1. Introduction

2. Modeling details

3. Results and discussion

3.1. Structural stability and electron-phonon properties of ZrO₂.

At the first stage of modeling, the structural-energy relaxation of pure ZrO₂ phases was carried out using the VASP package. To find the optimal cutoff energy for the ENCUT plane wave basis functions and the corresponding number of k-points in the Brillouin zone, we tested the convergence of the total unit cell energy as a function of ENCUT and KPOINTS.

The results of the k-point convergence test for ZrO₂ cubic phases are shown in Figure 3. 2 and performed to build a grid from k-point data with an initial value of ENCUT = 1.3*ENMAX. Based on the results obtained, it can be concluded that for a 4x4x4 k-point grid with the Monkhrost-Pack scheme, it is optimal for the geometric relaxation of ZrO2. However, when calculating the electronic structure of these compounds, the number of k-points was at least doubled in order to obtain a better density of states (DOS).

Figure 2. The total energy of the c-ZrO₂ unit cell as a function of the number of k-points under the condition ENCUT=1.3*ENMAX.

Figure 2. The total energy of the c-ZrO₂ unit cell as a function of the number of k-points under the condition ENCUT=1.3*ENMAX.

Similar tests were carried out to establish the cutoff energy, which shows that the choice of 600 eV is suitable for calculations, and a further increase in this energy increases the cost of the calculation without affecting its accuracy (Figure 3). Therefore, all further calculations were carried out at ENCUT = 600 eV.

Figure 3. Total energy of the c-ZrO₂ unit cell as a function of the cutoff energy (4 × 4 × 4 k-points).

Figure 3. Total energy of the c-ZrO₂ unit cell as a function of the cutoff energy (4 × 4 × 4 k-points).

Similar convergence tests were also carried out for the tetragonal and monoclinic phases of ZrO₂ using the GGA potential. In Table 1 compares the calculated values of the crystal lattice constants of the ZrO₂ phase, obtained from two exchange-correlation potentials, with the literature results.

According to the results given in Table 1, it can be seen that during the transition from the high-temperature phase to the low-temperature phase, the lattice distortion leads to a displacement of O ions in the c direction by the value of dz, expressed in relative units. As a result of distortion in the tetragonal phase, all Zr-O bonds will become nonequivalent. According to Table 3, the SCAN functionality describes the geometry much better than the standard GGA-PBE. However, it is also seen from the available data that GGA and SCAN almost identically describe the energy difference between the monoclinic and tetragonal phases of ZrO₂. Since the SCAN exchange-correlation functional describes the structural properties well, we decided to use this functional in the future when describing the geometry of other systems.

Table 2 compares the total energies calculated by the GGA method for systems of the monoclinic, tetragonal, and cubic phases of ZrO₂. It can be seen that among all systems, the most stable phase with the lowest energy is m-ZrO₂, that is, in fact, in terms of the field energy at low temperatures, the stable phase is monoclinic with the space group P21/c.

Table 2. GGA- calculated total electronic energies of c-ZrO₂, t-ZrO_2, m-ZrO₂.

Table 2. GGA- calculated total electronic energies of c-ZrO₂, t-ZrO_2, m-ZrO₂.

System	Energy	ΔE
m-ZrO2	-115.179	0
t-ZrO2	-114.754	0.425
c-ZrO2	-114.346	0.833

Table 3. Calculated and experimental band gap of c-ZrO₂, t-ZrO_2, m-ZrO₂ in eV.

Table 3. Calculated and experimental band gap of c-ZrO₂, t-ZrO_2, m-ZrO₂ in eV.

System	This work			Experiment [55]
	GGA	SCAN	HSE06	VUV
m-ZrO2	3.9	3.8	5.288	5.78
t-ZrO2	4.42	4.37	5.898	5.83
c-ZrO2	4.03	3.93	5.140	6.10

Further, using the Phonopy code in the VASP package, the thermodynamic properties and phonon spectra of the ZrO₂ phase were calculated for a more detailed discussion and substantiation of the structural stability of the ZrO₂ monoclinic phase. Figure 4 shows the change in the entropy of the unit cells of the ZrO₂ phase as a function of temperature.

According to Figure 4, as the transition from monoclinic to tetragonal and cubic phases, the entropy of these compounds decreases, which corresponds to the criterion of inverse dependence of enthalpy or direct dependence of entropy and stability of solid systems [53]. Thus, the monoclinic phase is the most stable with the highest entropy among other ZrO₂ phases. This pattern can be clearly observed after analyzing the pattern of phonon frequencies of the three phases of ZrO₂ (Figure 5a–c), from which it is clearly seen that the monoclinic phase has the smallest negative modes than the other two phases.

Figure 6(a–c) show the temperature dependence of the free energy, entropy, and heat capacity of a 12-atom supercell for m-ZrO₂, t-ZrO₂, and m-ZrO₂.

The results of calculations of the density of phonon states presented in Figure 7(a–c) indicate that as the transition from monoclinic to tetragonal and cubic phases, the density of electronic states increases, and they also agree well with the results shown in Figure 5 and confirm that the monoclinic phase is the most stable among the other phases of ZrO₂. This is also confirmed by the result of the Energy/Volume diagram presented in Figure 8. Therefore, for further stabilization by doping with Y₂O₃, it is reasonable to choose a monoclinic phase.

