Introduction

Topological insulators (TIs) have attracted extensive research interest due to the exotic properties and potential applications in spintronics and quantum computing1,2,3,4,5,6. 2D TIs are considered to be more promising materials than three-dimensional (3D) TIs for spin transport applications because of the robustness of conducting edge states from backscattering7,8,9. However, to date the quantum spin Hall (QSH) effect of existing 2D TIs occurs only at very low-temperatures, below 10 K for HgTe/CdTe7 and InAs/GaSb10 quantum wells and below 10−2 K for graphene11,12, due to the small band gaps. Therefore, the search for 2D TIs with large band gaps has intensified in recent years for both fundamental and practical interests9,13,14,15,16,17,18.

Since the discovery of graphene19, 2D materials have been attracting great interest due to the various applications in optoelectronics, spintronics, solar cells20,21. Most of 2D materials, such as hexagonal BN, metal chalcogenides, transition metal dichalcogenides, etc., have 3D layered parent counterparts exhibiting strong in-plane covalent bonding and weak out-of-plane van der Waals (vdW) interactions, thus facilitating isolation of single layers via mechanical or chemical exfoliation. Furthermore, there are several materials beyond 3D layered materials that have stable 2D structures and exhibit a wide range of interesting properties. For instance silicene has been theoretically predicted22,23,24,25,26 as a buckled honeycomb arrangement of Si atoms and having an electronic dispersion resembling that of relativistic Dirac fermions. Recently, silicene has been successfully grown on Ag27,28,29,30, Ir31 and ZrB232 substrates and is considered as one of the most promising materials for the next-generation electronic devices25,30,33. Interestingly, recent ab initio and phonon-mode calculations have shown that the 2D honeycomb structure of many group-IV elements and III-V binary compounds are stable in planar or buckled geometries24,25. Thus, the honeycomb lattice of these materials offers a unique playground for searching of novel QSH insulators with large band gap.

The 3D HgSe and HgTe with the space group are semimetals having the unique property that their band structure exhibits an inversion of the Γ6 and Γ8 band ordering at the Brillouin zone (BZ) center7,8,34,35, which is the origin of the QSH effect in 2D HgTe/CdTe quantum wells7,8. Consequently, HgSe and HgTe are also expected to have Dirac-like surface states which however are always coupled to metallic bulk states due to their semimetallic nature and hence difficult to be observed. However, applied strain induces a gap between the light- and heavy-hole bands rendering the systems 3D TIs1,36,37. In contrast to 2D TIs from group-IV elements, the mercury chalcogenides are composed of heavy elements having large spin orbit coupling (SOC) which combined with the 2D quantum confinement can lead to significant changes in the electronic band structure and the frontier band ordering compared with their 3D parent material. In the present work based on ab initio structure and phonon-mode calculations we predict that 2D monolayers of HgTe and HgSe are stable with low-buckled (LB) honeycomb geometry. By determining the uniaxial strain evolution of the band inversion in the band structure and of the topological invariant we demonstrate that LB HgTe undergoes a trivial insulator to topological insulator transition at ε > 2.6% uniaxial strain. Furthermore, the band gap of the TI phase can be tuned over a wide range from 0 eV to 0.2 eV as the tensile strain increases from 2.6% to 7.4%. In addition, we predict a similar strain-induced topological phase transition for the LB HgSe single layer at 3.1% strain, where the topological band gap increases to 0.05 eV at about 4.6%. The LB structure allows the formation of two different topological edge states in the zigzag and armchair edges.

Results and Discussions

HgTe monolayer

Figure 1(a) displays the variation of the total energy as a function of the lattice constant of the 2D HgTe monolayer in planar (PL) and buckled (B) geometries shown in Fig. 1(c–e), respectively. One can see that the planar honeycomb structure has a single minimum, while the buckled structure has two minima corresponding to high-buckled (HL) and low-buckled (LB) honeycomb geometries separated by an energy barrier of about 0.3 eV. We find that the most stable structure is the LB with a lattice constant of 4.616 Å and buckling distance Δ = 0.47 Å. The phonon dispersion of the LB structure in Fig. 1(b) clearly shows that the acoustical branches and optical branches are well separated with a frequency gap. Furthermore, the absence of an imaginary frequency strongly suggests that the structure has a local minimum in its energy landscape. The stability of 2D monolayer HgTe under strain was corroborated by calculations of the phonon dispersion and the formation energies.

