The CuBS₂ compound belongs to the space group I-42d and possesses a non-centrosymmetric tetragonal structure. The experimental lattice constants of this compound, as referenced in [4], are a = 5.044 Å and c = 8.947 Å, respectively. In this configuration, a represents the lattice constant along the x/y axes, and c represents the lattice constant along the z axis. As demonstrated in Fig. 2, the conventional unit cell of CuBS₂ comprises two molecular formulas. The Wyckoff positions of Cu, B, and S are denoted by 4a (0, 0, 0), 4b (0, 0, 0.5) and 8d (0.308, 0.25, 0.125), respectively.

Using the experimental lattice parameters of CuBS₂ [4] as initial values, geometric optimization was carried out, and the process was completed when the convergence condition was met. The calculated lattice constants a and c are 5.013 and 9.067 Å, respectively, and have deviations of 0.6% and 1.3% compared to the experimental values, which is a favorable agreement by DFT standards. The calculated lattice parameters, along with other experimental and theoretical values from the literature for comparative analysis, are listed in Table 1. Using the optimized crystal structure of CuBS₂, we can further explore the elastic, lattice dynamic, and thermodynamic properties of the material.

The elastic tensor of a solid material is important in two respects. Firstly, it characterises the material’s response to applied stress. Secondly, it provides a criterion for assessing structural stability. In consideration of the structural symmetry of the material, the elastic stiffness tensor of the chalcopyrite-structured compound CuBS₂ is comprised of six independent components, denoted by C₁₁, C₃₃, C₄₄, C₆₆, C₁₂, and C₁₃, according to Young’s notation. It is evident that, as of the present moment, there is an absence of experimental data concerning the elastic constants of the CuBS₂ crystal. The six independent elastic constants of CuBS₂ have been obtained via first-principles calculations and are listed in Table 2. It is evident that the elastic constants of CuBS₂ exhibit substantial anisotropy, which is attributed to the disparity in bond strength between the Cu-S and B-S chemical bonds. For the chalcopyrite structure, the Born stability criteria for the CuBS₂ crystal are given by C₁₁, C₃₃, C₄₄, C₆₆ > 0, C 11 > C 12 , C 11 C 33 > C 13 2 and ( C 11 + C 12 ) C 33 > 2 C 13 2 . It is evident that the calculated elastic constant values satisfy the mechanical stability requirements for chalcopyrite crystals [19], indicating that CuBS₂ is mechanically stable.

The isotropic elastic moduli of the polycrystalline aggregate were calculated from the anisotropic single-crystal elastic stiffness tensor using the Voigt and Reuss approximations [20,21]. (1) B V = 1 9 2 C 11 + C 33 + 2 C 12 + 4 C 13 (2) B R = C 11 + C 12 C 33 − 2 C 13 2 C 11 + C 12 + 2 C 33 − 4 C 13 (3) G V = 1 15 ( 2 C 11 + C 33 − C 12 − 2 C 13 + 6 C 44 + 3 C 66 ) (4) G R = 15 / ( 8 s 11 + 4 s 33 − 4 s 12 − 8 s 13 + 6 s 44 + 3 s 66 ) where s = C⁻¹ is the elastic compliance matrix.

Since the Voigt and Reuss isotropic average values of the bulk modulus and shear modulus represent the upper and lower bounds, respectively. Hill [22] proposed that the bulk modulus and shear modulus of polycrystalline materials should be approximated as the arithmetic mean of these two values, that is:

The Young’s modulus can be expressed as follows: E = (9B_HG_H)/(3B_H + G_H). After substituting the obtained values of B_H and G_H into the formula, we can yield that the Young’s modulus is 151.48 GPa. The Poisson’s ratio ν is then determined by the formula: ν = (3B_H − 2G_H)/(6B_H + 2G_H). Similarly, the Poisson’s ratio ν of this material can be obtained as 0.27. According to reference [23], it can be concluded that this material is an ionic material. According to Pugh’s rule [24], when the B/G ratio exceeds 1.75, the material exhibits ductility; conversely, when the B/G ratio is below 1.75, it becomes brittle. Calculations yield a B/G ratio of 1.87 for CuBS₂, indicating this material possesses excellent ductility. This characteristic has an indirect connection to Young’s modulus and elastic modulus anisotropy. A comparison was made between the elastic parameters B_H, G_H, E, and ν of the CuBS₂ crystal and those of the chalcopyrite crystal CuInS₂ (82.4 GPa, 33.8 GPa, 89.2 GPa, and 0.32, respectively) [25]. It can be observed that CuBS₂ demonstrates a higher degree of resistance to deformation.

The anisotropy of the crystalline material can be directly observed by displaying the Young’s modulus (E), the linear compressibility (β), the shear modulus (G), and the Poisson ratio (v) of the two- and three-dimensional graphs. Using the ELATE code [26], a free program, we may create 2D and 3D graphs of different elastic parameters based on the computed elastic constant matrix. Elastic isotropy is indicated by circular and spherical plots, however, the greater the disparity between 2D graphs and circles, or 3D graphs and spheres, the more pronounced the elastic anisotropy becomes. As illustrated in Fig. 3 and Fig. 4, the elastic parameters E, β, G, and ν of CuBS₂ crystals with a tetragonal crystal system exhibit anisotropic behaviour when viewed from both three-dimensional and two-dimensional perspectives (xy-, xz-, and yz-planes). The minimum and maximum values of the parameters are denoted by the green and blue curves, respectively. A thorough examination of the two-dimensional and three-dimensional images of the elastic parameters of CuBS₂ crystals was conducted, resulting in the discovery that the degree of anisotropy for each elastic parameter increases in the following order: β < E < G < ν.

