Xiuguo Chen

Estimation of the convergence order of rigorous coupled-wave analysis for binary gratings in optical critical dimension metrology

In most cases of optical critical dimension metrology, when applying rigorous coupled-wave analysis to optical modeling, a high order of Fourier harmonics is usually set up to guarantee the convergence of the final results. However, the total number of floating point operations grows dramatically as the truncation order increases. Therefore, it is cri- tical to choose an appropriate order to obtain high computational efficiency without losing much accuracy in the meantime. We show that the conver- gence order associated with the structural and optical parameters is esti- mated through simulation. The results indicate that the convergence order is linear with the period of the sample when fixing the other parameters, both for planar diffraction and conical diffraction. The illuminated wave- length also affects the convergence of a final result. With further investiga- tions concentrated on the ratio of illuminated wavelength to period, it is discovered that the convergence order decreases with the growth of the ratio, and when the ratio is fixed, convergence order jumps slightly, especially in a specific range of wavelength. This characteristic could be applied to estimate the optimum convergence order of given samples to obtain high computational efficiency.

Accurate characterization of nanoimprinted resist patterns using Mueller matrix ellipsometry.

In order to control nanoimprint lithography processes to achieve good fidelity, accurate characterization of structural parameters of nanoimprinted resist patterns is highly desirable. Among the possible techniques, optical scatterometry is relatively ideal due to its high throughput, low cost, and minimal sample damage. Compared with conventional optical scatterometry, which is usually based on reflectometry and ellipsometry and obtains at most two ellipsometric angles, Mueller matrix ellipsometry (MME) based scatterometry can provide up to 16 quantities of a 4 × 4 Mueller matrix in each measurement and can thereby acquire much more useful information about the sample. In addition, MME has different measurement accuracy in different measurement configurations. It is expected that much more accurate characterization of nanoimprinted resist patterns can be achieved by choosing appropriate measurement configurations and fully using the rich information hidden in the measured Mueller matrices. Accordingly, nanoimprinted resist patterns were characterized using an in-house developed Mueller matrix ellipsometer in this work. We have experimentally demonstrated that not only more accurate quantification of line width, line height, sidewall angle, and residual layer thickness of nanoimprinted resist patterns can be achieved, but also the residual layer thickness variation over the illumination spot can be directly determined, when performing MME measurements in the optimal configuration and meanwhile incorporating depolarization effects into the optical model. The comparison of MME-extracted imprinted resist profiles has also indicated excellent imprint pattern fidelity.

Measurement configuration optimization for accurate grating reconstruction by Mueller matrix polarimetry

Abstract. Due to the rich information provided by the Mueller matrices when the most general conical diffraction configuration is considered, the Mueller matrix polarimetry has demonstrated a great potential in semiconductor manufacturing. As the configurations of the incidence and azimuthal angles have different influences on the measurement accuracy, it is necessary to select an optimal one among the multitude of possible options. We introduce the norm of a configuration error propagating matrix to assess the measurement accuracy for different measurement configurations. The optimal configuration for a Si grating sample was achieved by minimizing the norm of the configuration error propagating matrix. Experimental results show the agreement between the theoretically predicted optimal configuration and the experimental exhibited one obtained by using a dual-rotating compensator Mueller matrix polarimeter and thus demonstrated the validity of the proposed optimization method.

Depolarization effects from nanoimprinted grating structures as measured by Mueller matrix polarimetry

Mueller matrix polarimetry (MMP) is introduced to characterize nanoimprinted grating structures, and noticeable depolarization effects from measured data are observed. We demonstrate that these depolarization effects are mainly induced by the finite bandwidth and numerical aperture of the instrument, as well as the residual layer thickness variation of the measured sample. After incorporating the depolarization effects into the optical model, not only improved accuracy can be achieved for the line width, line height, and residual layer thickness measurement but also the residual layer thickness variation over the illumination spot can be directly determined by MMP.

Robust solution to the inverse problem in optical scatterometry

In optical scatterometry, the least squares (LSQ) function is usually used as the objective function to quantify the difference between the calculated and measured signatures, which is based on the belief that the actual measurement errors are normally distributed with zero mean. However, in practice the normal distribution assumption of measurement errors is oversimplified since these errors come from the superimposed effect of different error sources. Biased or inaccurate results may be induced when the traditional LSQ function based Gauss-Newton (GN) method is used in optical scatterometry. In this paper, we propose a robust method based on the principle of robust estimation to deal with the abnormal distributed errors. An additional robust regression procedure is used at the end of each iteration of the GN method to obtain the more accurate parameter departure vector. Simulations and experiments have demonstrated the feasibility of our proposed method.

Improved measurement accuracy in optical scatterometry using correction-based library search.

