Atsushi Hiraiwa

Discrete power spectrum of line width roughness

The variation of device characteristics is a challenge to present and future large-scale integrations. One of the origins of the variation is line width roughness (LWR). To facilitate the efforts to cope with LWR, we developed a method to accurately characterize LWR basing on the analysis of power spectral densities (PSDs). Because experimental PSDs are intrinsically discrete, we derive simple analytic formulas of the discrete PSDs by assuming that the autocorrelation function (ACF) exponentially decays with distance. The PSDs calculated by using the formulas agree excellently with experimentally obtained PSDs of photoresist LWR. From the result we find that the photoresist LWR of this study has a standard deviation of 2.5 nm and an exponentially-decaying autocorrelation function with a correlation length of 35 nm. Although the experimental PSDs inevitably contain a component produced by scanning-electron-microscope (SEM)-image noise, the two components of LWR and noise are separately determined by the me...

Statistical-noise effect on discrete power spectrum of line-edge and line-width roughness

The control of line-edge roughness (LER) and line-width roughness (LWR) is a key issue in addressing the growing challenge of device variability in large-scale integrations. The accurate characterization of LER and LWR forms a basis for this effort and mostly hinges on reducing the effects of noise inherent in experimental results. This article reports how a power spectral density (PSD) is affected by a statistical noise that originates from the finiteness of the number N L of available samples. To achieve this, the authors numerically generated line-width data using the Monte Carlo(MC) method and assuming an exponential autocorrelation function (ACF). By analyzing the pseudoexperimental PSDs obtained using the MC data, they found that the standard deviation η of normalized analysis errors was determined by the total number N ALL of width data used in each analysis, regardless of N L and the number N of width data in each line segment. The authors found that η decreased with N ALL approximately in inverse proportion to N ALL 3 / 4 . It is noteworthy that they could obtain accurate results even in the case of N L = 1 as long as N ALL was sufficiently large, although the distribution of PSDs was large due to a large statistical noise. This resulted from the fact that the PSD distribution was not completely irregular, but centered at the true value and that the best-fitted PSD accordingly approached the true one with an increasing N . On the other hand, η at a fixed N ALL decreased with the ratio Δ y / ξ of an interval Δ y of width data to a correlation length ξ , approximately in inverse proportion to ( Δ y / ξ ) 3 / 8 . As a result, N ALL at a specified η decreased with Δ y / ξ in inverse proportion to the square root of Δ y / ξ in the case when Δ y / ξ was 0.3 or smaller. Beyond this threshold of Δ y / ξ , the authors needed to increase N ALL markedly to achieve the same accuracy of analyses. This comes from a decrease in the range of the PSD with an increasing Δ y / ξ and a subsequent loss of sensitivity of the PSD to the change of ξ . Based on these results, they established guidelines for accurate analyses as follows: Δ y / ξ ≤ 0.3 and N ALL ≥ A η − 4 / 3 ( Δ y / ξ ) − 1 / 2 , where A is 1.8 × 10 2 for ξ and 7.2 × 10 1 for the variance of widths, respectively. Equivalently in terms of the total measurement length L ALL , instead of N ALL , the guidelines are given in Δ y / ξ ≤ 0.3 and L ALL / ξ ≥ A η − 4 / 3 ( Δ y / ξ ) 1 / 2 using the same A ’s as those of N ALL . Being expressed in universal forms like these, the guidelines of this study can be applied to many practical problems beyond LER and LWR to accurately analyze PSDs, as long as the stochastic processes have exponential ACFs.

