A point defect model for YBa₂Cu₃O₇ from density functional theory

Defect energies for use in the point defect model were calculated using DFT implemented in the CASTEP simulation package [23, 24]. Simulations employed the Generalised Gradient Approximation (GGA) exchange correlation functional of Perdew et al [25]. Atoms were represented using a plane wave basis and ultrasoft pseudopotentials generated using the on-the-fly pseudopotential generator in CASTEP (version 17). The electrons considered as valence and hence treated explicitly in the simulations are presented in table 1. The plane wave expansion was truncated at a cutoff energy of 600 eV.

To demonstrate the efficacy of the DFT model the Brilloiun Zone (BZ) was sampled using a Monkhorst−Pack grid of 8 × 8 × 4 k-points in the unitcell and the lattice parameters and elastic constants were compared with available experimental data. Table 2 demonstrates the efficacy of the DFT model to reproduce the fundamental properties of the YBa₂Cu₃O₇ lattice.

The electronic density of states (DOS) for the reference YBa₂Cu₃O₇ unitcell is presented in figure 2. The total DOS is in excellent agreement with previous works that utilise local or semi-local exchange correlation functionals [28–30]. Previous comparisons between the band structures obtained from the local density approximation (LDA) and ARPES measurements indicates that even relatively simple functionals can accurately reproduce the electronic structure observed experimentally [31]. As a consequence, the PBE exchange correlation functional is considered appropriate for the current work. Also shown in figure 2 is the partial or projected DOS that shows that the region close to the Fermi level is dominated by oxygen 2p and Cu 3d states also in agreement with previous work by Wang et al [32]. Mulliken analysis of the ions indicates charges of 0.37∥e∥ and 0.34∥e∥ on the Cu1 and Cu2 respectively. It is noted that this relatively small difference may be due to limitations with localisation due to the GGA functional.

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It is essential that the DFT simulations accurately represent all of the subsystems of YBa₂Cu₃O₇ used in the construction of the point defect model. Therefore, the ability of the DFT model to reproduce the properties of these was also tested. Table 3 shows the lattice parameters for the various binary oxides and the metallic elements are in excellent agreement with the existing experimental data.

For the calculation of defect formation energies the simulation supercells were constructed by taking 4 × 4 × 1 repetitions of the unitcell resulting in a cell containing 208 atoms that is sufficiently large to ensure any finite size effects are negligible. For reasons of computational tractability and given the large number of defects considered here the k-point density was reduced such that the supercell is sampled using the gamma point, which corresponds to a sampling equivalent to 4 × 4 × 1 in the unitcell. The impact of the reduction in the BZ sampling density on the defect formation energies is found to be negligible. Defects are introduced into the cell by adding and removing atoms and performing energy minimisation under fixed volume conditions until the force on each atom converged to 0.01 eV Å⁻¹.

The concentration of any given defect, i, is proportional to the free energy required to form the defect, G_f ⁱ , as shown below:

$$\begin{eqnarray}&&\left[{c}_{i}\right]\propto {m}_{i}\exp \left(\displaystyle \frac{-{\rm{\Delta }}{G}_{f}^{i}}{{k}_{B}T}\right)\end{eqnarray} \tag{ 1 }$$

where m_i is the multiplicity of the defect (defined per formula unit), k_B is the Boltzmann constant and T is the temperature. At relatively low temperatures, the difference in vibrational contributions to the free energy between perfect and defect supercells can be safely neglected, so we can approximate ${\rm{\Delta }}{G}_{f}^{i}$ by ${\rm{\Delta }}{E}_{f}^{i}$, calculated from the total energies of the perfect and defective cells according to the formalism of Zhang and Northrup [39]. The defect formation energy is then given by:

