The Effects of Chlorine Doping on the Mechanical Properties of Bi₂O₂Se

Abstract

In this work, first-principle calculations based on density functional theory are employed to investigate how chlorine doping influences the elastic moduli, ductility, and lattice thermal conductivity of Bi₂O₂Se, aiming to explore an effective method to improve its mechanical properties for its applications under thermal stress. Our findings reveal that chlorine(Cl) doping significantly affects the electronic structure and mechanical properties of Bi₂O₂Se. The electrons are distributed on the Fermi level, and the Cl-doped Bi₂O₂Se exhibits metal-like properties. In addition, Cl doping enhances the ductility and toughness of Bi₂O₂Se and reduces its lattice thermal conductivity. These results suggest that Cl doping is an effective approach for tuning the mechanical properties of Bi₂O₂Se.

1. Introduction

Presently, in the face of the escalating global energy crisis and increasing environmental pollution, it is imperative to develop green and sustainable energy resources [1,2]. Thermoelectric power generation (TEG) has become one of the hot research topics due to its advantages of being green and efficient [3]. The assembly, manufacturing process, and reliable operation of TEG devices necessitate thermoelectric materials with excellent mechanical performance. Inferior mechanical properties can lead to crack formation and subsequent performance degradation under thermal stress, especially considering the operational conditions involving cyclic temperature gradients [4,5]. Therefore, it is essential for materials to possess favorable mechanical properties in order to meet the requirements of practical applications.

In recent years, Bismuth oxyselenide (Bi₂O₂Se), in which weak electrostatic interactions exist between adjacent layers, has attracted widespread attention and become an emerging structural material with great development potential in high-performance electronic, optoelectronic, and flexible devices due to its high electron mobility and excellent air stability [6,7]. So far, experimental and theoretical investigations on the mechanical properties of Bi₂O₂Se are still in their infancy stage. Theoretically, Liu et al. [8] investigated the electronic structure, bulk modulus, shear modulus, and thermal conductivity of Bi₂O₂Se using density functional theory (DFT), and revealed its high anisotropy and potential as a mechanical material. Pang et al. [9] studied the mechanical response of Bi₂O₂Se under uniaxial and biaxial tension using first-principles calculations. They found that the band gap of Bi₂O₂Se decreases with increasing tensile strain, and a phase transition from semiconductor to metal occurs eventually, due to the evolution of electron localization function. Wang et al. [10] studied the mechanical properties of Bi₂O₂Se under a pressure of 50 GPa using first-principles calculations and found that the Bi₂O₂Se exhibits mechanical stability at least below 50 GPa. Furthermore, the Bi₂O₂Se exhibits anisotropic and malleable properties within the pressure range, with the anisotropy becoming increasingly significant as the pressure increases. Zhang et al. [11] studied the elastic properties of Bi₂O₂Se_1−xTe_x by the first-principles calculations and proposed that substituting Se for Te in Bi₂O₂Se can optimize its mechanical properties. Despite these efforts, further improvements on the mechanical properties of Bi₂O₂Se are necessary in order to meet the requirements of thermoelectric generator device assembly and manufacturing processes, as well as to ensure reliable operation.

Experimentally, the halogen element Cl has been widely employed to enhance the mechanical properties of materials. Piriz et al. [12] investigated the mechanical properties of Cl-doped edge zigzag graphene nanoribbons (ZGNRs) and found that Cl-doped ZGNRs exhibit better mechanical properties as compared to pure graphene. Zhang et al. [13] prepared Cl-doped carbon nanotube sheets via floating catalyst chemical vapor deposition and studied the effects of Cl doping on the mechanical properties of carbon nanotube sheets. They found that the doped sandwich composite film displays excellent mechanical properties, with a tensile strength of 90 MPa. Theoretically, Park et al. [14] investigated the effect of Cl doping on the mechanical properties of amorphous carbon films using DFT and revealed that the introduction of Cl atoms into amorphous carbon films could significantly reduce the bulk modulus, thereby softening the films. These experimental and computational studies demonstrate that Cl doping is an effective approach for improving the mechanical properties of materials.

