A Fast Shell-Based Framework for Predicting Chucking-Induced In-Plane Distortion in Silicon Wafers from Measured Out-of-Plane Geometry

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Reviewer recommendation: The paper will be ready for publication after some corrections that must be made by the authors.

Reviewer comments:

• The paper discusses issues encountered in the fabrication of advanced semiconductors, where in-plane distortion (IPD) of silicon wafers induced by vacuum clamping during photolithography can compromise overlay accuracy and pattern fidelity;

• It is argued that although out-of-plane distortion (OPD) can be accurately measured using high-resolution optical methods such as wavefront phase imaging (WFPI), predicting the resulting IPD remains a challenge, especially when anisotropic material properties are involved;

• This study presents a finite element approach for modeling IPD in silicon wafers based on measured OPD data, where the mechanical behavior of the wafer is simulated using the ANDES (Assumed Natural Deviatoric Strain) formulation, implemented in custom triangular wafer bending elements with 9 degrees of freedom, as this method allows for a transparent and flexible simulation framework, avoiding the limitations of analytical solutions and the “black box” nature of commercial solvers;

• The adopted model incorporates realistic geometries and material anisotropy, providing a more accurate estimation of in-plane displacements, and the results demonstrate that even a small out-of-plane curvature can lead to measurable in-plane displacements when the wafer is flattened, highlighting the need to consider IPD effects in high-precision processes;

• It is argued that the proposed methodology provides a generalizable and physically grounded tool for improving overlap control and guiding the design of fixture systems and metrology strategies;

• The framework assumes "perfect contact" between the wafer and a rigid chuck. In real photolithography tools, vacuum or electrostatic chucking introduces significant friction and pinning effects at the pins/burls interface. The authors should discuss or simulate how slippage or friction might restrict the relaxation of the in-plane distortion (IPD) field;

• The authors praise the custom 9-DOF ANDES element for overcoming the "black-box" nature of commercial solvers. However, they should validate their code's accuracy against a standard commercial tool (ANSYS or ABAQUS) using standard 3D shell (like S4R) or solid elements on at least one free-form wafer to prove their custom element performs as intended;

• Silicon's orthotropic anisotropy means its Young's Modulus changes drastically based on the in-plane angle (φ) relative to the axis. However, industrial metrology maps often don't explicitly output the notch/flat orientation data. The authors need to explain how they align the physical wafer's crystal axes with their finite element mesh coordinates in practical application;

• It lacks a sensitivity analysis regarding metrology noise. The authors should add a synthetic noise test to demonstrate how standard nanometer-scale height measurement errors in Δ W affect the predicted sub-nanometer IPD values;

• The custom mesh uses 1410 nodes, which translates to a spatial resolution of roughly one node every few millimeters on a 300 mm wafer. While this works well for low-frequency global bow or warp, advanced packaging or localized thermal stress causes high-frequency local distortions. The authors should explicitly define the spatial frequency cutoff limits of their current mesh scheme.

Comments for author File: Comments.pdf

Author Response

A document containing our responses to your comments is attached. We thank you for your time and for the careful reading of the manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors This manuscript develops a finite‑element prediction pipeline using ANDES triangular shell elements combined with static condensation to estimate chucking‑induced in‑plane distortion purely from measured out‑of‑plane wafer geometry, incorporating monocrystalline‑silicon elastic anisotropy. Nevertheless, multiple questions remain require detailed replies from the authors. 1. The paper compares conceptually against gradient‑type analytical models from Turner and Jiang, but provides no quantitative head‑to‑head numerical comparison between the proposed FEM framework and these well‑established analytical formulations on identical wafer OPD inputs. 2. Why is a general‑purpose commercial shell‑FEM solver not included as a numerical benchmark on the same meshes and input OPD datasets? 3. The paper highlights the benefit of anisotropic silicon constitutive modelling. Please add direct ablation. 4. In real vacuum‑chuck hardware, partial vacuum holes, seal rings and non‑ideal contact may locally prevent full flattening. How would partial or spatially‑incomplete chuck contact alter the predicted IPD fields? 5. Raw WFPI metrology point data must be interpolated onto the unstructured triangular FEM mesh to obtain nodal prescribed deformation.

Author Response

A document containing our responses to your comments is attached. We thank you for your time and for the careful reading of the manuscript.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Pérez et al. presented a numerical methodology to assess the in-plane distortion (IPD) of anisotropic Si wafers from measured OPD through a FEM framework with static condensation and showed results in 4 measured wafers. The mathematical exposition is detailed, and the steps of the method are explicit with equations, which favors conceptual reproducibility. The manuscript provided a mesh convergence for validation. However, there are opportunities to improve the method implementation such as, how to go from a real OPD map to prescribed DOFs, what preprocessing is applied, how exactly the geometry of the chuck is defined in the cases shown, and how it would be quantitatively validated. In addition, experimental validation of the IPD is lacking and in fact, the authors recognize this as future work. I recommend major revision with the following observations.

 

Minor

Please enlarge the font in Fig 1, Fig 2. Some annotations are too small making them difficult to read.

 

Major

The abstract should be quantitative.

The originality of the approach depends on demonstrating that this proposed pipeline provides verifiable advantages vs. alternatives (analytical or commercial FEM): real speed, accuracy, robustness to noise, and/or improvements by Si anisotropy. In the current state, "novelty" is argued more than it is quantified comparatively.

The title includes Fast as a descriptor, but through the manuscript this is not quantified (CPU, memory, latency per wafer, factorization cost vs. evaluation or another metric). I recommend backing up "fast" with benchmarks.

Because IPD is in nm and OPD in μm, noise sensitivity can be critical. The authors should discuss tje WFPI measurement variability/noise and how it propagates to IPD. Furthermore, the implicit conclusions on "accurate estimation" should be moderated or supported with external validation.

Author Response

A document containing our responses to your comments is attached. We thank you for your time and for the careful reading of the manuscript.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

It can be accepted in this form.

Author Response

Dear Reviewer,

Thank you very much for taking the time to review our manuscript and for your positive assessment.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors submitted a revised version answering the concerns raised in the first revision.

I recommend it for publication

Author Response

Dear Reviewer,

Thank you very much for taking the time to review our manuscript and for your positive assessment.