Electroformed MIM structures: dependence of the current-voltage characteristics on the standard deviation of a normal distribution of filamentary radii

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Abstract
In electroformed metal-insulator-metal (MIM) structures, voltage controlled differential negative resistance (VCNR) is frequently observed. Such electrical characteristics are frequently modelled in terms of a matrix of conducting filaments spanning the insulating layer between the electrodes. The filaments are assumed to switch off at varying voltages Vp, depending on their individual resistances ρ, ultimately leading to the VCNR behaviour. The detailed current-voltage characteristics depend both on the nature of the relationship between Vp and ρ, and the probability density function P(ρ) of filament resistances. Often it is assumed that filaments switch off when their local temperatures reach some predetermined value; if the heat produced is exclusively by Joule heating, then Vp = βρ* where β is a constant depending on the geometry of the filaments and the thermal conductivity of the insulator. Previously, triangular and parabolic probability density functions have been investigated while more recently it has been argued that these are both approximations to the more fundamental normal distribution. In the present work it is argued that it is equally likely that the radii r of the filaments could be normally distributed, with mean rm and standard deviation σ. The corresponding distribution in terms of resistance is then obtained as a function of rm and a, and the resulting current-voltage characteristics are derived for various values of σ. The results obtained are compared to those obtained previously using other resistance distributions.