#### Abstract

Arsenic trisulfide (As_{2}S_{3}) has been found to be an excellent glass former at high temperature and pressure. However, there is still some scarcity for the elastic and phonon behavior of the orpiment phase. By using the Dreiding force field of the geometry optimization computations, we investigated the elastic constants, mechanical moduli, and the phonon dispersion of orpiment As_{2}S_{3} under the pressure from 0 to 5 GPa. Some results of the elastic parameters of orpiment-As_{2}S_{3} at 0 GPa are consistent with the experimental data. The phonon dispersions for orpiment As_{2}S_{3} under pressure are also reasonable with previous calculations.

#### 1. Introduction

The computational simulation of materials has been a successful and rapid tool to understand the complex physical problems of materials. In addition, predicting the good elastic and mechanical properties of materials is a necessary demand for recent solid-state physics and science. And these mechanical properties at high pressure and temperature are also important for the development of modern technologies [1–4].

Arsenic trisulfide (As_{2}S_{3}) is the most studied compound due to its properties like large glass-forming ability, excellent infrared radiation transmission, and structural and mechanical properties [5, 6]. Under normal conditions, the stable phase is an orpiment layered structural semiconductor material. In its lattice, each As atom is surrounded by three S atoms, whereas each S atom is surrounded by two As atoms. At normal conditions, no other crystalline metastable phases of As_{2}S_{3} could be artificially synthesized for a long time. The structure properties of arsenic sulfides were studied by the density functional theory [7], tight binding method [8], and nonempirical localized approach method [9]. The density of valence states of crystalline arsenic sulfides was investigated by using X-ray photoemission spectroscopy [10]. The optical and transport properties and electronic structure of Ni-doped arsenic sulfides were also studied [11]. Only one new phase of As_{2}S_{3} named anorpiment has been naturally discovered recently [12]. It has low symmetry with triclinic structure and is made of covalently bonded layers of As and S atoms with a similar coordination arranged in a different stacking sequence by van der Waals force.

The high-pressure technique implies the active use of pressure as the fundamental thermodynamic variable for innovative material design. M. Güler and E. Güler reported that the classical force field can describe the surface, elastic, and phonon properties of wurtzite ZnS accurately, and the calculated data were in good agreement with the related experimental results [4]. Another previous work [13] pointed out that bonding disorder can be expected under high pressure in As_{2}S_{3} glass, changing structural, vibrational, and optical properties. For example, using high-precision compressibility measurement and in situ high-energy X-ray diffraction to high pressure, the enhanced refractive index of the recovered As_{2}S_{3} glass is promising for optical applications with improved functionalities [14]. A nearly permanent red shift in optical absorption after decompression can be used in high-impact-force optical sensors. In order to expand its application in the industrial field, it is necessary and interesting to study the elastic constants, mechanical moduli, and the phonon dispersion of orpiment As_{2}S_{3} under high pressure.

In this paper, we provided the details of theoretical calculations with the employed interatomic potential in Section 2. Then, we discussed our results with the available experimental results in Section 3. Finally, we presented the main findings in Conclusions.

#### 2. Computational Methods

The most significant feature of material modeling is the choice of the interatomic potentials. A proper interatomic potential can describe the physical properties of materials sufficiently and accurately. The empirical potentials are the modelling tools for materials because they can yield reasonable results. These kinds of potentials can provide reasonable results of the structures, energies, and elastic and mechanical properties of alloys [15, 16] and many other materials [17, 18].

In this work, all the theoretical calculations were carried out with the General Utility Lattice Program (GULP) molecular dynamics code [19]. This versatile code allows the concerned structures to be optimized at constant pressure (all internal and cell variables are included) or at constant volume (unit cell remains frozen). The Dreiding force field [20] was chosen to be the empirical potentials for simulations. It is widely used in research describing the properties of organic and inorganic materials [21, 22]. To avoid the constraints, constant pressure optimization was applied to the geometry of o-As_{2}S_{3} cells with the Newton-Raphson method based on the Hessian matrix calculated from the second derivatives. In Figure 1, the cell geometry of o-As_{2}S_{3} was assigned as , , , and with space group PNMA. In the geometry optimization calculations, the Hessian matrix was recursively updated with the BFGS [23] algorithm. The phonon and associated properties of o-As_{2}S_{3} were also analyzed as a function of pressure within the ranges between 0 GPa and 5 GPa in steps of 1 GPa. We captured the phonon density of states and dispersions for a material after specifying a shrinking factor with phonon computations. Furthermore, phonons are described by calculating their values at points in a reciprocal space within the first Brillouin zone of the given crystal. To achieve the Brillouin zone integration and obtain the PDOS, we used a standard and reliable scheme developed by Monkhorst and Pack [24] with -point mesh.

