Joule heating in single-walled carbon nanotubes (CNTs) using a quantum mechanical approach is presented in this paper. The modeling is based on the energy transfer between the electrons and both acoustic and optical phonons. In this formulation, only the knowledge of the full energy dispersion relation, phonon dispersion relation, and the electron-phonon coupling potential is required for the calculations. For verification of the proposed model, the current-voltage relation for extremely long nanotubes is calculated and the results are compared with the experimental data. The electric field dependence of the amount of energy generated by Joule heating is plotted. Moreover the effect of the thermal environment on the behavior of Joule heating is studied. The formulation proposed in this paper can also be used for structures other than CNTs. Computations indicate that, contrary to popular opinion, metallic CNT does not follow Joule’s law of $P=IV$ . Joule heating in CNT is significantly less than what is predicted with Joule law $\left(P=IV\right)$ , which would make it a perfect candidate to replace copper as interconnect material in electronics.

Reliability of solder joints under Electromigration (EM) and Thermomigration (TM) has drawn increasing attention in recent years, however current understanding regarding degradation physics of solder alloys under time varying current loading is still quite limited. As most integrated circuit (IC) connectors carry Pulse Direct Current/AC signals under normal operating condition instead of DC, investigation of EM/TM failure under time varying current becomes essential. In this paper, detailed formulation is presented to characterize material damage due to electromigration and thermomigration. Entropy based damage model is adopted to characterize the material degradation physics. Thermal gradient is observed to reach 1000 K/cm in current crowding regions, which is high enough to induce TM damage. Most of the time solder joint failure is due to combined action of EM and TM effects, therefore failure under pulsed current loadings due to combined action is investigated. Vacancy accumulation during current on period, and healing during current off period is modeled. In order to simulate EM/TM degradation of solder joints under pulsed current, ABAQUS user defined subroutines (UEL and UMAT) are used.

Degradation, damage evolution, and fatigue models in the literature for various engineering materials, mostly metals and composites, are reviewed. For empirical models established under the framework of Newtonian mechanics, Gurson–Tvergaard–Needleman (GTN) type model, Johnson-Cook (J-C) type damage model, microplasticity model, some other micro-mechanism based damage models, and models using irreversible entropy as a metric with an empirical evolution function are thoroughly discussed. For Physics-based models, the development and applications of unified mechanics theory is reviewed.

A fatigue model based on entropy generation is presented and validated through experiments. This model is purely physical and combines statistical mechanics with thermodynamic laws applied at a local scale. The model does not require an empirical damage surface or phenomenological constitutive modeling constants. Damage evolution parameter varies from 0 to 1. As it is for the irreversible internal entropy production, this parameter is a non-decreasing quantity that increases with the degradation of a material.