However, the research on the mechanical properties of Cl-doped Bi₂O₂Se has not been reported thus far. Therefore, in this work, we employ first-principles methods to investigate the effects of Cl doping on the geometrical, electronic, and mechanical properties of Bi₂O₂Se. The geometrical structure, density of state distributions, bulk modulus B, Young’s modulus E, shear modulus G, Debye temperature ${\theta }_D$, and lattice thermal conductivity ${\kappa }_{\iota }$ are all determined for Bi₂O₂Se before and after Cl doping. It is shown that Cl doping results in redistribution of electrons in Bi₂O₂Se, and the Cl-doped Bi₂O₂Se exhibits metallic properties. In addition, Cl doping affects the mechanical strength, toughness, ductility, and lattice thermal conductivity of Bi₂O₂Se considerably. As compared with the pristine Bi₂O₂Se, the Cl-doped system exhibits better ductility, better toughness, and reduced lattice thermal conductivity, which is beneficial to restrain heat transfer and reduce the heat loss of energy. This study, thus, provides a new method to tune the physical properties of Bi₂O₂Se and has important implications for promoting related investigations.

2. Computational Details

Our calculations are implemented in the Ab initio Simulation Package (VASP) code (Vienna, Austria) within the framework of DFT [15]. The interaction between electrons and ions is described by the projector augmented-wave (PAW) pseudopotential approximation [16]. In addition, the Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA) [17] and the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional [18] are used to deal with the exchange-correlation potential between electrons. The Monkhorst-package scheme [19] with a 6 × 6 × 6 k-point sampling for the Brillouin-zone is used to perform all the calculations. Besides, the cut-off energy for the plane wave basis sets is 500 eV. In this work, the considered systems are Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125. The elastic constants of the material are calculated by using the stress-strain method, in which six finite distortions of the lattice are performed, and the relationship between the stress component ${\sigma }_i$ (i = 1–6) and the applied strain $\epsilon$_j (j = 1–6) at a small deformation is described as ${\sigma }_i={\displaystyle \sum }_{j=1}^6{C}_{ij}{\epsilon }_j$ [20]. In the calculation, the energy and force convergence criteria are set to be 1 × 10⁻⁵ eV/atom and 1 × 10⁻² eV/Å, respectively. The specific flowchart of the calculation process is shown in Figure 1.

3. Results and Discussions

3.1. Structural Properties of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125

The bulk phase of Bi₂O₂Se exhibits a tetragonal I4/mmm structure (No. 139) with a unit cell containing 10 atoms. Within Bi₂O₂Se, the curved [Bi₂O₂]²⁺ layers and [Se]²⁻ layers stack alternately along the c-axis through weak electrostatic interactions [21]. Figure 2 presents the optimized geometrical structure of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125, both containing 40 atoms. The lattice constants, volumes, and bond lengths of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 are presented in

Table 1. Comparison of lattice constants a₀ (Å) and c₀ (Å), volume (ų), and bond length (Å) of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 with experimental and other theoretical results.

		a0	c0	Volume	<Bi-O>	<Bi-Se>
Bi2O2Se	Our cal.	3.917	12.357	189.592	2.337	3.311
	Exp. [23]	3.88	12.16	183.06	---	---
	Other cal. [22]	3.90	12.39	188.45	---	---
Bi2O2Se0.875Cl0.125	Our cal.	3.946	12.287	191.320	2.339	3.336

. For undoped Bi₂O₂Se, the calculated lattice constants are a₀ = b₀ = 3.917 Å and c₀ = 12.357 Å, which show good agreements with other computational results of a₀ = b₀ = 3.90 Å and c₀ = 12.39 Å [22], as well as experimental data of a₀ = b₀ = 3.88 Å and c₀ = 12.16 Å [23]. Additionally, the <Bi-O> and <Bi-Se> bond lengths of Bi₂O₂Se are determined to be 2.337 Å and 3.311 Å, respectively. These values are comparable with the previously calculated results of 2.312 Å and 3.272 Å reported by Wu et al. [24]. As compared with Bi₂O₂Se, the calculated lattice constants a₀ and b₀ (3.946 Å) of Bi₂O₂Se_0.875Cl_0.125 are increased by 0.74%, while the c₀ (12.287 Å) is decreased by 0.57%, resulting in a 0.91% volume expansion. Meanwhile, the calculated <Bi-O> bond length (2.339 Å) of Bi₂O₂Se_0.875Cl_0.125 is increased by 0.09% and <Bi-Se> bond length (3.336 Å) is increased by 0.76%. This is mainly because the following reactions occur after Cl doping [25]:

$${\mathrm{Cl}}^{-}\stackrel{{\mathrm{Se}}^{2-}}{\to }{\mathrm{Cl}}_{\mathrm{Se}}^{.}+{\mathrm{e}}^{\prime },$$

(1)

where extra electrons are produced, resulting in increased repulsive interaction between electrons and an expansion of volume.

3.2. The Influence of Cl Doping on the Electronic Structure of Bi₂O₂Se

We further explore the influence of Cl doping on the electronic structure of Bi₂O₂Se by investigating the density of state (DOS) distributions of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125. The total and projected DOS distributions around the Fermi level of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125, obtained by the standard DFT method, are plotted in Figure 3a,b, respectively. For Bi₂O₂Se, the valence bands ranging from 4 to 5.53 eV are predominantly dominated by Se 4p orbitals hybridized with O 2p and Bi 6s orbitals. The conduction bands ranging from 6.02 to 8 eV are mainly characterized by Bi 6p orbitals and O 2p orbitals. The obtained band gap between the valence band maxima (contributed by Bi 6s) and the conduction band minima (contributed by Bi 6p) of 0.49 eV for Bi₂O₂Se is consistent with previous theoretical values (0.41 eV [26], 0.43 eV [27], and 0.472 eV [28]). In addition, the strong hybridization between these orbitals will lead to higher stability in Bi₂O₂Se [29]. The DOS distributions of Bi₂O₂Se_0.875Cl_0.125 shown in Figure 3b indicate that the Fermi levels cross the valence bands and conduction bands, and many electrons are observed at the Fermi level. These features imply metallic properties for Bi₂O₂Se_0.875Cl_0.125, which differ from the results reported by Tan et al. [25], primarily because our doping concentration of 2.5% is higher than the dissolution limit of 1.5% in the literature.

Due to the underestimation of band gap values in standard DFT calculations, we further use the hybrid DFT method to calculate the total and projected DOS distributions around the Fermi level of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125, and the calculated results are displayed in Figure 3c,d. For Bi₂O₂Se, the calculated band gap between the valence band maxima (contributed by Bi 6s) and the conduction band minima (contributed by Bi 6p) is 1.05 eV, which is consistent with the previous computational values of 0.9 eV [30], 0.99 eV [8] and 1.01 eV [31]. Nonetheless, our hybrid DFT value is larger than the corresponding experimental value of 0.85 eV [24]. We note that due to the inclusion of a mixture of Hartree-Fock (HF) and DFT exchange terms, hybrid DFT calculations may overestimate or underestimate material band gaps [32,33], leading to higher or lower calculated values relative to their experimental counterparts [34]. The hybrid DFT, thus, represents a practical, although not perfect, solution to reproduce experimental band gaps. For Bi₂O₂Se_0.875Cl_0.125, it can be seen that the electron distribution obtained by the hybrid DFT method is more delocalized, as compared with the standard DFT results, and a certain number of electrons are still distributed at the Fermi level, i.e., the doped Bi₂O₂Se shows metal-like properties.

3.3. Mechanical Properties of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125

3.3.1. Elastic Constants of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125

The elastic constants ($C_{ij}$) of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 are first calculated based on the optimized geometrical structures. The Bi₂O₂Se possesses a tetragonal crystal structure, resulting in six independent elastic constants, namely C₁₁, C₁₂, C₁₃, C₃₃, C₄₄ and C₆₆ [8]. The calculated elastic constants are shown in

Table 2. The calculated elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄ and C₆₆ in GPa) for Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125.

