Exploring the Physical Properties of Tungsten Diselenide (WSe 2 ): A First Principle Approach

The physical properties of crystalline compounds are highly essential for predicting their applications in various nanodevices. Density functional theory is very useful in that regard. Herein, some physical properties such as structural stability, and electronic, optical and mechanical properties of two-phase bulk WSe 2 have been comparatively investigated by first principle calculations. Band structure calculations elucidate that the materials are indirect band gap semiconductors having band gap energy in the range 1-1.5 eV. It was observed that the hexagonal crystal has higher absorption coefficients compared to the trigonal phase. Analysis of elastic constants of the materials carried out using the Born criteria method indicates that all systems are mechanically stable at room temperature. Our findings are in close agreement with the previous experimental and theoretical results, this further suggests that the materials are promising in optoelectronics applications.


I. INTRODUCTION
ungsten Diselenide (WSe2) is an inorganic crystalline compound belonging to the family of transition metal dichalcogenide (TMDC) [1] represented with general formula MX2, where M stands for transition metal and X is chalcogenides (S, Se and Te) [2].WSe2 is chemically stable [3], p-type in nature and has an indirect band gap [4,5].It takes hexagonal and trigonal phases.Recently research on TMDC materials has shown that dimensionality plays a very important role in determining the basic physical properties of these compounds [6].In this regard, a monolayer or twodimension, bilayer and a few layers of TMDCs have been successfully synthesized [7].The bulk and few layers of WSe2 have found applications in many devices and electronic components, such as solar cells, rechargeable lithium batteries, photodetectors, transistors and light emitting diode (LED) [8].It is also used as a lubricant.Among TMDCs WSe2 has been a very attractive crystalline compound in recent years due to its precious physical properties such as good electron mobility, structural stability, and band gap in the visible light spectrum (400 to 700 nm) [9].Reference [10] reported a GW and PBE method to examine the band structure of bulk WSe2.More recently, [6] investigated the electronic and optical properties of hexagonal bulk WSe2 (p63/mmc) using DFT, utilizing PBE-GGA.They confirmed the semiconducting nature of this compound.Though the physical properties of the bulk hexagonal phase of WSe2 are well studied, the trigonal phase has received far less attention and its properties are not completely explored.
In this paper, we used first-principles density functional theory (DFT) to comparatively investigate the structural, electronic, elastic and optical properties of the hexagonal with space group (p63/mmc), group number 164 and trigonal-phase having space group (3 1) and group number 187 bulk WSe2.

T II. COMPUTATIONAL DETAILS
In this study, we used the projector augmented plane wave (PAW) numerical method to solve the Kohn-Sham equation within the framework of density functional theory (DFT) implemented in Quantum ESPRESSO code [11] to investigate the hexagonal and trigonal phases of bulk tungsten Diselenide (WSe2) with 6 and 3 atoms respectively.First principle functional theory non-spin polarized calculations [12] were carried out in the generalized gradient approximation for crystals as given by Pardew, Burke and Ernzerhof (GGA-PBE) [13].For integrals smearing has been employed and precisely the Maxfesse-Paxton smearing method [14], for both the two crystalline compounds.Monkhost-Pack Scheme has been adopted for the Brillouin zone integration [15] with 12x12x3 k-points grids for WSe2, in the electronic properties calculation.The WSe2 structures were optimized using Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithms [16].The atomic positions were fully relaxed until the forces acting on the atoms were below 0.001 eV/Å.

A. Structural properties
The equilibrium structural ground state parameters of the two-phase bulk WSe2 compounds, such as the lattice constants, the structural optimization in the quantum espresso code has been carried out using Perdew Burke Ernzerhof generalized gradient approximation (PBE-GGA) by minimizing the total energy with respect to the volume of the unit cell.The geometric structures of the compounds are hexagonal and trigonal as shown in Fig. 1a and b, Fig. 1c shows the Brillouin zone used for k-path in band structure calculations for the considered crystal compounds.Fig. 2 depicts the energy as a function of the volume of the hexagonal and trigonal phases of the considered compounds.The calculated total energies are fitted to Murnaghan's equation of state to obtain the ground state equilibrium lattice constant.The total ground state energies for the hexagonal and trigonal structures are -106.207316and -53.103372Ry per supercell which confirm the structural stability of the considered systems.The calculated and experimental lattice constants, bulk modulus and its pressure derivative are listed in Table I.It is seen that the lattice constants obtained with PBE-GGA agree with other similar calculations.The calculated bulk modulus in Hexagonal WSe2, 59.7 GPa, is 15% smaller than that of trigonal.Table I.Calculated equilibrium lattice constants for Bulk Hexagonal and Trigonal WSe2 structures by using PBE-GGA.

