A derivation for the steady-state current J produced by a large homogeneous electric field E0 in the presence of a concentration gradient is presented which includes explicitly the effects due to lattice discreteness. The resulting equation is J=4aν exp(−W/kBT) sinh(ZeE0a/kBT)[C(L)−C(0) exp(ZeE0L/kBT)]/[1−exp(ZeE0L/kBT)],where C(0) and C(L) are the boundary concentrations of the diffusing species at the interfaces of the planar film at positions x=0 and x=L; e, the electronic-charge magnitude; Ze, the charge per particle of the diffusing species; 2a, the distance between adjacent potential minima; v, the frequency at which the ion attempts energy barriers which have height W in zero field; kB, the Boltzmann constant; and T, the absolute temperature. A derivation valid in the limit of a continuum model is also presented, and the results are compared numerically. The equations for the discrete and continuum models reduce to the results predicted by the ordinary linear diffusion equation for electric fields below approximately 105 V/cm. The relevance of the equations to the phenomena of anodic and thermal oxidation and to thin-film current-voltage devices is briefly described.
A model is given for the resistive-conductive transition in an amorphous or crystalline bistable resistance switch. The model is based on the observation that the current path in the switch is filamentary. The hypothesis is that the switching current heats the filament, and the resistive-conductive transition occurs only after the filament has achieved a critical temperature Tc. The predictions of the model are in excellent agreement with experimental observations of the resistive-conductive transition in single-crystalline Cu2O.
Wagner 1 based a theory for the formation of coherent oxide (and other tarnish) layers on metals on the hypothesis that particle transport through the layer occurs by the separate diffusion of ions and electrons by lattice-defect mechanisms, the driving force being the electrochemical potential gradient in the layer. The formation rate was considered to be limited by either ions or electrons, depending upon which had the smaller partial conductivity since equal magnitude charge currents were assumed. This model has been verified for a number of tarnishing processes, 1 one example apparently being very thick copper-oxide films (1.25 xl0~2 cm) formed during the high-temperature oxidation of copper. 2 Equal charge concentra-p. 109. 10 G. Dolling and R. A. Cowley, Proc. Phys. Soc. (London) ^8, 463 (1966); G. Dolling, private communication. u The Ge frequency-distribution function given in Ref. 10 is for 90°K. Following a suggestion of Dolling (private communication) based on the work of B. N. Brockhouse and B. A. Dasannacharya [Solid State Commun. 1. , 205 (1963)], the distribution at 300°K was obtained from the one at 90°K by multiplying all vibrational frequencies by 0.983. 12
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