Structural, Mechanical, Electronic, Optical, and Thermodynamic Properties of New Oxychalcogenide A₂O₂B₂Se₃ (A = Sr, Ba; B = Bi, Sb) Compounds: A First-Principles Study

Abstract

The structural, mechanical, electronic, and optical characteristics of Alkali chalcogenide and oxychalcogenides, i.e., A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba; B = Bi, Sb), were investigated using density functional theory (DFT). After full relaxation, the obtained structural parameters are in good agreement with the experimental parameters. Furthermore, the calculated elastic stiffness $C_{ij}$ shows that all of the studied compounds followed the mechanical stability criteria. Ductility for these compounds was analyzed by calculating Pugh’s ratio; we classified the Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ as ductile, and the Ba${}_2$O${}_2$Sb${}_2$Se${}_3$ as brittle. The Debye temperature and acoustic velocity were estimated. In addition, electronic and chemical bonding properties were studied from the analysis of the band structure and density of state. The main features of the valence and conduction bands were analyzed from the partial density of states. Electronic band structures are mainly contributed to by Se-$4p$ and Bi-$6p$/Sb-$5p$ states. Direct band gaps are 0.90, 0.47, and 0.73 eV for Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Sb${}_2$Se${}_3$, respectively. The Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ compound has an indirect band gap of 1.12 eV. Furthermore, we interpreted and quantified the optical properties, including the dielectric function, absorption coefficient, optical reflectivity, and refractive index. From the reflectivity spectra, we can state that these compounds will be useful for optical applications.

1. Introduction

Materials based on transition metal ions have attracted a lot of attention due to their unusual electrical, magnetic, and structural features [1,2]. Mixed-anion compounds containing oxide and chalcogenide anions have been extensively studied since the discovery of superconductivity in F-doped LaOFeAs at critical temperatures T∼26 K [3]. This result has accelerated studies on new layered materials. One of the most important compounds, layered oxychalcogenides, are mixed-anion compounds; chalcogenide and oxide anions are indirectly bounded via one or more cations, creating a stack of alternating oxide and chalcogenide layers [4,5].

The oxychalcogenides have attracted much interest owing to their rich and diverse chemistry, and are characterized by the coexistence of ionic oxide anions and more covalent chalcogenide anions. For example, oxide chalcogenides with the chemical formula A${}_2$MO${}_2$X${}_2$Ch${}_2$ (where A = Sr, Ba; X = Cu, Ag; Ch = S, Se and M = first-row transition metal) were first reported by Zhu et al. [6,7] and are isostructural with Sr${}_2$Mn${}_3$Sb${}_2$O${}_2$ structure. It was also found that it is possible to synthesize compounds with different thicknesses of the oxides and chalcogenide layers by varying the element ratios and heating conditions [8].

Currently, the most explored of such materials are bismuth chalcogenide compounds, particularly BiCh${}_2$ (Ch = S, Se)-based compounds; many of them possess layered or tunneled structures. Bi${}_2$O${}_2$Ch (Ch = S, Se) have been exploited for their photocatalytic activities and thermoelectric properties [9,10]. The quaternary oxychalcogenides with 1111 stoichiometry BiMOCh (Ch = S, Se; M = Cu, Ag), due to their outstanding mechanical, chemical, electrical, and optical properties, have gained extensive attention as ionic and transparent conductors. The BiS${}_2$-based superconductive compounds YO${}_{1-x}$F${}_x$BiS${}_2$ (Y = La, Nd, Pr, and Ce), with the highest Tc of 10 K, have then been discovered and NdO${}_{1-x}$F${}_x$BiS${}_2$ (x = 0.1–0.7) was found to have a maximum Tc of less than 5.6 K [11].

