1. Introduction

Titanium sulphide is a semi-metallic in its ground state. Yan-Bin et al. (^{Yan-Bin et al, 2007}) reported the metallic behaviour in the self-intercalated TiS₂ where the coulomb interaction with metal Ti-atom decreases the overlapping of orbitals. The Ti(3d)-S(3p) hybridization is responsible for the covalent bonding in TiS₂. Transition metal chalcogenides show interesting behaviour due to their layered and two dimensional symmetric structures. The metallic species can be easily intercalated into the pure material which is possible due to weak interlayer van der waals attraction (vdW). Strong hybridization occurs among M(3d), Ti(3d) and S(3p) with intercalation with transition metals (M). These chalcogenides have been useful in forming a family of intercalcated compounds (^{Bardhan, Kirczenow & Irwin, 1985}; ^{Tonti, Pettenkofer & Jaegermann, 2004}; ^{Whittingham, 1987}). TiS₂ has been utilised for oxygen catalyst for fuel cells (^{Baresel, Sarholz, Scharner, Schmitz & Bunsen Ges, 1974}) whereas Li-TiS₂ has been explored as a future material for high energy batteries (^{Abraham & Alamgir, 1991}). These observations have been supported by angle resolved inverse photoemission spectroscopy (ARIPES), angle resolved resonant photoemission spectroscopy (ARPES), X-ray photoemission spectroscopy (XPS) and high field magnetization measurements (^{Matsushita, Suga, Kimuta, Negishi, & Inoue, 1999}; ^{Ueda et al., 1986}).

The bonding behaviour in intercalcated TiS₂ has been studied theoretically (^{Cui et al., 2006}; ^{Kim, Li, Tanaka, Koyama & Adachi, 2000}; ^{Suzuki, Yamasaki & Motizuki, 1988}; ^{Yamasaki, Suzuki & Motizuki, 1987}). It has been pointed out that the transport properties of MTiS₂ are unusual and are dependent on those of the guest atom (^{Inoue, Moneta, Neigh & Sasaki, 1986}; ^{Inoue, Sadahiro & Negishi, 1991}; ^{Negishi, Yamada, Yuri & Inoue, 1997}; ^{Tazuke et al., 1988}; ^{Takase et al., 2000}).

Recently, ^{Sharma, Shukla, Dwivedi and Sharma (2015)}, have studied the electronic and transport properties of MTiS₂ (M = Cr, Mn and Fe). However, they have not addressed the mechanical properties of these compounds. In the present work the electronic, structural and mechanical properties of unpolarised TiS₂, TiS₂Cr and spin-polarised TiS₂Cr have been investigated in the framework of density functional theory (DFT).

2. Methodology

All the calculations have been performed by Abinit software based on norm conserving pseudopotential method within density functional theory (DFT) (^{Gonze et al., 2002}; ²⁰⁰⁵; ²⁰⁰⁹; ^{Hohenberg & Kohn, 1964}; ^{Kohn & Sham, 1965}; http://www.abinit.org). The generalized gradient approximation (GGA) using Perdew Burke Ernzerhof (PBE) formulation (^{Perdew, Burke & Ernzerhof, 1996}) has be utilized to include exchange-correlation effects. The wave functions from real to reciprocal lattice have been converted using fast fourier transform algorithm (^{Goedecker, 1997}). The wave functions have been determined self-consistently using conjugate gradient algorithm approach (^{Gonze, 1996}; ^{Payne, Teter, Allan, Arias, & Joannopoulos 1992}). The Monkhorst-Pack scheme has been used for the integration over the Brillouin zone (BZ) (^{Monkhorst & Pack, 1976}). The convergence with respect to plane wave cut-off energy and k-points has been checked. Accordingly, a plane wave cut-off energy of 80 Ry and k-mesh of 8x8x8 have been used. The lattice constants have been relaxed to get the minimum energy. In order to get convergence, system is relaxed until the differences between energies or forces in two consecutive steps are less than 27x10^-6 meV and 2.6 meV/Å, respectively.

Abinit software has been utilized to calculate the elastic constants using the linear-response method. The linear response method is based on second order derivative of the total energy with respect to the strain. For the visualization of the crystal structure and 2D charge density, Xcrysden (^{Kokalj, 1999}) software has been used.

