The structural, mechanical, electronic, optical and thermodynamic properties of t-X$_{3}$As$_{4}$ (X $=$ Si, Ge and Sn) by first-principles calculations

The structural, mechanical, electronic, optical and thermodynamic properties of the t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ (X $=$ Si, Ge and Sn) with tetragonal structure have been investigated by first principles calculations. Our calculated results show that these compounds are mechanically and dynamically stable. By the study of elastic anisotropy, it is found that the anisotropic of the t-Sn$_{\mathrm{3}}$As$_{\mathrm{4}}$ is stronger than that of t-Si$_{\mathrm{3}}$As$_{\mathrm{4}}$ and t-Ge$_{\mathrm{3}}$As$_{\mathrm{4}}$. The band structures and density of states show that the t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ (Si, Ge and Sn) are semiconductors with narrow band gaps. Based on the analyses of electron density difference, in t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ As atoms get electrons, X atoms lose electrons. The calculated static dielectric constants, $\varepsilon_{1} (0)$, are 15.5, 20.0 and 15.1 eV for t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ (X $=$ Si, Ge and Sn), respectively. The Dulong-Petit limit of t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ is about 10 J mol$^{\mathrm{-1}}$K$^{\mathrm{-1}}$. The thermodynamic stability successively decreases from t-Si$_{\mathrm{3}}$As$_{\mathrm{4}}$ to t-Ge$_{\mathrm{3}}$As$_{\mathrm{4}}$ to t-Sn$_{\mathrm{3}}$As$_{\mathrm{4}}$.


Introduction
Liu and Cohen predict the β-C 3 N 4 possesses the property of low compressibility. The structural and electronic properties of IV 3 V 4 compounds have attracted more and more attention [1]. The hardness of the cubic silicon nitride was experimentally determined to be 35.31 GPa as the third hardest material after diamond and cubic boron nitride [2]. The thermal stability of the cubic silicon nitride was studied by X-ray powder diffraction and scanning electron microscopy which shows that the material is stable up to 1673 K in air. So, the material is suitable for engineering superhard ceramics for high temperature structural applications [2]. IV 3 V 4 compounds have important potential applications in technical and scientific fields. The group IV nitrides have low compressibility and high hardness [3]. They have a wide application prospect in cutting [3]. Ching predicts the properties of the group IV nitrides using first principles theory [1]. Feng investigates the properties of pseudocubic-X 3 P 4 (X = C, Si, Ge and Sn) by using the first principle calculations [1]. They find that the modulus decreases with the atom change from C to Sn [1].
The structural and electronic properties of pseudocubic-X 3 As 4 are investigated by using first principles method [1]. The pseudocubic-C 3 As 4 is predicted to be metallic, pseudocubic-Si 3 As 4 , Ge 3 As 4 and Sn 3 As 4 are semiconductors [4]. Although X 3 As 4 have a wide range of applications in physics and chemistry, little research has been conducted regarding its mechanical and optical properties. Due to the lack of relevant experimental data at present, if we want to understand the application of tetragonal X 3 As 4 , further theoretical and computational research on its properties should be done.
In this work, the structural parameters, mechanical, electronic, optical and thermodynamic properties of the tetragonal X 3 As 4 are calculated by using the first-principles method based on plane waves and pseudo-potentials. By consulting the literature, we find that nobody has ever conducted experimental studies of the properties of the X 3 As 4 (X = Si, Ge and Sn). Therefore, we do not have the experimental data to compare with.

