Thermal Disorder Intrinsic Defect Concentrations

The formation of lattice defects requires energy, yet their presence means a certain entropy. Therefore according to equilibrium thermodynamics at all temperatures above absolute zero there are a certain number of defects. The concentration of defects [w] at equilibrium can be derived from the equilibrium constants and has the usual exponential temperature dependence  

Thermal disorder results in defects such as vacancies or interstitials. Such thermal defects are often formed in pairs as explained below. As long as there is no matter exchange with the surroundings of the system the stoichiometry of the compound does not change by heat-induced disorder. 

There are different types of formation reactions and equilibria, depending on the type of lattice and the type of defect. The types of disorders are known as Schottky, Frenkel, and anti-Frenkel, 

Ks = KJLVa ] and in NiO 0 V, + Vo = MKo]. Schottky equilibrium leads to vacancy pairs and this type of disorder dominates in alkaline halides and in oxides with a halite (NaCl) structure. 

Frenkel disorder a pair of defects involving only cations, with the pair 

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The defect concentrations that are the result of thermal disorder are small in most oxides. The formation enthalpy of vacancy pairs in MgO is 7 eV, which gives a vacancy concentration of 10 ppm at 1000°C. In most oxides the bandgap is also large (>4eV) and at 1000°C the charge carrier concentration is lower than 10 ppm. Now, oxides can be made with an impurity concentration of at best 10-100 ppm. The concentration of impurities contributes much more to the defect concentration than the thermal disorder at these low formation equilibrium constants and the thermal (intrinsic) contribution to the defect concentration can usually be disregarded. 

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). 

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... 

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