along two directions with opposite spins, which makes 2D TIs more suitable for low-power-consuming applications than 3D TIs. Unfortunately, the material realization of 2D TIs is challenging: while 3D TIs have been discovered in many materials, such as the Bi2Se3 family,5, 6 the experimental realization of 2D TIs is thus far limited to the HgTe/CdTe7 and InAs/GaSb8 quantum wells. Moreover, the QSH effect in these two quantum wells can occur only at ultralow temperature (<10 K) because of their extremely small bulk band gaps, and this limitation greatly obstructs their possible applications. The search for new 2D TIs with large band gaps that could support room temperature applications has thus become critically important.
Considerable effort has been devoted thus far to search for new 2D materials or new schemes to realize 2D TIs, and a variety of large-gap 2D TIs have been proposed theoretically. These large-gap 2D TIs include silicene,9 Bi(1 1 1) bilayers,10 III-Bi bilayers,11 BiF 2D crystals,12 Bi4Br4 single layers,13 chemically modified Ge/Sn14, 15, 16 and Bi/Sb17, 18 honeycomb lattices, ZrTe5/HfTe519 and 2D transition metal dichalcogenides.20, 21, 22, 23 In terms of geometrical motifs, square, hexagonal and pentagonal rings are considered to be the three basic building blocks of 2D materials. However, among all predicted or synthesized 2D TIs,9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 square and hexagonal rings are almost the only building blocks that have been found. Such a limitation mostly arises from the fact that the presence of a gapless band structure in these 2D materials makes them promising for harboring QSH states. Therefore, the common expectation is that the QSH effect cannot survive in pentagonal ring-based 2D crystals. To the best of our knowledge, extended 2D TIs composed exclusively of pentagonal rings have never been reported at all, although recent efforts have been conducted to search for such pentagonal ring-based 2D materials.24, 25
The purpose of this work is to explore novel 2D topological phases, especially at room temperature, in the supposedly ordinary pentagonal ring-based 2D crystals. The practical means of doing so will not only significantly enrich our fundamental understanding of the topological phenomena but also greatly broaden the possibilities for realistic applications of the QSH effect. After extensive research, we show that 2D TIs, consisting entirely of pentagonal rings, can indeed exist, namely in the family of SnX2 2D crystals, in which X=S, Se or Te. These SnX2 systems are identified to be room-temperature 2D TIs with sizeable nontrivial band gaps ranging from 121 to 224 meV, and they form robust QSH systems. These new QSH insulators present single Dirac cone edge states crossing the bulk band gap, which is ideal for dissipationless transport. More notably, by sandwiching the SnTe2 2D crystal between two BiOBiS2 sheets, we propose a promising quantum well configuration in which the considered 2D crystal remains topologically nontrivial with a sizeable band gap, thus inducting the robustness of its band topology against the effect of the substrate.
Density functional theory (DFT) calculations are performed using the plane wave basis Vienna ab initio simulation package.26, 27 The exchange-correlation potential is described by the Perdew–Burke–Ernzerhof28 version of the generalized gradient approximation.29 The plane-wave cutoff is set to 500 eV, and the convergence threshold for energy is set to 10−6 eV. Monkhorst–Pack k-point meshes30 of 9 × 9 × 1 and 13 × 13 × 1 are used for geometry optimization and self-consistent electronic structure calculations, respectively. The 2D crystals are modeled by the periodic slab approach with a vacuum layer of more than 18 Å to ensure decoupling between periodic images. All structures, including the lattice parameters and the internal ion coordinates, are fully optimized with a residual force tolerance of 0.01 eV Å−1. The spin-orbit coupling (SOC) is included in the self-consistent electronic structure calculations. For the quantum well structure, the van der Waals interaction is included by using the density functional theory-D2 approach.31
The schematic crystal structures of SnX2 2D crystals are shown in Figures 1a and b. We refer to these two configurations as α- and β-SnX2, respectively. β-SnX2 is more stable than α-SnX2 by 0.7, 22.8 and 25.8 meV per atom in energy for X=S, Se and Te, respectively. Supplementary Table S1 gives the corresponding structural details. From the top view, we can see that both α- and β-SnX2 2D crystals resemble the structure of experimentally identified layered silver azide32 and are composed entirely of the pentagonal rings; they present an amazing pattern that is well known as Cairo pentagonal tiling.33 The crystal structure of β-SnX2 displays P-421m symmetry (point group D2d) and a square lattice that contains two Sn and four X atoms in one unit cell. Its unit cell can be considered a triple-layer structure with an Sn layer sandwiched between two X layers, as shown in Figure 1b. α-SnX2 2D crystals also present a square lattice, but the symmetry is P4/mbm (point group D4H). This indicates that α-SnX2 2D crystals have inversion symmetry, whereas the inversion symmetry does not hold for the β-SnX2 2D crystals. Figure 1c shows the first Brillouin zone of SnX2 2D crystals, which is also square with four X points on the side centers and four M points on the corners. These systems predicted here are different from the known SnX2 2D crystals (labeled as T-SnX2). The T-SnX2 2D crystals display the CdI2-type structure with a hexagonal lattice instead of a square lattice. The symmetry space group of T-SnX2 2D crystals is P-3m1 (point group D3D), and it contains one Sn and two X atoms in the unit cell. For more details, please see Supplementary Figure S1 and the corresponding discussion.