Compounds		C11	C12	C13	C33	C44	C66
Bi2O2Se	Our Cal.	159.39	73.55	44.24	121.28	13.41	57.71
	Other Cal. [8]	155.48	71.56	43.47	119.01	11.23	56.41
Bi2O2Se0.875Cl0.125	Our Cal.	143.17	63.39	46.38	106.87	11.96	53.74

and compared with other literature values [8]. The calculated elastic constants for Bi₂O₂Se, i.e., C₁₁ = 159.39 GPa, C₁₂ = 73.55 GPa, C₁₃ = 44.24 GPa, C₃₃ = 121.28 GPa, C₄₄ = 13.41 GPa, and C₆₆ = 57.71 GPa, are in good agreement with previous computational results of C₁₁ = 155.48 GPa, C₁₂ = 71.56 GPa, C₁₃ = 43.47 GPa, C₃₃ = 119.01 GPa, C₄₄ = 11.23 GPa, and C₆₆ = 56.41 GPa [8]. Moreover, the calculated elastic constant C₁₁ (159.39 GPa) is found to be larger than C₃₃ (121.28 GPa), indicating that the Bi₂O₂Se exhibits higher resistance to deformation along the <100> direction as compared to the <001> direction. Moreover, the calculated value of C₁₃ (44.24 GPa) is smaller than that of C₁₂ (73.55 GPa), which indicates that the Bi₂O₂Se exhibits more significant contraction along the <001> direction as compared to the <010> direction when applying the same normal stress along the <100> direction. Furthermore, our results show that C₆₆ (57.71 GPa) is larger than C₄₄ (13.41 GPa), indicating that shear deformation along the (001) plane is more difficult to occur as compared to shear deformation along the (100) plane for Bi₂O₂Se [35,36].

As compared with the values of Bi₂O₂Se, the calculated elastic constants C₁₁ = 143.17 GPa, C₁₂ = 63.39 GPa, C₃₃ = 106.87 GPa, C₄₄ = 11.96 Gpa, and C₆₆ = 53.74 GPa for Bi₂O₂Se_0.875Cl_0.125 are reduced by 10.18%, 13.81%, 11.88%, 10.81%, and 6.88%, respectively, while the calculated value of C₁₃ (46.38 GPa) is increased by 4.84%. These results suggest that Cl doping has significant effects on the mechanical properties of Bi₂O₂Se. Furthermore, the mechanical stability of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 are determined by employing the following equation:

$${\mathrm{C}}_{11}>0,{\mathrm{C}}_{33}>0,{\mathrm{C}}_{44}>0,{\mathrm{C}}_{66}>0,({\mathrm{C}}_{11}-{\mathrm{C}}_{12})>0,$$

(2)

$$({\mathrm{C}}_{11}+{\mathrm{C}}_{33}-2{\mathrm{C}}_{13})>0,[2({\mathrm{C}}_{11}+{\mathrm{C}}_{12})+{\mathrm{C}}_{33}+4{\mathrm{C}}_{13}]>0.$$

(3)

Our calculations demonstrate that both Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 satisfy the above mechanical stability criteria.

3.3.2. Elastic Moduli of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125

The bulk modulus (B), shear modulus (G) and Young’s modulus (E) of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 can be calculated by using the Voigt–Reuss–Hill (VRH) approximation method [37]. In VRH approximation, the Voigt [38] and Reuss [39] approximations correspond to the upper and lower limits of the modulus, respectively. According to the Voigt approximation, the bulk modulus (B_V) and shear modulus (G_V) can be calculated by the following equations:

$${\mathrm{B}}_{\mathrm{V}}=\frac{1}{9\left[2{\mathrm{C}}_{11}+2{\mathrm{C}}_{12}+{\mathrm{C}}_{33}+4{\mathrm{C}}_{13}\right]},$$

(4)

$${\mathrm{G}}_{\mathrm{V}}=\frac{1}{15}\left[({\mathrm{C}}_{11}+{\mathrm{C}}_{22}+{\mathrm{C}}_{13}\right)-({\mathrm{C}}_{12}+{\mathrm{C}}_{23}+{\mathrm{C}}_{31})+3\left({\mathrm{C}}_{44}+{\mathrm{C}}_{55}+{\mathrm{C}}_{66}\right)].$$

