Two-dimensional topological insulators with tunable band gaps: Single-layer HgTe and HgSe

Two-dimensional (2D) topological insulators (TIs) with large band gaps are of great importance for the future applications of quantum spin Hall (QSH) effect. Employing ab initio electronic calculations we propose a novel type of 2D topological insulators, the monolayer (ML) low-buckled (LB) mercury telluride (HgTe) and mercury selenide (HgSe), with tunable band gap. We demonstrate that LB HgTe (HgSe) monolayers undergo a trivial insulator to topological insulator transition under in-plane tensile strain of 2.6% (3.1%) due to the combination of the strain and the spin orbital coupling (SOC) effects. Furthermore, the band gaps can be tuned up to large values (0.2 eV for HgTe and 0.05 eV for HgSe) by tensile strain, which far exceed those of current experimentally realized 2D quantum spin Hall insulators. Our results suggest a new type of material suitable for practical applications of 2D TI at room-temperature.

The large band gap of 2D LB HgTe and HgSe monolayers make this type of material suitable for practical applications at room-temperature.
2][3][4] 2D TIs are considered as more promising materials than three-dimensional (3D) TIs for spin transport applications because of the robustness of conducting edge states from backscattering. 5,6However, the quantum spin Hall (QSH) effect of most 2D TIs occurs only at very low-temperature, such as 10K for HgTe quantum well 5 and less than 10 −2 K for graphene, 7,8 owing to the small band gaps.[11] Since the discovery of graphene, 12 2D materials have been attracting great interest due to the various applications in optoelectronics, spintronics, solar cells. 13,14Up to now, most of the 2D materials are from van der Waals (vdW)-bonded solids, for example, 2D hexagonal BN, metal chalcogenides, transition metal dichalcogenides.It has been shown that some compounds beyond 3D layered structures have stable 2D structures as well.6][17][18][19] Recently, S. Ciraci et al. predicted that 22 honeycomb structures of group-IV and III-V binary compounds can be stable in planar or buckled geometries, such as Si, Ge, InAs, InSb and so on, based on the structure optimization and phonon-mode calculations.
Furthermore, 2D honeycom silicon, named as silicene, has been successfully grown on Ag, [20][21][22][23] Ir 25 and ZrB 2 24 substrates.These 2D structures have attracted enormous attention due to their fantastic properties.For example, silicene processes the so-called Dirac electronic dispersion and quantum spin Hall effect (QSHE) and is considered as one of the most promising materials for next generation electronic devices. 18,23,26This type of 2D materials has expanded a new field for low dimensional nanostructures.The II-VI mercury telluride (HgTe) and mercury selenide (HgSe) with the space group F43m are important prototypical TI materials because of the inverted band structures. 5,6,27,283D HgTe (HgSe) can become a topological insulator by opening the zero-band gap with external strain. 1,29,30rthermore, 2D TI state was theoretically predicted and experimentally confirmed in the CdTe/HgTe/CdTe quantum well as the thickness of HgTe is larger than 6.3nm. 5,6In this work, we theoretically predict that HgTe and HgSe are stable in 2D monolayers with low buckled honeycomb geometry, based on ab initio structure and phonon-mode calculations.By identifying the band inversion in the band structures and calculating the evolution of the Z 2 topological invariants, we demonstrated that monolayer LB HgTe undergoes a transition to a topological nontrivial phase under tensile strain ε > 2.6%.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%.Similarly, the topological phase transition of monolayer HgSe is induced by strain and SOC as the strain ε >3.1% and the topological band gap can be 0.05 eV as the strain increases to about 4.6%.
Fig. 1 (a) displays the variation of the total energy with respect to the lattice constant of the 2D HgTe monolayer in planar and buckled geometries.It can be seen 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.The most stable structure is the LB structure as it has the lowest total energy compared with that of HL and planar (PL) structures.
The lattice constant of LB structure is 4.616 Å and the buckling distance ∆=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 any imaginary frequency strongly suggests that the structure has a local minimum in its energy landscape.
For the calculation of topological invariant Z 2 , the accurate band gap and correct band order are necessary because Z 2 is strongly dependent on the electronic structure.We employed the modified Becke-Johnson exchange potential and local-density approximation (MBJLDA) for the correlation potential in the electronic structures calculations, implemented in the package of WIEN2K [33]. 31e LDA and MBJLDA band structures of bulk and HgTe monolayer are shown in Fig. 2. It can be seen that bulk HgTe is a zero-gap semiconductor as the quartet degenerate p-like Γ 8 states touch at the Γ point and processes the inverted band characteristic.The s-like Γ 6 of LDA band structure is about 1.20 eV below Γ 8 and 0.44 eV below Γ 7 , which gives the wrong band gap and band order.
The main reason is the poor treatment of the p − d hybridization by LDA, which in turn shifts the Te 5p states to higher energies. 32Our MBJLDA calculations gave correct band structure with 66 eV, which are in agreement with the experimental and theoretical values. 5,27,28,33,34For 2LB HgTe monolayer, the LDA band structure gives wrong band gap and band order.As shown in Fig. 2 (c), the LDA band structure processes an inverted band characteristic with the s-like Γ 4 state above p-like Γ 5,6 states.The inverted band gap at Γ point is about 0.27 eV and this results in a topological nontrivial phase with Z 2 =1.The artificial inverted bands by LDA have been also found in other systems, such as ScAuPb and YPdBi, and can be corrected by MBJLDA. 33As shown in Fig. 2 (d), the s-like Γ 4 states by MBJLDA are about 0.31 eV higher than the p-like Γ 5,6 states and Z 2 =0, indicating that the LB HgTe monolayer at equilibrium structure is not a TI.These results show that it is very important to use MBJLDA for the accurate band structure calculations.
