A hybrid multiscale kinetic Monte Carlo method for simulation of copper electrodeposition

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Abstract

A hybrid multiscale kinetic Monte Carlo (HMKMC) method for speeding up the simulation of copper electrodeposition is presented. The fast diffusion events are simulated deterministically with a heterogeneous diffusion model which considers site-blocking effects of additives. Chemical reactions are simulated by an accelerated (tau-leaping) method for discrete stochastic simulation which adaptively selects exact discrete stochastic simulation for the appropriate reaction whenever that is necessary. The HMKMC method is seen to be accurate and highly efficient.

Introduction

Electrodeposition processes are used in the fabrication of microelectronic devices and in nanotechnology applications [1], [2], [3]. Copper electrodeposition with complex additive systems is used to form interconnections in microelectronic devices [4]. The electrodeposition process involves the diffusion, migration, and convection of solution species in the bulk electrolyte, as well as the reaction and diffusion of surface species at the interface of the solid and the liquid. The surface morphology evolution is controlled via electrolyte additives, some of which are present in very small concentrations, that react with the copper species in solution and on the surface.

In addition to experimental studies [5], [6], [7], [8] of the electrodeposition process, computer simulation provides a powerful tool for studying and understanding complex multiscale phenomena. Continuum computational methods [9], [10], [11], [12], typically in the form of differential equations, can be used provided that the characteristic length is significantly greater than the molecular scale, and that the reactant concentrations are large. A stochastic algorithm, the kinetic Monte Carlo (KMC) [13], [14] method, has been used to study the molecular features of copper electrodeposition and has been coupled with continuum methods to form a multiscale approach [15], [16], [17]. In the KMC approach, the surface reactions and the surface diffusion are modeled as discrete events. However, the time scales of these events may be widely different, resulting in high computational cost.

A number of different approaches have been explored for developing a faster KMC algorithm and a more efficient multiscale approach, including coarse-grained KMC methods [17] and spatially adaptive coarse-grained KMC methods [18], [19] which can solve the problem at the mesoscopic scale. Chatterjee et al. [20] proposed a continuum mesoscopic model, which is based on a deterministic partial differential equation (PDE) approximation of the reaction–diffusion system. Multiscale KMC simulations are also available for epitaxial growth applications [21], [22], [23], [24].

In the present paper we develop a hybrid multiscale kinetic Monte Carlo (HMKMC) method for efficient simulation of copper electrodeposition in the presence of additives. The challenge of computational efficiency is addressed by partitioning the different reaction channels according to the speed of the reaction and concentration of the reactants. The slower reactions with reactants present in very low concentrations are treated by a detailed stochasatic simulation algorithm (SSA). Reactions with larger reactant concentrations are treated by a non-negativity-preserving tau-leaping SSA method [25], [26], [27], [28]. Since diffusion rates on metal surfaces can be very fast with respect to other reaction dynamics, the surface distribution of copper atoms is approximated deterministically by a continuum partial differential equation. The simulation method efficiently solves the surface kinetics of the copper additive chemistry, accurately resolving the surface concentrations of additive complexes, even for species with very small populations.

This paper is organized as follows. We begin in Section 2 with a brief review of the KMC and tau-leaping methods. The detailed HMKMC algorithm is presented in Section 3. In Section 4 we describe our heterogeneous diffusion model for monolayer diffusion of copper. Numerical results are provided in Section 5, and Conclusions are given in Section 6.

Section snippets

The kinetic Monte Carlo method

In this subsection we briefly describe the KMC method and introduce some notation. We use N to denote the number of chemical species in the reaction–diffusion system and M to denote the number of reaction channels. An Nx×Ny grid is used for the discretization of the interacting surface, where the size of each grid cell is ΔL×ΔL.

A pseudo-particle in the simulation is modeled as a cube of size ρ, which is usually much smaller than ΔL. A pseudo-particle corresponds to a site in a grid cell.

Hybrid multiscale kinetic Monte Carlo method

The proposed simulation algorithm is a hybrid method which draws on ideas from Haseltine and Rawlings [31], Rao and Arkin [32], Cao et al. [33], [34] and achieves its efficiency by combining the adaptive non-negativity-preserving tau-leaping SSA method [27], [28] with a deterministic (PDE) approximation of the surface diffusion, which is by far the fastest process in the system. The main idea is as follows: to achieve high efficiency in the simulation, reaction channels are partitioned into

Heterogeneous surface diffusion modeling

As discussed above, the site-blocking effects of additives on the surface require a heterogeneous treatment of surface diffusion. In this section we derive the relationship between the heterogeneous diffusion coefficient and the surface concentration of the diffusing species and the additives.

We define two grid cells k and l on the discretized space, for the two scenarios shown in Fig. 2, Fig. 3. In the first simplified scenario, i.e., Fig. 2, the pseudo-particle in grid cell k can diffuse to

Adsorption–desorption kinetics

The HMKMC method is first applied to the following reversible adsorption–desorption reactions:Aaq[ka]kdBads,where Aaq indicates that A is an aqueous solution species, Bads indicates that B is an adsorbed surface species, and ka and kd are the forward adsorption rate constant and backward desorption rate constant, respectively. In this test problem, diffusion is not considered and the reactions are treated by the adaptive non-negativity-preserving tau-leaping SSA method in the HMKMC method. For

Conclusions

In this paper a hybrid multiscale kinetic Monte Carlo (HMKMC) method has been introduced to accelerate the simulation of copper electrodeposition. The reactions are simulated by an adaptive non-negativity-preserving tau-leaping stochastic simulation algorithm (SSA), where an appropriate step-size is selected adaptively by the algorithm for best speed-up while retaining a desired accuracy. The fast diffusion events are solved deterministically with a heterogeneous diffusion model which considers

Acknowledgments

We thank Mohan Karulkar for the ODE model of the copper electrodeposition mechanism.

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    This work was supported by the National Science Foundation under NSF Award NSF/ITR CCF-0428912.

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