(5)

The bulk modulus (B_R) and shear modulus (G_R) can also be calculated by the Reuss approximation:

$${\mathrm{B}}_{\mathrm{R}}=\frac{\left({\mathrm{C}}_{11}+{\mathrm{C}}_{12}\right){\mathrm{C}}_{33}-2{\mathrm{C}}_{13}{}^2}{{\mathrm{C}}_{11}+{\mathrm{C}}_{12}+2{\mathrm{C}}_{33}-4{\mathrm{C}}_{13}},$$

(6)

$${\mathrm{G}}_{\mathrm{R}}=\frac{15}{18{\mathrm{B}}_{\mathrm{V}}/\left[({\mathrm{C}}_{11}+{\mathrm{C}}_{12}\right){\mathrm{C}}_{33}-2{\mathrm{C}}_{13}{}^2]+6/({\mathrm{C}}_{11}-{\mathrm{C}}_{12})+6/{\mathrm{C}}_{44}+3/{\mathrm{C}}_{66}}$$

(7)

Then, the elastic modulus is the arithmetic average of the Voigt and Reuss approximations, which can be expressed as:

$$\mathrm{B}=\frac{{\mathrm{B}}_{\mathrm{V}}+{\mathrm{B}}_{\mathrm{R}}}{2},$$

(8)

$$\mathrm{G}=\frac{{\mathrm{G}}_{\mathrm{V}}+{\mathrm{G}}_{\mathrm{R}}}{2}$$

(9)

Further, the Young’s modulus (E) can be given by:

$$\mathrm{E}=\frac{9\mathrm{BG}}{3\mathrm{B}+\mathrm{G}}.$$

(10)

The calculated results are shown in

Table 3. The calculated bulk modulus B (GPa), shear modulus G (GPa) and Young’s modulus E (GPa) for Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125.

Compounds		B	G	E
Bi2O2Se	Our Cal.	83.21	29.59	79.36
	Other Cal. [8]	81.42	27.38	73.56
	Other Cal. [10]	---	---	---
	Exp. [40]	---	---	66
	Exp. [41]	71.5	---	---
Bi2O2Se0.875Cl0.125	Our Cal.	77.05	26.20	70.59

, along with other results [8,40,41] for comparison. For Bi₂O₂Se, the calculated elastic moduli are B = 83.21 GPa, G = 29.59 Gpa, and E = 79.36 GPa, which agree well with the calculated results of B = 81.42 GPa, G = 27.38 GPa, and E = 73.56 GPa [8]. Besides, the calculated results of B = 83.21 GPa and E = 79.36 GPa are comparable to the experimental values of B = 71.5 GPa reported by Mahan et al. [41] and E = 66 GPa reported by Sagar et al. [40].

The B of Bi₂O₂Se_0.875Cl_0.125 is calculated to be 77.05 GPa, which is 7.40% lower than that of pure Bi₂O₂Se, indicating that the introduction of Cl dopant reduces the resistance of Bi₂O₂Se towards uniform compression. Moreover, Cl doping reduces the G of Bi₂O₂Se from 29.59 GPa to 26.20 GPa, indicating that the chemical bonds in Bi₂O₂Se_0.875Cl_0.125 are less strong and directional than those in Bi₂O₂Se, rendering it more susceptible to plastic deformation [42]. Furthermore, as compared with the calculated E of 79.36 GPa for Bi₂O₂Se, the calculated E = 70.59 GPa for Bi₂O₂Se_0.875Cl_0.125 is 11.05% smaller, i.e., the incorporation of Cl dopant in Bi₂O₂Se results in reduced tolerance to elastic deformation, thereby facilitating crack prevention under extreme temperature gradients [43]. In the literature, Zhang et al. [11] have reported that Te doping can reduce the elastic moduli of Bi₂O₂Se, which is similar to the phenomenon of Cl doping. In addition, the smaller B or E, the better the elastic compliance of materials. The presented results suggest that Cl doping enhances the elastic compliance of Bi₂O₂Se.

