Density Functional Theory Study of Pressure-Dependent Structural and Electronic Properties of Cubic Zirconium Dioxide

Abstract

In this study, the structural, electronic, and elastic properties of cubic zirconium dioxide (c-ZrO₂) were investigated using the Density Functional Theory (DFT) approach. Lattice parameter optimization revealed that the lattice constant is 5.107 Å, the Zr–O bond length is 2.21 Å, and the unit cell density is 6.075 g/cm³ for the B3LYP functional. The bandgap width was determined to be 5.1722 eV. The investigation of the elastic properties of the cubic ZrO₂ crystal determined the Young’s modulus, bulk modulus, Poisson’s ratio, and hardness, which were found to be 315.91 GPa, 241 GPa, 0.282, and 13 (Hv), respectively, under zero external pressure. These results confirm the mechanical stability of ZrO₂.

1. Introduction

Zirconium dioxide (ZrO₂) is a functional material with unique mechanical, thermal, dielectric, and electronic properties [1,2,3,4,5,6,7]. Due to these characteristics, it is widely used in industry, including in the production of electrochemical devices, oxygen sensors, photocatalytic systems, and optical instruments [8,9,10,11,12]. Furthermore, the high radiation resistance of ZrO₂ makes it an important material for nuclear technologies, including reactors, inert matrix fuel, radioactive waste disposal systems, and containers for storing radioactive substances [13,14,15,16,17,18].

At atmospheric pressure, ZrO₂ exists in three crystalline modifications: monoclinic (P2₁/c), which is stable at temperatures below 1150 °C, tetragonal (P4₂/nmc), existing in the range of 1150–2350 °C, and cubic (Fm3m, fluorite-like structure), which forms at temperatures above 2350 °C [19,20,21,22,23]. The transition between these phases is accompanied by changes in volumetric and mechanical properties, playing a crucial role in adjusting the characteristics of ZrO₂ [24,25,26]. However, upon cooling, the high-temperature phases of ZrO₂ undergo a martensitic transformation, accompanied by a volume change (~3–5%), which can lead to material cracking. To prevent this effect, ZrO₂ stabilization is carried out by introducing dopants of divalent, trivalent, and tetravalent oxides, such as CaO, MgO, Y₂O₃, La₂O₃, Yb₂O₃, and CeO₂ [27].

Depending on the stabilization conditions, partially stabilized zirconia (PSZ), which contains a mixture of cubic and tetragonal phases and is in demand for mechanical and structural applications, and fully stabilized zirconia (FSZ), which has a purely cubic structure and is used in oxygen sensors, heating elements, and fuel cells, can be distinguished [24].

ZrO₂-based materials possess high strength and resistance to external influences. For example, the electro-consolidation method enables the production of ZrO₂ composites with a high reproducibility of mechanical properties due to the precise control of sintering temperature and pressure. It has been established that the stability of elastic characteristics (Young’s modulus, hardness) makes such materials promising for aerospace technologies [28].

Additionally, composite materials based on ZrO₂, WC, and Al₂O₃, obtained using lamination and oscillatory pressing methods, exhibit high hardness (30.06 GPa) and fracture toughness (12.04 MPa·m¹/²), which is explained by crack growth inhibition and deflection effects [29].

Besides elastic properties, ZrO₂ exhibits interesting electronic characteristics under pressure. Studies by Wu et al. [30] have shown that phase transitions in ZrO₂ are accompanied by changes in bandgap width, opening the possibility of tuning its electronic properties. It has been found that at pressures of 17 GPa, the monoclinic phase transitions into the tetragonal phase, and at 44 GPa, this phase transitions into the cubic phase, which is due to dynamic lattice instability.

Hybrid DFT calculations (PBE0) by Eklund et al. [31] have revealed important structural trends confirming the high technological potential of ZrO₂. Additionally, it has been established that in the cotunnite phase of ZrO₂, high stiffness (bulk modulus of 254 GPa) is realized, which is associated with the dense atomic packing and ninefold coordination of Zr [32].

