## Abstract

The electron structure, elastic constant, Debye temperature and anisotropy of elastic wave velocity for cubic WO_{3} are studied using CASTEP based on density functional theory. The optimized structure is consistent with previous work and the band gap is obtained by computing the electronic structure; the top of the valence band is not at the same point as the bottom of the conduction band, which is an indirect band-gap oxide. Electronic properties are studied from the calculation of band structure, densities of states and charge densities. The bulk and shear moduli, Young's modulus, hardness and Poisson's ratio for WO_{3} are studied by the elastic constants. We calculated acoustic wave velocities in different directions and estimated the Debye temperature from the acoustic velocity. The anisotropy of WO_{3} was analysed from the point of view of a pure wave and quasi wave.

## 1. Introduction

As a type of excellent semiconductor material, tungsten trioxide (WO_{3}) has been widely used in multiphase catalysis, electroluminescence, photodegradation, high-temperature superconductivity and new energy fields [1–10]. The ideal cubic WO_{3} primitive cell is a kind of octahedral structure; W and O occupy the central and six corners of the octahedron (figure 1), respectively.

Tungsten trioxide experiences different structure transitions in the temperature range of −180 to 900°C; these phase transitions are not the recombination of tungsten and oxygen atoms, but the distortion of tungsten atoms in the original ideal crystal structure. The phase transition, from low to high temperature; is irreversible; a stable phase that forms at high temperature is also stable at low temperatures. As the temperature increases, the sequence of WO_{3} phase transitions is as follows: monoclinic (at low temperature) → triclinic → monocline (at room temperature) → orthorhombic phase → tetragonal phase. Of course, there is also hexagonal phase WO_{3}. Although cubic WO_{3} has not been observed at high temperatures, in many works it is considered as a reference structure, with many reports on experimental and theoretical studies of WO_{3}.

Since the 1970s, when Randin [11] found tungsten oxide with photochromic performance for the first time, many research works both at home and abroad have conducted a number of theoretical and experimental studies on photocatalysis, capacitance performance, photosensitivity and gas sensitivity of tungsten oxide [12–21]. Yan *et al*. [22] published an article on hydrogenation of WO_{3}. They synthesized tungsten oxide single-crystal nanosheets via the exfoliation of layered tungstic acid to tungsten oxide nanosheets and subsequent introduction of oxygen vacancies (figure 2). In Taiwan, Hsieh *et al*. [23] studied the growth along the [001] direction of triclinic WO_{3} nanowires, and the optical absorption of nanowires by the red shift phenomenon. The results showed the influence of oxygen vacancy on the red shift, and the influence of crystallinity and particle size distribution of the nanowires on the forbidden bandwidth. Hao Lai *et al.* [24] fabricated mesoporous WO_{3} nanofibres and tube-like nanofibres by the gas-filled assistant sol–gel immersion method with porous anodic alumina membrane confinement, which has a porous structure (figure 3). Wang *et al*. [25] studied the structural and electronic properties of WO_{3} by various hybrid functionals combined with both plane wave and localized basis sets. Levy & Pagnier [26], through the establishment of sheet WO_{3} to simulate a nanoscale with the required stoichiometric ratio and oxygen-containing defects, computed the position of the atoms, electron density and the electron state density distribution, and pointed out that the WO_{3} surface could absorb oxygen atoms, which is due to the existence of the acceptor level at the bottom of the surface valence band. Chatten *et al*. [27] studied the WO_{3} electron structure of the oxygen vacancy model in different crystal systems and its forbidden band gap, and found that the polarizations of the bonding and anti-bonding states were not only related to the location of the crystal system but also to the position of oxygen vacancy.

By reviewing the previous work, it has been found that the study of tungsten oxide experimentally and theoretically has made great advancements, but systematic studies of its elastic properties, Debye temperature and anisotropy are rare. It is well known that elastic properties can be used to provide information about the potential of atoms and relate to a variety of basic solid-state phenomena. Therefore, this paper is based on density functional theory (DFT) to study the cubic WO_{3} electronic structure, elastic properties, Debye temperature and anisotropy in different directions.