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{f}={E}_{\mathrm{defect}}^{T}-{E}_{\mathrm{perf}}^{T}+\displaystyle \sum _{\alpha }{n}_{\alpha }{\mu }_{\alpha },\end{eqnarray} \tag{ 2 }$$

where, ${E}_{\mathrm{defect}}^{T}$ and ${E}_{\mathrm{perf}}^{T}$ are the DFT total energies of the system with and without the defect, n_α is the number of atoms added/removed of each atomic species α and μ_α is the chemical potential of each species. The chemical potential represents the availability of the different elements and can be directly related to the experimental conditions present during fabrication through the use of elementary thermodynamics. From a thermodynamic perspective we assume that YBa₂Cu₃O₇ can be fabricated from a series of binary oxides and oxygen, i.e.:

$$\begin{eqnarray}&&3{\mathrm{Cu}}_{2}{{\rm{O}}}_{\left(s\right)}+4{\mathrm{BaO}}_{\left(s\right)}+{{\rm{Y}}}_{2}{{{\rm{O}}}_{3}}_{\left(s\right)}+2{{{\rm{O}}}_{2}}_{\left(g\right)}\to 2{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{{{\rm{O}}}_{7}}_{\left(s\right)},\end{eqnarray} \tag{ 3 }$$

while noting this is not the traditional route for synthesis of YBa₂Cu₃O₇. Under any given conditions, the sum of the chemical potentials per formula unit of the constituent species must equal the total Gibbs free energy of the YBa₂Cu₃O₇ product, i.e.:

$$\begin{eqnarray}&&3{\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}\left({\rho }_{{O}_{2}},T\right)+4{\mu }_{\mathrm{BaO}}\left({\rho }_{{O}_{2}},T\right)+{\mu }_{{{\rm{Y}}}_{2}{{\rm{O}}}_{3}}\left({\rho }_{{O}_{2}},T\right)+2{\mu }_{{{\rm{O}}}_{2}}\left({\rho }_{{O}_{2}},T\right)=2{\mu }_{{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{{{\rm{O}}}_{7}}_{\left(s\right)}}\end{eqnarray} \tag{ 4 }$$

where, ${\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}\left({\rho }_{{O}_{2}},T\right)$, ${\mu }_{\mathrm{BaO}}\left({\rho }_{{O}_{2}},T\right)$, and ${\mu }_{{{\rm{Y}}}_{2}{{\rm{O}}}_{3}}\left({\rho }_{{O}_{2}},T\right)$ are the chemical potentials of the constituent binary oxides as a function of the oxygen partial pressure and the temperature, ${\mu }_{{{\rm{O}}}_{2}}\left({\rho }_{{O}_{2}},T\right)$ is the chemical potential for the oxygen molecule and ${\mu }_{{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{{{\rm{O}}}_{7}}_{\left(s\right)}}$ is the chemical potential for YBa₂Cu₃O₇. Note that under standard conditions we assume that ${\mu }_{{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{{\rm{O}}}_{7}}\left({\rho }_{{O}_{2}}^{\circ },{T}^{\circ }\right)={\mu }_{{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{{{\rm{O}}}_{7}}_{\left(s\right)}}$, where the chemical potential of the YBCO is taken from DFT calculations.

It should be noted that only neutral defects are considered. This is because the Fermi level in YBa₂Cu₃O₇ falls within the band ensuring that it is metallic. As a consequence, any electrons added or removed from the system will be added/removed from the electron reservoir which will remain essentially fixed (except at very high defect concentrations).

Under equilibrium conditions, the chemical potentials of the constituents of a crystal cannot exceed the Gibbs free energy of any phase made of the constituent elements. For example, the chemical potential of Cu₂O in YBa₂Cu₃O₇ cannot exceed that of solid Cu₂O, otherwise a Cu₂O precipitate would form. By considering a range of different competing phases, it becomes possible to define a region of the phase diagram where the host compound is stable that bounds the possible chemical potentials as illustrated by Huang and Johannes [40]. However, this approach assumes that the host compound is itself stable, which is not the case for YBa₂Cu₃O₇ as it decomposes according to [41]:

$$\begin{eqnarray}&&{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{{\rm{O}}}_{7}\to {\mathrm{BaCuO}}_{2}+{\mathrm{Ba}}_{2}{\left({\mathrm{Cu}}_{2}\right)}_{3}+{\mathrm{Ba}}_{2}{\rm{Y}}{\left({\mathrm{CuO}}_{2}\right)}_{4}+{{\rm{Y}}}_{2}{{\rm{O}}}_{3}.\end{eqnarray} \tag{ 5 }$$

As it is not possible to define a region where the YBa₂Cu₃O₇ is stable the approach adopted here is to set the upper bounds for the chemical potentials to that of the binary oxides as determined from DFT, i.e.