One such development is the discovery of new superconductivity compounds that exhibit zero resistance below a critical temperature (Tc). This progress requires the synthesis of new products with novel and fascinating properties. Recently, alkali chalcogenide and oxychalcogenides, i.e., A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba; B = Bi, Sb) materials were synthesized by direct combination of SrO or BaO with Bi${}_2$Se${}_3$ or Sb${}_2$Se${}_3$ [12]. Insulating behavior has been revealed for all compounds. The crystal structure and chemical composition of these compounds were determined by X-ray powder diffraction (XRPD). Their structures consist of electronically active quasi-one-dimensional Sb-Se or Bi-Se ribbons isolated from one another by SrO units. Moreover, the structures of A${}_2$O${}_2$B${}_2$Se${}_3$ compounds share structural features with numerous bismuth and antimony chalcogenides, such as the ternary AB${}_2$X${}_4$ (A = Sr, Ba; B = Bi, Sb; X = S, Se). Since their structural and electronic similarity with the LnOBiX${}_2$ (X = S, Se and Ln = La, Nd, Ce, Pr, Yb) superconductors [11,12], this new family provides a one-of-a-kind opportunity to consider the impacts of dimensionality on superconductivity.

The modeling of physical properties (by means of DFT techniques) has become a very useful tool for understanding the structural, electronic, mechanical, optical, and thermodynamic properties of various materials. In the present work, we would make a theoretical study of the electronic structure, elastic and optical properties of A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba and B = Bi, Sb) materials using first-principle calculations. In Section 2, we provide the computational details of the calculations. Our results are presented in Section 3. A summary of the results is provided in Section 4.

2. Computational Methodology

The first-principles calculations were performed with the Quantum espresso code [13] using the generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) [14] to describe the exchange–correlation functional (XC). The plane waves of the electronic wave functions were expanded on the basis of a plane wave set with an energy cut-off of 40 Ry. the irreducible Brillouin zone was integrated with Monkhorst–Pack [15] 4 × 4 × 4 k-point mesh. For the density of state (DOS) calculations, we used a mesh of 12 × 12 × 12 with the tetrahedron method integration in order to obtain the high-quality charge density. An ultrasoft pseudo-potential [16] was adopted to describe the ionic cores and the valance electron interactions. In order to improve the convergence of the solution of the self-consistent Kohn–Sham equations, the energy levels were broadened by the Methfessel–Paxton [17] smearing with a Gaussian spreading $\sigma$ = 0.01 Ry. The total energy convergence in the iterative solution of the Kohn–Sham equations [18] was set at 1.0 × 10${}^{-7}$ Ry to obtain well-converged ground state energy. All structures were fully relaxed by using the BFGS algorithm [19] with a threshold force of 10${}^{-3}$ Ry/Bohr.

3. Results and Discussion

3.1. Crystal Structure

A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba; B = Bi, Sb) materials crystallized in monoclinic structures with the $P{2}_{1}C$ (No. 14) space group as displayed in Figure 1. The structures of these materials are described in detail by Jessica et al. [12], consisting of double-chain–quasi-one-dimensional ribbons of edge-linked BSe${}_4$O square pyramids, connected to SrO fragments by the apical B-O bond.

Table 1. Calculated lattice parameters a, b, c (Å), $\beta$ (${}^{\circ }$), formation energy, E${}_{form}$ (eV) of the studied A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba and B = Bi, Sb) materials compared with the available experimental data from Reference [12]. $\alpha$ = $\gamma ={90}^{\circ }$.

	Sr${}_2$O${}_2$Bi${}_2$Se${}_3$	Sr${}_2$O${}_2$Sb${}_2$Se${}_3$	Ba${}_2$O${}_2$Bi${}_2$Se${}_3$	Ba${}_2$O${}_2$Sb${}_2$Se${}_3$
$a_{calc.}$	9.200	9.199	9.463	9.729
$a_{exp.}$	9.499	9.424	9.820	-
$b_{calc.}$	4.005	3.957	4.095	4.122
$b_{exp.}$	4.084	4.057	4.190	-
$c_{calc.}$	13.111	12.987	13.535	13.723
$c_{exp.}$	13.437	13.341	13.904	-
${\beta }_{calc.}$	122.828	123.000	123.298	124.040
${\beta }_{exp.}$	122.861	121.955	123.692	-
E${}_{form}$	−2.145	−2.173	−2.139	−2.147

