Objective Hard templates are widely used in nanoimprint lithography (NIL) because their high stiffness and dimensional stability enable high-fidelity, large-area pattern transfer. However, under the combined effects of mechanical loading and thermal stress, hard templates remain susceptible to in-plane deformation, which can accumulate over multiple process steps and degrade pattern uniformity and overlay accuracy. As imprint fields expand and device dimensions continue to shrink, controlling such deformation has become increasingly important for high-yield manufacturing. Existing compensation strategies often rely on full-field metrology, restrictive boundary assumptions, or specialized structural designs that are difficult to integrate into practical production systems. There is therefore a clear need for a low-dimensional, actuation-efficient active compensation framework based on minimal displacement feedback and compatible with practical implementation.

Methods An active compensation framework driven by a force-actuator array was developed for a fused-silica NIL template. Using the displacements of four reference points at the corners of the patterned region as feedback, the global in-plane deformation was represented by five representative low-order modes: X-direction magnification, Y-direction magnification, diagonal distortion, X-direction trapezoidal distortion, and Y-direction trapezoidal distortion. Because rigid-body motions, including global translation and in-plane rotation, can be compensated by the wafer stage, only these five deformation modes were considered in the template-actuation model.

Offline finite-element calibration was performed by sequentially applying unit forces with a 16-actuator array distributed symmetrically along the four template edges. The resulting reference-point displacement responses were used to construct a force-displacement response matrix, which was combined with the geometric mapping from reference-point displacements to modal coefficients to establish a linear mapping from actuator forces to deformation modes.

Force allocation was formulated as a constrained optimization problem. Tikhonov-regularized least squares was employed to minimize modal residuals while penalizing excessive actuation forces. In-plane static-equilibrium constraints were further imposed to eliminate net force and net moment that could compromise vacuum chucking stability or induce rigid-body slip. The regularization parameter was selected using the L-curve criterion, and closed-loop iteration was performed until convergence of the residual deformation.

To improve actuation efficiency without sacrificing correction accuracy, a centrally located cylindrical cavity was introduced on the non-patterned side of the template. By reducing the local effective thickness and in-plane tensile stiffness near the patterned region while preserving structural symmetry, the cavity lowered the force required to generate a given in-plane correction. This mechanism can be interpreted from an equivalent strain-energy perspective.

Finite-element simulations were conducted under process-relevant boundary conditions, including vacuum-chuck constraints on the top contact region and a normal-displacement constraint within the patterned region to represent idealized wafer contact. Solid and cavity templates were evaluated under identical deformation targets to compare correction accuracy and actuation-force demand.

Results and Discussions The proposed closed-loop method effectively suppressed global in-plane deformation for all five representative modes. After correction, the root-mean-square error at the four reference points was reduced to no more than 4.3 nm, and the maximum residual displacement remained below 5.5 nm, demonstrating nanoscale correction accuracy with only four displacement measurements. The displacement vector fields showed that the dominant low-order deformation components were strongly suppressed after compensation, whereas the remaining residuals were concentrated mainly near the template boundaries and actuator contact regions. This behavior is consistent with discrete force application, boundary constraints, and the omission of higher-order components from the five-mode basis.

The five-mode representation provided a practical balance between model simplicity and descriptive capability for typical NIL in-plane distortion. It captures the leading low-order components that dominate overlay error, whereas the remaining discrepancy can be attributed to higher-order deformation, local compliance near loading interfaces, and unavoidable edge effects. These observations suggest that further accuracy gains could be achieved by enriching the modal basis while retaining the low-dimensional control structure.

The centrally located cavity significantly improved actuation efficiency. Relative to the solid-template baseline, the cavity reduced the average actuation force by approximately 39%–56% and the peak actuation force by about 50% at comparable correction accuracy. It also produced a more uniform force distribution, thereby reducing localized overload and improving the robustness of actuator design and control. These results are consistent with the strain-energy-based interpretation that lowering the effective thickness in the deformation-sensitive region reduces the in-plane stiffness that must be overcome by the actuator array.

The method also offers practical advantages for implementation. Because the feedback is limited to four displacement measurements, metrology complexity is reduced and integration with compact sensing hardware is simplified. The explicit static-equilibrium constraints improve physical realizability under vacuum chucking conditions, while regularization enhances numerical stability when the response matrix is ill-conditioned or the measurements are noisy.

Within the calibration and correction range considered here, the deformation remained within the small-deformation regime, and the in-plane strain stayed within the linear elastic range of fused silica, supporting the validity of the linear model. For larger deformations or time-varying operating conditions, geometric nonlinearity and drift in the response matrix may require online identification and periodic recalibration.

Conclusions A multi-actuator active compensation strategy was developed for hard-template in-plane deformation in NIL. The method combines four-point displacement feedback, five-mode deformation decomposition, finite-element-based offline calibration, and Tikhonov-regularized force allocation under static-equilibrium constraints, with the regularization parameter selected using the L-curve criterion. Finite-element simulations demonstrated nanoscale residual error across representative deformation modes, with an RMSE of no more than 4.3 nm and a maximum residual displacement below 5.5 nm.

A centrally located cylindrical cavity further improved force efficiency without compromising correction accuracy, reducing the average actuation force by 39%–56% and the peak force by approximately 50% while also improving force-distribution uniformity. Overall, the proposed framework provides a low-dimensional, energy-efficient, and practically viable route for controlling large-area hard-template deformation in NIL. Future work will focus on extending the modal basis to account for higher-order deformation, experimental validation, and adaptive updating of the response matrix under time-varying thermal and boundary conditions.