Rapid Detailed Temperature Estimation for Highly Coupled IC Interconnect

Steady-state temperature due to Joule self-heating for highly coupled integrated-circuit interconnect can be found rapidly on individual interconnect segments during electromigration reliability verification. It has previously been shown that the dc electric current solution on each interconnect segment of a net may be modified to form the analytical solution of the 1-D time-independent heat equation along the entire net. A symbolic solution of the network equations (requiring O(P³) operations, where P is the number of nodes) is first evaluated to solve the electrical problem and then evaluated again to solve the resulting Joule heat problem (each evaluation requiring O(P) operations). The symbolic solution is extended here to couple each interconnect segment to the weighted average temperature of the segments on neighboring nets. The temperature over the entire set of nets may be found by iterating until convergence, which does not require a significant overall increase in operations. The accuracy of the temperature trajectories is principally dependent on the validity of the assumptions that the temperature background seen by each individual interconnect segment is uniform and that vias conduct heat only along their lengths. The estimated temperature of self-heated nets is 110% of the finite-element result for a realistic layout example. The net-based solution is well suited to distributed processing and identifying problematic layout.