3.1.

First Study: Adding Line Edge Roughness to the Graphic Data System

This first study permits investigation of the effects of wafer noise by applying a well-defined LER to the nominal design of the 9-nm gate IDA through its Graphic Data System (GDS) file. This LER is characterized by its rms roughness, ${\sigma }_{\mathrm{LER}}$, and its correlation length, $\xi$. Simulation inputs were generated using a procedure⁵ laid out by Crimmins for applying a correlation length to a random sequence. The correlation length is invoked using a target autocorrelation function:

This modified random sequence was added to the sides and ends of each line segment within the GDS file including the defect. Specifically, one randomization yielded an LER with $({\sigma }_{\mathrm{LER}},\xi )$ for a defect-free reference, while a separate randomization was used to generate LER with the same magnitude $({\sigma }_{\mathrm{LER}},\xi )$ for the line arrays of the “Bx” and “By” defects. This process ensured a mismatch between the edge geometries of the reference and defect.

Simulation of the LER rms roughness on the subnanometer scale requires multiple assessments of one’s computational capabilities. Values were derived from the ITRS, with an LER rms roughness ${\sigma }_{\mathrm{LER}}=0.65\mathrm{nm}$ (one standard deviation) obtained from current LWR values (treated as uncorrelated), while assigning $\xi =8\mathrm{nm}$ which is forecasted for a 9-nm dynamic random-access memory ½ pitch.¹⁰ ITRS LER and LWR values are $3\sigma$ values, a coverage factor $k=3$; in this present treatment, ${\sigma }_{\mathrm{LER}}$ is the $1\sigma$ value ($k=1$). For this study, the rms roughness was not equally forward-looking given the memory requirements and evaluation time required for the FDTD model at these length scales. In an initial assessment, FDTD cubic grid sides were varied from 0.75 to 2.25 nm to test the convergence for ${\sigma }_{\mathrm{LER}}=0.65\mathrm{nm}$. A second investigation varied the FDTD domain size from $1\times 3$ unit cells ($0.72\mu \mathrm{m}\times 0.72\mu \mathrm{m}$ in $xy$) up to $4\times 12$ unit cells ($2.88\mu \mathrm{m}\times 2.88\mu \mathrm{m}$ in $xy$). After evaluating tradeoffs among speed, accuracy, and computational capabilities, the domain size used here was $1\times 3$ with a grid size of 1.125 nm, yielding lines 8 grids wide for these 9-nm gate designs.

The defect metric and SNR for the data in Fig. 3 are provided in Table 1. The intensity threshold in this study is $I>4{\sigma }_{\mathrm{I}}$. With this threshold, in each case, the defect is easily identified and separable from the wafer noise. Quantitatively, the “By” bridge defect is relatively stronger than that for the “Bx” defect in either linear polarization. (The exact same LER sequence is applied to the nominal patterns for both “Bx” and “By.”) While some wafer noise subvolumes are present in the data, none persist through focus on par with the main defect optical scattering signal. The strongest defect signal is for the TE polarization at the specified azimuthal angles. For the “Bx” defect, TE polarization at $\varphi =90\mathrm{deg}$ corresponds to $X$ polarization with respect to the sample, and for “By,” TE polarization at $\varphi =90\mathrm{deg}$ corresponds to $Y$ polarization.

Table 1

Defect metric and signal-to-noise (SNR) estimates for the 9-nm gate intentional defect array (IDA) with σLER=0.65  nm.

As ${\sigma }_{\mathrm{LER}}=0.65\mathrm{nm}$ yielded wafer noise that left the defect signal largely unperturbed, these simulations were then used as a baseline to evaluate the effects of increasing the LER rms roughness on defect detectability. Figure 4 displays the portions of the unit cell and defect from the GDS for defect “By,” with LER increasing from left to right, and the data presented illustrate potential difficulties in defect detection with increased LER. Visual comparison of the thresholded volumetric images shows that the wafer noise has a more profound impact as the LER is increased from ${\sigma }_{\mathrm{LER}}=0.65$ to 1.95 nm. As the ${\sigma }_{\mathrm{LER}}=0.65\mathrm{nm}$ case clearly delineates the $(x,y,z)$ position of the defect’s optical scatter and the nominal shape of its subvolume, we can identify not only false negatives, but also “false” positives that are unrelated to the actual defect scattering. Both types of misclassifications are related to wafer noise. In Fig. 4, there is a false negative for ${\sigma }_{\mathrm{LER}}=1.95\mathrm{nm}$ and an incorrectly positioned “false” positive for ${\sigma }_{\mathrm{LER}}=1.63\mathrm{nm}$. The defect scattering was obscured in each case by increased wafer noise. For the “Bx” defect (not shown), ${\sigma }_{\mathrm{LER}}=1.30\mathrm{nm}$ yields a false negative, while both ${\sigma }_{\mathrm{LER}}=1.63$ and 1.95 nm yield similarly shaped “defect” subvolumes completely below the substrate, in stark contrast to the ${\sigma }_{\mathrm{LER}}=0.65\mathrm{nm}$ case shown in Fig. 3(b).

