Optoelectronic Properties of TiS2: A Never Ended Story Tackled by

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Optoelectronic Properties of TiS2: A Never Ended Story Tackled by Density Functional Theory and Many-Body Methods Adrien Stoliaroff, Steṕ hane Jobic, and Camille Latouche* Institut des matériaux Jean Rouxel, Université de Nantes, CNRS, 2 rue de la Houssinière, BP 32229, 44322 Nantes, Cedex 3, France

Inorg. Chem. Downloaded from pubs.acs.org by EASTERN KENTUCKY UNIV on 01/16/19. For personal use only.

S Supporting Information *

ABSTRACT: Herein is reported a thorough computational investigation on the bulk TiS2 material with the CdI2 structure type and the ideal 1:2 Ti:S stoichiometry. Computations were performed using some of the most refined models, e.g., a hybrid functional together with dispersion effects (Grimme’s), the GW ansatz, and the Bethe− Salpether equation for the optical properties. We showed that switching from Perdew− Berke−Enzerhof (PBE) to PBE0 leads to a gap opening. Moreover, our results demonstrate unambiguously that van der Waals interactions must be properly treated with dispersion effects in order to retrieve the experimental crystal structure and the appropriate c/a ratio. Indeed, the calculations prove that when one uses a highly accurate computational protocol, the bulk hexagonal TiS2 is a semiconductor with a small gap, whereas using the generalized gradient approximation (GGA) PBE functional leads to a semimetal. Furthermore, the band structure is significantly modified when dispersion parameters are taken into account. Pressure effects were also investigated, and they fully describe the previously simulated electronic transition behavior of the material, e.g., TiS2 becomes semimetallic under strain.

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INTRODUCTION In the past decade, researches on inorganic materials have brought major advances in new technologies (nanomaterials, devices, etc.). Among all of the available techniques, theoretical calculations have allowed the breaking of many locks thanks to the growth of computer resources, theoretical advances, and software development.1−16 Despite the decisive improvement of computer codes, material behavior modeling still remains a complicated task, and many systems divide the theoretical community. In that framework, titanium disulfide, a material with the very simple CdI2 structure with only three atoms per unit cell, is a matter for debate. Conflicts concern its real electrical behavior.15−33 In other terms, does stoichiometric TiS2 have to be described as a semimetal or a low-gap semiconductor? Many theoretical investigations argue that the compound should be a semiconductor34−36 other calculations lead to a semimetallic behavior.23,26,31,37,38 From the experimental point of view, this question still remains open because of the reported propensity of the material to be naturally slightly off-stoichiometric with the insertion of Ti atoms in the van der Waals gap, i.e., between two 2/∞[TiS2] slabs built upon [TiS6] octahedra sharing edges. This singularly complicates the assignment of the TiS2 electrical behavior because the experimentalist cannot certify the exact chemical composition of the material on which the physical properties are collected. If TiS2 received much attention in the 1970s because of its ability to be used as a positive electrode in lithium and sodium batteries,39 its interest was recently renewed for its thermoelectric properties.40 In that context, the Seebeck coefficient S has to be regarded as a crucial parameter for which the © XXXX American Chemical Society

mathematical expression differs versus the semiconducting (or insulating) and semimetallic (or degenerated semiconducting) character of the material. Consequently, the exact nature of the electrical properties of stoichiometric TiS2 has to be elucidated to go further. In that context, the electronic competition between the sp anionic and d cationic levels here also has to be clarified once and for all. Thus, we have embarked on the reinvestigation of the electronic structure of TiS2 via the use of state-of-the-art methods in the domain of ab initio calculations of the electronic structures. Van der Waals interactions are vital to ensuring cohesion between the layers. Thus, the lack of dispersion effects (i.e., accounting for London, Debye, and Keesom forces) in most calculations may be, in part, responsible for the erratic results. One of the modern methods used by computationally oriented scientists to overcome this issue consists of keeping the experimental cell parameters without relaxing the structure.41 In that context, the c/a ratio is kept identical with the experiment (a value that does not necessarily correspond to the TiS2 ideal composition). However, several conceptual issues are associated with this method. The most important is linked to the difficulty to synthesize stoichiometric TiS2. Such an optimization does not ensure the discovery of the global energy minimum. A recent density functional theory (DFT) investigation based on the DFT-D2 (Grimme’s dispersion)42 set of parameters demonstrates the high efficiency of such a method to reproduce cell parameters, although a semimetallic behavior is still simulated.38 Received: October 11, 2018