#### 3. Results and Discussion

As people know the volume of the crystal materials decreases, the density increases straightly under pressure. The density curve of o-As_{2}S_{3} under pressure is shown in Figure 2. Under zero temperature, o-As_{2}S_{3} has the lowest density value of 3.38 g/cm^{3} at 0 GPa and has the highest density value of 3.80 g/cm^{3} at 5 GPa. These results are consistent with the experimental value of 3.49 g/cm^{3} [25]. So our prediction crystal structure of o-As_{2}S_{3} under pressure is accurate.

Then, we continued to compute the elastic and mechanical properties of o-As_{2}S_{3} under pressure, such as the elastic constants (), bulk modulus (), shear modulus () and Young moduli (), and elastic wave velocities ( and ). The elastic constants were obtained by the second derivatives of the energy density with respect to strain. And the mechanical parameters were obtained by the linear combination of elastic constants using Voigt, Reuss, and Hill (VRH) approximations [26, 27].

Elastic constants deliver clear perceptions about the mechanical and other associated properties of materials. Though the elastic constants we obtained belong to the monocrystalline, these results also have important guiding significance for polycrystalline materials [28, 29]. For orthorhombic crystals, seven well-known elastic constants exist, which are specified as , , , , , , and [30]. Figure 3 shows the results of , , , , , , and constants for o-As_{2}S_{3} under pressures between 0 GPa and 5 GPa. The calculated elastic constants are in the range of . The elastic constants and represent the longitudinal elastic character, and they increase under pressure for the reason of elastic wave propagation occurs easily under pressure. The values of and are similar to each other at 0 GPa, and the difference between them exists as the pressure increases. The elastic constant represents the shear elastic response to retarded wave propagation, and it also increases with the increasing pressure. In Table 1, our results for , , and of o-As_{2}S_{3} at 0 GPa are consistent with the experimental results of McNeil et al. [31], but and are larger than the data in Ref. [31]. The reason is due to the anisotropy of o-As_{2}S_{3} and the measuring technology of Brillouin scattering and the size of samples. McNeil’s measurement sample was made a natural crystal from Macedonia, and it was in the form of a thin plate roughly 5 mm square and 1 mm thick with only two good-quality surfaces (both perpendicular to the [010] axis). However, we used an orthorhombic crystal cell to do the simulation. So the anisotropy of o-As_{2}S_{3} would influence the results of elastic parameters between us and the McNeil’s.

According to stability, the mechanical stability criterion for orthorhombic crystals is ,, , and [30]. Our calculated results of elastic constants of o-As_{2}S_{3} obey the mechanical stability criterion, which consequently indicates that the o-As_{2}S_{3} is mechanically stable in its ground state.

The bulk modulus represents an essential elastic constant connected to the bonding strength and is used as a primary parameter for the calculation of a material’s hardness. The shear modulus represents the resistance to reversible deformations upon shear stress, while Young’s modulus is defined as the ratio of the tensile stress to the corresponding tensile strain. Figure 4 shows the , , and of o-As_{2}S_{3} under pressure, and they all increase with the increasing pressure. However, the increase of and is obviously stronger than that of . This result points that o-As_{2}S_{3} is an isotropic material, so the external pressure produces a uniform volume compression effect and much less effect on the shear deformation of the material. Usually, brittle materials display a considerable resistance to the deformation before fracture, whereas ductile materials would be deformed easily. Ductility is a mechanical property of the material that represents the ability to plastically deform before it undergoes the fracture. Brittleness refers to the nature of fracture failure of materials without obvious plastic deformation when the external force reaches a certain limit. Ductility and brittleness are important for the production of the desired materials, so we continued to study the ductile and brittle behaviors of o-As_{2}S_{3} under pressure. The Pugh ratio evaluation () [32], C for ductile and brittle performances of materials, was considered in this section. A low value is associated with ductility, whereas a higher one corresponds to the brittle nature. The critical value which separates ductile and brittle materials is around 0.5. Our results show that values decrease from 0.42 () to 0.32 () at zero temperature for o-As_{2}S_{3} as inserted in Figure 4. Therefore, o-As_{2}S_{3} is a kind of a ductile material for the entire pressure range.