3.3.3. Ductility and Elastic Anisotropy of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125

Ductility is an important index to measure the mechanical properties of a material. It represents the ability of a material to exhibit significant plastic deformation without breaking under mechanical stress [44]. Based on the calculated elastic moduli, the B/G values and brittle-ductile behavior of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 are further explored. The critical value for the brittle-ductile transition of the material is 1.75 [45,46]. When the B/G ratio is less than 1.75, the material exhibits brittleness; otherwise, it demonstrates ductility. The calculated B/G ratio for Bi₂O₂Se is 2.81, which is in close agreement with the value of 2.97 reported by MacIsaac et al. [47]. This result indicates that Bi₂O₂Se is a ductile material. For Bi₂O₂Se_0.875Cl_0.125, the calculated B/G value is 2.94, which is 4.63% higher than that of Bi₂O₂Se, suggesting that Cl doping enhances the ductility of Bi₂O₂Se, possibly due to the presence of metallic bonding in Bi₂O₂Se_0.875Cl_0.125.

Elastic anisotropy is another important mechanical property of materials, which is related to the appearance of microcracks [48,49]. The universal anisotropic index (A_U) of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 is calculated using the following formula [8]:

$${\mathrm{A}}_{\mathrm{U}}=\frac{5{\mathrm{G}}_{\mathrm{V}}}{{\mathrm{G}}_{\mathrm{R}}}+\frac{{\mathrm{B}}_{\mathrm{V}}}{{\mathrm{B}}_{\mathrm{R}}}-6.$$

(11)

When the value of A_U is not equal to 0, the material exhibits elastic anisotropy. For Bi₂O₂Se, the calculated A_U value is 2.51, which indicates that Bi₂O₂Se is mechanically anisotropic. This finding is consistent with the conclusions drawn by Liu et al. [8]. Moreover, for Bi₂O₂Se_0.875Cl_0.125, the calculated A_U value is 2.47, which is close to the A_U value (2.51) of Bi₂O₂Se, indicating that Cl doping has slight impacts on the mechanical anisotropy of Bi₂O₂Se. The ELATE tool [50] can be utilized to generate directional plots of Young’s modulus for Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125. A more isotropic modulus will resemble a sphere in shape. As illustrated in Figure 4, the plot of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 deviates from sphere shape, indicating that the Young’s modulus of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 is anisotropic.

3.3.4. Debye Temperature of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125

The Debye temperature $\mathsf{\theta }$ is the characterization of binding force between atoms, and the specific formula is [51]:

$$\mathsf{\theta }=\frac{\mathrm{h}}{{\mathrm{k}}_{\mathrm{B}}}{\left(\frac{3\mathrm{n}}{4\mathsf{\pi }\mathsf{\Omega }}\right)}^{1/3}{\mathrm{v}}_{\mathrm{a}},$$

(12)

where h and k_B are the Planck constant and Boltzmann constant, respectively. The $\mathsf{\Omega }$ represents cell volume, n represents number of atoms in the cell, and ${\mathrm{v}}_{\mathrm{a}}$ is the average sound wave velocity. The ${\mathrm{v}}_{\mathrm{a}}$ can be calculated by [52,53]:

$${\mathrm{v}}_{\mathrm{a}}={\left[\frac{1}{3}\left(\frac{1}{{\mathrm{v}}_{\mathrm{L}}^3}+\frac{2}{{\mathrm{v}}_{\mathrm{T}}^3}\right)\right]}^{-1/3}.$$

(13)

Here, the ${\mathrm{v}}_{\mathrm{L}}$ and ${\mathrm{v}}_{\mathrm{T}}$ are the longitudinal and transverse sound wave velocity, respectively, which can be calculated by [54]:

$${\mathrm{v}}_{\mathrm{L}}=\sqrt{\frac{\mathrm{E}\left(1-\mathrm{v}\right)}{\mathsf{\rho }\left(1+\mathrm{v}\right)\left(1-2\mathrm{v}\right)}}=\sqrt{\frac{\mathrm{B}+4/3\mathrm{G}}{\mathsf{\rho }}},$$

