Reliable 3-D Clock-Tree Synthesis Considering Nonlinear Capacitive TSV Model With Electrical–Thermal–Mechanical Coupling

A robust physical design of 3-D IC requires investigation on through-silicon via (TSV). The large temperatures and stress gradients can severely affect TSV delay with large variation. The traditional physical model treats TSV as a resistor with linear electrical-thermal dependence, which ignores the fundamental device physics. In this paper, a physics-based electrical-thermal–mechanical delay model is developed for signal TSVs in 3-D IC. With consideration of liner material and also stress, a nonlinear model is established between electrical delay with temperature and stress. Moreover, sensitivity analysis is performed to relate the reduction of temperature and stress gradients with respect to dummy TSVs insertion. Taking the design of 3-D clock tree as a case study, we have formulated a nonlinear optimization problem for clock-skew reduction. By allocating dummy TSVs to reduce the temperature and stress gradients, the clock skew introduced by signal TSVs and drivers can be minimized. A number of 3-D clock-tree benchmarks are utilized in experiments. We have observed that with the use of dummy TSV insertion, clock skew can be reduced by 61.3% on average when the accurate nonlinear electrical-thermal–mechanical delay model is applied.