Point defects chemical species

Analyses of the defect chemistry and thermodynamics of non-stoichiometric phases that are predominately ionic in nature (i.e. halides and oxides) are most often made using quasi-chemical reactions. The concentrations of the point defects are considered to be low, and defect-defect interactions as such are most often disregarded, although defect clusters often are incorporated. The resulting mass action equations give the relationship between the concentrations of point defects and partial pressure or chemical activity of the species involved in the defect reactions. 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

Point defects, electrons, and holes as chemical species... 

Concentration equilibrium among A , A , A , and h is discussed on the assumption that these equations can be treated as chemical equilibrium ones. (Similarly, D", D, (donor levels), and e are regarded as chemical species, see Fig. 1.24(c).) We have a reasonable reason for regarding these species as chemical species. As is well known, the electrical properties of metals and alloys are independent of the concentration of point defects or imperfections existing in their crystals, because the number of electrons or holes in metals or alloys is roughly equal to that of the constituent atoms. For the case of semiconductors or insulators, however, the number of electrons or holes is much lower than that of the constituent atoms and is closely correlated to the concentration of defects. In the latter case, electrons and holes can be considered as kinds of chemical species, for a reason similar to that discussed above for the case of point defects. Let us consider the chemical potential, which is most characteristic of chemical species. Electrochemical potential of electrons is written as... 

Thus, lattice defects such as point defects and carriers (electrons and holes) in semiconductors and insulators can be treated as chemical species, and the mass action law can be applied to the concentration equilibrium among these species. Without detailed calculations based on statistical thermodynamics, the mass action law gives us an important result about the equilibrium concentration of lattice defects, electrons, and holes (see Section 1.4.5). 

Fick s second law states the conservation of the diffusing species i no i is produced (or annihilated) in the diffusion zone by chemical reaction. If, however, production (annihilation) occurs, we have to add a (local) reaction term r, to the generalized version of Fick s second law c, = —Vjj + fj. In Section 1.3.1, we introduced the kinetics of point defect production if regular SE s are thermally activated to become irregular SE s (i.e., point defects). These concepts and rate equations can immediately be used to formulate electron-hole formation and annihilation... 

Defects which have extent of only about an atomic diameter also exist in crystals—the point defects. Vacant lattice sites may occur—vacancies. Extra atoms—interstitials—may be inserted between regular crystal atoms. Atoms of the wrong chemical species—impurities—also may be present. 

The equilibrium concentrations of point defects can be derived on the basis of statistical mechanics and the results are identical to those obtained by a less fundamental quasi-chemical approach in which the defects are treated as reacting chemical species obeying the law of mass action. The latter, and simpler, approach is the one widely followed. 

Solid-state diffusion, which is involved in the release of oxygen, proceeds generally through the movement of point defects. The vacancy mechanism, the interstitial mechanism, and the interstitialcy mechanism can occur depending on the distortion of the solid lattice and the nature of the diffusing species. When one of the steps 1-5 is the slowest step representing the major resistance, that step is the rate-controlling one, which is not necessarily the chemical reaction (step 3). 

A population of vacancies on one subset of atoms created by displacing some atoms into normally unoccupied interstitial sites constitute a second arrangement of paired point defects, termed Frenkel defects (Figure 2(b), (c)). Because one species of atom or ion is simply being redistributed in the crystal, charge balance is not an issue. A Frenkel defect in a crystal of formula MX consists of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. Equally, a Frenkel defect in a crystal of formula MX2 can consist of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. As with the other point defects, it is found that the free energy of a crystal is lowered by the presence of Frenkel defects and so a popnlation of these intrinsic defects is to be expected at temperatures above 0 K. The calculation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects. The appropriate chemical equilibrium for cation defects is ... 

One of the problems, where the use of cluster models is more appealing, is in the study of point defects. These centers are often the most interesting ones from the point of view of the physical and chemical properties of a material. Several chemical reactions taking place at an oxide surface are directly or indirectly connected to the presence of point and extended defects. Unfortunately, defect centers are elusive species because of their low concentration even in the bulk material, and their identification by spectroscopic methods can be rather difficult. Furthermore, the distinction of surface from bulk defects may be extremely subtle. For all these reasons, the theoretical modeling of defect centers at the surface of oxides is attracting an increasing interest. 

