Diffusion via native defects, and the appropriate choice of independent thermodynamic variables in both quasi-equilibrium and nonequilibrium experimental designs

ABSTRACTIt is often incorrectly assumed that temperature alone defines the equilibrium state of a crystal. However, the Gibbs phase rule shows that the number of independent thermodynamic variables required to define the equilibrium state depends upon the experimental design. In most practical cases, this means that temperature and components of the external phase or phases in proximate contact with the sample will determine the equilibrium state, e.g., the equilibrium native defect concentrations, impurity solubility, etc. If native defect concentrations approach their equilibrium values in a time which is short compared to the time of an experiment, then impurity diffusion can be described well by a thermodynamic model and impurity diffusion is analogous to classic tracer diffusion. If native defect concentrations require a long time to approach their equilibrium values, then diffusivities will exhibit significant time dependence and simple models do not apply. However, multi-phase, multi-component systems generally have several possible equilibrium regions within their phase diagram. Relating the phase diagram to a given experimental design allows one to qualitatively understand how the native defect concentrations change as a crystal in a nonequilibrium state relaxes toward one (of several possible) well-defined equilibrium states. Examples will focus largely on the diffusion of impurities in single crystal GaAs. Evidence will be presented that native defect concentrations can rapidly approach equilibrium in a limited group of experimental designs. We shall show why a description of diffusivity in terms of carrier concentration, a dependent thermodynamic variable, sometimes succeeds but often fails. Examples of commonly used experimental designs, in which there is inadequate control over the independent variables, will demonstrate some large variations in measured diffusivity. An enormous range of reported diffusivities, in GaAs covered by SiO2 or Si3N4 encapsulant layers, will demonstrate the difficulty and the futility of measuring diffusion when the solid is cut off from the external phases which define the equilibrium state.