Methodology for evaluating the information distribution in small angle scattering from periodic nanostructures

Abstract

Optimizing the extraction of information from x-ray measurements while minimizing exposure time is an important area of research in a variety of fields. The semiconductor industry is reaching a point where the traditional optical metrologies need to be augmented in order to better resolve the critical dimensions of structures with feature sizes below 10 nm. Critical dimension small angle x-ray scattering (CDSAXS) is one measurement technique that is capable of characterizing detailed features of periodic nanostructures. As currently implemented, the measurement utilizes the combined scattering from up to 60 different angles. Reducing the number of angles would dramatically improve the feasibility of CDSAXS for implementation in a fabrication setting, but currently there are no clear guidelines as to which angles provide the most information to minimize the uncertainty in the shape of the target structure while maximizing the throughput. In order to develop guidelines for optimizing the angle selection, simulation studies were conducted on a wide variety of structures with subsets of the full angular range to identify which angles minimized the overall shape uncertainty. Analyzing sets of two angle pairs (including all combinations between 0 deg and 60 deg) provides guidance on which angles best constrain the samples. For select samples, higher numbers of angles were included to explore the impact of additional information on the model uncertainty. In general, low angles (<3 deg) best contributed to minimizing the line-width uncertainty, while higher angles near high curvature regions of the scattering profile best constrained the height of the structure. The minimum uncertainty was generally achieved with combinations of the two. This simulation approach can be used to minimize the number of angles measured on real samples and significantly reduce the measurement time.

1. Introduction

There is a wide variety of x-ray-based measurements where it is highly desirable to minimize the measurement time. For measurements such as computed tomography, the obvious motivation is to minimize the radiation dose the patient receives.^1,2 Measurements on organic materials, such as protein crystallography, need to take into consideration beam damage to the sample.³ X-ray measurements for process monitoring have fewer concerns about sample damage, but instead desire minimal measurement time in order to optimize the product throughput.^4,5 A prime example of this challenge is in the semiconductor industry, which has begun to experience significant metrology challenges due to the feature sizes of the new technology nodes. As a result, there is a critical need for fast and accurate methods to monitor sub-nm level changes in structure sizes and spacings. X-ray scattering measurements are being developed as complements to optical scatterometry for monitoring the critical dimensions (CD) and structural features of integrated circuits (IC).⁶ CD small angle x-ray scattering (CDSAXS) in particular has significant promise due to its ability to accurately determine the dimensions of IC structures to sub-nm accuracy and to characterize pitch walking.⁷ CDSAXS can be used to measure periodic structures, such as FinFETS, metal lines, photoresist patterns, and contact holes. CDSAXS measurements are performed by rotating a sample through a series of incident angles and taking a diffraction pattern at each angle. A reciprocal space map is reconstructed from those patterns and analyzed to evaluate the structure. With a synchrotron x-ray source, it is straightforward to collect as many angles as desired. However, in a production setting using a compact x-ray source, collection of 60+ incident angles becomes prohibitively time intensive. It has been previously experimentally demonstrated that accurate determination of key structural features can be extracted with as few as five incident angles.⁸ Each incident angle samples a small portion of Fourier space (reciprocal space) and provides incomplete information on the original shape. The information describing the original shape is spread nonuniformly across Fourier space. In some cases, different angles will contain redundant information. These angles will not increase the total information known and only increase the signal to noise. Other angles will contain unique information about the structure and will increase the information known. In addition, certain angles could contain information about specific aspects of the original shape. The total information content of all the measured angles will determine the uncertainty of the final shape reconstruction. There is no simple approach for determining which incident angles will minimize the uncertainty of the shape parameters of interest due to the nonlinear relationship between the real space structure and the reciprocal space scattering pattern. The relationships will also be highly sample dependent. Here, we demonstrate a simulation methodology which can be used to evaluate the unique information content in individual incident angles of a rotational small angle x-ray scattering (SAXS) measurement and determine which angles or combinations of angles minimize the structural uncertainty of the parameters of interest.

