Equilibrium thermodynamics of point defect formation

It is to be established that Eq. (5.1) has not been very precisely formulated, since it is possible to add species from outside to the perfect solid (or vice versa), so that, in addition to the re-arrangement , it is also possible to encoimter substitution , addition and elimination processes . In all cases it is possible to describe, with thermodynamic precision, the real solid as a superposition of defect building units and the constituents of a perfect solid (the monomeric units are sometimes termed lattice molecules ), as we will now see in some detail. 

Let us take a simple example, namely the introduction of a vacancy into an elemental crystal, for which it is already possible to demonstrate the relevant points. Let us first consider a typical representative of an ideal covalent crystal, namely diamond, the ground structure of which was shown in Fig. 2.22. 

The creation of a carbon vacancy means the transfer of an internal C atom to the surface. This leads to a crystal with slightly reduced density. In order to remove an internal carbon atom from its crystal assembly it is necessary to break 4 bonds and, thus, to provide energy equal to 4ec-c- Since this C atom is not moved to infinity but to the surface of the crystal, the energy 2ec-c is recovered. The net loss of 2ec-c is exactly the lattice energy (4ec-c/2, see Eq. (2.45)). If the energy loss is estimated in this manner we get 

It is obvious that it is the static dielectric constant that must be inserted (for dilute defects) a typical value for alkali halides is 5. 

If we tentatively identify d with b in Eq. (2.37), it follows for the formation energy 

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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... 

In order to be able to calculate the concentrations of point defects at thermodynamic equilibrium, it is necessary to know the change in free energy of the crystal which accompanies the formation of point defects, since the equilibrium is determined by the minimization of the free energy when the pressure, the temperature, and the other independent thermodynamic variables are given. A theoretical calculation of the free energy of formation of defects is still one of the most difficult problems in solid state physics and chemistry. The methods of calculation for each group of materials - metals, covalent crystals, ionic crystals - are all very... 

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. 

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 ... 

The concentrations of point defects, and therefore the direction and degree of nonstoichiometry, are determined by the thermodynamic activities of external phases in equilibrium with the nonstoichiometric compound. For example, the concentration of X vacancies may be controlled by maintaining the crystal in equilibrium with another phase containing X at a definite activity. The external phase may be X2 gas (e.g., O2). The equation for formation of X vacancies then may be written... 

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... 

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... 

The concentration of defects can be derived from statistical thermodynamics point of view, but it is more convenient treat the formation of defects as a chemical reaction, so that equilibrium constant of mass action can be applied. For a general reaction, in which the reactants A and B lead to products C and D, the equation is given by ... 

Following the structural description of these defects, we should make some brief remarks regarding their energies of formation. If the defect energies and their spatial distribution are known, then thermodynamics can be used to give information regarding concentrations and stabilities. In this connection, however, it is observed that only point defects are thermodynamically stable. That is, only point defects are in an equilibrium state which is uniquely determined by the specification of the requisite number of independent variables such as pressure, temperature, and composition. The concentrations and configurations of all other defects depend upon the manner in which they were introduced into the crystal. That is, in the final analysis, they depend upon the method of preparation. 

Despite their relatively high enthalpy of formation, point defects are the only defects which exist in thermodynamic equilibrium in any appreciable concentrations. As shown in section 3.3.1, about the same amount of energy is required per atom on a dislocation line as is required for a point defect. However, one dislocation line can contain about 10 such atoms, and the thermodynamic probability of defects being lined up along a dislocation line, as opposed to being freely distributed in the crystal, is very small. Thus, it can easily be appreciated that the probability of the occurrence of equilibrium dislocations is negligible. 

On a scale a thousand times smaller, particles in dense colloidal suspensions also form two- or three-dimensional lattices, both of which can contain dislocations. There is a fundamental difference between dislocations in two- and three-dimensional lattices. In the former case, dislocations are point defects, which can be in thermodynamic equilibrium and can lead to phase transformations. Then-appearance into the hexagonal close-packed crystal leads to the loss of translational order and the formation of the hexatic phase. Their subsequent dissociation into dischnations leads to loss of the oriental order and formation of the Uquid phase [6-8]. Colloidal systems are well suited for experimental exploration of two-dimensional dynamic systems [9,10], and the results have played an important role in the development of this field [11,12]. 

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