$$\begin{eqnarray}&&{\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}^{{\mathrm{Cu}}_{2}{\rm{O}}-\mathrm{rich}}={\mu }_{{\mathrm{Cu}}_{2}{{\rm{O}}}_{\left(s\right)}}^{{DFT}}\end{eqnarray} \tag{ 6 }$$

where, ${\mu }_{{\mathrm{Cu}}_{2}{{\rm{O}}}_{\left(s\right)}}^{{DFT}}$ is the energy of a formula unit of Cu₂O determined using DFT. To calculate the lower bound for the chemical potential for a constituent, we assume all other binary oxides must be at their upper bound and thus:

$$\begin{eqnarray}&&{\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}^{{\mathrm{Cu}}_{2}{\rm{O}}-\mathrm{poor}}=\displaystyle \frac{2{\mu }_{{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{{\rm{O}}}_{{7}_{\left(s\right)}}}^{{DFT}}-4{\mu }_{{\mathrm{BaO}}_{\left(s\right)}}^{{DFT}}-{\mu }_{{{\rm{Y}}}_{2}{{\rm{O}}}_{{3}_{\left(s\right)}}}^{{DFT}}-2{\mu }_{{{\rm{O}}}_{2}}\left({\rho }_{{O}_{2}},T\right)}{3}.\end{eqnarray} \tag{ 7 }$$

The values for the chemical potentials can fall anywhere in this range, therefore we define a fraction, f, for each binary oxide ensuring that the chemical potential becomes:

$$\begin{eqnarray}&&{\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}^{f}=f{\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}^{{\mathrm{Cu}}_{2}{\rm{O}}-\mathrm{rich}}\left(1-f\right){\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}^{{\mathrm{Cu}}_{2}{\rm{O}}-\mathrm{poor}}\end{eqnarray} \tag{ 8 }$$

Within this framework a further limitation is placed on the possible fractions, f, such that:

$$\begin{eqnarray}&&\displaystyle \sum _{j}{f}^{j}=n-1\end{eqnarray} \tag{ 9 }$$

where, n is the number of cations in the system, in order to ensure that the chemical potentials do not exceed that of YBa₂Cu₃O₇. This approach enables the exploration of the defect chemistry under conditions that contain either an excess or deficit of each of the cation species.

The chemical potentials for the binary oxides can be further decomposed into their constituents, for example for Cu₂O;

$$\begin{eqnarray}&&2{\mu }_{\mathrm{Cu}}\left({\rho }_{{O}_{2}},T\right)+\displaystyle \frac{1}{2}{\mu }_{{{\rm{O}}}_{2}}\left({\rho }_{{O}_{2}},T\right)={\mu }_{{\mathrm{Cu}}_{2}{\rm{O}}}\end{eqnarray} \tag{ 10 }$$

where, ${\mu }_{\mathrm{Cu}}\left({\rho }_{{O}_{2}},T\right)$ and ${\mu }_{{{\rm{O}}}_{2}}\left({\rho }_{{O}_{2}},T\right)$ are the chemical potentials of Cu and O in Cu₂O in equilibrium with O₂ gas. From equations (7) and (10) it is clear that the chemical potential for the oxygen molecule is necessary. The chemical potential for oxygen may be determined using DFT, however, there are known issues surrounding the calculation of small dimer molecules using semilocal functionals such as the GGA employed here. Therefore, we follow the approach of Finnis et al [42] and reference the known formation energy of the oxide under standard conditions, i.e.:

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{f}^{{\mathrm{Cu}}_{2}{\rm{O}}}\left({\rho }_{{{\rm{O}}}_{2}}^{\circ },{T}^{\circ }\right)={\mu }_{{\mathrm{Cu}}_{2}{{\rm{O}}}_{\left(s\right)}}^{{DFT}}-2{\mu }_{{\mathrm{Cu}}_{(s)}}^{{DFT}}-{\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}\left({\rho }_{{{\rm{O}}}_{2}}^{\circ },{T}^{\circ }\right).\end{eqnarray} \tag{ 11 }$$

