Calculation of the Current Density in the Contacts of a Thin Film Resistor

Abstract

The two-dimensional boundary value problem appropriate to current flow in a film resistor is examined. A simple closedform solution for current density into the contact is found to exist for the important case of a thin film resistor with extended lands. The spatial dependence of the current density into the contact is found to be similar to that obtained by Kennedy and Murley for the diffused resistor, with film thickness entering the functional dependence in a role analogous to the diffusion length of the dopant ion in the diffused resistor.where the constants have been chosen so that the origin in the t plane is transformed into the origin in the Zl plane, and the point t = -jt is transformed into infinity in the Zl plane. The equipotentials between~= d/2 and = (d/2) + L, TJ = 0, are transformed into equipotentials between points a and (3 in the Zj plane by the substitutions positive and negative~directions. Contacts of infinite conductivity and finite length L are applied as shown, and it is desired to solve the boundary value problem for J(x), the current density into the contact through the plane TJ = O. The origin in the t system is at 0', and the origin for the coordinate x is at 0, which is the inner edge of the contact along the positive~axis. The resistor and its lands have finite width W in the direction perpendicular to the plane of the figure. Since the corresponding two-dimensional boundary value problem has been considered in the literature," we shall first outline the solution and then apply suitable simplifications.The strip in the t plane is mapped into the upper half of the Zj plane, Fig. l(b), by the transformation

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