Figure 1
figure 1

Structural properties of monolayer HgTe.

(a) Total energies of 2D HgTe monolayer as a function of lattice constant for the planar (blue curve) and buckled (black curve) geometries, respectively. The low- and high-buckled geometries are denoted by LB and HB, respectively. (b) Phonon dispersion of the low-buckled honeycomb structure along high symmetry lines. (c,d,e) Top and side views of the 2D low-buckled and planar honeycomb structures, where the buckling distance Δ denotes the interlayer distance between Hg and Te planes. The unit cell is denoted by the shaded area.

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The LDA and MBJLDA band structures of bulk HgTe along two symmetry directions are shown in Fig. 2(a,b), respectively. Both yield a semimetal which has an inverted band ordering (p-derived Γ8 bands are above the s-derived Γ6 band) where the heavy-hole and light-hole Γ8 bands are degenerate at the BZ center. LDA places the s-type Γ6-band 1.20 eV below the Γ8-band and 0.44 eV below the spin-orbit split Γ7-band, thus yielding the wrong band ordering between the Γ6 and Γ7 bands. On the other hand, the MBJLDA predicts the correct band ordering with E0 = E(Γ6) − E(Γ8) = 0.32 eV and Δ0 = E(Γ8) − E(Γ7) = 0.66 eV, in good agreement with experiment and previous ab initio calculations7,30,31. The underlying origin is the poor treatment of the p − d hybridization within LDA between the semicore Hg 5d states and the Te-p valence states, which in turn shifts the Te 5p states to higher energies38.

Figure 2
figure 2

Band structures of bulk and monolayer HgTe.

(a) Band structures of bulk HgTe employing the LDA and (b) MBJLDA exchange correlation functionals. Band structures of the 2D LB HgTe monolayer using the (c) LDA and (d) MBJLDA exchange correlation functionals. The Fermi level is denoted by the dashed lines at 0 eV. The red and blue lines denote the bands containing the s-like and p-like states at Γ point, respectively.

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The LDA and MBJLDA band structures of the LB HgTe monolayer along two symmetry directions are shown in Fig. 2(c,d), respectively. Note, that the LDA, used commonly in the literature to determine the band topology, yields the incorrect ordering of the frontier bands at Γ where the s-derived Γ4 state lie above the p-derived Γ5,6 states with an inverted band gap of 0.27 eV. Consequently, the LDA predicts incorrectly that the LB HgTe monolayer is a topological insulator with  = 1. Similar artificial inverted bands have been also found by LDA calculations in other systems, such as ScAuPb and YPdBi and can be corrected by MBJLDA39. In sharp contrast, the MBJLDA results predict that the s-derived Γ4 states are about 0.31 eV higher than the p-derived Γ5,6 states and  = 0, indicating that the LB HgTe monolayer at equilibrium is a trivial insulator. These results demonstrate the necessity to use MBJLDA for accurate determination of the band topology.

In order to demonstrate the strain-induced band inversion process, we present in Fig. 3(a) the MBJLDA band energies of the s-derived E(Γ4), p-derived E(Γ5,6), the band inversion strength, EI ≡ E(Γ5,6) − E(Γ4) and the absolute value of the global band gap |Eg| as a function of in-plane tensile strain where the LB geometry is the ground state up to 7.4%. We find that the even though both energies E(Γ4) and E(Γ5,6) decrease with tensile strain, the strain-induced change of the former is larger than that of the latter. Thus, we predict that the 2D HgTe honeycomb monolayer is a trivial insulator (EI < 0) for 0 < ε < 2.6% and undergoes a transition to a topological phase at 2.6% strain where EI vanishes and hence the band gap (Eg = |EI|) closes. For 2.6% < ε < 7.4% the band ordering between the frontier s-derived Γ4 and p-derived Γ5,6 bands is inverted (EI > 0) and EI increases with strain. It is important to emphasize that for ε > 2.6% Eg < |EI| due to the fact that the s-derived Γ4 valence band develops a camel-back-shape (Fig. 4(c)) where the valence band maximum occurs in the vicinity of the BZ center. The topologically nontrivial band gap Eg increases from 0 eV to 0.20 eV for 2.6% < ε < 6.0% and decreases to 0.14 eV for 6% < ε < 7.4% due to the enhancement of the camel-back-shape of the valence band. The large intrinsic band gap values of 0.0 ~ 0.2 eV in a wide range of tensile strain of 2.6% ~ 7.0% makes the 2D LB HgTe monolayer viable for room temperature applications.

