Preamble

Study on the Impact of X-ray Energy and Beam Size on CD-SAXS Measurement Precision Xuyang Qin, 1, 2, 3 Bing Guo, Nan Pan, 1, 2, 3 Xinhao Gao, 1, 2, 3 Shumin Yang, Chunxia Hong, Ying Wang, Xiuhong Li, 1, 2, 3 Chunming Yang, 1, 2, 3, and Fenggang Bian 1, 2, 3, 1 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201204, China University of Chinese Academy of Sciences, Beijing, 100049, China With the development of the semiconductor industry below the 7 nm node, Critical Dimension Small Angle X-ray Scattering (CD-SAXS) has emerged as a powerful tool for quantitatively measuring nanoscale deviations.

In this work, the effects of X-ray beam sizes and photon energies on the accuracy of critical dimension mea- surements were investigated. Critical dimensions measured by different spot sizes of beam showed different deviations from the expected values. A too large or too small beam size was found to not improve confidence intervals. As the incident energy increases, the X-ray transmission rate increases while the scattering cross section decreases, resulting in a gradual decrease in the signal-to-noise ratio of diffraction peaks, thereby re- ducing the accuracy of CD-SAXS measurements. An optimal accuracy was obtained at 12 keV with a smaller beam size. Using an effective trapezoid model, the results yielded an average pitch of nm, width nm, height of nm, and a sidewall angle below . These results can provide crucial guidance for the future development of CD-SAXS laboratories and the construction of X-ray machines, offering robust support for research in related fields.

Keywords

critical dimension small angle X-ray scattering, nonlinear fitting, beam size, X-ray energy, chip

INTRODUCTION

In the rapidly developing fields of microelectronics and 2

nanotechnology, the chip industry is undergoing a significant 3

transformation. Semiconductor manufacturers are continu- ously striving to enhance chip performance, add functional- ities, and integrate more components onto chips [ ]. With the improvement of integrated circuits, planar integrated cir- cuits can no longer meet the needs of development. vanced processes, such as 3D transistor designs and intricate

patterning techniques, have been developed. The fin-based 10

field-effect transistor (FinFET) remained the mainstream de- vice option until 2021, when gate-all-around (GAA) designs emerged and dominated at smaller dimensions owing to their

better electrostatics control. Meanwhile, techniques such as 14

3D stacking and 3D very large-scale integration (3DVLSI)

have introduced a significant number of additional steps to 16

the production processes of integrated circuits [ ]. In these processes, the grating serves as a crucial microstructure, play- ing a vital role in chip manufacturing. With technological ad- vancements, the critical dimensions of chips have gradually reduced, requiring more accurate and high-resolution mea-

surement techniques. The size of gratings is rapidly decreas- 22

ing, which poses considerable challenges for precise mea- surement.

In the methods used for the measurement of grating di-

mensions, Scanning Electron Microscopy (SEM) and Atomic 26

Force Microscopy (AFM) are capable of detecting local crit- ical dimensions in gratings [ ]. Optical Critical Dimension (OCD) is another technology that is widely applied, espe- cially in the study of devices like non-planar FinFETs on sil- icon insulators [ Small angle X-ray scattering has been extensively em- ployed in a multitude of scientific disciplines, including physics, chemistry, materials science, and life sciences, as a

highly effective technique for examining structural character- 35

istics at the nanoscale [ ]. It enables the non-destructive analysis of the dimensions, morphology, and distribution of nanoparticles [ ], the investigation of the structure of bio- logical macromolecules [ ], the measurement of the size, shape, and size distribution of nanoparticles, and the analysis of the microstructure of thin films for solar cells [ ]. No- tably, the CD-SAXS (Critical Dimension Small Angle X-ray

Scattering) technique, developed by scientists at NIST, stands 43

out as a promising measurement platform for revealing the three-dimensional configuration of regularly spaced arrays of nanostructured surface elements [ ]. CD-SAXS holds the potential to serve as a future alternative to OCD measure-

ment methods. This advanced X-ray scattering technique can 48

be used to explore critical dimensions, morphology, structure,

and organization of materials at the nanoscale. Being non- 50

destructive, CD-SAXS enables the measurement of the angle and intensity distribution of X-ray scattering in materials. Its key advantage lies in revealing detailed microstructural in- formation at the nanoscale. Precisely measuring critical di- mensions such as line width and spacing, CD-SAXS can con- tribute to process control and optimization, offering precision

at the nanometer scale while remaining non-destructive. 57

In the field of advanced 3D microelectronics architecture, 58

the outstanding performance of CD-SAXS in providing high- precision critical dimension measurements has been well- established [ ]. CD-SAXS has been effectively utilized to investigate the widely popular FinFET structures of gratings and has also validated its measurement capabilities on po- tential future grating structures [ The roughness criti- cally impacts transistor performance, and CD-SAXS has the