The electronic band structures of SnX2 2D crystals without and with the inclusion of SOC are shown in Figure 2. Evidently, the valence and conduction bands away from the Γ point are well separated. Thus, we only need to focus on the bands around the Γ point. By projecting the bands onto different atomic orbitals, we find that the bands around the Fermi level are mainly contributed by pz−, degenerate pz+, px, y+ and degenerate px, y− orbitals of the Sn/X atoms. Especially for α-SnSe2/SnTe2 2D crystals at the Γ point, when excluding SOC, the Fermi level separates the (px, y+, degenerate px, y−) orbitals from the (pz−, degenerate pz+) orbitals, with the (px, y+, degenerate px, y−) orbitals located above the (pz−, degenerate pz+) orbitals. Furthermore, for these two 2D crystals without including SOC, one energy band with the px, y− orbital character overlaps with another energy band with the pz− orbital character around the Fermi level, as shown in Figures 2b and c. This band overlap leads to the appearance of two band crossings located in very close proximity to the Fermi level near the Γ point for α-SnSe2/SnTe2 2D crystals. Such band crossings can be understood as a result of the symmetry incompatibility of these two bands. Alternately, compared with α-SnSe2/SnTe2 2D crystals, the bands around the Fermi level are markedly modified in β-SnSe2/SnTe2 2D crystals. For β-SnSe2/SnTe2 2D crystals, the energy band with the pz− orbital character is located far above the Fermi level, which leaves the degenerate px, y− orbitals on the Fermi level. As a result, without including SOC, β-SnSe2/SnTe2 2D crystals are gapless, and the valence band maximum and conduction band minimum degenerate at the Fermi level at the Γ point.
After including SOC, for α-SnX2 2D crystals, the band gap opens at the band crossing; see Figures 2a–c. The systems have indirect band gaps of 130, 224 and 121 meV for X=S, Se and Te, respectively (see Figure 1d). To understand the role of SOC on the band structures, we show the schematic diagram of the band evolution at the Γ point of α-SnSe2 and α-SnTe2 2D crystals under SOC in Figure 1e. λ, ξ and δ in Figure 1e denote the band gap without SOC, the band splitting between degenerate px, y− orbitals, and the band gap with SOC, respectively. We focus on the pz−, degenerate pz+, px, y+ and degenerate px, y− orbitals around the Fermi level and neglect other atomic orbitals; see Figure 1e. Here, the superscripts ‘+’ and ‘−’ represents the bonding and antibonding states, respectively. For α-SnSe2/SnTe2 2D crystals, without including SOC, one pz− orbital lies below the degenerate px, y− orbitals, and the Fermi level stays between them; see Figure 1e. When switching on SOC, the degenerate px, y− orbitals split, and the order of one px, y− orbital and the pz− orbital is exchanged. According to the band overlap features shown in Figures 2b and c, it seems that such nontrivial band inversion before including SOC is induced by the inversion between the pz− and px, y− orbitals near the Fermi level at the Γ point; however, this is not that case. We know that a band inversion that can affect the band topology, namely a nontrivial band inversion, refers to the inversion between bands with different parities. Otherwise, if a band inversion occurs between bands with the same parity, then it is a trivial band inversion; therefore, it could not affect the band topology. Our parity analysis shows that the pz− and px, y− orbitals all display ‘−’ parity, which indicates that a nontrivial band inversion can never be induced by the inversion between these orbitals. In other words, here, the band overlap features near the Fermi level, as shown in Figures 2b and c, have nothing to do with the intrinsic nontrivial band inversion. Actually, the nontrivial band order before including SOC in these systems stems from the band inversion between the (pz−, degenerate pz+) and (px, y+, degenerate px, y−) orbitals, which is caused by the crystal field. To elucidate the underlying band inversion mechanism explicitly, we display the orbital evolution at the Γ point under the crystal field effect in Figure 1e (namely, from (I) to (II)). The crystal field effect is estimated by comparing the orbital orders with those of the same crystal that is strained to 170%, as suggested by Zhou et al.34 and Huang et al.35 Without including the crystal field effect (see (I) of Figure 1e), the (pz−, degenerate pz+) orbitals are located above the Fermi level, whereas the (px, y+, degenerate px, y−) orbitals are located below the Fermi level. At this moment, the band topologies of α-SnSe2/SnTe2 are trivial. Under the crystal field effect (see (II) of Figure 1e), a band inversion occurs between the (pz− orbital with ‘−’ parity, degenerate pz+ orbitals with ‘+’ parity) and (px, y+ orbital with ‘+’ parity, degenerate px, y− orbitals with ‘−’ parity), indicating a topological phase transition from a trivial insulator to a nontrivial insulator. Obviously, the subsequent SOC-induced band exchange between the pz− and px, y− orbitals does not contribute to the band inversion because they both display ‘−’ parity. This is different from the situation in the previously reported systems, such as Bi2Te3,36 where the SOC-induced band exchange usually changes the band topology.