Figure 3
figure 3

Evolutions of the band structures and topological invariant of monolayer HgTe.

(a) Variation of the energies of the s-like Γ4, p-like Γ5,6, EI ≡ E(Γ5,6) − E(Γ4) and band gap Eg as a function of tensile strain. The shaded and white areas denote the trivial and nontrivial topological phases, respectively. (b,c) Evolutions of Wannier function centers along ky for 2D HgTe monolayer under (c) 0.0% and (d) 5.0% tensile strain, respectively. The evolution curves cross any arbitrary reference line parallel to ky (for example, the blue dashed line) an odd (even) number of times, yielding  = 1 (0).

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Figure 4
figure 4

Edge states of LB 2D HgTe in the topological phase for (a) Hg- and (b) Te-terminated zigzag edges, respectively; and for (c) armchair edges.

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The topological phase transition can be further confirmed by calculating the topological invariant before and after the band inversion. Figure 3(b,c) show the evolution lines of the Wannier function centers along ky for LB HgTe monolayer under in-plane tensile strain of ε = 0% and ε = 5%, respectively. The system is TI (trivial insulator) if the evolution curves of the Wannier function centers cross an arbitrary reference line parallel to ky (blue dashed line) an odd (even) number of times yielding  = 1 (0). We find that under zero (5%) strain the evolution curves (blue) cross any arbitrary line parallel to the horizontal axis (for example, the red dotted line) an even (odd) number of points, thus yielding  = 1, verifying the TI phase under strain higher than 2.6%. An important feature of topological insulators is the topologically protected gapless edge states. Therefore, we have calculated the band structure of the semi-infinite ribbon employing the Green’s function method with tight-binding parameters from VASP and wannier90. In Fig. 4 we show the band structure along the 2D symmetry lines of the (a) Hg-terminated zigzag (ZZ-Hg) edges, (b) Te-terminated zigzag (ZZ-Te) edges and (c) armchair (AC) edges. Since the zigzag ribbon has an electric polarization, the Hg (Te) side Dirac-point is shifted towards the valence (conduction) band. The intrinsic misalignment of Dirac-points at opposite edges implies that the LB 2D HgTe has potential in spintronic applications since the dominant current channel among two (Hg and Te) edges can be easily tuned by the chemical potential.

Figure 5 shows the change of the frontier bands of the LB HgTe monolayer near the BZ center under in-plane tensile strain without and with SOC, respectively. Under zero strain and in the absence of SOC, the s-derived Γ1 conduction band is about 0.54 eV above the four-fold degenerate p-derived Γ3 valence band. In the presence of SOC, the Γ1 band becomes Γ4 and the Γ3 band splits into two double-degenerate bands, the p-derived Γ5,6 and bands. The SOC energy, , is about 0.43 eV which in turn reduces the band gap to 0.31 eV. Thus, there is no band inversion and the 2D HgTe monolayer is topologically trivial. At 2.6% strain and in the absence of SOC, the anti-bonding Γ1 state shifts down with respect to the bonding Γ3 state and hence the band gap deceases to 0.22 eV compared to its 0.54 eV value at zero strain. The SOC shifts the valence Γ5,6 band up in energy resulting in the touching of the valence and Γ4 conduction band at the Fermi energy. Note that in the absence of SOC the gap is not closed for 2.6% < ε < 4.2% [Fig. 5(c)], while the system has already undergone a transition to a TI phase in the presence of SOC. This demonstrates that the SOC is the main driving force for the band inversion in the range of 2.6% < ε < 4.2% strain. In sharp contrast, for ε > 4.2% the band inversion shown in Fig. 5(d) already takes place without spin-orbit coupling but the system is semimetal due to the fact that the heavy- and light-hole p-derived Γ3 bands are degenerate at the BZ center. SOC then opens up the gap for ε > 4.2% rendering the LB HgTe a TI. Therefore, the strain and SOC have synergistic effects: In the range of 2.6% < ε < 4.2%, strain reduces the band gap so that band inversion can be induced by SOC, while for ε > 4.2% the SOC reopens the band gap of the already topologically inverted bands induced by strain.