Small angle x-ray scattering (SAXS) is a key tool in evaluating the shape and periodicity of a wide range of structures with length scales on the order of 5 to 100 nm. This includes block copolymers, ^9–11 micelles,^12–14 and structures of interest to the lithographic community such as line gratings and contact holes.^15–20 Typical transmission SAXS measurements are performed at normal incidence, and as a result the scattering pattern primarily contains information on the in-plane structure. Changing the angle between the incident beam and sample results in information on the out-of-plane structure being included in the scattering pattern as well. This allows complete reconstruction of the two- or three-dimensional optical constant profile of a periodic structure.^8,21,22 Grazing scattering geometries enable fast collection of data from both in-plane and out-of-plane scattering vectors simultaneously, but have the trade-off of requiring large sample area and challenging data analysis.^17,23–27

CDSAXS is most commonly applied to a lithographically patterned line grating, which forms the foundation of the IC.^28,29 Figure 1(a) shows the measurement geometry for this type of sample. The line gratings are aligned parallel to the axis of rotation and measurements are taken at a series of different incident angles. As the sample is rotated further away from normal incidence, the magnitude of the out-of-plane scattering vector (q_z), relative to the in-plane scattering vector (q_x) increases, providing more information about the vertical profile of the sample. A representative scattering pattern from simulated data at a 32-nm pitch is shown in Fig. 1(b). The spacing between the diffraction spots in the q_x direction is proportional to the longest periodic length scale in the sample. This spacing will shrink if the length scale of the unit cell increases. The changes in intensity along the q_z axis can be fit using an inverse, iterative method to evaluate the structure of the target sample.

Fig. 1.

(a) Scattering geometry for the CDSAXS measurement. The line grating is aligned parallel to the axis of rotation (φ); (x, y, z) represent the real space coordinates and (q_x, q_y, q_z) represent the scattering vectors. Here, q_x and q_y are the in-plane scattering vectors perpendicular and parallel to the grating, respectively; q_z is the out of plane scattering vector, and 2θ represents the scattering angle. All coordinates are defined relative to the sample. (b) Simulated data for a 32-nm pitch grating using 1 deg steps between −60 deg and 60 deg.

X-ray measurements of line gratings have been shown to be sensitive to both the shape of the line and to the spacing between lines. An example of this can be pitch walking between adjacent lines; the effective unit cell is now double the original spacing and as a result the scattering peaks occur at half the distance. Previous results have shown that the intensity of the half-order peaks correlates to magnitude of the offset from the average pitch.³⁰ As a result, the measurement is extremely sensitive to the magnitude of any offsets within a periodic pattern. The additional peaks contain information about the shape of the grating, which could reduce the uncertainty of the structure. Monitoring this type of pitch walking is particularly important for structures produced by self-aligned multiple patterning methods.³¹ This was studied for samples prepared with self-aligned spacer quadrupling.⁷ The pitch walking results in both half- and quarter-order peaks. Sub-nm shifts in the spacings were found to result in significant changes to the intensity of the half- and quarter-order peaks, enabling accurate determination of the pitch errors.

There has been some effort to examine the distribution of information as a function of angle in a rotational SAXS measurement.²⁰ A sample patterned via the spacer quadrupling approach, similar to what was discussed in the last paragraph, was measured at 1 deg steps from 60 deg to −60 deg. The results of this measurement were then resampled at uniform intervals for various ranges of the data (i.e., every 5 deg out to ±30 deg). It was found that for this sample, the in-plane parameters could be determined with similar accuracy to what was found in the full reconstruction for four different angles, at a maximum angle of only 30 deg. The lower angles generally have more information on the in-plane structure, as higher angles begin to provide more information on the vertical structure. As a result, it was necessary to include higher angles in the resampling to approach the same level of uncertainty for the height as was found in the full reconstruction. When the largest sampled angle was below 10 deg, there was both an order of magnitude increase in the uncertainty and a significant deviation of the best fit from the known structure. This shows that even for relatively short gratings (height ≈ 32 nm), some measurements at higher angles are required to enable an accurate fit. The results of this study demonstrated that the angular range could be significantly reduced from the full range of ±60 deg that is typically measured but did not attempt to determine which angles provided the most maximum information to the data fitting. There have been similar efforts to understand how to constrain optical scatterometry measurements in order to yield the highest parameter sensitivity. Scatterometry uses the change in intensity and polarization of visible light (~400 to 800 nm) to evaluate the shape of gratings, and the sensitivity of the measurement varies as a function of the range chosen for the incident angle and the azimuthal rotation.^32,33 Optimizing the measurement conditions is particularly important for interrogation of modern IC features which are considerably smaller than the optical wavelengths involved in the measurement.