## 2. Computational details

In recent years, the methods used for theoretical study of metal oxides have been mainly based on LDA, GGA, LDA + U and GGA + U [28–34]; Hubbard parameter U is known as the On-site Coulomb interaction energy [35]. However, a strong association effect is not considered in the calculation of LDA and GGA, that is, the unoccupied d and f orbits. Furthermore, because of the complexity of the electron cloud diffusion, the multibody effect of an orbit is difficult to accurately describe by LDA and GGA. In this system, the energy band structure of the metal is often given by LDA or GGA, and the bands across the Fermi level are often d or f orbits. In fact, the band structure of the system has semiconductor characteristics, there is a clear band gap between the bonding and the anti-bonding states, and the d or f orbit is tightly confined to the nucleus and does not show delocalization. To better describe the strong association system, it is necessary to surpass the traditional LDA or GGA approximation. In this respect, the more successful methods of improvement are LDA + U and GGA + U. Compared with the traditional DFT, the calculation of LDA(GGA) + U does not increase obviously, and the calculation results can be improved significantly with the proper parameters.

In this work, we use DFT to study electronic structure, elastic properties, Debye temperature and anisotropy of cubic WO_{3}; the calculations have been performed by the CASTEP code [36,37]. By comparison of different approximation methods, LDA + U [38] is finally determined for structural optimization. W 5d^{4}6s^{2} electrons and O 2s^{2}2p^{4} electrons are explicitly treated as valence electrons. According to the results of the convergence (figure 4), the cut-off energy of the plane wave is taken as *E*_{cut}* *= 700 eV, the *k-*point integral in the Brillouin zone is set to 7 × 7 × 7 and the self-consistent convergence method is used to optimize the structure of cubic WO_{3}.

## 3. Results and discussion

We optimized at the LDA + U (U = 5 eV [35]) level the lattice constant of the cubic WO_{3} to be 3.82 Å, which agrees quite well with those of previous self-consistent calculations: *a* = 3.73–3.84 Å [39–41], as well as the experimental value of *a* = 3.71–3.75 Å [42]. Optimized results by GGA + U are larger than those of the above theoretical and experimental studies (figure 5), which indicates that LDA + U is more suitable for structural optimization than GGA + U in this work.

### 3.1. Electronic structure

The calculated band gap, *E*_{g} = 0.544 eV, is in line with the previous theoretical studies, *E*_{g} = 0.3–0.6 eV [43,44], which is much smaller than the experimental value of 2.62 eV (for the monoclinic phase) [45] due to the well-known DFT error. The results can be modified by using the scissors operator [46], and the band-gap width of WO_{3} is *E*_{g} = 2.70 eV, which does not affect our next step of calculation.

As shown in figure 6, the WO_{3} band gap can attain the maximum value at the R point (the top of the valence band) and the minimum value at the G point (bottom of the conduction band), so WO_{3} is an indirect band-gap semiconductor. The valence band consists of 12 levels, which can be divided into four bands: three bonding and one non-bonding bands of hybridized O-2s, -2p and W-5d, -6s states. Based on the crystalline field theory, it can be seen that the lowest three states form *x ^{2}–y^{2}*) and 5d(

*z*). Three states of

^{2}*z*) and 5d(

^{2}*x*) with O-2p orbitals. The energy of

^{2}–y^{2}*t*band, which is generated by the interaction between W-5d (

_{2 g}*d*) and O-2p orbitals. Three levels near the top of the valence band form the

_{xy,}d_{xz}, d_{yz}As can be seen in figure 7, the calculated density of states (DOS) of WO_{3} is divided into three groups. The low-valence band (i), that is, with a bandwidth of 2.93 eV, is mainly composed of O-2s, and a minority of W-5d. Group (ii), near the top of the valence band, with a bandwidth of 7.63 eV, is formed by hybridization between O-2p and W-5d, and the contribution of tiny amounts of W-5p, -6 s. Group (iii) is located in the conduction band with a bandwidth of 13.87 eV. In the bottom of the conduction band, we found that there is a mixture of dominant O-2p with W-5d _{3} instability. Furthermore, the total charge densities of cubic WO_{3} is presented in figure 8; it shows that the bonds between W and O are covalent due to hybridization, which agrees well with our analysis of DOS.

