Physics-Based Compact TDDB Models for Low- $k$ BEOL Copper Interconnects With Time-Varying Voltage Stressing

Abstract

Time-dependent dielectric breakdown (TDDB) is one of the important failure mechanisms for copper (Cu) interconnects. This problem becomes more severe as the pitch between wires is shrinking and low-k dielectric materials (low electrical and mechanical strength) are used. Many TDDB models have been proposed based on different physics kinetics in the past. Recently, a physics-based TDDB model, which is based on the breakdown concept of electric path generation, has been proposed and has shown advantage over widely accepted existing electrostatic field-based TDDB assessment. However, determination of the time-to-failure from this model includes time-consuming finite-element method (FEM). In this paper, we try to mitigate this problem by developing fast time to failure evaluation method based on the closed form solution of the ion diffusion partial differential equations. We show that the location of the minimum concentration can be determined by the dominant terms with sufficient accuracy and the time to failure can also be computed with a few dominant terms. On top of this, we also consider the time-varying stressing voltages, which is commonly seen in practical VLSI chips. We propose to develop the equivalent dc stressing voltage, which is parameterized in terms of amplitude, duty cycle, and period for periodic stressing voltage waveforms using regression-based method. We further validate the proposed analytic TDDB concentration and time to failure formula, and the equivalent dc stressing voltage compact model against the results of an FEM analysis using COMSOL. Numerical results further show that the new compact TDDB model can lead to three orders of magnitude speedup with less than 1% error against the existing FEM results.