Properties of Cmc2₁-X₂As₂O (X = Si, Ge, and Sn) by First-Principles Calculations

3.1 Structural Properties

The compounds X₂As₂O (X = Si, Ge, and Sn) have an orthorhombic structure with Cmc2₁ space group (No. 36). Figure 1 shows the crystal structure of Cmc2₁-X₂As₂O at 0 GPa. The compounds Cmc2₁-Si₂As₂O, Cmc2₁-Ge₂As₂O, and Cmc2₁-Sn₂As₂O are all centro-symmetric structures (0, 0, 0) with 10 atoms per unit cell. The lattice parameters and volumes of Cmc2₁-X₂As₂O at 0 GPa were calculated on the basis of the geometry optimisation and are presented in Table 1. The density values of Cmc2₁-X₂As₂O (X = Si, Ge, and Sn) are 3.56, 4.60, and 4.90 g/cm³, respectively.

Figure 1:

Crystal structures of the orthorhombic-X₂As₂O (X = Si, Ge, and Sn) (Cmc2₁, No. 36).

3.2 Mechanical Properties and Young’s Moduli

The elastic constants and other mechanical properties were calculated by the GGA-PBE method. The nine independent elastic constants for Cmc2₁-X₂As₂O were calculated and are presented in Table 2. The mechanical stabilities of Cmc2₁-X₂As₂O can be estimated by the elastic constants. In this work, the mechanical stabilities and mechanical moduli of Cmc2₁-X₂As₂O were approximated by the corresponding relationships for orthorhombic crystal class [4], [13], [14], [15]. The elastic stability criteria of the orthorhombic phase are as follows:

Based on Table 2 and judging the stability, the crystal structures of Cmc2₁-X₂As₂O are stable. Cmc2₁-Si₂As₂O exhibits the largest elastic constants C₁₁, C₂₂, C₃₃. For Si₂As₂O and Ge₂As₂O, the calculated C₁₁ is larger than C₃₃ and C₂₂. Hence, the mechanical strength in the [100] direction is higher than that in the [001] and [010] directions. Moreover, C₄₄, C₅₅, and C₆₆ denote the shear moduli in the (100), (010), and (001) crystal planes, respectively. From Table 2, we find that there is a slight difference in C₄₄, C₅₅, and C₆₆, so the shear moduli of Si₂As₂O and Ge₂As₂O are almost isotropic [13].

For Cmc2₁-Sn₂As₂O, C₁₁ is larger than C₂₂and C₃₃, and the C₂₂ is smaller than C₃₃. Hence, the mechanical strength in the [100] direction is higher than that in the [001] and [010] directions. The mechanical strength in the [010] direction is less than that in the [001] direction. The calculated C₅₅ is smaller than C₄₄ and C₆₆.

C₁₂, C₁₃, and C₂₃ are the mechanical moduli corresponding to the biaxial strain (stress); they are the shear moduli of the (110) plane in the [110] direction ((110)[110]), ((101)[101]), and ((011)[011]), respectively [16].

Based on the elastic constants, other mechanical parameters of Cmc2₁-X₂As₂O were calculated by using the Voigt-Reuss-Hill (VRH) method, such as the bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (v). For Cmc2₁-X₂As₂O, based on the elastic constants, the Reuss shear modulus (G_R) and the Voigt shear modulus (G_V) are as follows [4]:

The Reuss bulk modulus (G_R) and the Voigt bulk modulus (G_V) are defined as

where S_ij and C_ij are related by

In the above formulas, C_ij represents the elastic stiffness matrix and S_ij represents the elastic flexibility matrix. Hill proposed that the upper and lower limits of the actual polycrystalline modulus are represented by the arithmetic mean of the Voigt and Reuss moduli, which are expressed as

Young’s modulus (E) and Poisson’s ratio (v) can be calculated by the equations

Based on (3)–(11), the calculated physical quantities are presented in Table 3.

