3.1. Structural, electronic and phonon properties of ZrO2
At the first step of the study, the geometry of pure ZrO
2 phases was optimized using the VASP package. To find the optimal value of the cutoff energy (ENCUT) and the appropriate 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 convergence test for the number of k-points for the ZrO2 cubic phase are shown in
Figure 2, performed to build a k-point grid with a starting value of ENCUT = 1.3*ENMAX. Based on the results, it can be concluded that a 4x4x4 k-point grid with the Monkhrost-Pack scheme is optimally suited 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 the most accurate density of states (DOS).
Similar tests were carried out to establish the cutoff energy, from which it can be seen that the choice of ENCUT = 600 eV is suitable for modeling ZrO
2, and further increase in this energy only increases the calculation time without affecting its accuracy (
Figure 3). Therefore, all further calculations were carried out at 600 eV. Similar convergence tests have also been carried out for the tetragonal and monoclinic phase of ZrO
2 using the GGA potential, which are also consistent with the results for the cubic phase.
Table 2 compares the calculated values of the lattice parameters of the ZrO
2 phase with the results from the literature.
According to the results given in
Table 2, it can be seen that as the transition from a high-temperature phase to a lower-temperature phase, lattice distortion leads to a displacement of O ions in the c direction by a 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 2, the SCAN functionality describes the geometry much better than the standard GGA-PBE. Since the SCAN exchange-correlation functional describes the structural properties well, we decided to use this functional further when describing the geometry of other systems. The X-ray diffraction patterns of the initial structures obtained by us after their final relaxing are compared with the data from the literature, from which it can be seen that the results obtained by us are in good agreement with the experimental data, except for an imperceptible difference in the position of the X-ray peaks depending on the Bregg angles for the tetragonal system (
Figure 4).
Table 3 compares the total energy calculated by the GGA method for the monoclinic, tetragonal, and cubic ZrO
2 phases, from which it can be seen that m-ZrO
2 is the most stable phase in terms of energy values compared to other phases, that is, the monoclinic ZrO
2 phase with space group P21/ c is the most stable at low temperatures.
Next, using the Phonopy code in the VASP package, we simulated the thermodynamic properties and phonon spectra of the ZrO
2 phase for a more detailed discussion of the structural stability of the ZrO
2 monoclinic phase.
Figure 5 shows the change in the entropy of the unit cells of the ZrO
2 phase as a function of temperature.
According to
Figure 5, as the transition from the monoclinic to the tetragonal and cubic phases, the entropy of these compounds decreases, which corresponds to the criterion of inverse dependence of enthalpy or direct dependence of the entropy and stability of solid systems [
28]. This pattern can also be clearly observed after analyzing the pattern of phonon frequencies of the three phases of ZrO
2 (
Figure 6 (a-c)) from which it is clearly seen that the monoclinic phase has the smallest negative modes than the other two phases.
Figure 7(a-c) shows the temperature dependence of free energy, entropy and heat capacity for m-ZrO
2, t-ZrO
2 and m-ZrO
2.
Рисунoк 7.
Temperature dependence of free energy, entropy and heat capacity for: m-ZrO2 (a), t-ZrO2 (b), and m-ZrO2 (c).
Рисунoк 7.
Temperature dependence of free energy, entropy and heat capacity for: m-ZrO2 (a), t-ZrO2 (b), and m-ZrO2 (c).
The results of calculations of the density of phonon states are shown in
Figure 7 (a-c) indicate that as the transition from the monoclinic to the tetragonal and cubic phases, the density of electronic states increases, and they also correspond well with the results given in
Figure 6 and confirm the monoclinic phase as the most stable among other ZrO
2 phases. This is also confirmed by the result of the Energy/Volume diagram shown in
Figure 8 and is in good agreement with literature data [
29]. Therefore, for further stabilization by doping with Y
2O
3, the choice of the monoclinic phase is appropriate.
Next, using the well-optimized structures of the three phases of ZrO
2, we studied the electronic properties of these systems. Using the GGA, SCAN functionals and the HSE06 hybrid functional, the band gap values of these systems were found (
Table 4), their orbital structure was analyzed, and the change in the position of the Fermi level in these systems was modeled.
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 [
31], which traditionally overcomes underestimation of the band gap well. Given the suitability of HSE06 for estimating the band gap energy, we further used this particular hybrid functional to describe all problems related to the electronic properties of the systems under study.
Next, using the ZrO
2 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 ZrO
2 and the transport characteristics of electronic devices based on it.
According to
Figure 9, the density of electronic states for c-ZrO
2 is slightly overestimated compared to other phases. Moreover, secondary energy gaps are observed in the energy diagram of the tetragonal and cubic phases. Also, this gap increases with the transition from the tetragonal to the cubic phase.
Рисунoк 10.
Conduction (red) - and valence (green)-band change for c-ZrO2, t-ZrO2, m-ZrO2. Position of the Fermi level corresponds to the maximum of the valence band at each of the sites.