We simulated the scattering patterns for a series of different line grating structures to develop guidelines for the relationship between target structure and the incident angles which will minimize the uncertainty. Subsets of angles from the simulated scattering patterns were fit with a Markov chain Monte–Carlo (MCMC) algorithm in order to investigate the uncertainty of the model for each subset of angles. These results were analyzed for the different structural parameters to develop general guidelines for selection of incident angles with greatest information content for arbitrary structures.

2. Methods

The simulation of scattering patterns from CDSAXS has been described extensively in the literature^7,8,34 and will be discussed briefly here along with the specific details for the scattering study. The scattering is simulated according to the Born approximation and is calculated through the Fourier transform of the scattering length density profile [Eq. (1), ρ(r) is the shape function including contrast, P is the pitch, * represents a convolution]. The grating structure is defined by a stack of trapezoids, which have a computationally efficient analytical solution to the Fourier transform. Debye–Waller factor [DW, Eq. (2)] is used to account for roughness and was set to 1.5 nm for all samples; this includes the impact of line edge and line width roughness. Once the intensity is calculated, a background term [B(q)] and Poisson noise [N(q)] are added to the system as shown in Eq. (3). B(q) is generated from a Gaussian distribution centered at the average background value, while N(q) is generated as a Poisson distribution centered at I(q). Both noise terms were calibrated to match the noise level in previously collected experimental data.^7,8,30 This approach accurately approximates the quality of data obtained by synchrotron measurements.

$$I_o(\mathbf{q})={\mid {\int }_V\rho (\mathbf{r})\ast \sum _n\delta (x-\mathit{nP})e^{-i\mathbf{qr}}\mathrm{d}\mathbf{r}\mid }^2,$$ (1)

$$I(\mathbf{q})=I_o(\mathbf{q})\times e^{-q^2{\mathit{DW}}^2},$$ (2)

$$I(\mathbf{q})=I(\mathbf{q})+N(\mathbf{q})+B(\mathbf{q}).$$ (3)

The uncertainty in the data is probed using an MCMC algorithm. The MCMC algorithm generates a population of models which can then be used to calculate the uncertainty in the profile of the sample. This is accomplished by starting with an initial model (M_init, generally the best known fit) and randomly generating perturbations to the model, which are either accepted or rejected based on the change in fit quality, which is characterized by the change in the goodness of fit (GF) metric described in Eq. (3) [I(q) refers to the intensity simulated by the target structure and I(q)^S refers to the intensity simulated during the MCMC algorithm). The random perturbations generate a candidate model (M_i, with a GF_i), if GF_i < GF_i−1 then the model is always accepted into the population and if GF_i > GF_i−1 then the probability (P_i) of the model being accepted is calculated according to Eq. (5). A random variable α [0,1] is generated and if α < P_i then M_i is accepted into the model population. If α > P_i then M_i is rejected and a new candidate model is generated from the previous position. This process is repeated until enough models have been accepted into the population to reach equilibrium, typically 24 chains with 400,000 steps were utilized and the chains were resampled every 20 steps to remove interchain correlations. The chains were tuned so that they had an acceptance probability between 30% and 45%. The metrics for the uncertainty in this study will be as follows: the ratio of the uncertainty area (A_R), defined as the ratio of the area between the perimeter of the 95% confidence interval (A₉₅) around the fit to the sample and the area of the sample (A) as defined by Eq. (6). The relative line-width at half the height (W_R) and relative height (H_R) uncertainties as defined in Eqs. (7) and (8) (respectively) (W₉₅ is the 95% confidence interval of the line-width at half the height and H₉₅ is the 95% confidence interval of the overall height of the line:

$$\mathrm{GF}=\mid \log [I(\mathbf{q})]-\log [I{(\mathbf{q})}^S]\mid ,$$ (4)

$$P_i=e^{-0.5({\mathrm{GF}}_i-{\mathrm{GF}}_{i-1})},$$ (5)

$$A_R=\frac{A_{95}}{A},$$ (6)

$$W_R=\frac{W_{95}}{W},$$ (7)

$$H_R=\frac{H_{95}}{H}.$$ (8)

