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

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.

Figures (10)

• Figure 1: (a) Cross-sectional view of the initial domain configuration showing the top wafer $\Omega_{top}$ and the bottom wafer $\Omega_{bot}$ separated by an air gap of height $h_0$, and (b) schematic of the six stages of the wafer-to-wafer bonding process.
• Figure 2: Quarter-domain $\omega_q$ exploiting four-fold symmetry of the circular wafer mid-surface $\omega$. (a) Red boundaries: dashed lines denote symmetry edges $\Gamma_x,\Gamma_y$ (zero normal slope); solid arc denotes the outer boundary $\Gamma_R$ (Dirichlet conditions). The striker region $\omega_S$ is shown near the origin. (b) The corresponding finite-element mesh ($42{,}449$ triangular elements) used in the DOLFINx computation.
• Figure 3: Top wafer displacement $w(r,t)$ versus time at a fixed radial location $r = \textrm{mid-region}$ for $h_0 = 30$, $70$, and $100\,\mu$m. Solid lines (orange) show the finite-element solution; dashed lines (blue) show the experimental measurement.
• Figure 4: Snapshots of the displacement field $w$ at nine time instances during bonding ($h = 70\,\mu$m gap). The out-of-plane displacement is warped by a factor of 500 to enhance the visualisation. The bottom wafer shape is represented by the gray shape, and the colormap indicates the out-of-plane displacement of the top wafer.
• Figure 5: Displacement $w(r)$ (left) and pressure $p(r)$ (right) at the final time step ($t = 50$ s) for a $70\,\mu$m gap. The average pressure $\bar{p}$ computed by integrating the Reynolds pressure over the bonded area ($r \leq R_{\mathrm{pin}}$) equals the applied pressure $|\widetilde{p}|$, verifying force equilibrium across the bonding front. This confirms that the Reynolds pressure field correctly captures the contact mechanics at the bond front.