Phase polymorphism and electronic structures of TeSe2

Tekalign Terfa Debela a and Hong Seok Kang *b
aInstitute for Application of Advanced Materials, Jeonju University, Chonju, Chonbuk 55069, Republic of Korea
bDepartment of Nano and Advanced Materials, College of Engineering, Jeonju University, Chonju, Chonbuk 55069, Republic of Korea. E-mail: hsk@jj.ac.kr; jjhskang@gmail.com

Received 5th July 2018 , Accepted 13th August 2018

First published on 14th August 2018


Structure prediction complemented by density functional theory (DFT) calculation indicates that TeSe2 is the most stable among the various Te(1−x)Sex compounds. Different from the case of bulk Te, the material can equally adopt three different crystal structures: HγT, MH, and Mβα phases, which are similar to 1T-TiSe2, trigonal Te, and orthorhombic Te, respectively. These phases can be transformed from one to another by uniaxial tensile and shear stress of less than 1 MPa; they can be even transformed to their chiral mirror images. Band structure calculations including spin–orbit coupling (SOC) show that all three phases are semiconductors. The band gap (= 0.43 eV) increases with density, being the largest (= 1.86 eV) in the Mβα phase with the highest density. The HγT phase exhibits a hidden spin texture because of centrosymmetry. The other two phases display chiral spin texture due to the lack of symmetry, in that two spin components of frontier bands can split by more than 100 meV in opposite directions.


Introduction

Bulk trigonal Te is an indirect gap material with a band gap of 0.35 eV.1,2 Apparently, its chemical structure is strongly anisotropic in that quasi-one-dimensional (1D) helical chains with three atoms in a primitive cell interact with the four nearest chains via noncovalent bonding mediated by lone-pair electrons. As a result, anisotropic strain responses were detected by Raman spectroscopy measurement.3 For the same reason, Te-based nanomaterials grow into anisotropic structures, such as nanowires,4 nanorolls,5 and nanotubes.6–8 However, the electrical conductivity along the corresponding directions is nearly isotropic due to the delocalization of lone-pair electrons,9,10 which is in strong contrast with the cases of MoS2.11 In regard to this observation, it has been noted that elemental Te is different from other group VI elements in that the secondary bonding effect is also important in Te-containing compounds in addition to the covalent bonding.12

The trigonal Te was shown to exhibit a thermoelectric figure of merit as high as 1.0 due to the nestification of the valence bands.13 High photoresponse was measured in a trigonal Te nanobelt grown by van der Waals epitaxy.14 Te nanowires were shown to possess unique applications in gas sensing,15,16 photoconductivity, light detectors, and switching devices for optoelectronic applications.17,18 Te nanowires and nanorods also serve as effective templates for fabricating other thermoelectric materials.19–21 The ultrathin quasi-1D Te nanowire was proposed for a high performance piezoelectric nanogenerator.22 Recent calculations predicted that Te would be a strong topological insulator under strain23 and have multiple Weyl nodes near the Fermi level.24

In addition to the quasi-1D phase, Te-based two-dimensional (2D) layered phases have been investigated using both experimental and theoretical methods. All of these phases basically correspond to reconstructions of the trigonal 1D structure, in which the coordination number of each atom changes from two to three. A substrate-free solution process for large-scale synthesis of the 2D crystal of Te was presented, in which the material was air-stable and the thickness was tunable from monolayers to tens of nanometers. In the work, Te has also been demonstrated to exhibit a hole mobility two or three times higher than that of black phosphorus.9 Various distinct layered phases of Te with various band gaps were predicted, such as 1T-MoS2-like γ, 2H-MoS2-like ε, and tetragonal β phases depending upon the orientation of the cleavage plane.25 The prediction was partially verified in the same work, in that multilayers of the β phase were grown on highly oriented pyrolytic graphite. Monolayer and few-layer β Te films were epitaxially grown on the graphene/6H-SiC(0001) surface, for which the measured band gap increases from 0.49 to 0.92 eV as the number of layers varies from 13 to 1.26 In another experiment, a few-layer Te film with a different morphology was grown on highly oriented pyrolytic graphite.27