Spectral analysis of line edge and line-width roughness with long-range correlation

Large-scale integrations (LSIs) are facing an ever-growing problem of device variability. One of the origins that cause the variability is line-width roughness (LWR) caused by line edge roughness (LER). Accurate characterization of the LWR plays an essential role in controlling the LWR. To do this, we report a methodology, named the “assembly method,” that enables to analyze LWR statistics beyond the conventional correlation length limit, basing on the previous “patchwork” method and recent discrete power spectral density (PSD) method. The methodology virtually assembles a long line by gathering line segments that are randomly scattered on a single line or equally processed different lines. The virtual lines are repeatedly assembled by randomly changing the combination of the segments and the order of the gathered segments while permitting overlaps of the segments between the assembled lines. Squared Fourier transforms of their widths are averaged over the assembled lines to obtain the PSD. By these steps, the statistical noise, which is inherent to experimental PSDs, is markedly reduced. Furthermore, to extract LWR statistics by comparing experimental and theoretical PSDs, we derived an analytic formula of the assembled-line PSD. In the derivation, the randomness of the segment collections played a key role. The PSDs calculated using the formula almost completely fitted experimental PSDs that were obtained by the assembly method. The parameters used in the best-fitted calculation revealed that the photoresist LWR of this study contained a component that had a correlation length of 2780 nm in addition to the previously reported LWR of 35 nm. The LWR variance of the component accounted for approximately 10% of the total variance. The formula also enabled us to evaluate the accuracy of experimentally obtained averages of widths. We find two distinct features in the PSDs by the assembly method. One is the oscillatory structure that shows up in the case when the correlation length is larger than half the length of the segments. A trace of this structure was actually observed in the experimental PSDs of this study. The other is the spikes that are periodically observed as a function of wave number. The spikes originate from a nonstochastic width variation that exists in all the segments in common. Their intensity is proportional to the number of gathered segments in the assembled lines. Because the spikes are excluded from the analysis, the LWR parameters determined by the assembly method are not affected by the nonstochastic variation, unlike the conventional methods. By all these results, we confirm that the assembly method of this study extends the upper limit of analyzable correlation lengths by a factor of approximately 20 and enhances the accuracy as well. This feature also has a practical significance that the widely observed LWR with a correlation length of approximately 35 nm can be analyzed by the assembly method using a conventional critical-dimension scanning-electron-microscope, without resorting to a specially designed one. Accordingly, the method will be a key tool for investigating LER and LWR in developing and manufacturing LSIs. It will also help analyze other stochastic processes in many research and development settings.

Statistically accurate analysis of line width roughness based on discrete power spectrum

We established guidelines for accurately analyzing line-edge and line-width roughness (LER and LWR) basing on the recent discrete power-spectral-density (PSD) method. Extraction of correlation length ζ requires a plateau of PSD in a small-wave-number region. This requirement is met by letting a ratio of inspection length L to ζ be larger than 4π. Analysis errors caused by scanning-electron-microscope image noise are determined by ratios of measurement interval Δy to ζ and of noise-induced variance var(φ) to LWR variance var(w). The ratios need to be at most 20/35 and 1, respectively. var(φ) is reduced by averaging image pixels perpendicularly to lines. This averaging does not smooth LWR, unlike parallel averaging. Statistical noise, i.e. jaggy of PSDs, is another noise source that is caused by a finiteness of the number NFT of Fourier transforms averaged to obtain PSDs. The jaggy level decreases with NFT and with a decrease in Δy. Under the above Δy, NFT should preferably be 50 or larger. The total variance of this study was larger than the sum of var(w) and var(φ). The additional roughness results from a long-range correlation that exceeds the limit of this study. It will be analyzed in our forthcoming report.

Statistical- and image-noise effects on experimental spectrum of line-edge and line-width roughness

The accuracy of estimated line-edge-roughness and line-width-roughness (LER and LWR) statistics is mostly determined by the noise inherent in experimental power spectral densities (PSDs). One type of noise is statistical noise, a kind of jagged structure, that is caused by the finiteness of a number NL of line segments used in analyses. To keep the estimation error below 5%, the ratio of sampling interval to correlation length should be 0.3 or smaller, and NL needs to be larger than 100 under the condition that the length of line segments is 2000 nm or larger, in compliance with the Semiconductor Equipment and Materials International standard. Another noise type is scanning-electron-microscope image noise. It causes edge-detection errors and induces an additional variation in LER/LWR. This variation raises the minima of PSDs and accordingly enhances the errors. The factor of the error enhancement is suppressed below 1.5 by controlling the ratio of image-noise-induced LER/LWR variance to the true variance below 0.6. This is achieved by averaging image pixels perpendicularly to fine lines, and is free from any appreciable drawbacks. The experimental results agree well with analytical approximations to Monte-Carlo results that are separately obtained. This leads us to obtain more general guidelines for accurate analyses by using the analytical formulas.

