An investigation of electromigration induced void nucleation time statistics in short copper interconnects

Abstract

The stress evolution model ͑SEM͒ of Korhonenet al., is used to calculate the void nucleation time in a large number of short interconnects ͑lengths up to 50 m͒. Finite element calculations show that the effect of the nonlinearity in the SEM model is small, and that a mesh size of the order of the grain size is quite adequate to give accurate simulation results. Via failure is the only mode considered in the current calculations, however the gain in simulation time over other solution methods means that more complex situations, possibly including void dynamics, may be modeled in future in this way. Using normal mass-lumping methods the analysis is isomorphic to the voltage development on a random RC chain, so standard methods from very large scale integrated static timing analysis may be used to obtain dominant time constants at each mesh point. This allows the distribution of nucleation times to be obtained as a function of the distributions of line parameters. Under the assumption of a lognormal grain size distribution and a normal distribution of diffusion activation energies, the nucleation time distribution is shown to be close to lognormal.

Electromigration-induced failure in operando characterization of 3D interconnects: microstructure influence

“…However, due to the test time (up to months), only few samples can be analyzed with this method. Indeed, as microstructure is mainly responsible for the lifetime variations in EM studies [9,10], in operando results are strongly influenced by the microstructure of each sample, so that blind statistical studies on large numbers of devices remain necessary.…”

Analysis of critical-length data from Electromigration failure studies

“…Estimating L cr using data from ref [11]. for variations in D a with individual grain orientations, is necessary to interpret EM effects [17]. Consequently, short lines with fewer grains will have a wider variation.…”

Diffusivity variation in Electromigration failure

“…As in ref. [10] this is converted into a finite element expression based on chapeau shape functions and a mesh X k (k=1,2,…,N+1) consisting of the set of grain boundary points chosen according to some …”

“…Values for the physical parameters in eqn (3) are as given in ref. [10]. Eqn (3) may be used to investigate the nucleation, growth and shrinkage of voids (e.g.…”

“…Naturally, as the diffusivity values determine the rates of stress build up and void growth, a distribution of diffusivity values f D (D eff ) will lead to a distribution of failure times, all other things being equal. Consequently f D (D eff ) will account for some portion of the lognormal standard deviation observed in virtually all failure time distributions [10]. We shall show that, in order to simulate such void nucleation and void dynamics, a broad distribution f D (D eff ) is required.…”