Computational methods
The structural optimization, mechanical, electronic, optical and thermodynamic properties of t-X 3 As 4 are calculated by using the density functional theory (DFT) with the generalized gradient approximation (GGA) parameterized by Perdew, Burke and Ernzerrof (PBE) in the CASTEP code [5]. The band structures, electronic density of states (DOS) are calculated by PBE0 hybrid functional. The phonon spectra and phonon density of states (PHDOS) are calculated by using first-principles linear response method. Plane wave cut-off energy is set at 610 eV. The convergence test and energy cut-off analysis of k-point mesh samples show that the convergence of the Brillouin zone sampling and the kinetic energy cut-off are reliable and satisfy the computational requirements [6,7]. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme is used in geometry optimization. The value of self-consistent field tolerance threshold is set as 5.0 × 10 −6 eV/atom. The maximum Hellmann-Feynman force, the maximum stress and the maximum displacement are set to be 0.01 eV/Å, 0.02 GPa and 5.0 × 10 −4 Å in geometry optimization, respectively. The Monkhorst-Pack k-points in the first irreducible Brillouin zone are given as 10×10×13 for t-X 3 As 4 , respectively. The mechanical, electronic, optical and thermodynamic properties of t-X 3 As 4 are computed according to the optimized crystal structures and parameters.

Structural properties
The t-X 3 As 4 has a tetragonal structure with I-42M (No. 121). Figure 1 shows the crystal structures of t-X 3 As 4 at 0 GPa. The t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 are all centro-symmetry structures (0, 0, 0) with 14 atoms/unit cell. The lattice parameters and volumes of t-X 3 As 4 at 0 GPa are calculated on the basis of geometry optimization and are presented in table 1. The density values of t-X 3 As 4 are 4.15015 g/cm 3 , 5.19889 g/cm 3 and 5.46287 g/cm 3 , respectively.

Mechanical properties
The nine independent quantities of the elastic constants for t-X 3 As 4 are calculated and presented in table 2. The mechanical stabilities of t-X 3 As 4 can be estimated by elastic constants. In this work, the mechanical stabilities and mechanical moduli of t-X 3 As 4 are approximated by the corresponding relationships for a tetragonal crystal class [8][9][10][11]. The elastic stability criteria of tetragonal phases are given as follows [12]: As indicated in table 2, the calculated elastic constants of the t-X 3 As 4 do satisfy the criteria, indicating that they are mechanically stable. The t-Si 3 As 4 exhibits the largest elastic constants of C 11 , C 22 , C 33 . For the t-Si 3 As 4 and the t-Sn 3 As 4 , the calculated C 11 and C 22 are equal and they are larger than C 33 . Hence, the mechanical strength in [100] and [010] directions is stronger than that in [001] direction. Moreover, C 44 , C 55 and C 66 denote the shear moduli in (100), (010) and (001) crystal planes, respectively. For t-Ge 3 As 4 , the values of C 11 and C 22 are the same and smaller than C 33 . Hence, the mechanical strength in [100] and [010] directions are smaller than that in [001] direction. From table 2, the C 44 , C 55 of t-X 3 As 4 are equal and they are larger than C 66 . Therefore, the shear moduli at (100) and (010) crystal planes are larger than that of (001) crystal planes.
