Oxygen vacancies in cubic zirconia result in a calculated displacement pattern as shown in Fig. 9 . In this figure, the vacancy is depicted as a small cube, the oxygen
Fig. 8 Charge density in the plane through ZrA, OA, and O., and a schematic diagram of a neutral oxygen interstitial (О,) near a triple-bonded oxygen (OA) in zirconia. Charge density is in 0.1 eV A-1 and all distances are in A  (reprinted with permission)
Fig. 9 Displacement pattern of atoms around an isolated vacancy in a 95 atom supercell  (reprinted with permission)
atoms occupy the sites at the corners of the cubes, and metal cations occupy half of the sites at the center of the cubes. The six oxygen neighbors nearest to the vacancy move along <100> by 0.024 nm, while the zirconium atoms move outward along <111> by 0.018 nm. The oxygen atoms nearest to the zirconium, but not nearest to the vacancy, follow the displacement of the cation and move outwards along <111> by 0.004 nm. The oxygen atoms in the outermost right corner of the figure move inwards by 0.004 nm along <111>.
Tetragonal zirconia contains anion vacancies and may be written as ZrO2_x, with x varying from 0.001 at 1,925°C to 0.052 at 2,410°C . To accommodate these vacancies, surrounding ions move toward the vacancy to reduce its size. The two zirconium ions move by an approximate amount of 0.008 nm and the oxygen ions by 0.013 nm. The energy gain due to the relaxation of these ions is 0.22 eV. These values are, of course, different depending on the charge of the vacancy .
For the case of a singly-charged vacancy, the structural distortion results in the movement of surrounding oxygen ions by an approximate amount of 0.022 nm toward the vacancy and the zirconium ions by 0.009 nm away from the vacancy. These values are modified for the case of a doubly-charged vacancy to 0.033 nm for the oxygen ions and 0.022nm for the zirconium ions. Obviously, the higher the positive charge of the vacancy, the greater the distortion towards or away from it. The energy gains due to the formation of singly – and doubly-charged vacancies are 1.0 and 3.3 eV, respectively.
Oxygen vacancies in monoclinic zirconia can occur in both the triple-planar and tetragonal geometries. When the vacancy is neutral, these vacancies have formation energies of 8.88 eV and 8.90 eV, respectively. Once the vacancy is singly charged positively (i. e., V+) and in a tetrahedral position, the atomic relaxation energy is 0.47 eV. Creation of a doubly-charged positive vacancy (i. e., V2+) in a tetrahedral position causes further displacement of the four surrounding zirconium ions away from the vacancy by an additional 0.01 nm. This leads to a further decrease in energy of 0.74 eV. Creation of a singly-charged negative vacancy (i. e., V-) in the same tetrahedral position causes minimal displacement of the surrounding zirconium ions (by less than 0.002nm) and an energy decrease that is less than 0.1 eV, which clearly points to the fact that the additional electron is only weakly localized in the vicinity of the vacancy and, hence, has little influence on the surrounding ions. The lattice relaxation and formation energies in the case of a neutral zirconium vacancy are about 1.4 and 24.2 eV, respectively. The oxygen ions surrounding this type of vacancy are displaced outwards from their equilibrium positions by about 0.01-0.02 nm.
At higher temperatures (i. e., 1,000°C) and excess partial pressure of oxygen (i. e., 10-6to 1 atm.), monoclinic zirconia contains completely ionized zirconium vacancies . At 1,000°C, zirconia is stoichiometric at a pressure of 10-16 atm. At this point, the concentration of oxygen vacancies is equal to twice the concentration of zirconium vacancies. As the partial pressure of oxygen increases, the stoichiometry changes such that for ZrO^ with the d value defined by:
d = 6 x 10-3 Vx0, (2)
where po2 is the oxygen partial pressure in atm.