3. Results and Discussion

The samples are labeled as Wxx_Hxx, where W indicates the line-width of the grating at half the height of the line and H indicates the total height of the grating, all units are in nm. The parameter space for potential structures in this study is enormous and was therefore paired down to representative samples in order to develop a methodology for choosing incident angles which minimize the uncertainty in the shape of the target structure. The aspect ratio and sidewall angle are key parameters for determining the location of the major features in the scattering profiles for simple structures. The scattering is sensitive to the size of the structure relative to the unit cell, as a result, simulations with various line-widths and heights at a given pitch will be sufficient to develop guidelines on the information distribution for an arbitrary sample pitch. For all samples examined here, the pitch was selected to be 32 nm. Changes in the pitch will only change the q_x positions which are sampled and will not impact the general guidelines developed for optimizing angle selection. The heights were chosen to cover a representative range of structures of interest to the semiconductor industry, from short trenches to taller features which may be present in finFETs. The simulations were scaled to previous measurements taken on silicon oxide lines. For single component structures, the type of materials the lines are composed of will not impact the shape of the scattering curves, only the scattering intensity, and therefore will not impact the conclusions of this analysis. Underlying layers lacking periodicity will also not impact the scattering. A_R, W_R, and H_R were chosen as simple descriptors of the grating, but this process could also have been used to evaluate any other parameter of interest, such as the sidewall angle and top width. The line gratings were approximated via a stack of three trapezoids, the first sample set that was investigated utilized a constant W of 12 nm and heights of 25, 50, and 100 nm. Structural details for all target structures in this paper are presented in Table 1. These structures are shown in Fig. 2 Also shown in Fig. 2(a) are the outer bounds of the algorithm where the limit on the line-width is defined by the pitch and the limit on the height is defined by 2x the line height. These limits will be applied to all models in the study unless otherwise noted. The bounds were chosen to maintain physically realistic structures and to be large enough as to not artificially constrain the results. To investigate the information distribution throughout the angular range subsets of the full angular distribution were investigated. The MCMC algorithm was nm on pairs of angles and all possible pairings were analyzed, i.e., 1 deg was analyzed in conjunction with all other angles between 0 deg and 60 deg and then iterated until all possible combinations were studied. Negative angles were omitted as the scattering is symmetric and therefore the positive angle data will be equivalent to the negative angle data. The simulated data for the three structures are shown in Fig. 3, along with the resulting angle-angle correlation maps for A_R, W_R, and H_R.

Table 1.

Parameters for the structures in Fig. 2.

W12H25		W12H50		W12H100
Z position (nm)	Line-width (nm)	Z position (nm)	Line-width (nm)	Z position (nm)	Line-width (nm)
0	18	0	18	0	18
3	14	6	14	10	14
23	10	46	10	92	10
25	6	50	6	100	6

Fig. 2.

Structures of (a) W12H25, (b) W12H50, and (c) W12H100, showing two unit cells of each structure for visual clarity. Parameters for each model are listed in Table 1. The dotted lines in part (a) show an arbitrary representation of the uncertainty area (U_A) determined by the MCMC algorithm. The line height (H) and the line-width at half the height (W) are also defined. The area inside the blue-dashed outline represents the limits of the area that the algorithm can search, with the limit on the line-width being equal to the pitch and the limit on the height being equal to 2× the height of the structure. The correlation plot analysis to evaluate the information distribution in the scattering pattern from these three samples is shown in Fig. 3.

Fig. 3.

Simulated intensity versus angle plots for the first to fourth peak orders, along with the corresponding angle–angle correlation plots for A_R, W_R, and H_R for the three samples shown in Fig. 2. The results for W12H25 are shown in (a)–(d), for W12H50 in (e)–(h), and W12H100 in (i)–(l).

