IEEE ELECTRON DEVICE LETTERS, VOL. 30, NO. 12, DECEMBER 2009 1359 Analytical Cell-Based Model for the Breakdown Statistics of Multilayer Insulator Stacks Jordi Suñé, Senior Member, IEEE, Santi Tous, and Ernest Y. Wu, Member, IEEE Abstract—A fully analytical cell-based model is proposed to describe the breakdown (BD) statistics of multiple-layer insula- tor stacks. The model is shown to be completely equivalent to the full percolation model by comparing with recently published kinetic Monte Carlo simulations. The role of an initial density of native defects in the as-grown oxide and the time-dependent generation of inert defects are successfully addressed. The effect of the generation of interface states on the stack BD statistics is also incorporated into the model. Index Terms—Dielectric breakdown (BD), MOS devices, reliability theory. I. I NTRODUCTION I T IS WIDELY accepted that the generation of defects in the gate oxide finally triggers oxide breakdown (BD) by forming of a defect-related conduction path. This is the basis of the widely accepted percolation theory of BD [1], [2]. However, the first model relating the generation of defects to the BD statistics was a 2-D cell-based picture [3], which successfully explained the scaling of the BD distribution with oxide area and justified the Weibull model. Nevertheless, the scaling with oxide thickness was not captured until the concept of percolation path was introduced [1], [2]. Subsequently, a 3-D cell-based approach provided an analytical model for the BD distribution and its scaling with oxide area and thickness [4], [5]. Across many technological generations, SiO 2 was the material of choice, but having reached its practical scaling limit (∼1–1.2 nm), effective scaling is pursued through the use of high-k (HK) dielectrics, which nevertheless require a SiO 2 interface layer (IL) to separate the HK layer from the substrate. Thus, SiO 2 /HK multilayer stack dielectrics are the gate insulators of present interest. Recently, the percolation model was successfully applied to the BD statistics of SiO 2 /HK stacks using kinetic Monte Carlo (kMC) simulations [6]. In this letter, we generalize the cell-based approach and provide a fully analytical model for the BD statistics of stack insulators formed by an arbitrary number of layers. Manuscript received July 16, 2009; revised August 23, 2009. Current version published November 20, 2009. The work of J. Suñé and S. Tous were supported in part by IBM Microelectronics, by the Spanish Ministry of Science and Tech- nology under Project TEC2009-09350, and by the Departament d’Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya, under Project 2009SGR-783. The review of this letter was arranged by Editor B.-G. Park. J. Suñé and S. Tous are with the Departament d’Enginyeria Elec- trònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain (e-mail: Jordi.Sune@uab.es; santi.tous@uab.es). E. Y. Wu is with the IBM Microelectronics Division, Essex Junction, VT 05452 USA (e-mail: eywu@us.ibm.com). Digital Object Identifier 10.1109/LED.2009.2033617 II. SINGLE-LAYER CELL-BASED MODEL First, the insulator volume is divided into an ordered array of cells. The area of the cells is σ, and the thickness is a 0 . In previous works [4], a cubic cell was considered (i.e., σ = a 2 0 ), but there is no need to limit ourselves to this particular case. Thus, the oxide area A OX is divided into N CELL = A OX /σ sites, and the thickness is divided into n = T OX /a 0 layers of cells. A cell is defective if it contains one or more defects. A percolation path is formed when there is a connection of defective cells between the electrodes. Krishnan showed that the number of possible paths with minimum number of cells (n) departing from one cell of the bottom layer is 5 n−1 [5]. However, if diagonal connections between cells are allowed (as in [6]), the number of possible paths becomes 9 n−1 , and the probability that one cell of the bottom layer is connected to the top electrode by a percolation path is P path =(1/9)λ n , with λ being the probability that there is at least one defective cell among the nine neighbor cells that might contribute to the percolation path formation in each layer of cells. According to the weakest link property of the BD, the survival function is (1 − F BD )=(1 − P PATH ) N CELL so that the BD cumulative distribution is F BD = 1 −  1 − λ n 9  N CELL . (1) Relating this equation to the experiments requires a model for the time dependence of λ, which is related to the density of defects per unit of volume N OT (t) that is known to follow a power law of time, i.e., N OT = ξt α [1]–[6]. The average num- ber of defects in a set of nine cells is n DEF = 9σa 0 ξt α ≡ At α , and according to the Poisson distribution λ = 1 − exp(−At α ). (2) Notice that, as required for a variable defined as a probability, λ ranges from 0 (at t = 0) to 1 for t →∞. In [4], λ ≈ At α was considered for small defect densities, thus identifying the BD distribution as a Weibull distribution with shape factor (Weibull slope) β = αn and scale factor T BD =(9/N A n ) (1/αn) . This approximation is almost universally valid for single-layer BD, but the explicit consideration of the exponential saturation of λ(t) is required to deal with multiple-layer stack insulator BD. III. CELL-BASED MODEL FOR I NSULATOR STACKS Let us first consider the case of a two-layer insulator stack. The structure area is divided into N CELL cells of area σ. The IL and HK layers are assumed to be composed by n IL and 0741-3106/$26.00 © 2009 IEEE