More recently, a few layers of a binary group VI material, i.e., β TeSe2, were theoretically investigated as an isoelectronic compound of β Te. A monolayer of β TeSe2 was shown to exhibit indirect-to-direct transition of the band gap under biaxial strain.28 In their work, they simply substituted two thirds of the atoms with Se atoms in the underlying β Te. However, TeSe2 may display more diverse morphological and electronic properties than the monoelemental Te in that (1) the two kinds of atoms have different electronegativities and (2) the spin–orbit interaction (SOC) will be more pronounced on heavier Te atoms than on Se atoms. Motivated by all these findings and expectations, we first show that there are stable polymorphic phases for bulk TeSe2 based on rigorous first-principles calculations, such as density functional theory (DFT) and phonon spectra calculations. Next, we investigate possible phase transformations among them and their electronic structures, focusing on their spin-texture for possible applications in spintronics.

Computational methods

Structure prediction for crystalline Te(1−x)Sex is performed using the CALYPSO (crystal structure predictions by particle swarm optimization) program.29,30 We explore the atomic structure of the crystal at various compositions in 0 < x < 1, i.e., at x = 1/3, 1/2, and 2/3. At each stoichiometry of Te(1−x)Sex, ∼600 initial structures that include 1–4 formula units in the unit cell are generated, and the total energy of each structure is calculated based on DFT as implemented in the Vienna ab initio simulation package (VASP).31,32 The projected augmented plane wave (PAW) approach is employed with a plane-wave kinetic energy cutoff of 520 eV and reciprocal-space k-point meshes of 2π × 0.045 Å−1.33 The ten most stable structures are selected, for which more accurate structure optimizations are performed with denser k-point meshes at a resolution of 2π × 0.03 Å−1. During the optimization, the structures are refined until the average force is <0.01 eV per (Å atom) and the final energy change is <10−5 eV per atom.

Attractive van der Waals interactions are taken into account by employing the Grimme's D3 correction (PBE-D3).34 In specific cases described in the main text, attractive van der Waals interactions are taken into account by employing the nonlocal optB88-vdW exchange–correlation functional.35,36k-point sampling is done so that total energy is accurate within 1 meV compared to the case when the sampling is three times denser. The band structure calculations are performed employing the modified Becke–Johnson local density approximation (MBJLDA)37 with the SOC effect. Phonon dispersion calculations are performed using the density functional perturbation theory (DFPT),38 as implemented in the phonopy program39 with the VASP used as the force-constant calculator.40 Force calculations are performed using a supercell with converged k-point sampling meshes.

Results and discussion

First, we calculate the energy of formation (Ef) of Te(1−x)Sex in the range 0 < x < 1, which is obtained from the relation: Ef = ET(Te1−xSex) − (1 − x)ET(Te) − xET(Se), where ET(Te1−xSex) is the total energy of Te1−xSex per atom. ET(Te) represents the total energy of the Te crystal in the trigonal phase. Our optimized lattice parameters are a = b = 4.42 and c = 5.92 Å, which are in good agreement with experimental values: a = b = 4.45 and c = 5.93 Å.41ET(Se) represents the total energy of the Se crystal in the monoclinic phase. The optimized lattice parameters of ET(Se) are a = 9.10, b = 9.15 and c = 11.89 Å in our calculation, which are also in good agreement with the experimental values a = 9.05, b = 9.08 and c = 11.60 Å.41 At each x, the crystal structure of Te(1−x)Sex is obtained from the structure prediction method in combination with DFT calculations. Fig. 1 shows that the formation energy is the lowest at x = 0.66, indicating that Se-rich TeSe2 corresponds to the most stable stoichiometry.
image file: c8tc03295e-f1.tif
Fig. 1 The formation energy (Ef) of bulk Te1−xSex as a function of x.

