Kroger-Vink

The formation of the combination of defects may be described as a chemical reaction and thermodynamic equilibrium conditions may be applied. The chemical notations of Kroger-Vink, Schottky, and defect structure elements (DSEs) are used [3, 11]. The chemical reactions have to balance the chemical species, lattice sites, and charges. An unoccupied lattice site is considered to be a chemical species (V) it is quite common that specific crystal structures are only found in the presence of a certain number of vacancies [12]. The Kroger-Vink notation makes use of the chemical element followed by the lattice site of this element as subscript and the charge relative to the ideal undisturbed lattice as superscript. An example is the formation of interstitial metal M ions and metal M ion vacancies, e.g., in silver halides ... 

This notation by Kroger-Vink is very intuitive. However, the laws of thermodynamic equilibrium may not be applied to these symbols because the elements are not independent of each other as required by thermodynamics. For example the formation of the interstitial metal ion re-... 

Thermodynamic equilibrium laws are applicable to the Schottky building units. However, there is a loss in intuition. In view of that conflict, the DSEs [11] make use of Kroger-Vink structural elements with the meaning of Schottky building units. This conversion is easily achieved by omitting all ideal lattice elements such as M on M sites, M M, and interstitial vacancies, V . This reads, for the example of Eqs. (5H7),... 

The DSEs thus combine the advantages of both descriptions — Kroger-Vink and Schottky. 

This reaction reads, in Kroger-Vink nota-... 

Kamlet scale 458 kinetics, lithium alloys 366 ff kinks, lithium deposition 345 Kroger-Vink notation 529... 

TABLE 25.1 Kroger-Vink Notation for Point Defects in Crystals... 

Another source of departure from stoichiometry occurs when cations are reduced, as for example in the reduction of zinc oxide to yield an oxygen-defective oxide. The zinc atoms which are formed in this process dissolve in the lattice, Zn+ ions entering interstitial sites and the corresponding number of electrons being released from these dissolved atoms in much the same manner as was found when phosphorus was dissolved in the Group IV semiconductors. The Kroger-Vink representation of this reduction is... 

Point defect populations profoundly affect both the physical and chemical properties of materials. In order to describe these consequences a simple and self-consistent set of symbols is required. The most widely employed system is the Kroger-Vink notation. Using this formalism, it is possible to incorporate defect formation into chemical equations and hence use the powerful methods of chemical thermodynamics to treat defect equilibria. 

In the Kroger-Vink notation, empty atom positions, that is, vacancies, are indicated by the symbol V. Acknowledging that V is the chemical symbol for the... 

TABLE 1.2 Kroger-Vink Notation for Defects in Crystals"... 

The main features of the Kroger-Vink notation are summarized in Table 1.2 and are illustrated with respect to point defects in a crystal containing Ni2+ and O2- ions in Figure 1.13. 

Defects are often deliberately introduced into a solid in order to modify physical or chemical properties. However, defects do not occur in the balance of reactants expressed in traditional chemical equations, and so these important components are lost to the chemical accounting system that the equations represent. Fortunately, traditional chemical equations can be easily modified so as to include defect formation. The incorporation of defects into normal chemical equations allows a strict account of these important entities to be kept and at the same time facilitates the application of chemical thermodynamics to the system. In this sense it is possible to build up a defect chemistry in which the defects play a role analogous to that of the chemical atoms themselves. The Kroger-Vink notation allows this to be done provided the normal mles that apply to balanced chemical equations are preserved. 

When Y3+ cations are used to substitute Zr4 at the corresponding lattice sites, they also create vacancies in the oxygen sublattice since Y3+ cations have a lower valence than Zr4+. The vacancy production can be shown in Kroger-Vink notation similar to Equation 1.1. 

When we consider the defect charges on the species involved, using the Kroger- Vink notation in which the superscripts , and refer to positive, negative and neutral species, the above equation may be rewritten ... 

Write balanced defect reaction equations using Kroger-Vink notation. 

It is theoretically possible for cations to occupy anion sites, and vice versa. Kroger-Vink notation, then, dictates that an M atom on an X site be designated as Mx and that an X atom on an M site be designated as Xm- Recall that we can have defect clusters, such as a Frenkel defect. Defect clusters are enclosed in parentheses—for example, (VmVx) or (X Xm)—to indicate that the individual defects are associated with one another. Impurity atoms are also coded as to lattice position. If we introduce a metal impurity atom L into our compound MX, it might occupy a metal cation site, and is thus designated as Lm- Similarly, Sj is an S impurity atom on an interstitial site. 

The conductivity of these materials can be controlled by the number of defects. In a /7-type semiconductor such as CU2O, in which vacancies are formed in the cation lattice when the oxygen partial pressure is increased, we can develop relationships between conductivity and oxygen partial pressure. The overall reaction for the formation of vacancies and electron holes can be written in Kroger-Vink notation (cf. Section 1.2.6.1) as... 

Approximated solution of eqns (I.204)-(1.216) (Kroger-Vink diagram, see Figs 1.63 and 1.64)... 

Fig. 1.63 Kroger-Vink diagram of semiconductive compound MX for the case Kg < (see Table 1.6).

Fig. 1.67 Kroger-Vink diagrams of (a) Bi-doped PbS, (b) pure PbS, and (c) Ag-doped PbS. Log[ ] denotes log of defect concentration. Thick curves show the Ps. dependence of carrier concentrations (measured). Thin lines are calculated ones.

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