Accurate calculations of the high-frequency impedance matrix for VLSI interconnects and inductors above a multi-layer substrate:

Impedance characterization of Interconnects and intentional Inductors in the broad frequency domain that extends from near DC to 100’s of GHz in integrated circuits is full of unachieved goals. Existing computational methods are near the end of their usefulness, since accurate characterization of the Impedance matrix Z(ω) at high-frequencies with existing methods can only be applied to small structures. We present a computationally inexpensive approach that extends the ability for accurate characterization to problem sizes that are between one and two orders of magnitude larger, opening the door to the validation of high frequency wireless circuits in terms of real time simulation, rather than the less desirable alter-native of validation by manufacturing and testing. The starting point in our approach is an integral representation of the Green’s function for the magnetic vector potential in classical Electromagnetic theory. The intermediate computation involves a least-square fit to refection coefficients in terms of linear combinations of complex exponentials, so as to render integrable the coordinate space representation of the Green’s function. The end result is an analytical description of derivative quantities, including the matrix elements of the serial Impedance matrix of the interconnect configuration, for all frequencies of interest. We study the problem in two and three dimensions. Among the alternative least square fits, we found that those utilizing VARPRO in the complex domain – an extension of VARPRO created specifically to attack this problem – give the best results. The levels of accuracy (errors less than 3%) and efficiency (better than an order of magnitude lower computational cost than existing methods) have a major impact on nano-electronic circuit design.

Keywords: High-frequency impedance matrix; VLSI interconnects; VLSI inductors; multi-layer substrate; high-frequency wireless circuits; real time simulation; complex exponentials