Table 1 indicates that bulk TeSe2 can equally adopt three different crystal structures shown in Fig. 2(a–c), i.e., hexagonal (HγT) and two different monoclinic phases of MH and Mβα. In fact, both the PBE-D3 and optB88-vdW calculations indicate that the energy difference among them is less than 7 meV per atom. For the bulk Te, our calculation shows that the HγT is appreciably (0.03 eV per atom) less stable than the trigonal phase. Throughout this work, different phases are denoted by different Greek subscripts, depending upon their correspondence to those (α, β, γ, and ε) for Te defined in ref. 42. In the HγT phase, TeSe2 adopts a chemical structure similar to that of γ-Te. γ-Te was found to be the most stable Te monolayer.25 However, it was also found to be significantly less stable than β-Te reconstructed after being cut along the [10[1 with combining macron]0] plane or α-Te for multilayers. Different from that of the β-Te, the α-Te adopts the AB stacking pattern without centrosymmetry.42

Table 1 Calculated structural and energetic properties of various phases of bulk TeSe2. Lengths and energies are in units of Å and eV per atom, respectively
Phase Space group E rel (a, b, c) l(Te–Se), l(Se–Se)e θ(Se–Te–Se), θ(Se–Se–Te)f ρ (g cm−1)
a The relative energy per atom of the phase with respect to the HγT phase. Values inside and outside parentheses represent those obtained from the PBE-D3 and optB88-vdW calculations, respectively. b Values inside the parentheses represent the corresponding parameters in γ Te and 1T-TiSe2 in sequence. c Values inside the parentheses represent the corresponding parameters in the trigonal Te. d Values inside the parentheses represent the corresponding parameters in bulk α Te. e Te–Se and Se–Se bond lengths obtained from the PBE-D3 calculation. Only l(Te–Se) is shown for the HγT and HεH phases since there is no Se–Se bond. f Se–Te–Se and Se–Se–Te bond angles obtained from the PBE-D3 calculation. g Values inside the parentheses represent the corresponding parameters in ε Te and 2H-MoS2 in sequence.
HγT P[3 with combining macron]m1(164) 0.000 (0.007) 3.99, 3.99, 5.97 2.84 90.9° 5.04
α = β = 90°, γ = 120° (4.41, 4.41, 5.94)b 44.6°
(3.54, 3.54, 6.01)
MH C2(2) 0.002 (−0.005) 7.40, 4.17, 5.40 2.66, 2.48 100.9°, 103.5° 5.69
α = γ = 90°, β = 88.6° (7.66, 4.24, 5.92)c 2.90 101.3°
Mβα C1[1 with combining macron] 0.002 (−0.004) 5.40, 4.16, 7.40 2.66, 3.21 79.5°,100.8° 5.70
α = γ = 90°, β = 88.6° (5.94, 4.41, 7.64)d 4.10 50.2°, 39.6°
HεH P63 (194) 0.227 (0.220) 3.83, 3.83, 6.51 2.87 83.7° 4.96
α = β = 90°, γ = 120° (4.03, 4.03, 6.55)g 48.2°
(3.16, 3.16, 6.15)



image file: c8tc03295e-f2.tif
Fig. 2 Top and side views of the optimized atomic structures of bulk TeSe2 in the HγT (a), MH (b) and Mβα (c) phases. The black dashed line indicates the unit cell of each structure. The red and blue balls represent Te and Se atoms, respectively.

The HγT TeSe2 is similar to the 1T-TiSe2 or 1T-MoS2 structure; however, its a constant is approximately 11% larger than that of 1T-TiSe2, as shown in Table 1. Our separate calculation indicates that it is slightly (1 meV per atom) more stable than the 1T′-like phase (HγT′) observed in MoS2.43 Later, we found that the HγT phase has the lowest density among all the stable phases investigated. In the MH phase, the material represents prismic tube bundles quite similar to those of trigonal Te. However, its β angle is 88.6°; hence, the material adopts the monoclinic phase. Our separate comparison indicates that it is more stable than the corresponding orthorhombic phase in which the β angle is 90° by 1 meV per atom. As shown in Table 1, its lattice constants are slightly smaller than those in the trigonal Te: (a, b, c) = (7.40, 4.17, 5.40) instead of (7.66, 4.24, 5.92) in Å units, where the latter set of data are reduced to those of the orthorhombic cell from the trigonal cell.

