13.diff.mech - T Y Tan 13 DIFFUSION MECHANISMS In chapter...

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T. Y. Tan 1 13. DIFFUSION MECHANISMS In chapter 12 we have discussed the basic reason for diffusion to occur in crystalline matters as well as the basic governing expressions. In the present chapter, the atomistic mechanisms re- sponsible for the diffusion processes are discussed. In crystalline matters, point defects diffuse by themselves, the crystal self-atoms diffuse utilizing point defects as diffusion vehicles, and an impurity species may either be diffusing by itself or utilizing point defects as diffusion vehicles. These various different ways of diffusion are exhibited in the diffusion profile shapes as well as the diffusivity values. For constant D cases, it has been well established that the D values exhibit the Arrhenius behavior as functions of 1/T: D(T) = A exp - H k B T , (13.1) where H is an activation enthalpy, see Fig. 13.1 for a schematic illustration. D(T) is of the form of Eq. (13.1), because of the existence of the needed free energy of activation. This free energy of activation has two parts: (i) the free energy of migration; and (ii) the free energy of point de- fect formation. The free energy of migration is involved for all atomic species and point defects, the point defect free energy of formation is involved for substitutional impurity and self-atoms which must utilize point defects as their diffusion vehicles. In the following we discuss the dif- fusion mechanisms in accordance with whether point defects are involved. 13.1 Diffusion Mechanism Involving No Point Defects 13.1.1 Interstitial Mechanism Certain impurities reside in a crystal on interstitial positions, e.g., C in Fe; H in metals and in semiconductors; all noble gas atoms in metals and in semiconductors; O, Cu, Fe, Ni, etc. in Si. Almost all interstitial impurity atoms interact only weakly with the host crystal atoms via secon- dary bonding, i.e., of the van der Waals type. During the migration of these impurities, no point defects are involved, see Fig. 13.2 for a schematic illustration. Their diffusivities are described by D i = α a 2 ν exp - g i m k B T , (13.2)
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T. Y. Tan 2 where g i m = h i m - Ts i m , (13.3) is the free energy of migration, which is a barrier the impurity atom must surmount in order to make a jump. In Eq. (13.3), h i m is the enthalpy of migration and s i m the entropy of migration. In Eq. (13.2) α is a dimensionless geometric constant specifying the host crystal lattice structure and the number of paths the impurity interstitial can make the jump to a neighboring position, a is the lattice constant of the host crystal, and ν is the vibrational frequency of the atoms which is on the order of 10 13 s -1 . The free energy barrier g i m exists, since, during migration, the impurity atom must temporarily adopt a most unstable position before it reaches the neighboring stable position. The free energy difference between the unstable and stable positions of the atom consti- tutes g i m , see Fig. 13.3 for a schematic illustration. The probability that the atom can surmount the free energy barrier and therefore will successfully make the jump in one attempt is exp -g i m /k B T . Thus, for the atom attempting the jump with a vibration frequency
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13.diff.mech - T Y Tan 13 DIFFUSION MECHANISMS In chapter...

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