Fig. 4

Differential volumetric analysis for the “By” defect using TE polarization, $\theta =21\mathrm{deg}$, $\varphi =0\mathrm{deg}$ for five different line edge roughness (LER) rms roughnesses. The defect is correctly identified for ${\sigma }_{\mathrm{LER}}\le 1.30\mathrm{nm}$.

Table 2 shows the defect metrics, SNR, and $I>4{\sigma }_{\mathrm{I}}$ intensity thresholds for both the “Bx” and “By” defects. The SNR for the “By” defect decreases as the LER increases from ${\sigma }_{\mathrm{LER}}=0.65$ to 1.30 nm. For ${\sigma }_{\mathrm{LER}}\ge 1.63\mathrm{nm}$, wafer noise is sufficiently strong to frustrate clear identification of the optical defect signal for either bridge direction. Defect signals misclassified as false negatives or incorrectly positioned false positives are shown in Table 2 with their defect metric and SNR marked with an asterisk (*) to demonstrate that these observed values do not correspond to an actual defect signal.

Table 2

Intensity threshold, defect metric, and signal-to-noise (SNR) estimates as the rms roughness, σLER, of the line edge roughness (LER) increases from 0.65 to 1.95 nm. The defect metric is the summed absolute defect intensity above the intensity threshold. Subvolumes misidentified as defects (“false positives”) and false negatives are marked with an asterisk (*) and do not correspond to an actual defect signal.

This simulation study confirms that the wafer noise increases with LER rms roughness and that this wafer noise can be distributed as bright regions of intensity that are located throughout the through-focus volumetric image. Subvolumes of wafer noise for rougher LER samples can be comparable in intensity and spatial extent with the signal from the defect itself. As newer patterning layouts are developed with narrower pitches, smaller line widths, and smaller LER rms roughness, such evaluations will need to be repeated to gauge the relative volume and intensity of the defect signal with respect to the wafer noise as this scattering cannot be overcome by additional measurements at a single plane. In this sense, this study is a first step in evaluating the extensibility and possible necessity of volumetric processing given the challenges of wafer noise.

3.2.

Second Study: Scanning Electron Microscopy Data in the Finite-Difference Time Domain

However, achieving a fuller understanding through simulation of the measured experimental data presented earlier as Fig. 2 and subsequently in Sec. 4 may be achieved through use of the as-printed layout, which varies from its original design. The second simulation study compares the wafer noise scattered from the SEMATECH 9-nm gate IDA by using SEM data to define the simulation geometry. Instead of relying upon the nominal design from the GDS file as modeling input, the simulation was based entirely upon SEM images acquired from two specific dies on the IDA chosen for their relative patterning quality. For each die, images of the “Bx” and “By” defects were collected, then correlated, and cropped to $3\times 4$ unit cells with the defect near the center of the image. A third cropped image for each die was taken from the periphery of the image of the “By” defect as a reference defect-free pattern. These six images in all were converted using intensity thresholding from grayscale SEM images to binary masks. For these normalized SEM images, this threshold was $0.35I_{\max }$ and thus the apparent CDs fed into the FDTD are slightly larger than that would be expected from a more traditional 50% threshold. The binary mask in Fig. 5(a) comes from the “well-patterned die,” while Fig. 5(b) shows a binary mask derived from the “imperfect die.” These two separate dies offer unprogrammed variations in LER, LWR, and line fidelity. The sidewalls of the simulated structures were assumed to be normal. Perspective views of the defects appear in Fig. 6.

Fig. 5

Comparisons of the simulation inputs for the “well-patterned die” versus the “imperfectly patterned die.” (a) The “By” defect binary mask for the well-patterned die. (b) The “By” defect binary mask for the imperfectly patterned die. (c) Line width roughness (LWR) calculated for the “Bx” and “By” defects.

