Atomic Flux Divergence-Based AC Electromigration Model for Signal Line Reliability Assessment Zhong Guan and Malgorzata Marek-Sadowska Electrical and Computer Engineering, UC Santa Barbara UC Santa Barbara, Santa Barbara CA 93106-9560 {zhong, mms}@ece.ucsb.edu Abstract In this paper, we develop an AC electromigration (EM) model for signal lines manufactured with copper dual damascene process. For the first time, the healing factor of AC EM is quantitatively modeled. To measure EM reliability of interconnects considering timing margins we introduce AC EM functional lifetime. We also develop an atomic flux divergence (AFD)-based void growth model to explain the resistance curves of measured results and calculate the functional EM lifetime of AC signal lines without extracting parameters from experiments. We demonstrate fidelity of the proposed model with measured results for both the healing factor and the rate of resistance change. 1. Introduction Electromigration (EM) is a gradual material transport due to the momentum transfer between conducting electrons and diffusing metal ions. In advanced technology nodes, EM is considered the dominant failure mechanism that limits reliability of VLSI interconnects. Conventionally, an interconnect line is said to fail due to EM when its resistance increases (usually by 10%). Interconnect lines affected by EM will generate voids but they can still function correctly for some time referred to as EM lifetime. In his pioneering work [1], J. R. Black developed a formula to statistically model the mean time to fail (MTTF) of a line with constant DC. In Black’s equation (1), A is a constant, j is current density, Ea is activation energy, k is Boltzmann's constant, T is temperature and n is a measurement-extracted model-constant with a value between 1 and 2. ) / ( exp kT E j A MTTF a n   (1) Black’s equation has been verified by many experiments and the majority of previous research on EM lifetime is based on it. This equation was initially proposed to model the lifetime of a line with constant DC before the void spans the whole trench. It has been observed [2][3] that lines with pulsed DC and AC have much longer MTTF than those with constant DC. However, as dimensions of interconnect lines shrink, lines with pulsed DC and AC have been reported to suffer from EM failures with comparable lifetimes to constant DC cases [4]. In other words, not only power grid lines (with DC load) but also signal lines (with pulsed DC or AC) may experience EM reliability problems. The existing works on pulsed DC and AC apply current conversion and determine the EM reliability equivalent DC in order to be compatible with Black’s equation. However, consensus among researchers has only been reached for high frequency pulsed DC in long lines, and simple via to via interconnect structure [5]. In such cases, the average current over time is used as the equivalent current. In general, the conversion methods from AC to DC are still researched. Unlike pulsed DC case, a simple conversion from AC to DC using the average current or root mean square (RMS) current either obtains too optimistic or too pessimistic MTTF. To fit the measured data, the concept of healing factor was proposed to facilitate conversion from AC to DC [6]. The EM reliability equivalent current density for AC case is then expressed as (2), where jAC, avg+ and jAC, avg- denote the time average current density including only positive or negative pulses and  is the healing factor.      , , avg ， AC avg ， AC EQ ， DC j j j  (2) This equation can correctly model the AC EM. However, the healing factor itself needs to be extracted from measurements and depends on several parameters such as frequency, current waveform, and physical structure of interconnect lines [4]. Thus, healing factor varies from sample to sample and previous research does not provide quantitative model for it. Signal lines such as clock can still function properly when a void spans the conducting trench due to the barrier layer conduction. Even with higher resistivity than copper, barrier layer can still pass the signal as long as there is enough timing margin. However, as the void keeps growing due to EM, the timing requirement may no longer be satisfied due to continuously increasing resistance. Therefore, the EM lifetime of such signal lines with AC EM should be decided by the resistance evolution curve of EM rather than the Black’s equation. Existing physics papers [7][8] all report similar resistance evolution curves obtained from physical measurements but do not provide a theoretical model. In this paper, we start from 3D RC model of interconnect line and propose to use atomic flux divergence (AFD) as the criterion to convert AC to its EM lifetime equivalent DC. Healing factor can be then quantitatively modeled and determined through simulation. In order to decide EM lifetime of those signal lines that can properly function when their resistance increases more than the percentage defining the EM failure, we propose an AFD-based three-step void growth model that can best fit resistance evolution curve. We conclude the paper presenting a general way of determining MTTF for AC signal line electromigration. The rest of this paper is organized as follows. In Section 2, we introduce the background knowledge on AC EM, healing factor, and EM reliability equivalent DC. In Section 3, we introduce atomic flux divergence and propose to use it for current conversion. In Section 4, we use a 3D distributed RC SPICE model and perform simulation to calculate healing factor and compare it with measurement results. In Section 5, we propose a new definition for the signal line EM lifetime, introduce the resistance evolution curve and its corresponding 978-1-4799-8609-5/15/$31.00 ©2015 IEEE 2155 2015 Electronic Components & Technology Conference