From the calculation results, the bulk modulus (B), shear modulus (G), and Young’s modulus (E) of Cmc2₁-Si₂As₂O are significantly larger than those of Cmc2₁-Ge₂As₂O and Cmc2₁-Sn₂As₂O. Cmc2₁-Si₂As₂O has the largest value of bulk modulus among Cmc2₁-X₂As₂O, which indicates that Cmc2₁-Si₂As₂O has lower compressibility. The shear modulus of Cmc2₁-Sn₂As₂O is the smallest among Cmc2₁-X₂As₂O. Cmc2₁-Sn₂As₂O has the smallest Young’s modulus among Cmc2₁-X₂As₂O, which indicates that the hardness of Cmc2₁-Sn₂As₂O is less than that of Cmc2₁-Si₂As₂O and Cmc2₁-Ge₂As₂O.

Poisson’s ratio represents the stability of the shear strain of the crystal. For a typical metal, the value should be 0.33. For an ionic-covalent crystal, the value ranges from 0.2 to 0.3. Poisson’s ratio of a strong covalent crystal is relatively small, usually below 0.15 [4]. For Cmc2₁-Si₂As₂O and Cmc2₁-Sn₂As₂O, Poisson’s ratios are close to that of covalent-ionic crystals. For Cmc2₁-Ge₂As₂O, the calculated Poisson’s ratio shows that it is a covalent crystal.

The ratio B/G reflects the brittleness or ductility of a material, and the critical value is close to 1.75 [17]. Below 1.75, the material shows brittleness; otherwise it shows toughness. The calculated B/G values of Cmc2₁-Si₂As₂O, Cmc2₁-Ge₂As₂O, and Cmc2₁-Sn₂As₂O are 1.31, 1.12, and 1.45, respectively. It suggests that Cmc2₁-Ge₂As₂O is more brittle than the other two.

The anisotropy of the mechanical properties of a crystal structure can be characterised by many different ways, such as the universal anisotropic index (A^U) and the percent anisotropy (A_G and A_B for shear and bulk moduli) [18]. In contrast to many anisotropic indexes or factors, the universal anisotropic indexes or factors and the universal anisotropic index and percent anisotropy can be applied to any of these crystal structures. The equations used for the calculations are

The shear anisotropic factor for the (100) shear planes between the [011] and [010] directions is

for the (010) shear planes between [101] and [001] directions, it is

and for the (0 0 1) shear planes between [110] and [0 10] directions, it is

In Table 4, we show the values of some of the anisotropic parameters mentioned above. For an isotropic structure, we expect that A^U = 0, A_G = 0, and A_B = 0.

The anisotropic index and the shear anisotropic factors are shown in Table 4. Based on the values of A_G, A_B, and A^U, Cmc2₁-Si₂As₂O shows stronger mechanical anisotropy than Cmc2₁-Ge₂As₂O and Cmc2₁-Sn₂As₂O. The calculated A_G and A^U values of the Cmc2₁-Si₂As₂O are higher than those of the other two structures, which indicates that the shear modulus of Cmc2₁-Si₂As₂O has stronger anisotropy than those of Cmc2₁-Ge₂As₂O and Cmc2₁-Sn₂As₂O. The calculated A_B of Cmc2₁-Ge₂As₂O is larger than those of the other two structures, which indicates that the bulk modulus of Cmc2₁-Ge₂As₂O shows stronger anisotropy than those for Cmc2₁-Si₂As₂O and Cmc2₁-Sn₂As₂O.

The anisotropic 3D graph of Young’s modulus is an easier way to determine anisotropy. For the orthorhombic crystal class, the directional dependence of Young’s modulus (E) can be written as

In (18), S_ij is the compliance matrix; l₁, l₂, and l₃ are the direction cosines (which are given as l₁ = sinθcosφ, l2=sinθsinφ, and l3=cosφ in spherical coordinates) [4]. The directional dependences of Young’s moduli of Cmc2₁-X₂As₂O are shown in Figure 2, and they show strong anisotropy. For Cmc2₁-Si₂As₂O, the anisotropy is weaker than those of Cmc2₁-Ge₂As₂O and Cmc2₁-Sn₂As₂O.