A

DOI: 10.1021/acs.inorgchem.8b02883 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry However, performing calculations as such artificially keeps the geometry induced by weak interactions (dispersion) without properly taking them into account in the electronic structure, which may lead to an incorrect description of the electronic properties. In order to better understand the impact of these interactions on the TiS2 structural arrangement and electronic structure (and particularly on the band gap), calculations including Grimme’s GD3-BJ dispersion effects have been performed on the primitive cell using the generalized gradient approximation (GGA)-Perdew−Berke−Enzerhof (PBE) functional and the hybrid-PBE0 functional. D3 is a dispersion correction scheme proposed in 2010 and was further improved in 2011 by adding Becke−Johnson (BJ) damping terms (hence, the acronym GD3-BJ, hereafter labeled “D”).43,44 Many-body (GW) corrections (on top of the perfect cell optimized at the GGA level of theory) have also been carried out to correct as accurately as possible the positioning of the valence-band maximum (VBM) and conduction-band minimum (CBM). To explore the electronic properties further, we also provide simulations of the relative permittivity (ε1 and ε2), together with the refractive index (n) and extinction coefficient (k) using the hybrid functional and pure GGA, with and without the inclusion of empirical dispersion effects. According to our results, the ideal TiS2 material is unambiguously a semiconductor. Nevertheless, an explanation of the calculated, and sometime observed, semimetallic behaviors is also furnished. Moreover, in the vein of a previous investigation on MoS2,45 pieces of evidence of the relationship between the pressure and electronic properties are given and remarkably explain the high difficulty to characterize the TiS2 electronic behavior on an experimental point of view. Finally, the complex dielectric function of TiS2 is calculated and compared to the experimental results.

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Table 1. GD3-BJ Parameters Used in the Calculations Depending on the Functional56 functional/parameter

S8

S6

A1

A2

PBE PBE0

0.7875 1.2177

1.000 1.000

0.4289 0.4145

4.4407 4.8593

Hereafter, we distinguish the functional used with respect to the inclusion, or not, of dispersion effects. PBE and PBE0 acronyms are used for computations without any dispersion effects. PBE-D and PBE0-D correspond to calculations including empirical dispersion effects. Many experimental and computational studies relate the pressure effects on the electronic properties of TiS2.18,24,26,31,38,57 Therefore, calculations have also been carried out by taking into account such constraints. To mimic isotropic negative and positive pressures, computations have been performed by adding 10% and removing 4% on each previously refined cell parameter, followed by their refinement at constant volume to depress and press the system (increase and decrease of the volume cell of ∼33% and ∼10%, respectively).

