Frenkel disorder, intrinsic

In this section we are concerned with the properties of intrinsic Schottky and Frenkel disorder in pure ionic conducting crystals and with the same systems doped with aliovalent cations. As already remarked in Section I, the properties of uni-univalent crystals, e.g. sodium choride and silver bromide which contain Schottky and cationic Frenkel disorder respectively, doped with divalent cation impurities are of particular interest. At low concentrations the impurity is incorporated substitutionally together with an additional cation vacancy to preserve electrical neutrality. At sufficiently low temperatures the concentration of intrinsic defects in a doped crystal is negligible compared with the concentration of added defects. We shall first mention briefly the theoretical methods used for such systems and then review the use of the cluster formalism. 

Intrinsic Frenkel disorder, in which some of the oxygens are displaced into normally unoccupied sites, is responsible for the oxide ion conduction in, for example, Zr2Gd207, Fig. 2.11. The interstitial oxygen concentration is rather low, however, and is responsible for the low value of the preexponential factor and for the rather low (by -Bi203 standards ) conductivity. 

Let us finally estimate the relaxation times of homogeneous defect reactions. To this end, we analyze the equilibration course of a silver halide crystal, AX, with predominantly intrinsic cation Frenkel disorder. The Frenkel reaction is... 

It is proposed that intrinsic Frenkel disorder adds to the interstitial oxygen atoms resulting from the variation of oxygen pressure. From these assumptions, the diffusion coefficient in region I may be written... 

The suggestion of Mott [190], that photodecomposition of Ba(N3)2 occurs by the same mechanism in silver halides, was disputed by Tompkins and coworkers on the basis of additional observations [80,191,206]. In particular, the photoconductivity was found to be too small to account for the electron motion necessary for the formation of barium colloids [80]. More recently, Marinkas and Bartram were unable to detect photoconductivity in anhydrous crystals [49]. In addition, measurements of the dark conductivity indicated that if it is due to Ba ", it is much too small to account for the observed rate of photodecomposition [80,206]. As a further indication that the photodecomposition of Ba(N3)2 does not take place by the silver hahde process, the energy of formation of a barium interstitial was estimated and found to be much greater than the estimated energy for vacancy formation, thus indicating the possibility of Schottky disorder rather than Frenkel disorder as intrinsic to Ba(N3)2 [206]. Interstitial metal ions are required for the Mott-Gurney mechanism discussed above [167]. 

At this point of the discussion it is worthwhile to distinguish between two different kinds of disorder. If the concentrations of the majority defect centers, which constitute the disorder type, are independent of the component activities and are only determined by P and 7, then we speak of thermal disorder or intrinsic disorder (e. g. Frenkel disorder in silver bromide). However, the concentrations of minority defect centers do depend upon the component activities even in the case of a crystal with thermal disorder. This will be discussed more explicitly later. On the other hand, if the concentrations of the majority defects are dependent upon the component activities, then we speak of activity-dependent disorder or extrinsic disorder (e. g. cation vacancies and electron holes in transition metal oxides). 

Fig. 4-3. The concentrations of point defects (/) in the ternary crystal AB2O4 at constant as a function of the activity of B2O3. The intrinsic disorder is assumed to be Frenkel disorder in the B-ion sublattice (By ) ai (Vb").

Defect Chemistry in Solid State Ionic Materials, Fig. 3 Defect structure of E2O3 doped, stoichiometric MO with the Frenkel disorder as the majority, not to scale. Small triangles denote the slopes. Note the intrinsic regime at the high temperatures and extrinsic regime at the low temperatures... 

In view of the many types of point defects that may be formed in inorganic compounds and that each type of defect may have varying effective charge, numerous defect reactions may in principle be formulated. In the following, a few simple cases will be treated as examples. First, we will consider defect stmcture situations in stoichiometric compounds (Schottky, Frenkel and intrinsic electronic disorders) and then defect structure situations in nonstoichiometric oxides will be illustrated. Finally, examples of defect reactions involving foreign elements will be considered. 

At or close to stoichiometry two alternative hmiting conditions must be considered, namely dominance by intrinsic electronic ionisation or by anion Frenkel disorder. 

The intrinsic defects fall into two main categories, i.e., Schottky disorder and Frenkel disorder. As these point defects do not change the overah composition, they are also referred to as stoichiometric defects. Their thermal generation will be exemphfied for a metal oxide MO using the Kroger-Vink notation, and assuming that activities of point defects are equal to their concentrations. Hence, the law of mass action is apphcable to these equilibria... 