Despite the widespread use of zirconium dioxide (ZrO₂), its behavior under high pressure—particularly regarding the elastic and electronic characteristics of the cubic phase—has not been sufficiently studied. This is especially true for the pressure range of 40–80 GPa, where phase stability, mechanical robustness, and electronic tunability are of considerable technological interest. While previous DFT investigations have largely focused on basic structural properties or bandgap calculations, they often overlook the mechanical response under pressure, including stress–strain behavior, which is challenging to measure experimentally, especially in the low-pressure regime.

In this study, we present a comprehensive first-principles analysis of the structural, elastic, and electronic properties of undoped cubic ZrO₂ under hydrostatic pressures up to 80 GPa using the hybrid B3LYP functional. A detailed evaluation of mechanical parameters—such as Young’s modulus, Poisson’s ratio, bulk modulus, and hardness—is performed, along with an investigation of brittleness and plasticity indicators, allowing us to characterize the material’s mechanical performance under extreme conditions.

A key novelty of this work is the application of stress tensor analysis to detect the onset of non-linear mechanical behavior in cubic zirconia. Specifically, we identify a critical deviation from linear elasticity at pressures of approximately 2.4–2.6 GPa, which we interpret as a theoretical yield point—analogous to the transition from elastic to pre-yield regimes commonly observed in mechanical testing. To the best of our knowledge, this is the first density functional theory (DFT)-based quantification of such a transition for undoped c-ZrO₂. Furthermore, we conduct a high-resolution pressure-dependent study (1 GPa increments) covering the range of 1–24 GPa, with particular attention to the less-explored low-pressure regime. This approach enables the detection of subtle anomalies in the elastic and electronic properties that are typically overlooked in studies with coarser pressure increments. Combined with the accurate reproduction of the experimental band gap (~5.8 eV), these results demonstrate the robustness of our computational framework and offer new insights into the intrinsic mechanical and electronic response of c-ZrO₂ under compression. This work thus advances the understanding of the material’s behavior in technologically relevant pressure regimes and complements existing experimental studies on the strength and deformation characteristics of zirconia-based ceramics.

The findings of this study have significant implications for the design and application of high-performance ZrO₂-based materials in various fields. These include aerospace components that demand high strength and thermal stability, nuclear materials requiring radiation resistance and mechanical reliability, as well as electronic and optoelectronic devices where bandgap tunability under pressure is a desirable feature. Thus, the present work not only deepens our fundamental understanding of pressure-induced effects in cubic ZrO₂ but also broadens its applicability in advanced functional materials.

2. Materials and Methods

The development of functional materials with desired properties requires either enhancing or suppressing certain characteristics of the original material. Achieving such results is impossible without modern experimental and theoretical approaches. Therefore, it is essential to develop computational models that accurately describe the electronic properties of materials and analyze their structural characteristics. This allows for the precise control and tuning of functional material properties.

One such material, cubic ZrO₂ (c-ZrO₂), is a binary inorganic compound composed of oxygen and zirconium. It crystallizes in a face-centered cubic lattice belonging to the Fm3m space group, where oxygen atoms occupy tetrahedral positions. The elementary rhombohedral unit cell has an edge length of 3.61 Å with angles α = β = γ = 60° and consists of three atoms with the following crystallographic coordinates: Zr (0, 0, 0), O₁ (0.25, 0.25, 0.25), and O₂ (0.75, 0.75, 0.75). The Zr–O bond lengths are equal, measuring 2.21 Å. The calculated lattice parameter is a = 5.10 Å [33], while the experimental bandgap is reported as 6.1 eV [34]. This lattice parameter was used as the initial value for structural optimization.

