Inadequacy of Markov Model in Modeling of Electromigration-induced Resistance Degradation Aparna Adhikari Department of electronics West Bengal State University Barasat, Kolkata, India 700126 adhikariaparna61@gmail.com Arijit Roy * Department of electronics West Bengal State University Barasat, Kolkata, India 700126 * Corresponding Author arijitroy@live.com Abstract—Electromigration-induced resistance change carries lot of information about the failure phenomenon and is an important aspect of the degradation process. Hence modeling electromigration-induced resistance degradation is of paramount importance, especially for submicron dual- damascene Cu interconnects. On the other hand, Markov model is extensively used in reliability engineering. This study focuses on the nature of the time-domain discrete states in the failure process. We argue about the memoryless discrete states in Markovian model to predict the electromigration-induced resistance degradation. The physics behind the electromigration failure does not support the application of Markovian model in electromigration and the inadequacy of such application is described. In contrast to the memoryless states, the resistance change behavior can be better explained by considering very generic and dependent discrete states. Whenever required, simulations are performed to obtain the resistance change behaviors. Our findings are concurrence with the experimental observations. Keywords—Copper interconnects, Electromigration, Markov model, Reliability. I. INTRODUCTION Due to desperate scaling and excessive complexity, electromigration (EM) remains an important reliability concern; especially in submicron Cu dual damascene interconnects. EM reliability depends on many parameters which include process variation, purity of material, geometry of interconnect line, line microstructure, surrounding materials and their fabrication processing history etc. [1-2]. Hence, EM reliability evaluation is routinely conducted by Si- industry for every batch of their product. It is to be noted here that optimal design is possible for optical interconnect [3]. For copper interconnect, gouging vias [4] and fabrication process [5-6] can minimize the migration effect. EM-induced resistance degradation profile together with physical failure analysis carries lots of information about the actual failure process. A wealth of information is obtained on the physics of EM-induced failure [1-2]. In fact, various types of models are presented to understand the behavior of EM- induced resistance degradation [7-14]. The type of these models is very different such as analytical, finite element, empirical, consideration of the failure process into a number of discrete states in time-domain (Markovian model) etc. When a given state is not dependent on its previous state, then we call the failure model as memoryless model or independent model. On the other hand, when a given state depends on its previous state, then it is referred as dependent model. Markovian model is memoryless in nature whereas cumulative or analytical models are dependent in nature. The failure models based on the physics-of-failure approach generally predict the time to failure by analysing root-cause of failure mechanisms that are governed by fundamental mechanical, electrical, thermal and chemical processes [15]. We argue about the electromigration failure whether it is dependent or independent in nature, keeping in mind that the failure process is governed by physical parameters. In this regard, simulation is performed considering dependent and independent states. In contrary to the independent state model, we found that dependent state models are more appropriate in explaining the EM-induced resistance degradation. The various shortcomings of the application of Markovian model (independent state) in EM-induced resistance degradation are discussed. II. MARKOV MODEL A. Introduction to Markov model ‘Markov model’ is named after the mathematician Andrei Markov, refers to the mathematical models where the future state of a system does not depend on the past state but depends only on the present state [21]. Markovian property which is basically memoryless suggests that all transitions from one state to another takes place at constant rates. Markov models are practically important for reliability analysis.. Markov models play an important role for reliability estimation, especially for a system consisting of many repairable and non-repairable components. It is a special type of stochastic process where the future probability behavior is determined by its present state. The Markov model is a stochastic process with a discontinuous (or discrete) state space and discontinuous (or discrete) time space. A Markov process undergoes transitions from one state to another, among a countable number of feasible (plausible) states. Solving Markov model analytically with time- dependent transition rates is a tedious job [16]. Then, various approximation methods are employed. In one of these approximation methods, the component lifetime is discretised into intervals and a constant transition rate is assumed [16-18]. In a recent work an attempt was made to obtain an appropriate distribution and to model the resistance degradation pattern, the range of different resistance values are divided into a finite number of different states, and thus a multistate Markov model is ultimately developed [7,19]. In short, Markov model is extensively used in vast area of reliability engineering [20-22] and is found to be suitable to 2018 IEEE Electron Device Kolkata Conference (EDKCON), 24-25 November, 2018, Kolkata, India 142