Figure 5
figure 5

Evolutions of the frontier bands of monolayer HgTe.

(a–d) are the band structures of the frontier bands of 2D LB HgTe under in-plane tensile strain of ε = 0%, 2.6%, 4% and 5%, respectively. The band structures without and with SOC are shown in the top and bottom panels, respectively. The red and blue bands denote the s- and p-derived states at Γ.

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To illustrate the origin of the strain-induced topological transition near Γ, we show schematically in Fig. 6(a–c) the evolution of the energy levels of the atomic levels under in-plane tensile strain including the effects of chemical bonding and SOC. The states near Fermi level are mainly Hg s-derived and Te p-derived orbitals which hybridize and split into the bonding and antibonding states, denoted by s+ (p+) and s− (p−) with different parities. The SOC splits the p+ state into the and with a total angular momentum of and , respectively. In the absence of strain, because of the strong hybridization of the Hg- and Te-derived orbitals, the energy of s− is much higher than that of the p+ shown in Fig. 6(a) and the SOC splitting (Δsoc) of p+ is insufficient to induce the band inversion. With increasing in-plane tensile strain, the He-Te interlayer bond length increases and hence the Hg- and Te-derived orbitals hybridization decreases leading to a reduction of the splitting between the bonding and antibonding states. Under tensile strain in the range of 2.6% ~ 4.2%, the energy difference between s− and p+ decreases but the energy of s− is still higher than that of the p+ without SOC. Turning on the SOC, the energy of states becomes higher than that of s− inducing the band inversion and the phase transition to the topologically nontrivial phase as shown in Fig. 6(b). Further increasing the tensile strain, the energy of s− becomes lower than that of p+ and the band inversion is achieved solely by the strain. However, there is no band gap as the Fermi level crosses the degenerate p+ states. By including SOC, the topological band gap is reopened by the splitting of p+ into the and and the 2D monolayer HgTe becomes TI. The change of the band order by strain in 2D monolayer HgTe is consistent with the previous theoretical study on the strain tuning of topological band order in cubic semiconductors, as increasing the lattice constant leads to a decrease of the coupling potentials40.

Figure 6
figure 6

Schematic diagram of topological transition.

The evolution of the energy levels at Γ point for the 2D HgTe monolayer with increasing in-plane tensile strain. The red and blue solid lines denote the s-derived and p-derived states. The superscripts “+” and “−” denote the bonding and antibonding states, respectively. The states are split by the SOC (Δsoc) into the and state. The Fermi level is shown by the horizontal black dashed line.

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We have also examined the origin of band splitting in Fig. 5(c) which can be either of Rashba41 and/or Dresselhaus42 spin-orbit coupling since both structural and bulk-inversion symmetries are broken. As was recently demonstrated by Fu43, under C3v symmetry (which is the point group symmetry for HgTe) the Dresselhaus-type spin-orbit coupling vanishes along the Γ-M symmetry direction. Analysis of the k-resolved energy difference between the two top-most valence bands in Fig. 5(c) shows that further away from the Γ point the Dresselhaus-type spin-orbit coupling is dominant because the band splitting diminishes along the Γ-M point symmetry line. On the other hand, in the vicinity of Γ point the Rashba effect becomes dominant. We find that different orbital character of the valence bands (Hg-s and Te-p) induces distinct spin-orbit coupling effects.

HgSe monolayer

These results invite the intriguing question whether the 2D single layer of HgSe undergoes a similar TI transition under tensile strain. We find that the HgSe honeycomb monolayer has stable low-buckled structure with a = 4.35 Å and Δ = 0.368 Å. The stability of 2D monolayer HgSe is confirmed by the phonon dispersion as shown in Fig. 7(e), even there are very small imaginary frequencies near Γ point, presumably these small imaginary frequencies will be removed using larger supercell sizes26. Figure 7 shows the MBJLDA band structure of the frontier bands of the LB HgSe monolayer under tensile strain ε = 0%, 3.1% and 4% without and with SOC, respectively. Under zero strain and without SOC, the s-derived Γ1 band is about 0.42 eV higher than the p-derived Γ3 band. Turning on SOC reduces the band gap between the s-like Γ4 and p-like Γ5,6 to 0.36 eV. As the tensile strain increases to 3.1%, the band gap without SOC decreases significantly to 0.07 eV. Inclusion of SOC closes the band gap with the Γ4 conduction band touching the Γ5,6 valence band. For strain greater than 3.3% the HgSe LB monolayer exhibits a band inversion [Fig. 7(c)] even in the absence of SOC. However, the SOC facilitates the reopening of the band gap. For 3.1% < ε < 4.6% the band gap of the topological phase increases to 0.05 eV, which is much larger than that for graphene and CdTe/HgTe/CdTe QW. For strain greater than 4.6% HgSe monolayer becomes planar and will be discussed elsewhere.