3. Results and Discussion

TiS₂ has a CdI2 type hexagonal crystal structure with space group P3-ml (164). The Wyckoff positions for Ti is 1a (0, 0, 0); the two S atoms are placed at 2d (1/3, 2/3, 1/4) and (2/3, 1/3, 3/4). TiS₂ exhibits strong covalent bonding between Ti and S atoms and weak Van der Waals attraction between the layers. The crystal structures of TiS₂ and TiS₂Cr compounds are shown in Fig. 1. Due to weak attractive vdW contacts between the slabs, TiS₂ can be easily cleaved along planes and hence can be easily intercalated by guest species. The guest Cr atom occupies the 1b (0, 0, 0.5) position, the structure of Cr intercalated in TiS₂ is shown in Fig. 1(b).

The lattice constants ‘a’ and ‘c’ of unpolarized TiS₂, TiS₂Cr and spin-polarized TiS₂Cr compounds have been computed and given in Table 1. The equilibrium lattice constant has been achieved by changing lattice constant and computing the energy equivalent to each lattice constant. The lattice constant corresponding to lowest energy is considered as an equilibrium lattice constant. The lattice constant values obtained in present study for unpolarized TiS₂ and TiS₂Cr are in agreement with available theoretical values (^{Fang, De Groot & Hass, 1997}; ^{Yamasaki et a., 1987}). However, the slight variation in the lattice constant values between present results and pervious values are attributed due to the differences in the calculation schemes.

Density of states (DOS) and projected density of states (PDOS) of the unpolarized TiS₂, TiS₂Cr and spin-polarized TiS₂Cr compounds are given in Figs. 2-4. The DOS for TiS₂ has a minimum at Fermi level (E_F) compared to a high peak in both unpolarized and spin polarised TiS₂Cr.

The DOS for the TiS₂ (Fig. 2) can be divided in three energy regions from -5 to 0, 0 to 5 and 5 to 10 eV. In the low energy region, there are two major peaks around -4.5 and -2.7 eV because of p and d orbital contribution of S and Ti atoms, respectively. In the 0-5 eV region, three peaks are found around 0.7, 1.8 and 3.3 eV mainly because of S(p) and Ti(d) orbital contributions whereas in high energy region (5-10 eV), a continuous energy band are seen. Similar behaviour of the DOS and peaks at similar positions are found in other published reports (^{Fang, Groot & Hass, 1997}; ^{Reshak, Kityk & Auluck, 2008}; ^{Sharma et al., 1999}; ^{Sharma et al., 2015}). For unpolarised TiS₂Cr, the DOS can be divided in three energy regions namely, -7.5 to -2.5, -2.5 to 4 and 4 to 10 eV (Fig. 3). In the low energy region, two peaks are found around -7.0 and -5.0 eV. The peak at -5 eV is because of Cr(d), Ti(d) and S(p) orbital contributions. A continuous band found close to Fermi level having peaks at 0.2, 2.0 and 2.8 eV mainly because of Cr(d) and Ti(d) orbital contributions. In high energy region, again a continuous band with a high peak at 7.6 eV is found mainly because of Cr(s) orbital contribution.

Additionally, DOS for the spin-polarised TiS₂Cr in spin up and spin down has been shown in Fig. 4. The DOS for spin up can be divided in three energy regions namely, -8.0 to -2.8, -2.8 to 4 and 4 to 10 eV. For low energy region, two peaks are found at -7.1 and 5.2 eV mainly due to Cr(d), Ti(d) and S(p) orbital contribution, whereas continuous band having peaks at -1.0, 1.8 and 2.5 eV are found close to Fermi level mainly because of Cr(d) and Ti(d) orbital contributions. In high energy region, a continuous band with one major peak at 6.2 eV is found mainly because of Cr(s) orbital contribution. Slightly shifted peaks are found in down spin. Further, the nature and location of peaks found in the present study are in good agreement with the reported peaks by ^{Sharma et al. (2015)}. Two peaks (at 6.2 eV in up spin and 9.6 eV in down spin) which are originated due to s-orbital contribution of Cr-atom are not found in Sharma et al. (^{Sharma et al., 2015}). Also, the PDOS shows strong Ti(d)-S(p) bonding for TiS₂ whereas strong Cr(d)-S(p), Cr(d)-Ti(d) and Ti(d)-S(s) bonding in unpolarised and spin-polarised TiS₂Cr (Figs. 2-4).

Further, the electronic band structures of unpolarised TiS₂, unpolarised TiS₂Cr and spin-polarised TiS₂Cr have been shown in Figs. 5-7. These band structures are plotted in the high symmetry Γ-M-K- Γ-A directions in the irreducible brillouin zone (IBZ). It is to be noted that Fermi level (E_F) is set to 0 eV.

