Patent Publication Number: US-2023134093-A1

Title: System and method for determining and/or predicting unbiased parameters associated with semiconductor measurements

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This patent application is a continuation-in-part of U.S. application Ser. No. 17/583,982, filed Jan. 25, 2022 titled “System and Method for Generating and Analyzing Roughness Measurements and Their Use for Process Monitoring and Control”, which is a continuation of U.S. application Ser. No. 17/097,593 filed Nov. 13, 2020 titled “System and Method for Generating and Analyzing Roughness Measurements and Their Use for Process Monitoring and Control,” which is a continuation of U.S. application Ser. No. 16/716,131 filed Dec. 16, 2019, titled “Edge Detection System” (now U.S. Pat. No. 11,004,653), which is a continuation of U.S. application Ser. No. 16/222,668 filed Dec. 17, 2018 titled “Edge Detection System,” (now U.S. Pat. No. 10,510,509), which is a continuation of U.S. application Ser. No. 15/892,080 filed Feb. 8, 2018 titled “Edge Detection System” (now U.S. Pat. No. 10,176,966). The Ser. No. 15/892,080 application claims priority to U.S. Provisional Patent Application Ser. No. 62/602,152, filed Apr. 13, 2017 and also titled “Edge Detection System.” All applications are incorporated by reference herein as if reproduced in full below. 
     Further, as a continuation-in-part of U.S. application Ser. No. 17/583,982, which is a continuation of U.S. application Ser. No. 17/097,593, this patent application is also a continuation-in-part of U.S. application Ser. No. 16/730,393 filed Dec. 20, 2019 titled “System and Method for Generating and Analyzing Roughness Measurements”, which is a continuation of U.S. application Ser. No. 16/218,346 titled “System and Method for Generating and Analyzing Roughness Measurements” (now U.S. Pat. No. 10,522,322), which is a continuation-in-part of and claims priority to U.S. application Ser. No. 15/892,080 filed Feb. 8, 2018 titled “Edge Detection System” (now U.S. Pat. No. 10,176,966). This application further claims priority to U.S. Provisional Patent Application No. 62/739,721 filed Oct. 1, 2018 titled “System and Method for Generating and Analyzing Roughness Measurements” and U.S. Provisional Patent Application No. 62/678,866 filed May 31, 2018 titled “System and Method for Removing Noise From Roughness Measurements.” Further, as a continuation of U.S. application Ser. No. 16/218,346, this application claims priority to U.S. Provisional Patent Application Ser. No. 62/602,152, filed Apr. 13, 2017 and titled “Edge Detection System.” All applications are incorporated by reference herein as if reproduced in full below. 
    
    
     BACKGROUND 
     The disclosures herein relate generally to roughness measurements of pattern structures, and more particularly, to roughness measurements of pattern structures in noise-prone images, such as in images formed when using a scanning electron microscope (SEM) or other imaging apparatus that produce images including undesired noise, and even more particularly, to analyzing such roughness measurements to remove unwanted artifacts (spikes) and measure desired features (bumps). Further, the disclosures generally relate to controlling a manufacturing process through the use of such measurements. 
     BRIEF SUMMARY 
     In one embodiment, a method includes determining, by a processor, a measurement of edge detection noise, and receiving a measurement of a biased parameter including measurement noise. Based on the measurement of edge detection noise and a number of measurement points, the method also includes determining a contribution of edge detection noise to the biased parameter. The method also includes determining an unbiased parameter by subtracting the contribution of noise from the biased parameter including the measurement noise, and outputting the unbiased parameter. 
     In one embodiment, a method includes receiving an image of a semiconductor device, and determining, based on the image, one or more measurements of unbiased power spectral density data for a first feature included in the image. The first feature may be associated with a first property. The method may also include predicting, based on the one or more measurements of unbiased power spectral density data for the first feature, an unbiased parameter for a second feature associated with a second property, wherein the first property is associated with a value greater than the second property. The method may also include outputting the predicted unbiased parameter for the second feature. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The appended drawings illustrate only exemplary embodiments of the invention and therefore do not limit its scope because the inventive concepts lend themselves to other equally effective embodiments. 
         FIG.  1 A  is a representation of a pattern structure that exhibits parallel line features with spaces in between the lines. 
         FIG.  1 B  is a representation of a pattern structure that includes contact hole features. 
         FIG.  2    shows four different rough edges, all with the same standard deviation. 
         FIG.  3    is a representation of power spectral density (PSD) vs. frequency on a log-log scale. 
         FIG.  4    is a graphic representation of power spectral density (PSD) plotted vs. frequency and depicting roughness parameters PSD(0), correlation length, and roughness exponent. 
         FIG.  5    shows two power spectral densities (PSDs) corresponding to respective edges of a feature on a pattern structure. 
         FIG.  6    is a graphic representation of the tradeoff of within-feature variation and feature-to-feature variation as a function of line length. 
         FIG.  7    is a block diagram of a scanning electron microscope (SEM) coupled to an information handling system (IHS) that together form one embodiment of the disclosed edge detection apparatus. 
         FIG.  8 A  is a representation of a feature disposed on a substrate that depicts an electron beam impinging on the center of the feature. 
         FIG.  8 B  is a representation of a feature disposed on a substrate that depicts an electron beam impinging on the feature near its edge. 
         FIG.  9    shows a gray scale image representation on top with a corresponding grayscale linescan along one horizontal cut being graphically plotted immediately below. 
         FIG.  10    shows an example of a pattern structure including a feature situated atop a substrate with varying numbers of electrons escaping from the pattern structure depending on where the electron beam impinges on the pattern structure. 
         FIG.  11    shows a predicted linescan of a resist step on a pattern structure such as a silicon wafer. 
         FIG.  12    shows another representative predicted linescan of a pattern of resist lines and spaces on a silicon wafer. 
         FIG.  13 A  is an original SEM image of a pattern structure without using the disclosed edge detection apparatus and method. 
         FIG.  13 B  is the same SEM image as  FIG.  13 A  except using the disclosed edge detection apparatus and method. 
         FIG.  14    is a Raw (Biased) linewidth roughness plot vs. threshold settings showing both a prior art result (using a filter with conventional threshold edge detection), and a result using no filter and an inverse linescan model (ILM). 
         FIG.  15 A  is a power spectral density (PSD) vs. frequency plot of the right and left edges of a feature shown before noise subtraction. 
         FIG.  15 B  is a power spectral density (PSD) vs. frequency plot of the right and left edges of a feature shown after noise subtraction. 
         FIG.  16    shows portions of three SEM images of nominally the same lithographic features taken at different SEM electron doses. 
         FIG.  17 A  shows a typical linescan for a line feature on a wafer for a case when there is an extremely large number of electrons so that the pixel noise is negligible. 
         FIG.  17 B  shows the 1-sigma uncertainty in edge detection position for perfectly smooth features in the presence of grayscale noise, for three different X pixel sizes. 
         FIG.  17 C  shows grayscale images as an example of using a simple threshold edge detection algorithm with image filtering in the right image, and without image filtering in the left image. 
         FIG.  18    is a plot of linewidth roughness (LWR) PSD vs. frequency that shows the impact of two different image filters on a collection of 30 images. 
         FIG.  19    is a power spectral density plot vs. frequency that shows the noise subtraction process of the disclosed edge detection apparatus and method. 
         FIG.  20    shows PSDs of a particular resist feature type on a given wafer, measured with different frames of integration in the SEM. 
         FIG.  21    shows the biased and unbiased values of the 3σ linewidth roughness (LWR) measured as a function of the number of frames of integration in the SEM. 
         FIG.  22 A  shows biased linewidth roughness (LWR) power spectral densities (PSDs) as a function of different pixel sizes and magnifications employed by the SEM. 
         FIG.  22 B  shows unbiased linewidth roughness (LWR) power spectral densities (PSDs) as a function of different pixel sizes and magnifications employed by the SEM. 
         FIG.  23    is a flowchart that depicts a representative overall process flow that the disclosed SEM edge detection system employs to detect edges of a pattern structure. 
         FIG.  24 A  is a grayscale representation of a pattern structure of vertical lines and spaces that the disclosed metrology tool analyzes. 
         FIG.  24 B  shows a single linescan at one Y-pixel position. 
         FIG.  24 C  shows the averaged linescan that is generated by averaging over all Y-pixels. 
         FIG.  25 A  shows a PSD that includes high-frequency spike artifacts. 
         FIG.  25 B  shows the PSD with spike artifacts removed. 
         FIG.  26    shows a PSD that includes mid-frequency spike artifacts and harmonics. 
         FIG.  27 A  shows the impact of mid-frequency spike artifacts on the modeling and interpreting of the PSD. 
         FIG.  27 B  shows the impact of removing mid-frequency spike artifacts on the modeling and interpreting of the PSD. 
         FIG.  28 A  shows a PSD dataset that exhibits a type of bump behavior. 
         FIG.  28 B  shows an additional PSD dataset that exhibits a type of bump behavior. 
         FIG.  29 A  shows the modeling and analysis of a low frequency bump of type I. 
         FIG.  29 B  shows the modeling and analysis of a low frequency bump of type II. 
         FIG.  30    is a flowchart that depicts a representative process flow to detect undesired spikes in a PSD dataset, and to remove the spikes from the PSD dataset and obtain roughness parameters for a feature. 
         FIG.  31    is a flowchart that depicts another representative process flow to model bumps in a PSD dataset, and to obtain unbiased roughness parameters for a feature. 
         FIG.  32    is a flowchart that depicts a representative process flow to determine unbiased parameters from biased parameters and measurement of edge detection noise. 
         FIG.  33    is a flowchart that depicts a representative process flow to predict unbiased parameter for short features from measurement of unbiased power spectral density for long features. 
     
