Wafer-to-Wafer Bonding: Part: I -- The Coupled Physics Problem and the 2D Finite Element Implementation

TL;DR

This paper presents a coupled physics model for wafer-to-wafer bonding, integrating wafer deformation and air film dynamics, implemented with high-performance finite element methods to analyze bonding behavior and sensitivities.

Contribution

It develops a reduced-order, mathematically consistent model coupling wafer bending and air film flow, solved monolithically using FEniCSx with automatic differentiation.

Findings

Simulations match experimental probe-displacement histories.

Force equilibrium verified at the bond front.

Bonding front kinetics are sensitive to initial gap, air viscosity, and interfacial energy.

Abstract

Wafer-to-wafer (WxW) bonding is a key enabler for three-dimensional integration, including hybrid bonding for fine-pitch Cu-Cu interconnects. During bonding, wafer deformation and the air entrapped between the wafers interact through a strongly coupled, time-dependent fluid-structure interaction (FSI) that can produce non-intuitive bonding dynamics and process sensitivities. This paper develops a mathematically consistent reduced-order model for WxW bonding by deriving a Kirchhoff-Love plate equation for wafer bending from three-dimensional linear elasticity and coupling it to a Reynolds lubrication equation for the inter-wafer air film. The resulting nonlinear plate-Reynolds system is discretized and solved monolithically in the high-performance FEniCSx framework using a $C^0$ interior-penalty formulation for the fourth-order plate operator, standard continuous Galerkin discretization for the pressure field, implicit time integration, and a Newton solver with automatic differentiation. Simulations reproduce experimentally reported probe-displacement histories for multiple initial gaps and verify force equilibrium at the bond front, where the Reynolds pressure acts as an effective contact reaction. Parametric studies reveal nonlinear, and in some cases non-monotonic, sensitivities of bonding-front kinetics to the initial gap, air viscosity, and interfacial energy, providing actionable trends for process optimization.