ability to measure roughness information and has exhibited robust characterization potential under varied conditions, in- cluding synchrotron radiation sources and laboratory environ- ments [ ]. Previous research indicates that the precision of grating critical dimensions is affected by the energy of the X- rays, highlighting a need for further studies on the influence of X-ray energies on measurement accuracy. The technology

has marked significant advancements in addressing grating 73

roughness issues, providing important insights for optimiz- ing transistor performance [ ]. Notably, the CD-SAXS experiments conducted under synchrotron radiation sources not only provide highly accurate data for grating roughness but also lay the foundation for the design and optimization of future grating structures. Furthermore, the successful ap- plication of CD-SAXS in a laboratory environment has pro- moted the development and spread of CD-SAXS technology.

In summary, CD-SAXS demonstrates remarkable potential in

3D microelectronics architecture, offering powerful tools and 83

methods for the study of grating structures. Currently, CD-SAXS primarily utilizes synchrotron radia- tion as its X-ray source. Although synchrotron radiation can meet the requirements of high brightness and sensitivity, CD- SAXS applications under laboratory X-ray sources still face certain limitations. To address the challenges associated with the high cost and constrained experimental conditions of syn- chrotron sources and to expand the application domains of the technology, it is crucial to establish laboratory X-ray sources.

The selection of target materials for laboratory X-ray sources

is a critical step in determining the X-ray energy spectra to 94

meet the specific requirements of grating measurements. Ad- ditionally, the influence of beam size on measurement pre- cision has not been systematically investigated in previous studies. In this work, the effects of varying beam size and X-ray photon energies on CD-SAXS experimental accuracy were examined.

EXPERIMENTAL AND CALCULATION METHODS CD-SAXS measurements were conducted at the small- angle X-ray scattering beamline of the Shanghai Synchrotron Radiation Facility (SSRF). Measurements were conducted with 12 keV photons at the center position of the sample, uti- lizing four different beam sizes. Additionally, a further series of measurements were performed at the same sample position with the same beam size, employing three different photon energies.

Critical Dimension Small Angle X-ray Scattering CD-SAXS measurements were conducted on a trapezoid sample, detailed in Figure S1, which illustrates a trapezoidal profile grating model characterized by several geometrical parameters: pitch , height , width , the sidewall angle (SWA) , and the roughness of the sidewall.

This study utilized transmission small-angle X-ray scatter- ing through a silicon substrate with a thickness of 0.7 mm.

The measurement geometry, as elaborated in Figure S2, spec- ifies the vectors for both the beam and the detector. The configuration aligns the -direction with the incoming beam, while the - and -directions are set perpendicular to both the primary beam and each other. The sample was placed on a rotating platform with the line direction of the grating aligned in parallel with the rotation axis. Its rotation range was from with a step of , and the data collection time at each angle was 10 seconds. To facilitate the visualization of the diffraction data from all the tilt conditions simultaneously, a reciprocal space map (RSM) was constructed. The recipro- cal space map representation was prepared by extracting the diffracted intensities from each individual tilt condition into a one-dimensional scattering profile.

The scattered intensity from the system was recorded as a function of the scattering vector , whose modulus value can be obtained from Equation ( ), where is the angle of diffraction (see Figure S2) and is the X-ray wavelength.

The diffraction peak separation is given by Equation ( where is the period of the grating sample, thus facilitat- ing the straightforward acquisition of pitch information. Crit- ical dimensions describing the nanostructure are determined by the relative intensity and profile of the peaks at different rotation angles relative to the incidence.

| q | = 4 π λ sin θ (1) 142

∆ q = 2 π L (2) 144

The three-dimensional configuration of the lines is de- picted through a sequence of form factors that have been convoluted with the structure factors from an ideal grating.

The general form of the scattered intensity in CD-SAXS is calculated from Equations ( ), where is the shape function [ is the pitch, and denotes the convolu- tion operation. The Debye-Waller factor describes the interfacial roughness. In application, the Fourier transform of each trapezoid in the model is computed, and results are aggregated to determine the amplitude , which is then used in the calculation of . The structure factor of the grating reflects the periodic information of the grating nanos- tructure. Since the Voigt profile serves as a theoretically nat- ural description for the shape of diffraction peaks, the Voigt

function’s profile can be adjusted by tuning its parameters to 159

match the shape of each diffraction peak in the CD-SAXS ex- perimental results [ ]. Using the structure factor, each Voigt profile is constrained to periodic positions along the direc- tion defined by the pitch . In practice, the structure factor can be constructed by convoluting the Voigt profile with a one-dimensional lattice of Dirac delta functions along the direction.