where, ${\mu }_{{\mathrm{Cu}}_{(s)}}^{{DFT}}$ is the chemical potential of metallic copper determined using DFT. Oxygen chemical potentials at standard temperature and pressure as calculated from the three binary oxides all fall within 0.1 eV of each other, therefore, ${\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}\left({\rho }_{{{\rm{O}}}_{2}}^{\circ },{T}^{\circ }\right)$ is simply calculated as a weighted average of the contributions by the fractions of binary oxides present, i.e.:

$$\begin{eqnarray}&&{\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}\left({\rho }_{{{\rm{O}}}_{2}}^{\circ },{T}^{\circ }\right)=\displaystyle \frac{1}{2}\displaystyle \sum _{j}{f}^{j}{\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}^{j}\left({\rho }_{{{\rm{O}}}_{2}}^{\circ },{T}^{\circ }\right)\end{eqnarray} \tag{ 12 }$$

where the summation runs over all binary oxides, j.

The oxygen chemical potential under the desired conditions, ${\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}\left({\rho }_{{{\rm{O}}}_{2}},T\right)$, is extrapolated from ${\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}\left({\rho }_{{{\rm{O}}}_{2}}^{\circ },{T}^{\circ }\right)$ according to equation (13):

$$\begin{eqnarray}&&{\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}\left({p}_{{{\rm{O}}}_{2}},T\right)={\mu }_{\tfrac{1}{2}{{\rm{O}}}_{2(g)}}\left({\rho }_{{{\rm{O}}}_{2}}^{\circ },{T}^{\circ }\right)+{\rm{\Delta }}\mu (T)+\displaystyle \frac{1}{2}{k}_{B}T\mathrm{log}\left(\displaystyle \frac{{\rho }_{{{\rm{O}}}_{2}}}{{\rho }_{{{\rm{O}}}_{2}}^{\circ }}\right)\end{eqnarray} \tag{ 13 }$$

where Δμ(T) can be be determined from the real heat capacities for the O₂ molecule, i.e.:

$$\begin{eqnarray}&&{\rm{\Delta }}\mu (T)=\displaystyle \frac{1}{2}\left(G\left({\rho }^{0},T\right)-G\left({\rho }^{0},{T}^{0}\right)\right)\end{eqnarray} \tag{ 14 }$$

where,

$$\begin{eqnarray}&&G\left({p}^{0},T\right)=A\left(T-T\mathrm{ln}\left(T\right)\right)-\displaystyle \frac{1}{2}{{BT}}^{2}-\displaystyle \frac{1}{6}{{CT}}^{3}-\displaystyle \frac{1}{12}{{DT}}^{4}-\displaystyle \frac{E}{2T}+F-{GT}.\end{eqnarray} \tag{ 15 }$$

The parameters A-G are taken from Johnston et al [43].

The determination of the oxygen chemical potential enables determination of the chemical potentials of the binary oxides and YBa₂Cu₃O₇ as shown in table 4. These can then be used to define of a region of chemical potential space representing the availability of the individual cations. Figure 3 shows the relationship between μ_Cu and μ_Y at an oxygen partial pressure of 10⁻⁴ atm and a temperature of 700 K. Varying the fractions of each binary oxide f^j enables the examination of this region of the phase diagram.

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The defect formation energies for multicomponent compounds exhibit a complex dependency on the values chosen for the chemical potentials. When a chemical potential for a given specie is at its upper bound it will have the effect of minimising the formation energy of an interstitial defect for the corresponding specie and maximising the vacancy formation energy. Therefore, any formation energy presented will depend on the choice of reference chemical potential. In order to enable a simple discussion of the relative energies for the different defect sites and allow comparison with previous studies we begin our discussion by presenting the defect formation energies for all defects considered at both the rich and poor limits (referenced to their elemental states) in the reference YBa₂Cu₃O7. A discussion of the prevalence of the different defect species based on the conditions present during fabrication will be illustrated using the point defect model later.

4.1.1. Vacancy defects

4.1.2. Interstitial defects

Here we consider seven symmetrically distinct interstitial sites as illustrated in figure 4. The energies for the different interstitial defects are reported in table 6. For the oxygen interstitial defect the lowest energy interstitial site is in the yttrium layer (indicated as interstitial site 5 in figure 4). Other interstitial defects that lie close to interstitial site 5 (sites 6 and 7) are not stable and relax to give the same structure as the interstitial site 5, hence the almost identical formation energies. Surprisingly, this interstitial defect is lower in energy than the interstitial 2 site where the oxygen atoms are located in tetragonal YBa₂Cu₃O₈.