Figure 2:

Directional dependences of Young’s moduli: (a) Cmc2₁-Si₂As₂O, (b) Cmc2₁-Ge₂As₂O, and (c) Cmc2₁-Sn₂As₂O.

3.3 Phonon Spectra

The calculated phonon dispersion spectra of Cmc2₁-X₂As₂O at 0 GPa are shown in Figure 3. The absence of any imaginary phonon frequencies in the entire BZ confirms that Cmc2₁-X₂As₂O are dynamically stable [19].

Figure 3:

Phonon band structures: (a) Cmc2₁-Si₂As₂O, (b) Cmc2₁-Ge₂As₂O, and (c) Cmc2₁-Sn₂As₂O.

The PHDOS of Cmc2₁-X₂As₂O were calculated by CASTEP and are shown in Figure 4. In Figure 4a, there are five main regions of phonon bands for Cmc2₁-Si₂As₂O. When the frequency is in the range 0–6.21 THz, the PHDOS is mainly from the As atoms. When the frequency is in the ranges 6.21–9.0 and 25.10–27.97 THz, the PHDOS mainly originates from the O atoms. Within the ranges 10.64–14.05 and 18.37–19.36 THz, the PHDOS is mainly from the Si atoms. From Figure 4b, there are five main regions of phonon bands for Cmc2₁-Ge₂As₂O. Within the ranges 0–4.70 and 6.9–9.29 THz, the PHDOS is mainly from the Ge and As atoms. In the range 4.7–6.9 THz, the PHDOS mainly originates from the O and As atoms. Within the range 13.1–13.89 THz, the PHDOS is mainly from the O and Ge atoms. In the range 18.5–20.43 THz, the PHDOS mainly comes from the O atoms. From Figure 4c, there are four main regions of phonon bands for Cmc2₁-Sn₂As₂O. Because the Sn and As atoms are heavier than the O atom, the vibrational frequencies of the Sn and As atoms are obviously lower than that of the O atom [20].

Figure 4:

Phonon DOS: (a) Cmc2₁-Si₂As₂O, (b) Cmc2₁-Ge₂As₂O, and (c) Cmc2₁-Sn₂As₂O.

3.4 Band Structures and Densities of States

The electronic properties of the Cmc2₁-X₂As₂O structures were analysed at 0 GPa. The DOS was calculated to get further insight into the bonding characteristics of Cmc2₁-X₂As₂O. The structural stability of a compound is related to its bonding electron orbits.

For Cmc2₁-Si₂As₂O, the valence band maximum is at the highly symmetric S-point and the conduction band minimum is at the Z-points, which indicates Cmc2₁-Si₂As₂O is an indirect bandgap semiconductor (Figure 5a); and the calculated bandgap is 2.024 eV. The total and partial DOS of Cmc2₁-Si₂As₂O are presented in Figure 5b. The main bonding peaks are in the range –8 to 8 eV. The total DOS in the valence band mainly comes from As p and O p orbitals and partially from Si p. The latter contributes most to the total DOS in the conduction band, with partial contributions from As p and Si s. In Figure 5b, the p states of the As atoms and the Si atoms have strong hybridisation. The result shows that interactions of covalent Si-As exist in Cmc2₁-Si₂As₂O.

Figure 5:

Band structure and densities of states: (a) Cmc2₁-Si₂As₂O band structure and (b) DOS; (c) Cmc2₁-Ge₂As₂O band structure and (d) DOS; (e) Cmc2₁-Sn₂As₂O band structure and (f) DOS.