### 3.2. Elastic and mechanical properties

According to Hooke's law, there are only three independent elastic constants *C*_{11}, *C*_{12} and *C*_{44} for cubic crystal structures (*C*_{11} = *C*_{22} *=* *C*_{33}, *C*_{12} *=* *C*_{13} *=* *C*_{23}, *C*_{44} *=* *C*_{55} *=* *C*_{66}). Here. *C*_{11} = 546, *C*_{12} = 35 and *C*_{44} = 71. Based on the Born's stability restrictions [47]:
_{3} are satisfied with the above stability conditions, so the cubic WO_{3} is stable in mechanics.

We further calculated the phonon dispersion of WO_{3}, whose imaginary frequency can be found from figure 9, which indicates that the cubic WO_{3} structure is unstable, which is not a contradiction with the above equation (3.1). In cubic WO_{3}, the obtained results by the Born's stability restrictions show that the stability is affected by the external force, that is, from the point of view of macroscopic mechanics. While the imaginary frequency of the phonon dispersion shows that the atomic arrangement is unstable at present, from the point of view of microscopic atomic structure, atoms in the WO_{3} will be rearranged by the lattice vibration to form a new and stable structure.

The bulk and shear modulus are two key parameters for the characterization of material hardness. The bulk modulus can be described as the ability of the material to resist the change of bulk, and is also understood as the mean value of the bond strength. The shear modulus can be described as the ability of the material to resist the shape change caused by shear force, and the ability to resist the change of the bond angle. The calculation of the bulk and shear modulus can be obtained by *C _{ij}*, and there are two different methods to do so. One is the calculation of the strain continuity on grain boundaries by Reuss [48], and another is Voigt's [49] proposed stress continuity on the grain boundary. Hill [50] proves that the calculations of the Reuss and Voigt models are the lower and upper limit of the elastic constants, respectively, so the Hill model calculates the arithmetic mean of the results of the Reuss and the Voigt models:

The Young's modulus *E* of cubic crystal material can be expressed as the corresponding coefficient:
*x* represents Reuss, Voigt and Hill.

The comparison of our calculated data and available theoretical studies [51–54] has been presented in table 1. It is obvious that our calculation constants are in accordance with previous work. According to the criterion of Pugh [55], *B*_{X}/*G*_{X}, which is defined as the shear modulus corresponding to the plastic deformation, and the bulk modulus are related to the fracture resistance. Pugh proposed that if *B*_{X}/*G*_{X} exceeds the critical value (1.75), the material will have ductility. In our calculation, the value is less than 1.75, which indicates that cubic WO_{3} has little ductility, which is also explained in the next study on Poisson's ratio.

Hence, we introduced the study of the Vickers hardness, which is an empirical formula for evaluating the hardness of the material.
*k* is the Pugh ratio, which is the ratio of the shear *G* to the bulk *B* modulus, *k = G/B*.

Compared to other metal oxides in table 2, we can see that WO_{3} has greater hardness. This is due to the fact that the bulk modulus is not directly related to the hardness, and the shear modulus is more capable of characterizing the hardness of the material. Furthermore, with regard to the Poisson ratio, it reflects the strength of the covalent bond in the material. In general, the Poisson ratio 0.1 ∼ 0.28 represents the covalent property of the material, while 0.29 ∼ 0.33 represents the metal characteristics. The Poisson ratios for the above metal oxides, except for WO_{3} and MgO, are more than 0.29, which means that they have metal properties along with their mechanical properties, and their hardness is less than that of WO_{3}.

