Consideration on a boundary condition for impulse thermal assumption

Abstract

In general, the equation of thermal breakdown is given as follows: Cv (∂T/∂t) − div (κ grad T) = j F, (1) where Cv is the heat capacity of the material per unit volume, κ the thermal conductivity, j the current density, and F the electric field. Consider the case when the heat conduction from the dielectric surface to the surroundings is ideally complete. Provided the voltage duration to breakdown tB is sufficiently shorter than the thermal time constant of dielectric tth, that is tB << tth (2) the second term of the left-hand side in eq.(1), which is the heat conduction term, can be ignored. In such a case, the impulse thermal assumption can be expected to be valid [1]. However, many reports have been published which insist the impulse thermal mechanism even under the condition tth < tB [2,3,4,5]. It is surprising that this case appears in the breakdown of plasma-polymerized styrene thin films with about 3000Å thickness [5]. These experimental results imply that provided an unsufficient heat conduction from the dielectric material is taken into account, the impulse thermal breakdown form is possibly established even though the condition tB << tth is not fulfilled. In this paper, a quantitative examination on this relationship is made by solving the fundamental equation of thermal breakdown numerically under the boundary condition which obeys Newton's law of cooling for various values of heat transfer coefficient A from the dielectric surface to the ambient. As an example, assuming ionic conduction and varying the ionic conduction parameters, the electric strength is calculated as a function of λ. From these results, the validity of approximation to the impulse thermal assumption is examined quantitatively. λ is also discussed.