(14)

$${\mathrm{v}}_{\mathrm{T}}=\sqrt{\frac{\mathrm{E}}{2\mathsf{\rho }\left(1+\mathrm{v}\right)}}=\sqrt{\frac{\mathrm{G}}{\mathsf{\rho }}}.$$

(15)

The calculated results are shown in

Table 4. The calculated transverse wave velocity ${\mathrm{v}}_{\mathrm{T}}$ (m/s), longitudinal wave velocity ${\mathrm{v}}_{\mathrm{L}}$ (m/s), average wave velocity ${\mathrm{v}}_{\mathrm{a}}$ (m/s), and Debye temperature θ (K) for Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125.

Compounds		${\mathbf{v}}_{\mathbf{L}}$	${\mathbf{v}}_{\mathbf{T}}$	${\mathbf{v}}_{\mathbf{a}}$	$\mathbf{\theta }$
Bi2O2Se	Our Cal.	3638	1787	2007	224.1
	Other Cal. [8]	3584	1727	1939	--
	Other Cal. [10]	1440	2630	1600	181.0
Bi2O2Se0.875Cl0.125	Our Cal.	3513	1699	1909	212.5

, along with other results [8,10] for comparison. The calculated average sound velocity for Bi₂O₂Se is 2007 m/s, which is comparable to the value of 1939 m/s calculated by Liu et al. [8], but different from the calculated results of 1600 m/s reported by Wang et al. [10]. This is mainly because in the calculations of Wang et al. they used the CASTEP code with ultrasoft pseudopotentials, whereas we employ the VASP code with PAW pseudopotentials. Therefore, their calculated Debye temperature of 224.1 K is different from our calculated value of 181.0 K [10]. For Bi₂O₂Se_0.875Cl_0.125, our calculated Debye temperature $\theta$ (212.5 K) is 11.6 K smaller than that of pure Bi₂O₂Se (224.1 K), indicating that the atomic binding force in Bi₂O₂Se is reduced by Cl doping, due to the weakened covalency of chemical bonds [55], which will be beneficial to improve the toughness and processing performance of Bi₂O₂Se.

3.3.5. Lattice Thermal Conductivity ${\mathsf{\kappa }}_{\mathrm{l}}$ of Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125

The Slack equation [56] can be used to calculate the lattice thermal conductivity ${\mathsf{\kappa }}_{\mathrm{l}}$:

$${\mathsf{\kappa }}_{\mathsf{\iota }}=\mathrm{A}·\frac{\overline{\mathrm{M}}{\mathsf{\Theta }}^3\mathsf{\delta }}{{\mathsf{\gamma }}^2{\mathrm{Tn}}^{\frac{2}{3}}}.$$

(16)

Here, A is a constant, and the calculation formula is given by:

$$\mathrm{A}=2.43\times \frac{{10}^{-6}}{1-\frac{0.514}{\mathsf{\gamma }}+\frac{0.228}{{\mathsf{\gamma }}^2}}.$$

(17)

In this equation, $\overline{\mathrm{M}}$ represents the average atomic mass, ${\mathsf{\delta }}^3$ is the volume of each atom, γ is the Grüneisen parameter, Θ is the Debye temperature, and T is the absolute temperature. The Slack model has been demonstrated to provide lattice thermal conductivity results that agree well with experimental measurements, and has been extensively used in the calculation of lattice thermal conductivity of materials [30,57]. Figure 5 shows the temperature-dependent results of ${\mathsf{\kappa }}_{\mathrm{l}}$. It is observed that the lattice thermal conductivity of both Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 decreases with increasing temperature. For Bi₂O₂Se, the calculated ${\mathsf{\kappa }}_{\mathrm{l}}$ at 300 K is 1.68 W/mK, which agrees well with the experimental data of ${\mathsf{\kappa }}_{\mathrm{l}}$ = 1.8 W/mK and other calculated results of ${\mathsf{\kappa }}_{\mathrm{l}}$ = 1.2 W/mK [31]. Furthermore, Cl doping reduces the lattice thermal conductivity of Bi₂O₂Se. For example, the calculated lattice thermal conductivity of Bi₂O₂Se_0.875Cl_0.125 at 300 K is 1.29 W/mK, which is 23.21% smaller than that of Bi₂O₂Se (1.68 W/mK). This is mainly due to the low elastic constants of Bi₂O₂Se_0.875Cl_0.125. The reduced lattice thermal conductivity will be beneficial for suppressing heat transfer and minimizing thermal energy loss. Bi₂O₂Se_0.875Cl_0.125 exhibits metallic character and electrons should also contribute to the thermal transport of Bi₂O₂Se. Considering that the Bi₂O₂Se is a semiconductor and its thermal conductivity is mainly contributed by phonons, and this work mainly focuses on the effect of Cl doping on the thermal transport properties of Bi₂O₂Se, the electronic thermal conductivity of Cl-doped Bi₂O₂Se, thus, is not considered in this work.