summarizes our calculated structural parameters (a, b, c, and angle $\beta$) and the formation energies for all compounds compared with the available experimental data. It appears from our results that optimized equilibrium parameters for Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ agree well with the experimental values reported in Reference [12]. For Ba${}_2$O${}_2$Sb${}_2$Se${}_3$, there are no corresponding experimental data for the lattice parameters. Moreover, by examining the structural parameters, the substitution of Sr by Ba and Sb by Bi tends to increase the lattice parameters, which may be interpreted in terms of the larger ionic radii of Ba${}^{2+}$(0.134 Å) and Bi${}^{3+}$(0.96 Å) compared to that of Sr${}^{2+}$(1.12 Å) and Sb${}^{3+}$(0.76 Å).

The relative stability of the A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba; B = Bi, Sb) materials was examined by calculating their formation energies, E${}_{form}$, defined as:

$$E_{form}={\displaystyle \frac{E\left(A_2{O}_2{B}_{2}S{e}_3\right)-2E\left(A\right)-2\times \frac{1}{2}E\left(O_2\right)-2E\left(B\right)-3E\left(Se\right)}{9}}$$

(1)

where E(A${}_2$O${}_2$B${}_2$Se${}_3$) is the total energy of one unit cell of the A${}_2$O${}_2$B${}_2$Se${}_3$ compound. E(A), E(B), and E(Se) are the energies of each atom for A, B, and Se in their stable bulk phases, respectively. E(O${}_2$) is the total energy per O${}_2$ molecule.

The calculated formation energies of the different compounds are regrouped in Table 1. As seen, the calculated values are all negatives, confirming the relative stability of all compounds.

3.2. Mechanical Properties

The mechanical properties of these compounds are important for their potential technological and industrial applications. Elastic constants, needed to determine the mechanical stability of these compounds, were calculated. They give an overview of the mechanical and dynamical characteristics, especially the stability and stiffness of the present materials.

For monoclinic crystal [20,21], 13 independent elastic constants, i.e., $C_{11}$, $C_{12}=C_{21}$, $C_{22}$, $C_{13}=C_{31}$, $C_{23}=C_{32}$, $C_{33}$, $C_{44}$, $C_{15}=C_{51}$, $C_{25}=C_{52}$, $C_{35}=C_{53}$, $C_{55}$, $C_{46}=C_{64}$, and $C_{66}$ remained with the explicit form of the tensor reduced to:

$$\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& C_{15}& 0\\ C_{12}& C_{22}& C_{23}& 0& C_{25}& 0\\ C_{13}& C_{23}& C_{33}& 0& C_{35}& 0\\ 0& 0& 0& C_{44}& 0& C_{46}\\ C_{15}& C_{25}& C_{35}& 0& C_{55}& 0\\ 0& 0& 0& C_{46}& 0& C_{66}\end{array}\right)$$

In

Table 2. The calculated elastic constants of the A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba and B = Bi, Sb) compounds in the units of GPa.

	Sr${}_2$O${}_2$Bi${}_2$Se${}_3$	Sr${}_2$O${}_2$Sb${}_2$Se${}_3$	Ba${}_2$O${}_2$Bi${}_2$Se${}_3$	Ba${}_2$O${}_2$Sb${}_2$Se${}_3$
$C_{11}$	106.86	107.04	102.65	54.30
$C_{22}$	133.74	137.47	125.26	108.80
$C_{33}$	122.92	126.51	118.39	109.62
$C_{44}$	39.56	35.86	51.37	43.83
$C_{55}$	32.48	34.53	31.00	29.42
$C_{66}$	26.02	28.87	26.00	28.07
$C_{12}$	31.42	31.87	32.16	22.79
$C_{13}$	35.93	36.20	35.23	26.23
$C_{23}$	56.48	55.99	57.86	46.94
$C_{15}$	5.37	4.85	6.93	4.14
$C_{25}$	2.10	1.05	5.57	11.66
$C_{35}$	10.66	12.69	6.35	9.04
$C_{46}$	0.48	2.40	2.10	3.14

, we present the calculated thirteen independent elastic constants $C_{ij}$ for the monoclinic lattice structures.

