The mass changes of pure ZnO samples equilibrated in all studied thermodynamic conditions were very small, clearly indicating that the deviation from stoichiometry of zinc oxide is lower than 10^{−4}. As an example, the results of such calculations obtained at 900 °C are presented in Figure 2. These results are in qualitative agreement with data reported in literature [3, 13, 14].

Figure 2: Pressure dependence of deviation from stoichiometry in pure ZnO obtained at 900 °C.

On the other hand, the deviation from stoichiometry in chromium-doped zinc oxide is relatively high, as shown in Figures 3–5, which illustrate the deviation from stoichiometry in this material as a function of oxygen pressure at different temperatures.

Figure 3: Pressure dependence of deviation from stoichiometry in chromium-doped ZnO obtained at 700 °C.

Figure 4: Pressure dependence of deviation from stoichiometry in chromium-doped ZnO obtained at 800 °C.

Figure 5: Pressure dependence of deviation from stoichiometry in chromium-doped ZnO obtained at 900 °C.

These figures indicate that the deviation from stoichiometry increases with oxygen pressure at constant temperature, which strongly suggests that the predominant ionic defects in this material are cation vacancies and chromium-doped zinc oxide is a metal-deficient oxide (Zn_{1-y}O). This conclusion is the direct result of comparing the theoretical point defect situation illustrated in Figure 1 with the experimentally obtained results presented in Figures 3–5. Thus, the results of nonstoichiometry studies obtained in this work from studying chromium-doped zinc oxide can be used to determine point defect concentration in pure Zn_{1-y}O. From the analysis of experimentally obtained data of nonstoichiometry in chromium-doped zinc oxide, in terms of the theoretical diagram presented in Figure 1, it can be concluded that the concentration of zinc vacancies is much higher than that of zinc in interstitial positions. This denotes that the concentration of interstitial zinc can be neglected during analysis of the point defect situation in chromium-doped zinc oxide. Thus, concentrations of three types of point defect (i. e. zinc vacancies, electron holes and quasi-free electrons) can be calculated as a function of temperature and oxygen pressure from experimentally obtained results (Figures 3–5).

The formation of zinc vacancies as well as electron holes is presented by eq. (1). On the other hand, quasi-free electrons can be formed as a result of intrinsic ionization of electrons:
$\mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}\phantom{\rule{thinmathspace}{0ex}}\leftrightarrow \phantom{\rule{thinmathspace}{0ex}}{\mathrm{h}}^{\bullet}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{{\prime}^{}}$(7)

The application of the mass action law to predominant defect equilibriums described by eqs (1) and (7), together with the appropriate electroneutrality condition for chromium doped zinc oxide to form a Cr_{2}O_{3}-Zn_{1-y}O solid solution, lead to the following system of equations:
$\{\begin{array}{l}\text{K=}\left[{\text{V}}_{\u2033}\right]\cdot {\left[{\text{h}}^{\u2022}\right]}^{\text{2}}\cdot {\text{p}}_{{\text{O}}_{\text{2}}}^{{\text{-}}^{\text{1}}\text{/2}}\\ {\text{K}}_{\text{e}}=\left[{\text{h}}^{\u2022}\right]\cdot [\text{e}\prime ]\\ \left[{\text{Cr}}_{\text{Zn}}^{\u2022}\right]\text{+}\left[{\text{h}}^{\u2022}\right]\text{=}[\text{e}\prime ]\text{+2}\left[{\text{V}}_{\u2033}\right]\end{array}$(8)

where K and K_{e} denote constants of the reactions presented by eqs (1) and (7), square brackets denote concentrations of appropriate point defects.

In the above set of equations, only oxygen pressure and concentration of vacancies (experimental results presented in Figures 3–5) can be treated as known, while reaction constants as well as concentrations of electronic defects and chromium ions incorporated substitutionally into the cation sub-lattice of Zn_{1-y}O are unknowns and should be computed. Using simple algebra, this set of equations (eq. (8)) can be rearranged and reduced to the following equation:
$\begin{array}{l}\left[C{r}_{Zn}^{\u2022}\right]+{\text{K}}^{\text{1/2}}\cdot {\left[{\text{V}}_{\u2033}\right]}^{\text{-1/2}}.{\text{p}}_{{\text{O}}_{\text{2}}}^{\text{1/4}}\\ {\text{=K}}_{\text{e}}\cdot {\text{K}}^{\text{-1/2}}\cdot {\left[{\text{V}}_{\u2033}\right]}^{\text{1/2}}\cdot {\text{p}}_{{\text{O}}_{\text{2}}}^{\text{-1/4}}\text{+2}\left[{\text{V}}_{\u2033}\right]\end{array}$(9)

Next, eq. (9) can be approximated using experimental results and, consequently, at each three experimental points obtained at the same constant temperature: ($\mathrm{P}{\mathrm{o}}_{2}(1);\phantom{\rule{thinmathspace}{0ex}}\left[{\mathrm{V}}^{\prime \prime}{\mathrm{Z}\mathrm{n}}_{}\right]$(1)),

($\mathrm{P}{\mathrm{o}}_{2}(2);\phantom{\rule{thinmathspace}{0ex}}\left[\phantom{\rule{thinmathspace}{0ex}}{{\mathrm{V}}^{\prime \prime}}_{\mathrm{Z}\mathrm{n}}\phantom{\rule{thinmathspace}{0ex}}\right](2)$), ($\mathrm{P}{\mathrm{o}}_{2}(3);\phantom{\rule{thinmathspace}{0ex}}\left[\phantom{\rule{thinmathspace}{0ex}}{{\mathrm{V}}^{\prime \prime}}_{\mathrm{Z}\mathrm{n}}\phantom{\rule{thinmathspace}{0ex}}\right](3)$), all of the unknowns can be determined.