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Table 6. Defect formation energies for the intrinsic interstitial defects in YBa₂Cu₃O₇.

Site	O		Y		Ba		Cu
	Rich	Poor	Rich	Poor	Rich	Poor	Rich	Poor
1	2.14	5.48	−0.31	23.05	1.84	13.53	1.52	9.31
2	0.26	3.59	−1.94	21.43	0.55	12.23	2.60	10.39
3	2.27	3.61	−1.61	21.76	2.73	14.41	1.33	9.12
4	0.27	3.61	−1.31	22.05	0.55	12.23	1.81	9.60
5	0.11	3.45	−1.63	21.73	3.06	14.74	2.91	10.70
6	0.1	3.44	1.65	25.01	3.89	15.57	2.49	10.28
7	0.11	3.45	3.01	26.38	5.55	17.23	3.27	11.06

By contrast the interstitial 2 site is predicted to have the lowest formation energies for the large cations Y and Ba. This is likely due to this site minimising the interaction of these cations with the other Y and Ba atoms in the structure. This is exemplified by the high energies predicted for the large cation defects in the BaO and Y layers. For the Cu interstitial the lowest interstitial site is in the BaO layer (site 3 in figure 4). As was the case for the other cation species the interstitial sites close to the Y and Ba ions are higher in energy due to repulsive interactions with other cations.

4.1.3. Antisite defects

The defect formation energies presented above can be used to calculate the reaction energies for the intrinsic defect processes. For the calculation of the reaction energies the chemical potentials appear on both sides of the equation and therefore cancel, making interpretation unambiguous. The reaction energies for the intrinsic defect processes are presented in table 8. Table 8 shows that the lowest energy process is the oxygen Frenkel process, which correlates with the relatively high level of oxygen ion diffusion anticipated in YBa₂Cu₃O₇ even when the material is close to stoichiometric. By contrast, the cation Frenkel processes all have relatively high reaction energies which appears to agree well with previously very high activation energies predicted for Y and Ba diffusion in YBCO. Of particular note is the relatively low energy for swapping the Y and Ba cations. This implies the as fabricated material will likely exhibit high levels of these antisite defects. The reaction energy for the Schottky process is predicted to be very high at 13.26 eV, however, this corresponds to a normalised reaction energy of 1.02 eV per defect formed, making it the third lowest energy defect process in YBa₂Cu₃O₇.

The chemical formula for YBCO is frequently written YBa₂Cu₃O_7−x indicating that it can accommodate a significant degree of oxygen non-stoichiometry. This reduction of the oxygen stoichiometry is accommodated in the structure by oxygen vacancy defects. As the concentration of these defects increases the defects begin to interact and form clusters, therefore, the isolated oxygen vacancy defect will not offer the best description of the defect chemistry. Therefore, defect formation energies of pairs of oxygen vacancies in the Cu-O chains were calculated. Note that given the large number of possible vacancy cluster configurations we have ignored clusters in the CuO₂ planes as these defects have significantly higher formation energies.

Plotted in figure 5 is the defect formation energy per vacancy defect for the oxygen vacancy in YBa₂Cu₃O₇ as a function of the separation between vacant sites. The horizontal dashed line represents the formation energy of an isolated oxygen vacancy defect on the O4 site, which was shown to have the lowest formation energy above. Figure 5 indicates that there are a number of defect clusters that have lower formation energies per defect than the isolated vacancy on the O4 site. Importantly, it appears the lowest energy oxygen vacancy defect clusters occur in the CuO chains on the O1 site, which is in good agreement with existing experimental data. An example of one of these low energy clusters showing two vacancies on the O1 sites is illustrated in figure 6.

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The results indicated in figure 5 suggest that at low deviations from stoichiometry the oxygen vacancy defects will prefer to occupy the O4 site, however, as the oxygen substoichiometry increases there will be an increase of vacancies on the O1 site with some expected on the O4 site, in agreement with experiment. Figure 7 shows the binding energy between the vacancy defects as a function of their separation. The plot shows that there is a repulsive interaction between the vacancy defects at short separations. In general, the binding energy decreases as the separation increases, such that it is negligible at about 10 Å, however, there are a series of configurations with significant binding energies representing the most favourable cluster configurations.