    
    
     DETAILED DESCRIPTION 
     Measuring the roughness of a pattern is complicated by that fact that noise in the measurement system is difficult to differentiate from the roughness being measured. It is common to using an imaging tool, such as a microscope, to create a detailed image of an object to be measured and then analyze the information on that image to measure and characterize the roughness of one or more features of the object. In this case, noise in the acquired image can appear to be roughness of the features in the image. Described below are techniques useful to separate the noise in the image from the actual roughness of the features in order to produce more accurate measurements of the roughness of the features. 
     Measuring the roughness of a pattern is further complicated by that fact that undesirable artifacts at specific spatial frequencies can be present in the images. These artifacts might be caused by imaging anomalies such as jitter in the scanning used to acquire the image. These artifacts might also be caused by physical aspects of the object to be measured, such as regular topographical structures lying below the features to be measured, that interfere with the measurement of those features. 
     As an example, scanning electron microscopes (SEMs) are very useful for studying the features of pattern structures, such as semiconductor devices, for example. Unfortunately, measuring feature roughness of these structures is often challenging because of the noise that is inherent in SEM images. Filtering (smoothing) of the SEM image is typically needed to achieve accurate edge detection, but such filtering undesirably changes the feature roughness that is measured. An edge detection approach is needed that reliably detects edges in very noisy SEM images without the use of image filtering (or at least without any filtering that would change the feature roughness that is measured). 
     Pattern roughness is a major problem in many fields. Many if not all techniques for creating patterns of various shapes produce roughness on the edges of those patterns, at least on the near molecular scale if not larger scales. For example, in advanced lithography for semiconductor manufacturing, especially for extreme ultraviolet (EUV) lithography but for other lithography methods as well, roughness of the printed and etched patterns can cause many negative effects. Reduction in roughness requires a better understanding of the sources of stochastic variation, which in turn requires better measurement and characterization of rough features. Prior art roughness measurement approaches suffer from severe bias because noise in the image adds to the roughness on the wafer. The disclosures herein provide a practical approach to making unbiased roughness measurements through the use of a physics-based inverse linescan model. This enables accurate and robust measurement of roughness parameters over a wide range of SEM metrology conditions. 
     Before discussing embodiments of the disclosed technology that address the SEM image noise problem, this disclosure first discusses lithography of pattern structures and the frequency dependence of roughness. 
     1. Stochastic Effects in Lithography 
     Lithography and patterning advances continue to propel Moore&#39;s Law by cost-effectively shrinking the area of silicon consumed by a transistor in an integrated circuit. Besides the need for improved resolution, these lithography advances should also allow improved control of the smaller features being manufactured. Historically, lithographers focused on “global” sources of variation that affect patterning fidelity (e.g., exposure dose and focus variations, hotplate temperature non-uniformity, scanner aberrations) by attempting to minimize the sources of these variations and by developing processes with minimum sensitivity to these variations. Today&#39;s small features, however, also suffer from “local” variations caused by the fundamental stochastics of patterning near the molecular scale. 
     In lithography, light is used to expose a photosensitive material called a photoresist. The resulting chemical reactions (including those that occur during a post-exposure bake) change the solubility of the resist, enabling patterns to be developed and producing the desired critical dimension (CD). For a volume of resist that is “large” (that is, a volume that contains many, many resist molecules), the amount of light energy averaged over that volume produces a certain amount of chemical change (on average) which produces a certain (average) amount of dissolution to create the pattern. The relationships between light energy, chemical concentration, and dissolution rate can be described with deterministic equations that predict outputs for a given set of inputs. These models of lithography are extremely useful and are commonly used to understand and control lithography processes for semiconductor manufacturing. 
     This deterministic view of a lithography process (certain inputs always produce certain outputs) is only approximately true. The “mean field theory” of lithography says that, on average, the deterministic models accurately predict lithographic results. If we average over a large number of photons, a single number for light energy (the average) is sufficient to describe the light energy. For a large volume of resist, the average concentration of a chemical species sufficiently describes its chemical state. But for very small volumes, the number of atoms or molecules in the volume becomes random even for a fixed “average” concentration. This randomness within small volumes (that is, for small quantities of photons or molecules or numbers of events) is generally referred to as “shot noise”, and is an example of a stochastic variation in lithography that occurs when the region of interest approaches the molecular scale. 
     A stochastic process is one in which the results of the process are randomly determined. At the atomic/molecular level, essentially all processes are stochastic. For semiconductor patterning at the 20-nm node and below (with minimum feature sizes below 40 nm), the dimensions of interest are sufficiently small that stochastic effects become important and may even dominate the total variations that affect the dimensions, shapes, and placements of the patterns being fabricated. These stochastic effects can also be important for larger feature sizes under some circumstances. 
     The most prominent manifestation of stochastic variations in lithography (as well as etch and other parts of the patterning process) is that the patterns being produced are rough rather than smooth ( FIG.  1 A ). In the pattern structure shown in  FIG.  1 A , nominally parallel vertical lines appear as bright vertical regions, while spaces appear as dark vertical regions between the lines. The roughness of the edge of a feature is called line-edge roughness (LER), and the roughness of the width of a feature is called linewidth roughness (LWR). The roughness of the centerline of the feature (the midpoint between left and right edges) is called pattern placement roughness (PPR). Another important consequence of these stochastic variations is the random variation of the size, shape, and placement of features, which are especially evident for contact hole features ( FIG.  1 B ). 
     Stochastic effects in patterning can reduce the yield and performance of semiconductor devices in several ways: a) Within-feature roughness can affect the electrical properties of a device, such as metal line resistance and transistor gate leakage; b) Feature-to-feature size variation caused by stochastics (also called local CD uniformity, LCDU) adds to the total budget of CD variation, sometimes becoming the dominant source; c) Feature-to-feature pattern placement variation caused by stochastics (also called local pattern placement error, LPPE) adds to the total budget of PPE, sometimes becoming the dominant source; d) Rare events leading to greater than expected occurrence of catastrophic bridges or breaks are more probable if error distributions have fat tails; and e) Decisions based on metrology results (including process monitoring and control, as well as the calibration of optical proximity correction (OPC) models) can be poor if those metrology results do not properly take into account stochastic variations. For these reasons, proper measurement and characterization of stochastic-induced roughness is critical. 
     Many other kinds of devices are also sensitive to feature roughness. For example, roughness along the edge of an optical waveguide can cause loss of light due to scattering. Feature roughness in radio frequency microelectromechanical systems (MEMS) switches can affect performance and reliability, as is true for other MEMS devices. Feature roughness can degrade the output of light emitting diodes. Edge roughness can also affect the mechanical and wetting properties of a feature in microfluidic devices. Roughness of the features in a wire grid polarizer can affect the efficiency and transmission of the polarizer. 
     Unfortunately, prior art roughness measurements (such as the measurement of linewidth roughness or line-edge roughness using a critical dimension scanning electron microscope, CD-SEM) are contaminated by measurement noise caused by the measurement tool. This results in a biased measurement, where the measurement noise adds to the true roughness to produce an apparent roughness that overestimates the true roughness. Furthermore, these biases are dependent on the specific measurement tool used and on its settings. These biases are also a function of the patterns being measured. Prior art attempts at providing unbiased roughness estimates often struggle in many of today&#39;s applications due to the smaller feature sizes and higher levels of SEM noise. 
     Thus, there is a need for a new approach to making unbiased roughness measurements that avoids the problems of prior art attempts and provides an unbiased estimate of the feature roughness that is both accurate and precise. Further, a good pattern roughness measurement method should have minimum dependence on metrology tool settings. CD-SEM settings such as magnification, pixel size, number of frames of averaging (equivalent to total electron dose in the SEM), voltage, and current may cause fairly large changes in the biased roughness that is measured. Ideally, an unbiased roughness measurement would be independent of these settings to a large degree. 
     Additionally, bias in the measurement of roughness degrades the quality and usefulness of the roughness measurements themselves. The results of roughness measurement can be used in many ways to make many types of decisions, for example, in an integrated circuit manufacturing process. A measurement of pattern structure roughness can be used to assess the quality of the device being fabricated, and to predict yield and performance of the devices being fabricated. If the predicted yield or performance is sufficiently poor, a decision could be made to stop further processing of those devices and scrap the specific devices, the integrated circuit in which the devices were found, the wafer or substrate on which the integrated circuit was found, or the manufacturing lot or batch in which the wafer was found. 
     A further use for the results of a measurement of pattern roughness is in the assessment of the quality of the manufacturing process or manufacturing tools that were used to make the measured patterns. Such an assessment could be used to affect changes in the manufacturing process or tool to improve pattern quality. For example, pattern roughness is sensitive to the focus setting of a lithography tool used to print the patterns. An increase in measured pattern roughness could be used to trigger a focus check and focus adjustment of the lithography tool, thus improving the quality of subsequently printed patterns. Degradation in the quality of the roughness measurements could result in degradation in the quality of the process decisions made using those measurements, such as making a focus change when none is needed, or not making a focus change when one is needed. 
     The use of unbiased measurements of roughness would improve the quality of the decisions made, for example, the decision to scrap a wafer or lot, or the decision to change a process or tool setting. Since the bias in a roughness measurement can change from measurement to measurement even if the true roughness of the pattern structure does not change, the use of biased roughness measurements for these and other decisions is problematic. 
     A further use for the results of a measurement of pattern roughness is in the assessment of the quality of the metrology used to make the measurements. If, for example, a change in the bias in the measurements were detected, adjustment to the measurement tool (for example, the SEM) could be made to improve the quality and reliability of the measurements. 
     2. The Frequency Dependence of Line-Edge Roughness (LER), Line-Width Roughness (LWR), and Pattern Placement Roughness (PPR) 
     Rough features are most commonly characterized by the standard deviation of the edge position (for LER), linewidth (for LWR), or feature centerline (for PPR). But describing the standard deviation is not enough to fully describe the roughness.  FIG.  2    shows four different rough edges, all with the same standard deviation. The prominent differences visible in the edges make it clear that the standard deviation is not enough to fully characterize the roughness. Instead, a frequency analysis of the roughness is required. The four randomly rough edges depicted in  FIG.  2    all have the same standard deviation of roughness, but differ in the frequency parameters of correlation length (ξ) and roughness exponent (H). More specifically, with respect to  FIG.  2   , in case a) ξ=10, H=0.5; in case b) ξ=10, H=1.0; in case c) ξ=100, H=0.5; and in case d) ξ=0.1, H=0.5. 
     The standard deviation of a rough edge describes its variation relative to and perpendicular to an ideal straight line. In  FIG.  2   , the standard deviation describes the vertical variation of the edge. But the variation can be spread out differently along the length of the line (in the horizontal direction in  FIG.  2   ). This line-length dependence can be described using a correlation function such as the autocorrelation function or the height-height correlation function. 
     Alternatively, the frequency f can be defined as one over a length along the line ( FIG.  3   ). The dependency of the roughness on frequency can be characterized using the well-known power spectral density (PSD). The PSD is the variance of the edge per unit frequency ( FIG.  3   ), and is calculated as the square of the coefficients of the Fourier transform of the edge deviation. The low-frequency region of the PSD curve describes edge deviations that occur over long length scales, whereas the high-frequency region describes edge deviations over short length scales. Commonly, PSDs are plotted on a log-log scale as used in  FIG.  3   . 
     The PSD of lithographically defined features generally has a shape similar to that shown in  FIG.  3   . The low-frequency region of the PSD is flat (so-called “white noise” behavior), and then above a certain frequency it falls off as a power of the frequency (a statistically fractal behavior). The difference in these two regions has to do with correlations along the length of the feature. Points along the edge that are far apart are uncorrelated with each other (statistically independent), and uncorrelated noise has a flat power spectral density. But at short length scales the edge deviations become correlated, reflecting a correlating mechanism in the generation of the roughness, such as acid reaction-diffusion for a chemically amplified resist. The transition between uncorrelated and correlated behavior occurs at a distance called the correlation length. 
       FIG.  4    shows that a typical PSD curve can be described with three parameters. PSD(0) is the zero-frequency value of the PSD. While this value of the PSD can never be directly measured (zero frequency corresponds to an infinitely long line), PSD(0) can be thought of as the value of the PSD in the flat low-frequency region. The PSD begins to fall near a frequency of 1/(2πξ) where ξ is the correlation length. In the fractal region, we have what is sometimes called “1/f” noise and the PSD has a slope (on the log-log plot) corresponding to a power of 1/f. The slope is defined as 2H+1 where H is called the roughness exponent (or Hurst exponent). Typical values of H are between 0.5 and 1.0. For example, H=0.5 when a simple diffusion process causes the correlation. Each of the parameters of the PSD curve has important physical meaning for a lithographically defined feature as discussed in more detail below. The variance of the roughness is the area under the PSD curve and can be derived from the other three PSD parameters. The exact relationship between variance and the other three PSD parameters depends on the exact shape of the PSD curve in the mid-frequency region (defined by the correlation length), but an approximate relationship can be used to show the general trend, as per EQUATION 1 below: 
     
