Equilibrium thermodynamics of electronic defects

The fundamental electronic disorder reaction is equivalent to the overcoming of the gap between valence band and conduction band and, hence, represents the generation of conduction electrons e (in the conduction band) and holes h (in the valence band). Relations between the (electro-) chemical potentials of defects (e, h) and components (e ) are analogous to the ionic case (see below). 

If we initially consider solids at high temperatures, that possess adequately large band gaps, low doping levels and a sufficient number of available electronic levels, Fermi-Dirac statistics can be replaced by Boltzmann statistics and the formalism is substantially that applied to ions. Here, too, there is an expression of the form of Eq. (5.15) whereby the number of positions available has a somewhat more complex meaning and can differ for conduction electrons (excess electrons) and holes ( electron vacancies ). We will leave this discussion for later and name these reservoir quantities, the effective numbers of states for the conduction (Nc) and for the valence band (Ny). Conditions under which we cannot neglect correlations (i.e. energy levels, in particular, become dependent on occupation) will also be considered later. The chemical potentials are then obtained in the expected form 

On the right hand side they have been expressed as logarithmic functions of [e ] = or [h ] = by rescaling the /x s to /i° s accordingly. As with ions it is e tlie 

As already discussed in Section 1.1 we can describe the electronic transitions from the valence into the conduction band as 

This is the fundamental electronic disorder reaction for which because of 

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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). 

If the pressure P and temperature T are fixed, then the total concentrations of point defects at thermodynamic equilibrium in a compound with n components can only be dependent upon the n — 1) independent chemical potentials fXi of the components (i.e. upon the (n — 1) component activities <2.). For example, the concentration of electronic defects in silicon that has been doped with aluminum (or phosphorus) is uniquely determined by the activity of the dopant element, since this is a binary system. In a completely analogous way, the concentrations of electron holes and cation vacancies in NiO are uniquely dependent upon the oxygen partial pressure as long as overall equilibrium can be assumed. 

At the beginning of the century, nobody knew that a small proportion of atoms in a crystal are routinely missing, even less that this was not a mailer of accident but of thermodynamic equilibrium. The recognition in the 1920s that such vacancies had to exist in equilibrium was due to a school of statistical thermodynamicians such as the Russian Frenkel and the Germans Jost, Wagner and Schollky. That, moreover, as we know now, is only one kind of point defect an atom removed for whatever reason from its lattice site can be inserted into a small gap in the crystal structure, and then it becomes an interstitial . Moreover, in insulating crystals a point defect is apt to be associated with a local excess or deficiency of electrons. 

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... 

Thermodynamics of this situation at equilibrium Metallurgy of defects on metal surfaces Crystallography of surface Quantum mechanics of transfer of electrons through barrier at interface Pick s second law diffusion theory of time dependence of concentration... 

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. 

Because oxidation and reduction are reversed reactions each other, which are actually the same process from the thermodynamic point of view, their reaction equations are dependent mutually. For example, if the reduction reaction of Eq. (5.21) is combined with the intrinsic electronic defect equilibrium ... 

Chebotin s scientific interests were characterized by a variety of topics and covered nearly all aspects of solid electrolytes electrochemistry. He made a significant contribution to the theory of electron conductivity of ionic crystals in equilibrium with a gas phase and solved a number of important problems related to the statistical-thermodynamic description of defect formation in solid electrolytes and mixed ionic-electronic conductors. Vital results were obtained in the theory of ion transport in solid electrolytes (chemical diffusion and interdiffusion, correlation effects, thermo-EMF of ionic crystals, and others). Chebotin paid great attention to the solution of actual electrochemical problem—first of all to the theory of the double layer and issues related to the nature of the polarization at the interface of the solid electrol34e and gas electrode. 

As in any semiconductors, point defects affect the electrical and optical properties of ZnO as well. Point defects include native defects (vacancies, interstitials, and antisites), impurities, and defect complexes. The concentration of point defects depends on their formation energies. Van de WaHe et al. [86,87] calculated formation energies and electronic structure of native point defects and hydrogen in ZnO by using the first-principles, plane-wave pseudopotential technique together with the supercell approach. In this theory, the concentration of a defect in a crystal under thermodynamic equilibrium depends upon its formation energy if in the following form ... 

Besides electronic effects, structure sensitivity phenomena can be understood on the basis of geometric effects. The shape of (metal) nanoparticles is determined by the minimization of the particles free surface energy. According to Wulffs law, this requirement is met if (on condition of thermodynamic equilibrium) for all surfaces that delimit the (crystalline) particle, the ratio between their corresponding energies cr, and their distance to the particle center hi is constant [153]. In (non-model) catalysts, the particles real structure however is furthermore determined by the interaction with the support [154] and by the formation of defects for which Figure 14 shows an example. 

FIGURE 17.1 Number of species contributing to the effect of radiation on matter. PI, Primary interaction occurs in very short time CE, the cascade excitation P, the electron cloud plasma, electron relaxation LR, lattice relaxation TD, thermodynamic equilibrium PD, point-defect and DL, dislocations loops. 

Equations (14.14) and (14.18) can be used as starting point for generating equations describing O2 and H2 permeation within single-phase perovskite membranes. Key to these equations is the nature of the boundary conditions at the feed/membrane and permeate/membrane surfaces. To this aim, one needs to address appropriate defect point thermodynamics to establish equilibrium and surface exchange relations for all potential species that can play a role during permeation. As a general rule, the law of mass action can be used to predict the concentration of ionic vacancies, protons, electrons, and electron holes in the membrane. Below we describe a series of models that can be deduced for ID steady-state permeation within perovskite and extensively other MIEC membranes. 

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