B. Electronic Properties
We use the optimized structures of the considered materials to carry out the electronic properties' calculations.In electronic band structure calculation, we used PBE-GGA.The band structures of the two compounds were calculated along high-symmetry directions of the Brillouin zone Г-M-K-Г and the fermi energy level is set at 0 eV.The calculated band structures for the considered crystals are shown in Fig. 3.Both phases of bulk WSe2 are indirect band gap semiconductors.The calculated band gap energies are 1.4 and 1.5 eV for hexagonal and trigonal respectively, these values together with experimental values are listed in Table II.These results show that DFT has overestimated the band gaps.This is due to the inherent drawback of GGA/LDA functional courses by self-interaction errors [18,19].Fig. 4 illustrated the density of state plots for the considered compounds, they all confirmed the band gap energies of the materials as observed in band structure plots.
Electron acquires masses, different from its rest mass (mo) when accelerated in a magnetic or electric field.This mass is known as effective mass (m*) and has a negative value in the valence band and a positive value in the conduction band.By using the curve fitting technique, the effective mass of the electron in conduction band minimum (CBM) and hole in valence band maximum (VBM) were obtained from the E-K plot (Band structure) of the compounds by applying (1).
Table III shows the calculated effective masses of electrons and holes for the two phases in the unit of electron rest mass (mo).Reference [20] has reported small effective masses of electron holes for perovskite/HfS2 and attributed it to the efficient separation of photon-generated excitons.This is what makes them attractive for photovoltaic applications.

C. Optical Properties
The crystal's response to incident radiation determined its application for optoelectronic devices and nanoelectronic components [21].This prompted the study of the optical properties of crystalline materials, such as absorption coefficient, optical conductivity, refractive index, reflectivity etc.These parameters can be determined from the dielectric tensor ɛ(ω) given by [22].
The imaginary part ɛ (ω)) can be calculated from the Kubo-Greenwood equation expressed as (3) [23], While the real part ɛ (ω) of the dielectric tensor can be obtain from the imaginary component using the Kramer-Kronig equation given by, Where ω is the frequency of phonon, Ω is the unit cell volume, u is the unit vector, e is the electron charge,  and  are the wavefunctions for conduction and valence band electrons respectively at k, () is the absorption coefficient, () optical conductivity and n is the refractive index of the material.
We investigated the optical properties in the incident photon energy range of zero to 20 eV.Fig. 5a and b show the plots of the real part of the dielectric tensor ɛ1(ω) against energy and imaginary part ɛ2(ω) against energy for the hexagonal system respectively.The graphs show that there is no significant difference along the x and y-directions for the values of both components.This is also observed with trigonal WSe2 structure Fig. 5c and d, this could be attributed to their lattice constants, where a is equal to b but not equal to lattice parameter c.The values of the static real part of the tensor ɛ1(0) in the x and y-direction are 5.93 while in the z-direction is 2.40 for the hexagonal compound, as recorded in Table IV.The trigonal phase ɛ1(0) in the x and y-direction has a value of 4.19, and 2.35 in the z-direction.This confirmed that both structures are anisotropic.
Table IV.Calculated static dielectric ε1(0) for Hexagonal and Trigonal WSe2 bulk structures in three polarization directions.
As we compared to the hexagonal, the dielectric behaviour for trigonal shown in Fig. 5 is slightly the same.However, the values of the dielectric tensor for hexagonal are much larger than trigonal.The zero-frequency limit ε1(0) shown in the Fig. 5a gives the static dielectric tensor.The calculated static dielectric functions of both systems for different polarization directions are given in Table IV, the calculated dielectric tensors of the hexagonal are a little bit higher than trigonal.Fig. 6a shows the variation of the refractive index and 6b absorption coefficient in the energy range from zero to 20 eV for the hexagonal compound.Whereas Fig. 6c and 6d show the same for the trigonal phase.In the absorption plot in Fig. 6b and 6d the first maxima occur at 1.48 eV and 1.53 eV corresponding with band gap energies of the materials earlier stated in Table II.Table V. Calculated static refractive index n (0) for Hexagonal and Trigonal WSe2 bulk structures in three polarization directions.
Fig. 6c and d depict the plots of refractive index with photon energy.From these graphs, we extract the static refractive indices (n (0)) of the two-phase WSe2 as seen in Table V.