       
         
           
             
               
                 
                   
                     σ 
                     2 
                   
                   ≈ 
                   
                     
                       PSD 
                       ⁡ 
                       ( 
                       0 
                       ) 
                     
                     
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             H 
                           
                           + 
                           1 
                         
                         ) 
                       
                       ⁢ 
                       ξ 
                     
                   
                 
               
               
                 
                   EQUATION 
                   ⁢ 
                       
                   1 
                 
               
             
           
         
       
     
     The differences observed in the respective four rough edges of  FIG.  2    can now be easily seen as differences in the PSD behavior of the features.  FIG.  5    shows two PSDs, corresponding to edge a) and edge c) from  FIG.  2   . While these two edges have the same variance (the same area under the PSD curve), they have different values of PSD(0) and correlation length (in this case the roughness exponent was kept constant). Although the standard deviations of the roughness of edge a) and edge c) are the same, these edges exhibit different PSD behaviors. As discussed below, the different PSD curves will result in different roughness behavior for lithographic features of finite length. 
     3. Impact of the Frequency Behavior of Roughness 
     The roughness of the lines and spaces of pattern structures is characterized by measuring very long lines and spaces, sufficiently long that the flat region of the PSD becomes apparent. For a sufficiently long feature the measured LWR (that is, the standard deviation  6  of the measured linewidths along the line) can be thought of as the LWR of an infinitely long feature, σ LWR (∞). But pattern structures such as semiconductor devices are made from features that have a variety of lengths L. For these shorter features, stochastics will cause within-feature roughness, σ LWR (L), and feature-to-feature variation described by the standard deviation of the mean linewidths of the features, σ CDU (L). This feature-to-feature variation is called the local critical dimension uniformity, LCDU, since it represents CD (critical dimension) variation that is not caused by the well-known “global” sources of error (scanner aberrations, mask illumination non-uniformity, hotplate temperature variation, etc.). 
     For a line of length L, the within-feature variation and the feature-to-feature variation can be related to the LWR of an infinitely long line (of the same nominal CD and pitch) by the Conservation of Roughness principle given in EQUATION 2 below: 
       σ CDU   2 ( L )+σ LWR   2 ( L )=σ LWR   2 (∞)  EQUATION 2
 
     The Conservation of Roughness principle says that the variance of a very long line is partitioned for a shorter line into within-feature variation and feature-to-feature variation. How this partition occurs is determined by the correlation length, or more specifically by U. Using a basic model for the shape of the PSD as an example, it is seen that: 
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       CDU 
                       2 
                     
                     ( 
                     L 
                     ) 
                   
                   = 
                   
                     
                       
                         PSD 
                         ⁡ 
                         ( 
                         0 
                         ) 
                       
                       L 
                     
                     [ 
                     
                       1 
                       - 
                       
                         
                           ξ 
                           L 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               e 
                               
                                 
                                   - 
                                   L 
                                 
                                 / 
                                   
                                 ξ 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   EQUATION 
                   ⁢ 
                       
                   3 
                 
               
             
           
         
       
     
     Thus, EQUATIONS 1-3 show that a measurement of the PSD for a long line, and its description by the parameters PSD(0), ξ, and H, enables one to predict the stochastic influence on a line of any length L. It is noted that the LCDU does not depend on the roughness exponent, making H less important than PSD(0) and ξ. For this reason, it useful to describe the frequency dependence of roughness using an alternate triplet of parameters: σ LWR (∞), PSD(0), and ξ. Note that these same relationships apply to LER and PPR as well. 
     It is also noted that, examining EQUATION 3, the correlation length is the length scale that determines whether a line of length L acts “long” or “short”. For a long line, L&gt;&gt;ξ and the local CDU behaves as per EQUATION 4 below: 
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       CDU 
                     
                     ( 
                     L 
                     ) 
                   
                   ≈ 
                   
                     
                       
                         
                           PSD 
                           ⁡ 
                           ( 
                           0 
                           ) 
                         
                         L 
                       
                     
                     ⁢ 
                         
                     when 
                     ⁢ 
                         
                     L 
                   
                   ≫ 
                   ξ 
                 
               
               
                 
                   EQUATION 
                   ⁢ 
                       
                   4 
                 
               
             
           
         
       
     