### 3.3. Debye temperature

Debye temperature is a very important thermodynamic parameter reflecting thermodynamic properties, which is related to many physical properties of solids, such as acoustic velocity, specific heat capacity and thermal expansion coefficient. According to the above bulk and shear modulus, we calculated wave velocity *v _{s}*, longitudinal wave velocity

*v*and average wave velocity

_{p}*v*, and further studied the Debye temperature of WO

_{m}_{3}by equations (3.9) and (3.10). The relationship between the Debye temperature and the average wave velocity is as follows:

*h*is the Planck constant,

*k*is the Boltzmann constant,

*N*

_{A}is the Avogadro constant,

*M*is the molecular weight and

*ρ*is the density (table 3).

The Debye temperature is calculated to be 582 K; unfortunately, there are no data available for comparison in the literature, as far as we know. To analyse the results, we only compared other functional materials, such as GaN (390), ZnO (303) and Al_{2}O_{3}(370) [61,62]; it can be found that WO_{3} has a higher Debye temperature, which shows that the interatomic binding force, melting point, hardness and thermal expansion coefficient of WO_{3} are larger than those of the above materials.

### 3.4. Anisotropy of elastic wave velocity

For cubic WO_{3}, the propagation velocity of longitudinal and transverse waves along different crystal directions is related to the elastic constants *C _{ij}*. In table 4, the equations relating velocities of propagation and elastic constants used for the calculations on WO

_{3}are given.

From table 4, we can draw a conclusion: for cubic WO_{3}, *v*_{1} is the largest of longitudinal wave velocities; the wave is a pure longitudinal wave in the direction of [100], while in [110] and [111], waves of *v*_{3} and *v*_{6} are quasi-longitudinal waves. In a pure longitudinal wave, the direction of the particle vibration is perpendicular to the forward direction of the wave, whereas for the quasi-longitudinal wave it is not so, and the velocity component will be produced in other directions, and therefore *v*_{1}> *v*_{3}, *v*_{6}. The velocities for *v*_{3}, *v*_{4} and *v*_{5} are all different; transverse waves, which are formed by the degeneracy split into *v*_{4} and *v*_{5}, have different velocities. Based on the above-mentioned reasons, we can conclude that it is anisotropy in this direction.

## 4. Conclusion

In this paper, we have performed first-principles calculations on cubic WO_{3}, including structural parameters as well as the band structure, density of state, elastic constants, Debye temperature and the acoustic wave velocity in different directions. The calculated results show that the cubic WO_{3} is an indirect band-gap oxide; the valence band is mainly composed of O-2p, and the bottom of the conduction band is mainly contributed by W-5d and a few of O-2p. There is no doubt about the importance of elastic constants in this work; based on the elastic constants, the bulk, shear and Young's modulus are also further studied, which shows that the shear modulus is determined according to the stability of cubic WO_{3}. Vickers' hardness and Poisson ratio are also investigated by obtaining the bulk and the shear modulus; there is no research on the hardness of WO_{3}, and we can only compare it with other oxides.

The calculation of Debye temperature is further obtained by elastic constants, as 582 K, which is useful not only for the potential application of WO_{3} on thermoelectric and thermal resistance materials but also for the development of thermoelectric materials in future. Finally, we investigated the acoustic wave velocities in different directions. Based on the concept of pure and quasi wave, and the calculated result of wave velocity, it can be shown that cubic WO_{3} acoustic waves have anisotropy in the [110] direction.

## Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

## Authors' contributions

L.X. carried out all simulations and wrote the first draft of the manuscript under the supervision of H.-Q.F. The manuscript correction and revision have been carried out by both authors.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by the National Natural Science Foundation (51672220) and 111 Program (B08040) of MOE, the National Defense Science Foundation (32102060303), the Xi'an Science and Technology Foundation, the Shaanxi Provincial Science Foundation and NPU Gaofeng Project (17GH020824) of China.

## Footnotes

This article has been edited by the Royal Society of Chemistry, including the commissioning, peer review process and editorial aspects up to the point of acceptance.

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4117070.

- Received November 15, 2017.
- Accepted May 14, 2018.

- © 2018 The Authors.

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.