1.4.4 Test Data Analysis and Lifetime Extrapolation

The primary interest for performing an accelerated electromigration test is to ensure the reliability of the interconnect by analyzing its lifetime under realistic use conditions [149]. The data resulting from the electromigration experiments should therefore be extrapolated to the device operating conditions, which are usually at room temperature and at a current density stress below 5mA/μm² [79].

The extrapolation is generally performed based on a lognormal statistical distribution of the measured individual lifetimes obtained for each structure of the sample [51,56,44,75]. The sample estimates for the MTTF and the standard deviation σ of a lognormal distribution are calculated from the average and the standard deviation of the natural logarithm of the individual lifetimes data, respectively, obtained from several electromigration stress tests with different stress conditions, as follows [52]

\[\begin{equation} \ln MTTF= \cfrac{\sum_{k=1}^{S} \ln TTF_{k}}{S} \end{equation}\] (1.4)

and

\[\begin{equation} \sigma= \sqrt{\cfrac{\sum_{k=1}^{S} (\ln TTF_k-\ln MTTF)^{2}}{S-1}}, \end{equation}\] (1.5)

where S is the sample size and TTF_k is the lifetime of the interconnect test structure of the sample. The lifetime lognormal distribution is characterized by a probability density function (PDF) in the form [73]

\[\begin{equation} PDF(t)= \cfrac{1}{\sigma t \sqrt{2 \pi}}\ \text{exp}\left(-\cfrac{(\ln t-\ln MTTF)^{2}}{2\sigma^{2}}\right) \end{equation}\] (1.6)

and the corresponding cumulative density function (CDF), which represents the probability that a randomly chosen part of the sample will fail by a time t^', is given by

\[\begin{equation} CDF(t)= \int_{0}^{t{'}} \cfrac{1}{\sigma t \sqrt{2 \pi}}\ \text{exp}\left(-\cfrac{(\ln t-\ln MTTF)^{2}}{2\sigma^{2}}\right) \text{d}t. \end{equation}\] (1.7)

Solving the integral from equation (1.7), we obtain

\[\begin{equation} CDF(t)= \Psi \left(\cfrac{\ln (t/MTTF)}{\sigma}\right), \end{equation}\] (1.8)

where ψ is the integral of the Gaussian function [29,73]. Once the MTTF and the deviation standard are determined, the extrapolation of the electromigration lifetime from testing (stress) to operative (oper) conditions is calculated, based on Black's equation, as follows [75]

\[\begin{equation} TTF_{oper} =MTTF_{stress}\left(\cfrac{j_{stress}}{j_{oper}}\right)^{n}\ \text{exp}\left[\cfrac{E_\text{a}}{k_\text{B}}\left(\cfrac{1}{T_{stress}}-\cfrac{1}{T_{oper}}\right)+\Psi^{-1}(f_{max})\sigma\right], \end{equation}\] (1.9)

where ψ^-1 is the inverse of the normal cumulative distribution function at a given maximum tolerable percentile of cumulative failure f_max at real operating conditions. The last exponential term in equation (1.9) allows to extrapolate the failure of the accelerated test (50%) to a small percentage accepted under operative conditions (0.01%) [40]. The lifetime under use conditions can be controlled by adjusting the parameters governing the lognormal distribution in equation (1.9). Its value is intensified by increasing the MTTF as well as reducing the standard deviation.

Electromigration lifetimes are usually described by a lognormal distribution, but multi-lognormal [160] and bimodal distributions [57] can also be used to statistically describe and analyze the lifetimes obtained from experimental tests.

The goal of the extrapolation methodology is to obtain the parameters relating the dominant physical effects which affect the electromigration induced failure. These parameters are the current density exponent n, related to the dominant phase of failure, and the activation energy E_a, related to the dominant mechanism of material transport in the interconnect line [40]. These parameters are extracted from the accelerated tests and are also valid at real operating conditions. The procedure for the estimation of the current density exponent n and its impact on the lifetime prediction is described in Section 1.4.3. The activation energy E_a can be determined in a similar manner [52]; its influence on the electromigration lifetime is related to the dependence of the transport of the material caused by electromigration on different available diffusivity paths in the interconnect.