In the Mβα phase, the material adopts a layered α structure similar to that in the α-Te, in which a primitive cell contains two staggered layers in Fig. 2(c). As in the case of bulk Te, the centrosymmetric β phase is metastable and transforms into the α phase upon structure optimization. Again, Table 1 shows that the Mβα adopts a monoclinic structure, which is more stable than the corresponding orthorhombic structure by 2 meV per atom. Hence, its β angle is 88.6°.

Our calculation shows that the Mβα phase is marginally less stable than the trigonal phase for the bulk Te, suggesting that the phase could be experimentally identified for both Te and TeSe2. As shown in the table, its lattice constants are slightly smaller than the corresponding ones in the reconstructed phase from the trigonal Te: (a, b, c) = (5.40, 4.16, 7.40) instead of (5.94, 4.41, 7.64) in Å units. As the table shows, the Mβα phase has the highest density among all the phases investigated in this work. Another hexagonal phase (HεH) shown in Fig. S1 (ESI), which adopts a chemical structure similar to that of ε Te in the 2H phase, is far less (= 0.23 eV within the PBE-D3 in Table 1) stable than the three stable phases described above.

In addition, phonon spectra shown in Fig. 3(a–c) also indicate that all three phases would be dynamically stable. The phonon calculation was done with the displacement step size of 0.01 Å for 2 × 2 × 2, 2 × 4 × 3, and 3 × 4 × 2 supercells of HγT, MH, and Mβα phases, respectively. For the Mβα phase, a small imaginary phonon mode (−0.11 THz) is observed near the Γ point. This mode could be fictitious instability due to numerical error since the frequency of an out-of plane mode (ZA) approaches zero quadratically. This small imaginary frequency in the transverse acoustical phonon branch is similar to that of other materials, such as Ge,44 borophene,45 and GeTe.46


image file: c8tc03295e-f3.tif
Fig. 3 Phonon spectra of bulk HγT (a), MH (b) and Mβα (c) phases. All three phases are dynamically stable without noticeable soft phonon modes.

Here, we will focus on the possible transformations among the three stable phases. First, we apply uniaxial tensile stress (TS) to the MH phase along the c axis in Fig. 2(b), during which two other lattice parameters are relaxed freely. For simplicity, the β angle is fixed to that (= 88.6°) of the MH phase in this calculation because the angle is different from that of the final phase only by a marginal amount (= 1.4°). In this regard, we recall that the phase is more stable than the orthorhombic phase by only 1 meV per atom. As shown in Fig. 4(a), we observe a phase transformation from the MH phase to the HγT phase at ∼7.4% strain, at which the applied stress amounts to ∼0.53 MPa. Here, HγT represents a metastable phase (see ESI, Table S1) whose underlying chemical structure is similar to that of the HγT phase except that the β angle is 88.6°. Table 1 shows that the latter phase becomes increasingly stable as the c parameter further increases, finally reaching an energy minimum at 10.4% strain. The compressive stress (CS) at which the reverse transformation occurs is 0.32 MPa, which is smaller than that for the forward transformation. In this regard, we highlight that such a transformation can be easily applied experimentally. Experimental phase changes were realized in carbon nanotubes at much higher (>104 times) compression in diamond anvil cells.47–50


image file: c8tc03295e-f4.tif
Fig. 4 Phase transformations under (a) tensile and (b) shear stresses. To comprehend the stability from an energy perspective, ΔE is given as the difference in energies between the undistorted and distorted structures.