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RESULTS AND DISCUSSION Optimized Structures. We first discuss the cell parameters of the relaxed geometries, and we compare them to the experimental ones used as a reference (Table 2).15 TiS2 crystallizes in a CdI2 structure (Figure 1) within the P3̅m1 space group [Ti at (0, 0, 0) and S at (1/3, 2/3, z) and (2/3, 1/3, −z) with 1a and 2d Wyckoff positions, respectively].15 Computations started from the experimental structure taken as a reference15 (keeping the local D3d symmetry). Because of the difficulty in taking into account van der Waals interactions occurring between the layers, the cell parameters of bulk TiS2 are generally not allowed to relax and the c/a ratio is maintained constant and equal to the experimental one.15 Herein, we optimized independently the cell parameters in order to demonstrate, first, the impact of the functional (PBE vs PBE0) and, second, the impact of the empirical dispersion for the PBE and PBE0 functionals. The a parameter is almost equally described with and without the inclusion of dispersion corrections. Indeed, these later induce a systematic decrease of ∼0.05 Å, which is best fitted with the PBE functional. In contrast, the c parameter is badly described in PBE (6.527 Å) and PBE0 (6.317 Å) when one compares it to the experimental one (5.695 Å)15 (errors of +15 and +11%, respectively) but becomes satisfactory for PBE-D and PBE0-D (5.536 and 5.492 Å, respectively) with corresponding errors lowered to −3 and −4%. Moreover, it is worth noting that the Ti−S bond lengths of [TiS6] octahedra are systematically lowered by increasing the level of accuracy. Indeed, going from PBE to PBE0 led to a decrease of ca 0.02 Å. The same trend (and deviation) is observed when one takes into account dispersion effects, i.e., from PBE to PBE-D and from PBE0 to PBE0-D. Notice that interslab S−S distances are strongly overestimated for PBE and PBE0 calculations, while calculations with dispersions yield interatomic distances in much better agreement with the experiment. The intraslab distances are all in good agreement with respect to the experiment, although the switch from PBE to PBE0 (PBE-D to PBE0-D, respectively) tends to show a small diminution of this distance. Finally, the cell volume is also discussed. Using classic PBE and PBE0 functionals, the disagreement between the experiment and simulations is more than 10% (overestimation). However, this disagreement falls around 6% (underestimation) when dispersion effects are added to the computations. In short,

COMPUTATIONAL DETAILS

All of the computations were performed using the Vienna Ab Initio Simulation Package (VASP).46−48 The GGA-PBE and hybrid-PBE0 functionals were used within a projector-augmented-wave (PAW) scheme.49,50 Hybrid functionals replace part of the DFT exchange with the Hartree−Fock exchange to correct the band positions. For both, the bulk TiS2 material was optimized (with and without dispersion) with the cutoff energy set to 400 eV. The atomic relaxation was carried out with a convergence criterion of 0.01 eV/Å. Different k meshes were used to optimize the structures and to check the convergence. In GGA, the best ratio of computational time versus accuracy was when one optimizes in 7 × 7 × 7 k-point mesh and performs density of states (DOS) calculations in 17 × 17 × 11, leading to 318 k points in the irreducible Brillouin zone for the DOS. Despite the relatively small size of the cell, using the hybrid functional remains very computationally demanding. Therefore, the cells were optimized with 5 × 5 × 5 k meshes and DOS calculations performed in 7 × 7 × 7. A test calculation using spin−orbit coupling has been performed. According to our results, the impact of the spin−orbit in this compound is negligible (ca. 0.01 eV). DOS, band structures, and optical indices were plotted using the PyDEF program.51,52 On top of the optimized PBE0-D geometry, a partially self-consistent GW calculation has been performed by iterating only the Green function part G but keeping the screened Coulomb interaction W and the orbitals fixed to the initial DFT level,53 correcting the band-edge positions and band gap. In order to get a more realistic simulation of the optical properties, a Bethe−Salpeter equation (BSE) computation has also been performed after the aforementioned GW calculation using a 5 × 5 × 5 k mesh.54,55 A good description of the c/a lattice parameter ratio is critical for such material. Therefore, the impact of the so-called GD3-BJ empirical dispersion effects proposed by Grimme et al. was tested using the parameters given in Table 1.43,44,56 B

DOI: 10.1021/acs.inorgchem.8b02883 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry Table 2. Relevant Parametric Data of Bulk TiS2 a (Å) c (Å) c/a Ti−S (Å) S−S(interslab) (Å) S−S(intraslab) (Å) V (Å3)

PBE

PBE0

PBE-D

PBE0-D

expt15

3.390 (−0.5%) 6.527 (15%) 1.93 (16%) 2.427 (−0.1%) 4.150 (20%) 3.472 (0.3%) 64.97 (14%)

3.402 (−0.1%) 6.317 (11%) 1.86 (−0.6%) 2.41 (−0.8%) 4.042 (17%) 3.407 (−1.6%) 63.3 (11%)

3.340 (−2.0%) 5.536 (−3%) 1.66 (−0.6%) 2.407 (−0.9%) 3.282 (−5%) 3.466 (0.1%) 53.48 (−6.6%)

3.356 (−1.5%) 5.492 (−3.5%) 1.64 (−1.8%) 2.389 (−1.6%) 3.319 (−4%) 3.402 (−1.7%) 53.57 (−6.4%)