From Figure 5c, the valence band maximum and the conduction band minimum are near the Z-point, which indicates that Cmc2₁-Ge₂As₂O is a direct narrow-bandgap semiconductor. The bandgap is 1.487 eV. Figure 5d shows the total and partial DOS. Below the Fermi level, the total DOS is mainly from As p and partially from Ge p and O p. Above the Fermi level, the total DOS originates mainly from the Ge p and As p. The p states of As and Ge have a hybrid effect.

The band structure of Cmc2₁-Sn₂As₂O is shown in Figure 5e. In the band structure, the valence band maximum is located at the S-point and the conduction band minimum at the G-point. Cmc2₁-Sn₂As₂O is an indirect bandgap semiconductor. The bandgap of Cmc2₁-Sn₂As₂O is 1.512 eV. From Figure 5f, the total DOS in the valence band originates mainly from As s state. The total DOS in the conduction band originates mainly from O s state. From Figure 5f, some hybridisation from the s states of the As atoms and the O atoms can be found. The results show that there are covalent As-O interactions in Sn₂As₂O.

3.5 Optical Properties

Dielectric function is the most common characteristic of materials. It can characterise the response of materials to incident electromagnetic waves [21]. Optical properties of materials can usually be evaluated on the basis of a complex dielectric function, which depends on the frequency. The complex dielectric function is given by [22]

where ε1(ω) is the real part and ε2(ω) is the imaginary part. ε2(ω) is calculated on the basis of the momentum matrix elements between the occupied and unoccupied wave functions as follows [23]:

where V is the volume of the unit cell, e is electronic charge, p is the momentum operator, |kn⟩ is for crystal wave function, f(kn) denotes the Fermi distribution function, and ℏω is the energy of the incident photon. The real part ε1(ω) is given by Kramers-Kroning relationship

where M is the principal value of the integral. Generally, ε1(ω) is connected with the electric polarisation characteristics of the material.

The absorption coefficient A(ω) can be calculated on the basis of the complex dielectric function ε(ω), as follows [20]:

The optical conductivity usually plays an important part in exploring the electronic structures of a compound and its response to external conditions such as temperature and pressure. So, it is necessary to calculate the optical conductivity. The optical conductivity σ(ω) is expressed by the complex dielectric function as follows:

Other optical constants, such as the absorption coefficient I(ω), the optical reflectivity R(ω), and energy-loss spectrum L(ω), can be calculated using ε(ω). These expressions are as follows: [24], [25], [26]:

The optical properties were calculated and are presented in Figure 6 for the energy range up to 40 eV. In Figure 6a, the real parts of the calculated dielectric functions are shown for Cmc2₁-X₂As₂O in the [100], [010], and [001] directions. The static dielectric constants of Cmc2₁-X₂As₂O are presented in Table 5. The static dielectric constants of Cmc2₁-Si₂As₂O and Cmc2₁-Sn₂As₂O in the [001] polarisation direction are higher than those in the [100] and [010] directions. The static dielectric constant of Cmc2₁-Ge₂As₂O in the [001] polarisation direction is lower than those in the [100] and [010] directions. For Cmc2₁-X₂As₂O, the real parts of dielectric functions in the [001] direction increase with increasing photon energy and achieve the highest values (16.2, 16.9, and 14.9 F/m) at 2.64, 2.14, and 2.58 eV, respectively. The real parts of the dielectric functions of X₂As₂O in the [010] direction increase with the increasing photon energy and achieve the highest values (11.1, 15.2, and 11.6 F/m) at 2.15, 1.12, and 1.76 eV, respectively. The real parts of dielectric functions in the [100] direction achieve the highest values (11.1, 15.2, and 11.6 F/m) at 2.15, 1.12, and 1.76 eV, respectively.

Figure 6:

(a) Real parts of dielectric function, (b) imaginary parts of dielectric function, (c) real parts of conductivity, (d) absorption, (e) reflectivity, and (f) loss function for Cmc2₁-X₂As₂O in the [100], [010], and [001] directions.