Figure 3(a) shows the first- to fourth-order peaks as a function of angle for W12H25. The angle-angle correlation plots are shown in Figs. 3(b)–3(d) for A_R, W_R, and H_R, respectively. From this analysis, the pair of angles that contained the lowest A_R can be identified, along with general correspondence between regions of lower A_R and the intensity versus angle curves. The A_R correlation plot [Fig. 3(b)] shows that the fits with the lowest A_R were obtained for pairs including an angle below 3 deg, with most of the lowest A_R pairs including 1 deg. The A_R generally increased with larger angles, with the exception of specific regions, such as those near 25 deg and 45 deg. For reference, the upper limit on the color bar (A_R ≈ 6) shows that the data do not constrain the algorithm at all within the search bounds. The regions that result in reduced A_R correspond with either the angle or the intersection of angles near minima in the scattering peaks, particularly minima in the first-order peak. These are regions of high curvature in the scattering curves and changes in intensity in those regions result in more significant penalty to the GF compared to changes around angles such as 10 deg, where the slopes of all the peak orders are relatively flat. The angle pair with the lowest overall A_R was 1 deg/46 deg, this makes intuitive sense based on the understanding of the information distribution in the scattering pattern from theory and previous experiments.²⁰ Low angles have small out-of-plane (q_z) components and large in-plane (q_x) components, while the opposite is true for the higher angles; therefore, a combination of small and large angles allows information from both components to contribute to the analysis. 46 deg is in the immediate vicinity of a minimum in the first-order peak intensity, as a result changes in the structure result in rapid changes to the GF in that region. The next best combination was 1 deg/20 deg, which again corresponds to the combination of a low angle with a higher angle in the vicinity of a minimum for the first-order peak intensity. Similar correlation maps for W_R [Fig. 3(c)] and H_R [Fig. 3(d)] can be examined to interrogate how W_R and H_R are constrained by the inclusion of different angular regions. The line-width angle-angle correlation map shows a region of lower W_R for pairs of angles below 3 deg, with significantly larger W_R for higher angles pairs which do not include one angle less than 3 deg. This is consistent with the reduction of the q_x component at higher sample angles. In this case, including higher angles near the first-order peak intensity minimum constrains the W_R relative to other high angles, but this constrained W_R does not fully compensate for the reduction in the in-plane information content in the scattering and the pair which results in minimal W_R is 1 deg/2 deg. The angle-angle correlation map for H_RR shows minima in the regions near the first-order peaks at 25 deg and 45 deg. While the combination of these two angles might be expected to minimize H_R, the pairs that produce the lowest H_R include a lower angle. For example, the pair that minimizes H_R is 1 deg/44 deg. Unlike minimizing W_R, which included two low angles, the lower overall intensity from the combination of two high angles does not sufficiently constrain the algorithm to minimize H_R.

These results can be compared to taller structures where the relative heights of all three trapezoids are scaled equivalently and the line-width at half height remains identical. The results of the analysis on a 50-nm tall grating are shown in Figs. 3(e)–3(h) and on a 100-nm tall grating in Figs. 3(i)–3(l). The 50-nm sample had two pairs with similar A_R, 1 deg/14 deg and 1 deg/23 deg, corresponding to combinations of a low angle and an angle near the minima in the intensity of the first-order peak (23 deg) or second-order peak (14 deg). The 1 deg/23 deg pair resulted in lower H_R relative to 1 deg/14 deg, while the latter had a lower W_R. The A_R for the 100-nm sample was minimized at 1 deg/11 deg, again with the second angle in the pair corresponding to the minima in the first-order peak intensity. These results show a clear trend where the high angle pair in the combination that minimizes A_R trends lower with increasing aspect ratio of the structure. This trend follows the shift in the location of the first-order minimum toward lower angles for higher aspect ratio structures. It allows additional contributions from the q_x component of the scattering to be included while still providing sufficient contributions to constrain the line height. The A_R value at the minimum also trends lower with increasing aspect ratio, due to a combination of higher scattering intensity (the scattering power of sample is proportional to the square of the sample volume) and this shift in the location of the first-order minimum. Similar to the 25-nm sample, the angle pairs that minimize W_R are combinations of angles below 3 deg, while the shift in minimum position means that the H_R minima no longer includes a low angle, but rather pairs of angles near the minimum of either the first- or second-order peak intensities (14 deg/23 deg for the 50-nm tall sample and 5 deg/11 deg for the 100-nm tall sample).