The electrical properties of the titanate-based pyrochlores can be described by point defect models in which the acceptor (A) and donor (D) impurities are compensated by oxide ion vacancies, or oxide ion interstitials, respectively. The principal defect reactions inclnde the redox reaction, Equation (5.67), the Frenkel disorder, Equation (5.57), dopant ionization, intrinsic electronic disorder, Equation (5.60), and the electroneutrality relation. For these compounds the total electroneutrality condition given by Equation (5.55) is, on the one hand, reduced, taking Equation (5.57) into account, and is, on the other hand, extended to include acceptor and donor impurities. In addition, defect association is included explicitly. 

For a doped oxide M2O3 exhibiting anti-Frenkel disorder, Colomban and Novak present a sehematie Kroger-Vink diagram of the extrinsic and intrinsic point defects as a function of the partial water pressure. With regard to electrical properties, the proton conductivity in the binary metal oxides is usually much lower than in the perovskite-type oxides. ... 

Frenkel disorder enthalpy causes the defect concentration to increase steeply with temperature. One can also see, however, that the defect concentration at room temperature is so low that other effects, the so-called extrinsic effects, which are discussed in the coining Sections 5.5.2, 5.6 will certainty dominate here . The analogy with aqueous chemistry is also evident There corresponds to the intrinsic proton or hydroxide ion concentration and also there, the charge carrier effects are generally extrinsically controlled at room temperature. 

Fig. 5.37 Coupling of the ionic and the electronic energy level pictures for an anti-Frenkel disordered material M+X" through MX = 5MX2 = MX- - Me-- According to the position of the electrochemical potentials the materied is in the I-regime on the r.h.s. ([h l > [e ]j [X(] [Vx]) of the intrinsic point (cf. Fig. 5.38). Note that jiie- =Jle = —Mh-and Mx- = Mx = Mvx> see Sections 5.2, 5.3). Instead of fix[ (or Mi) and (or Mv) it should more precisely read Jlx. -Wi and Mvx-Xx- (Cf- also Fig. 5.9 on page 127.) From Ref. [168].

Intrinsic disorder is observed in conditions of perfect stoichiometry of the crystal. It is related to two main defect equilibria Schottky defects and Frenkel defects. 

Equation 4.75 finds its application in the region of intrinsic disorder (a similar equation can be developed for Frenkel defects), where Schottky and Frenkel defects are dominant with respect to point impurities and nonstoichiometry. 

To explore the factors that determine how far nonstoichiometry may be observed in practice, we may consider the straightforward case of an oxide, MO, in which the intrinsic lattice disorder is of Frenkel type. It will be involved in simul-... 

At that date, palladium hydride was regarded as a special case. Lacher s approach was subsequently developed by the author (1946) (I) and by Rees (1954) (34) into attempts to frame a general theory of the nature and existence of solid compounds. The one model starts with the idea of the crystal of a binary compound, of perfect stoichiometric composition, but with intrinsic lattice disorder —e.g., of Frenkel type. As the stoichiometry adjusts itself to higher or lower partial pressures of one or other component, by incorporating cation vacancies or interstitial cations, the relevant feature is the interaction of point defects located on adjacent sites. These interactions contribute to the partition function of the crystal and set a maximum attainable concentration of each type of defect. Conjugate with the maximum concentration of, for example, cation vacancies, Nh 9 and fixed by the intrinsic lattice disorder, is a minimum concentration of interstitials, N. The difference, Nh — Ni, measures the nonstoichiometry at the nonmetal-rich phase limit. The metal-rich limit is similarly determined by the maximum attainable concentration of interstitials. With the maximum concentrations of defects, so defined, may be compared the intrinsic disorder in the stoichiometric crystals, and from the several energies concerned there can be specified the conditions under which the stoichiometric crystal lies outside the stability limits. 

There are three possible thermally generated intrinsic disorder reactions in ceria that do not involve exchange with the gas phase. These defects, which are of the Schottky (Eq. 2.2) and Frenkel (Eqs. 2.3 and 2.4) types, can be represented using the Kroger and Vink defect notation, which will be used throughout this chapter ... 

As the incorporation reaction proceeds, the other defect equilibria such as the creation of Schottky or Frenkel defects are still present. Supposing that the intrinsic defects in MgO consist of Schottky disorder, then following Eq. (7.16), we can write... 

While intrinsic disorder of the Schottky, Frenkel, or anti-Frenkel type frequently occurs in binaiy metal oxides and metal halides, i.e., Equations (5.1), (5.3), and (5.5), Schottky disorder is seldomly encountered in temaiy compounds. However, in several studies Schottky disorder has been proposed to occur in perovskite oxides. Cation and anion vacancies or interstitials can occur in ternary compounds, but such defect stractures are usually to be related with deviations from molecularity (viz. Sections II.B.2 and II.B.3), which in fact represent extrinsic disorder and not intrinsic Schottky disorder. From Figures 5.3 and 5.4 it is apparent that deviations from molecularity always influence ionic point defect concentrations, while deviations from stoichiometry always lead to combinations of ionic and electronic point defects, as can be seen from Figures 5.2 and 5.5. 

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