Uncertainty Quantification of Bonding Wave Velocity in Direct Wafer Bonding
Direct wafer bonding is a key process for advanced 3D integration, in which the propagation of the bonding front strongly influences bonding efficiency and process stability [1]. Bonding wave velocity is governed by the coupled effects of wafer deformation, interfacial fluid transport, adhesion, material properties, and process conditions [2]. However, most numerical studies are based on deterministic parameter sets, while the influence of parameter uncertainty on bonding wave propagation remains insufficiently quantified. In this work, an uncertainty quantification method for bonding wave propagation is used based on the fluid-structure interaction (FSI) using the basic PDE module in COMSOL Multiphysics. Wafer deformation is described by thin-plate deformation, with the flexural rigidity governed by the Young's modulus E, Poisson's ratio nu, and wafer thickness tw [3]. Gas flow is described by the Reynolds lubrication equation, where rho is the wafer density and miu is the dynamic viscosity of the interfacial gas. The initial gap profile is controlled by the blade thickness tb. Interfacial attraction is characterized by the adhesion energy Ea and the adhesion interaction length beta. Patterned cavities are described by their depth p, inner start radius rin, outer radius rout, and angular width dphig. In this model, the cavity depth modifies the local effective gap and therefore the viscous drainage resistance. A sensitivity analysis is employed to investigate the uncertainty of bonding wave velocity. The Morris one-at-a-time (MOAT) method is first used to screen a high dimensional input space and identify parameters with dominant effects as well as potential nonlinearities and interactions [4]. The dominant parameters identified by the MOAT analysis are then selected for inverse uncertainty quantification, in which their prior distributions are updated using observed bonding wave data through Bayesian inference. This inverse-UQ procedure further constrains the uncertain model parameters and provides posterior distributions that are consistent with the observed bonding behavior. Using the updated posterior distributions as inputs, Sobol sensitivity analysis is subsequently performed to quantitatively decompose the variance of bonding wave velocity into first order and interaction contributions [5]. The first-order index SI measures the variance contribution of an individual input alone, whereas the total index ST additionally includes all interactions involving that input. The MOAT map in Fig. 1 places the adhesion energy Ea, and adhesion interaction length beta in the high sensitivity region, while the cavity depth p, rout, tw, and dphig show intermediate effects. The other parameters have comparatively small elementary effects over the investigated ranges. The Sobol results in Fig. 2 further identify the cavity depth as the dominant source of global velocity variance, with SI approximately 0.52 and ST approximately 0.63. The ST indices of Ea, dphi_g, tw, rout, and beta are approximately 0.24, 0.21, 0.14, 0.10, and 0.04, respectively. The gaps between SI and ST demonstrate that interactions are non-negligible, particularly for adhesion and cavity related parameters. The strong influence of p is physically consistent with its direct modification of the local flow gap. Together, MOAT screening, inverse UQ, and Sobol decomposition extend the COMSOL model from deterministic prediction to uncertainty evaluation of bonding wave dynamics and provide a basis for identifying robust parameter ranges for direct wafer bonding.