3.407 5.693 1.67 2.429 3.460 3.462 57.26

major correction remains the one made by the nature of the functional. Furthermore, as sketched in Figure 2, the inclusion of dispersion effects tends to diminish the sharpness of the DOS of Ti atoms in the conduction band. For instance, the very sharp peak at ∼1.7 above the Fermi level in PBE is decreased and widened when one uses PBE-D. From GW calculations with the same number of k points, an affordable and accurate band-edge correction was determined. Then, according to our simulations, the band gap is slightly increased from the starting point (i.e., 1.20 vs 0.95 eV). Let us now focus our attention on the band structures. Figure 3 depicts the first Brillouin zone of the hexagonal structure type, and the following reciprocal paths were chosen: Γ−M−K−Γ− A−L−H−A|L−M|K−H. Some of the high-symmetry points, e.g., A, L and H, are directly related to the c lattice parameters and account for the interslab interactions. As a matter of fact, the c/a ratio is the critical point to cleanly simulate the bulk TiS2 band structure. The band diagrams of TiS2 with GGA and hybrid functionals are sketched in Figures 4 and 5, respectively. As illustrated in Figure 4, the top of the valence band (anionic band) is in the vicinity of the Γ high-symmetry point (hole pocket), whereas the bottom of the conduction band (cationic band) is located at L (electron pocket). At that point in GGA, the energy level of this band is below E − EF (zero energy), i.e., below the calculated Fermi level, meaning that a semimetallic behavior is achieved. This trend is enhanced when one adds dispersion effects into the calculation. Indeed, the bottom of the conduction band at L lies around −0.35 eV in PBE-D versus −0.15 eV in PBE only. Furthermore, the patterns of the band diagrams in PBE and PBE-D are almost identical (vide infra for a discussion of the differences). On these grounds, we performed the same calculations using hybrid functionals with and without taking into account dispersion effects (Figure 5). Furthermore, it should be pointed out that a small hole pocket shows up in Γ, which fully explains the simulated semimetallic behavior of TiS2. As one can see, the overall pattern remains quite similar on going from pure GGA to a hybrid functional, but a true (indirect) band gap shows up. This one is estimated at ∼1.05 eV in PBE0 and ∼0.70 eV in PBE0-D, with the maximum of the valence band being at Γ and the minimum of conduction band being at L as previously (Figure 5). This result is in excellent agreement with the experimental data reported by Ohno, where a band gap close to 0.5 eV was determined by reflection electron energy loss spectroscopy (REELS).63 Including dispersion effects slightly affects the overall band structure. Nevertheless, in the Γ−A segment (along the c* parameter, i.e., dispersion perpendicular to the 2/∞[TiS2] layers), the valence bands have a nonnegligible trend to decrease in energy when dispersions effects are included. In contrast, the lowest conduction bands are not affected by the inclusion of dispersion effects in this segment.

Figure 1. TiS2 primitive cell.

dispersion effects favor a slight shrinkage of the cell parameters compared to the experiments, while regular PBE and PBE0 calculations lead to an accurate a value but to an unrealistic overrated c parameter. These geometrical results fully emphasize that enforcing dispersion effects is mandatory for retrieving accurate structures when van der Waals interactions are present in the compound modeled. One should mention that test computations using other dispersion corrective schemes, together with the GGA-PBE functional, have been performed, namely, the so-called dDsC,58−61 Tkatchenko−Scheffler, and Tkatchenko−Scheffler with iterative Hirshfeld partitioning.62 All of the optimizations with the aforementioned dispersion schemes gave the same trend of results as the one exposed in Table 2. Because Grimme’s dispersion effects are efficiently implemented in the VASP program for both the PBE and PBE0 functionals and the c/a ratio is well-reproduced, these corrections have been chosen for the following study. Electronic Structure. On top of the optimized geometries, the DOSs have been plotted (Figure 2). For PBE calculations, in agreement with previous reports on this material with such a level of theory, no band gap exists. Nevertheless, two bands where atomic contributions are clearly distinguishable (S 3p and Ti 3d, respectively) can be differentiated. The GGA functionals are known to severely underestimate the band gap, and this behavior is expected here. The inclusion of dispersion effects does not induce significant changes to the electronic structure because the two aforementioned bands still slightly overlap. When one uses the PBE0 hybrid functional, the d levels of titanium are pushed higher in energy, leading to an opening of the band gap. In this study, the band gaps with hybrid functionals were found to be 1.25 and 0.95 eV for PBE0 and PBE0-D, respectively. In such a case, the TiS2 material is more likely a semiconductor with a low gap. As one can notice, the inclusion of empirical dispersion effects leads to a decrease in the band gap (−0.30 eV) when using a hybrid functional. This means that the correct description of the c/a ratio is critical for a correct rationalization of the material optoelectronic properties, but the C

