[Paper Review] Grain Boundary Diffusion in Copper under Tensile Stress

Kevin Crosby|arXiv (Cornell University)|Jul 2, 2003

Microstructure and mechanical properties3 citations

TL;DR

This study uses molecular dynamics simulations to investigate how uniaxial tensile stress affects self-diffusion at the $Σ$3 twin grain boundary in copper. It proposes a theoretical model showing that tensile strain reduces the effective activation energy for diffusion by 5 eV per unit strain, with the activation volume $V^* = 0.6\Omega$, confirming a vacancy-dominated diffusion mechanism.

ABSTRACT

Stress enhanced self-diffusion of Copper on the $Σ$3 twin grain boundary was examined with molecular dynamics simulations. The presence of uniaxial tensile stress results in a significant reduction in activation energy for grain-boundary self-diffusion of magnitude 5 eV per unit strain. Using a theoretical model of point defect formation and diffusion, the functional dependence of the effective activation energy $Q$ on uniaxial tensile strain $ε$ is shown to be described by $Q(ε)=Q_0-E_0V^*ε$ where $E_0$ is the zero-temperature Young's modulus and $V^*$ is an effective activation volume. The simulation data agree well with this model and comparison between data and model suggests that $V^*=0.6Ω$ where $Ω$ is the atomic volume. $V^*/Ω=0.6$ is consistent with a vacancy-dominated diffusion mechanism.

Motivation & Objective

• To understand the effect of tensile stress on grain boundary self-diffusion in copper, a critical factor in interconnect reliability.
• To determine the functional dependence of effective activation energy $Q(\epsilon)$ on uniaxial tensile strain $\epsilon$ in the $Σ$3 twin boundary.
• To identify the dominant diffusion mechanism (vacancy vs. interstitial) at the grain boundary under strain.
• To validate a theoretical model linking strain, activation energy, and effective activation volume $V^*$.

Proposed method

• Molecular dynamics (MD) simulations were performed to calculate the parallel diffusion coefficient $D_{\parallel}(\epsilon)$ at various tensile strains $\epsilon$.
• Arrhenius plots of $D_{\parallel}$ vs. inverse temperature were used to extract the effective activation energy $Q(\epsilon)$.
• A theoretical model was applied: $Q(\epsilon) = Q_0 - E_0 V^* \epsilon$, where $E_0$ is the zero-temperature Young’s modulus and $V^*$ is the effective activation volume.
• The model was fitted to simulation data to determine $V^*$, with $\Omega$ as the atomic volume of fcc copper.
• Comparison of $V^*/\Omega$ with theoretical values for vacancy and interstitial mechanisms was used to infer the dominant diffusion mechanism.
• The analysis assumed negligible contribution from migration volume $V^{*}_{xx}$, focusing on relaxation volume effects.

Experimental results

Research questions

• RQ1 How does uniaxial tensile stress influence the activation energy for self-diffusion at the $Σ$3 twin grain boundary in copper?
• RQ2 What is the functional form of the effective activation energy $Q(\epsilon)$ as a function of tensile strain $\epsilon$?
• RQ3 What is the value of the effective activation volume $V^*$, and what does it reveal about the dominant diffusion mechanism?
• RQ4 Is the theoretical model $Q(\epsilon) = Q_0 - E_0 V^* \epsilon$ consistent with the simulation data?
• RQ5 Is the observed $V^*/\Omega = 0.6$ consistent with a vacancy-mediated diffusion mechanism?

Key findings

• The effective activation energy $Q(\epsilon)$ decreases linearly with tensile strain, with a slope of $-4.9$ eV per unit strain.
• The zero-strain activation energy $Q(0) = 0.65$ eV is lower than bulk copper’s 1.3 eV, consistent with grain boundary diffusion being faster than in the bulk.
• The effective activation volume was determined to be $V^* = 0.6\Omega$, where $\Omega = 1.2 \times 10^{-29}$ m³ is the atomic volume of fcc copper.
• The value $V^*/\Omega = 0.6$ is consistent with a vacancy-dominated diffusion mechanism and inconsistent with an interstitial mechanism.
• The theoretical model $Q(\epsilon) = Q_0 - E_0 V^* \epsilon$ fits the simulation data well, with $E_0 = 116$ GPa used to derive $V^*$.
• Tensile strain of $\sim 10^{-3}$ reduces activation energy by approximately 0.005 eV, effectively doubling the diffusion coefficient at room temperature.