From the calculated elastic constants, other mechanical parameters such as bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (v) can be derived using Voigt-Reuss-Hill (VRH) approximation [8]. For t-X 3 As 4 , based on elastic constants, the Reuss shear modulus (G R ) and the Voigt shear modulus (G V ) are as follows: The Reuss bulk modulus (B R ) and the Voigt bulk modulus (B V ) are defined as B V = (C 11 + C 22 + C 33 )/9 + 2(C 12 + C 13 + C 23 )/9, where the relationship between S i j and C i j is shown as In the above formulae, the C i j represents the elastic stiffness matrix and the S i j represents the elastic flexibility matrix. The Hill's averages are taken from the averages of the two [8] The Young's modulus, E, and Poisson's ratio, v, can be calculated by the equations Table 2. Calculated independent elastic constants of the t-X 3 As 4 (X = Si, Ge and Sn).
Species  Table 3. Calculated mechanical moduli of t-X 3 As 4 , including bulk modulus (B V , B R and B), shear modulus (G V , G R and G), Young's modulus (E) and Poisson's ratio (v), B/G and H V , in GPa.
From (4) to (10), the calculated physical quantities are presented in the table 3. From the calculated results, t-Si 3 As 4 has the largest value of bulk modulus among t-X 3 As 4 , which indicates that t-Si 3 As 4 has a lower compressibility. The shear modulus of t-Si 3 As 4 is the largest in t-X 3 As 4 , indicating that t-Si 3 As 4 has a strong rigidity. The Young's modulus values of t-X 3 As 4 show that t-Si 3 As 4 has a larger hardness.
The Poisson's ratio represents the stability of the shear strain of the crystal. For a typical metal, the value should be 0.33. For the ionic-covalent crystal, the value ranges from 0.2 to 0.3. Poisson's ratio of the strong covalent crystal is relatively small, usually below 0.15 [8]. For t-X 3 As 4 , the Poisson's ratios show that they are ionic-covalent crystal.
The ratio B/G reflects the brittleness or ductility of a material, and the critical value is close to 1.75 [13]. Below 1.75, the material shows brittleness; otherwise it shows toughness. As indicated in table 3, the B/G values for t-Si 3 As 4 and t-Ge 3 As 4 are smaller than 1.75, and the B/G value for t-Sn 3 As 4 is higher than 1.75. Hence, t-Si 3 As 4 and t-Ge 3 As 4 show brittleness and t-Sn 3 As 4 shows toughness.
It is well known that elastic anisotropy plays an important role in engineering science and crystal physics. The shear anisotropic factor for the (100) shear planes between [011] and [010] directions can be written by [14]: For the (010) shear plane between [101] and [001] direction is: For the (001) shear planes between [110] and [010] direction is: For isotropic crystal, the factors A 1 , A 2 and A 3 must be 1, while the deviation from 1 is a measure of the degree of the elastic anisotropy. Furthermore, since the compound is tetragonal, rather than cubic, the shear anisotropic factors are not sufficient to describe the elastic anisotropy. Therefore, the anisotropy of the linear bulk modulus should be considered. The anisotropy of the bulk modulus along a and c axes relative to the anisotropy along b axis can be estimated using the following equations: When the value is 1, it represents an elastic isotropy, but if it is not equal to 1, it is an elastic anisotropy. Where B a , B b and B c are the bulk moduli along different crystal axes, defined as