In this study, we employed density functional theory (DFT) as implemented in the CRYSTAL23 code [35], which utilizes Kohn–Sham Hamiltonians. DFT calculations serve as a powerful research tool in the design and optimization of new materials. This work presents a comparative study of the structural, electronic, and elastic properties of ZrO₂, using various functionals, including GGA-based PBE [36], PBEsol [37], HSE06 [36], B3LYP [38], B3PW [39], PWGGA [40], and Meta-GGA (SCAN) [41]. All calculations were performed on a Lenovo ThinkStation P620 workstation equipped with an AMD Ryzen™ Threadripper™ PRO 5995WX processor (Manufactured by Lenovo, China, purchased from Server IT LLP, which ordered through the distribution company Lenovo Kazakhstan).

To accurately describe the structural and electronic properties of c-ZrO₂ under hydrostatic pressure, we performed full structural relaxation, optimizing both atomic positions and lattice parameters. The full geometric optimization approach ensured precise lattice configurations. The convergence criteria were set as follows: an energy threshold of 10⁻⁶ eV, a gradient threshold of 4.5·10⁻⁴ eV/Å, and a maximum atomic displacement of 1.8·10⁻³ Å. The final optimized lattice parameters were a = b = c = 10.2589 Å, α = β = γ = 90°. These optimized parameters serve as a solid foundation for further investigations into the mechanical and electronic properties of c-ZrO₂. To ensure reproducibility and transparency, we provide the original optimization files.

All quantum-chemical calculations were performed using the CRYSTAL23 software package (CRYSTAL23 for Unix/Linux, version v1.0.1; license purchased on 20 December 2023 with an unlimited term of use) and the hybrid density functional method B3LYP, which is widely used in computational materials science [42,43,44,45,46,47,48,49,50,51]. Hybrid functionals represent an approximate method for calculating system energy by combining Hartree–Fock exchange energy with exchange–correlation energy derived from density functional approaches. This hybrid methodology improves the accuracy of parameters that standard DFT functionals often struggle to describe accurately.

In this case, the exchange energy is expressed through Kohn–Sham orbitals rather than electron density, which is why such functionals are referred to as implicit. The application of solutions obtained using the Hartree–Fock method improves the accuracy of calculated parameters that are poorly described by ab initio functionals. Typically, such functionals are constructed as a linear combination of the exchange functional and a set of exchange–correlation functionals operating on electron density. The most-used functional of this type is B3LYP (the three-parameter Becke, Lee–Yang–Parr functional). It is defined by the following expression [52,53]:

$${\mathrm{E}}_{\mathrm{X}\mathrm{C}}^{\mathrm{B}3\mathrm{L}\mathrm{Y}\mathrm{P}}={\mathrm{E}}_{\mathrm{X}\mathrm{C}}^{\mathrm{L}\mathrm{D}\mathrm{A}}+\mathrm{a}\left({\mathrm{E}}_{\mathrm{X}}^{\mathrm{H}\mathrm{F}}-{\mathrm{E}}_{\mathrm{X}}^{\mathrm{L}\mathrm{D}\mathrm{A}}\right)+\mathrm{b}\left({\mathrm{E}}_{\mathrm{X}}^{\mathrm{G}\mathrm{G}\mathrm{A}}-{\mathrm{E}}_{\mathrm{X}}^{\mathrm{L}\mathrm{D}\mathrm{A}}\right)+\mathrm{c}\left({\mathrm{E}}_{\mathrm{C}}^{\mathrm{G}\mathrm{G}\mathrm{A}}-{\mathrm{E}}_{\mathrm{C}}^{\mathrm{L}\mathrm{D}\mathrm{A}}\right)$$

(1)

where a = 0.2, b = 0.72, c = 0.81, B3—Becke functional, and LYP—Lee–Yang–Parr correlation functional.

In further calculations, the second-order elastic constants and bulk moduli for polycrystalline samples with optimized geometry [54] were obtained using an automated procedure in CRYSTAL.

The elastic properties of the material, such as the bulk modulus (B), shear modulus (G), Young’s modulus (Y), their ratio (B/G), and Poisson’s ratio (ν), can be estimated using the Voigt–Reuss–Hill (VRH) approximations. The universal elastic anisotropy index (AU) [53], a semi-empirical formula for assessing mechanical hardness (Hv) [54], and the intrinsic resistance to deformation were also considered. The formulas for these parameters are presented in the literature [54,55,56,57].

