Thermodynamics point defect formation

Walter Haus Schottky (1886-1976) received his doctorate in physics under Max Planck from the Humboldt University in Berlin in 1912. Although his thesis was on the special theory of relativity, Schottky spent his life s work in the area of semiconductor physics. He alternated between industrial and academic positions in Germany for several years. He was with Siemens AG until 1919 and the University of Wurzburg from 1920 to 1923. From 1923 to 1927, Schottky was professor of theoretical physics at the University of Rostock. He rejoined Siemens in 1927, where he finished out his career. Schottky s inventions include the ribbon microphone, the superheterodyne radio receiver, and the tetrode vacuum tube. In 1929, he published Thermodynamik, a book on the thermodynamics of solids. Schottky and Wagner studied the statistical thermodynamics of point defect formation. The cation/anion vacancy pair in ionic solids is named the Schottky defect. In 1938, he produced a barrier layer theory to explain the rectifying behavior of metal-semiconductor contacts. Metal-semiconductor diodes are now called Schottky barrier diodes. 

Thermodynamics of Point Defect Formation in Elemental Crystals... 

The creation of single, unassociated point defects in an elemental, crystalline solid increases the internal energy of the system and the enthalpy of the defect formation is positive. But the configurational entropy of the system also increases, and the equilibrium concentration of the defects will be reached when the Gibbs energy of the system is at minimum. Thermodynamically, point defects will thus always be present in a crystal above 0 K. 

Table 5.1 Thermodynamic standard parameters for point defect formation.

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Point defects are always present in every material in thermodynamic equilibrium. Considering the formation of n vacancies, the increase in configuration entropy is determined by the number of different possible ways of taking n atoms out of the crystal comprising N atoms in total. This number, c1, is given by... 

We will be considering primarily inorganic solids but must keep in mind that the same principles also apply to organic solids. Therefore, we intend to examine the nature of point defects in terms of their thermodynamics, equilibria and the energy required for their formation. It will be seen that point defects follow the same physical chemistry laws that apply to inorgcuiic compounds and physical properties in general. 

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. 

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... 

Defect clustering is the result of defect interactions. Pair formation is the most common mode of clustering. Let us distinguish the following situations a) two point defects of the same sort form a defect pair (B + B = B2 = [B, B] V+V = V2 = [V, V]) and b) two different point defects form a defect pair (electronic defects can be included here). The main question concerns the (relative) concentration of pairs as a function of the independent thermodynamic variables (P, T, pk). Under isothermal, isobaric conditions and given a dilute solution of B impurities, the equilibrium condition for the pair formation reaction B + B = B2 is 2-pB = The mass balance reads NB + 2-NBi = NB, where NB denotes the overall B content in the matrix crystal. It follows, considering Eqns. (2.39) and (2.40), that... 

Therefore, the related defect concentration depends on the impurity level and or temperature. When we consider the thermodynamics for the formation of point defects, vacancies are important, i.e., = N (T). 

When we consider continuous scale growth, we can expect that the mobile species from the metal (cations diffusing out) will be supplied by alloy grain-boundaries, bulk defects, and dislocations. These diffusivities are quite different from each other D(bulk) D(dislocation) < D(grain boundary) < D(surface). Therefore, we expect the formation of voids around the alloy grain boundaries and dislocations as the scale continues to grow. The chief concerns, here, is How can we prepare an inert state (kinetically and thermodynamically) for the point defects, for the grain boundaries, and especially for the dislocations in the alloy substrate ... 

Shear Plane-Point Defect Equilibria.—The question of the existence of point defects in compounds where extended defects are known to occur has been controversial. Indeed, it has occasionally been claimed that point defects cannot form in such phases and that they will always be eliminated with the formation of extended structures. We reject these latter arguments as thermodynamically unsound. From a thermodynamic standpoint, the formation of extended defects can be viewed as a special mode of point defect aggregation as such, shear planes will be in equilibrium with point defects, with the position of the equilibrium depending on both temperature and the extent of the deviation from stoicheiometry. Thus, if we assume, as is suggested by our calculations, that anion vacancies are the predominant point defects in reduced rutile (a further point of controversy as mentioned above) then there will exist an equilibrium of the type... 

A proper description of electronic defects in terms of simple point defect chemistry is even more complicated as the d electrons of the transition metals and their compounds are intermediate between localized and delocalized behaviour. Recent analysis of the redox thermodynamics of Lao.8Sro,2Co03. based upon data from coulometric titration measurements supports itinerant behaviour of the electronic charge carriers in this compound [172]. The analysis was based on the partial molar enthalpy and entropy of the oxygen incorporation reaction, which can be evaluated from changes in emf with temperature at different oxygen (non-)stoichiometries. The experimental value of the partial molar entropy (free formation entropy) of oxygen incorporation, Asq, could be... 

Although the latter occurs of course always to some extent just on entropy grounds, there will at any rate be an enrichment of oxygen atoms in the easier to relax interstitial sites in the immediate near-surface fringe. And one could expect this deformation argument to hold even more for subsurface sites close to even lower coordinated atoms, i.e., interstitials in the vicinity of point defects, steps, or dislocations. The latter are in fact frequently believed to be the nucleation centers for surface oxide formation, and kinetic arguments like an easier O penetration are often put forward as explanation. Yet, we see that the thermodynamic deformation cost factor could also favor an initial oxygen accommodation close to such sites. 

The rules for quasi-chemical reactions are the same as for the normal chemical reactions, namely mass balance and electroneutrahty conditions one extra requirement appears, however, for crystalline solids where the ratio of sites in the crystal structure should be constant and should satisfy to stoichiometric formula. This means that if, for instance, in the AB2 crystal one site for A atom is formed, then automatically two B-sites appear as well, regardless of their occupancy. It should be noted that the point defects and/or the processes of their formation can also classified into two groups, namely stoichiometric and nonstoichiometric. The first type of process does not disturb the stoichiometric ratio of components constituting the crystal, which is a closed thermodynamic system the second type leads to nonstoichiometric compounds by exchanging components between the... 

Point defects such as vacancies are thermodynamically stable entities 18) meaning, in effect, that their concentration increases with increasing temperature because their free energy of formation is negative. A crystal that is quenched from high temperatures will, at lower temperatures, contain a supersaturation of point defects. But the excess... 

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