Fig. 6

Differential volumetric analysis of the scattering from two dies and two defect types from the 9-nm gate intentional defect array (IDA) using optimal linear polarizations. Defect SEVD and critical dimension (CD) values are shown in Table 3.

From these masks, the LER, LWR, and line fidelity were computed from the middle 80% of the patterned lines. The LER varied among defects within each die. Unplanned imperfections and line breaks in the imperfectly patterned die contributed to the approximate 1-nm increase in LWR relative to the well-patterned die as shown in Fig. 5(c). The LWR may be difficult to visualize in these panels as the pixel resolution is nominally $2.48\mathrm{nm}/\mathrm{pixel}$. The continuity of the well-patterned die is nearly 100%, but is as low as 98% for targets on the imperfectly patterned die.

The 2.48-nm SEM pixel resolution provided the lower limit for the cubic FDTD grid size and enabled many more FDTD simulations to be performed relative to the first simulation study, in which a 1.125-nm grid size was used, thus 80 plane waves were simulated to account for the finite aperture of the National Institute of Standards and Technology (NIST) 193-nm microscope.¹¹ The polar angle $\theta$ was varied from 11 deg to 45 deg with the entire range of azimuthal angles $\varphi$ simulated to cover the conjugate to the back focal plane of that microscope’s objective lens. Assuming incoherent illumination and Köhler illumination, these several plane wave simulations have been combined to simulate annular illumination for linearly polarized light in Fig. 6 for the two dies and two defects of interest. For this study, the intensity threshold was increased to $I>5{\sigma }_{\mathrm{I}}$ in order to isolate a single defect in each volumetric image. As in the first simulation study in the previous subsection, there proved to be a preferential direction for measuring these defects that corresponded to the alignment of the linear polarization along the axis of the bridge defect. The defect metric, SNR, sphere-equivalent volume diameter (SEVD), and the CDs of the defects simulated are shown in Table 3.

Table 3

Defect metric and SNR comparing two dies of differing patterning quality, with the critical dimension (CD) of the defect simulated using finite-difference time-domain (FDTD). Though based on scanning electron microscopy (SEM) measurement, the defect CDs are fixed inputs to the model; the specific SEM intensity threshold used overestimates the CD compared with the SEM CD measurements in Table 4.

Several trends can be identified from this study. The center $z$ positions of the wafer noise roughly coincide with those of the identified defects. Although there are more wafer subvolumes in the scattering from defects “Bx” on the imperfect die and “By” on the well-patterned die, there is more total intensity in the wafer noise subvolumes for both defects for the imperfect die. Despite similar SEVD values between the two dies per defect, the SNR values are larger for the well-patterned die, especially the “Bx” defect. This defect better illustrates the anticipated difficulties in defect detection as dimensions continue to decrease. The spatial extent along the $z$-axis from the wafer noise is 500 nm for the imperfect die, comparable with the 700-nm extent of the defect. Although the smaller $xyz$ extent of the wafer noise observed aids in isolating the defects, more sophisticated discriminants against false positives from wafer noise will likely be needed in future evaluations. At present, with volumetric processing, the defect metrics are comparable between the well-patterned die and the imperfect die, though the SNR values are more favorable for the well-patterned die. These trends will be evaluated experimentally in Sec. 4.

4.1.

Comparison of Dies on the SEMATECH 9-nm Intentional Defect Array

Simulation studies have shown that the extra wafer noise is to be expected in the 3-D volumetric analysis of samples with LER and specifically for the imperfectly patterned die. These trends must be confirmed using measurement. In Fig. 2, direct comparison between the imperfectly patterned die and another well-patterned die was hampered by substantially different illumination conditions. In this experiment, “Bx” and “By” defects are measured on two dies on the 9-nm IDA using bright-field illumination using the NIST 193-nm microscope. The full-field effective illumination numerical aperture (NA) is annular due to a catadioptric objective and ranges from $\mathrm{NA}=0.11$ to 0.74. As discussed in Sec. 1, the “defect” and “reference” volumes are collected sequentially and instrumentation noise effects should be largely mitigated, allowing investigation of the wafer noise.

