Vacancies equilibrium processes

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

It is a straightforward but rather lengthy exercise to write down and evaluate the flux equations jA, jA, jB, jB under the assumption of local (vacancy) equilibrium (A v = 0). We find that five independent L-ti are needed to fully describe the transport in such a system. However, only four experimental parameters DA, DB, DA, and Db are available from flux measurements. Since DA = DB, jA jB in the solid solution crystal. Lattice site conservation requires that the sum of the fluxes /a + 7b + 7v = 0, that is,, /v = 0, despite X = 0. The external observer of the A-B interdiffusion process therefore sees the fluxes... 

Oxygen deficiency in pure quartz glasses is ordinarily associated with the presence of the two types of diamagnetic centers in glass oxygen vacancies (=Si-0)3Si-Si(0-Si=)3 and SCs (=Si-0)2Si . The enthalpy of the equilibrium process ... 

In addition to a vacancy-driven process, diffusion under irradiation may also be enhanced by the formation and diffusion of other point defects, such as selfinterstitials, divacancies, and other defect aggregates, which are not present under equilibrium conditions. A general statement for the atomic diffusion coefficient can be written in terms of the various point defects as... 

However, a second equilibrium process can occur in which the molecular form of the eluent solute is adsorbed by the resin matrix. This adsorption process also shifts the acid-base equilibrium so that a greater proportion of the eluent is in the molecular form. When a sample is injected that contains no benzoate or benzoic acid, a new equilibrium is established in which some of the adsorbed benzoic acid passes from the stationary phase into the mobile phase. After the zone occupied by the sample has passed, the resin re-equilibrates with the mobile phase to replace the adsorbed benzoic acid it has lost. This creates a vacancy in the mobile phase that contains a lower total concentration of benzoate plus benzoic acid. In the example described, a decrease in the detector signal is observed when the vacancy passes through the detector. This is commonly called the system peak even if it is in the negative rather than the positive direction. 

In diffusional processes, such as the classic Kirkendall effect of interdiffusion in a bulk diffusion couple of A and B, the atomic flux of A is not equal to the opposite flux of B. If we assume that A diffuses into B faster than B diffuses into A, we might expect that there will be a compressive stress in B, since there are more A atoms diffusing into it than B atoms diffusing out of it. However, in Darken s analysis of interdiffusion, there is no stress generated in either A or B. But Darken has made a key assumption that vacancy concentration is in equilibrium everywhere in the sample. To achieve vacancy equilibrium, we must assume that lattice sites can be created and/or annihilated in both A and B, as needed. Hence, provided that the lattice sites in B can be added to accommodate the incoming A atoms, there is no stress. The addition of a large number of lattice sites implies an increase in lattice planes if we assume that the mechanism of vacancy creation and/or annihilation is by dislocation climb mechanism. It further implies that lattice planes can migrate. 

The vacancy is very mobile in many semiconductors. In Si, its activation energy for diffusion ranges from 0.18 to 0.45 eV depending on its charge state, that is, on the position of the Fenni level. Wlrile the equilibrium concentration of vacancies is rather low, many processing steps inject vacancies into the bulk ion implantation, electron irradiation, etching, the deposition of some thin films on the surface, such as Al contacts or nitride layers etc. Such non-equilibrium situations can greatly affect the mobility of impurities as vacancies flood the sample and trap interstitials. 

Let us consider the typical mechanisms of spontaneous processes that decrease /. The direction and driving force of such mechanisms are determined by the laws of equilibrium thermodynamics, and the rate is proportional to diffusion in gases, viscosity in liquids, and transfer of atoms, vacancies, and other defects in solids. 

The first process produces doubly ionized positive oxygen vacancies and electrons (el) and the second produces doubly ionized positive cation interstitials and electrons. The equilibrium constants of the two processes are given by... 

Diffusion in general, not only in the case of thin films, is a thermodynamically irreversible self-driven process. It is best defined in simple terms, such as the tendency of two gases to mix when separated by a porous partition. It drives toward an equilibrium maximum-entropy state of a system. It does so by eliminating concentration gradients of, for example, impurity atoms or vacancies in a solid or between physically connected thin films. In the case of two gases separated by a porous partition, it leads eventually to perfect mixing of the two. 

In equilibrium, impurities or vacancies wiU be distributed uniformly. Similarly, in the case of two gases, as above, once a thorough mixture has been formed on both sides of the partition, the diffusion process is complete. Also at that stage, the entropy of the system has reached its maximum value because the information regarding the whereabouts of the two gases has been minimized. In general, it should be remembered that entropy of a system is a measure of the information available about that system. Thus, the constant increase of entropy in the universe, it is argued, should lead eventually to an absolutely chaotic state in which absolutely no information is available. 

More generally, co is independent of the external gas pressure k is the Boltzmann constant (1.38 x 10 erg deg ) and T is the temperature in Kelvin. Furthermore, the equilibrium between co and a collapsed CS plane fault is maintained by exchange at dislocations bounding the CS planes. Clearly, this equilibrium cannot be maintained except by the nucleation of a dislocation loop and such a process requires a supersaturation of vacancies and CS planes eliminate supersaturation of anion vacancies (Gai 1981, Gai et al 1982). Thus we introduce the concept of supersaturation of oxygen point defects in the reacting catalytic oxides, which contributes to the driving force for the nucleation of CS planes. From thermodynamics. 

Equations (1.206) and (1.207) describe the ionization of neutral vacancies (Vx, Vm). We assume here that the ionization of V and Vm to Vx and Vm does not take place. In a crystal in thermal equilibrium, electrons and holes will be formed by thermal excitation of electrons from the valence band to the conduction band, and the reverse process is also possible. This process can be expressed by eqn (1.210) as a chemical reaction, (see eqn (1.136)). Such reactions are called creation-annihilation reactions. Equations (1.208) and (1.209) describe the creation-annihilation reactions of neutral vacancies and charged vacancies in a crystal. Equation (1.211) shows the formation reaction of MX from constituent gases. It is to be noted that of these eight equations two are not independent. For example, the equilibrium constants Ks and K x in eqns (1.209) and (1.211) are expressed in terms of the other Ks as... 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

Let us refer to Figure 5-7 and start with a homogeneous sample of a transition-metal oxide, the state of which is defined by T,P, and the oxygen partial pressure p0. At time t = 0, one (or more) of these intensive state variables is changed instantaneously. We assume that the subsequent equilibration process is controlled by the transport of point defects (cation vacancies and compensating electron holes) and not by chemical reactions at the surface. Thus, the new equilibrium state corresponding to the changed variables is immediately established at the surface, where it remains constant in time. We therefore have to solve a fixed boundary diffusion problem. 

The vacancy flux and the corresponding lattice shift vanish if bA = bB. In agreement with the irreversible thermodynamics of binary systems i.e., if local equilibrium prevails), there is only one single independent kinetic coefficient, D, necessary for a unique description of the chemical interdiffusion process. Information about individual mobilities and diffusivities can be obtained only from additional knowledge about vL, which must include concepts of the crystal lattice and point defects. 

ANNEALING. The process of holding a solid material at an elevated temperature for a specified length of time in order that any metastable condition, such as frozen-in stains, dislocations, and vacancies may go into thermodynamic equilibrium. This may result in re-crystallization and polygonization of cold-worked materials. 

At sufficiently long times and high temperatures, equation 29 is fulfilled (33, 34). However, a substantial amount of time may be required to reach dynamical equilibrium (34, 35). This observation suggests that vacancy-selfinterstitial recombination is an activated process. In addition, under conditions in which point defects are injected, equation 29 may not be valid. 

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