To quantitatively analyze the elastic anisotropy of CuBS₂ semiconductor, we employed the formula from reference [27] to calculate the elastic anisotropy index (A^U, A_B, A_G) and the shear anisotropy factors (A₁, A₂, and A₃) for the <100>, <010>, and <001> crystal planes of CuBS₂,

Because CuBS₂ semiconductor is tetragonal symmetry, hence, C₁₁ = C₂₂. The results are presented in Table 2. Here, A^U represents the overall anisotropy degree of the material, while A_B and A_G denote the percentage of compressive anisotropy and shear anisotropy, respectively. The calculated elastic anisotropy index A^U, A_B, and A_G, along with the shear anisotropy factors A₁, A₂, and A₃ of CuBS₂ are all given in Table 3. As is widely known, If the elastic anisotropy index satisfies A^U = A_B = A_G = 0, this means the material is isotropic. The elastic anisotropy indexes shown in Table 3 are all non-zero, indicating that CuBS₂ has a strong degree of anisotropy.

It is found that A_B is much smaller than A_G, indicating that the compressive anisotropy is very small, and the anisotropy of the material mainly originates from the shear modulus. For a material in tetragonal symmetry, the shear anisotropy factors A₁ = A₂. From Table 3, A₁, A₂ and A₃, all are greater than 1. This indicates that on these crystal planes, the shear resistance is low, making sliding easy, and the material exhibits ductility.

The phonon spectrum serves as a bridge that links the microscopic atomic vibrations of a material to its macroscopic properties. By using the finite-displacement method based on DFT, we obtained the phonon spectrum along high-symmetry points in the Brillouin zone of CuBS₂. Fig. 5 and Fig. 6 illustrate the phonon band structure and phonon density of states of CuBS₂, respectively. Fig. 5 shows that there are no imaginary frequencies throughout the entire Brillouin zone, indicating that the tetragonal CuBS₂ is dynamically stable. Two phonon band-gaps are present across the entire vibrational spectrum, located in the frequency ranges of 120–160 cm⁻¹ and 416–533 cm⁻¹, respectively. One can see that, the phonon spectrum reveals no distinct bandgap between the acoustic vibration branch (black line) and the low-frequency optical vibration branch (red line), in fact, the two branches even partially overlap, indicating that energy exchange between them is readily achievable. As shown in Fig. 6, the high-frequency vibrational peaks (>533 cm⁻¹) are primarily contributed by B atoms. The low-frequency acoustic and optical branches primarily originate from the vibrations of Cu atoms, while the mid-frequency optical branch is mainly due to the contribution of S atoms, with some contributions from B and Cu atoms.

According to the lattice dynamics theory, if the primitive cell of a material contains n atoms, its vibrational spectrum contains 3n vibrational branches. The CuBS₂ semiconductor material has 8 atoms in the primitive cell. The vibrational spectrum of this crystal contains 24 vibrational modes, of which 3 branches are acoustic modes and the remaining 21 branches are optical vibrational modes. At the wave vector K = 0, the irreducible representation of these 24 vibrational modes is classified as: Γ = 1A₁ + 2A₂ + 3B₁ + 4B₂ + 7E. The three acoustic modes are B₂ and E, which have a frequency of 0 at the Γ point. Among the remaining 21 optical modes, there are 9 infrared-excited modes (3B₂ and 6E) and 19 Raman-excited modes (A₁, 3B₁, 3B₂, and 6E). It can be seen that 9 of these modes, namely 3B₂ and 6E, are simultaneously Raman and infrared excited. The results of our calculations for the optical vibration branches, along with the theoretical values from other researchers, are listed in Table 3. As shown in Table 3, the frequency variation patterns of the optical vibration modes calculated at the Γ point in the Brillouin zone are consistent with those reported in reference [9], and each mode exhibits slightly lower frequencies, this phenomenon may be due to the temperature-dependent volume change was not considered in the calculations.

Based on the calculated results of the phonon density of states, we employed the quasi-harmonic Debye model to compute the Debye temperature Θ_D of CuBS₂, along with the temperature dependence of the free energy, enthalpy, entropy, and heat capacity, within a temperature range of 0–1000 K. Fig. 7 depicts the Debye temperature curve of the CuBS₂ crystal calculated in this study. It can be noted that in the low-temperature region (0–100 K), the Debye temperature increases sharply. After surpassing 100 K, the increase in the Debye temperature tends to level off. In the high-temperature region, the Debye temperature stabilizes around 760 K. Unfortunately, there are no experimental values for the Debye temperature of CuBS₂ crystal. For comparison, we examine the Debye temperature Θ_D = 284.9 K [26] of chalcopyrite crystal CuInS₂, it is large than the Debye temperature Θ_D of CuBS₂ in the 0 K.

As illustrated in Fig. 8, the free energy of the CuBS₂ crystal material exhibits a monotonic decrease with increasing temperature, while the enthalpy demonstrates a monotonic increase with rising temperature. Entropy, which defines the degree of disorder in a system, increases with rising temperature due to the enhanced lattice vibrations in the CuBS₂ material. The trend in the product of temperature and entropy (TS), as illustrated in Fig. 8, further substantiates this conclusion.

As illustrated in Fig. 9, under low-temperature conditions, the specific heat capacity (Cv) of the CuBS₂ crystal exhibits a nearly proportional relationship with the cube of the temperature (T), thereby aligning with the Debye (T³) law. Within the temperature range of 200 K to 500 K, the growth rate of specific heat capacity gradually decelerates. It can be found that specific heat capacity converges to a constant value at elevated temperatures, which aligns with the predictions of the classical Dulong–Petit law. This phenomenon indicates the reasonableness and reliability of the computational results presented in this study.