A f ( q ) = �

) exp(

I f ( q ) = | A f ( q ) | 2 (4) 169

I ( q ) = I f ( q ) exp � − i q 2 σ 2 DWF � (5) 171

The analysis of CD-SAXS involves an inverse and itera- tive approach. This method compares the calculated scatter- ing from a hypothesized form function with the actual scat- tering data. The trial shape is adjusted iteratively until the calculated scattering aligns with the observed scattering data.

For the trapezoidal grating sample in this trial, the influence of its various parameters on the form factor is illustrated in figure S3. From these graphs, it can be observed that differ- ent critical dimensions of the grating have varying effects on the shape factor. Comparing the influences of each parame- ter aids in adjusting their ranges to fit experimental data more accurately.

Curve Fit

Curve fitting is a prevalent technique in the sciences and 185

engineering to approximate the functional relationship repre- sented by discrete data points [ ]. Through mathemati- cal methods, these discrete data points are represented using continuous curves or more densely packed discrete equations.

This enables mathematical calculations, theoretical analysis of experimental results, and even estimation for unmeasur- able locations. This method plays a crucial role in solving many problems that rely solely on sampling or experimenta- tion to acquire discrete data.

The fitting of experimental data is framed as an optimiza- tion problem, with the fitting function’s undetermined param- eters, such as the critical dimensions in CD-SAXS. The op-

timization objective aims to minimize a certain error metric 198

between the observed experimental data and the function val- ues of the fitting function. A typical optimization goal is to

minimize the sum of squares of the errors between the values 201

of the fitting function and the observed data. Alternatively, in

cases where the significance or distribution of observed data 203

varies, a weighted sum of squares of errors can be employed as the optimization objective [ The least squares approach is the fundamental method for

data fitting. For observed data ( x i , y i ) , ( i = 1 , 2 , . . . , n ) , the 207

objective function of the optimization problem is to minimize 208

the sum of squared errors between the observed experimental and the calculated values of the fitting func- , as illustrated in Equation ( , . . . , p repre- sent the undetermined parameters within the fitting function.

i =1 [ y i − f ( x i )] 2 (6) 213

min J ( p 1 , . . . , p m ) =

Theoretical computation of CD-SAXS involves parame- ters related to several critical dimensions, along with consid- the high-dimensional parameter space formed by the param- eters to be fitted, navigating for optimal solutions within vast

vanced curve fitting approach was employed, combining a 220

stepwise fitting strategy with an iterative optimization pro-

cess. Initially, traditional gradient descent methods were 222

tested but proved inadequate for locating the global optimum

ting approach was adopted, beginning with data visualisation 225

and focusing on analysing the impact of each parameter on formed on specific parameters without considering the influ- ence of others, thereby reducing the dimensionality of the pa- rameter space. Subsequently, an iterative optimization strat- egy is applied, where multiple starting points are randomly selected within the parameter space, and the loss function imum value of the loss function is identified as the optimal starting points is fixed, and the fitting process is repeated mul- tiple times to confirm consistency. This combined approach

has been shown to significantly improve the accuracy and re- 238

liability of the fitting results, thereby enabling efficient navi- gation of the high-dimensional parameter space in CD-SAXS measurements.

Measurement Error Analysis To ensure the reliability of the measurements, all experi- mental conditions other than the variables of interest (beam size and X-ray energy) were carefully controlled. The sample environment, detector settings, and other instrumental param- eters were kept constant throughout the experiments. Data collection began only after the system had stabilized follow- ing changes in beam size or X-ray energy. This rigorous con-

trol of experimental conditions minimized potential interfer- 250

imental data were pre-processed to ensure accuracy and re- liability. Outliers, such as those caused by detector defects (e.g., bad pixels), were identified and removed. Given the large number of data points involved in CD-SAXS fitting, the

removal of outliers did not significantly impact the overall 256

dataset. Additionally, the data were normalized to facilitate background scattering subtraction, which reduced the influ- ence of background noise on the fitting process. This nor- malization step is critical for improving the precision of the subsequent analysis.

In small-angle X-ray scattering experiments, Equation ( provides a fundamental tool for describing the relationship between scattering angle , scattering wavelength , and scat- tering vector . Simultaneously, Equation ( ) offers a method for analyzing the uncertainty of the scattering vector. In this equation, represents the scattering angle error caused by the pixel size, with each pixel size being fixed for the detector used in CD-SAXS measurements. As the energy of the X- rays increases, their scattering wavelength becomes shorter, resulting in a decrease in the corresponding scattering angles for each scattering vector . In such cases, the impact of the scattering angle error induced by pixel size in

will increase. This results in an increased uncertainty of the scattering vector. This suggests that increased X-ray energy corresponds with elevated signal error detection. This analy- sis provides insight into potential sources of errors when con- ducting small-angle X-ray scattering experiments at different energies. It is important to take appropriate measures to max- imize the accuracy and reliability of experimental data.