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In addition, the percentage of elastic anisotropy for bulk modulus A B and shear modulus A G in polycrystalline materials can be used as follows: The implication of the definition is that the value of zero corresponds to elastic isotropy and the value of 100% identifies the largest elastic anisotropy.
The calculated results are listed in table 4. It is seen that t-X 3 As 4 are elastic anisotropic. The t-Sn 3 As 4 shows to be more anisotropic than t-Si 3 As 4 and t-Ge 3 As 4 . For t-X 3 As 4 , the bulk moduli along a axis are equal to that along the b. For t-Si 3 As 4 , the bulk modulus along c axis is smaller than that along a axis and b axis. For t-Ge 3 As 4 and t-Sn 3 As 4 , the bulk moduli along c axis are larger than that along a axis and b axis. In addition, we also noticed that the percentage bulk moduli anisotropy A B is smaller than shear modulus anisotropy A G , suggesting that the structures are anisotropic in compressibility.

Phonon spectra
The calculated phonon dispersion spectra of t-X 3 As 4 at 0 GPa are shown in figure 2. No any imaginary phonon frequencies in the entire Brillouin zone confirms that t-X 3 As 4 are dynamically stable [15,16]. Since there are seven atoms in the protocell of t-X 3 As 4 , the crystal should have twenty one dispersion relationships in theory. That is, there should be twenty one phonon spectra calculation curves, which shows that the calculated results in this paper are consistent with the theoretical conclusions. Some of the twenty one phonon spectral curves overlap in some range of wave vectors. There are three acoustic branches and eighteen optical branches. Lattice vibrational modes play a major role in Raman scattering and infrared absorption at point Γ. From figure 2 (a) and figure 3 (a), we can see that an optical gap exists in the phonon dispersion spectrum of t-Si 3 As 4 . This is due to the large quality difference between Si atom and As atom. From figure 2 (b) and figure 3 (b), we can observe that the optical band gap of t-Ge 3 As 4 is smaller than that of t-Si 3 As 4 , which is due to the mass difference between Ge and As being smaller. A similar situation occurs in t-Sn 3 As 4 .
The phonon density of states (PHDOS) of t-X 3 As 4 is calculated by CASTEP and shown in figure 3, respectively. For t-Si 3 As 4 , we can observe that in the range of 0.8−6.8 eV, the total PHDOS is mainly from As, which indicates that the vibration of As atoms is dominant in this frequency band. In the range of 9.5−13.0 eV, the total PHDOS is mainly from Si, which indicates that the vibration of Si atoms is dominant in this frequency band. For t-Ge 3 As 4 , in the range of 0.2−6.1 eV, the total PHDOS is mainly from As, which indicates that the vibration of As is dominant in this frequency band. In the range of 6.2−8.8 eV, the total PHDOS is from Ge and As, which shows that Ge and As have a similar vibrational probability. For t-Sn 3 As 4 , in the range of 0.2−4.0 eV, the total PHDOS is from Sn and As, which shows that Sn and As have a similar vibrational probability. And in the range of 4.2−7.6 eV, the total PHDOS is mainly from As and partial from Sn, which indicates that the vibration of As is dominant in this frequency band.

Band structures and densities of states
The electronic properties of t-X 3 As 4 are analyzed at 0 GPa. The DOS is calculated to have a further insight into the bonding characteristics of t-X 3 As 4 .
As indicated in figure 4, for t-Si 3 As 4 , the valence maximum is at high symmetric Γ-point and the conduction band minimum is at Z-point, which indicates that t-Si 3 As 4 is an indirect band semiconductor with a band gap of 1.438 eV. For t-Ge 3 As 4 , the valence maximum is at high symmetric Γ-point and the 43601-6

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conduction band minimum is at Z-point, which indicates that t-Ge 3 As 4 is an indirect band semiconductor with a band gap of 0.952 eV. For t-Sn 3 As 4 , the highest point of the valence band and the lowest point of the conduction band are at the Γ point. It shows that t-Sn 3 As 4 is a direct band semiconductor with a band gap of 1.73 eV.
The density of states of t-X 3 As 4 are presented in figure 5. For t-Si 3 As 4 , the main bonding peaks are in the range of −15−10 eV. The total DOS in the valence band mainly comes from As s and As p, with partially from Si s and Si p. Si p and As p contribute most to the total DOS in the conduction band, with partially from As s and Si s. For t-Ge 3 As 4 , the total DOS in the valence band mainly comes from As s, As p and Ge s, with partially from Ge p. In the conduction band, the total DOS mainly come from Ge p and As p with partially from Ge s and As s. For t-Sn 3 As 4 , the total DOS in the valence band mainly comes from As s, As p and Sn s, with partially from Sn p. In the conduction band, the total DOS mainly come from Sn p and As p with partially from Sn s and As s.

Electron density difference and Mulliken charge population
In order to investigate the chemical bonding, the electron density differences are calculated and the results are shown in figure 6. The electron density differences are the discrepancy between the electron densities of the total system and the unperturbed electron densities of X (Si, Ge and Sn) and As [16]. The contour plots show the electron density differences (between −0.2 and 0.2) due to the formation of chemical bonds in the I-42M lattice, which are relative to the electron density in an isolated atom. It is helpful to analyze how chemical bonds are formed. The electron density differences are useful for identifying the types of chemical bonds. From figure 6, we can see that As atoms get electrons, X (Si, Ge and Sn) lose electrons. The electrovalent bond is formed between X (Si, Ge and Sn) atoms and As atoms. The combination of X and As atoms is dependent on the ionic effect of Coulomb attraction. In order to further investigate the bonding behaviour of the three materials, we also obtain the Mulliken charge population. The Mulliken population results are given in table 5. Table 5 gives a quantitative determination of the number of loss electrons per X (Si, Ge and Sn) atom, the number of acquired electrons per X (Si, Ge and Sn) and the charger of per X atom and per As atom after the formation of Si 3 As 4 . From table 5, it is shown that the bonding behaviour of t-X 3 As 4 is a combination of covalent and ionic nature.