To understand the difference in band structures between α-SnS2 and α-SnSe2/SnTe2 2D crystals, we further display the orbital evolution at the Γ point for α-SnS2 2D crystal in Supplementary Figure S2. As shown in Supplementary Figure S2, similar to α-SnSe2/SnTe2 2D crystals, the (pz−, degenerate pz+) orbitals also lie above the (px, y+, degenerate px, y−) orbitals before including the crystal field effect, and then a band inversion occurs between the (pz−, degenerate pz+) and (px, y+, degenerate px, y−) orbitals after including the crystal field effect, thus driving the α-SnS2 2D crystal into a topological phase. Here, the SOC also does not contribute to the band inversion. It should be noted that the crystal field effect shifts the (pz−, degenerate pz+) orbitals of the α-SnS2 2D crystal far above the Fermi level with respect to the case of α-SnSe2/SnTe2, as shown in Supplementary Figure S2. Therefore, the subsequent SOC-induced band exchange between the pz− and px, y− orbitals observed in α-SnSe2/SnTe2 does not occur in the α-SnS2 2D crystal (see Figure 1e and Supplementary Figure S2).
For β-SnSe2 and β-SnTe2 2D crystals, SOC also lifts the degeneracy of the bands at the Fermi level, and the conduction band shifts upwards while the valence band shifts downwards, which forms a band gap of 146 and 160 meV, respectively. The corresponding schematic diagram of the band evolution at the Γ point under the crystal field effect and SOC in sequence is plotted in Figure 1f. Before including the crystal field effect (see (I) of Figure 1f), the (pz−, degenerate pz+) and (px, y+, degenerate px, y−) orbitals are located above and below the Fermi level, respectively. After including the crystal field effect (see (II) of Figure 1f), a band inversion occurs between the degenerate pz+ and (px, y+, degenerate px, y−) orbitals, indicting a topological phase transition from a normal insulator to a TI. This leaves the Fermi level right on the degenerate px, y− orbitals; subsequently, after turning on SOC, the degeneracy of the two px, y− orbitals is lifted; see the SOC part of Figure 1f. Therefore, β-SnSe2/SnTe2 2D crystals also display a nontrivial band order before including SOC, which also arises from the crystal field effect. In these systems, the SOC-induced band exchange also does not contribute to the band inversion, and the only effect of SOC is to create energy gaps at the touching points. By comparing Figures 1e and f, we can see that, without including the crystal field, the orbital order of the β-SnSe2/SnTe2 2D crystals is similar to that of α-SnSe2/SnTe2, that is, the (pz−, degenerate pz+) orbitals are located above the (px, y+, degenerate px, y−) orbitals. This is due to the similar elemental composition of α- and β-SnSe2/SnTe2 2D crystals. However, as shown in Figures 1a and b, the geometric structures of the α and β phases are very different, which indicates that the crystal field effects in the α- and β-SnSe2/SnTe2 2D crystals are very different. Thus, after including the crystal field effect, a band inversion occurs between the degenerate pz+ and (px, y+, degenerate px, y−) orbitals for β-SnSe2/SnTe2 2D crystals, whereas the (pz−, degenerate pz+) and (px, y+, degenerate px, y−) orbitals are inverted for the α phase (see Figures 1e and f). As a result, in contrast to the α phase in which the pz− orbital shifts below the degenerate px, y− orbitals, the pz− orbital of β-SnSe2/SnTe2 2D crystals remains above the degenerate px, y− orbitals after including the crystal field effect. Moreover, by comparing Figures 2d–f, we notice that SOC is responsible for a significant spin splitting of both the valence and conduction bands of β-SnX2 2D crystals around the Fermi level. The origin lies in the break of inversion symmetry in β-SnX2 2D crystals, which induces the local dipole field and thus leads to the spin splitting. Alternately, because α-SnX2 2D crystals possess inversion symmetry, bands in Figures 2a–c are thus spin degenerate in the presence of SOC. For the β-SnS2 2D crystal, the scenario of the band evolution under SOC is very special. This scenario is different from the other five systems because after including SOC, the band gap does not open, and the entire system is metallic. Therefore, for the remainder of our discussion, we will focus on the α-SnX2, β-SnSe2 and β-SnTe2 2D crystals.
