15.defects_metal - T. Y. Tan 15. NATIVE POINT DEFECTS IN...

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T. Y. Tan 1 15. NATIVE POINT DEFECTS IN METALS AND IONIC CRYSTALS In this chapter we first discuss various aspects associated with the thermal equilibrium con- centrations of the native point defects in metals, which are the atomic defects vacancies V and self-interstitials I . The fundamental aspects of the discussion apply also to V and I in any other elemental crystals. Metals contain free electrons with a concentration equals to one to a few times of the density of atoms in the crystal. Thus, for metals whether the point defect species are charged or not is unimportant, because there is little possibility that in an experiment the electron concentration is altered. Furthermore, the interaction of point defects in metals is not as impor- tant as that in semiconductors, since this high density of electrons effectively screen one defect from another. In ionic crystals, native point defects always occur in pairs, with one being posi- tively and the other negatively charged. 15.1 Thermal Equilibrium Concentrations of Vacancies and Self-Interstitials in Elemental Crystals The physical reason for the existence of native point defects is that their presence lowers the Gibbs free energy of the crystal. For these defects, a sizable thermal equilibrium concentration exists for each species. For elemental crystals, including metals, the equilibrium pressure of its own vapor phase has no influence on the thermal equilibrium concentrations of the atomic point defects V and I . Let the Gibbs free energy of formation of such a defect be g f , then at a given temperature we have G = Cg f - TS m , (15.1) where G is the excess Gibbs free energy of the crystal due to the presence of the defect species, C is the defect concentration, and S m is the crystal entropy of mixing due to the presence of the defects: S m = k B ln C o ! (C o -C)! C! , (15.2) with C o being the number of possible site density the defect may occupy. For V , C o is the lattice site density. For I , C o has a further dependency on the specific type of sites they occupy in the
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T. Y. Tan 2 crystal, but it is always in the same order of that of the lattice site density. For example, for I oc- cupying the bond-centered interstitial sites in Si, C o is twice the lattice site density. It is specifi- cally noted that, at a given temperature, g f is a constant that differs from the chemical potential μ of the defect species (e.g., as given by Eq. (15.6) below) for which the role of the entropy of mix- ing has been also included. Using the Sterling approximation that ln(x!) xlnx-x for x>>1 holding, the function ln{C o !/[C!(C o -C)!]} is evaluated to be ln C o ! C! C o -C = C o ln C o - C ln C - C o -C ln C o -C . Noting that C/C
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15.defects_metal - T. Y. Tan 15. NATIVE POINT DEFECTS IN...

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