Optical properties
Dielectric function is the most common characteristic of materials. It can characterize the response of materials to incident electromagnetic waves [17]. Optical properties of materials can usually be evaluated on the basis of a complex dielectric function, which depends on the frequency. Complex dielectric function is shown as follows [18]: Figure 6. (Colour online) Electron density difference distribution for (a) t-Si 3 As 4 , (b) t-Ge 3 As 4 , (c) t-Sn 3 As 4 . where ε 1 (ω) is the real part, ε 2 (ω) is the imaginary part. The real and imaginary parts of the dielectric function are given in figure 7 (a). The calculated static dielectric constants, ε 1 (0), are 15.5, 20.0 and 15.1 eV for t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 , respectively. For t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 , the real parts of dielectric function enhance with an increasing photon energy and get to the highest values at about 1.94, 1.30 and 1.48 eV, respectively. The imaginary part curves increase with an increase of photon energy and get to the highest values at about 3.69, 3.54 and 3.97 eV. Refractive index n(ω) and extinction coefficient k(ω) of t-X 3 As 4 can be obtained from ε 1 (ω) and ε 2 (ω):

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The static refractive indices are found to be 3.94, 4.48 and 3.89 eV for t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 , respectively. The values of n(ω) increase with an increasing photon energy in the visible light region, and in the ultraviolet band, reach the peaks at about 2.24, 1.40 and 1.67 eV for t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 , respectively. In figure 7 (c), the reflective coefficients are displayed. The main peaks lie at 11.1, 6.55 and 10.8 eV, respectively.
Vibrational Raman spectroscopy is one of the widely used optical techniques in materials science. It is a standard method for quality control of a production line. It is very effective in determining the occurrence of new phases or structural changes at extreme conditions. Moreover, it can be used in the absence of a long-range structural order as for liquid or amorphous materials. Raman spectroscopy can link Raman lines to specific microstructures. In order to further verify the crystal structure, the micro-Raman spectra of t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 were calculated by first-principles. Our Raman intensity results excited by the 514.5 nm incident light are presented in figure 7 (d). The Raman line 43601-9 shape is assumed to be Lorentzian, and the line-width is fixed at 10 cm −1 FWHM. The peaks mainly locate at 103, 161, 220, 336, and 361 cm −1 for t-Si 3 As 4 , 83.8, 109, 151, 214 and 239 cm −1 for t-Ge 3 As 4 , 92.8, 145, 191 and 219 cm −1 for t-Sn 3 As 4 .

