Crystal, defect, point equilibrium

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. 

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

Let us first examine what happens to a crystal when we remove, add, or displace an atom in the lattice. We will then describe how a different atom, called an impurity (regardless of whether or not it is beneficial), can fit into an established lattice. As shown by Eq. (1.36), point defects have equilibrium concentrations that are determined by temperature, pressure, and composition. This is not true of all types of dimensional defects that we will study. 

Figure 11.3 is a plot of nAE, AS, and AG. From this plot you can see that introducing vacancies lowers the free energy of the crystal until an equilibrium concentration is reached adding too many vacancies increases G again. At higher temperatures the equilibrium number of vacancies increases. The implications are important. In pure crystals we expect to hnd point defects at all temperatures above OK. Since these defects are in thermodynamic equilibrium, annealing or other thermal treatments cannot remove them. 

According to Nernsfs postulate, completely ordered solid phases (i.e. ideal crystals) occur when equilibrium is attained at 0 K. However, at finite temperatures the concentrations of point defects are fixed through the condition of minimization of the free energy. For given values of pressure, temperature, and component activities, these concentrations are dependent upon the magnitude of the free energy of formation of the defects. As an example, let us briefly consider the temperature dependence of the vacancy concentration in a crystal composed only of A atoms. If Ni are the mole fractions, then the free enthalpy (7 per mole of A lattice sites is given by where / stands for atoms A and for vacancies V, and where /if is the partial... 

From eqs. (4-5) and (4-8) it can be seen that formal mass action equations may be written for the concentrations of defects in equilibrium when P, T, and the activity or partial pressure of a component are given. As has already been pointed out, the formulation of the reaction equation connecting the defect concentrations and the component activities is to some degree arbitrary, as long as the previously mentioned rules are properly obeyed. This results from the existence of internal defect equilibria which are always maintained if the crystal is in thermodynamic equilibrium. In the present case, for instance, we can write several other reaction equations. For example, we could write ... 

Intrinsic defects such as lattice vacancies or interstitials are present in the pure crystal at thermodynamic equilibrium. The simplest of these crystalline defects involve single or pairs of atoms or ions and are therefore known as point defects. Two main types of point defect have been identified Schottky defects,in which an atom or ion pair are missing from the lattice (Figure 3.35a), and Frenkel defects, in which an atom or ion is displaced from its ideal lattice position into an interstitial site (Figure 3.35b). 

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 ... 

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... 

Figure 2.1 Change in Gibbs energy, AG, of a crystal as a function of the number of point defects present (a) Variation of AG with number of point defects (schematic) at equilibrium, neq defects are present in the crystal (b) calculated variation of AG for hy = 0.6 eV, kT = 0.1, N = 1000 the equilibrium number of point defects is 2.5 per 1000.

At all temperatures above 0°K Schottky, Frenkel, and antisite point defects are present in thermodynamic equilibrium, and it will not be possible to remove them by annealing or other thermal treatments. Unfortunately, it is not possible to predict, from knowledge of crystal structure alone, which defect type will be present in any crystal. However, it is possible to say that rather close-packed compounds, such as those with the NaCl structure, tend to contain Schottky defects. The important exceptions are the silver halides. More open structures, on the other hand, will be more receptive to the presence of Frenkel defects. Semiconductor crystals are more amenable to antisite defects. 

The interesting point is that thermodynamically we do not expect a crystalline solid to be perfect, contrary, perhaps to our commonsense expectation of symmetry and order At any particular temperature there will be an equilibrium population of defects in the crystal. 

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