It is known that the longitudinal () and shear elastic () waves may arise in solids at low temperatures due to vibrational excitations originating from the acoustic modes [33]. And the pressure behavior of the and of o-As_{2}S_{3} pressure at is shown in Figure 5. The indicates the longitudinal elastic wave velocity, and the indicates the shear wave velocity. We can see that has a significant increment compared to , and this is the most common case for materials. The reason is that the bulk modulus and Young’s modulus compared to the shear modulus have the same trend.

The ground state phonon dispersions of o-As_{2}S_{3} were analyzed by using Dreiding potential quantitatively and compared in Reference [34] with theoretical results. We calculated the partial ground state phonon density of states (PDOS) of o-As_{2}S_{3} to explain the contribution of As and S elements to the total phonon density of state (TDOS) of the material and showed them in Figure 6. The PDOS and TDOS appear with some separated regions corresponding to the longitudinal acoustic mode, transverse acoustic mode, longitudinal optic, and transverse optic mode of o-As_{2}S_{3}. Besides, the contribution of the As element to acoustic phonon modes is the same as that of S element, but in the high-frequency area, the case is different, and the contribution of the S element is dominant to optical modes. There is a big gap of 200 cm^{-1} (between the frequencies 300 cm^{-1} and 500 cm^{-1}) originating from the mass differences of As and S elements of o-As_{2}S_{3}.

In addition, Figure 7 shows the phonon dispersion of o-As_{2}S_{3} along the chosen -A path (the same as with original Ref [29]) in a reciprocal space for pressures 0 GPa and 5 GPa. It is seen that the pressure at 5 GPa shifts the phonon dispersion curves to higher frequency values. The reason is the atoms of o-As_{2}S_{3} move towards each other and have to locate in the steeper potential wells under high pressures. In Figure 8, the corresponding PDOS curves of five pressures (0 GPa, 1 GPa, 2 GPa, 3 GPa, 4 GPa, and 5 GPa) are presented. The increasing pressure increases the PDOS peaks of o-As_{2}S_{3} under the high pressures, and the gap between acoustic and optical modes also shifts to higher frequencies from 350 cm^{-1} to 500 cm^{-1}.

In summary, the calculated parameters of o-As_{2}S_{3} in this work are consistent with experiments and those of some published theoretical data. And some of the elastic constants and phonon properties of o-As_{2}S_{3} at 0 GPa demonstrate fair accordance with the experimental and theoretical data. We hope that our results add value to the forthcoming researches about o-As_{2}S_{3} under pressure.

#### 4. Conclusions

In this paper, we applied the Dreiding interatomic potential for the geometry optimization on the structural properties of o-As_{2}S_{3} and studied both the ground state and pressure-dependent elastic, mechanical, and phonon dispersions of o-As_{2}S_{3}. The present results for the ground state of o-As_{2}S_{3} were former experiments for the elastic constants, and the phonon characteristics were consistent with the published theoretical data. Moreover, we presented the effect of pressure on o-As_{2}S_{3} and obtained some reasonable results. The mechanical properties of bulk modulus, shear modulus, Young’s modulus, and longitudinal elastic character were increments under pressure. The o-As_{2}S_{3} exhibits ductile character under high pressure. At last, the increasing pressure increases the PDOS peaks of o-As_{2}S_{3} under the high pressures, and the gap between acoustic and optical modes also shifts to higher frequencies from 350 cm^{-1} to 500 cm^{-1}.

#### Data Availability

The research data used to support the findings of this study are included in the article.

#### Conflicts of Interest

We declare that we have no conflict of interest about the manuscript.

#### Acknowledgments

This research was supported by the National Science and Technology Major Project (Grant No. 2017-VI-0012-0084), the Fundamental Research Funds for the Central Universities (Grant No. FRF-GF-19-029B), and the National Natural Science Foundation of China (Grant No. 51471022).