     This long-line result provides a useful interpretation for PSD(0): It is the square of the LCDU for a given line times the length of that line. Reducing PSD(0) by a factor of 4 reduces the LCDU by a factor of 2, and the other PSD parameters have no impact (so long as L&gt;&gt;ξ). Typically, resists have yielded correlation lengths on the order of one quarter to one half of the minimum half-pitch of their lithographic generation. Thus, when features are longer than approximately five times the minimum half-pitch of the technology node, we are generally in this long line length regime. For shorter line lengths, the correlation length begins to matter as well. 
     EQUATIONS 1-3 show a trade-off of within-feature variation and feature-to-feature variation as a function of line length.  FIG.  6    shows an example of this relationship. For very long lines, LCDU is small and within-feature roughness approaches its maximum value. For very short lines the LCDU dominates. However, due to the quadratic nature of the Conservation of Roughness, σ LWR (L) rises very quickly as L increases, but LCDU falls very slowly as L increases. Thus, there is a wide range of line lengths where both feature roughness and LCDU are significant. 
     Since the Conservation of Roughness principle applies to PPR as well, short features suffer not only from local CDU problems but also from local pattern placement errors (LPPE) as well. For the case of uncorrelated left and right edges of a feature, the PSD(0) for LWR is typically twice the PSD(0) of the LER. Likewise, the PSD(0) of the LER is typically twice the PSD(0) of the PPR. Thus, in general, the LPPE is about half the LCDU. When left and right feature edges are significantly correlated, these simple relationships no longer hold. 
     The above equations allow a measurement of a very long pattern structure (where measurements are approaching the value one would obtain for an infinitely long feature) to predict the within-feature and feature-to-feature variation of shorter features of any length. For example, the feature length for such a prediction can be chosen to match the feature length of device features of interest. Predictions of the variations of this shorter feature could then be used to predict the yield and/or performance of a device that used such a feature using well-known models of device performance. Such models include, for example, TCAD models that predict the electrical behavior of a single transistor, or circuit timing models that predict the timing and skew of an entire circuit or sub-circuit. 
     4. Measurements of the Roughness of Pattern Structures with a Scanning Electron Microscope (SEM) 
     A common way to measure feature roughness for small features is the top-down critical dimension scanning electron microscope (CD-SEM). Typical light microscopes have magnifications up to 1000× and resolutions down to a few hundred nanometers. Scanning electron microscopes use electrons to create very small spots (near 1 nm in width) that can be used to create high-resolution images, with magnifications above 20,000×. CD-SEMs are SEMs that have been optimized for measuring the dimensions of a wide range of features found on semiconductor wafers. They can measure the mean critical dimension of a rough feature with high precision, but have also proven very useful for measuring LER, LWR, PPR, and their PSDs as well. However, there are errors in the SEM images that can have large impacts on the measured roughness and the roughness PSD while having little impact on the measurement of mean CD. For this reason, the metrology approach needed for PSD measurement may be quite different than the approach commonly used for mean CD measurement. 
       FIG.  7    shows a block diagram of one embodiment of the disclosed measurement system  700  that determines feature roughness. The pattern structure sample  800  and the electron imaging optics ( 710 ,  715 ,  720 ,  725 ) are situated in a vacuum chamber  703  that is evacuated by vacuum pump  702 . Electrons are generated from a source such as an electron gun  705  to form an electron beam  707 . Common electron beam sources include a heated tungsten filament, a lanthanum hexaboride (LaB6) crystal formed into a thermionic emission gun, or a sharp-tipped metal wire formed to make a field emission gun. The emitted electrons are accelerated and focused using electromagnetic condenser lenses  710 ,  715 , and  720 . The energy of the electrons striking the pattern structure sample  800  is generally in the 200 eV to 40 keV range in SEMs, but more typically 300 eV to 800 eV for CD-SEMs. Final condenser lens  720  employs scanning coils  725  to provide an electric field that deflects electron beam  707  toward pattern structure  800  as a focused spot. Scanning coils  725  scan the focused spot across the pattern structure  800  through final lens aperture  735  in a raster scan fashion to expose a specific field of view on the pattern structure  800 . SEM  701  includes a backscatter electron detector  740  that detects backscatter electrons scattering back from pattern structure sample  800 . SEM  700  also includes a secondary electron detector  745 , as shown in  FIG.  7   . Prior to imaging pattern structure  800 , the user places pattern structure  800  on a pattern structure receiver  732  that supports and positions pattern structure  800  within SEM  700 . SEM  700  includes a controller (not shown) that controls the raster scanning of pattern structure  800  during imaging. 
     Referring now to  FIGS.  8 A and  8 B , the electrons of electron beam  705  that strike pattern structure sample  800  undergo a number of processes that depend on the energy of the electron and the material properties of the sample. Electrons scatter off the atoms of the sample material, release energy, change direction, and often generate a cascade of secondary electrons by ionizing the sample atoms. Some of these secondary electrons may escape from the pattern structure ( 805 ) and others may remain inside the pattern structure. Pattern structure  800  includes a substrate  810 , such as a semiconductor wafer. A feature  815  is disposed atop substrate  810 , as shown in  FIG.  8 A . Feature  815  may be a metallic line, a semiconductor line, a photoresist line or other structures on substrate  810 . Feature  815  may have other shapes such as a pillar or a hole, or more complicated shapes. Feature  815  may be repeating or isolated with respect to other features on the pattern structure. The space surrounding feature  815  may be empty (vacuum or air) or may be filled with a different material. Pattern structure  800  may be a liquid crystal or other flat panel display, or other pattern semiconductor or non-semiconductor device. Feature  815  includes edges  815 - 1  and  815 - 2 . The region of feature  815  where electron beam  705  interacts with feature  815  is the interaction volume  820  that exhibits, for example, a tear-droplet-like shape as depicted in  FIG.  8 A . 
     Occasionally electrons ricochet backwards off the atom nucleus and exit out of the sample (called backscatter electrons). Some of the lower energy secondary electrons can also escape out of the sample  805  (frequently through the edges of a feature, see  FIG.  8 B ). The way in which a SEM forms an image is by detecting the number of secondary electrons and/or backscatter electrons that escape the sample for each beam position. 
     As the electron beam is scanned across pattern structure sample  800  during one linescan, it “dwells” at a specific spot for a specific time. During that dwell time, the number of electrons detected by either the backscatter detector  725  or secondary electron detector  740 , or both, is recorded. The spot is then moved to the next “pixel” location, and the process is repeated. The result is a two-dimensional array of pixels (locations along the surface of the sample) with detected electron counts digitally recorded for each pixel. The counts are typically then normalized and expressed as an 8-bit grayscale value between 0 and 255. This allows the detected electron counts to be plotted as a grayscale “image”, such as those images shown in  FIG.  1   . While the image coming from a SEM reminds a viewer of an optical image as perceived through the eye, it is important to note that these grayscale images are actually just convenient plots of the collected data. 
     A CD-SEM measures the width of a feature using the SEM image. The first step in measuring feature width is to detect the edges of the features. For pixels near an edge of a feature, higher numbers of secondary electrons escape through the feature edge, producing bright pixels called “edge bloom” (see  FIG.  8 B  and  FIG.  9   ). It is this bright edge bloom that allows the feature edge to be detected. For example, in the grayscale image representation in the upper portion of  FIG.  9   , such edge blooms are observed at edges  905  and  910  of feature  915 . A linescan is essentially a horizontal cut through a 2D SEM image that provides a grayscale value as a function of horizontal pixel position on the feature, as in the graph shown in the bottom half of  FIG.  9   . 
     The data from a single horizontal row of pixels across the sample is called a “linescan”. Note that the term linescan is used here broadly enough to include cases where an image is formed without the use of scanning. The positions of the edges of a feature can be detected from a single linescan, or from a collection of linescans representing the entire image, such as shown in the upper portion of  FIG.  9   . These same edges appear as peaks  905 ′ and  910 ′ in the grayscale value vs. pixel position graph in the lower portion of  FIG.  9   . Once the edges of a particular feature have been determined, the width of the particular feature is the difference between the positions of these two edges. 
     5. Linescan Models 
     Images are created through a physical process based on the microscope or other imaging tool used to acquire the image of a structure. Often these images are two-dimensional arrays of data, where the image can be thought of as a data set derived from the structure. A single one-dimensional cut through the image is called a linescan. A model of the imaging tool can predict the image for a given structure being imaged. For example, a model that describes a scanning electron microscope can predict the image that would be obtained by a SEM when imaging a given structure. 
     A CD-SEM converts a measured linescan or a series of measured linescans into a single dimension number, the measured CD. To better understand how the linescan relates to the actual dimensions of the feature being measured, it is important to understand how the systematic response of the SEM measurement tool to pattern structures impacts the shape of the resulting linescan. Rigorous 3D Monte Carlo simulations of SEM linescans can be extremely valuable for this purpose, but they are often too computationally expensive for day-to-day use. Thus, one approach is to develop a simplified analytical linescan model (ALM) that is more computationally appropriate to the task of quickly predicting linescans. The ALM employs the physics of electron scattering and secondary electron generation, and each term in the model has physical significance. This analytical linescan expression can be fit to rigorous Monte Carlo simulations to both validate and calibrate its use. 
     The general application for the ALM has been the typical forward modeling problem: Given material properties (for the feature and the substrate) and a geometric description of the feature (width, pitch, sidewall angle, top corner rounding, footing, etc.), the ALM predicts the linescan that would result. The mathematical details of the ALM are found in the publications: Chris A. Mack and Benjamin D. Bunday, “Analytical Linescan Model for SEM Metrology”,  Metrology, Inspection, and Process Control for Microlithography XXIX, Proc ., SPIE Vol. 9424, 94240F (2015), and Chris A. Mack and Benjamin D. Bunday, “Improvements to the Analytical Linescan Model for SEM Metrology”,  Metrology, Inspection, and Process Control for Microlithography XXX, Proc ., SPIE Vol. 9778, 97780A (2016), the disclosures of both publications being incorporated herein by reference in their entireties. Other models with similar inputs and outputs can also be used. 
     The analytical linescan model (ALM) is briefly reviewed below. The mathematical modeling begins by assuming the interaction of the electron beam with a flat sample of a given substance produces an energy deposition profile that takes the form of a double Gaussian, with a forward scattering width and a fraction of the energy forward scattered, and a backscatter width and a fraction of the energy deposited by those backscattered electrons. The model also assumes that the number of secondary electrons that is generated within the material is in direct proportion to the energy deposited per unit volume, and the number of secondary electrons that escape the wafer (and so are detected by the SEM) are in direct proportion to the number of secondary electrons near the very top of the wafer. 
     The secondary electrons that reach the detector will emerge some distance r away from the position of the incident beam. From the assumptions above, the number of secondary electrons detected will be a function as given in EQUATION 5. 
         f ( r ) ae   −r     2     /2σ     f       2     +be   −r     2     /2σ     b       2     EQUATION 5
 
     where σ f  and σ b  are the forward and backscatter ranges, respectively, and a and b are the amounts of forward scattering and backscattering, respectively. 
     SEMs detect topography because of the different number of secondary electrons that escape when the beam is in the space between features compared to when the beam is on top of the feature.  FIG.  10    shows that secondary electrons have trouble escaping from a space (especially if it is small), making spaces appear relatively dark. When an electron beam is focused to a spot in a space between lines, scattered electrons interact with feature  815  which absorbs some of the escaping secondary electrons. The detected secondary electron signal is reduced as the beam approaches the feature edge within the space. 
     The absorption by the step (i.e. feature  815 ) can be modeled to produce a prediction of the shape of the linescan in the space region. If a large feature has a left edge  815 - 1  at x=0, with the feature  815  to the right (positive x), the detected secondary electron signal as a function of position (SE(x)) will be given by EQUATION 6 below: 
     
       
         
           
             
               
                 
                   
                     
                       For 
                       ⁢ 
                           
                       x 
                     
                     &lt; 
                     0 
                   
                   , 
                   
                     
                       
                         S 
                         ⁢ 
                         
                           E 
                           ⁡ 
                           ( 
                           x 
                           ) 
                         
                       
                       
                         S 
                         ⁢ 
                         
                           E 
                           ⁡ 
                           ( 
                           
                             - 
                             ∞ 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           α 
                           j 
                         
                         ⁢ 
                         
                           e 
                           
                             x 
                             / 
                               
                             
                               σ 
                               f 
                             
                           
                         
                       
                       - 
                       
                         
                           α 
                           b 
                         
                         ⁢ 
                         
                           e 
                           
                             x 
                             / 
                             
                               σ 
                               b 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQUATION 
                   ⁢ 
                       
                   6 
                 
               
             
           
         