Next, using the well-optimized structures of the three phases of ZrO₂, we performed calculations to study their electronic properties. Using the GGA and SCAN functionals and the HSE06 hybrid functional, we found the band gaps of these systems (Table 3), analyzed their orbital structure, and modeled the change in the position of the Fermi level in these systems.

According to the results presented in Table 3, the GGA and SCAN functionals showed a rather small band gap compared to the HSE06 hybrid functional [56], which makes it possible to overcome the underestimation of the band gap. On the other hand, it is obvious that the standard SCAN and GGA functionals greatly underestimate the band gap. Given the suitability of HSE06 for estimating the band gap energy, we further used this hybrid functional to describe all the problems associated with the electronic properties of the systems under study.

Next, for ZrO₂ structures relaxed using the SCAN functional, calculations were made of the density of available electronic states at the Fermi level (Figure 9), which is crucial for interpreting the electronic properties of ZrO₂ and the transport characteristics of electronic devices based on this.

According to Figure 9, the density of electronic states for c-ZrO₂ is somewhat overestimated compared to other phases. In addition, secondary energy gaps are observed in the energy diagram of the tetragonal and cubic phases. Also, this gap increases during the transition from the tetragonal to the cubic phase.

Next, the position of the Fermi level in ZrO₂ crystals and the shift of this level during their phase transformation were determined. As can be seen from Figure 10, if we take the position of the Fermi level (maximum of the valence band) for the monoclinic phase as a reference point, then during the m-t phase transformation of ZrO₂, this level first shifts by 0.125 eV towards higher energies (towards the valence band), and then , in the t-c section, decrease by 0.08 eV. This is also observed in detail from the band stacking results for the orbital analysis, which are shown in Figure 11 for the three phases of ZrO₂.

It can be seen that as the transition from the monoclinic to the tetragonal and cubic phases, the contribution of the p orbitals becomes more significant in CB, and the s orbitals make a small contribution, while the d state shows a different trend. It is assumed that this behavior may be associated with a change in the crystal field and covalence of ZrO₂ during the phase transformation..

3.2. Structural and energy properties of Y₂O₃ doped with m-ZrO₂. Electronic properties of YSZ.

Next, supercells with a size of 2x2x2 of 96 atoms were created to simulate the effect of Y₂O₃ on the stability and electronic properties of the most stable (monoclinic) modification of ZrO₂. To dope yttrium, it was necessary to replace some formula units of Y₂O₃ with ZrO₂ in a 2x2x2 supercell, with each replacement creating one oxygen vacancy. A schematic description of the generation of YSZ structures is given below:

$${xZrO}_2+k{Y}_2{O}_3\to {Zr}_x{Y}_{2k}{O}_{2x+3k}+V_{o_k}$$

$$\%Y_2{O}_3=\frac{k}{x+k}\times 100\%,$$

which can be considered as the union of x ZrO₂ formula units with k Y₂O₃ formula units located on the initial lattice of x + k ZrO₂ units, which leads to the formation of m oxygen defects. Based on this, we determine the percentage of vacancies equal to the percentage of yttrium units in the final structure. Thus, starting with a pure 96-atom ZrO₂ supercell, we mainly focused on 4 different concentrations of Y₂O₃ in our calculations (Table 4).

After the final preparation of the YSZ structures, geometric optimization and doping relaxation of the Y₂O₃ supercell were performed using the GGA and SCAN potentials. Figure 12 shows a diagram of the dependence of the change in the enthalpy of formation of YSZ on the concentration of Y₂O₃, calculated by Formula 4:

$$\Delta H=\frac{E_{YSZ}-[x{E}_{Zr{O}_2}+k{E}_{Y_2{O}_3}]}{x+k}$$

(4)

from which it is clearly seen that doping with Y₂O₃ reduces the enthalpy and leads to the stabilization of zirconium dioxide. The empirical formula obtained by the least squares method says that the enthalpy of formation energy decreases linearly according to the law ΔН = -1.0407x + 63.532, where x is the concentration of Y₂O₃ in YSZ.

Thus, with an increase in the Y₂O₃ concentration, the number of oxygen vacancies in YSZ increases, and the growth of these O vacancies is considered as a stabilizing mechanism of the monoclinic zirconium phase, as evidenced by a decrease in the enthalpy of formation.

Numerical values of the enthalpy formation energy are given in Table 6. Table 5 shows the geometric parameters of the ZrO₂ and YSZ supercells at various Y₂O₃ concentrations after thorough relaxation using the SCAN functional.

Table 5. Lattice parameters of 2x2x2 ZrO₂ and YSZ supercells at various Y₂O₃ concentrations.

Table 5. Lattice parameters of 2x2x2 ZrO₂ and YSZ supercells at various Y₂O₃ concentrations.