D. Mechanical Properties
Elastic constant and elastic moduli provide the ground for understanding the mechanical properties of crystalline solids such as brittleness [12], stability, hardness, ductility, and other mechanical characteristics.The calculated elastic constant for the considered bulk structures within the framework of PBE GGA functional is presented in Table VI.These results are compared with the theoretical and experimental findings.The mechanical stability of bulk materials under isotropic pressure can be analyzed using the popular Born stability criteria [9] given as,  > 0,  > 0,  > 0 (8) The independent elastic constant is given by, All the studied materials satisfy the Born mechanical stability limitations as illustrated by ( 8)- (11).Furthermore, the table demonstrates that the elastic constants for all the materials are all positive indicating that they are all stable [9] [24].
Table VI.Elastic Constant of Bulk Hexagonal and Trigonal WSe2 within GGA-PBE Functional.

1) Hardness and Elastic moduli
The mechanical properties of a material can be determined from its calculated elastic constants.The machinability of a material can be calculated by dividing its bulk modulus by its elastic constant C44.It was established that the higher the value of the ratio the better the machinability [9].Herein, we have calculated the bulk modulus (B), the shear modulus (G) and the Young's modulus (Y) for the individual  for all the considered materials.According to Pugh's rule, the ratio (G/B) can be used to determine the brittleness or ductility of the material compound.This rule states that if > 0.57 then the material is ductile else it behaves as brittle.In this work, the calculated values of indicate that all the materials have shown a ductile nature.Reference [25] reported that material ductility can be determined using Poisson's ratio.They elucidated that Poisson's ratio greater than 0.25 means the material is ductile, and anything less than 0.25 indicates a brittle material.For hexagonal structures, the bulk (Bv) and shear (Gv) moduli can be obtained from ( 12) and ( 13) using the elastic constant,  = [(2 +  ) +  +  ] (12) Synthesized systems are more often studied in the form of ceramic polycrystalline, hence, it is important to calculate the elastic parameters of the investigated materials systems in their polycrystalline state.To that effect, we use the Vogt-Reuss-Hill approximation, according to which two other essential approximations are utilized, i.e., the Voigt and Reuss approximations.The expressions for the shear and bulk moduli are given by the Reuss model as The effective modulus of anisotropic polycrystalline crystals is calculated using bulk modulus and the shear modulus, as approximated by Hill [26] is given by,  = ( +  ) = ( +  ) (17) The average and the actual effective moduli above can be used to obtain bulk and shear moduli as approximated by Voigt-Reuss-Hill given by  = Now, the Poisson's ratio(v) for polycrystalline material can be given by the expression,  = ( ) 2) Crystal Anisotropic Index The orientation dependency of the elastic moduli is called crystal elastic anisotropy [27,28].Anisotropy of a material can be calculated using elastic constant C44 as shown in (20) [9].The anisotropy index has been assigned to crystals for its importance in the design of crystals in science and engineering.Reference [9] reported that  = 1.0 shows an isotropic crystal, while  > 1.0   < 1.0 indicates anisotropy.In this work, the results presented in Table III suggest that all our crystals are anisotropic by nature. = IV. CONCLUSION This paper reports a computational description and comparison of structural, electronic, optical and mechanical characteristics of hexagonal and trigonal phase WSe2 bulk systems using DFT calculations.Our results elucidate shows that the two Polymorphs of the crystal WSe2 are stable, they are both semiconductors with an indirect band gap between 1-1.5 eV, which agrees well with previous theoretical and experimental findings.The results show that the bandgap of the trigonal is larger than hexagonal.The complex dielectric tensor, absorption spectrum and refractive index were calculated and discussed.Our findings further reveal that the hexagonal's real and imaginary parts are larger than the real and the imaginary parts of the dielectric function of trigonal.The hexagonal has a higher absorption coefficient and refractive index in all three polarization directions.The results further suggested that the two materials are anisotropic and brittle.

Fig. 1 Fig. 2
Fig. 1 Primitive cells of Bulk (a) Trigonal, (b) Hexagonal WSe2 structures and (c) Brillouin zone used for k-path in band structure plots.

Fig. 5
Fig. 5 Dielectric tensors of Bulk (a and b) Hexagonal and (c and d) Trigonal WSe2 crystalline compounds.

Table II .
Calculate band gap for Hexagonal and Trigonal bulk WSe2 Structures.

Table III .
Calculated effective mass of electrons and holes for Hexagonal and Trigonal bulk WSe2 Structures.