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This final section will examine the defect chemistry of YBa₂Cu₃O₇ under different conditions. Initially, the Y₂O₃, BaO and Cu₂O-poor limits were considered. When the conditions are Y₂O₃-poor the defect formation energy for the Ba_Y antisite defect was found to be negative for all temperatures and partial pressures of interest. This indicates that these conditions would not be accessible in a real system. However, it is noted that YBa₂Cu₃O₇ is largely fabricated with excess Y₂O₃ to introduce nanoparticles that can act as flux pinning centres, thereby improving superconductivity [48]. Due to the combination of these two factors, the results for Y₂O₃-poor YBa₂Cu₃O₇ are not presented here.

4.4.1. Cu₂O-poor YBa₂Cu₃O₇

Defect concentrations for Cu₂O-poor YBa₂Cu₃O₇ are presented as a function of oxygen partial pressure at a temperature of 700 K in figure 8. Such plots are commonly referred to as Brouwer diagrams. In materials with a bandgap the gradient of the line indicates the specific reaction that ensures overall charge neutrality in the system, for example a ${{\rm{V}}}_{{\rm{O}}}^{2+}$ defect charge compensated for by 2e^1- would have a gradient -1/6. However, the defects considered here are charge neutral and there is, consequently, no requirement to ensure charge neutrality and therefore, there is no similar relationship to the defect charges. Note that for clarity all defects of the same type have been summed and plotted together (i.e. [V_O]=${\sum }_{i}{[{V}_{{\rm{O}}}}_{i}$]).

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Figure 8 shows that the defect chemistry is dominated by oxygen defects as would be expected. As the oxygen partial pressure decreases the concentration of oxygen vacancy defects increases rapidly resulting in a substantial increase in x in YBa₂Cu₃O_7−x . This overall trend is in good agreement with the experimental data of Yamaguchi et al [49]. Concomitant with an increase in the concentration of V_O defects is an increase in the number of these defects contained in clusters and consequently on the V_O4 site, also in agreement with experimental observations.

In addition to the oxygen vacancy defects there are a number of other important classes of defects in Cu₂O-poor YBa₂Cu₃O₇. Interestingly, there is a relative high concentration of oxygen interstitial defects present. The concentration of O_i defects increases as the oxygen partial pressure increases and is almost equal to the vacancy concentration under atmospheric pressure. Also evident are significant concentrations of the Y_Ba and Ba_Y antisite defects as predicted above. In the Cu₂O-poor limit the paucity of Cu is reflected in the high concentration of V_Cu and Y_Cu defects. The concentration of these defects is sensitive to the oxygen partial pressure with the V_Cu predicted to be of increasing importance at higher partial pressures while the concentration of Y_Cu defects shows a slight decrease. All of the remaining defect species are predicted to have concentrations below 10⁻⁸ per formula unit and therefore, these are unlikely to play a significant role in determining the macroscopic properties of the material.

As demonstrated by Yamaguchi et al the deviation in oxygen stoichiometry increases linearly as the oxygen partial pressure decreases, however, as x approaches unity defect-defect interactions and competition for lattice sites leads to the loss of this linear relationship. The results presented in figure 9 shows an almost to linear relationship between x and log$[{P}_{{{\rm{O}}}_{2}}]$, although the point defect model applied does not consider the defect-defect interactions and as a consequence is not applicable at the highest vacancy concentrations where such interactions would inevitably occur.

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4.4.2. BaO-poor YBa₂Cu₃O₇

Figure 10 shows the concentrations of the defects in BaO-poor YBa₂Cu₃O₇ as a function of the oxygen partial pressure at 700 K. The results show that once again the oxygen defects have very high concentrations, however, this time the Y_Ba antisite defect is dominant at the highest oxygen partial pressures. The reason for this is that the system is both poor in Ba and rich in Y. Although of much lesser importance the Cu_Ba defect is also present under these conditions.

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It should be noted that the dominant defects under BaO-poor conditions have high concentrations that are not best represented using the point defect model applied here and consequently the results in this limit should be treated with a degree of caution.