During the transformation, the (a, b, c) parameters (= 7.40, 4.17, 5.40 in Å units) of the MH change to equivalent ones (= 6.90, 4.00, 5.97 in Å units) in the HγT phase, showing Poisson's contractions along the a and b directions. Table S1 (ESI) shows that the HγT is slightly more elongated (= 3.54 Å) in the metastable HγT phase before being broken. The phase change is accompanied by the formation of four inter-strand Te–Se bonds with the sacrifice of one intra-chain Se–Se bond per Te atom.

Moreover, we find that the Mβα can be obtained from the HγT upon application of shear stress (SS) along the a direction on the ac plane. For convenience, the stress is applied to the 2 × 1 × 1 equivalent orthorhombic supercell of the latter phase with (a, b, c) = (6.90, 4.0, 5.97) in Å units. As shown in Fig. 4(b), the transformation is observed at the shear strain of ∼1.10% with the applied SS of 0.12 MPa, whose magnitude is somewhat smaller than the CS described in the previous paragraph. The reverse transformation occurs with the SS of 0.20 MPa, which is comparable to that for the forward transformation. In the figure, Mβα′ represents a metastable phase (see ESI, Table S1) from which the stable Mβα suddenly emerges. The initial (a, b, c) parameters change to (7.40, 4.16, 5.40) in the Mβα. In our convention, these lattice parameters are redefined to (c, b, a), indicating that the 2 × 1 × 1 supercell of the HγT is transformed to a primitive cell of Mβα. The lattice dimension on the ab plane is slightly expanded during the transformation, whereas the opposite is true for the c parameter, and β is changed to 88.6°. All Te–Se bonds (Te–Se2) along the b direction, which amount to two of the total six Te–Se bonds per Te atom, are broken during the transformation so that the original a axis becomes normal to the 2D layers in the Mβα.

To summarize, the HγT → Mβα phase transformation is accompanied by the breaking of two Te–Se bonds per Te atom and the formation of one Se–Se covalent bond. The shear stress required for the forward and reverse transformations (0.12 and 0.20 MPa, respectively) are of the same order of magnitude as that for the tensile stress in the MH ↔ HγT transformation. For comparison, we consider the similar transformations of the bulk Te. The H → γ transition occurs at ∼7.4% tensile strain under the tensile stress of ∼0.78 MPa, which is comparable to that for the MH → HγT transition. The γ → α transformation can also be easily achieved by applying a shear stress of ∼0.2 MPa.

Finally, Fig. S2 (ESI) shows that Mβα can be transformed into MLβαvia shear stress along the b direction, where the latter phase is the mirror image of the former. The transition state corresponds to the Mβ phase, in which all layers adopt a centrosymmetric stacking pattern. The activation barrier of 22 meV per atom can be easily surmounted by thermal energy at room temperature, for which the required shear stress of 0.1 MPa is also of the same magnitude as the other transformations described above. We can easily understand that MLβα ↔ HγT ↔ MLH transformations can be realized by applying the appropriate shear and tensile stresses. Here, MLH denotes another phase which constitutes the mirror image of the MH.

Next, we focus on the electronic structures of bulk TeSe2 in the three phases. In the HγT phase, the band structure in Fig. 5(a) calculated from the MBJLDA suggests that the material is a semiconductor with an indirect gap of 0.43 eV and 0.54 eV with and without inclusion of SOC, respectively. As a result, the valence band maximum (VBM) and the conduction band minimum (CBM) are located at the A′′ along the AH path and A points, respectively. The second and third VBMs are located at the A′ along the AL path and K′ points along the ΓK path, respectively. Their energy eigenvalues are higher than that at the VBM only by thermal energy, i.e., 0.01 and 0.05 eV at the A′ and K′ points, respectively. There is no spin splitting due to the inversion symmetry of the system. The energy eigenvalues of the three VBMs are not affected by the SOC since their charge densities are predominantly concentrated on lighter Se atoms in Fig. S3(a–c) (ESI). Alternatively, two degenerate CBs at the A point of time reversal symmetry are stabilized by the SOC; this behavior can be ascribed to appreciable charge densities on the heavier Te atoms. Fig. S3(d) (ESI) indeed shows this behavior for one of the two CB states, indicating that they are mainly composed of pz(Te) and linear combinations of two spz(Se) orbitals. As previously noted for the 1T phase MX2, type-2 Rashba (R-2) hidden spin polarization will be observed.51,52


image file: c8tc03295e-f5.tif
Fig. 5 (a) The MBJLDA band structure of bulk TeSe2 in the HγT phase without and with SOC (black and red lines, respectively). (b) The 1st Brillion zone of bulk TeSe2 in the HγT phase.