It is known that, for the monoclinic structure, mechanical stability requires elastic constants satisfying the following Born’s criteria [22]:

$$\begin{array}{}{C}_{ii}>0(i=1,6), (C_{11}+C_{22}+C_{33}+2(C_{12}+C_{13}+C_{23}))>0, (C_{33}{C}_{55}-C_{35}^2)>0,\hfill & (C_{44}{C}_{66}-C_{46}^2)>0,\hfill & (C_{22}+C_{33}-2C_{23})>0, [C_{22}(C_{33}{C}_{55}-C_{35}^2)+2C_{23}{C}_{25}{C}_{35}-C_{23}^2{C}_{55}-C_{25}^2{C}_{33}]>0,\hfill \end{array}$$

and

$$\begin{array}{}\{2[C_{15}{C}_{25}(C_{33}{C}_{12}-C_{13}{C}_{23})+C_{15}{C}_{35}(C_{22}{C}_{13}+C_{12}{C}_{23})+C_{25}{C}_{35}(C_{11}{C}_{23}-C_{12}{C}_{13})]\hfill & -[C_{15}^2(C_{22}{C}_{33}-C_{23}^2)+C_{25}^2(C_{11}{C}_{33}-C_{13}^2)+C_{35}^2(C_{11}{C}_{22}-C_{12}^2)]+g{C}_{55}\}>0,\hfill \end{array}$$

where $g=C_{11}{C}_{22}{C}_{33}-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-C_{33}{C}_{12}^2+2C_{12}{C}_{13}{C}_{23}$. Therefore, these four compounds are structurally and mechanically stable.

The macroscopic mechanical properties of the different crystals, namely Young’s modulus, bulk modulus, Poisson’s ratio, and shear modulus, can be determined by the obtained elastic constants using the equations presented in Reference [23].

The bulk modulus (B) expresses the response of a material to a volume change of the hydrostatic pressure, whereas the shear modulus (G) reflects the resistance of a material to a shape change, and is deduced from elastic and compliance constants. Voigt [24] proposed to express the polycrystalline bulk modulus $B_V$ and shear modulus $G_V$ via the combinations of elastic constants $C_{ij}$. Similarly, Reuss and Angew [25] determined the bulk modulus $B_R$ and shear modulus $G_R$ expressions in terms of compliance constants $S_{ij}$. Moreover, Hill [26] calculated B and G from the average of the Voigt and Reuss bounds as:

$$B=\frac{B_V+B_R}{2},G=\frac{G_V+G_R}{2}.$$

The polycrystalline Young’s modulus (E) and Poisson’s ratio ($\nu$) are calculated using the relationships [26]

$$E=\frac{9BG}{3B+G},\nu =\frac{3B-2G}{6B+2G}.$$

The values of these polycrystalline elastic constants are listed in

Table 3. The calculated polycrystalline elastic constants (B (GPa), G (GPa), and E (GPa)), and Poisson’s ratio $\nu$ and Pugh’s ratio (D) for A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba and B = Bi, Sb) materials.

	B	G	E	$\mathit{\nu }$	D	Type
Sr${}_2$O${}_2$Bi${}_2$Se${}_3$	66.523	34.755	88.801	0.277	1.914	ductile
Sr${}_2$O${}_2$Sb${}_2$Se${}_3$	67.333	35.566	90.723	0.275	1.893	ductile
Ba${}_2$O${}_2$Bi${}_2$Se${}_3$	64.817	35.201	89.417	0.270	1.841	ductile
Ba${}_2$O${}_2$Sb${}_2$Se${}_3$	46.913	30.764	75.737	0.230	1.525	brittle

. We can see from this table that all compounds exhibit good mechanical properties. The Sr${}_2{\mathrm{O}}_2\mathrm{S}{\mathrm{b}}_2\mathrm{S}{\mathrm{e}}_3$ compound has larger bulk and shear modulus, indicating that it presents better mechanical properties compared to the others, which mainly attributes to the more stable Sr-O-Sb bond than Sr-O-Bi, Ba-O-Bi, and Ba-O-Sb. As also seen, the bulk modulus B is larger than the shear modulus G for all compounds, implying that G limits their stabilities.