To study the structural properties of c-ZrO₂, a supercell consisting of 96 atoms was used.

3. Results

3.1. Structure of C-ZrO₂

Lattice parameter optimization was performed, and the volume, density, and valence bond lengths between atoms in the unit cell were determined. Based on the results obtained from the geometric optimization calculations, a 2 × 2 × 2 supercell was created for the c-ZrO₂ structure, consisting of 96 atoms, i.e., 32 Zr atoms and 64 O atoms.

The structure of cubic zirconium dioxide is shown in Figure 1. It was observed that the deviation of the lattice parameters is less than 2% from the reference value of 5.1 Å, depending on the functional used (

Table 1. Supercell lattice parameters of c-ZrO₂.

Parameters	Functional
c-ZrO2	B3LYP	B3PW	PBE	PBE SOL	LDA	HSE06	SCAN	Others	Experiment
a = b = c	5.107	5.075	5.104	5.056	5.020	5.067	5.076	5.11 [59] 3.58 [33]	5.11 [60] 3.6012 [61]
V, Ả3	33.31	32.68	33.24	32.32	31.63	32.53	32.69	32.5850 [62]	32.723 [61]
ρ, g/cm3	6.075	6.192	6.088	6.263	6.398	6.220	6.190	6.081 [63]	6.151 [63]

). The obtained results were compared with known experimental and theoretical studies [33,58,59,60,61,62,63,64].

After optimization, the smallest density error was observed with the PBE functional (6.088 g/cm³), with an absolute deviation of 0.007. The minimum error in the Zr–O valence bond length was –0.02 Å, which was calculated using the B3PW (2.22 Å) and SCAN (2.2 Å) functionals.

Thus, the geometric structure calculations demonstrated that the results obtained using the DFT method differ only slightly from experimental data [52].

3.2. Band Structure of C-ZrO₂

DFT calculations were used to determine the bandgap width corresponding to the equilibrium lattice constant, as well as the electronic structure and density of states (DOS). The calculated bandgap values obtained within the DFT framework using exchange–correlation potentials from the LDA and GGA approximations were compared with experimental and computational data from other authors [36,53,65].

The bandgap width of c-ZrO₂ was found to have the following values depending on the standard hybrid functionals used (

Table 2. Band gap of c-ZrO₂.

Parameters	Functionals
c-ZrO2	B3LYP	B3PW	PBE	PBEsol	LDA	HSE06	SCAN	PWGGA
Eg, eV (this work)	5.1722	5.2065	3.3195	3.3530	3.3953	5.0088	3.9392	3.2911
Experimental [34]	6.10
LDA [66]	3.349
GGA [67]	3.329

). According to our calculations, the most accurate functionals were the three-parameter Becke functional combined with nonlocal correlation PWGGA (B3PW) and the three-parameter Becke functional combined with nonlocal correlation LYP (B3LYP) [52]. The bandgap values obtained using B3LYP (5.17 eV), B3PW (5.2 eV), and HSE06 (5.0 eV) functionals were found to be the closest to the experimental value [50].

In quantum physics, the energy band parameter is used to describe electronic states, characterizing the distribution of allowed electron energy levels within a crystal. Figure 2 presents the calculated band structure of c-ZrO₂ within the first Brillouin zone, indicating high-symmetry points (Γ, L, W, X, Γ) and directions. Due to the symmetry of the Brillouin zone, an energy bandgap is formed, with a width of 5.17 eV for c-ZrO₂. This confirms that c-ZrO₂ is a dielectric with a direct bandgap along the same symmetry point (Γ-Γ).

It is well known that the density of states in an energy band corresponds to the number of allowed electron states per energy interval. Figure 3 shows the total electronic density of states for ZrO₂, where the dashed line represents the Fermi level.

