Vacancy equilibrium concentration

According to Sieradzki and Friedersdorf [63], the vacancy equilibrium concentration could be increased by normal stress. They suggested the following equation for the crack velocities that result from surface atom mobility ... 

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. 

Consider now the system Cu/CujO in oxygen gas at a pressure p (X signifies the oxide/oxygen interface in Fig. 1.75). Ignoring space charges, x the equilibrium concentration of cation vacancies or positive holes at the CujO/Oj interface, is given by... 

In thermodynamic equilibrium, the free energy has a minimum. Accordingly, F does not change with the number of introduced vacancies n. Feeding Eq. (2) into Eq. (3) results in the following equilibrium concentration of vacancies ... 

A Schottky defect in a crystal consists of a cation and anion vacancy combination that ensures overall electroneutrality in the crystal (Section 1.9). The estimation of the configurational entropy change in creating a population of Schottky defects in a crystal can be obtained in the same way as that of a population of vacancies in a monatomic crystal. The method follows that given in Section 2.1 for the equilibrium concentration of vacancies in a monatomic crystal and is set out in detail in Supplementary Material S4. 

The most important application to be considered under this heading is the calculation of intrinsic defect concentrations in dilute solid solutions. If the solution is so dilute that only the leading terms in the various cluster expansions need be retained then the results required are slight generalizations of those above and follow at once from the notation for the general results. For example, the equilibrium concentration of vacancies in a dilute solution of a single solute, s, is found from Eqs. (74a) and (75) to be... 

The results indicate that a supersaturation of vacancies, c/co — 10, at the catalyst s surface is required to nucleate CS planes in M0O3 catalysts. CS planes are formed by the elimination of anion vacancies in supersaturation (where the supersaturation is defined relative to the background concentration of anion vacancies in equilibrium with CS planes, as described earlier) (Gai 1981, Gai et al 1982). The driving force for the nucleation of the CS fault is the difference between the chemical stress due to the supersaturation of anion vacancies of the faulted (defective) structure and the force required to create the fault. The estimate of Co is consistent with the equilibrium concentration of anion vacancies found in electron beam heating studies of M0O3 in vacuum (Bursill 1969). 

Table 1.13 Formation Energy of Vacancies for Selected Elements and Equilibrium Concentrations at Various Temperatures...

The resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

Of particular interest in kinetics is the non-conservative dislocation motion (climb). The net force on a dislocation line in the climb direction (per unit length) consists of two parts Kei is the force due to elastic interactions (Peach-Koehler force), Kcbcm is the force due to the deviation from SE equilibrium in the dislocation-free bulk relative to the established equilibrium at the dislocation line. Sites of repeatable growth (kinks, jogs) allow fast equilibration at the dislocation. For example, if cv is the supersaturated concentration and c is the equilibrium concentration of vacancies, (in the sense of an osmotic pressure) is... 

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. 

The smoothing of a rough isotropic surface such as illustrated in Fig. 3.7 due to vacancy flow follows from Eq. 3.69 and the boundary conditions imposed on the vacancy concentration at the surface.12 In general, the surface acts as an efficient source or sink for vacancies and the equilibrium vacancy concentration will be maintained in its vicinity. The boundary condition on cy at the surface will therefore correspond to the local equilibrium concentration. Alternatively, if cy, and therefore Xy, do not vary significantly throughout the crystal, smoothing can be modeled using the diffusion potential and Eq. 3.72 subject to the boundary conditions on a at the surface and in the bulk.13... 

There is a sufficient density of sources and sinks for vacancies so that the vacancies are maintained at their local equilibrium concentration everywhere. 

The current majority opinion is that both types of point defects are important. Thermal equilibrium concentrations of point defects at the melting point are orders of magnitude lower in Si than in metals. Therefore, a direct determination of their nature by Simmons-Balluffi-type experiments (26) has not been possible. The accuracy of calculated enthalpies of formation and migration is within 1 eV, and the calculations do not help in distinguishing between the dominance of vacancies or interstitials in diffusion. The interpretation of low-temperature experiments on the migration of irradiation-induced point defects is complicated by the occurrence of radiation-induced migration of self-interstitials (27, 28). 

Point Defect Models of Diffusion in Silicon. Under conditions of thermal equilibrium, a Si crystal contains a certain equilibrium concentration of vacancies, C v°, and a certain equilibrium concentration of Si self-interstitials, Cz°. For diffusion models based on the vacancy, Cv° Cf and the coefficients of dopant diffusion and self-diffusion can be described by equation 27 (15)... 

It is estimated that the energy to form an interstitial atom in a metal is four times as great as that to form a vacancy. Estimate the equilibrium concentration of interstitials just below the melting point. 

Cic is the equilibrium concentration of cation interstitials CVc is the total concentration of cation vacancies... 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

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