Vacancy mechanism concentration-dependent diffusion

The diffusion of Si was studied in <100> GaAs which had been implanted with 40keV 30si+ ions. Profiles were determined by means of secondary-ion mass spectrometry and nuclear resonance broadening techniques. The implanted samples were subjected to annealing in Ar at 650 to 850C. Two independent Si diffusion mechanisms were observed. A concentration-independent diffusion, seen as a broadening of the initial implanted distribution, was very slow and was attributed to Si atoms that diffused interstitially. A concentration-dependent diffusivity with low solubility, was quantitatively explained in terms of the diffusion, via vacancies, of Si atoms on the Ga and As sub-lattices. The concentration-independent diffusion of Si was described by ... 

It is clear Ifom Fig. 7 that the silicon self-diffusion coefficient in undoped crystals is lower than in doped crystals. An opposite dependence was observed for the carbon self-diffusion coefficient. According to the law of acting masses, the silicon vacancy (acceptor) concentration increases with increasing donor dopant concentration (nitrogen). AU this leads to an increase in the diffusion coefficient in the presence of a vacancy mechanism of self-diffusion. Carbon vacancies are donors, and their concentration decreases when the donor concentration is increased. This leads to a decrease in the carbon self-diffusion coefficient. 

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. 

For diffusion by a vacancy mechanism, the temperature dependence of dilfusivity will depend on both the migration enthalpy A// and the energy required to form the vacancies if the latter are thermally activated i.e., the concentration of intrinsic defects is much greater than the concentration of extrinsic defects. If, however, A is fixed by doping, it becomes a constant independent of temperature. The activation energy for diffusion in the latter case will only depend on A/f, . 

Correlations of Nemst-Haskell [9] for electrolytes were mentioned. The effect of concentration, that is, dilnte versus concentrated solutions were separately discussed. Correlations of Wilkee-Chang, Siddiqi-Lucas, and Haydeek-Minhas were described. The diffusion mechanism in solids was discussed. The various mechanisms of diffusion such as vacancy mechanism, interstitial mechanism, snbstitu-tional mechanism, and crowd ion mechanism were outlined. Knudsen diffusion, when the mean free path of the molecule is greater than the diffusion path, as in pore diffusion, was discussed. Diffusion in polymers and the Arrhenius dependence of the diffusion coefficient with temperature were discussed. 

Diffusion in NijAl has been studied by few investigators - in particular Chou and Chou (1985) and Hoshino et al. (1988) -and has been reviewed and discussed with respect to mechanisms and defects (Bakker, 1984 Wever et al., 1989 Stoloff, 1989). The constitutional defects are antistructure atoms on both sides of stoichiometry, i.e. Al on Ni sites and Ni on Al sites, and the concentration of constitutional, i.e. ather-mal, vacancies is very small. The vacancy content of 6 x 10 at the melting temperature and the vacancy formation enthalpy of 1.60 eV correspond to the respective values for Ni, i.e. the vacancy behavior of NijAI is similar to that of pure metals (Schaefer et al., 1992). The diffusion of Ni in NijAl is not very different from that in pure Ni and at high temperatures it is insensitive to deviations from stoichiometry. The diffusion of Al in NijAl is less well studied because a tracer is not readily available. Defects may interact with dissolved third elements which affects diffusion. In particular vacancies interact with B which is needed for ductilization , and this leads to a complex dependence of the Ni diffusion coefficient on the Al and B content of NijAl (Hoshino etal., 1988). Data for the diffusion of the third elements, Co, Cr, or Ti, in Nij Al are available (Minamino etal., 1992). 

Extensive experiments have been carried out on the effect of impurity ions on the kinetics of decomposition, the optical properties, and the temperature dependence of ionic conductivity of several azides in an attempt to determine the nature and concentration of the species in the material. Torkar and colleagues studied the kinetics and conductivity of pure and doped sodium azide [97] and observed that cationic impurities and anionic vacancies speed up decomposition by acting as electron traps which facilitate the formation of nitrogen from N3. They also found that the activation energy for ionic conductivity was close to that for decomposition, implying a diffusion-controlled mechanism of decomposition. These results are qualitatively in accord with the microscopic observations of decomposition made by Secco [25] and Walker et al. [26]. 

In general, the existence of a Kirkendall effect in the sense of a displacement of markers does not permit a conclusion to be drawn as to a definite diffusion mechanism as, for example, a vacancy diffusion mechanism. To give a very improbable example - even if a ring mechanism is the predominant diffusion mechanism, a displacement of markers in the external system could nevertheless occur because lattice sites can be locally created or destroyed at sites of repeatable growth as a result of the fact that the concentration of vacancies, which are always present, depends upon the composition of the alloy. 

The diffusivity of a constituent such as the host metal or oxygen ions by an interstitial mechanism is not only proportional to the probability that the interstitial defect jumps, but also to the probabiUty that a constituent ion is interstitial, i.e., the fractional concentration of interstitials. Thus the diffusion coefficient of the constituent contains the temperature and oxygen pressure dependencies of the concentration of interstitials in addition to the temperature dependency of the mobility of these defects. As in the case of vacancy diffusion, the fixation of the defect concentration by doping or freezing as well as association and trapping of defects apply also to interstitial diffusion. 

This flux will be dependent upon the surface vacancy concentration, the surface electron concentration, and the dissociation rate of the dioxygen molecule however, at present, the rate-limiting step has yet to be identified. Kilner et al. [11] have derived a simple relationship for the surface exchange coefficient in terms of the bulk and surface vacancy concentrations, in an attempt to explain the apparent correlation found between the activation enthalpy for the surface exchange coefficient and the diffusion coefficient, in a number of La-based perovskites. Adler et al. [12] have also arrived at a similar relationship for k, by consideration of the AC electrode behavior of symmetrical cells with a double cathode structure. As already mentioned, the exact mechanisms of oxygen surface exchange remain elusive however, the vacancy concentration is clearly a very important parameter. 

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