       
     
     where α f  is the fraction of forward scatter secondary electrons absorbed by the step and α b  is the fraction of backscatter secondary electrons absorbed by the step. 
     However, when the beam is on top of feature  815 , the interaction of the scattered electrons with the feature is very different, as accounted for in EQUATION 7 below. As illustrated in  FIG.  8   , two phenomena occur as when the beam is closer to the edge compared to further away. First, secondary electrons from both forward and backscattered electrons can more easily escape out of the edge  815 - 1 . This causes the edge bloom already discussed above. To account for this effect, a positive term α e e −x/σ     e    is added to account for the enhanced escape of forward-scattered secondary electrons where σ e  is very similar to the forward scatter range of the step material. Additionally, the interaction volume itself decreases when the beam is near the edge  815 - 1 , so that there are fewer secondary electrons being generated. Thus, the term α v e −x/σ     v    where σ v &lt;σ e  is subtracted to give EQUATION 7 below which is the linescan expression for the top of the large feature  815 : 
     
       
         
           
             
               
                 
                   
                     
                       For 
                       ⁢ 
                           
                       x 
                     
                     &lt; 
                     0 
                   
                   , 
                   
                     
                       
                         S 
                         ⁢ 
                         
                           E 
                           ⁡ 
                           ( 
                           x 
                           ) 
                         
                       
                       
                         S 
                         ⁢ 
                         
                           E 
                           ⁡ 
                           ( 
                           ∞ 
                           ) 
                         
                       
                     
                     = 
                     
                       1 
                       + 
                       
                         
                           α 
                           e 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               x 
                             
                             / 
                               
                             
                               σ 
                               e 
                             
                           
                         
                       
                       - 
                       
                         
                           α 
                           v 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               x 
                             
                             / 
                             
                               σ 
                               v 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQUATION 
                   ⁢ 
                       
                   7 
                 
               
             
           
         
       
     
       FIG.  11    shows an example of the result for this model. More specifically,  FIG.  11    shows a predicted linescan of a left-facing resist step  815  (large feature with left edge  815 - 1  at x=0) on a substrate such as a silicon wafer. The calibrated model  1105  is superimposed on the rigorous Monte Carlo simulation results  1110 . The calibrated model  1105  agrees so closely with the Monte Carlo simulation results  1110  that the two curves appear together almost as one line. 
     The above discussion involves modelling an isolated left-facing edge  815 - 1 . Adapting the model to include a right-facing edge involves translating and reversing the edge and adding the resulting secondaries (i.e., secondary electrons). Some complications arise if the two edges are close enough to interact, resulting in additional terms. Additionally, the impact of non-vertical sidewalls and rounded corners at the top and bottom of the feature edge may be included in the model ( FIG.  12   ). 
       FIG.  12    shows a representative predicted linescan of a pattern of resist lines and spaces on a silicon wafer. The calibrated model  1205  is superimposed on the rigorous Monte Carlo simulation results  1210 . Again, the calibrated model  1205  agrees so closely with the Monte Carlo simulation results  1110  that the two curves appear together almost as one line. A final model (ALM) includes 15 parameters that depend on the properties of the materials of the wafer and feature, and the beam voltage. To validate the model and to calibrate these parameters, rigorous first principle Monte Carlo simulations can be used to generate linescans for different materials and feature geometries. The ALM can then be fit to the Monte Carlo results, producing best-fit values of the 15 unknown parameters. 
     6. Inverse Linescan Model 
     Linescan or image models, such as the analytical linescan model (ALM) discussed above, predict an image or the shape of an image linescan for a particular pattern structure (such as a feature on a wafer). The ALM solves a forward modelling problem wherein the model receives geometry information for the particular feature as input, and provides the predicted shape of a respective SEM linescan of the particular feature as output. 
     In contrast to ALM, the disclosed edge detection system  700  includes a reverse model that receives as input “measured linescan information” from SEM  701  that describes a particular feature on the wafer. In response to the measured linescan information describing the particular feature, edge detection system  700  employs its reverse model to generate as output “feature geometry information” that describes the feature geometry that would produce the measured linescan. Advantageously, edge detection system  700  has been found to be effective even when the measured linescan information from SEM  701  includes a significant amount of image noise. In one embodiment, the outputted feature geometry information includes at least feature width. In another embodiment, the outputted feature information includes feature width and/or other geometry descriptors relative to the geometry of the particular feature, such as sidewall angle, feature thickness, top corner rounding, or bottom footing. It is noted that a feature disposed on a semiconductor wafer is an example of one particular type of pattern structure to which the disclosed technology applies. 
     Like many models of imaging systems, the ALM is inherently nonlinear. To address the nonlinear nature of the ALM, edge detection system  700  numerically inverts the ALM or a similar forward model and fits the resulting inverse linescan model to a measured linescan to detect feature edges (e.g. to estimate the feature geometry on the wafer). The disclosed edge detection system apparatus and edge detection process include the ability to detect and measure feature roughness. The disclosed apparatus and methodology may apply as well to other applications in general CD metrology of 1D or 2D features, such as the precise measurement of feature width (CD) and edge position or placement. 
     It is first noted that the ALM (and similar models as well) has two types of input parameters, namely material-dependent parameters and geometry parameters. Material-dependent parameters include parameters such as forward and backscatter distances, while geometry parameters include parameters such as feature width and pitch. In one embodiment, for a repeated edge detection application, the material parameters will be fixed and only the geometry parameters will vary. In the simplest case (that is, for simple edge detection), it is assumed that only the edge positions for the feature are changing, such that sidewall angle, corner rounding, etc., are assumed to be constant. Thus, the use of a linescan model for edge detection in edge detection system  700  involves two steps: 1) calibrating the parameters that are assumed to be constant across the entire image, and then 2) finding the feature edge positions that provide a best fit of the measured linescan to the linescan model for each measurement. 
     In one embodiment, in the first step, calibration is accomplished by comparing the linescan model to rigorous Monte Carlo simulations. The goal in this step is to find material parameters over the needed range of applications, and to ensure the fitting is adequate for the needed range of feature geometries. When finished, this calibrated linescan model can serve as the starting point for the generation of an inverse linescan model. The Inverse Linescan Model (ILM) should be calibrated to the specific SEM images that are to be measured. Since image grayscale values are only proportional to secondary electron signals, at the very least a mapping to grayscale values is required. In real-world applications, material properties in the experimental measurement will not be identical to those assumed in the Monte Carlo simulations such that some calibration of those parameters will also be required. 
     7. Calibration of the Inverse Linescan Model 
     Before using the ILM for edge detection, the ILM is first calibrated. Some parameters of the model (such as material-dependent parameters) are assumed to be constant for the entire image. However, geometry parameters, such as the positions of the edges, feature width and pitch, are assumed to vary for every linescan. The goal of ILM calibration is to determine the parameters that are constant for the whole image, regardless of the exact positions of the feature edges. It is a further goal of ILM calibration to accurately determine these parameters in the presence of image noise. These goals are accomplished by averaging along an axis of symmetry for the feature being measured, thus averaging out both the image noise and the actual feature roughness. 
     By averaging the linescan along an axis of symmetry (such as the direction parallel to a long line or space feature), information about the actual edge positions is lost, but information about the material parameters of the linescan model remain. Further, noise in the image is mostly averaged out in this way. Calibrating the ILM to the average linescan produces a set of material parameters (or any parameters assumed constant throughout the image) specific to this image. 
     Many features to be measured exhibit an axis of symmetry appropriate for ILM calibration. For example, a vertical edge has a vertical axis of symmetry. Averaging all pixels in a vertical column of pixels from the image will average away all vertical variation, leaving only horizontal information, in a direction perpendicular to the edge of the feature. The result of this averaging is a one-dimensional linescan called the average linescan. Likewise, a nominally circular contact hole or pillar is ideally radially symmetric. Averaging through polar angle about the center of the feature will produce an average linescan that removes noise and roughness from the image. An elliptical hole shape can also be so averaged by compressing or expanding the pixel size in one direction in proportion to the ratio of major to minor axes of the ellipse. Other axes of symmetry exist for other features as well. 
     One measured image (for example, one SEM image) may contain one or more features in the image. For example,  FIG.  1 A  shows multiple vertical line features and multiple vertical space features.  FIG.  1 B  shows multiple contact holes. For such a case, each feature can be separately averaged along an axis of symmetry to form an average linescan for that feature. For the example of  FIG.  1 A , the SEM image can be partitioned into vertical stripes, each stripe containing only one line feature, where the stripe extends horizontally from approximately the center of one space to approximately the center of the next space. For the example of  FIG.  1 B , the image can be partitioned into separate rectangular regions, each containing exactly one contact hole with the center of the contact hole approximately coinciding with the center of the rectangular region. The averaged linescan for that contact hole is then determined from that rectangular region of the image. Alternately, each of the averaged linescans from each feature in an image can themselves be averaged together to form a single averaged linescan applicable to the entire image. 
     For a repeated edge detection application (such as the detection of all the edges on a single SEM image), the material parameters will be fixed and only the geometry parameters will vary. In the simplest case (that is, for simple edge detection), one can assume that only the edge positions for the feature are changing, so that feature thickness, sidewall angle, corner rounding, etc., are assumed constant. Thus, the use of the ILM for edge detection will involve two steps: calibrating one time for the parameters that are assumed to be constant (i.e., material and fixed geometry properties) using the average linescan, and then finding the feature edge positions that provide a best fit of the measured linescan to the linescan model for each linescan. Optionally, calibration is first accomplished by comparison of the linescan model to rigorous Monte Carlo simulations, as has been previously described. The goal of this initial step is to find material parameters over the needed range of applications, and to ensure the model is adequate for the needed range of feature geometries. When finished, this partially calibrated linescan model must still be fully calibrated to the specific SEM images that are to be measured using the average linescan. 
     Once the ILM has been calibrated to the given SEM image or sets of images, it is then used to detect edges. Due to the non-linear nature of linescan models such as the ALM model, numerical inversion is needed, for example using non-linear least-square regression to find the values of the left and right edge positions that best fit the model to the data. For simpler linescan models, a linear least-squares fit may be possible. Other means of “best fit” are also known in the art. The ILM as an edge detector allows the detection of edges in a high noise environment without the use of filters.  FIGS.  13 A and  13 B  demonstrate the reliable detection of edges for a very noisy image without the use of any filtering or image smoothing. More particularly,  FIG.  13 A  is an original SEM image of a pattern structure that exhibits 18 nm lines and spaces before edge detection with an ILM.  FIG.  13 B  is the same image after edge detection using an ILM. 
     Gaussian filters are common image smoothing filters designed to reduce noise in an image. Other filters such as box filters and median filters are also commonly used for this purpose. To illustrate the impact of image filtering on roughness measurement, TABLE 1 below shows the measured 3σ linewidth roughness (LWR) as a function of Gaussian filter x- and y-width (in pixels). For each case, the ILM edge detection method was used, so that the difference in the resulting LWR is only a function of the image filter parameters. The range is almost a factor of two, showing that many different roughness measurements can be obtained based on the arbitrary choice of filter parameters. In all cases, the ILM edge detection was used. If a conventional threshold edge detection method is used, the range of resulting 3σ roughness values is much greater (TABLE 2). Similar results are obtained if other filter types (box or median, for example) are used. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 The raw (biased) 3σ LWR (nm) as a function of Gaussian 
               
               
                 filter x- and y-width (in pixels), using ILM edge detection. 
               