In fact, the degenerate CBs have opposite spin polarizations, as manifested in the three components of the spin magnetic moment of (0.32, 0.16, 0.24) and (−0.32, −0.16, −0.24) in the Bohr magneton, respectively. Similar hidden spin texture is also observed for doubly degenerate CB-1 and CB-2 lying above the CB in Fig. 5(a).

We turn our attention to the electronic structure of the bulk MH TeSe2. Fig. 6(a) shows that it is a semiconductor with an indirect gap (1.00 and 1.11 eV with and without the SOC, respectively) within the MBJLDA. The gap is larger than that of the HγT phase with the lower density described above. In addition, it is also significantly larger than that (= 0.25 eV) of the trigonal Te obtained from our calculation due to the difference in the electronegativity of Te and Se atoms. [The latter gap is in a good agreement with an experimental (0.33 eV) result2 and other calculations with the SOC of 0.251 and 0.21 eV23 based on the MBJLDA, and 0.32 eV from the hybrid HSE06 functional calculation.53] The figure shows that the VBM of the phase is located at the Z point. The VBM and CBM show interesting spin textures caused by the SOC. For example, the CBM is located at the Z and Z′ points for different spin components, where Z′ is located slightly off from the Z point to Z′ along the ZΓ path. Consequently, the band gaps (1.04 and 1.11 eV at the Z and Z′ points, respectively) are different for the two spin components. As shown in Fig. 6(b), the second VBM at the E point does not show splitting under the SOC, whereas the third VBM at the D′ point along the BD direction splits by 140 meV. The splitting is slightly larger than that at the H point in the trigonal Te, which was cited to be 110 and 112 meV from the MBJLDA calculation1 and an experimental mesurement,2 respectively. The second VBM is within 0.15 eV below the VBM, and the third VBM is within 0.40 and 0.25 eV for the two spin components. Similarly, Fig. 6(c) shows that the second CBM at the D′ point also splits into two under the SOC, being separated by a similar amount from (= 132 meV) each other. Hence, the material exhibits quasi-direct gaps of ∼1.11 and ∼1.40 eV at the Z and D′ points under the SOC, respectively. Note that charge carriers will be transported along the two directions, i.e., the helical direction and another off direction. The charge density plots at various k-points are shown in the ESI, Fig. S4(a–e). In the VBM at the Z point, the charge densities are delocalized over the whole space. In contrast, the densities are highly localized around Te atoms at the D′ point; this localization appears to be the reason for the large spin splitting at the k-point. In addition, they are mostly localized on the Se atoms at the E point, where the spin splitting is absent. Fig. S4(d and e) (ESI) show that the spin splitting at the Z′ point (CBM) and the D′ point of the second CBM can be ascribed to the appreciable charge densities on the heavier Te atoms. In addition, Fig. S5(a and b) (ESI) show that MH and MLH display opposite spin polarizations at the k-points of spin splitting depending upon the chirality.


image file: c8tc03295e-f6.tif
Fig. 6 (a) The MBJLDA and MBJLDA + SOC (black and red lines) band structures of bulk TeSe2 in the MH phase. (b) Valence band and (c) conduction band states near the Fermi energy, showing the spin splitting (red and blue) corresponding to non-centrosymmetry. (d) The 1st Brillion zone of bulk TeSe2 in the MH phase.