The upper part of the valence band is predominantly formed by the 2p electrons of oxygen atoms. The 2p electrons of oxygen correspond to the energy range of 0 to –5 eV. In the energy range of 0 to –5 eV in the valence band of zirconium atoms, the partial density of electronic states, which is partially associated with 4d electrons, is significantly lower than that of oxygen atoms (see Figure 3).

The conduction band of c-ZrO₂ is primarily formed by the electronic states of zirconium atoms, with only a minor contribution from the oxygen atoms. From the partial density of states of zirconium atoms, in the energy range of 5.6 to 11 eV, the density of states is associated with unoccupied 4d orbitals of zirconium.

The bandgap width calculated using the B3LYP functional is 5.17 eV for both the primitive cell and the supercell. These results are in good agreement with the findings of other authors [1] on the electronic structure and elastic properties of ZrO₂.

3.3. Elastic Properties of C-ZrO₂

Elastic constants provide essential information about the mechanical properties of materials and their structural stability. These properties play a key role in assessing the strength of a material and determining its behavior under applied pressure [66].

Cubic symmetry is characterized by three independent elastic stiffness constants (ESC), namely C₁₁, C₁₂, and C₄₄. According to Hooke’s law, at sufficiently small stresses, the deformation is proportional to the applied stress magnitude [56,57,67]:

$${\mathsf{\sigma }}_{\mathrm{i}}={\mathrm{C}}_{\mathrm{i}\mathrm{j}}{\mathsf{\epsilon }}_{\mathrm{j}}$$

(2)

where i, j = 1,2,3…6.

C_ij These constants represent elastic stiffness coefficients, where σᵢ denotes the applied stress and εⱼ corresponds to the resulting strain.

Our calculated values of the elastic constants, bulk modulus, Young’s modulus, and shear modulus for c-ZrO₂ confirm its mechanical stability under hydrostatic pressure ranging from 0 to 80 GPa. The obtained values at zero pressure are in good agreement with results derived from first-principles calculations [59,66,67,68].

The relationship C₁₁ > C₁₂ > C₄₄ indicates that c-ZrO₂ exhibits the highest resistance to linear compression under hydrostatic pressure. Additionally, C₄₄ and G define the intrinsic hardness of the structure under pressure. Both parameters show a consistent trend over the 0–80 GPa pressure range, as illustrated in Figure 4a,b. The increase in G and C₄₄ values suggests enhanced hardness and resistance to shear deformation under pressure.

Furthermore, as shown in Figure 4b, C₁₁ and C₁₂ increase linearly with pressure compared to C₄₄, indicating their greater sensitivity to high-pressure conditions. Figure 4a demonstrates that B exhibits a linear dependence on applied pressure, increasing from 241.01 GPa at 0 GPa to 547.36 GPa at 80 GPa, confirming its greater hardness and ability to withstand volumetric deformation and failure under high pressure.

Young’s modulus (Y) characterizes the stiffness of the material and the ratio of tensile stress to strain under pressure [23]. Figure 4a illustrates that Y increases from 315.91 GPa at 0 GPa to 490.23 GPa at 80 GPa, indicating higher stiffness and enhanced resistance to tensile deformation at elevated pressures.

The calculated Cauchy constants are positive and smoothly increase within the pressure range from 6.5 GPa (at 0 GPa) to 221 GPa (at 80 GPa) (Figure 5a). The ductility of c-ZrO₂ is confirmed by the calculated Cauchy difference (C₁₂–C₄₄) [69,70]. Positive values indicate plasticity, while negative values suggest brittle behavior.

As shown in Figure 5b, the B/G ratio increases from 1.95 (at 0 GPa) to 3.01 (at 80 GPa), confirming the plasticity of c-ZrO₂ according to Pugh’s criterion [68]. If a material has B/G < 1.75, it behaves as a brittle material, whereas B/G > 1.75 corresponds to plastic materials.

Poisson’s ratio (ν) is 0.282 at 0 GPa and increases to 0.351 at 80 GPa (Figure 6a), further supporting the plasticity of the crystalline structure [69]. Poisson’s ratio values of 0.25 and higher indicate an ionic character of bonding between atoms [70]. Our calculations suggest a significant ionic contribution to the intra-atomic bonding of c-ZrO₂ up to 80 GPa.