            
           
           
               
               
               
               
               
            
               
                   
                 y-width = 1 
                 y-width = 2 
                 y-width = 3 
                 y-width = 4 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                 x-width = 1 
                 4.99 
                 4.67 
                 4.03 
                 3.82 
               
               
                 x-width = 3 
                 4.92 
                 4.02 
                 3.48 
                 3.28 
               
               
                 x-width = 5 
                 4.85 
                 3.82 
                 3.28 
                 3.00 
               
               
                 x-width = 7 
                 4.79 
                 3.69 
                 3.13 
                 2.84 
               
               
                 x-width = 9 
                 4.73 
                 3.59 
                 3.08 
                 2.80 
               
               
                 x-width = 11 
                 4.68 
                 3.54 
                 3.07 
                 2.80 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 The raw (biased) 3σ LWR (nm) as a function of Gaussian filter 
               
               
                 x- and y-width (in pixels), using conventional threshold edge detection. 
               
            
           
           
               
               
               
               
               
            
               
                   
                 y-width = 1 
                 y-width = 2 
                 y-width = 3 
                 y-width = 4 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                 x-width = 1 
                   
                 11.17 
                 8.52 
                 7.28 
               
               
                 x-width = 3 
                 9.58 
                 5.22 
                 4.02 
                 3.72 
               
               
                 x-width = 5 
                 8.12 
                 4.62 
                 3.83 
                 3.49 
               
               
                 x-width = 7 
                 7.44 
                 4.50 
                 3.78 
                 3.42 
               
               
                 x-width = 9 
                 7.03 
                 4.45 
                 3.77 
                 3.41 
               
               
                 x-width = 11 
                 6.77 
                 4.44 
                 3.77 
                 3.41 
               
               
                   
               
            
           
         
       
     
     While the arbitrary choice of image filter parameters has a large impact on the measurement of roughness of the pattern structure, the impact of threshold value depends on the specific edge detection method used. For the case of a simple threshold edge detection after image filtering, there is one threshold value that minimizes the 3σ roughness measured, with other values changing the roughness quite dramatically (see  FIG.  14   ). For the case of the ILM, the choice of threshold has almost no impact on the measured LWR (in  FIG.  14   , the LWR varies from 5.00 nm to 4.95 nm as the threshold is changed from 0.25 to 0.75). Thus, for the conventional prior art method of detecting edges the arbitrary choice of threshold value can cause a large variation in the measured roughness. For the ILM, there are essentially no arbitrary choices that affect the measurement of roughness. 
     While the disclosed ILM system achieves accurate detection of edges in the presence of high levels of noise, the noise still adds to the measured roughness. For a linescan of a given edge slope, uncertainty in the grayscale values near the line edge translates directly into uncertainty in the edge position. A major difference, though, is that the impact of noise can be measured for the case without filtering. The noise floor of an unfiltered image can be subtracted out from the PSD (power spectral density), producing an unbiased estimate of the PSD (and thus the roughness). For the case of a filtered image, the noise floor is mostly smeared away, so that it cannot be detected, measured, or removed. 
       FIGS.  15 A and  15 B  show LER power spectral densities from many rough features with right and left edges combined separately. More specifically,  FIG.  15 A  shows raw PSDs after edge detection using the disclosed ILM technology, while  FIG.  15 B  shows PSDs after noise subtraction. 
     Consider the results shown in  FIG.  15 A , where the line-edge roughness (LER) for the left and right edges of a feature on a pattern structure are compared. The raw PSDs indicate that the two edges behave differently. However, these differences are an artifact of the SEM, caused by a scan-direction asymmetry (such as charging) that makes the right linescan slope lower than the left linescan slope. In fact, there is no difference between right and left edge on the wafer for this sample. By measuring the noise floor for each edge separately, subtracting the noise produces a common left/right LER ( FIG.  15 B ) that is an unbiased estimate of the true PSD. 
     Once the noise has been subtracted, reliable analysis of the PSD can lead to reliable estimates of the important roughness parameters, such as the zero-frequency PSD(0), the correlation length ξ, and the roughness exponent H. The unbiased 3σ roughness can also be obtained. Without removing the noise, extraction of these parameters from the empirical PSD is problematic and prone to systematic errors. 
     8. Unbiased Measurement of PSD 
     The biggest impediment to accurate roughness measurement is noise in the CD-SEM image. Among other noise sources, SEM images suffer from shot noise, where the number of electrons detected for a given pixel varies randomly. For the expected Poisson distribution, the variance in the number of electrons detected for a given pixel of the image is equal to the expected number of electrons detected for that pixel. Since the number of detected electrons is proportional to the number of electrons that impinge on the sample location represented by that pixel, relative amount of noise can be reduced by increasing the electron dose that the sample is subjected to. For some types of samples, electron dose can be increased with few consequences. But for other types of samples (such as photoresist), high electron dose leads to sample damage (resist line slimming, for example). Other types of samples, such as biological specimens, can also suffer from electron damage. Thus, to prevent sample damage electron dose is kept as low as possible, where the lowest dose possible is limited by the noise in the resulting image. 
       FIG.  16    shows portions of three SEM images of nominally the same lithographic features taken at different electron doses. More specifically,  FIG.  16    shows portions of SEM images of nominally identical resist features with 2, 8, and 32 frames of integration (respectively, from left to right). Doubling the frames of integration doubles the electron dose per pixel. Since the dose is increased by a factor of 4 in each case, the noise goes down by a factor of 2. 
     SEM image noise adds to the actual roughness of the patterns on the wafer to produce a measured roughness that is biased higher. Typically, we obtain a biased roughness as given by EQUATION 8A. 
       σ biased   2 =σ unbiased   2 +σ noise   2   EQUATION 8A
 
     where σ biased  is the roughness measured directly from the SEM image, σ unbiased  is the unbiased roughness (that is, the true roughness of the wafer features), and σ noise  is the random error in detected edge position (or linewidth) due to noise in the SEM imaging and edge detection. EQUATION 8A assumes that the noise is statistically independent of the roughness on the feature being measured. If this is not the case, more complicated noise models can be used, as further described below. Since an unbiased estimate of the feature roughness is desired, the measured roughness can be corrected by subtracting an estimate of the noise term. 
     Pixel noise in the SEM creates edge detection noise depending on the shape of the expected linescan for the feature. For example,  FIG.  17 A  shows a typical linescan (grayscale value versus horizontal position, g(x)) for a line feature on a wafer when there is an extremely large number of electrons so that the pixel noise is negligible. The result is the “expected” linescan, that is, the expectation value of the linescan signal from a statistical perspective. By defining a threshold grayscale level, the edge position can be determined. But noise in the grayscale values results in noise in the detected edge position. For a given grayscale noise σ gray , the edge position uncertainty σ noise  will depend on the slope of the linescan at the edge dg/dx. For small levels of noise, 
     
       
         
           
             
               
                 
                   
                     σ 
                     noise 
                   
                   ~ 
                   
                     
                       σ 
                       gray 
                     
                     
                       d 
                       ⁢ 
                       g 
                       / 
                       dx 
                     
                   
                 
               
               
