A Percolative Approach to Electromigration Modelling
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A Percolative Approach to Electromigration Modelling C. Pennetta1,2, L. Reggiani1,2, Gy. Trefán2, F. Fantini3, A. Scorzoni4, I. DeMunari5 1 Lecce University, Dept. of Innovation Engineering, Lecce, Italy 2 National Institute for Material Science, INFM, Italy 3 Modena University, Dept. of Engineering Sciences, Modena, Italy 4 Perugia University, Dept. of Electronic and Information Engineering, Perugia, Italy 5 Parma University, Centro MTI, Parma, Italy ABSTRACT We present a stochastic model which simulates electromigration damage in metallic interconnects by biased percolation of a random resistor network. The main features of experiments including Black's law and the log-normal distribution of the times to failure are well reproduced together with compositional effects showing up in early stage measurements made on Al-0.5%Cu and Al-1%Si lines. INTRODUCTION AND MODEL We present a stochastic approach which simulates electromigration (EM) damage in metallic interconnects in terms of percolation in a random resistor network [1]. The proposed approach has the inherent novelty of exhibiting specific stochastic features during degradation and healing processes associated with a current stress [2]. Here we present an extension of the previously proposed model [3] which allows us to reproduce most of the compositional effects (CE), often acting during the early stages of EM [4,5]. Typical examples of CE for Al-0.5%Cu lines under standard Median Time to Failure (MTF) measurements and for Al-1%Si lines under high resolution measurements made with the ratio of resistance (ROR) technique [6] will be reported in the next section and compared with the numerical results of the model. We describe a metallic line as a two-dimensional square-lattice network of resistors (denoted as "regular" resistors, rreg) laying on an insulating substrate at temperature T0. The resistance of each n-th resistor depends on temperature as: rreg,n (Tn ) = rref [1 + α (Tn - Tref )]
(1)
Here α is the temperature coefficient of resistance (TCR), Tn the local temperature, Tref and rref the reference values for the TCR. When Joule heating is negligible, the resistors are all equal to r0=rreg(T0). To save computational time, instead of using long rectangular networks we perform the calculations on a square N x N network, where N determines the linear sizes of the studied region, with the total number of resistors being Ntot = 2 N2. Our network thus represents the region of dominant void growth of the film. As the line length is taken f times greater than the linear size of the studied region, the relative resistance variation of the whole line is obtained by multiplying the relative resistance variations of the network by the factor 1/(1+f). The network is contacted at the left and right hand sides to perfectly conducting bars, acting as electrical
D2.7.1
contacts, through which a constant stress current I is applied. The Joule heating induced by I is taken into account by defining Tn as: B Tn = T0 + A rnin2 + N neigh
N neigh
∑ (r
m=1
2 m,n m,n
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