As shown in Fig. 7, there is more wafer noise from the imperfectly patterned die than from the well-patterned die. In Fig. 7(b), two copies of the “By” defect are clearly identified with the noise subvolumes mostly confined to a narrow $z$-range. For the same defect on the imperfect die shown in Fig. 7(d), two false positives have been detected and the noise subvolumes are approximately 250 nm in the $z$ extent. For “By,” the $xyz$ extents of the defect signals differentiate the defect from the noise. Contrast this with Figs. 7(a) and 7(c), where the labeled noise subvolumes often are positioned in the same $z$-range as the intentional defect. From the first simulation study, LER effects did not produce such wafer noise. A number of the noise subvolumes in Fig. 7(c) also have copies due to the shift between the “defect” and “reference” images, thus it is likely that these are unintended defects, such as the line breaks observed by SEM in Fig. 5(b), with intensity noise on par with the scattering from the “Bx” defect. Similar events are less readily observed in Fig. 7(d) due to the relatively large intensity of the “By” defect.

Fig. 7

Experimental differential volumetric defect detection comparing measured intensities from the well-patterned die for defects (a) “Bx” and (b) “By” versus intensities from the imperfectly patterned die for defects (c) “Bx” and (d) “By” on the 9-nm IDA. Thresholds are consistent for a given defect: the “Bx” defect was analyzed using a $7{\sigma }_{\mathrm{I}}$ intensity threshold with a $160\mathrm{nm}\times 160\mathrm{nm}\times 550\mathrm{nm}$ spatial extent, the “By” defect was analyzed using a $6{\sigma }_{\mathrm{I}}$ intensity threshold with a $160\mathrm{nm}\times 160\mathrm{nm}\times 700\mathrm{nm}$ spatial extent. In both cases, wafer noise is larger on the imperfectly patterned die.

The experimental defect metric and SNR are provided in Table 4. For these experiments, the defect metric is the mean intensity per pixel in the defect subvolume(s). The differences in the defect metrics and SNR for the two “Bx” defects are minimal. Values for the two “By” defects indicate a greater detectably for the imperfectly patterned die. There is a general decrease in the SNR between the second, SEM-based simulation study and the measurement, which indicates that all sources of noise from the samples and instrument have yet to be fully integrated into the simulation.

Table 4

Experimental defect metric and SNR comparing two dies of differing patterning quality, with the CD of the defect as measured using SEM. SEM CD values are averages of the defect’s width as measured between 25% and 75% of its length using a 0.42 normalized intensity threshold where Imin=0 and Imax=1. SEM uncertainties are 1σ values (k=1).

4.2.

Extensibility of Through-Focus Techniques

Bringing this metrology capability from the laboratory to industry is currently an unmet challenge. While the measurements above demonstrate volumetric processing for samples with wafer noise, the added data acquisition required to assemble such volumes is currently impractical for full-time manufacturing process control. In this present work, the time required for a measurement was not optimized and the dataset at left in Fig. 8 is comprised of 47 different $z$ slices. This dataset can be reduced to determine how few slices are required to obtain similar results. In the center and right panels of Fig. 8, the $z$ resolution is reduced to evaluate the effectiveness of volumetric methods with less data.

Fig. 8

Measured differential volumetric defect detection as the $z$ resolution is reduced from left to right as shown. Although 3-D filtering and interpolation were performed at the highest $z$ resolution, the intensity and spatial filtering were performed for each new $z$ resolution.

Varying the $z$ resolution, $\mathrm{\Delta }z$, from 50 to 300 nm in steps of 50 nm, the number of required slices is reduced from 47 to 8, with each iteration having an SNR comparable with the SNR of the highest resolution with only the $\mathrm{\Delta }z=200\mathrm{nm}$ case (not shown) yielding a false positive. Otherwise, only the two copies of the defect were identified.

Given that several images will be necessary to form any volume, at this time volumetric processing may best be applied intermittently. Possibilities include identification of a favorable focal position where defects may be identified with little noise, or application after conventional single-focus inspection to measure a marginally flagged defect more closely before defect review. A commercial tool would have to further optimize image collection time through the $z$-direction. Possible directions include utilizing several collection paths to obtain images simultaneously at several $z$-heights^12,13 or the use of ultrahigh-speed adaptive optics to quickly image along the $z$-axis.¹⁴ The SNR would be affected negatively in these fast-scanning techniques, but with decreasing dimensional sizes, the wafer noise may necessitate a new balance among measurement time, SNR, and multiplane image acquisition to maintain yield.