                 
                   ( 
                   
                     8 
                     ⁢ 
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
     Thus, the level of edge detection noise is a function of the pixel grayscale noise and the slope of the linescan at the feature edge. 
     This equation 8B is strictly only valid for small levels of noise and an infinitely small pixel size. To explore the impact of greater amounts of noise and a non-zero pixel size, simulation of SEM images was employed. Perfectly smooth lines and spaces (25 nm width, 50 nm pitch) were used as inputs to the Analytical Linescan Model in order to create synthetic SEM images. Then the resulting grayscale values (which range from 0 to 255) of each pixel were treated as the mean of a normal distribution with a given standard deviation (σ gray ) and a random grayscale number was assigned to each pixel drawn from this normal distribution. These SEM images were then treated as experimental SEM images and measured using an inverse linescan model to detect the edge positions of each feature. The 1-sigma LER measured from these images is the detected edge position uncertainty due to the grayscale pixel noise.  FIG.  17 B  shows the 1-sigma uncertainty in edge detection position for these perfectly smooth features in the presence of grayscale noise. In this graph, the edge detection noise, for three different X pixel sizes, is plotted as a function of grayscale noise for simulated synthetic SEM images (average of 100 images, each with 20 dense lines/space features of width 25 nm and pitch 50 nm). The edge detection used an inverse linescan model and the resulting line-edge roughness of the features was considered to be the edge detection noise. The result is somewhat nonlinear, with higher levels of pixel noise producing ever greater edge detection noise. Further, smaller X pixel sizes produce lower levels of edge detection noise. In fact, the edge detection variance a σ noise   2  is directly proportional to the X pixel size for low levels of grayscale noise. 
     Pixel noise is not the only source of edge detection noise. During operation the electron beam is scanned from left to right using beam steering electronics. Errors in the beam steering can place the beam at an incorrect position, which produces an edge error. Charging of the sample during electron exposure will deflect the beam to an incorrect position. While some of the charging effects will be systematic, there will also be random or pseudo-random components that will appear as random variation in the detected edge position. 
     While several approaches for estimating the SEM edge position noise and subtracting it out have been proposed in the prior art, these approaches have not proven successful for today&#39;s small feature sizes and high levels of SEM image noise. The problem is the lack of edge detection robustness in the presence of high image noise. More particularly, when noise levels are high, edge detection algorithms often fail to find the edge. The solution to this problem is typically to filter the image, smoothing out the high frequency noise. For example, if a Gaussian 7×3 filter is applied to the image, then for each rectangular region of the image 7 pixels wide and 3 pixels tall, the grayscale values for each pixel are multiplied by a Gaussian weight and then averaged together. The result is assigned to the center pixel of the rectangle. Box (mean) filters and median filters can also be used and produce similar results. This smoothing makes edge detection significantly more robust when image noise is high.  FIG.  17 C  shows an example of using a simple threshold edge detection algorithm with image filtering in the right image and without image filtering in the left image. Without image filtering, the edge detection algorithm is mostly detecting the noise in the image and does not reliably find the edge. 
     The use of image filtering can have a large effect on the resulting PSD and measured roughness.  FIG.  18    shows the impact of two different image filters on the PSD obtained from a collection of 30 images, each containing 12 features. All images were measured using an inverse linescan model for edge detection. The power spectral densities were averaged from these 360 rough features with images preprocessed using a 7×2 or 7×3 Gaussian filter, or not filtered at all, as labelled in the drawing. As can be appreciated, the high-frequency region is greatly affected by filtering. But even the low frequency region of the PSD shows a noticeable change when using a smoothing filter. Filtering in the y-direction smoothes out high-frequency roughness. Filtering in the x-direction lowers the slope of the linescan, which can affect measured low-frequency roughness. As will be described next, the use of image filtering makes measurement and subtraction of image noise impossible. 
     If edge detection without image filtering can be accomplished, noise measurement and subtraction can be achieved by contrasting the PSD behavior of the noise with the PSD behavior of the actual wafer features. We expect resist features (as well as after-etch features) to have a PSD behavior as shown in  FIG.  19    as the “True PSD” (and also shown earlier in  FIG.  4   ). Correlations along the length of the feature edge reduce high-frequency roughness so that the roughness becomes very small over very short length scales. SEM image noise, on the other hand, can often be assumed to be white noise, so that the noise PSD is flat over all frequencies. Other models of the SEM image noise are also possible, for example using linescan-to-linescan correlation to describe the noise, as further described below. Thus, at a high enough frequency the measured PSD will be dominated by image noise and not actual feature roughness (the so-called “noise floor”). Given the grid size along the length of the line (Δy), SEM edge detection white noise affects the PSD according to EQUATION 9 below: 
       PSD biased ( f )=PSD unbiased ( f )+σ noise   2   Δy   EQUATION 9
 