The constant of reaction (1) calculated via the previously described method is presented as a function of temperature by the following equation:
$\mathrm{K}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}7.07\cdot {10}^{-7}\cdot exp\left(-\frac{81\phantom{\rule{thinmathspace}{0ex}}\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}}{R\mathrm{T}}\right)$(10)

In the case of pure Zn_{1-y}O oxide, the electroneutrality condition assumes the following form:
$\phantom{\rule{thinmathspace}{0ex}}{\left[\phantom{\rule{thinmathspace}{0ex}}{\mathrm{h}}^{\bullet}\phantom{\rule{thinmathspace}{0ex}}\right]}_{}^{}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}2\phantom{\rule{thinmathspace}{0ex}}{\left[\phantom{\rule{thinmathspace}{0ex}}{{\mathrm{V}}^{\prime \prime}}_{\mathrm{Z}\mathrm{n}}\phantom{\rule{thinmathspace}{0ex}}\right]}_{}^{}$(11)

And, consequently, the concentration of predominant ionic and electronic defects is the following function of temperature and oxygen pressure:
$\begin{array}{rl}\left[\phantom{\rule{thinmathspace}{0ex}}{{\mathrm{V}}^{\prime \prime}}_{\mathrm{Z}\mathrm{n}}\phantom{\rule{thinmathspace}{0ex}}\right]\phantom{\rule{thinmathspace}{0ex}}& =\phantom{\rule{thinmathspace}{0ex}}\frac{1}{2}\left[\phantom{\rule{thinmathspace}{0ex}}{\mathrm{h}}^{\bullet}\phantom{\rule{thinmathspace}{0ex}}\right]\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}0.63\cdot \phantom{\rule{thinmathspace}{0ex}}{p}_{{O}_{2}}^{1/6}\phantom{\rule{thinmathspace}{0ex}}\cdot {\mathrm{K}}^{1/3}{\phantom{\rule{thinmathspace}{0ex}}}_{}^{}\phantom{\rule{thinmathspace}{0ex}}\\ & =\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}5.6\cdot {10}^{-3}\cdot \phantom{\rule{thinmathspace}{0ex}}{p}_{{O}_{2}}^{1/6}\phantom{\rule{thinmathspace}{0ex}}\cdot exp\left(-\frac{27\phantom{\rule{thinmathspace}{0ex}}\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}}{R\mathrm{T}}\right)\end{array}$(12)

These results are presented, once again, in Figure 6, which illustrates the concentration of double-ionized zinc vacancies in Zn_{1-y}O oxide.

Figure 6: Pressure dependence of the concentration of doubly ionized zinc vacancies in p-type ZnO for different temperatures, determined indirectly.

It should be noted that the data presented in Figure 6 were obtained under the assumption that the double-ionized zinc vacancies are the predominant ionic defects and thus the concentration of interstitial zinc was neglected. However, at relatively low oxygen pressures (i. e. up to about 10^{4}–10^{6} Pa) the predominant defects in the discussed oxide are zinc interstitials. Unfortunately, there is no agreement in literature concerning zinc interstitial concentration as a function of oxygen pressure and temperature. As a consequence, it is impossible to define the inversion point of predominant disorder in ZnO. Therefore, the pressure dependence of doubly ionized zinc vacancies, presented in Figure 6, in a low oxygen pressure range can be different, due to interaction between zinc vacancies and zinc interstitials.

It should be also mentioned that the calculated concentration of trivalent chromium addition was virtually independent of temperature and equal to 1.9×10^{−3} (cation fraction). This value is about one order of magnitude lower than the concentration of chromium used to prepare Zn_{0.99}Cr_{0.01}O. The main reason for this situation seems to be the application of zinc nitrate hexahydrate Zn(NO_{3})_{2} ·6H_{2}O with relatively low purity as a starting material. Consequently, different aliovalent impurities present in zinc nitrate hexahydrate can have the opposite influence on the concentration of zinc vacancies in chromium-doped zinc oxide. In addition, the entire chromium addition (as well as all other impurities) does not have to be substitutionally incorporated into the crystal lattice of zinc oxide, thereby forming a solid state solution. Some chromium can form small Cr_{2}O_{3} grains or can be present in the grain boundary regions, which means that not all chromium in ZnO is “active” from the point of view of the Hauffe-Wagner theory of doping. This short discussion strongly suggests that the concentration of the chromium dopant should be calculated from experimental data and not determined via analytical experimental methods.

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