Thermal analytical model for analysis of pulsed DC electromigration results

Thermal calculations have been performed in order to evaluate the real operating temperature of an interconnection during a pulsed current stress. The developed thermal model gives a good approximation of the average temperature needed for the exploitation of pulsed electromigration tests, in function of the stress parameters : frequency, duty cycle and current density. Electrical measurements on single level AlCu metallization allowed us to verify the model and to point out some thermal effects in the high frequency region. Introduction In order to exploit electromigration results with the Black s equation [1] , the knowledge of the temperature of the failure site is needed. During DC tests, this temperature is considered being the average temperature of the line, is assumed to be constant and is calculated through resistance measurements. The problem becomes more complex when periodic currents are applied to the sample. Some authors have already faced this problem during their pulsed EM studies [2-4]. Hereafter, we develop a simple thermal model which allows us to calculate more precisely the real sample temperature in the case of unidirectional current pulses and which can be applied to bi-directional current stresses. Thermal model Our thermal model describes the sample as a first order system, which means the use of one thermal time constant [2,3]. This hypothesis has been verified via 2D simulations using a finite elements method. A purely resistive line is submitted to a periodic Joule heating. The power P(t) dissipated during the time interval dt is equal to the accumulated heat added to thermal losses (Equation 1). • P(t)dt dT (T T )dt s = + − α β (1) Ts : Substrate temperature α, β : Constants For one power step with a peak value P0, Equation 1 can easily be solved by assuming that T=T0 for t=0 : • T(t) T exp( )( exp( = − − − 0 1 β α β β α t) + ( P + T t)) 0 S (2) where α is the thermal capacitance, 1/β the thermal resistance. Thus the thermal time constant of the sample is : τth= α/β=Cth.Rth In the following, we set down Ts=0°C, DTm=P0/b. ∆Tm is then the DC Joule heating of the sample. By applying square power pulses, the temperature of the sample reaches a periodic regime and evolves in a temperature interval [T1 ; T2] defined by the stress conditions and the thermal time constant. Figure 1 shows the plot of Temperature vs Time for 2 values of τth at steady state.