Image-Noise Effect on Discrete Power Spectrum of Line-Edge and Line-Width Roughness

The authors investigated the effect of scanning-electron-microscope image noise on the accuracy of line-edge-roughness and line-width-roughness (LER/LWR) statistics extracted from power spectral densities (PSDs). To do this, they numerically prepared pseudo-experimental PSDs of LWR using the Monte Carlo (MC) method. The estimation error η decreased with the total number NALL of width data points in the same way as that observed in the absence of the image noise. η first increased gradually with the image-noise intensity R but markedly when R went beyond the threshold value Rth determined by the ratio of the sampling interval Δy to the correlation length ξ. The PSDs with these Rths had the same maximum-to-minimum ratio γ (= 10 in this study). The authors approximated η by BNALL-3/4(Δy/ξ)-3/8 [1+g(R,Δy/ξ,γ)], where B is 49. They also empirically determined the functional form of g(R,Δy/ξ,γ). Because these functions well fitted massive MC simulation results, they provide guidelines for setting up analysis conditions for securing arbitrarily prescribed accuracy.

Statistical-noise effect on autocorrelation function of line-edge and line-width roughness

Accurate characterization of line-edge roughness (LER) and line-width roughness (LWR) is essential to cope with the growing challenge of device variability in large-scale integrations. The accuracy is affected markedly by statistical noise, which is caused by the finiteness of a number of samples. The statistical noise produces random oscillatory fluctuations of autocorrelation function (ACF) of LER/LWR. These fluctuations are obstacles to estimating LWR statistics by comparing experimental and theoretical ACFs. Using the Monte Carlo (MC) method to prepare pseudoexperimental ACFs (MC-ACFs), the authors found that an error η of the estimates is minimized in the case when a ratio of a fitting-window size to a correlation length is 0.3 or smaller, being less affected by the statistical noise. η under a fixed sampling interval is determined by the total number N all of width data used to obtain the MC-ACF. This comes from the fact that the MC-ACF is obtained after averaging approximately for N all times. The authors also investigated the case when LWR consisted of two components that had different correlation lengths. They confirmed that η of both components increase with a decrease in their occupancies in the entire LWR. This, together with a large correlation length, makes it difficult to accurately characterize the longer-correlation component, which is mostly minor (small occupancy) in actual cases. This difficulty is also an obstacle to estimating the shorter-correlation component, because the statistics of the former are mostly the prerequisites for analyzing the latter. These facts make a stark contrast to a power-spectral-density (PSD) fitting method, where at least the shorter-correlation component is estimated with almost the same accuracy as in the case of a single component. Based on these results, the authors propose to investigate PSDs, rather than ACFs, in the case of multicomponent LWR.

Atomic Layer Control for Suppressing Extrinsic Defects in Ultrathin SiON Gate Insulator of Advanced Complementary Metal–Oxide–Semiconductor Field-Effect Transistors

By measuring the minimum supply voltage for normal operation of test random access memories, we detected low-density extrinsic defects in silicon-oxynitride (SiON) gate insulators that were formed by state-of-the-art technologies. The density of the detected defects had a strong correlation with optical thickness dopt, which was ellipsometrically measured, regardless of the processing conditions of the SiON films. We propose to maintain the dopt above a threshold value of 1.7 nm to suppress the problems caused by the defects. The optimization of post nitridation annealing (PNA) condition is promising for meeting the criterion without sacrificing device performance. By elaborate investigations based on the Clausius–Mosotti relation, we found that the optical thickness of SiON films is approximately proportional to the atomic area density in the films. On the basis of this finding, we developed a model, which is an extension of the conventional analytical cell-based model, to figure out the physical process of the extrinsic-defect formation. The results analyzed using the model revealed that the extrinsic defects are formed in the SiON films in the case when the number of normal cells in a vertical arrangement becomes equal to or smaller than the threshold value of 3 or 4.