The interest in physico-chemical properties of zinc oxide has continuously increased during last decades because of the growing potential possibilities for applying the oxide in modern electronics . Zinc oxide is a promising material for production of light emitting devices, flat-panel displays and solar cells [1, 2]. Over the major part of the phase field that corresponds to high temperatures, ZnO, which exhibits a Wurzite (zinc-blende) crystal structure, is a metal excess n-type semiconductor with the predominant defects, i. e. cation interstitials and quasi-free electrons (Zn1+yO) . On the other hand, several papers, published recently in literature, report a several possibilities for p-type ZnO (Zn1-yO) production [1, 4, 5, 6, 7]. It should be noted that p-type-doped ZnO, as well as p-n ZnO junctions, seem to be especially attractive for novel micro- and optoelectronics. However, for optimal application of this type of zinc oxide, knowledge of its point defect structure and transport properties is urgently needed. Unfortunately, there are considerable experimental difficulties in studying the point defect situation in pure p-type ZnO, because it is thermodynamically stable at high temperatures under oxygen pressures much higher than 105 Pa. Consequently, point defect concentration in metal-deficit ZnO could not have been determined so far by direct thermogravimetric or other techniques.
Thus, the aim of the present paper is an attempt to solve this important problem, using an indirect method, which allows for determination of native point defect concentration in metal-deficit p-type Zn1-yO oxide by analyzing the influence of an appropriate dopant, i. e. from the doping effect.
Recently, a novel method has been demonstrated, which enables an indirect way of determining very low defect concentration in transition metal oxides by studying the influence of aliovalent metallic additions on the oxidation kinetics of given metals, in the reaction product of which the nonstoichiometry is to be calculated (i. e. from doping effect) . The validity of the proposed method was illustrated on the example of nonstoichiometric nickel oxide, Ni1-yO, in which defect concentration and the mobility of defects have been calculated.
Unfortunately, the melting point of zinc is very low in comparison to that of zinc oxide. Consequently, the discussed indirect method  cannot be used in its proposed form to determine the point defect concentration in p-type ZnO, because oxidation of zinc would take place at temperatures much lower than Tammann’s temperature for ZnO. Therefore, it would not be possible to draw any rational conclusions concerning defect concentration in the investigated material. Thus, the previously proposed indirect method  was modified so that it could be possible to determine native point defect concentration in metal-deficit Zn1-yO.
The idea of the modified method consists of determining the nonstoichiometry (and consequently predominant point defect concentration) in pure p-type zinc oxide (Zn1-yO) by studying the influence of trivalent chromium dopant on defect concentration in this material as a function of temperature and oxygen pressure. To explain this possibility, the influence of chromium on the point defect situation in ZnO will be shortly discussed. It should be noted, that there are no data related to the defect concentration in Zn1-yO. However, assuming that the concentration of zinc vacancies is small (which seems to be true for oxygen pressures near the inversion point of predominant disorder) and, consequently that they are double ionized, the formation process of point defects in Zn1-yO can be written as follows:
If trivalent chromium ions are substitutionally incorporated into the cation sublattice of the Zn1+yO oxide, this process can be described by the following quasi-chemical reversible reactions:
On the other hand, the effect of chromium doping in the case of Zn1-yO can be presented as follows:
According to reaction (2), trivalent chromium ions substitutionally incorporated into the cation sublattice of Zn1+yO decrease the concentration of zinc interstitials. However, in the case of Zn1-yO, chromium addition increases the concentration of zinc vacancies (eq. (4)). As a consequence, the inversion point of predominant ionic disorder in the discussed chromium doped zinc oxide is shifted in the direction of lower oxygen pressures. Discussing situation is more visible in Figure 1, which shows the schematic dependence of ionic defect concentration on oxygen pressure for pure (dotted lines) and chromium doped (solid line) ZnO. From this figure it follows that the inversion point of predominant disorder may be considerable shifted by appropriate doping, which enables studies of zinc vacancy concentration in a much lower oxygen pressure range, i. e. at ambient oxygen pressures (dark polygon). It should be noted, that a situation, analogous to that presented in Figure 1, will be observed for a different degree of native point defect ionization in ZnO than that considered in this work. It denotes that studies of zinc vacancy concentration as a function of oxygen pressure are possible under atmospheric pressure in a reactive environment, which can be obtained with relative ease by mixing pure oxygen with noble gas. Experimentally determined values of zinc vacancies in chromium doped zinc oxide can be used to calculate the constant of reaction 1 and thereby the predominant defect concentration in pure Zn1-yO as well. Such calculations are described in the following section of this work.