To identify the topological nature of the insulating phases in these systems, we investigate the Z2 topological invariants within the density functional theory frame. Z2=1 characterizes a topologically nontrivial phase, whereas Z2=0 corresponds to a topologically trivial phase. For α-SnX2 2D crystals, the existence of inversion symmetry simplifies the calculations of Z2 topological invariants. According to the method developed by Fu and Kane,37 topological invariants can be easily obtained by evaluating the parity eigenvalues of each pair of Kramer’s degenerate occupied energy bands at the four time-reversal-invariant momenta points (one Γ, one M and two X) of the Brillouin zone. Owing to the square symmetry, the parities at the two X points are identical, δ(X)2≡1, which has no effect on the band topology; thus, the topological invariants can be reduced to (−1)ν=δ(Γ) δ(M). In α-SnX2 2D crystals, the product of the parity eigenvalues at the Γ point is −1, whereas it is +1 at the M point; thus, we conclude that α-SnX2 2D crystals are nontrivial 2D TIs with Z2=1. For more details, see Supplementary Table S2. For β-SnSe2 and β-SnTe2 2D crystals, the inversion symmetry is broken, and the Z2 topological invariants cannot be determined by the method mentioned above. Instead, we use the method in Fukui and Hatsugai38 to directly perform the lattice computation of the Z2 topological invariants in terms of the n-field configuration. The resultant Z2=1 verifies that β-SnSe2 and β-SnTe2 2D crystals are also 2D TIs. The room-temperature QSH effect can be readily realized in all these systems if one considers their nontrivial band gaps. We emphasize that these results go beyond all predicted or synthesized 2D TIs9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 in which square and hexagonal atomic rings are almost the only building blocks. We show, for the first time, that the QSH effect can also occur in the supposedly ordinary 2D crystals composed entirely of pentagonal rings. Our work opens new avenues for the discovery of additional large-gap 2D TIs in different and unexplored classes of systems.
To further reveal the nontrivial topological nature of these systems, we perform calculations to examine the topological edge states on the boundaries of α-SnX2, β-SnSe2 and β-SnTe2 2D crystals. They are calculated using a nanoribbon structure with symmetric edges, which leads to two energetically degenerate edge states located at opposite sides. The corresponding schematic ribbon model is shown in Figure 3a. All the dangling bonds of the edge atoms are saturated by hydrogen atoms. The widths of the nanoribbons are set large enough to avoid interactions between the two edges. Figures 3b–f presents the calculated band structures of α-SnX2, β-SnSe2 and β-SnTe2 nanoribbons. One can explicitly see that each edge has a single pair of helical edge states (yellow lines) for all five of these systems, and the helical edge states disperse in the bulk band gap and cross linearly at the point. These features further prove the nontrivial nature of these systems, which is consistent with the Z2 calculations. Remarkably, in the cases of α-SnS2, α-SnSe2, α-SnTe2 and β-SnTe2, the Dirac points formed by the helical edge states lie exactly at the Fermi level. Such placement of Dirac points is important for actual applications. Moreover, their sizeable bulk band gaps can stabilize the edge states against the interference of the thermally activated carriers, which are beneficial for observing the room-temperature QSH effect in these 2D crystals.