For TiS₂, the band structure can be divided into two energy regions namely, -5 to 0 eV and 0 to 5 eV. These two regions are also called as bonding (-5 - 0 eV) and antibonding regions (0 - 5 eV). In the lower energy region, the energy states are because of p and d orbital contribution of S and Ti atoms, respectively. In conduction band region (0 to 5eV) the energy states are mainly due to S(p) and Ti(d) orbital contributions. The valence band maximum (VBM) and the conduction band minimum (CBM) are found at Γ and M symmetry points, respectively. Similar behaviour of band structure for TiS₂ has been found in other published reported (^{Fang, et al., 1997}; ^{Reshak et al., 2008}; ^{Sharma, Naut, 1999}; ^{Sharma et al., 2015}).

For unpolarised TiS₂Cr, the band structure is divided into three energy regions. These are -7.5 to 0 eV, 0 to 5eV and 5 to 10 eV. In lower region, energy sates are of similar nature of parent TiS₂. However, the energy states are shifted towards lower energy side. These states are because of Cr(d), Ti(d) and S(p) orbital contribution. In the vicinity of E_F, near VBM and CBM additional states compared to TiS₂ are found mainly because of d-orbital contribution of Cr atom. In high energy region, apart from original states seen in TiS₂ few other states are also found due to Cr atom.

Further, for the spin-polarised TiS₂Cr, the spin up and spin down band structures are shown in Fig. 7. The band structure can be divided in three energy regions namely, -10 to -5 eV, -5 to 5 eV and 5 to 10 eV, respectively. In the lower region, slightly shifted and similar to TiS₂ energy states are found mainly because of Ti(d) and S(p) orbital contribution. Near E_F, a dense states due to mixing of different atomic states mainly due to d-orbital contribution of Cr and Ti atoms. In high energy region, a part from original states in TiS₂ few new states because of Cr atom are found. Finally, due to intercalation of Cr atom, the band structure modifies significantly and there is enhancement in the number of energy states in valence and conduction regions.

The corresponding 2D charge density distributions of the unpolarized TiS₂, TiS₂Cr and spin-polarized TiS₂Cr compounds are shown in Figs. 8-10. TiS₂ displays strong and weak electronic interactions for Ti-S and S-S bonds respectively. TiS₂Cr shows strong electronic interactions for Cr-S compared to Ti-S and S-S bonds. The spin-polarization density or magnetization charge density (spin up minus spin down density) for spin-polarized TiS₂Cr compound (Fig. 10) shows that net magnetization for S atom is nearly zero whereas finite magnetization for Cr and Ti atoms. The extent of magnetization in Cr atoms is higher compared to Ti atoms. This is also supported by the calculated magnetic moment values for spin-polarized TiS₂Cr. The magnetic moments for Ti (m_Ti), Cr (m_Cr) and (m_S) are -0.039 µ_B, 2.393 µ_B and -0.007 µ_B, respectively. The atomic magnetic moment for Ti (m_Ti) is anti-parallel to the Cr atom, indicating that spin-polarized TiS₂Cr compound is ferromagnetic.

The single crystal elastic properties of hexagonal structure are defined by five independent elastic constants namely, C₁₁, C₁₂, C₃₃, C₁₃ and C₄₄ and are given in Table 1. The nature of atomic bonding for hexagonal structures can be predicted based on Cauchy pressures (^{Pettifor & Aoki, 1991}). These are well-defined as

The negative Cauchy pressure shows directional bonding whereas positive value displays predominant metallic bonding. The computed values of Cauchy pressures of unpolarized TiS₂ and TiS₂Cr compounds are positive (Table 2) compared to negative values for spin-polarized TiS₂Cr. This clearly points towards the presence of major metallic bonds in unpolarized TiS₂ and TiS₂Cr whereas directional bonding in spin-polarized TiS₂Cr.

The values of elastic constants can also be utilized to predict the mechanical stability of these compounds. The Born stability criteria for hexagonal crystals are given in Equations (2) and (3) respectively (^{Born, 1940}; ^{Fedorov, 1968}).

and

The values for Born stability criteria suggest that these compounds (unpolarized TiS₂, TiS₂Cr and spin-polarized TiS₂Cr) are mechanically stable (Table 2). The anisotropy aspects (A) for hexagonal structure are defined below.