Our calculations showed that the anisotropy index (AU) at zero pressure is 1.4, which is 0.8 lower than the experimental value of 2.21 [71]. A high AU value can lead to reduced elasticity and an increased number of microcracks and microdefects in the material [72].

As pressure increases, the anisotropy of c-ZrO₂ decreases from 1.4 to 0.88 at 80 GPa, as shown in Figure 6b. This suggests that the elastic behavior of c-ZrO₂ at high pressure implies greater resilience and fewer structural defects and cracks.

Mechanical hardness plays a crucial role among elastic parameters when evaluating the intrinsic strength of a material. Materials are typically classified into three categories: hard, superhard, and ultrahard, with hardness values exceeding 10 GPa, 40 GPa, and 90 GPa, respectively [55].

To demonstrate the effect of pressure on the hardness of cubic ZrO₂, we present the calculated Hv values in Figure 6b.

For c-ZrO₂, the Hv graph (Figure 7a) shows irregular variations under high pressure. The initial hardness is 13 GPa at zero pressure, with a maximum hardness observed in the range of 5 to 10 GPa. After this, Hv decreases to 9.95 GPa at 40 GPa. The intrinsic hardness of c-ZrO₂ decreases from 13 GPa (0 GPa) to 10.3 GPa (80 GPa) under high pressure, classifying c-ZrO₂ as a hard material under compression. This indicates that c-ZrO₂ exhibits high resistance to compression under pressure.

The calculated bandgap width at zero pressure is in good agreement with the experimental value of 6.1 eV [51].

Contrary to expectations, crystal compression does not significantly increase electronic excitations, as shown in Figure 7b. The bandgap width of c-ZrO₂ increases with pressure, from 5.17 eV (0 GPa) to 5.98 eV (80 GPa), confirming its wide bandgap (WBG) behavior under high pressure. These results are in good agreement with experimental data [62]. Figure 8 illustrates that c-ZrO₂ compresses under applied pressure. The lattice parameter (a) decreases from 5.129 Å (0 GPa) to 4.787 Å (80 GPa) proportionally to the increase in pressure.

The values obtained at zero pressure align well with the results reported by Zhang et al. [67] and are significantly lower than the theoretical predictions by Muhammad et al. [70]. However, the experimental data from Kandil et al. [71], except for Poisson’s ratio, show significant discrepancies compared to our results.

A comprehensive analysis indicates that c-ZrO₂ becomes more resistant to compression under high pressures. The improvements in elasticity and hardness under pressure suggest their potential for applications in high-pressure environments.

These findings provide valuable insights into the mechanical behavior of c-ZrO₂, which may be crucial for various engineering and materials science applications.

3.4. Hidden Anomalies in the Elastic and Electronic Properties of C-ZrO₂ Under Low Pressure

During electronic and elastic structure calculations of cubic zirconia (c-ZrO₂) in the pressure range of 1–24 GPa with a step of 1 GPa, a set of pressure-dependent physical parameters was obtained, allowing a comprehensive assessment of the material’s behavior under external compression. The calculations showed that the lattice parameter (Figure 9a) decreases linearly with increasing pressure, indicating the uniform compression of the crystal structure without signs of a phase transition. The elastic constants C₁₁, C₁₂, and C₄₄ (Figure 9b), as well as the Young’s modulus Y, shear modulus G, and bulk modulus B (Figure 10a), also exhibit a monotonic increase, confirming the mechanical stability of the structure throughout the studied range. Moreover, the difference C₁₂−C₄₄ (Figure 10b) increases linearly, indicating a steady rise in stiffness against directional deformation.