This analysis can be extended to larger numbers of angles. For W12H50, the A_R values resulting from simulations with sets of 3, 4, and 5 angles, along with the visual representation, are shown in Fig. 4. For the set of three angles, the pair that led to the best fit for the two angle pair (1 deg/14 deg) was analyzed in combination with the remaining angles from 0 deg to 60 deg. The addition of a third angle produces the lowest_AR either below 10 deg or in the vicinity of the minima of the first-order peak (which is overlaid on the figure for reference), consistent with the information distribution determined by the two angle analysis. The lowest A_R results from the inclusion of 23 deg in the analysis, which results in a significant reduction in H_R relative to two angles. Progression to sets of four angles reduces the magnitude of the variation in the A_R reduction between different angles in the series, and for the set of five angles (which includes 1 deg/2 deg/14 deg/23 deg for all combinations) the relationship between A_R and the fifth angle is essentially flat.

Fig. 4.

(a) Visual representation of A_R for W12H50 as a function of increasing number of angle pairs (2,3,4,5) for the combination which yielded the lowest A_R for each number of angles, along with the model structure. A single unit cell is shown for clarity. (b) A_R as a function of angle combinations for three, four, and five angles including the angles which yielded the lowest A_R for the previous series (e.g., 1 deg/14 deg yielded the lowest A_R for two angle pairs, while 1 deg/14 deg/23 deg yielded the lowest A_R for three angle pairs). The right axis corresponds to the intensity of the first-order peak.

The initial sample series all had relatively small sidewall angles, particularly for the taller structures. Significant sidewall angles in a grating will shift the location of the maxima in the intensity profiles away from 0 deg and reduce the curvature of the minima for the first-order peak. To explore how this effect will change the information distribution, a series of samples with increasingly large sidewall angles were simulated and fit using the same approach as for the prior sample series. The structures used in this simulation all had the same height (50 nm) and are shown above the corresponding simulated scattering in Fig. 5, with the angle-angle correlation plots for A_R shown below. In comparison to W12H50 [intensity profile shown in Fig. 3(e)], the location of the minima for W11H50 [Fig. 5(a)] for the first-order peak is nearly identical, although shallower. The angle–angle correlation plot for W11H50 looks qualitatively similar to W12H50. There are quantitative variations though. For example, the reduced curvature for the first-order minima reduces the impact that portion of the data have on the density distribution and, as a result, contributions from the second- and third-order peak minima become more significant compared to W12H50. As the sidewall angle increases this trend continues. W9H50 and W8H50 now have minima for the third-order peak near an incident angle of 0 deg and the curvature of the first-order peak minima are significantly reduced. This significantly changes the impact of that first-order peak on the A_R map. The lowest A_R now clearly corresponds to the vicinity of the minimum of the second-order peak. The reduced contribution of the information content in the first-order peak of the scattering profile results in the minimum A_R for the two angle set shifting from 1 for W12H50 to 1.17 for W11H50 up to 3.12 for H8H50.

Fig. 5.

Intensity profiles and two-angle correlation series for A_R for (a, b) W11H50, (c, d) W9H50, and (e, f) W8H50. The structures used to simulate these profiles are shown as insets above the simulated intensities, and their parameters are listed in Table 2.

All of the structures explored so far consisted of a line grating with uniform composition. Many device structures in the semiconductor industry have multiple components in a structure. One common example is that of a capping layer on top of silicon. In order to investigate the impact of multiple components in a sample, analogous structures to W12H50 were simulated where the top 6 nm consisted of either a spin on carbon (SOC) or silicon nitride (SiN) (the relative electron density for an SOC was estimated at 0.5× that of silicon, while for the SiN it was 1.5× greater). The structure is shown in Fig. 6(e). The results for the analogous all-silicon sample are shown in Figs. 3(e) and 3(f). The simulated scattering for the SOC sample is shown in Fig. 6(a), and for the SiN in Fig. 6(c). The scattering from both samples with a capping layer as well as the all silicon grating closely resembles each other. The primary difference is a mild damping of the first-order minima for the carbon sample and several degree shifts in the location of the minima for all peak orders, with the SiN minima shifting to smaller angles relative to the all silicon grating. This change in the location of the minima is equivalent to scattering from a structure with a larger aspect ratio. Similarly, the minimum for the SOC sample shifts to larger angles, the equivalent of a shorter single component grating. The angles that produced the best fit remain in the vicinity of the low angle/ first-/second-order minima pairs. These are at 1 deg/15 deg for the SOC sample and 1 deg/22 deg for the SiN sample. Indicating that while the change in composition does impact the scattering profile, it was not significant enough to change the optimum measurement conditions. This raises the importance of including sufficient prior information in the analysis. A visual comparison of the two scattering profiles for the two different capping layers shows differences, but those would be difficult to differentiate accurately with a small number of angles unless additional information about the layer thickness, for example, were included in the model.