The curves of the imaginary parts of the dielectric functions in the [001], [010], and [100] directions are shown in Figure 6b. Cmc2₁-Si₂As₂O and Cmc2₁-Sn₂As₂O have only one peak in the [100], [010], and [001] directions. Cmc2₁-Ge₂As₂O has two peaks in the [100] and [010] directions and one peak in the [001] direction. Thus, Cmc2₁-Ge₂As₂O is more anisotropic than Cmc2₁-Si₂As₂O and Cmc2₁-Sn₂As₂O.

The curves of the real parts of the conductivities for Cmc2₁-X₂As₂O are plotted in Figure 6c. The real parts of the conductivities of Cmc2₁-Si₂As₂O in the [100], [010], and [001] directions increase with the increase in photon energy and reach the highest values (6.84, 11, and 11) at 5.91, 2.25, and 2.08 eV, respectively. For Cmc2₁-Ge₂As₂O, the real parts of the conductivities in the [100], [010], and [001] directions increase with the increase in photon energy and reach the highest values (6.72, 6.74, and 7.41) at 5.62, 5.71, and 4.43. For Cmc2₁-Sn₂As₂O, the real parts of the conductivities in the [100], [010], and [001] directions increase with the increase in photon energy and reach the highest values (6.15, 6.15, and 7.32) at 5.66, 5.77, and 4.31 eV.

The absorption coefficients in the [100], [010], and [001] directions are shown in Figure 6d. For Cmc2₁-Si₂As₂O, the peaks of the absorption spectra in the [100], [010], and [001] directions are located at 7.99, 8.06, and 5.62 eV, respectively. For Cmc2₁-Ge₂As₂O, the corresponding peaks are located at 6.8, 6.8, and 5.8 eV, respectively, and for Cmc2₁-Sn₂As₂O they are located at 6.83, 6.88, and 5.57 eV, respectively.

The curves of the reflectivity R(ω) of Cmc2₁-X₂As₂O are plotted in Figure 6e. For Cmc2₁-Si₂As₂O, the reflectivity spectra in the [100], [010], and [001] directions are characterised by two, two, and one peaks, respectively. In the [100] direction, these peaks are located at 8.4 and 11.9 eV, respectively. Similar peaks also appear in the [010] direction. In the [001] direction, the peak is located at 5.96 eV.

For Cmc2₁-Ge₂As₂O, the reflectivity spectra in the [100] direction are characterised by four peaks. These peaks are located at 1.63, 7.19, 11.4, and 14 eV, respectively. In the [010] direction, the four peaks are located at 1.72, 7.05, 11.6, and 14 eV, respectively. In the [001] direction, there is only one peak, which is located at 6.17 eV.

For Cmc2₁-Sn₂As₂O, the reflectivity spectra in the [100] direction are characterised by three peaks. These peaks are located at 2.2, 7.03, and 11.4 eV, respectively. In the [010] direction, the three peaks are located at 2.23, 7.06, and 11 eV, respectively. In the [001] direction, there is only one main peak, which is located at 13.4 eV.

The loss function L(ω) describes the energy loss of an electron passing through the material. For Cmc2₁-X₂As₂O, the calculated energy losses are shown in Figure 6f. We can see the characteristics of plasma oscillation from the peak of L(ω). And L(ω) also can describe the frequency of collective oscillation of the valence electrons. Above the plasma frequency, the material describes the dielectric behaviour, and below, the material shows the metallic property. The position of peak in the L(ω) spectrum indicates the change of the material from metallic to dielectric character [6]. For Cmc2₁-Si₂As₂O, in the [100], [010], and [001] directions, each peak of L(ω) is located at 17.7 eV. Similar are the situations in Cmc2₁-Ge₂As₂O in the [100], [010], and [001] directions. For Cmc2₁-Sn₂As₂O, the peaks of L(ω) in the [100], [010], and [001] directions are located at 16.4, 16.4, and 16.9 eV, respectively.