Modeling of Electromigration and Lock-In Thermography in Microelectronics and Microelectronics Packagin

As technologies in microelectronics going forward in the direction of scaling down from micrometer scale to nanometer scale, and from two dimensions to three dimensions, electromigration remains as one of the critical reliability issues at both device and package levels. In order to achieve the satisfactory level in the reliability of microelectronic packages, new or improved techniques to identity the location of failures are required. This work reports the results of modeling in the study of electromigration and Lock-in Thermography (LIT) technique.;First, the governing equations of electromigration and several variations of the existing theories are studied for comparison purpose. Their results and arguments are discussed, and a new theory that couples the diffusion with stress and temperature is presented. The present mathematical model displays excellent flexibility to study the phenomena observed in electromigration, such as Blech's length, the effective charge, the increase in the line resistance, and the stress evolution of interconnect line. The effect due to the temperature gradient in electromigration is also studied. There are four major driving forces in electromigration, which are electrical current, vacancy concentration gradient, mechanical stress gradient, and temperature gradient. Without considering the mechanical stress, the predicted time to failure was shown to be several orders of magnitude faster than the time to failure observed in experiments. It was found that the effect due to the mechanical stress led to the important differences in time scale. The effect due to the temperature gradient is found to be generally small compared to those caused by other driving forces. However, the parametric study shows that at high temperature gradient, which is the result of Joule heating effect at high current density, the mechanical stress generated by thermomigration can affect the time to failure of the interconnect line.;Next, the current density exponent n in relationship to time to failure is investigated. Results show that the current density exponent n equals to 2 if semi-infinite line configuration is assumed, which is in agreement with Black's empirical formula. For finite length line, the effect due to boundary condition becomes apparent that it can cause the current density exponent n to be greater than 2. At high current density, the effect due to temperature gradient is found to cause the current density exponent n to be less than 2. These findings propose a new angle to explain the experiment data where the current density exponent n is not equal to 2.;Finally, the dimensional and heat source effects in Lock-in Thermography (LIT) applied to semiconductor packages are studied. LIT is a powerful non-contact and non-destructive investigating technique that has recently emerged in semiconductor industry. The current LIT applications interpret LIT data using the conclusions drawn from the semi-infinite models. However, semiconductor package has finite geometry with convective heat transfer on surface. These conditions are not considered in semi-infinite model. This work studies the accuracy of the interpretation of LIT data based on the commonly used semi-infinite models. The effects of finite dimensions, heat source configuration, convective heat transfer coefficient, thickness of material, thermal diffusivity of material, and lock-in frequency are examined. Furthermore, the effects of heat source's shape and location, as well as the problems involving non-homogeneous materials are also investigated. It has been found that the difference in phase shift, defined as (Delta-phi), between the actual phase shift and the one predicted by semi-infinite model is notable at low lock-in frequency, and decreases to zero at high lock-in frequency. This (Delta-phi) results in an overestimate of the z-depth information when semi-infinite model is employed to interpret LIT data. Numerous errors may also occur when the semi-infinite models are extended to a system with multiple materials.