Рисунoк 10.
Conduction (red) - and valence (green)-band change for c-ZrO2, t-ZrO2, m-ZrO2. Position of the Fermi level corresponds to the maximum of the valence band at each of the sites.
Next, we determined the position of the Fermi level in ZrO
2 crystals and the shift of this level during their phase transformation. As can be seen from
Figure 10, if the position of the Fermi level (maximum of the valence band) for the monoclinic phase is taken as the reference point, then during the m-t phase transformation of ZrO
2, this level first drops by 0.125 eV towards lower energies and then descends again in the t-c section by 0.08 eV. This is also observed in detail from the results of the summarized bands for the orbital analysis, which are shown in
Figure 11 for the three phases of ZrO
2.
It can be seen from
Figure 11 that as we move from the monoclinic to the tetragonal and cubic phase, 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 ZrO
2 during the phase transformation.
3.2. Stabilization of m-ZrO2 and electronic properties of YSZ
After the final preparation of YSZ - structures (according to the scheme proposed in formula 1 and table 1), geometric optimization was carried out and Y
2O
3 - doped ZrO
2 supercells were relaxed using the GGA and SCAN functionals.
Table 5 shows the geometric parameters of the ZrO
2 and YSZ supercells at various Y
2O
3 concentrations after relaxation using the SCAN functional.
According to the results shown in
Table 5, it can be seen that the doping of yttrium oxide, namely the concentration of more than 14 mol. %Y
2O
3 stabilizes the monoclinic phase of zirconia to a cubic phase. However, in this case, the lattice parameters of Y
2O
3 - doped structures change non-linearly.
Figure 12 shows the volume vibration of ZrO
2 lattices as a function of Y
2O
3 concentration (x).
According to
Figure 13, with increasing doping concentration, the energy difference between the monoclinic phase with the tetragonal and cubic decreases, which indicates the possibility of doping-induced phase transitions of zirconium dioxide in the indicated Y
2O
3 doping concentration ranges.
Having obtained the optimized structures, we calculated the energy of formation (E
f) and enthalpy energy of formation (ΔH) for ZrO
2 and YSZ, as well as the energy of formation of a vacancy (E
df) for YSZ using the following formulas:
where E
tot is the total energy of the system, E
tot (x) is the total energy of individual components, and δ is the number of vacancies (defects) in the crystal. The calculated values of E
f and E
df per atom are given in
Table 6.
On fig.
Figure 14 shows a diagram of the dependence of the change in the enthalpy of formation of YSZ on the concentration of Y
2O
3, calculated by formula 4, from which it is clearly seen that doping with Y
2O
3 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 ΔH = -1.0407x + 63.532, where x is the concentration of Y
2O
3 in YSZ.
Thus, as the concentration of Y
2O
3 increases, the number of oxygen vacancies in YSZ increases, and the growth of these O vacancies is considered as a stabilizing mechanism for the monoclinic phase of zirconia, as evidenced by a decrease in the enthalpy of formation. The numerical value of the enthalpy formation energy is shown in
Table 6.
Figure 15 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 carried out to study the electronic structure of Y
2O
3 stabilized ZrO
2 supercells to reveal in detail the effects of doping on their density of states, Fermi energy behavior, and orbital states.
Figure 16 shows plots of the change in the density of electronic states YSZ for all doping concentrations of Y
2O
3.
According to the results presented in
Figure 16, it can be seen that after Y
2O
3 doping in TDOS structures, no new energy states arise due to the introduction of defects, that is, it does not lead to a change, except for the band gap shift, which can be considered in detail after orbital analysis (
Figure 17) and estimates of the shift of the Fermi level (
Figure 18). The band gap is 4.71 eV, 4.92 eV, 4.75 eV, and 4.72 eV, respectively, for ZrO
2 with dopid 3.23, 6.67, 10.34, and 16.15 mol% Y
2O
3.
According to
Figure 18, after doping with 3.23 mol% Y
2O
3 in pure m-ZrO
2, the Fermi level drops by 0.067 eV and then shifts by 0.007 eV towards the conduction band when doped with 6.67 mol% Y
2O
3. Then, at a doping concentration of 10.34 mol%, it still rises by 0.01 eV, being 0.017 eV higher than in the case of 3.23 mol% Y
2O
3. However, after doping with 16.15 mol% Y
2O
3, it drops to 0.012 eV. According to the PDOS diagram, one can interpret and observe the step pattern of the conduction band with the contribution of s-, p- and d-orbitals. Understanding these features make it possible to tune the Fermi energies in the band structure for specific tasks in modern materials science and instrumentation. The results obtained will help to interpret some of the features of the electronic properties of ZrO2 and solid materials [
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47,48], and also complement the base of scientific work carried out in the field of using biocompatible zirconium dioxide crystals and ceramics for generating green energy. The data can be used in the design of moisture-to-electricity converters and the creation of solid oxide fuels. cells based on ZrO
2 .