DOI: 10.1021/acs.inorgchem.8b02883 Inorg. Chem. XXXX, XXX, XXX−XXX

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Figure 3. Irreducible first Brillouin zone of the hexagonal structure type together with the reciprocal high-symmetry k-points path chosen for the band-structure calculations: Γ(0, 0, 0);

(

A 0, 0,

1 2

); H( 13 , 13 , 12 ); K( 13 , 13 , 0); L( 12 , 0, 12 ); M( 12 , 0, 0).

Figure 4. Band diagram of TiS2 at the PBE and PBE-D levels of accuracy (semimetallic behavior: the conduction band in L is lower than the valence band in Γ). See the main text for the partial occupancy at Γ point in GGA.

parallel to the layers), the band originating around −1 eV crosses the bands that were previously higher in energy when enforcing the hybrid functional. With the inclusion of dispersion effects, this is no longer the case. These results clearly indicate that the modification when one uses Grimme’s corrections has a huge impact on the band structure. Consequently, it is imperative to take them into account for a proper description of the electronic structure of the material. To go one step further in cross-validation of the calculations, the DOSs in the vicinity of the VBM were compared to the available experimental X-ray photoelectron spectroscopy (XPS) values collected with a monochromatized incident beam at 1486.6 eV (Al Kα) on a TiS2 supposed to be stoichiometric.64 The only calculations reported in Figure 6 are the ones including dispersion effects because it has been proven above that they are necessary to explain the structural behavior of the compound of interest. In order to get a picture closer to the experimental one to ensure a more realistic comparison, a moving average was

Figure 2. Simulated DOSs (total and projected on atomic species) in PBE (a), PBE + GD3-BJ (PBE-D) (b), PBE0 (c), PBE0 + GD3BJ (PBE0-D) (d), and GW@PBE0 + GD3-BJ (e) from top to bottom, respectively.

On the L−M segment in the reciprocal lattice (segment also perpendicular to the aforementioned layers), the lowest conduction band and highest valence band are very flat in PBE and PBE0, whereas they look dispersed with the inclusion of empirical dispersion effects. In the Γ−M segment (segment D

DOI: 10.1021/acs.inorgchem.8b02883 Inorg. Chem. XXXX, XXX, XXX−XXX

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better fit the experimental data, the calculation using the hybrid functional led to the good shape, although larger bands are obtained compared to the experiment. At the GW level of accuracy, on top of the PBE0-D geometry (orange curve), the wideness of the global maximum including the shoulder (2.2 eV) is accurately reproduced and the data are quantitatively comparable to those observed in XPS. This confirms the hierarchy in the electronic property predictions between the three levels of theory used in this investigation and the semiconducting behavior of stoichiometric TiS2. To sum up this section, one should say that using the GGA functional (with/without van der Waals corrections) leads to a semimetal. Conversely, when one enforces a global hybrid functional (with/without van der Waals corrections) or GW, a gap shows up. Optical Indices. In order to get more insight into the electronic behavior of the 1T TiS2 material, ε1, ε2, n, and k optical coefficients were calculated with the PBE0 hybrid functional with and without dispersion (see the Supporting Information). Calculations were cross-checked against available experimental data collected at 300 K (Figure 7) on a sample prepared by growth from constituent elements by a chemical vapor transport reaction in a sealed silica ampule.63

Figure 5. Band diagram of TiS2 at the PBE0 and PBE0-D levels of theory (semiconductor, indirect Γ−L band gap).

