Temperature gradient effects in electromigration using an extended transition probability model and temperature gradient free tests. I. Transition probability model

Temperature gradient effects incorporated in electromigration are examined via the movement of vacancies. To explain the movement of metal ions or vacancies, an extended transition probability model with temperature gradients and vacancy concentration gradients is compared with the usual drift-diffusion model including the equation of continuity and the Einstein relationship. The self-consistent constant vacancy source boundary condition for repeated SWEAT structure is proposed so as to solve the equation of continuity. A derived time-to-fail equation from our model with temperature gradients is similar to the original Black´s equation. Also temperature gradient effects are simulated using our temperature model and the vacancy concentration profile that will ultimately lead to failure is investigated for various current densities and substrate temperatures. Based on our simulation and experiments at extremely high current density (typical of SWEAT type tests), there is a failure site with very small standard deviation for both NIST and SWEAT when strongly controlled by temperature gradient effects. By adding a compensating heat flow structure under the interconnection region, a temperature gradient free test structure was designed and tested with various temperature gradients. Test results show dramatic changes in failure location and time to failure as the amount of the temperature gradient is varied