Figure 7
figure 7

Evolutions of the band structures of monolayer HgSe.

MBJLDA band structures of the frontier bands of the 2D LB HgSe under in-plane tensile strain of (a) ε = 0%, (b) 3.1% and (c) 4%. The band structures without and with SOC are shown in the upper and bottom panels, respectively. The Fermi level is at 0 eV. The red and blue bands denote the s- and p-derived states at Γ, respectively. (d) The band inversion strength EI and band gap Eg as a function of in-plane tensile strain. (e) The phonon dispersion of the low-buckled honeycomb HgSe at ε = 0%.

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Although HgTe and HgSe thin films have been experimentally studied extensively8,44,45,46, the atomic-layer thick 2D HgTe and HgSe monolayers have not been reported so far. Based on the successful fabrication of the aforementioned 2D atomic-layer materials, a possible way to synthesize 2D-LB HgTe and HgSe monolayers is to grow them on appropriate substrates. For example, the honeycomb ZnO bilayer47 and buckled silicene have been successfully synthesized on metal substrates such as Ag27,28,29,30, Ir31 and ZrB232. Other hybrid 2D superlattices have been successfully synthesized on 2d van der Waals substrates such as graphene/hexagonal boron nitride48,49 and graphene/MoS250,51,52. Therefore, the 2D vdW nanostructures are also potential candidate substrates for growing of 2D LB HgTe and HgSe monolayers.

Conclusion

In conclusion, based on ab initio structure and phonon-mode calculations we predict that the 2D LB HgTe and HgSe monolayers are stable. By examining the band inversion in the band structure and the evolution of the topological invariant, , we demonstrate that the 2D LB HgTe monolayer undergoes a transition from a topologically trivial to a topological insulator phase at ε > 2.4% in-plane tensile strain. The underlying origin of the topological transition is the interplay of the SOC and strain. Most remarkably, the band gap of the 2D-LB HgTe TI phase can be tuned over a wide range from 0 eV to 0.2 eV as the tensile strain increases from 2.6% to 7.4%. Furthermore, we predict that the 2D LB HgSe monolayer underdoes a similar strain-induced topological insulator phase transition at ε > 3.1% where the topological insulator band gap is 0.05 eV at about 4.6%. The large band gap of the LB HgTe and HgSe monolayers make these 2D quantum spin Hall insulators suitable for practical applications at room-temperature.

Methods

The calculations of the structural properties and the phonon dispersions were performed by using the Vienna Ab initio Simulation Package (VASP)53 and phonopy54 code, with the projector-augmented-wave55 approach and the local density approximation56. An accurate description of the electronic structure is a prerequisite in the search and discovery efforts for the next-generation TIs. Therefore, the ab initio electronic structure calculations employed the full-potential WIEN2K code57 with the SOC included and the modified Becke-Johnson Local Density Approximation (MBJLDA) functional58. The MBJLDA functional has been shown to yield accurate band gaps, effective masses and frontier-band ordering at time-reversal invariant momenta (TRIM) that are in very good agreement with the computationally more intense GW and hybrid-functional approaches59. The topological invariant which characterizes the global band topology of the occupied Bloch wave functions in the entire BZ is determined by calculating the evolution of the Wannier function center (WFC) in reciprocal space during a “time-reversal pumping” process37,60,61. The edge state of a semi-infinite ribbon is calculated by using the Green’s function method with the tight-binding parameters determined from wannier9062.

Additional Information

How to cite this article: Li, J. et al. Two-dimensional topological insulators with tunable band gaps: Single-layer HgTe and HgSe. Sci. Rep. 5, 14115; doi: 10.1038/srep14115 (2015).