q = 4 π sin θ λ (7) 281

∆ q = 4 π sin( θ + ∆ θ ) λ − 4 π sin θ λ (8) 282

The detector enables precise measurement of scattered sig- nal intensity. Processing the collected data allows for deter- mination of the peak center position. Typically, the position of the maximum intensity value in the collected data corre- sponds to the center of the peak. However, the pixels of the

detector have a finite size, which introduces an inherent error 288

in determining the peak center position within a pixel-sized 289

area. The error in the peak center position caused by pixel size can be described using the uncertainty of the scattered signal, as expressed in Equation ( Where is the scattering angle, is the beam divergence, is the size of the beam, is the point spread function of the small-angle detector, is the distance from the sample to the detector, and is the energy resolution of the incident X-ray.

In CD-SAXS measurement experiments, the distance be- tween the sample and the detector is maintained constant.

Notably, a decrease in the beam size , as depicted in Equa- tion ( ), results in a reduction in the value. This implies enhanced accuracy for the measurement results for each scat- tering vector . Therefore, a smaller beam size is associated with more precise measurement results. However, the influ- ence of pixel size on the results must be considered. When the beam size is small enough, the detector cannot provide the necessary resolution to accurately measure the scattering vector due to the constraint of the pixel size. If overly small, the beam size can narrow the full-width at half-maximum (FWHM) of the diffraction peak, thus challenging the detec- tor’s ability to accurately characterize the peak’s shape, ad- versely affecting result accuracy. Careful beam size selection is essential to ensure accurate measurement results and over- come the limitations of detector resolution.

As the scattering angle increases, the uncertainty of the

scattered signal decreases. Therefore, as the magnitude of 317

the scattering vector increases, the signal uncertainty de- creases. Utilizing the center position of the peak value of the first-order diffraction peak to determine the periodic informa- tion of the grating, the period of the sample is not accurate due to the influence of uncertainty. This uncertainty is fur- ther accentuated with an increase in the scattering vector The phenomenon is evidenced by the concordance between the theoretical peak position of lower-order diffraction peaks and the corresponding experimental data, while there is a dif- ference between the theoretical peak position and the experi- mental data peak position of higher-order diffraction peaks.

Considering the characteristics of the structure factor in CD-SAXS, the peak positions of higher-order diffraction peaks can be used to determine the grating period. In this process, the uncertainty in the scattering signal caused by a single pixel can be divided into multiple portions, effectively

reducing the overall uncertainty and consequently minimiz- 334

ing errors in fitting the grating period. Meanwhile, the gaps between pixels from some detectors may lead to undetected peak values in the diffraction pat- terns, rendering sole reliance on maximum intensity for de-

termining diffraction peak centers inaccurate. The shape and 339

positional information of each diffraction peak can be accu- rately determined through the method of fitting the diffrac- tion peaks. To mitigate this deviation, a multi-peak fitting approach is employed to fit the diffraction peaks in experi-

mental results, simultaneously obtaining position information 344

for each diffraction peak. The final fitting results align the peak centers of the structure factor with the diffraction peaks in experimental results, thereby enhancing the accuracy and reliability of the measurement results.

In the process of fitting diffraction peaks, the full-width at half-maximum (FWHM) determines the shape and sub- sequently helps ascertain the peak center position. Insuffi- cient data points can lead to imprecise measurements of the FWHM, affecting both the peak fitting and the final autocor- relation coefficient.

Although narrower diffraction peaks facilitate peak posi- tion localization, they offer fewer data, which limits the detail in peak information. This scarcity of data complicates dis- tinguishing background noise in the overall fitting process.

Therefore, selecting an appropriate beam size is crucial for achieving accurate fitting results.

RESULTS AND DISCUSSION Effect of Different Beam Sizes

results alongside other measurement techniques to evaluate 364

performance. Scanning Electron Microscopy (SEM) can pro- 365

vide information about the grating pitch without delving into the grating’s depth, thus preserving sample integrity. contrast, Atomic Force Microscopy (AFM) can theoretically measure the grating depth. However, due to the relatively

coarse nature of AFM probes concerning critical scales, cru- 370

cial information within surface topographic images might be obscured, leading to potential inaccuracies. Moreover, limita- tions may arise during grating depth measurements, restrict- ing the probe’s access to the bottom of grooves due to neigh- boring gratings. Consequently, AFM struggles to provide pre-

cise depth-oriented details, offering only periodic information within the scanned grating sample area.