Fig. 6.

Simulated scattering profiles and two-angle A_R correlation plots for line gratings with a capping layer consisting of either (a, b) SOC or (c, d) SiN. The structure of the lines is shown in (e), where the capping layer is differentiated from the remainder of the grating. The contrast of the carbon layer relative to the silicon grating is ~50% lower, whereas the contrast of the SiN layer is ~50% higher.

The previous simulations worked only under the assumption that the model was capable of describing the target structure, but did not include any meaningful limits on the structure. In a real process monitoring situation, there is likely to be a significant amount of additional information available to constrain the model. This additional information may shift the ideal set of angles for minimizing A_R. If, for example, the height of a grating is known based on layer thicknesses or alternative measurements, then the model could be adjusted to include that information. To demonstrate how restriction of a given parameter changes model sensitivity, we re-examined W12H50 under the assumption that the total height of the grating was known to within ±5%. The result of this restriction on the A_R correlation map is shown in Fig. 7. There are two significant changes to the comparison to the unrestricted sample: first the A_R is reduced nearly half on average. Second, the impact of the higher angle minima is reduced, but not eliminated. The lowest A_R is still observed for the 1 deg/14 deg pair, at an A_R of 0.42, which is less than half that of the unconstrained model.

Fig. 7.

Two-angle A_R correlation plots for W12H50 with the overall height of the structure restricted to ±5%.

4. Conclusions

Subsets of the complete scattering pattern from simulated results of CDSAXS measurements were analyzed in order to evaluate which combinations of incident angles result in the minimum uncertainty and to develop guidelines for selecting those angles. Reducing the number of incident angles required to accurately reconstruct the critical features of measured structures will allow for higher sample throughput. The demonstrations of CDSAXS in the literature take up to 121 measurements at angles between −60 deg and 60 deg and are highly oversampled. Efficiently reducing the number of incident angles requires determining which angles are most critical to reducing the CD uncertainty of the measurement. Utilizing the guidelines provided here, the number of angles collected could be reduced to 5 or less, potentially reducing the measurement time needed by 95%. A range of structures with varying aspect ratios, sidewall angles, and composition variations was investigated. The simulations demonstrate that a combination of angles at normal incidence and angles near minima with high curvature best constrain the shape uncertainty for simple line gratings. The simulation approach outlined here can be used to optimize the CDSAXS measurement for samples including complex features of interest to the semiconductor industry such as pitch walking, multiple material stacks, or three-dimensional structures such as gated finFETS and 3D-NAND.

Table 2.

Parameters for the structures in Fig. 5.

W11H50		W9H50		W8H50
Z position (nm)	Line-width (nm)	Z position (nm)	Line-width (nm)	Z position (nm)	Line-width (nm)
0	18	0	18	0	18
6	14	6	14	0	14
46	8	46	4	46	2
50	6	50	2	50	1

Biographies

Daniel F. Sunday received his BS degree in chemical engineering from Carnegie Mellon and his PhD in chemical engineering from the University of Virginia. He is a research scientist at NIST, where he researches x-ray characterization methods for nanostructures and thin films as well as the self-assembly of block copolymers.

R. Joseph Kline received his BS and MS degrees in material science from North Carolina State University and his PhD in materials science from Stanford University. He is the leader of the dimensional metrology for nanomanufacturing project at NIST. He researches x-ray-based dimensional metrology of nanostructures as well as x-ray structure measurements of soft matter systems. He has published more than 80 articles and 4 book chapters, and has given more than 40 invited presentations. In 2012, he received the Presidential Early Career Award for Science and Engineering.