Pure zinc oxide and ZnO with 1 at % of chromium (Zn0.99Cr0.01O) have been used in this work for analysis of zinc vacancy concentration. Both these materials were prepared using the EDTA-gel technique  (EDTA-Ethylenediaminetetraacetic acid), which is especially useful in this case due to the good conditional stability constant of complexes consisting of EDTA anions with Zn and Cr ions. For the EDTA gel precursor preparation, zinc nitrate hexahydrate Zn(NO3)2 ·6H2O (Sigma-Aldrich, purity 99 %) and chromia (III) nitrate nonahydrate Cr(NO3)3 · 9H2O (Sigma-Aldrich, purity 99.999 %) were used as the starting materials, while ethylenediaminetetraacetic acid, C10H16N2O8 (better known as EDTA) was applied as the chelating agent. First of all, salts with known dry matter content were used for the preparation of aqueous solutions containing 0.5 M (M – molar concentration mol · dm−3) Zn or Cr nitrate, which were subsequently mixed in the appropriate ratio to yield the desired stoichiometry. The nitrate aqueous solutions were mixed with the proper amount of 0.1 M EDTA acid (in the following ratio: 1 mol of EDTA acid per 1 mol of metal cations). The final pH of the Zn-Cr-EDTA as well as Zn-EDTA solutions was constant (pH = 8). It was maintained via proper addition of ammonia dropwise. Afterwards, the reaction mixtures containing zinc and chromia complexes were slowly heated and stirred to prevent salt precipitation at 70–90 °C in order to evaporate the water until transparent, glassy gels were obtained. Next, the pre-calcined powders were calcined at 550 °C for 2 h with a heating rate of 3 °C/min in static air. The resulting ZnO and Zn0.99Cr0.01O powders were ground in a rotary-vibratory mill in 2-propanol for 1 h and finally dried at room temperature. To prepare bulk samples in cylindrical shapes with a diameter of 11 mm and a height of approximately 1 mm powders were uniaxially dry-pressed in a steel die at 100 MPa, followed by isostatic pressing at 250 MPa. The green bodies were then sintered in static air at 1250 °C for 2 h, with a heating rate of 5 °C/min. The resulting dense disc-shaped samples (11 mm in diameter and 1 mm in thickness) were used to determine the predominant ionic defect concentration via thermogravimetric studies of the nonstoichiometry of the prepared oxides.
The nonstoichiometry of pure and chromium doped zinc oxide has been determined as a function of temperature (700–900 °C) and oxygen pressure (30–105 Pa) in the following way. A given oxide sample, suspended in the reaction zone of a microthermogravimetric apparatus was equilibrated at constant temperature and oxygen pressure. When a constant weight of the oxide sample had been reached, the nonstoichiometry, y, in a given material was calculated using the following empirical formula:
where mMe and mO denote masses of the metal in investigated oxide sample and consumed oxygen; and MMe and MO are atomic masses of the metal and oxidant, respectively.
The nonstoichiometry measurements have been carried out in a microthermogravimetric apparatus with the sensitivity of 10−7 g. The partial pressure of oxygen was obtained by using a suitable composition of ternary Ar-He-O2 gas mixture with density exactly equal to that of oxygen, thus eliminating the possible Archimedes effect during changes of gas composition. Details of the microthermogravimetric equipment and experimental procedure have been described elsewhere [11, 12]. The results of nonstoichiometry measurements have been used to calculate the predominant ionic defect concentration in pure p-type zinc oxide. This procedure has been described in the next section of this work.
Results and discussion
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].
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.
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 (Zn1-yO). 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 Zn1-yO. 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:
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 Cr2O3-Zn1-yO solid solution, lead to the following system of equations:
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 Zn1-yO are unknowns and should be computed. Using simple algebra, this set of equations (eq. (8)) can be rearranged and reduced to the following equation:
Next, eq. (9) can be approximated using experimental results and, consequently, at each three experimental points obtained at the same constant temperature: (
The constant of reaction (1) calculated via the previously described method is presented as a function of temperature by the following equation:
In the case of pure Zn1-yO oxide, the electroneutrality condition assumes the following form:
And, consequently, the concentration of predominant ionic and electronic defects is the following function of temperature and oxygen pressure:
These results are presented, once again, in Figure 6, which illustrates the concentration of double-ionized zinc vacancies in Zn1-yO oxide.
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 104–106 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 Zn0.99Cr0.01O. The main reason for this situation seems to be the application of zinc nitrate hexahydrate Zn(NO3)2 ·6H2O 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 Cr2O3 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.
The results of presented work clearly indicate that the doping effect can be used to indirectly calculate point defect concentration in transition metal oxides. It has been experimentally proved that chromium addition, substitutionally incorporated into the crystal lattice of zinc oxide, can change the predominant ionic disorder in this material. The experimentally obtained data pertaining to pressure dependence of nonstoichiometry in chromium-doped zinc oxide can be used to calculate zinc vacancy concentration in pure
Zn1-yO. The proposed method of investigating point defect concentration in metal oxides seems to be effective in the case of oxides stable at extremely high oxidant pressures with a relatively low concentration of native point defects.
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