CD-SAXS Pitch (nm) Width (nm) Height (nm) Sidewall angle /( Optical Critical Dimension (OCD) yields depth informa- tion but requires referencing results from AFM and SEM be- fore the data fitting process. Accurate period determination in

OCD measurements is critical for obtaining precise structural 381

size information. Figure S4 shows the measurement results of gratings by SEM, AFM, and OCD.

This study investigates the effect of various beam sizes on data acquisition and result precision during the experimen- tal process. Four distinct beam sizes were identified on the grating sample at the small-angle X-ray scattering beamline, with the smallest spot designated as P1 and the largest as P4. displays the shapes and dimensions of beams P1 to P4.

It is acknowledged that there is a degree of human error in- herent in the process of sample placement. Consequently, the result of each angle measured will be affected by an error in

the initial angle. To compensate for the error in the initial an- 394

gle, the symmetry of Figure can be utilized. Reconstructed images of CD-SAXS measurements from different angles and different beam sizes produce an intensity map as a function of , shown in Figure In CD-SAXS measurement, a silver behenate sample is used for calibrating the distance from the sample to the de- tector, which is set at approximately 2.95 meters.

maintaining a fixed sample-to-detector distance, a smaller 402

beam may result in some data loss near the full-width at half- maximum (FWHM) of the diffraction peak, as shown in Fig- . A detailed analysis of the results obtained from the P1 group shows that each peak exists relatively independently in the one-dimensional curve. The measured data points are mainly distributed around the peak values of the diffraction peaks and between the neighboring diffraction peaks.

Although this experiment rapidly determines the diffrac-

tion peak positions, a significant drawback is the shortage of 411

data points near the half-maximum of the diffraction peaks, which is a phenomenon evident in the P1 curve illustrated in Figure . In the process of fitting the diffraction peaks, FWHM was employed to determine peak shapes and subse- quently to determine the positions of the peak centers. The scarcity of data points results in large absolute values of the derivatives at the diffraction peaks’ half-maximum positions, making it difficult to accurately determine the FWHM of the peak shapes. Insufficient data points led to inaccurate mea- surements of the width at half-maximum, which in turn af- fected the fitting of the diffraction peaks and thus the final autocorrelation coefficient. points, particularly in the direction, where the number of data points is comparatively lower than in the other groups.

This scarcity of data points makes it challenging to discern the true trend, leading to a decrease in fitting accuracy and rendering the fitted values less reliable. Notably, the data at

q x = 0 . 252 nm − 1 are particularly susceptible to noise, which 430

hinders a comprehensive assessment of the overall trend of the experimental data. This increased difficulty in separating background signals complicates distinguishing between use- ful signals and background noise. Consequently, the fitting al-

gorithm may overfit to noise, leading to significant deviations 435

in fitted values from the expected experimental outcomes.

The results of CD-SAXS measurements using different beam sizes were summarized in Table . Observing the re- sults, measurements with the smallest beam size exhibit the smallest pitch confidence interval.

Although the results in this group show minor differences in grating width, height, and sidewall angle compared to the expected values (i.e., the known critical dimensions of the reference material (RM) used in our experiments), the confidence interval for the side- wall angle was notably large.

Even if completed many exper- iments, the result of the sidewall angle was still a higher level

of uncertainty in the P1 group. Significant sidewall angle un- 447

certainty for P1 derives from the small beam size, which re- sults in experimental data that does not accurately describe the shape of diffraction peaks. Therefore, it is challenging to provide precise information about the shape factors. A mod- erate increase in the beam size for group P2 notably narrows the confidence interval without evident deviations in fitting results. Conversely, further beam size increases in P3 and P4 elevated uncertainty in the critical dimensions. Specif- ically, the fitting results in the group P4 deviate noticeably from those of other groups and the expected values, substan- tially reducing measurement precision.

The deviation ob- served in the results can be attributed to the relatively large beam size, resulting in a set of diffraction peaks with com- paratively smooth and broadened profiles by the smear effect

of scattering. This broadening in the peak profiles leads to 462

increased uncertainty in confirming the exact locations of the peak centers. In the context of CD-SAXS data fitting, it is imperative to use the peak values of the diffraction peaks to discern the trend of the shape factor, thereby extracting cru- cial dimensional information about the grating sample. The increased uncertainty in peak center identification renders the fitting algorithm more vulnerable to nearby values, thus di-

minishing the emphasis on peak centers and adversely affect- 470

ing shape factor determination and fitting precision. These

findings highlight the significant impact of beam size on the 472

uncertainty and precision of results in CD-SAXS measure- ments. Therefore, appropriate beam size selection is crucial

for obtaining accurate and reliable measurement results, en- 475

suring the credibility and reproducibility of experimental re- sults.