In the bulk Mβα phase, Fig. 7(a) shows that TeSe2 is a semiconductor with the largest band gap among the three phases investigated. The gap is indirect and lies in the visible region. Its magnitude is 1.93 and 1.86 eV within the MBJLDA and MBJLDA + SOC, respectively, for the Z′′ → Z′′′ transition; Z′′ and Z′′′ are located around the center along the ΓZ path. Therefore, carrier transport will primarily occur along the inter-layer direction of the material. The second VBM at the Γ′ point and the second CBM at the Γ′′ point is 0.02 eV lower and 0.11 eV higher than the VBM and the CBM, respectively; Γ′ and Γ′′ are located slightly off from the Γ point along the ΓY path. Due to the absence of centrosymmetry, spin degeneracies are also lifted when the SOC is included, also displaying spin textures for both of the VBMs and CBMs in Fig. 7(b and c). In fact, Fig. S6(a–d) (ESI) show that there are appreciable charge densities on the heavier Te atoms at the VBMs and CBMs. Specifically, the splitting is the largest (= 150 meV) at the Z′′′ points of the CBM, for which the charge concentration on the Te atoms is quite pronounced. Finally, Fig. S7(a and b) (ESI) also show chiral spin splitting in that the Mβα and MLβα display opposite spin polarizations at specific k-points.


image file: c8tc03295e-f7.tif
Fig. 7 (a) The MBJLDA and MBJLDA + SOC (black and red lines) band structures of bulk TeSe2 in the Mβα phase. (b) Valence band and (c) conduction band states near the Fermi energy showing the spin splitting (red and blue) corresponding to non-centrosymmetry. (d) The 1st Brillion zone of bulk TeSe2 in the Mβα phase.

Conclusions

In conclusion, our structure prediction complemented by the calculation of the energy of formation indicated that TeSe2 is the most stable among various stoichiometries of bulk Te(1−x)Sex. In addition, the phonon spectra confirmed the dynamical stabilities of the polymorphic phases identified. The HγT and Mβα phases adopt layered structures, as in 1T-TiSe2 and orthorhombic Te, respectively. On the one hand, the MH phase consists of bundles of helical chains similar to trigonal Te. Different from the case of bulk Te, bulk HγT is as stable as bulk MH, being within 2 meV per atom within the PBE-D3. The three phases can be transformed not only from one to another but also to their mirror images (MLH and MLβα) via uniaxial tensile stress or shear stress of magnitude less than 1 MPa. The transformations are mediated by the breaking and forming of different chemical bonds.

Band structure calculations within the MBJLDA + SOC show that all three phases are semiconductors. The band gap (= 0.43 eV) is the smallest in the HγT phase with the lowest density. Its band structure displays Rashba's type-2 hidden spin texture because of the centrosymmetry, as exemplified in the different spin polarizations of the two degenerate CBMs. In the Mβα phase with the highest density, the band gap (= 1.86 eV) is the largest. The band gap (= 1.11 eV) of the MH phase is significantly larger than that (= 0.25 eV) of the homonuclear trigonal Te. Depending upon the charge density distribution, the band edge positions can be affected to different degrees by the SOC. The MH and Mβα phases exhibit conspicuous spin texture under the SOC due to the lack of centrosymmetry, in that two spin components of their frontier bands split by more than 100 meV. Moreover, the MLH and MLβα phases exhibit spin polarization opposite to the MH and Mβα, respectively. The polymorphic phase transformations may have applications not only in spin memory devices but also in real time spin computation via external gating of spin configurations.54 The few layers of the MH and Mβα phases can also be applied to photocatalyzed CO2 splitting because of the huge surface to volume ratio.55,56

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This study was supported by NRF-2017H1D3A1A01014082 and NRF-2018R1A2B2006474 funded by the Ministry of Science and ICT. This work was also supported by the National Institute of Supercomputing and the Network/Korea Institute of Science and Technology Information with supercomputing resources, including technical support (KSC-2018-C2-0022). We would also like to thank Jeonju University for providing partial financial support.

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Footnote

Electronic supplementary information (ESI) available. See DOI: 10.1039/c8tc03295e

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