Figure 6. Comparison between the DOSs in the vicinity of the VBM and experimental XPS values.64 Simulation results were smoothed using a moving average (PyDEF)52 post-treatment on the order of 300, 400, and 250 for PBE-D, PBE0-D, and GW calculations, respectively.

applied.52,65 Such a data post-treatment of the nth order on a given signal (s1, ..., sN) consists of averaging each value with the n − 1 previous points, resulting into a smoother signal. Mathematically, the pth value of the smoothed signal sp′ can be written as ∀ p ≥ n , sp′ =

1 n

p

∑

sk Figure 7. Comparison of simulated ε2 optical indices to the experimental data.63 Simulation results were smoothed using a moving average post-treatment on the order of 150 and 200 for PBE0-D and BSE calculations, respectively52 (top = simulated and experimental curves; bottom = shifted simulated curves to fit the experiment based on the position of the second peak around 5 eV).

k=p−n+1

The four curves were aligned horizontally on their maximum and normalized. One can notice that the general shape of the XPS measurements is well reproduced qualitatively within the three levels of theory. The intensity shows a first local maximum around 5 eV below the VBM, followed by a local minimum. It then exhibits a global maximum around 3 eV below the VBM and a shoulder 0.5 eV closer to the VBM. First, it should be mentioned that the global pattern is well depicted in all of the simulations, giving us confidence in the approach used. Second, at the GGA level of accuracy (blue curve), the simulation exhibits an increasing intensity near the VBM, in contradiction with the experiment that originates from the incorrectly predicted semimetallic behavior of TiS2 at the GGA level. This is in good agreement with our previous statements. Such an artifact is absent when one enforces hybrid and GW calculations. Because of the strong smoothing post-treatment step applied to

It should be recalled here that getting the exact stoichiometric TiS2 material is extremely difficult from the experimental point of view. Consequently, the recorded optical band gap is probably not measured on a perfect TiS2. Furthermore, while data are collected at room temperature, all of the simulations are performed at 0 K. Thus, it remains possible that some electrons fill the conduction band with respect to the Fermi−Dirac distribution. This makes a direct comparison between the experiment and simulations very difficult. E

DOI: 10.1021/acs.inorgchem.8b02883 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry First, it should be pointed out that including the dispersion increases the ε1 and n values by ca. 10−15% at the lowest energies. Still, at the lowest energies, the values of ε1 and ε2 along the z direction are almost doubled (see the figure in the Supporting Information; the optical indices n and k are also given). This variation can be fully explained by the diminution of the c-cell parameter when going from the PBE0 to the PBE0-D level of accuracy. However, it must be mentioned that, for nzz and ε1zz at the PBE0 level, this maximum is the global one. On the contrary, when one enforces the PBE0-D level of theory, this second maximum is below the first one by ∼10−15%. Second, we compare our data to the experimental ε2 values (Figure 7) obtained by REELS, followed by a Kramers−Kronig analysis. It should be mentioned that there are no data of recorded TiS2 optical indices below ∼2.31 eV. This complicates the interpretation of the simulated results with respect to the experiment. However, to enable a comparison with the experiment for this anisotropic compound, the traces of the imaginary dielectric tensors are plotted and the simulated curves are shifted to match the experimental peak at 5 eV (Figure 7, bottom). It must be recalled that the compound used to measure this spectrum was not proven to be stoichiometric and a small deviation can be expected.64 The shifts correspond to ca ∼1.6 and ∼0.4 eV for PBE0-D and BSE, respectively. The BSE calculation allows one to calculate the response functions including excitonic effects on top of the GW. The experimental spectrum only suggests two maxima that are nicely reproduced in the simulations. The very intense increase in ε2 below 3 eV is probably issuing from the Raleigh peak that prevents a precise record. In Figure 7 (bottom), one observes a very good match between the simulated BSE and experimental spectra. The two maxima around 2.5 and 5 eV match almost perfectly the experimental spectrum. Finally, it should be pointed out that if one fits the simulated PBE0-D and BSE optical curves to the experimental one, the results issuing from the hybrid DFT independent particle approximation and BSE are different. As a consequence, on the basis of the previous discussions, the electronic band gap of TiS2 is around 0.7 eV and thus fully explains the observed semiconductor-like behavior. On the other hand, the fact that the optical band gap is predicted near 0 eV in PBE0-D also explains why other experimental papers report that TiS2 is a semimetal. To distinguish the semimetallic behavior from the semiconductor one, experimental and theoretical reports also gathered their forces by investigating pressure effects on TiS2. In order to test the validity of our model, the next section is dedicated to the switch in behavior under pressure and depression constraints. Volume and Pressure Effects. MoX2 and TiX2 (X = S, Se, Te) are well-known materials for which strains may modulate the band structure, leading to the switch of the electronic properties, e.g., from semiconductor to semimetal.26,57 Moreover, a DFT (GGA) investigation shows that, by the application of uniform biaxial tensile strain, the model switches from semimetal to semiconductor.38 As explained in the Computational Details section, variations of the cell volume have been performed to mimic the pressure effects (Figure 8). As a recall, it is shown that using the PBE-D level of theory without pressure leads to the conduction-band bottom of ∼−0.35 eV. Moreover, when one adds exact Hartree−Fock exchange (25% in PBE0) together with dispersion, a band gap shows up (vide supra). If a positive pressure (contraction of the cell) is applied, the semimetallic behavior may be triggered. In contrast, a negative one (dilatation) reinforced the semi-