The values of anisotropic factors should be unity for an isotropic crystal whereas away from unity point towards the presence of anisotropy. The anisotropic values (A₁, A₂, A₃) obtained in present study for unpolarized TiS₂, TiS₂Cr are (1.52, 0.81, 0.69), (1.65, 1.23, 0.62) respectively whereas for spin-polarized TiS₂Cr is (1.80, 1.28, 0.51). These values of anisotropic factors are away from unity indicating the presence of anisotropy in these materials.

Further, in the present study, these anisotropy values are also related to atomic bonding between different layered atoms. For TiS₂ system, z axis atomic positions are Ti (z= 0.0 Å), S1 (z = 1.4648 Å) and S2 (z = 4.3943 Å), respectively. The three Ti-S1 bond lengths are 2.4982 Å and forth one is 4.3043 Å. The S1-S2 bond length is 3.5606 Å. For unpolarised TiS₂Cr, z axis atomic positions are Ti (z = 0.0 Å), S1 (z = 1.4751 Å), Cr (z = 2.9502 Å) and S2 (z = 4.4253 Å), respectively. The three Ti-S1, three Cr-S1 and the three Cr-S2 bond lengths are equal (2.5158 Å). The forth bond for Ti-S1, Cr-S1 and Cr-S2 are also same (4.3347 Å).

Finally, for spin-polarised TiS₂Cr, z axis atomic positions are Ti (z = 0.0 Å), S1 (z = 1.4753 Å), Cr (z = 2.9506 Å) and S2 (z = 4.4258 Å), respectively. The neighbouring bonds for Ti-S1, Cr-S1 and Cr-S2 are of equal magnitude (2.5160 Å). One bond for Ti-S1, Cr-S1 and Cr-S2 are of magnitude (4.3351 Å). Therefore, the magnitude of atomic bonding between Ti-S1, Cr-S1 and Cr-S2 is same after the intercalation of Cr atom in the TiS₂ system. Further, the elastic anisotropic values such as A1 and A2 increase after the intercalation of Cr atom in the TiS2 system (Table 2). This indicates that the atomic bonds along z-direction and shear plane perpendicular to z-direction are stiffer after intercalation of Cr atom in the TiS₂ system.

For polycrystalline hexagonal materials, ^{Voigt (1928)} and ^{Reuss (1929)} approximations can be utilized to estimate the upper and lower limit of elastic modulus. These are

Where B is bulk modulus while G_V and G_R are shear modulus values obtained by Voigt and Reuss approximations respectively.

The average value of these two estimates mentioned above is given by ^{Hill (1952)} approximation for hexagonal materials. The Voigt-Reuss-Hill (VRH) average values are given by

Where B (=B_H), G (=G_H), E and ν are bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio, respectively.

The bulk modulus, shear modulus and Young’s modulus (Table 3) exhibit maximum value for spin-polarized TiS₂Cr followed by unpolarized TiS₂Cr and TiS₂. The values of shear and bulk modulus can also be utilized for The brittle and ductile behaviour of materials can also be predicted by taking ratios of G and B (^{Pugh, 1954}). If the ratio (G/B) greater than 0.57, material is brittle otherwise related with ductile behaviour. Based on the G/B ratio, the spin-polarized TiS₂Cr is brittle whereas unpolarized TiS₂Cr and TiS₂ are ductile.

Debye temperature (θ_D) is one of the most significant parameter and it decides the thermal characteristics of the materials. The Debye temperature (θ_D) can be obtained from the mean sound velocity and can be calculated from the certain relations (^{Anderson, 1963}; ^{Sun, Ahuja & Schneider, 2004}).

where v_L, v_T and v_m are the longitudinal, transverse and mean sound velocities obtained using the shear modulus (G), bulk modulus (B) and the density (ρ).

The Debye temperature (θ_D) can be given as

where h is the Planck’s constant, k is the Boltzmann’s constant, N _A is the Avogadro’s number, ρ is the density, M is the molecular weight, n is the number of atoms in the unit cell, and v _m is the mean sound velocity.

The calculated values of longitudinal sound velocity (v_L), transverse sound velocity (v_T), mean sound velocity (v_m) and Debye temperature (θ_D) have been given in Table 4. The higher θ_D suggests the higher thermal conductivity associated with the material. Present study suggests that the value of θ_D is maximum for the spin-polarised TiS₂Cr followed by unpolarized TiS₂Cr and TiS₂.

4. Conclusions