Microstructure simulation of grain growth in Cu through silicon vias using phase-field modeling

Highlights

• We developed a phase-field model and finite element simulation of Cu TSVs.

• Phase-field model models the grain growth in TSV.

• The results are used in finite element model, to simulate microstructure size.

• Results capture the impact of anisotropy and grain size in Cu pumping.

Abstract

A computationally-efficient 3D phase-field model for simulating grain growth in through silicon vias (TSVs) is presented. The model is capable of simulating grain growth in the cylindrical shape of a TSV. The results generated from the phase-field simulations are used in a finite element model with anisotropic elastic and isotropic plastic effects to investigate the large statistical distribution of Cu pumping (i.e. the irreversible thermal expansion of TSV) experimentally seen. The model thus allows to correlate the macroscopic plastic deformation with the grain size and grain orientations.

Introduction

Through silicon vias (TSVs) are a key part of 3D System in Package (SIP) devices, enabling the vertical interconnection of stacked dies. Most often they are filled with electro-plated Cu in polycrystalline form. Due to the large difference in coefficient of thermal expansion with Si, the exposure to high temperatures during subsequent processing steps causes irreversible (plastic) extrusion of the Cu, referred to as ‘Cu pumping’. This results in a relatively high tensile stress inside the Cu at room temperature. The distribution of both Cu pumping and Cu stress values shows a large spread over TSVs of a single wafer [1], [2], [3], [4]. For Cu pumping this spread is clearly correlated to variations in the Cu microstructure [5]. As potential reliability issues related to Cu pumping or Cu stress will first occur at the TSVs with the highest values for either, any model aiming to predict this behavior should include a statistical spread in addition to a median value. Therefore, variations in the Cu microstructure between TSVs and during exposure to the high BEOL processing temperatures (grain evolution [4], [6]) must be taken into account.

Finite Element Models (FEM) for the study of reliability and failure mechanisms in TSVs encountered in literature assume homogeneous isotropic Cu properties [7], [8], [9]. We are developing a finite element model for the thermo-mechanical behavior of Cu TSVs incorporating Cu microstructure, in order to capture the resulting variations and build further understanding of the role of Cu microstructure. This paper presents a computationally-efficient model for simulating grain growth inside the TSV using the phase-field method.

Phase-field modeling is widely used to simulate grain-growth in heterogeneous materials on a mesoscale. This method allows to simulate the evolution of the polycrystalline structure in a Cu TSV and when coupled with a finite element model enables to include anisotropic properties as a function of grain orientation in the elasticity and plasticity models. In the phase-field method, different order parameters are assigned to the different grain orientations and grain boundaries are described as diffuse transitions in the values of these order parameters. Moreover, differential equations are derived from kinetic and thermodynamic principles, based on the assumption that a reduction in bulk energy, interfacial energy or elastic energy, is the driving force for grain evolution [10], [11]. An important advantage of the phase-field method is that, thanks to the diffuse-interface description, there is no need to track the grain boundaries during microstructure evolution and therefore it is mathematically feasible to simulate the evolution of complex grain shapes and connected grain structures in 3D [10], [11].

In this work a semi-implicit Fourier-spectral method is used to solve the differential equations. The implementation was adapted to treat the cylindrical shape of the TSVs. The time steps in the simulations are related to a physical time and temperature through the thermodynamic and kinetic properties of the material. The resulting grain structures were used in a finite element code, where the effect of anisotropic elasticity in the different grains in a TSV were analyzed as a function of the grain size.

The next section describes the theoretical background of the phase-field method with a focus on grain growth models. In the third section, we describe the modification made to the standard phase-field grain growth model in this work and show how it can be combined with a FEM to simulate grain growth and Cu pumping in a cylindrical TSV.