Library search is one of the most commonly used methods for solving the inverse problem in optical scatterometry. The final measurement accuracy of the conventional library search method highly depends on the grid interval selected for each parameter in the signature library, and the time cost of the parameter extraction increases dramatically when the grid interval is decreasing. In this paper, we propose a correction-based library search method to improve the measurement accuracy for a pregenerated signature library. We derive a formulation to estimate the error between the expected solution of the inverse problem and the actually searched solution obtained by the conventional library search method. Then we use the estimate of the error as a correction term to correct the actually searched solution to improve the measurement accuracy. Experiments performed on a photoresist grating have demonstrated that the proposed correction-based library search method can achieve much more accurate measurement with negligible computational penalty to the conventional library search method in the parameter extraction. It has also been observed that the correction-based library search method has higher measurement accuracy and less time cost than the interpolation-based library search method. The proposed correction-based library search method is expected to provide a more practical means to solve the inverse problem in state-of-the-art optical scatterometry.

Calibration of misalignment errors in composite waveplates using Mueller matrix ellipsometry

Composite waveplates consisting of two or more single waveplates are widely used in optical instruments, such as ellipsometry, polarimetry, cryptography, and photoelasticity. Accurate calibration of the misalignment errors in composite waveplates is of great importance (to minimize or correct the spurious artifacts in the final collected spectral data of these instruments induced by the misalignment errors). In this paper, we choose the fast axis azimuth and the rotary angle of composite waveplates as the detected characteristic parameters to calibrate the misalignment errors in composite waveplates. We first derive a general analytical model to describe the relationship between the mislignment errors and the characteristic parameters, and then propose an inverse approach to the calibration of the misalignment errors in composite waveplates. An experimental device based on the dual rotating-compensator Mueller matrix ellipsometry principle is set up to measure the characteristic parameters of composite waveplates. Both numerical simulations and experiments on an MgF(2)-MgF(2)-quartz triplate demonstrate the correctness and efficiency of the proposed approach. It is expected that the proposed approach can be readily extended to calibrate the misalignment errors in more complex composite waveplates.

Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes

Accurate measurement of the Mueller matrix over a broad band is highly desirable for the characterization of nanostructures and nanomaterials. In this paper, we propose a general composite waveplate (GCW) that consists of multiple waveplates with flexibly oriented axes as a polarization modulating component in the Mueller matrix ellipsometer (MME). Although it is a common practice to make achromatic retarders by combining multiple waveplates, the novelty of the GCW is that both the retardances and azimuths of fast axes of the single-waveplates in the GCW are flexible parameters to be optimized, which is different from the conventional design where single-waveplates are usually arranged in symmetrical layout or with their fast axes parallel or perpendicular to each other. Consequently, the GCW can provide many more flexibilities to adapt to the optimization of the MME over a broad band. A quartz triplate, as a concrete example of the GCW, is designed and used in a house-made MME. The experimental results on the air demonstrate that the house-made MME using the optimally designed quartz triplates has an accuracy better than 0.2% and a precision better than 0.1% in the Mueller matrix measurement over a broad spectral range of 200~1000 nm. The house-made MME exhibits high measurement repeatability better than 0.004 nm in testing a series of standard SiO2/Si samples with nominal oxide layer thicknesses ranging from 2 nm to 1000 nm.

Improved deep-etched multilayer grating reconstruction by considering etching anisotropy and abnormal errors in optical scatterometry

Accurate, fast, and nondestructive reconstruction of the etched nanostructures is important for etching process control to achieve good fidelity, as well as high manufacturing yield. In this work, we have demonstrated the improved deep-etched multilayer grating reconstruction by simultaneously considering the model error of nonuniform side-wall angle (SWA) because of different etching anisotropies of various materials and by suppressing the abnormally distributed measurement errors using a robust statistics based method in optical scatterometry. More specifically, we introduce an additional parameter to perfect the profile model under measurement, and use a robust estimation procedure at the end of each iteration of the Gauss-Newton (GN) method to obtain the more accurate parameter departure vector. By applying the proposed methods, more accurate reconstructed results can be achieved.

Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures

Mueller matrix ellipsometry (MME) is applied to detect foot-like asymmetry encountered in nanoimprint lithography (NIL) processes. We present both theoretical and experimental results which show that MME has good sensitivity to both the magnitude and direction of asymmetric profiles. The physics behind the use of MME for asymmetry detection is the breaking of electromagnetic reciprocity theorem for the zeroth-order diffraction of asymmetric gratings. We demonstrate that accurate characterization of asymmetric nanoimprinted gratings can be achieved by performing MME measurements in a conical mounting with the plane of incidence parallel to grating lines and meanwhile incorporating depolarization effects into the optical model. The comparison of MME-extracted asymmetric profile with the measurement by cross-sectional scanning electron microscopy also reveals the strong potential of this technique for in-line monitoring NIL processes, where symmetric structures are desired.