Figure 8. Schematic representation of the evolution of the unit cell under a hypothetical applied positive (ΔV/V = −11%) and negative (ΔV/V = +33%) pressure (cell parameters and volume from PBE0-D).

conducting behavior (Table 3).26,31 In the contracted case, the CBM falls below −0.2 eV with respect to the E − EF energy. On Table 3. Values of the Band Extrema (eV) at Specific highSymmetry k Points Γ M A L ΔΓ−L

CB VB CB VB CB VB CB VB gap

PBE0-D

PBE0-D(contracted)

PBE0-D(dilated)

1.48 0.00 1.07 −2.17 1.46 −0.30 0.72 −1.63 0.72

0.94 0.43 0.42 −3.17 0.89 −1.09 −0.27 −1.88 −0.70

1.78 0.00 1.75 −1.81 1.77 −0.01 1.67 −1.77 1.67

the contrary, when one dilates the investigated system, the conduction band is set to around 1.7 eV above the Fermi level (Figure 9). Consequently, the transport properties may be dramatically affected by the presence of pressure/depression.

Figure 9. Band structures of TiS2 cells under constraint at the PBE0-D level of accuracy (semimetallic behavior for ΔV/V = −11% and semiconductor for ΔV/V = +33%). F

DOI: 10.1021/acs.inorgchem.8b02883 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry

conductor behavior of TiS2 when one uses a global hybrid. When a dilatation is mimicked, TiS2 is clearly a semiconductor. This result nicely encompasses the observed monolayer TiS2 optoelectronic properties. Indeed, if one extrapolates the dilatation effect, the material may be described as a monolayer in the limit of an infinite dilatation.