Width/nm 50.0±0.6 Height/nm 131.3±0.4 Sidewall angle/ (°) 1.1±0.1 The four different beam sizes used in CDSAXS measurement experiments. (a) Picture (a) shows the initial beam size, and the experiment performed on this beam is named P1. The beam sizes corresponding to (b), (c), and (d) are, respectively, 2, 3, and 4 times the original beam size. The experiments corresponding to these groups are designated as P2-P4.

Effect of Different Sizes To thoroughly analyze the impact of X-ray photon energy on CD-SAXS measurement accuracy, this study performed CD-SAXS assessments on gratings using various X-ray pho-

ton energies while maintaining a constant beam size. In CD- 482

SAXS experiments, scientists at NIST recommend conduct- ing tests at energy levels above 17 keV. Higher-energy X-rays (a) is the q intensity map obtained by the CD-SAXS measurement of the P1 group, (b), (c) and (d) are the results of the P2-P4 group In CD-SAXS measurement, a silver behenate sample is used for calibrating the distance from the sample to the detector , which is set at about 2.95 meters. When maintaining a sample-to-detector distance, a smaller beam may result in some data loss near the FWFM of diffraction peak, as shown in figure 3 [FIGURE:3]. A detailed analysis of the results obtained from the P1 group shows that each peak exists relatively independently in the one-dimensional curve. The measured data points are mainly distributed around the

= 0 using the least squares method, the circles are raw data and the lines represent the fitting . In the experiments with the four different beam sizes, fit experimental data as a function of = 0 using the least squares method, the circles are raw data and the lines represent the fitting Parameter Pitch (nm) Width (nm) Height (nm) Sidewall angle /( facilitate the straightforward transmission through the sub- strate of the grating sample. However, this simplicity comes at the cost of reduced interaction between the X-rays and the els. When the X-ray photon energy is high, the absorption of the grating sample decreases, and the transmittance in- creases. This results in some of the background scattering getting through the grating and being received by the detec- tor. To perform CD-SAXS measurements using laboratory X-ray sources, the challenge of low intensity must be ad- dressed. Achieving high-quality scattering signals in a short time frame necessitates an improvement in photon utilization efficiency. Therefore, the careful selection of the metal target in the X-ray tube is crucial in the design of laboratory X-ray sources. In contrast, the synchrotron radiation sources pro- duce X-rays with exceptionally high brightness, resulting in a

significantly elevated photon flux per unit area. This charac- 502

teristic allows researchers to acquire a substantial amount of information from the detector within a given time period. The utilization of synchrotron radiation sources for CD-SAXS ex- periments at various X-ray photon energy levels can serve as a reference for the choice of metal targets when constructing laboratory X-ray sources.

Reconstructed images from CD-SAXS measurements pro- duce an intensity map as a function of , shown in tion peaks, and the peak centers of these diffraction peaks can provide information about the structure factor of the trape- As shown in Figure (b), a detailed examination of the Notably, the data at =0.252nm are particularly susceptible to noise, which hinders a comprehensive assessment of the overall trend of the experimental data. This increased difficulty in separating background signals makes it difficult to distinguish between useful signals and background noise. As a result, the fitting algorithm may overfit to noise, leading to significant deviations in fitted values from expected experimental outcomes.

. In the experiments with the four different beam sizes, (a)-(d) correspond to the results obtained from the experiments of groups P1-P4. The scattering intensity along vertical slices at evenly spaced intervals in , obtained from reciprocal space maps, the circles are raw data and the lines represent the fitting.

The scattering intensity along vertical slices at evenly spaced intervals in , obtained from reciprocal space maps, the circles are raw data and the lines represent the fitting.

( q z = 0 ) and over q z (across all peak orders). Lack of clear 517

spacing between diffraction peaks significantly increased mu- 518

tual interference, and excessively broad peaks reduced the resolution, complicating the identification of peak positions.

High resolution is crucial in CD-SAXS measurements to ac- curately determine the periodic information. Observation at

16 keV revealed a notable broadening of diffraction peaks that 523

caused adjacent peaks to obscure each other, with significant 524

intensity variations. The second-order diffraction peak was heavily influenced by the first-order peak, posing challenges

in accurate observation. Furthermore, peak widening con- 527

siderably decreased the signal-to-noise ratio, creating uncer- tainty in the relative intensity correlation between signals and noise. Consequently, this decreased the accuracy of the fitting results, markedly increasing the uncertainty.