Thermodynamic properties
For t-X 3 As 4 , we have studied the thermodynamic properties based on the phonon density of states at 0-1000 K temperature. According to CASTEP, the heat capacity is contributed by the lattice, C V is Debye temperature can be used to measure the properties of crystals, such as melting temperature, elastic constants and specific heat [19]. With the development of cryogenic technology, the deviation between Debye theory and practice increases. It is shown that the Debye temperature is different at different temperatures. According to the temperature dependence of Debye temperature at a constant volume, some general theoretical predictions can be made [20]. Heat capacity in Debye model is given by  where N is the number of atoms per cell. Thus, the value of the Debye temperature, θ D , at a certain temperature, T, is obtained by calculating the actual heat capacity according to equation (24), then by inverting equation (25). The relations of the Debye temperature with temperature are given as figure 8. The Debye temperature of t-Si 3 As 4 is larger than that of t-Ge 3 As 4 and t-Sn 3 As 4 between 0 and 1000 K. It is found that the Debye temperature decreases with an increase of temperature, and then increases after reaching a minimum, and that minimum values are 279, 216 and 181 K for t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 . While above 300 K, θ D of t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 show weak temperature dependence and approach the constant values. At 1000 K, values of θ D are 503 K, 338 K and 304 K, respectively. Heat capacity is an important parameter in condensed matter physics. It can also describe the vibrational properties in the heat transition process. The lattice (or phonon) contribution and the electron contribution are two main sources of heat capacity. The former contributes most at a low temperature, while the latter plays an important role at a high temperature. The relations of the constant volume heat capacity C V with temperature are presented in figure 9 for t-X 3 As 4 . It can be seen that these heat capacities increase rapidly with the temperature increase at a lower temperature. At a high temperature, heat capacities rise slowly and are close to the Dulong-Petit limit. It indicates that the atomic interactions in t-X 3 As 4 occur at a low temperature. The Dulong-Petit limit of t-X 3 As 4 is about 10 J mol −1 K −1 . The variations of the entropy, enthalpy and free energy with temperature at 0 Gpa are shown in figure 10. All values are given in the form of per t-X 3 As 4 formula unit. It is noted that the free energy decreases with the temperature increase, and the entropy increases more rapidly than that of enthalpy as temperature 43601-11  increases. The calculated values of the entropy, enthalpy and free energy of t-X 3 As 4 at room temperature are listed in table 6. All absolute values of the entropy, enthalpy and free energy for t-Sn 3 As 4 are larger than those of t-Si 3 As 4 and t-Ge 3 As 4 under the temperature range from 0 to 1000 K, which means that t-Sn 3 As 4 are less stable than t-Si 3 As 4 and t-Ge 3 As 4 . From the figure 10 and table 6, we can see that the descending order of thermodynamic stability is from t-Si 3 As 4 to t-Ge 3 As 4 to t-Sn 3 As 4 .

Conclusion
For the novel predicted t-X 3 As 4 , the structural, mechanical, electronic, optical and thermodynamic properties are studied by the first-principles calculations. It is found that t-X 3 As 4 are stable by elastic constants and phonons analysis. The t-Sn 3 As 4 shows to me more anisotropic than t-Si 3 As 4 and t-Ge 3 As 4. Due to a larger mass difference between As and Si atom, there is a larger optical band gap in the dispersion curves of t-Si 3 As 4 . By analyzing the density of phonon states in t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 , we can see that the vibration of As atoms is dominant in the frequency range of 0.8−6.8 eV, 0.2−6.1 eV and 4.2−7.6 eV, respectively. The vibration of Si and Ge atoms is dominant in the band of 9.5−13 eV and 6.2−8.8 eV, respectively. For t-Sn 3 As 4 , in the range of 0.2−4.0 eV, the total PHDOS is from Sn and As, which shows that Sn and As have the similar vibrational probability. The band structures and densities of state show that the t-X 3 As 4 (X = Si and Ge) are indirect band gap semiconductors with narrow band gaps of 1.438 and 0.952, respectively. The band structure of t-Sn 3 As 4 shows that it is a direct band gap semiconductor. By the analysis of electron density difference and Mulliken charge population, it is found that X (Si, Ge and Sn) atoms lose electrons, and As atoms acquire electrons. The static refractive indices are found to be 3.94, 4.48 and 3.89 eV for t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 , respectively. The ε 1 (0) are 15.5, 20.0 and 15.1 eV for t-Si 3 As 4 , t-Ge 3 As 4 and t-Sn 3 As 4 , respectively. The Debye temperature of t-Si 3 As 4 is larger than that of t-Ge 3 As 4 and t-Sn 3 As 4 . The Dulong-Petit limit of t-X 3 As 4 is about 10 J mol −1 K −1 . The thermodynamic stability of t-Si 3 As 4 is higher.