Accordingly, we calculate “Pugh’s criterion” ($D=B/G$), which is proposed to judge a metal’s ductility and brittleness; the critical value that separates brittle and ductile materials is around 1.75 [27]. As shown in Table 3, Pugh’s ratio is $D>1.75$ for Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Bi${}_2$Se${}_3$; thus, they behave in a ductile manner. The calculated D ratio for Ba${}_2$O${}_2$Sb${}_2$Se${}_3$ is less than 1.75, so it exhibits brittle behavior.

3.3. Electronic Structure

Band structure calculations are required to provide more information for the fabrication and development of electronic and optoelectronic devices. To explore the electronic properties of the A${}_2$O${}_2$B${}_2$Se${}_3$ compounds, the energy band structures and the partial density of states (PDOS) of the four compounds are illustrated in Figure 2.

One can notice that band structures show similar shapes but with different gap values. It can be clearly seen that Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Sb${}_2$Se${}_3$ show direct band gaps of 0.90, 0.47, and 0.73 eV, respectively, with the valence band maximum (VBM) and the conduction band minimum (CBM) lying at the $\Gamma$ point. However, Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ has an indirect band gap of 1.12 eV where the VBM is at the $\Gamma$ point and the CBM is at the Y point. The difference between the experimental value and the theoretical one can be attributed to some aspects, such as the experimental environment and the exchange–correlation description considered in the present study.

From the partial density of states (PDOS), it can be observed that the VBMs of all compounds mainly consist of Se-$4p$ states, while the CBM is composed of Bi-$6p$/Sb-$5p$ states. We can see the contribution of the A states at the lowest-lying states. A strong hybridization between Sb and Se states was observed in the conduction band for the Ba${}_2$O${}_2$Sb${}_2$Se${}_3$ compound.

3.4. Optical Properties

Optical response functions of the materials are described by the complex dielectric function $(\epsilon \left(\omega \right)={\epsilon }_1\left(\omega \right)+i{\epsilon }_2\left(\omega \right))$. It gives the optical response of the medium at all photon energies $E=\hslash \omega$ and it is closely related to the electronic structure of the material. The imaginary part of the dielectric function ${\epsilon }_2\left(\omega \right)$ is calculated using the matrix elements of occupied and unoccupied wave functions; it is given as follows [28]:

$${\epsilon }_2\left(\omega \right)=\frac{V{e}^2}{2\pi m^2{\hslash }^2{\omega }^2}\int d^{3}k\sum _{n,n^{\prime }}|<kn|p|k{n}^{\prime }{>|}^{2}f\left(kn\right)\times (1-f\left(k{n}^{\prime }\right))\delta (E_{kn}-E_{k{n}^{\prime }}-\hslash \omega ),$$

where n and $n\prime$ are the initial and final states, respectively, p is the momentum operator $(\hslash /i)\partial /\partial x,|kn>$ is a crystal wave function, $f\left(kn\right)$ is the Fermi function for the $n^{th}$ state, and $\hslash \omega$ is the energy of the incident photon. The real part ${\epsilon }_1\left(\omega \right)$ of the dielectric function can be obtained from the imaginary part ${\epsilon }_2\left(\omega \right)$ through the Kramers–Kroning relations [29,30],

$${\epsilon }_1\left(\omega \right)=1+\frac{2}{\pi }P{\int }_0^{\infty }\frac{{\omega }^{\prime }{\epsilon }_2\left({\omega }^{\prime }\right)}{{\omega }^{\prime 2}-{\omega }^2}d{\omega }^{\prime }.$$

P implies the principal value of the integral.