The CD-SAXS results obtained at 10 keV and 12 keV are shown in Figure . These results have been normalized to facilitate the analysis of the energy effect with various scat- tering cross-sections on the experimental results. Although the X-ray transmissivities at 10 keV and 12 keV were lower than at 16 keV, the diffraction peaks were clearly distinct and do not overlap. The clear shapes of diffraction peaks pro- vide advantages in data fitting. Additionally, the diffraction peaks were distinctly separated from the background scat-

tering. There are significant differences in peak heights and 541

background intensity, which makes it easy to distinguish be- tween them. Higher X-ray photon energy leads to increased background scattering, as shown in Figure . This decreased contrast can impair the accuracy of peak measurement and analysis. As the X-ray energy increases, the absorption ca- pacity of the grating sample decreases, while the penetration capacity increases. This implies that a higher number of X- The results of CD-SAXS measurements using different beam sizes were summarized in Table 2 [TABLE:2]. Observing the results, measurements with the smallest beam size exhibit the smallest pitch confidence interval. Although the results in this group show minor differences in grating width, height, and sidewall angle compared to the expected values, the confidence interval for the sidewall angle was notably large. Even if completed many experiments, the result of the sidewall angle was still a higher level of uncertainty in P1 group. Significant sidewall angle uncertainty for P1 derives from the small beam size, which results in experimental data that does not accurately describe the shape of diffraction peaks. Therefore, it is challenging to provide precise information about the shape factors. A moderate increasing in the beam size for group P2 notably narrows the confidence interval without evident deviations in fitting results. Conversely, further beam size increases in P3 and P4 elevated uncertainty in the critical dimensions.

Specifically, the fitting results in the group P4 deviate noticeably from those of other groups and the expected values, substantially reducing measurement precision. The

intensity map obtained from CDSAXS measurements at 16 keV. (b) In the =0cut, it is clearly visible that there is overlap between the diffraction peaks. intensity map obtained from CDSAXS measurements at 16 keV. (b) In the = 0 cut, it is clearly visible that there is overlap between the diffraction peaks.

ray photons can pass through the substrate, accompanied by 549

a corresponding enhancement in the background signals that pass through the sample and are received by the detector.

Concurrently, the scattering cross-section decreases, result-

ing in a weakening of the intensity of each scattering event. 553

Yet, an overall enhancement of the diffraction peak signal and background signal is observed, but the signal-to-noise ratio decreases. Figure S5 enables the calculation of the scatter- ing probability at each energy level; it can be demonstrated that as the energy increases, the scattering probability also in- creases. The enhancement of the background signal has the effect of reducing the signal-to-noise ratio of the diffraction pattern, which, in turn, results in the boundaries of the diffrac- tion peaks becoming blurred and increasing their width. The analysis of the grating structure becomes more complex due to the indistinct positions and peaks of the different diffrac- tion peaks.

In the experimental results at 10 keV and 12 keV, we can readily observe that the background scattering remains stable between the diffraction peaks, unaffected by the presence of the diffraction peaks. This reduces uncertainties introduced by noise during the fitting process. The CD values obtained at three different energies are listed in Table Parameter 10 keV 12 keV 16 keV Pitch (nm) Width (nm) Height (nm) Sidewall angle ( The above confirms the feasibility of CD-SAXS testing be- low 17 keV. As the energy of the X-ray increases, the param- eters of the grating achieve an optimal solution, which occurs when the energy of the X-ray is 12 keV. The mean values of height, sidewall angle, and pitch at 16 keV exhibit a notable discrepancy from the expected values. The mean value of width at 10 keV exhibits a considerable discrepancy from the expected value. Notably, the measurement results at 12 keV exhibit the smallest confidence interval. In comparison to the other two sets of results, the data at 12 keV are more accurate and relatively stable.

The dependences of CD on the X-ray beam size and photon energy are shown in Figure . Clearly, the changes in these conditions affect the average measurement values, allowing easy observation of the confidence intervals for each dataset.

By comparing the size of the grating standard sample, it is possible to select experimental conditions with high precision and small confidence intervals. This provides a basis for de-

termining the experimental conditions for future CD-SAXS 590

measurements.

CONCLUSION

In this work, the critical dimensions of a grating with a half-pitch of 50 nm were investigated using small-angle X-ray scattering with various spot sizes and incidence energies. We proposed an effective fitting method for reconstructing grat- ing structural information and obtained critical dimensions.

The optimized values of CD were achieved with an X-ray en- ergy of 12 keV and a spot size of 200 m, with precisions of 0.2 nm for the pitch, width, and height, as well as 0.1° for the

sidewall angle (SWA). This research demonstrates the signif- 601

icance of beam size and confirms the viability of CD-SAXS experiments within specific energy ranges, laying the foun- dation for future developments and providing valuable refer- ences for laboratory setups. In conclusion, our study makes

a significant contribution to CD-SAXS experimentation and 606

paves the way for further technological advancements.

Despite its advantages, CD-SAXS technology faces sev- eral challenges that limit its current applicability. One ma- Reconstructed images from CD SAXS measurements produce an intensity map as a function of , shown in figure 5 FIGURE:5. The 16 keV test results reveal numerous diffraction peaks, and the peak centers of these diffraction peaks can provide information about the structure factor of the trapezoidal grating sample.