System	Lattice parameters						Structure
	a (Å)	b(Å)	c (Å)	α(◦)	β (◦)	γ(◦)
0	10.382	10.491	10.757	90	99.64	90.00	m - YSZ
3.23 mol. %Y2O3	10.274	10.524	10.536	90.21	98.84	89.94	m - YSZ
6.67 mol. %Y2O3	10.512	10.544	10.603	89.90	90.12	89.62	t - YSZ
10.35 mol. %Y2O3	10.529	10.541	10.546	89.98	90.09	90.08	t - YSZ
16.15 mol. %Y2O3	10.540	10.541	10.543	90.08	90.00	90.02	c - YSZ

After obtaining the optimized structures, the energy of formation (Ef) for ZrO₂ and YSZ and the energy of formation of vacancies (Ev) for YSZ were calculated as:

$$E_f=E_{tot}-{\sum }_x{E}_{tot}\left(x\right)$$

$$E_{df}=E_{tot}^{{Zr}_{32-x}{Y}_x{O}_{64-\delta }}-E_{tot}^{{Zr}_{32}{O}_{64}}+\delta *E_{tot}^O$$

where $E_{tot}$- total energy of the system, $E_{tot}\left(x\right)$ – total energy of individual components, δ is the number of vacancies (defects) in the crystal. The calculated values of $E_f$and $E_{df}$for each atom are given in Table 6.

Table 6. GGA-calculated value of enthalpy (ΔН) and energy of formation ($E_f$) for ZrO₂ and YSZ. Oxygen vacancy formation energy ($E_{df}$) for YSZ.

Table 6. GGA-calculated value of enthalpy (ΔН) and energy of formation ($E_f$) for ZrO₂ and YSZ. Oxygen vacancy formation energy ($E_{df}$) for YSZ.

System	ΔН	${\mathit{E}}_{\mathit{f}}$	${\mathit{E}}_{\mathit{d}\mathit{f}}$
0	64.02917222	-4.747216667	0
3.23 mol. %Y2O3	59.91124404	-4.848422632	-1.874577368
6.67 mol. %Y2O3	56.13271879	-4.967857447	-3.739875532
10.35 mol. %Y2O3	52.7041267	-5.106527419	-5.596013441
16.15 mol. %Y2O3	47.00229139	-5.384704945	-9.220196154

Figure 13 shows the nature of the change in $E_f$and $E_{df}$ from the concentration of yttrium oxide, from which the regularity of their linear decrease is clearly visible.

Next, calculations were performed to study the electronic structure of Y₂O₃-stabilized ZrO₂ supercells to reveal the effect of doping on the density of states, the behavior of the Fermi energy, and the orbital components. Figure 14 shows plots of changes in the density of electronic states YSZ for all doping concentrations of Y₂O₃.

According to the results presented in Figure 14, it can be noted that after doping with Y₂O₃, new energy states do not appear in the TDOS patterns due to the introduction of defects, that is, there are no noticeable changes, except for a decrease in the band gap , which can be understood in detail after orbital analysis (Figure 16) and Fermi level mixing estimates (Figure 15). The band gap is 4.71 eV, 4.92 eV, 4.75 eV, and 4.72 eV, respectively, for ZrO₂ doped with 3.23, 6.67, 10.34, and 16.15 mol. %Y₂O₃.

Figure 15. Conduction (red) - and valence (green)-band change for ZrO₂ doped with 3.23, 6.67, 10.34 и 16.15 mol. %Y₂O₃. The position of the Fermi level corresponds to the maximum of the valence band in each section.

Figure 15. Conduction (red) - and valence (green)-band change for ZrO₂ doped with 3.23, 6.67, 10.34 и 16.15 mol. %Y₂O₃. The position of the Fermi level corresponds to the maximum of the valence band in each section.

Figure 16. Composite PDOS diagram showing the main contributions of the s-, p- and d-orbitals to the states forming the CB bottom for ZrO₂ doped with 3.23, 6.67, 10.34 and 16.15 mol. %Y₂O₃. Top VB (green), scaled to zero.

Figure 16. Composite PDOS diagram showing the main contributions of the s-, p- and d-orbitals to the states forming the CB bottom for ZrO₂ doped with 3.23, 6.67, 10.34 and 16.15 mol. %Y₂O₃. Top VB (green), scaled to zero.

According to Figure 15, after doping with 3.23 mol. %Y₂O₃ into pure m-ZrO₂, the Fermi level drops by 0.067 eV and then shifts by 0.007 eV towards the conduction band upon doping with 6.67 mol. %Y2O3. Then, at a doping concentration of 10.34 mol. %Y₂O₃, it still increases by 0.01 eV, which is 0.017 eV more than in the case of 3.23 mol. %Y2O3. However, after doping with 16.15 mol. %Y₂O₃, it drops to 0.012 eV. The PDOS diagram also interprets the stepped conduction band pattern in terms of the contribution of the s, p, and d orbitals. Understanding these features makes it possible to tune the Fermi energies in the band structure to solve the most important problems of materials science and instrumentation.

The problems of studying the influence of doping of yttrium oxide on the properties and stability of tetragonal and cubic zirconia remain the subject of our future research.

4. Conclusions

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