The temperature dependence of the diffusion coeffi­cient has an Arrhenius form:

D = D"exp( “її

where Ha is the activation enthalpy of diffusion, and D0 is the diffusion prefactor that contains all entropy terms and is related to the attempt frequency for migration. When diffusion involves only an interstitial migrating from one interstitial site to an adjacent interstitial site, the activation enthalpy of diffusion is composed mainly of the migration enthalpy. In com­parison, for vacancy-mediated diffusion, dopants are trapped in substitutional positions and form a cluster with one or more vacancies. In such a situation, diffu­sion requires the formation ofthe cluster that assists in diffusion, migration of the cluster, and finally, the dissociation of the cluster. It is common for experi­mental studies referring to vacancy-mediated diffu­sion to refer to the activation enthalpy of diffusion. The activation enthalpy is the sum of the formation enthalpy and the migration enthalpy. The formation energy represents the energetic cost to construct a defect in the lattice (which may well require a com­plete Frenkel or Schottky process to occur). The for­mation energy of a defect, Ef (defect), is defined by

Ef (defect) = E (defect) + qme — E

j

where Ef (defect) is the total energy of the supercell containing the defect; q, the charge state of the defect; me, the electron chemical potential with respect to the top of the valence band of the pure material; the number of atoms of type j; and m the chemical poten­tial of atoms of type j. It should be noted that in this definition, contributions of entropy and phonons have been neglected. The migration energy is the energy barrier between an initial state and a final state of the diffusion process. For a system with a complex poten­tial energy landscape, there are a number of different paths that need to be considered.

1.02.4 Summary

Point defects are ubiquitous: as intrinsic species, they are a consequence of equilibrium, but usually they are far more numerous incorporated as extrinsic spe­cies formed as a consequence of fabrication condi­tions. Slow kinetics mean that impurities are trapped in ceramic materials, typically once temperatures drop below 800 K, although this value is quite material-dependent. The intentional incorporation of dopants into a crystal lattice can be used to funda­mentally alter a whole range of processes: this includes the transport of ions, electrons, and holes. As a result, diffusion rates and electrical conductivity can be manipulated to increase or decrease by many orders of magnitude.1,2 Other mechanical or radia­tion tolerance-related properties can also be changed radically.

This chapter has provided the framework for understanding the properties of point defects. In par­ticular, the understanding of the concentration of equilibrium-intrinsic species, dopant ions and their interdependence, defect association to form clusters and nonstoichiometry. In each case, these defects alter the lattice surrounding them, with atoms being shifted from their perfect lattice positions in response to the specific defect type. Electronic defects have been described: not only electrons and holes formed by doping, but also states formed by excitation. Struc­tural defects and electronic defects are considered together through Brouwer diagrams. Finally, we have also considered the transport of ions through the lattice via different processes, all of which require the formation of point defects.