As shown in figure 5(b), a detailed examination of the data is achieved through one dimensional profile cuts over =0) and over (across all peak orders). Lack of clear spacing between diffraction peaks significantly increased mutual interference, and excessively broad peaks reduced the resolution, complicating the identification of peak positions. High resolution is crucial in CD AXS measurements to accurately determine the periodic information. Observation at 16keV revealed a notable broadening of diffraction peaks that caused adjacent peaks to obscure each other, with significant intensity variations. The second order diffraction peak was heavily influenced by the first order peak, posing challenging in accurate observation.

Furthermore, peak widening considerably decreased the signal noise ratio, creating uncertainty in the relative intensity correlation between signals and noise. Consequently, this decreased the accuracy of the fitting results, markedly increasing the uncertainty.

The CD SAXS results obtained at 10 keV and 12 keV were shown in the figure These results have been normalized to facilitate the analysis the energy effect with various scattering cross section on the experimental results. Although the X transmissivities at 10 keV and 12 keV were lower than at 16 keV, the diffraction peaks were clearly distinct and do not overlap. The clear shapes of diffraction peaks provide advantages in data fitting. Additionally, the diffraction peaks were distinctly separated m the background scattering. There are significant differences in peak heights and

ratio of the diffraction pattern, which, in turn, results in the boundaries of the diffraction peaks becoming blurred and increasing their width. The analysis of the grating structure becomes more complex due to the indistinct positions and peaks of the different diffraction peaks. jor limitation is the requirement for a high signal-to-noise ratio (SNR) to accurately infer key dimensions from the data.

Achieving high SNR often requires prolonged irra- diation times, which can be impractical for routine labo- ratory use. Additionally, the low flux of laboratory X-ray sources further restricts the efficiency of CD-SAXS measure- ments. To address these challenges, the Mo X-ray source emerges as a favorable choice due to its balanced perfor- mance characteristics. However, the Liquid-Metal-Jet X-ray source—particularly alloy-rich indium variants—presents a promising alternative, as it enhances flux density by an order

of magnitude while achieving smaller spot sizes (100 µ m) 621

compared to solid-target sources (500 m with multilayer- coated Montel mirrors). These advancements directly align with our CD-SAXS measurement accuracy results , where reduced spot size and enhanced flux synergistically improve precision. Future efforts should prioritize optimizing such high-brightness sources alongside algorithmic innovations to streamline data acquisition and analysis, ultimately accelerat- ing the adoption of CD-SAXS in both research and industrial metrology.

ACKNOWLEDGEMENTS The authors are deeply grateful to the National Natural Sci- ence Foundation of China (No. 12175295), and the National Key R&D Program of China (2021YFA1601000) for finan- cial support. This work was also supported by the User Ex- periment Assist System of Shanghai Synchrotron Radiation Facility (SSRF). The authors thank the beamline scientists at beamlines (BL16B1, BL19U2, BL10U1, and BL08U) of Shanghai Synchrotron Radiation Facility for support and dis- cussions.

SUPPORTING INFORMATION The Supporting Information shows the structure of the grating sample for CD-SAXS measurement, provides a schematic of the CD-SAXS measurement, and illustrates the effect of different critical dimensions on the shape of the form factor. In addition, the Supporting Information includes graphs of the measurement results from other comparative methods. Finally, it shows the scattering cross-section ver- sus transmission curves at different X-ray energies. . In the experiments with the three different energies, fit experimental data as a function of In the experimental results at 10 keV and 12 keV, we can readily observe that the background scattering remains stable between the diffraction peaks, unaffected by the presence of the diffraction peaks. This reduces uncertainties introduced by noise during the fitting process. The CD values obtained at three different energies as listed in Table The above confirms the feasibility of CD-SAXS testing below 17 keV. As the energy of the X-ray increases, the parameters of the grating achieve an optimal solution, which occurs when the energy of the X-ray is 12 keV. The mean values of height, sidewall angle, and pitch at 16 keV exhibit a notable discrepancy from the expected values. The mean value of width at 10 keV exhibits a considerable discrepancy from the expected value. Notably, the measurement results at 12 keV exhibit the smallest confidence interval. In comparison to the other two sets of results, the data at 12 keV are more accurate and relatively stable.

Width/nm 51.0±0.9 50.1±0.4 50.4±2.4 Height/nm 131.9±1.7 131.2±1.3 132.4±3.1 Sidewall angle/(°) 1.3±0.7 1.2±0.3 1.4±0.4 Measurement of critical dimensions of gratings using SAXS with varying energies and spot sizes. (a) shows the pitch information of gratings measured in each group of CD-SAXS measurements. (b) - (d) respectively show the width, height and side wall angle information of the grating.

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