Three dimensional analysis of integrated circuit resistors

In integrated-circuit resistors current never flows unidirectionally. Indeed, for many integrated-circuit resistors three spatial coordinates are needed to specify the flow lines at any point within the resistor. This occurs because electrical contacts are made on the top face rather than on the ends of the resistor and because often the contacts do not extend all the way to the bounding surfaces of the resistor. This paper describes a method for determining the resistance of integrated-circuit resistors that accounts for the three-dimensionality of the current flow. The method involves solving the three-dimensional boundary value problem for the electroslatic potential. From the gradient of the potential one then finds the current that flows for a given voltage applied between the terminals. The work in this paper contrasts sharply with previous treatments of integrated-circuit resistors, none of which has accounted for three-dimensional flow.