Normally, the point defects are expected to be in a local or global equilibrium state when the thermodynamic approach can be used. Within the framework of this approach, the defects and their simplest associates are treated as chemical species [9,15-19]. Therefore, the chemical potential of each structural element (p ), which may correspond to atoms (ions) in their regular positions or defects, and the Gibbs energy change for any process involving the i-type species (AG), can be written as... 

To relate the concentrations of point and electronic defects to temperature and externally imposed thermodynamic conditions such as oxygen partial pressures, the defects are treated as chemical species and their equilibrium concentrations are calculated from mass action expressions. If the free-energy changes associated with all defect reactions were known, then in principle diagrams, known as Kroger-Vink diagrams, relating the defect concentrations to the externally imposed thermodynamic parameters, impurity levels, etc., can be constructed. 

In the hypothesis of a perfect crystal lattice defined by a single pattern present on all of the lattice s nodes, it is difficnlt to imagine an atom or ion moving aroimd in the structure of the solid. Yet, experience shows scattering of chemical species in most materials. To explain this movement and to conceptualize some heterogenous reactions involving solid compoimds, it was necessary to imagine the existence of point defects in solids. 

Defects in solids are ubiquitous and can be found both in the bulk and at the surface of materials.Two classes can be distinguished point defects and extended defects. The former, also called local defects, produce a modification of the site environment of an otherwise perfect lattice for instance, the absence of an atom in a lattice position (vacancy), the presence of an atom in an interstitial position (interstitial defect), or the substitution of an atom for another atom of a different chemical species at a regular lattice site (substitutional defect). Figure 45 shows typical examples of local defects in an ionic solid. 

The concentrations of point defects in ionic compoxmds vary with the concentration of dopants. The defect concentrations can be predicted using defect chemistry where each defect species is considered to be a chemical species and the reactions among defects are expressed as chemical reaction equations. In expressing defect chemical reactions, some basic principles are applied. 

The concentrations of charged atomic defects—point defects—follow the law of mass action. The considerations of thermodynamic equilibria can be applied to disorder equilibria in solid crystalline compounds, the so-called ordered mixtures. Point defects can be regarded as quasi-chemical species with which chemical reactions can be formulated. This has led to the so-called imperfection chemistry. As an example, the disorder equilibrium between vacancies and interstitial particles—the so-called Frenkel equilibrium—will be regarded. 

The point defects, in turn, are classified as native (intrinsic) and substitution defects. The intrinsic point defects appear as a vacancy (the absence of an atom in a crystal lattice position) or as an interstitial defect (the presence of the host crystal atom in an interstitial position). The host crystal atoms can be substituted for another atom of a different chemical species at a regular lattice site or at the interstitial position (impurity center or substitution defect). The point defects can also be classified as neutral and charged relative to the host crystal lattice. The perturbation of a solid by... 

Crystalline solids - in the case of crystalline sohds, we can no longer identify components and chemical species (atoms, ions or molecules), because the same species can occupy different sites in the crystalline lattice, which thus constitutes two different components. In addition, to take accoimt of some of the properties of solids, we also need to take accoimt of certain entities which appear to be irregularities in the arrangement of the species in space. These irregularities constitute point defects. We then define what is known as structure elements, and it is these which play the role of components. We are then led to the method used for ionic solutions. 

All diversified point defects mentioned above, including intrinsic, impurity and electron defects, can be regarded as quasi-chemical species hke atoms or ions, which would exist or participate in chemical reactions as a composition of substance. 

To begin with, we can accept the hypothesis that a point defect does not affect the lattice s fundamental vibration frequency, and therefore the term v in relation [3.51] does not depend on the species involved in the quasi-chemical reaction. For ionic compounds, we sometimes choose one anionic vibration frequency and one cationic frequency. Certain authors, such as Mott [MOT 38], opt instead for a half frequency for the defect. We will now examine a few examples, with the vibration frequency being kept constant and unique. 

Consider, for example, a crystallized, binary solid made up of A and B elements. If this solid does not present any point defects, its composition is uniform, and if all other intensive parameters are also uniform, the solid is said to be in internal equilibrium. The diffusion of one of the species A or B can be considered only if the solid presents other structrrre elements than the oires of the ideal crystal, that is, point defects. Indeed, on the contrary there is no gradient of chemical potential, so atoms or molecules of A or B cotrld not diffuse. 

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