4. Conclusions

In this work, the effect of Cl doping on the structural, electronic, and mechanical properties of Bi₂O₂Se is investigated by using the first-principles calculations based on the DFT method. The main results are summarized as follows:

• The Cl doping results in an increase of 0.74% in the lattice constant a₀ (3.946 Å) of Bi₂O₂Se, while c₀ (12.287 Å) is decreased by 0.57% and the volume is expanded by 0.91%. At the same time, the calculated <Bi-O> bond lengths (2.339 Å) is increased by 0.09%, and the <Bi-Se> bond length (3.336 Å) is increased by 0.76%.
• The Cl doping has significant influences on the electronic structure of Bi₂O₂Se. Different from the semiconductor character of the pristine Bi₂O₂Se, the Bi₂O₂Se_0.875Cl_0.125 exhibits metallic properties, since a certain number of electrons are distributed at the Fermi level.
• As compared with the elastic constants of Bi₂O₂Se, the calculated elastic constants C₁₁ = 143.17 GPa, C₁₂ = 63.39 GPa, C₃₃ = 106.87 GPa, C₄₄ = 11.96 GPa and C₆₆ = 53.74 GPa for Bi₂O₂Se_0.875Cl_0.125 are reduced by 10.18%, 13.81%, 11.88%, 10.81%, and 6.88%, respectively, while the calculated value of C₁₃ (46.38 GPa) is increased by 4.84%. These results suggest that Cl doping has remarkable effects on the mechanical properties of Bi₂O₂Se. Both the doped and undoped Bi₂O₂Se satisfy the criteria of mechanical stability and exhibit elastic anisotropy.
• The Cl doping leads to a reduction of 7.40%, 11.46%, and 11.05% in the bulk modulus, shear modulus, and Young’s modulus of Bi₂O₂Se, respectively, suggesting that Cl doping leads to a more plastic deformation of Bi₂O₂Se, which could help to prevent cracking under extreme temperature gradients.
• The calculated B/G value of Bi₂O₂Se_0.875Cl_0.125 is 2.94, which is 4.63% higher than that of pure Bi₂O₂Se, indicating that Cl doping enhances the ductility of Bi₂O₂Se.
• The calculated Debye temperature θ of Bi₂O₂Se_0.875Cl_0.125 is 212.5 K, which is 11.6 K lower than that of pure Bi₂O₂Se (224.1 K). This lower Debye temperature helps to improve the toughness and processability of Bi₂O₂Se.
• The lattice thermal conductivity of both Bi₂O₂Se and Bi₂O₂Se_0.875Cl_0.125 decreases with increasing temperature. Cl doping reduces the lattice thermal conductivity of Bi₂O₂Se considerably. For example, the lattice thermal conductivity of Bi₂O₂Se_0.875Cl_0.125 at 300 K is 1.29 W/mK, which is 23.21% lower than that of pure Bi₂O₂Se (1.68 W/mK). The lower lattice thermal conductivity is beneficial in inhibiting heat transfer and minimizing thermal energy loss.

Generally, Cl doping leads to an improvement in the mechanical properties and a decrease in the lattice thermal conductivity of Bi₂O₂Se. Therefore, it is suggested that the Cl-doped Bi₂O₂Se can be used to develop TEG materials with good thermoelectric and mechanical properties.