It is interesting that the topological trivial 2D LB HgTe monolayer can be tuned into a strong topological insulator under the in-plane tensile strain with a remarkable band inversion characteristic.To investigate the band inversion, we define the band inversion strength E I by the difference between p-like Γ 5,6 states and s-like Γ 4 states, i.e.E I = E(Γ 5,6 ) − E(Γ 4 ).E I is positive when the band is inverted and negative otherwise.In Fig. 3 (a), we displayed the evolutions of E(Γ 4 ), ) and E I with respect to the tensile strain up to 7.4%, as it is found that the largest strain to keep the low buckled geometry is about 7.4%.From Fig. 3 (a), it can be seen that the energies of p-like Γ 5,6 and s-like Γ 4 states decrease as the tensile strain increases, but E(Γ 4 ) descends faster than that of E(Γ 5,6 ).As 0 < ε < 2.6%, Γ 4 is still above Γ 5,6 resulting the negative E I but E I increases from -0.31 eV to 0 eV.The topologically trivial band gap is closed at ε = 2.6% as Γ 4 states touch Γ 5,6 .The band inversion occurs as the tensile strain ε > 2.6%, indicating a transition from normal insulator to topological insulator at ε = 2.6%.The band inversion strength E I increases from -0.31 eV to 0.33 eV as strain increases from 0% to 7.4%.The evolutions of the global band gap E g as a function of the tensile strain are also shown in Fig. 3 (a).For 0 < ε < 2.6%, the band gap is determined by Γ 5,6 and Γ 4 , i.e.E g = |E I |.As 0ε > 2.6%, the global band gap is smaller than E I because of the camel near Γ point as shown in Fig. 2 (c).In the range of 2.6% < ε < 6.0%, the topologically nontrivial band gap E g increases from 0 eV to 0.20 eV as the band inversion strength E I increases.When 6% < ε < 7.4%, E g decreases to 0.14 eV because the camel near Γ increases significantly.The topological phase transition can be confirmed by calculating topological invariant Z 2 before and after the band inversion.The topological invariant Z 2 for the structure without inversion symmetry can be computed by the method based on the U(2N) non-Abelian Berry connection. 35e main idea is to calculate the evolution of the Wannier function center (WFC) directly during a time-reversal pumping process.The Z 2 topological number can be determined by the total number of crossing between any horizontal reference line and the WFC mod 2. In Fig. 3 (b) and (c), we display the evolution lines of Wannier function centers for LB HgTe monolayer at ε = 0% and ε = 5%, respectively.It can be seen that the WFC evolution curves cross any arbitrary reference lines even times, thus yielding Z 2 = 0 for ε = 0%.In contrast, the the WFC evolution ε = 5% results in Z 2 = 1 , which confirms the non-trivial topological phase.
), is about 0.43 eV, which significantly reduces the band gap to 0.31 eV.It is found that the tensile strain also reduces the band gap remarkably, because the splitting between bonding state and anti-bonding state reduces with the weakening of the Hg-Te bonding by strain.
At critical strain ε = 2.6% of TI transition, the anti-bonding state Γ 1 shifts down with respect to the bonding state Γ 3 and the band gap deceases to 0.22 eV, which is about 0.32 eV smaller than that with ε = 0% .After including SOC, the Γ 5,6 state touches Γ 4 state as SOC splitting shifts the Γ 5,6 up.In the range of 2.6% < ε < 4.2%, there is still a band gap of the band structure without SOC, as shown in Fig. 4 (c).However, by including SOC, the band inversion is achieved and LB HgTe monolayer changes into TI.Therefore, SOC is the main driving force for the band inversion and the topological phase transition during 2.6% < ε < 4.2%.For ε > 4.2%, is clear that the band inversion can be achieved by strain without SOC, but there is no topological band gap as the p-like Γ 3 states is double-degenerated at Γ point.LB HgTe monolayer is a TI with an ing gap induced by SOC for ε > 4.2%, as shown in Fig. 4 (d).Generally speaking, the TI phase transition is induced by the combination effects of strain and SOC.In the range of 2.6% < ε < 4.2%, the strain reduces the band gap so that the band inversion can be induced by SOC.For ε > 4.2%, strain enlarges the topological band gap opened by SOC.
Similarly, we find that HgSe monolayer has the stable low buckled structure with a=4.35 Å and ∆=0.368Å and the topological phase transition can be achieved by the tensile strain and SOC as well.Fig. 5 shows the MBJLDA band structure of LB HgSe monolayer under the tensile strain ε = 0%, 3.1% and 4%.For ε = 0% without SOC, the s-like Γ 1 state is about 0.42 eV higher than the p-like Γ 3 state.By including SOC, the band gap between the s-like Γ 4 and p-like Γ 5,6 reduces to 0.36 eV.As the tensile strain increases to 3.1%, the band gap without SOC decreases significantly to 0.07 eV and the band gap is closed by including SOC, which makes Γ 4 touche Γ 5,6 and the LB HgSe monolayer undergoes the transition to the TI phase with ε > 3.1%.As ε > 3.3%, the band inversion can be achieved by strain effects and the topological band gap is opened by SOC, as shown in Fig. 5 (c).During 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.As ε > 4.6%, HgSe monolayer becomes planar, which will be discussed elsewhere.
In conclusion, we have demonstrated that the 2D LB HgTe and HgSe monolayers can be stable based on ab initio structure, phonon-mode and finite temperature molecular dynamics calculations.By identifying the band inversion in the band structure and calculating the evolution of the topological invariants Z 2 , we have shown that 2D LB HgTe monolayer undergoes a transition from a topologically trivial phase to a topologically nontrivial phase as the in-plane tensile strain ε > 2.4%.The strain induced topological insulating behavior is due to the combination effects of strain and SOC.As 2.6% < ε < 4.2%, the strain reduces the topologically trivial band gap so that the SOC can induce the band inversion and a topologically nontrivial gap.As ε > 4.2%, the band inversion already is realized by strain but the topological gap is induced by SOC.Furthermore, the band gap of 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%.Similar situation occurs for 2D LB HgSe monolayer, the topological phase transition is induced by strain and SOC as the strain ε >3.1% and the topological band gap can be 0.05 eV as the strain increases to about 4.6%.The large band gap of 2D LB HgTe and HgSe make this type of material suitable for practical applications at room-temperature.

