Tekalign Terfa
Debela
^{a} and
Hong Seok
Kang
*^{b}
^{a}Institute for Application of Advanced Materials, Jeonju University, Chonju, Chonbuk 55069, Republic of Korea
^{b}Department 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
First published on 14th August 2018
Structure prediction complemented by density functional theory (DFT) calculation indicates that TeSe_{2} is the most stable among the various Te_{(1−x)}Se_{x} compounds. Different from the case of bulk Te, the material can equally adopt three different crystal structures: H_{γT}, M_{H}, and M_{βα} phases, which are similar to 1T-TiSe_{2}, 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 M_{βα} phase with the highest density. The H_{γT} 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.
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 strain^{23} 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-MoS_{2}-like γ, 2H-MoS_{2}-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 TeSe_{2}, were theoretically investigated as an isoelectronic compound of β Te. A monolayer of β TeSe_{2} 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, TeSe_{2} 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 TeSe_{2} 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.
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.^{35,36}k-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 program^{39} with the VASP used as the force-constant calculator.^{40} Force calculations are performed using a supercell with converged k-point sampling meshes.
Table 1 indicates that bulk TeSe_{2} can equally adopt three different crystal structures shown in Fig. 2(a–c), i.e., hexagonal (H_{γT}) and two different monoclinic phases of M_{H} and M_{βα}. 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 H_{γT} 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 H_{γT} phase, TeSe_{2} 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 [100] plane or α-Te for multilayers. Different from that of the β-Te, the α-Te adopts the AB stacking pattern without centrosymmetry.^{42}
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 H_{γT} 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-TiSe_{2} 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 H_{γT} and H_{εH} 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-MoS_{2} in sequence. | ||||||
H_{γT} | Pm1(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) | ||||||
M_{H} | 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° | |||
M_{βα} | C1 | 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° | |||
H_{εH} | P6_{3} (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) |
The H_{γT} TeSe_{2} is similar to the 1T-TiSe_{2} or 1T-MoS_{2} structure; however, its a constant is approximately 11% larger than that of 1T-TiSe_{2}, as shown in Table 1. Our separate calculation indicates that it is slightly (1 meV per atom) more stable than the 1T′-like phase (H_{γT}′) observed in MoS_{2}.^{43} Later, we found that the H_{γT} phase has the lowest density among all the stable phases investigated. In the M_{H} 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 M_{βα} 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 M_{βα} 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 M_{βα} 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 TeSe_{2}. 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 M_{βα} phase has the highest density among all the phases investigated in this work. Another hexagonal phase (H_{εH}) 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, M_{H}, and M_{βα} phases, respectively. For the M_{βα} 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}
Fig. 3 Phonon spectra of bulk H_{γT} (a), M_{H} (b) and M_{βα} (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 M_{H} 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 M_{H} 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 M_{H} phase to the H_{γ}′_{T} phase at ∼7.4% strain, at which the applied stress amounts to ∼0.53 MPa. Here, H_{γ}′_{T} represents a metastable phase (see ESI,† Table S1) whose underlying chemical structure is similar to that of the H_{γT} 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 (>10^{4} times) compression in diamond anvil cells.^{47–50}
During the transformation, the (a, b, c) parameters (= 7.40, 4.17, 5.40 in Å units) of the M_{H} change to equivalent ones (= 6.90, 4.00, 5.97 in Å units) in the H_{γT} phase, showing Poisson's contractions along the a and b directions. Table S1 (ESI†) shows that the H_{γT} is slightly more elongated (= 3.54 Å) in the metastable H_{γ}′_{T} 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 M_{βα} can be obtained from the H_{γT} 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, M_{βα}′ represents a metastable phase (see ESI,† Table S1) from which the stable M_{βα} suddenly emerges. The initial (a, b, c) parameters change to (7.40, 4.16, 5.40) in the M_{βα}. In our convention, these lattice parameters are redefined to (c, b, a), indicating that the 2 × 1 × 1 supercell of the H_{γT} is transformed to a primitive cell of M_{βα}. 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–Se^{2}) 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 M_{βα}.
To summarize, the H_{γT} → M_{βα} 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 M_{H} ↔ H_{γT} 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 M_{H} → H_{γT} 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 M_{βα} can be transformed into M^{L}_{βα}via shear stress along the b direction, where the latter phase is the mirror image of the former. The transition state corresponds to the M_{β} 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 M^{L}_{βα} ↔ H_{γT} ↔ M^{L}_{H} transformations can be realized by applying the appropriate shear and tensile stresses. Here, M^{L}_{H} denotes another phase which constitutes the mirror image of the M_{H}.
Next, we focus on the electronic structures of bulk TeSe_{2} in the three phases. In the H_{γT} 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 A → H path and A points, respectively. The second and third VBMs are located at the A′ along the A → L path and K′ points along the Γ → K 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 p_{z}(Te) and linear combinations of two sp_{z}(Se) orbitals. As previously noted for the 1T phase MX_{2}, type-2 Rashba (R-2) hidden spin polarization will be observed.^{51,52}
Fig. 5 (a) The MBJLDA band structure of bulk TeSe_{2} in the H_{γT} phase without and with SOC (black and red lines, respectively). (b) The 1st Brillion zone of bulk TeSe_{2} in the H_{γT} 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 M_{H} TeSe_{2}. 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 H_{γT} 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) result^{2} and other calculations with the SOC of 0.25^{1} and 0.21 eV^{23} 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 Z → Γ 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 B → D 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 calculation^{1} 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 M_{H} and M^{L}_{H} display opposite spin polarizations at the k-points of spin splitting depending upon the chirality.
In the bulk M_{βα} phase, Fig. 7(a) shows that TeSe_{2} 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 Γ → Z 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 Γ → Y 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 M_{βα} and M^{L}_{βα} display opposite spin polarizations at specific k-points.
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 H_{γT} 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 M_{βα} phase with the highest density, the band gap (= 1.86 eV) is the largest. The band gap (= 1.11 eV) of the M_{H} 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 M_{H} and M_{βα} 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 M^{L}_{H} and M^{L}_{βα} phases exhibit spin polarization opposite to the M_{H} and M_{βα}, 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 M_{H} and M_{βα} phases can also be applied to photocatalyzed CO_{2} splitting because of the huge surface to volume ratio.^{55,56}
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8tc03295e |
This journal is © The Royal Society of Chemistry 2018 |