Furthermore, it should be mentioned that, when going from the contracted structure to the depressed one, the energy difference between the VBM and CBM in A decreases (∼2 and ∼1.8 eV). Because the high-symmetry point A is located at (0, 0, 1/2) in reciprocal space, this phenomenon fully confirms all of our previous statements concerning the importance of accurately reproducing the structure and, more precisely, the c lattice parameter together with the c/a ratio. Furthermore, on going from Γ to A high-symmetry points, the most impacted level is not the CBM but the VBM. In the case of the dilated structure, the VBM does not vary between the two points, whereas the differences are around ∼0.3 and ∼1.5 eV for the usual and contracted cells, respectively. At L (1/2, 0, 1/2), all of the bands are affected, and contrary to A, the absolute energy differences between the VBM and CBM decrease with the addition of pressure. Consequently, this demonstrates that, depending on the pressure effect, the valence-band profile is modified. This implies a change in the hole effective mass (mh*) from high to low on going from depressed to pressed structures. On the other hand, the conduction band displays a smaller dispersive nature, leading to higher electron effective mass (me*). Performing calculations at the hybrid level of theory to study the impact of pressure on the electronic properties of the material, one notices that the general behavior is changed by the constraint applied. Indeed, pressured TiS2 exhibits a semimetallic behavior as the conduction-band energy dives to ∼−0.3 eV at L. On the contrary, dilating the cell results in a gap opening, with the band gap reaching ∼1.7 eV (indirect between Γ and L). The previously dispersed valence-band top on the Γ−A vertical segment is flattened by the dilatation and dramatically dispersed further by the contraction (∼−1.1 eV at A vs ∼−0.30 eV without constraints). Both the conduction and valence bands are flattened by the dilatation and show a greater dispersion under pressure. The vicinity of Γ in the vertical direction (Γ−A) is also perturbed by compression with the appearance of oscillations in the valence-band energies. Our calculations show that pressure can switch the electronic behavior of TiS2. Because it is difficult to obtain experimentally stoichiometric materials, this can explain the discrepancies found in the literature regarding the nature of TiS2, with point defects (intrinsic or extrinsic)17 being expected to affect the structure and to induce internal strains. On the basis of these calculations, it can be concluded that pressure plays a major role in the establishment of a semiconducting regime instead of a metallic one and vice versa. The semiconductivity of ideal TiS2 in normal pressure conditions probably originates from the attractive forces exerted on each 2/∞[TiS2] slab by the two adjacent layers that sandwich it and actually pulling the slab on both sides along the normal direction. Accounting for the van der Waals interactions (i.e., accounting for London, Debye, and Keesom forces) is thus crucial to understanding of the electrical behavior of lamellar materials. Beyond a gap to the 1:2 Ti:S ideal stoichiometry, these interactions may be strongly perturbed by the existence of intrinsic defects, explaining also possibly the large panel of electrical behaviors reported in the literature. Finally, because the pressure state on the 2/∞[TiS2] slab seems to dictate the electrical behavior of the material, substitution of Ti cations for isoelectronic ones to generate an internal pressure may be envisioned as a possible route to stabilizing the semimetallic state instead of the semiconducting one (and vice versa). To sum up this section, one should say that adding pressure (here by modifying the volume) switches on/off the semi-

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CONCLUSION To conclude this investigation, an unambiguous proof that computing TiS2 material without dispersion effects leads to a wrong structure and to unphysical optoelectronic properties. This fact is probably valuable for each structure exhibiting a van der Waals gap. Furthermore, the well-known and severe bandgap underestimation with the PBE functional leads here to negative band gaps with a typical semimetallic behavior. However, the addition of the dispersion effects, together with the use of a global hybrid, led us to think that ideal bulk, and perhaps hypothetical, TiS2 is a semiconductor with a low gap below 1 eV. In addition, the very small GW band-edge corrections fully confirm this assessment. Investigation of the pressure effects proves unambiguously that a small change of the structure can switch the electronic properties of TiS2 from semimetal to semiconductor and conversely. Such a modification is easily achievable thanks to the large van der Waals gap in this structure. It is well-known that controlling the sulfur atmosphere is experimentally difficult and an overstoichiometry in titanium is often expected. Consequently, this latter point also signifies that the presence of a low concentration of defects is dominating the properties of this material. The local modification arising from Frenkel or antisite defects would lead to severe property modifications, although keeping approximately the stoichiometric ratio. Therefore, an investigation of the impact of point defects in TiS2 is planned together with formation energies and concentrations.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.8b02883. Simulated ε1 and ε2 optical indices enforcing PBE0 and PBE0-D levels of theory and simulated n and k optical indices enforcing PBE0, PBE0-D, and BSE levels of theory (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Stéphane Jobic: 0000-0002-1900-0030 Camille Latouche: 0000-0002-3541-3417 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

C.L. thanks the Centre de Calculs Intensifs des Pays de la Loire for computational resources. A.S. thanks Region Pays de la Loire and CNRS for financial support. The authors thank Cédric Doutrieux for his help in creating the graphical abstract. G

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Inorganic Chemistry

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