Disorder Schottky

The compound will be stoichiometric, with an exact composition of MX10ooo when the number of metal vacancies is equal to the number of nonmetal vacancies. At the same time, the number of electrons and holes will be equal. In an inorganic compound, which is an insulator or poor semiconductor with a fairly large band-gap, the number of point defects is greater than the number of intrinsic electrons or holes. To illustrate the procedure, suppose that the values for the equilibrium constants describing Schottky disorder, Ks, and intrinsic electron and hole numbers, Kc, are... 

The notion of point defects in an otherwise perfect crystal dates from the classical papers by Frenkel88 and by Schottky and Wagner.75 86 The perfect lattice is thermodynamically unstable with respect to a lattice in which a certain number of atoms are removed from normal lattice sites to the surface (vacancy disorder) or in which a certain number of atoms are transferred from the surface to interstitial positions inside the crystal (interstitial disorder). These forms of disorder can occur in many elemental solids and compounds. The formation of equal numbers of vacant lattice sites in both M and X sublattices of a compound M0Xft is called Schottky disorder. In compounds in which M and X occupy different sublattices in the perfect crystal there is also the possibility of antistructure disorder in which small numbers of M and X atoms are interchanged. These three sorts of disorder can be combined to give three hybrid types of disorder in crystalline compounds. The most important of these is Frenkel disorder, in which equal numbers of vacancies and interstitials of the same kind of atom are formed in a compound. The possibility of Schottky-antistructure disorder (in which a vacancy is formed by... 

The equilibrium concentration for a Schottky disorder can be found in a similar manner. Recall that a Schottky defect is a cation-anion defect pair. For example, the migration of an MgO molecule to the surface in an MgO crystal can be described as follows ... 

Figure 2-2. a) Fraction /Vp of defect pairs (e.g., [B.V] in Schottky-disordered AX) as a function of the normalized temperature (R/ AGp )-T for various dopant concentrations Nb) NP as a function of at given T. The parameter Ks denotes the Schottky equilibrium constant FVv-A )-... 

Figure 18. Kroger-Vink diagrams of a Schottky disordered negatively doped oxide (trivalent cations assumed). The ordinate refers to a logarithmic concentration axis, the abscissa to a logarithmic partial pressure axis.110 Reprinted from J. Fleig, K.D. Kreuer and J. Maier, in Handbook of Advanced Ceramics. Volume II Processing and Their Applications, S. Somiya, F. Aldinger, N. Claussen, R. M. Spriggs, K. Uchino, K. Koumoto and M. Kaneno (eds.), Elsevier Academic Press (2003) p. 59. Copyright 2003 with permission from Elsevier.

Schottky defects, named after W. Schottky, consist of unoccupied anion and cation sites. A stoichiometric crystalline oxide having Schottky disorder alone contains charge-equivalent numbers of anion and cation vacancies. A Frenkel... 

Fig. 2. (a) Sketch of the relations between defect concentrations and partial pressure (Brouwer diagram) of a pure oxide MO In regime II the intrinsic Schottky disorder determines the concentration, whereas in I and III non-stoichiometry prevails, (b) Dependence of the hole and electron concentration on the frozen-in oxygen vacancy concentration in a negatively (acceptor) doped oxide. 

Considering different pairs of majority defects, all relationships between defect concentrations and partial pressure can be constructed from simplified situations, and this leads to so-called Brouwer diagrams. Figs. 2a and 3a show such Brouwer diagrams for a pure oxide MO with Schottky disorder, and for a Schottky-disordered oxide with a negative dopant. (Please notice that the exact curves calculated from the complete electroneutrality equation (Eq. (11)) exhibit smooth transitions rather than sharp bends.)... 

A major difference between crystals and fluids refers to the necessity of distinguishing between different sites. So the autoprotolysis in water could, just from a mass balance point of view, also be considered e.g. as a formation of a OH vacancy and a IT vacancy. In solids such a disorder is called Schottky disorder (S) and has to be well discerned from the Frenkel disorder (F). In the densely packed alkali metal halides in which the cations are not as polarizable as the Ag+, the formation of interstitial defects requires an unrealistically high energy and the dominating disorder is thus the Schottky reaction... 

The potential energy changes accompanying displacement from one lattice site to an adjacent site (a) and from a lattice site to an interstitial site (b) and from one interstitial site to an adjacent one (c) are depicted in Figure 26. One sees that the activation energy of a dijffusion or ionic migration in a Schottky-disordered... 

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]. 

Schottky Disorder Equal concentrations of cation vacancies and anion vacancies... 

In this case, AHs is in J mol and represents the enthalpy required to form 1 mole of Schottky defects, and R is the gas constant (JK mol ). The fraction of vacant sites in a crystal as a result of Schottky disorder is given by ... 

Calculations of defect formation energies [49] suggest that Schottky disorder... 

Fig. 2.10 Brouwer diagram for a metal oxide MO doped with a donor species (FO2, with F being a metal ion) in which Schottky disorder dominates. Also indicated are defect associates (marked by an asterisk), clusters of defects that are held together by electrostatic ftnces. Such clusters are known to form at high defect concentrations. From [27], with permission...

Figure 10.7. Calculated equilibrium defect diagrams for a binary oxide MO with Schottky defect pairs. In case (a) the equilibrium constant for vacancies is taken to be much larger than for electronic disorder (Kg = 10 Ki) case (b) gives the concentrations if the Schottky disorder is the smaller. The defect concentrations in regions I and III have a power dependence on the oxygen pressure with the exponent Region II in case (a), which includes the electrolytic domain has an exponent of the defect lines of in case (b) the exponent is...

The defects which disrupt the regular patterns of crystals, can be classified into point defects (zero-dimensional), line defects (1-dimensional), planar (2-dimensional) and bulk defects (3-dimensional). Point defects are imperfections of the crystal lattice having dimensions of the order of the atomic size. The formation of point defects in solids was predicted by Frenkel [40], At high temperatures, the thermal motion of atoms becomes more intensive and some of atoms obtain energies sufficient to leave their lattice sites and occupy interstitial positions. In this case, a vacancy and an interstitial atom, the so-called Frenkel pair, appear simultaneously. A way to create only vacancies has been shown later by Wagner and Schottky [41] atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal (voids, grain boundaries, dislocations). Such vacancies are often called Schottky defects (Fig. 6.3). This mechanism dominates in solids with close-packed lattices where the formation of vacancies requires considerably smaller energies than that of interstitials. In ionic compounds also there are defects of two types, Frenkel and Schottky disorder. In the first case there are equal numbers of cation vacancies... 

Fig. 3-1. Schematic representation of the disorder types in AgBr and in KCl. AgBr exhibits Frenkel disorder. KCl exhibits Schottky disorder. In both cases thermal disorder predominates.

The considerations presented up to this point can be easily extended to higher ionic crystals and compounds with more than two or three components [4]. Again, quite generally, the energetically favourable defects constitute the disorder type. For a binary ionic crystal without electronic majority defects there are, in principle, only four disorder types. These are the previously described Schottky and Frenkel types and their corresponding anti-types namely, cations and an equivalent number of anions in the interstices (anti-Schottky disorder), and anion vacancies with an equal number of anions in the interstices (anti-Frenkel disorder). However, for higher ionic crystals the number of possible disorder types increases considerably because of the greater number of components and sublattices. Therefore, in such crystals, it is much more difficult to uniquely determine the disorder type. 

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