The optical reflectivity spectra $R\left(\omega \right)$, the refractive index $n\left(\omega \right)$, and the absorption coefficient $\alpha \left(\omega \right)$ are derived from the dielectric function as follows [30,31,32]:

$$R\left(\omega \right)={\left|\frac{\sqrt{\epsilon \left(\omega \right)}-1}{\sqrt{\epsilon \left(\omega \right)}+1}\right|}^2,$$

$$n\left(\omega \right)={\left[\frac{{\epsilon }_1{\left(\omega )+({\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)\right)}^{1/2}}{2}\right]}^{1/2}$$

and

$$\alpha \left(\omega \right)={\left[2{\omega }^2\left(\sqrt{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)}-{\epsilon }_2\left(\omega \right)\right)\right]}^{1/2}$$

We note that this part of the Brillouin zone integration was performed using the tetrahedron method with denser k-points in the irreducible part of the Brillouin zone without broadening. Moreover, all the calculations are along parallel and perpendicular directions to the directions of propagation.

Since the dielectric function and the absorption coefficient play crucial roles in the characterization and optical applications of materials, we discuss the optical properties of the A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba; B = Bi, Sb) in this section.

The calculated imaginary part shows three different components of dielectrics, which predict the anisotropic nature of the materials. In Figure 3a, we present only imaginary part of the dielectric function for E‖a. The peaks were caused by transitions from the upper valence band to the lower conduction band. These correspond to transitions from $\mathrm{Se}$-$4$$p$ valence states to $\mathrm{Bi}$-$6$$p$/$\mathrm{Sb}$-$5$$p$ conduction band. It is noted that the ${\epsilon }_2$ of all studied materials have similar shapes but different amplitudes because of the similar band structures. Moreover, the ${\epsilon }_2$ of Sr${}_2$O${}_2$Bi${}_2$Se${}_3$ and Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ were much higher than that of the Sr${}_2$O${}_2$Sb${}_2$Se${}_3$ and Ba${}_2$O${}_2$Sb${}_2$Se${}_3$, showing remarkably enhanced absorption of the photons.

The absorption coefficient defines the region where a material absorbs energy. The energy dependence of the absorption spectrum of the present compounds for E‖a is given in Figure 3b. The absorption edge is away from 0 eV, which corresponds to the energy gaps. The absorption coefficient exhibits two prominent peaks indicating that they could absorb visible light. The first peaks of Sr${}_2$O${}_2$Bi${}_2$Se${}_3$ and Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ are present in the same region at around 3 eV and extend to 12 eV, the second peaks are located at 16 eV and 24 eV for Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ and Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, respectively. For Sr${}_2$O${}_2$Sb${}_2$Se${}_3$ and Ba${}_2$O${}_2$Sb${}_2$Se${}_3$, the same trend is found but in a different region of adsorption. Notably, they exhibited two prominent peaks.

The reflection spectra of the studied compounds are presented in Figure 3c. We can see that the spectra are mainly in the areas between 5 eV and 25 eV, after that, the reflectivity falls sharply to low values (high transparency) for higher energy ranges. The peaks of the reflectivity correspond to the dielectric peaks, which is the macroscopic expression of the inter-band transition behavior. Several obvious peaks are identified, which correspond to the transition from the valence bands to the conduction bands located at 7, 10, 11, 14, 15, 18, 21, and 24 eV.

The calculated refractive index of the studied materials is shown in Figure 3d. The refractive index spectra of all studied materials show similar features, first reaching a maximum value of around 3.3 at around 4 eV, falling at intermediate energies, and then the curve vanishes at higher energies. Indeed, beyond certain energy, the considered material absorbs high-energy photons and cannot behave as transparent material. The extracted static refractive indices are 2.51, 2.48, 2.50, and 2.47 for Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Ba${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Sb${}_2$Se${}_3$, respectively.

