A 3-D Minimum-order Boundary Integral Equation Technique to Extract Frequency-dependent Inductance and Resistance in ULSI

The frequency-dependent resistance and inductance can be calculated by solving an eddy current problem. In this paper, a model to describe such 3-D eddy current problem is proposed, which is called 3-D Omni-A model because both of the conducting and non-conducting regions are described in terms of magnetic vector potential A. Therefore, the induced voltages of the conductors may appear as the unknowns directly in the boundary integral equations (BIE). Compared with popular coupled circuit methods, the computational method based on 3-D Omni-A model has two advantages. First, it does not fix the current direction along the axis of conductor, so in this method the perpendicular conductors may have mutual impedance. It could be more accurate in deep submicron (0.1μm) chips at high speed (10G Hz). Second, it only discretizes the surfaces of the conductor, so it could be more efficient.