The Poisson’s ratio (Figure 11a) also increases with pressure, but a slight deviation from the smooth trend is observed in the 12–14 GPa range. Similar features are found in the B/G ratio (Figure 11b) and the universal elastic anisotropy A^U (Figure 12a), where subtle bends in the curves are also evident within this pressure interval. These deviations may be interpreted as signs of hidden structural or electronic rearrangements. The most pronounced anomaly appears in the bandgap plot E_g (Figure 12b), where a sharp increase of approximately 0.08 eV is observed between 12 and 14 GPa—clearly deviating from the otherwise gradual trend. This may indicate a reconstruction of the band structure associated with enhanced overlap between Zr 4d and O 2p orbitals, a redistribution of the electronic density of states, and a possible transition from an indirect to a direct bandgap character.

Further evidence of these deviations is supported by the non-linearity observed in the stress–strain diagram at stress levels of 2400–2600 MPa (Figure 13), which, according to the literature [73,74,75], may be attributed to local changes in electronic density, the accumulation of internal defects, and stress redistribution while maintaining the overall cubic symmetry of the crystal.

Thus, despite the apparent linearity of the main elastic parameters, more detailed analysis revealed subtle signs of electronic instability and potential structural rearrangements not previously reported in DFT studies of c-ZrO₂.

4. Discussion and Conclusions

The study of c-ZrO₂ using DFT provided detailed insights into its structural, electronic, and elastic properties. Lattice parameter optimization revealed that the lattice constant of c-ZrO₂ is 5.107 Å, and the Zr–O bond length is 2.21 Å, demonstrating good agreement with known experimental values. This confirms the effectiveness of the B3LYP functional in accurately describing the structural properties of c-ZrO₂.

A comprehensive DFT-based analysis of cubic zirconium dioxide (c-ZrO₂) was conducted to evaluate its structural, electronic, and elastic properties under hydrostatic pressure up to 80 GPa. Bandgap calculations showed that the hybrid functionals B3LYP and B3PW yielded values of 5.17 eV and 5.2 eV, respectively, closely matching the experimental value of 6.1 eV and confirming the accuracy of the chosen computational methods. Elastic property analysis indicated significant mechanical stability, with a bulk modulus (B) of 241 GPa, Young’s modulus (Y) of 315.91 GPa, and Poisson’s ratio (ν) of 0.282, suggesting good elasticity. The calculated hardness (Hv) of 13 GPa classifies c-ZrO₂ as a hard material, and the consistent increase in elastic constants under pressure confirms its enhanced stiffness and applicability under extreme mechanical loads.

Importantly, special attention was given to the low-pressure range (1–24 GPa) with high-resolution analysis (1 GPa step), allowing the detection of fine irregularities that are often missed in studies using larger pressure intervals. For the first time, a noticeable change in the bandgap behavior was observed around 12–14 GPa—a sharp increase of approximately 0.08 eV—suggesting a pressure-driven rearrangement of the electronic structure without a phase transition. This effect was accompanied by small shifts in Poisson’s ratio, the B/G ratio, and the elastic anisotropy factor A^U, indicating minor changes in interatomic bonding. Furthermore, stress–strain analysis revealed a deviation from linearity in the stress tensor at ~2.4–2.6 GPa, which can be interpreted as the beginning of non-linear mechanical behavior or a theoretical yield point. These findings provide a new understanding of how the electronic and elastic properties of undoped c-ZrO₂ respond to pressure and reveal features not previously reported in the stable cubic phase.

Overall, this study not only deepens the fundamental understanding of pressure effects on c-ZrO₂ but also expands its potential for practical use in advanced functional materials. The results confirm its suitability for applications in electrochemical devices, oxygen sensors, high-strength ceramics, and components for nuclear and aerospace technologies. Future research may focus on the influence of various dopants and defect structures on the material’s electronic and mechanical behavior, further broadening its application scope.

In addition, considering recent DFT-based investigations [76] on toxic gas adsorption over oxide surfaces [77,78,79,80,81], the mechanical robustness of c-ZrO₂ demonstrated in this study may also support its role as a stable platform for gas sensing and environmental remediation technologies. The combination of structural integrity under pressure and potential surface reactivity highlights its multifunctionality for emerging applications.