3.5. Thermodynamic Properties

Thermodynamic properties of materials, such as specific heat, melting temperature, and thermal conductivity [33] are most suitably described in terms of the Debye temperature (${\theta }_D$). It is a fundamental and very important parameter that helps to obtain the thermodynamic properties and stability of crystals and, thus, design and develop new materials. At a low temperature, the average sound velocities ($v_m$) and the Debye temperature ($\theta$) can be calculated from elastic constants taking into account the fact that the vibrational excitations arise solely from acoustic vibrations. We calculated the ${\theta }_D$ from the average sound velocity through the following equation $v_m$ [34]:

$${\theta }_D=\frac{h}{k_B}{\left[\frac{3n}{4\pi V_a}\right]}^{1/3},$$

where h is Planck’s constant, $k_B$ is Boltzmann’s constant, $V_a$ is the atomic volume, n is the number of atoms per formula unit, and $v_m$ can be obtained from [34]:

$$v_m={\left[\frac{1}{3}(\frac{2}{v_t^3}+\frac{1}{v_l^3})\right]}^{-1/3},$$

where $v_t$ is transverse velocity, $v_l$ is longitudinal velocity. $v_t$ and $v_l$ are calculated from Navier’s equation [35]:

$$v_t={\left[\frac{4G+3B}{3\rho }\right]}^{1/2}\mathrm{and}{v}_l={\left[\frac{G}{\rho }\right]}^{1/2}.$$

The calculated results of Debye temperatures, transverse, longitudinal, and average sound velocities are given in

Table 4. Calculated Debye temperature ${\theta }_D$ (K), average elastic wave velocities $v_m$ (ms${}^{-1}$), longitudinal $v_l$ (ms${}^{-1}$), and transverse $v_t$ (ms${}^{-1}$) for A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba, and B = Bi, Sb) compounds.

	${\mathit{\theta }}_{\mathit{D}}$	${\mathit{v}}_{\mathit{l}}$	${\mathit{v}}_{\mathit{t}}$	${\mathit{v}}_{\mathit{m}}$
Sr${}_2$O${}_2$Bi${}_2$Se${}_3$	260.55	4000.36	2219.89	2472.64
Sr${}_2$O${}_2$Sb${}_2$Se${}_3$	293.61	4463.99	2482.98	2765.16
Ba${}_2$O${}_2$Bi${}_2$Se${}_3$	250.38	3918.86	2189.84	2437.81
Ba${}_2$O${}_2$Sb${}_2$Se${}_3$	255.84	3833.41	2245.95	2490.15

. Sr${}_2$O${}_2$Sb${}_2$Se${}_3$ has a higher Debye temperature; thus, it has a greater micro-hardness. According to References [36,37,38], a larger $\theta$ suggests a higher normal vibration, which is associated with better thermal conductivity. Meanwhile, the Debye temperature can characterize the strength of the covalent bond for the solid. Unfortunately, to the best of our knowledge, there are no other theoretical or measured data available in the literature to compare with our results. Nevertheless, we hope that our results can support further experiments.

4. Conclusions

In summary, first-principles calculations were performed on the A${}_2$O${}_2$B${}_2$Se${}_3$ (A = Sr, Ba; B = Bi, Sb) compounds, including structural parameters, band structure, the density of state, elastic constants, and optical parameters. Our study shows that all studied materials are mechanically stable. By analyzing the band structures and the DOS, we show that the Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Sb${}_2$Se${}_3$ compounds have direct band gaps while Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ possesses an indirect band gap. In addition, this paper also shows that the Sr${}_2$O${}_2$Bi${}_2$Se${}_3$, Sr${}_2$O${}_2$Sb${}_2$Se${}_3$, and Ba${}_2$O${}_2$Bi${}_2$Se${}_3$ are ductile while Ba${}_2$O${}_2$Sb${}_2$Se${}_3$ is brittle. Moreover, the calculated refractive indices n(0) for all compounds are in the range of 2.47–2.51, making them good candidates for use as waveguides. Finally, this new family offers a unique chance to research the impact of dimensionality on superconductivity due to the structural and electronic similarities between them and the LnOBiX${}_2$ (X = S, Se, and Ln = La, Nd, Ce, Pr, Yb) superconductors.