Defect formation energies

In pure and stoichiometric compounds, intrinsic defects are formed for energetic reasons. Intrinsic ionic conduction, or creation of thermal vacancies by Frenkel, ie, vacancy plus interstitial lattice defects, or by Schottky, cation and anion vacancies, mechanisms can be expressed in terms of an equilibrium constant and, therefore, as a free energy for the formation of defects, If the ion is to jump into a normally occupied lattice site, a term for... 

In terms of the point defect energies so defined, our stoichiometry-conserving defects have formation energies given by ... 

Several points are worth noting about these formulae. Firstly, the concentrations follow an Arrhenius law except for the constitutional def t, however in no case is the activation energy a single point defect formation energy. Secondly, in a quantitative calculation the activation energy should include a temperature dependence of the formation energies and their formation entropies. The latter will appear as a preexponential factor, for example, the first equation becomes... 

Formula for the chemical potentials have been derived in terms of the formation energy of the four point defects. In the process the conceptual basis for calculating point defect energies in ordered alloys and the dependence of point defect concentrations on them has been clarified. The statistical physics of point defects in ordered alloys has been well described before [13], but the present work represents a generalisation in the sense that it is not dependent on any particular model, such as the Bragg-Williams approach with nearest neighbour bond energies. It is hoped that the results will be of use to theoreticians as well as... 

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. 

Most impurities can occur in different charge states we will see that H in Si can occur as H+, H°, or H. Which charge state is preferred depends on the position of the Fermi level, with which the defect can exchange electrons. Relative formation energies as a function of Fermi level position can be calculated and tell us which charge state will be preferred in material of a certain doping type. Section V will discuss charge states in detail. 

The formation energy of Schottky defects is described further in Chapter 2. 

There is no obvious reason why only one defect type should occur in a crystal, and several different species would be expected to be present. However, the formation energy of each defect type is different, and it is often a reasonable approximation to assume that only one or a small number of defect types will dominate the chemical and physical properties of the solid. 

Calculations of defect formation energies (Section 2.10) will allow estimates of the numbers of each kind of defect present to be made using formulas of the type ... 

What is the basis of atomistic simulation calculations of point defect formation energies ... 

Similar relationships hold if the Gibbs energy of formation per defect, Agy and the enthalpy of formation per defect, Afij, is used and R is replaced by k, the Boltzmann constant ... 

Some values for the enthalpy of formation of Schottky defects in alkali halides of formula MX that adopt the sodium chloride structure are given in Table 2.1. The experimental determination of these values (obtained mostly from diffusion or ionic conductivity data (Chapters 5 and 6) is not easy, and there is a large scatter of values in the literature. The most reliable data are for the easily purified alkali halides. Currently, values for defect formation energies are more often obtained from calculations (Section 2.10). 

Some experimental values for the formation enthalpy of Frenkel defects are given in Table 2.2. As with Schottky defects, it is not easy to determine these values experimentally and there is a large scatter in the values found in the literature. (Calculated values of the defect formation energies for AgCl and AgBr, which differ a little from those in Table 2.2, can be found in Fig. 2.5.)... 

Figure 2.5 Calculated variation of the formation energy of Schottky and Frenkel defects in the halides AgCl and AgBr as a function of temperature. [Redrawn from data in C. R. A. Catlow, Mat. Res. Soc. Bull., XIV, 23 (1989).]...

Calculations are now carried out routinely using a wide variety of programs, many of which are freely available. In particular, the charge on a defect can be included so that the formation energies, interactions, and relative importance of two defects such as a charged interstitial as against a neutral interstitial are now accessible. Similarly, computation is not restricted to intrinsic defects, and the energy of formation of... 

Results have shown that the properties of solids can usually be modeled effectively if the interactions are expressed in terms of those between just pairs of atoms. The resulting potential expressions are termed pair potentials. The number and form of the pair potentials varies with the system chosen, and metals require a different set of potentials than semiconductors or molecules bound by van der Waals forces. To illustrate this consider the method employed with nominally ionic compounds, typically used to calculate the properties of perfect crystals and defect formation energies in these materials. 

The energy of the solid containing the defect is minimized with respect to the various relaxations possible. The result is compared to that of the perfect defect-free structure to give a value for the defect formation energy. 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

An intrinsic defect is one that is in thermodynamic equilibrium in the crystal. This means that a population of these defects cannot be removed by any forms of physical or chemical processing. Schottky, Frenkel, and antisite defects are the best characterized intrinsic defects. A totally defect-free crystal, if warmed to a temperature that allows a certain degree of atom movement, will adjust to allow for the generation of intrinsic defects. The type of intrinsic defects that form will depend upon the relative formation energies of all of the possibilities. The defect with the lowest formation energy will be present in the greatest numbers. This can change with temperature. 

The formation energies of point defects in a pure metal are 1.0 eV (vacancies) and 1.1 eV (interstitials). The number of vacancies is ... 

The energy of formation of a defect in a typical metal varies from approximately 1 x 10 19 J to 6x 10—19 J. (a) Calculate the variation in the fraction of defects present, n([/ V, in a crystal as a function of the defect formation energy, (b) Calculate the variation in the fraction of defects present as a function of temperature if the defect formation energy is 3.5 x 10-19 J. 

The energy of formation of defects in PbF2 are anion Frenkel defect, 0.69 eV cation Frenkel defect, 4.53 eV Schottky defect, 1.96 eV. (a) What point defects do these consist of (b) What are (approximately) the relative numbers of these defects in a crystal at 300 K (Data from H. Jiang et al., 2000). 

The formation energy of Schottky defects in NiO has been estimated at 198 kJ mol-1. The lattice parameter of the sodium chloride structure unit cell is 0.417 nm. (a) Calculate the number of Schottky defects per cubic meter in NiO at 1000°C. (b) How many vacancies are there at this temperature (c) Estimate the density of NiO and hence the number of Schottky defects per gram of NiO. 

Calculations can provide information on the formation energy of these alternatives. Broadly speaking, the number of defects nd is related to the formation energy by an equation of the sort described in Chapter 2 ... 

In terms of formal point defect terminology, it is possible to think of each silver or copper ion creating an instantaneous interstitial defect and a vacancy, Ag and VAg, or Cu and Vcu as it jumps between two tetrahedral sites. This is equivalent to a high and dynamic concentration of cation Frenkel defects that continuously form and are eliminated. For this to occur, the formation energy of these notional defects must be close to zero. 

While zinc interstitials are possible, the formation energy for these defects is higher than that of oxygen vacancies. As in the case of NiO, continuing theoretical studies are needed to clarify the location of holes and electrons in these phases. 

These rules are not foolproof but should serve as a first approximation. A more rigorous approach must be used for more quantitative assessments. One possibility is to compute defect formation energies for the compound in question using atomistic simulations or quantum mechanical theory. These formation energies can be inserted into formulas similar to those described for simple defect populations (Chapter 2) ... 

Defect formation energy x 1019/J Defect formation energy x 1019/J... 

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