In the following section, we take α-SnTe2 2D crystal as an example and discuss the realistic possibility of experimentally realizing these 2D crystals. We find that the widely used square BiOBiS2 sheet39 can be used as a suitable substrate for an α-SnTe2 2D crystal in experiments. The in-plane lattice constant of an α-SnTe2 2D crystal is very close to that of the insulating 2 × 2 BiOBiS2 sheet. The lattice mismatch between them is only 2.3%. To this end, we propose a quantum well structure between the α-SnTe2 2D crystal and BiOBiS2 sheet. As shown in Figure 4a, in the BiOBiS2/α-SnTe2 quantum well, one α-SnTe2 layer is sandwiched between two BiOBiS2 sheets. Our calculated band structure displayed in Figure 4b shows that the states around the Fermi level mainly come from the pz− and px, y− orbitals of α-SnTe2. In the absence of SOC for the BiOBiS2/α-SnTe2 quantum well, which is similar to the case of a free-standing α-SnTe2 2D crystal, there is still one energy band with the px, y− orbital character that overlaps with another energy band with the px, y− orbital character around the Fermi level. However, such band overlapping only forms one Dirac point located at the Fermi level along the Γ–X direction. After including SOC, the Dirac point is deformed, and a band gap appears, which is a strong indication for the existence of a topologically nontrivial phase. Therefore, the BiOBiS2/α-SnTe2 quantum well may still be a 2D TI. To firmly confirm its band topology, we investigated the Z2 topological invariant. The corresponding results are listed in Supplementary Table S3. The result of Z2=1 indicates that a BiOBiS2/α-SnTe2 quantum well is indeed a QSH insulator. More remarkably, the nontrivial bulk band gap is 44 meV, which is still well above the energy scale of room temperature.
To deeply understand the effect of the insulating substrate BiOBiS2 sheet on the topological nature of an α-SnTe2 2D crystal, we investigated the related properties of an isolated α-SnTe2 layer that is taken directly from the BiOBiS2/α-SnTe2 quantum well without any further structure relaxation. The corresponding band structures are plotted in Figure 4c, and we can observe that the band structure near the Fermi level retains linear band crossing in the absence of SOC, which resembles that of the BiOBiS2/α-SnTe2 quantum well. When including SOC, a band gap of 214 meV is produced in the isolated α-SnTe2 layer. Our band topology analysis estimates that the topologically nontrivial state remains intact in the isolated α-SnTe2 layer. However, by comparing Figures 4b and c, it is important to notice that the nontrivial band gap is significantly reduced (from 214 to 44 meV) when the isolated α-SnTe2 layer is sandwiched between the BiOBiS2 sheets. This reduction reflects the band realignment effect when placing thin films on the substrate. The conduction band maximum and valence band maximum of the BiOBiS2/α-SnTe2 quantum well are contributed by the BiOBiS2 sheets and isolated α-SnTe2 layer, respectively. Obviously, these are the unique features of the type-II quantum well: the electron sub-band and hole sub-band are separated in two different layers. Considering these results, one can easily understand this reduction. Furthermore, these results also reveal the importance of the gap size of the substrate for maintaining the nontrivial topological states.
Finally, we wish to note that topological states have also been reported recently in IV–VI 2D crystals.40 Unlike the SnX2-based 2D TIs studied here, the topological states of IV–VI 2D crystals are protected by crystal symmetry instead of by time-reversal symmetry. Namely, these IV–VI 2D crystals are 2D topological crystalline insulators.40 Therefore, these two classes of 2D crystals with the same elements but different compositions/structures can lead to very distinct topological phases: 2D TI and 2D topological crystalline insulator. This finding is certainly interesting and suggests greater room in the design of topological materials beyond conventional structures.
In summary, we show by first-principles calculations that α-SnX2, β-SnSe2 and β-SnTe2 2D crystals, which are composed exclusively of pentagonal rings, are 2D TIs. This result challenges the current knowledge 2D TIs, that is, that square or hexagonal rings are the commonly building blocks. Their topologically nontrivial characteristics are confirmed by the nontrivial Z2-type topological invariants and the existence of topologically protected helical edge states. Sizeable nontrivial band gaps of 121–224 meV are obtained, suggesting that the QSH effect can be realized in these systems above room temperature. For β-SnSe2 and β-SnTe2 2D crystals, their bands exhibit the SOC-induced spin-splitting effect due to their broken inversion symmetry. α-SnX2 2D crystals are inversion symmetric; thus, no spin splitting can be observed. To probe the possibility of experimentally realizing these 2D crystals, we further propose the BiOBiS2/α-SnTe2 quantum well. Additionally, we find that the α-SnTe2 2D crystal remains topologically nontrivial with a sizeable gap (44 meV) when sandwiched between the BiOBiS2 sheets, thus suggesting that the 2D TIs predicted here can be sandwiched between appropriate nanostructures for realistic applications. Our work predicts that more pentagonal ring-based 2D TIs will be discovered in the future, which will greatly broaden the scientific and technological impact of the QSH effect.
Financial support from the European Research Council (ERC, StG 256962) and the Taishan Scholar Program of Shandong is gratefully acknowledged.
YM and TH conceived and designed this study, YM performed the calculations, and all authors discussed the results and contributed to the writing of the manuscript.
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Supplementary Information accompanies the paper on the NPG Asia Materials website (http://www.nature.com/am)