Figure 1 :
Figure 1: (Color online) (a) Total energies of 2D HgTe monolayer with respect to lattice constant for planar (blue line) and buckled (black line) geometries, respectively.(b) The phonon dispersion of the low buckled honeycomb structure.(c) and (d) are the top and side view of the low buckled honeycomb structure.

Figure 2 :
Figure 2: (Color online) Band structures of bulk and 2D LB HgTe monolayer.(a) and (b) are the LDA and MBJLDA band structures of bulk HgTe.The LDA and MBJLDA band structures of 2D LB HgTe monolayer are shown in (c) and (d).All the Fermi levels denoted by the dashed lines are at 0 eV.The red and blue lines denote the bands containing the s-like and p-like states at Γ point, respectively.

Figure 3 :
Figure 3: (Color online) (a) The variation of s-like Γ 4 , p-like Γ 5,6 , band inversion strength E I and band gap E g as a function of the in-plane tensile strain.The shaded and white areas denote the trivial and nontrivial topological phase, respectively.(b)and (c) are the evolutions of Wannier function centers for 2D HgTe monolayer under 0.0% and 5.0% in-plane tensile strain, respectively.The blue dashed lines are the horizontal reference lines.

Figure 4 :
Figure4: (Color online) Band structures of 2D LB HgTe under the tensile strain ε = 0%, 2.6%, 4% and 5%.The band structures without and with SOC are shown in the upper and bottom panels, respectively.The red and blue lines denote the bands containing the s-like and p-like states at Γ point.

Figure 5 :
Figure 5: (Color online) MBJLDA band structures of 2D LB HgSe under the tensile strain (a) ε = 0%, (b)3.1% and (c) 4%.The band structures without and with SOC are shown in the upper and bottom panels.All the Fermi levels denoted by the dashed lines are at 0 eV.The red and blue lines denote the bands containing the s-like and p-like states at Γ point, respectively.(d) The band inversion strength E I and band gap E g as a function of the in-plane tensile strain.