     Thus, measurement of the high-frequency PSD (in the absence of any image filtering) provides a measurement of the SEM edge detection noise.  FIG.  19    illustrates this approach for the case of a white SEM noise model. Clearly, this approach to noise subtraction cannot be used on PSDs coming from images that have been filtered, because such filtering removes the high-frequency noise floor (see  FIG.  18   ). 
     EQUATION 9 assumes a white noise model, where the noise found in any pixel of the image is independent of the noise found in any other pixel. This may not always be the case. For example, the noise in each pixel may be correlated somewhat with its nearest neighbors, affecting σ gray  in equation 8B. Alternately, the grayscale slope in equation 8B may be correlated from one row of pixels to the next, possibly caused by the interaction volume of the electrons as shown in  FIG.  8   . If a correlation model is assumed or measured, a suitable noise expression for the PSD can be used to replace EQUATION 9, as further described below. 
       FIG.  19    shows one embodiment of the noise subtraction process of the disclosed edge detection apparatus and method. In the disclosed edge detection method, the method first detects the positions of the edges using the ILM without the use of any image filtering (for example, using an inverse linescan method). From these detected edges a biased PSD is obtained, which is the sum of the actual wafer roughness PSD and the SEM noise PSD. Using a model for the SEM image noise (such as a constant white noise PSD), the amount of noise is determined by measuring the noise floor in the high-frequency portion of the measured PSD. The true (unbiased) PSD is obtained by subtracting the noise level from the as-measured (biased) PSD. The key to using the above approach of noise subtraction for obtaining an unbiased PSD (and thus unbiased estimates of the parameters σ LWR (∞), PSD(0), and ξ) is to robustly detect edges without the use of image filtering. This can be accomplished using an inverse linescan model. An inverse linescan model was used to generate the no-filter PSD data shown in  FIG.  18   . 
     An example method for subtracting white noise will now be described. First, edges are detected from a SEM image without using any image filtering (for example, using an inverse linescan model). The power spectral densities of one or more edges are calculated in the usual way. Since the PSD of a single edge is quite noisy, it is extremely valuable to measure many edges and average the PSDs. Often hundreds or thousands of edges are measured and their PSDs averaged. This averaged PSD is called the biased PSD. From the average biased PSD, the highest frequencies are inspected to determine if a flat noise floor is observed. Such a noise floor is observed whenever the y pixel size is sufficiently smaller than the correlation length of the true roughness. Typically, a y-pixel size that is 20% of the correlation length or smaller is adequate. If a noise floor is observed, the average PSD value in the flat region is calculated. This is the noise floor. This number is then subtracted from the biased PSD at every frequency to produce the unbiased PSD. The biased PSD is our best estimate of the true PSD of the roughness on the wafer. 
     Other SEM errors can influence the measurement of roughness PSD as well. For example, SEM field distortion can artificially increase the low-frequency PSD for LER and PPR, though it has little impact on LWR. Background intensity variation in the SEM can also cause an increase in the measured low-frequency PSD, including LWR as well as LER and PPR. If these variations can be measured, they can potentially be subtracted out, producing the best possible unbiased estimate of the PSD and its parameters. By averaging the results of many SEM images where the only common aspect of the measurements is the SEM used, determination of SEM image distortion and background intensity variation can be made. 
     Further, the SEM noise itself can vary across the SEM image field. Thus, unbiasing of the roughness measurement can also include the detection of noise that varies across the SEM image field and unbiasing different points in the SEM image field according to its measured noise bias at those points in the SEM image field. 
     9. Sensitivity to Metrology Tool Settings 
     Not all noise in measured PSDs is white noise. White noise occurs when the measurement noise of the edge position from each linescan is completely independent of all other linescans (and in particular, its nearest neighbors). White noise occurs in the absence of correlations that connect the errors in one linescan to the errors in the neighboring linescans. Any small correlations in edge errors along the length of the line would cause “pink noise”, a noise signature that is not perfectly flat over the entire frequency region. 
     The settings of the SEM metrology tool can impact the measured roughness of a feature in a pattern structure. These settings include the magnification and pixel size of SEM  701 . These two parameters can be changed independently by changing the number of pixels in the image (from 512×512 to 2048×2048, for example). Additionally, the number of frames of integration (the electron dose) when capturing an SEM image can be adjusted. To study the impact of this setting, the number of frames of integration can be varied from 2 to 32, representing a 16× variation in electron dose, for example. 
     Total electron dose is directly proportional to the number of frames of integration. Thus, shot noise and its impact on edge detection noise is expected to be proportional to the square root of the number of frames of integration.  FIG.  20    shows PSDs of a particular resist feature type on a given wafer, measured with different numbers of frames of integration. In this case, the PSDs correspond to 18 nm resist lines and spaces where only the number of frames of integration was varied. SEM conditions used were 500 eV, 49 images per condition, 21 features per image, pixel size=0.8 nm square, and image size=1024×1024 pixels. The cases of 8 or more frames of integration produce PSDs that exhibit a fairly flat high-frequency noise region. For 2 and 4 frames of integration the noise region is noticeably sloped. Thus, the assumption of white SEM noise is only approximately true, and becomes a more accurate assumption as the number of frames of integration increases and noise level decreases. This observation has been borne out in other circumstances: High noise cases are more likely to exhibit non-flat noise floors. 
       FIG.  21    shows the biased and unbiased values of the 3σ linewidth roughness measured as a function of the number of frames of integration. All conditions were the same as described in  FIG.  20   , and error bars represent 95% confidence interval estimates. The biased roughness varies from 8.83 nm at two frames of integration to 5.68 nm at 8 frames and 3.98 nm at 32 frames. The unbiased roughness, on the other hand, is fairly stable after 6 frames of integration, varying from 5.25 nm at two frames of integration to 3.25 nm at 8 frames and 3.11 nm at 32 frames. While the biased roughness is 43% higher at 8 frames compared to 32, the unbiased roughness is only 4% higher at 8 frames compared to 32. Since the assumption of white SEM noise is not very accurate at 2 and 4 frames of integration, the noise subtraction of the unbiased measurement using a white noise model is not completely successful at these very low frames of integration. A correlated noise model can produce better noise subtraction especially for the low frames of integration, as is more fully described below. While the results shown are for LWR, similar results are obtained for the measurement of line edge roughness (LER) and pattern placement roughness (PPR). 
     One possible cause of correlations in edge noise would be correlations in the pixel noise. To test this possibility, isolated edges were measured in the CD-SEM. The edge allows the SEM to perform its imaging functions in a typical way, but at a distance left or right from the edge the field is flat and featureless. In this region the only variation in pixel grayscale values comes from image noise. The correlation coefficient between neighboring pixels can then be calculated. Performing these calculations, the average correlation between neighboring pixels in the x-direction was 0.12, but the average correlation in the y-direction was only 0.01, essentially zero. These correlations coefficients were determined for edges measured at 2 to 32 frames of integration. There was little variation in the pixel-to-pixel correlation as a function of the number of frames of integration. Thus, correlated pixel noise is not responsible for the pink noise observed at low frames of integration. However, it is possible that the linescan slope in equation 8B is responsible for the noise correlations. 
     A possible cause of noise correlations in the linescan slope comes from the interaction of the beam with the sample. Electrons striking the sample undergo a number of processes that depend on the energy of the electron and the material properties of the sample. Electrons scatter off the atoms of the sample material, release energy, change direction, and often generate a cascade of secondary electrons by ionizing the sample atoms. Occasionally electrons ricochet backwards off the atom nucleus and exit out of the sample (called backscatter electrons). Some of the lower energy secondary electrons can also escape out of the sample (frequently through the edges of a feature, see  FIGS.  8 A and  8 B ). The way in which a SEM forms an image is by detecting the number of secondary electrons and/or backscatter electrons that escape the sample for each beam position. 
     When forming an image using an SEM, a small spot of electrons dwells at a specific point on the sample (i.e., a pixel) while the number of escaping secondary electrons is counted by the secondary electron detector. When the spot is a long way from a feature edge, as in  FIG.  8 A , the number of detected secondary electrons  805  is small (and the pixel is dark). When the spot is near a feature edge, as in  FIG.  8 B , secondary electrons  805  from the interaction volume readily escape from the feature edge producing a bright pixel. 
     The interaction volume of the electrons can be one to a few tens of nanometers in diameter, depending on the beam voltage and the sample material properties. This interaction volume means that electrons impinging on one spot on the sample are influenced by the sample shape over a range determined by the interaction volume. Thus, the slope of the linescan at one row of pixels will not be independent of the slope of the linescan at neighboring pixels whenever the interaction volume radius is greater than the y pixel size. This dependency can be the cause of correlations in the noise, with a noise correlation length affected by the electron beam interaction volume. 
     10. Detecting and Removing Spikes from a Power Spectral Density 
     In addition to noise interfering with the signal in typical images of rough features, other errors can be present in the images that have a very different frequency behavior as compared to white noise or pink noise, and as compared to the roughness being measured. Some such errors produce large but narrow spikes in a PSD.  FIG.  25 A  shows one example of high frequency “spikes” that intermittently are found in datasets. One cause for such spikes can be electrical interference in the scanning electronics of the imaging tool. If the interference is at a frequency in a range that allows one or more interfering events within a full scan of the image, this interference can result in a slight but regular “jitter” of the scanning beam position. For highly precise scanning, even a sub-nanometer jitter can result in one or more large spikes in the measured PSD. Depending on the mechanism, such interference spikes may be present in the line-edge roughness (LER) and pattern placement roughness (PPR) but not the linewidth roughness (LWR) PSD. Alternately, the interference may cause spikes at the same frequencies in all three PSDs. 
     For example, electrical interference at a frequency of 50 Hz or 60 Hz can cause noticeable spikes in a measured PSD when the measurement tool captures images at a standard “TV” scan rate or small multiples of this rate. Additionally, electrical interference at normal audio frequencies can cause spikes that are visible at higher PSD frequencies in typical measurement tool images. 
     The presence of spikes in the PSD can be undesirable for a number of reasons depending on their quantity, their amplitude, and their frequency. For the case of high-frequency spikes as seen in  FIG.  25 A , the spikes can affect the noise removal process described above, resulting in an overestimation of the amount of white or pink noise in the image. 
     PSD spikes can be caused by phenomenon other than electrical interference within the imaging tool. The object being measured may include periodic or semi-periodic structures other than the rough features that are to be measured. For example, a set of vertically oriented rough features of the object may be on top of a periodic set of horizontal features resulting in topography below the rough features that are slightly visible in the image. Such underlying topography can result in a mid-frequency spike to the PSD (with higher-frequency harmonics possible as well).  FIG.  26    shows an example of this phenomenon. 
     Another phenomenon that can give rise to spikes in the PSD would be the presence of grains of a small size range within the material of the features on the object to be measured. Grains of similar size packed tightly together can produce a nearly periodic appearance that results in a spike in the measured PSD. 
     Roughness measurements can also be performed on images taken of photomask features, where said photomasks are used in a lithography process. Photomasks are typically fabricated using a direct-write lithography tool with limitations such as a non-zero address grid and rectangular shots to make up the image. For some features, such as a line oriented at 45 degrees to the direction of the writing grid of the tool used to print the photomask, the result will be small, regularly spaced jogs along the edge of the photomask feature. These jogs will produce a spike (or a main spike plus harmonic spikes) in the PSD of the measured photomask roughness. 
     Spikes such as those found in  FIG.  26    can be very disadvantageous to the measurement of roughness parameters from the biased or unbiased PSD.  FIG.  27 A  shows how a PSD with spikes can alter the model that is fit to the unbiased PSD, including modeling parameters such as PSD(0), correlation length, and roughness exponent. In contrast,  FIG.  27 B  shows how a PSD with spikes removed can affect the model that is fit to the unbiased PSD, including modeling parameters such as PSD (0), correlation length, and roughness exponent. 
     For these and other reasons, it is desirable to remove spikes in the PSD when the cause of those spikes is thought to be from a mechanism different from the mechanisms that give rise to the roughness of the features being measured. In other words, it is desirable to separate the PSD artifacts caused by one mechanism (such as spikes caused by electrical interference) from the PSD artifacts caused by other mechanisms (such as the stochastic effects that give rise to roughness). This can be done much like the noise removal described above, by recognizing the different frequency signatures of the different mechanisms. 
     As mentioned above, white noise (or pink noise) can be separated from the true (unbiased) roughness PSD since the noise frequency signature (flat or near flat at high frequencies) is very different from the frequency signature of the true roughness (a power-law decreasing at high frequencies). Likewise these so-called spikes in the PSD have frequency signatures that are very different from the frequency signature of the feature roughness itself. In particular, a so-called spike has a high amplitude over a very narrow frequency range. 
     A procedure for detecting and removing spikes will now be described. First, the definition of a “spike” can be established as being a frequency response that rises and falls over a frequency range smaller than a threshold (the “threshold range”) and has a height great than a threshold (the “threshold height”). 
     Next, a baseline can be established as being the best estimate of the PSD without the spike. For example, the threshold range for spike detection can be set to three frequency increments in the PSD data (which typically is sampled at a constant frequency increment). Other threshold ranges are also possible. A baseline can be determined by smoothly connecting PSD values separated by the threshold range plus one increment (using a straight line on a linear or logarithmic scale, for example, or by using a model for the expected PSD behavior). This baseline is then subtracted from the actual PSD data within this threshold range to arrive at an estimate of the non-baseline PSD behavior within this frequency range. If the non-baseline PSD behavior rises to a value greater than the threshold height (expressed either in absolute terms or as a multiple of the baseline PSD value), then a spike has been identified. To remove the spike, the calculated baseline behavior can be used to replace the actual PSD values within the threshold range. A search for spikes can cover the entire PSD frequency range if desired. 
     The threshold range can be chosen in such a way as to only detect (and possibly remove) spikes that occur due to specific types of mechanisms. For example, interference at exactly a single frequency will most likely cause a spike in the PSD that is up to two frequency increments wide (since the spike is unlikely to be at a frequency that exactly coincides with the sampled frequencies of the PSD). A threshold range of two to three frequency increments wide will be effective in detecting such “single frequency” interference events. A wider threshold range will detect other, broader-band interference events. 
     The threshold height can also be adjusted based on the mechanisms that are desired to be detected. But the minimum threshold height is also a function of the overall noise in the PSD. Since a PSD measures, by definition, the randomness in a random rough sample, PSD measurement is inherently noisey. It is well known that the PSD of a single measured feature has a statistical uncertain of 100% (1-sigma). That is, the statistical uncertainty in any given PSD value at any given frequency is 100% for the measurement of a single feature. For that reason, many features are typically measured and averaged together so that the uncertainty in the PSD can be reduced by one over the square root of the number of features being measured. 
     But for any given number of features measured and averaged, the PSD will have a statistical uncertainty that is inherent in the sample size. The threshold height for spike detection should be chosen to be significantly higher than the inherent noise level of the PSD. Otherwise, the detection of spikes would be frequently triggered not by physical spikes but rather by noise in the PSD data. Alternately, the threshold height can be chosen to be a multiple of the measured or calculated PSD noise (for example, 5×). 
       FIG.  25 A  shows several PSDs (linewidth roughness (LWR PSD  2502 ), line-edge roughness (LER PSD  2504 ), and pattern placement roughness PPR PSD  2506 ) which exhibit several high-frequency spikes (spike artifacts  2507 ).  FIG.  25 B  shows the same PSDs (e.g., LWR PSD  2502  as LWR PSD  2508 , LER PSD  2504  as LER PSD  2510 , and PPR PSD  2506  as PPR PSD  2512 ) with the spikes removed using the procedure outlined in the previous paragraphs. For this removal, the threshold range was set to three frequency increments, and the threshold height was set to be three times the baseline PSD value. Effective removal of the spikes was accomplished using these settings. 
       FIGS.  27 A and  27 B  show another case of spike removal, this time for mid-frequency spikes. The left-hand graph  FIG.  27 A  shows the PSDs (biased and unbiased) before spike removal. The presence of the spikes has a deleterious effect on the modeling the PSD and the extraction of PSD measured values. The right-hand graph,  FIG.  27 B  shows the same PSDs with the spikes removed using the procedure outlined in the previous paragraph. For this removal, the threshold range was set to three frequency increments, and the threshold height was set to be three times the baseline PSD value. Effective removal of the spikes was accomplished using these settings. The resulting PSD modeling and PSD measurement more accurately reflects the feature roughness PSD behavior excluding the mechanism that gave rise to the spikes. 
     An alternate procedure of removing spikes will now be described. Spikes can be removed from a PSD by passing the PSD through a low-pass filter. Using well-known techniques, the PSD can be Fourier transformed, multiplied by a low-pass frequency filter, then inverse Fourier transformed. The cut-off frequency of the low-pass filter can be set to only filter away spikes narrower than a set limit. Other approaches to low-pass filtering known in the field can also be applied. 
     Other methods for detecting and removing spikes based on the different frequency characteristics of a spike compared to the more slowly varying true roughness PSD will be known to those skilled in the art. 
     Referring to  FIG.  7   , the Information Handling System  750  can be modified to include the detection and/or removal of spikes using one of the exemplary methods described here. Information about each detected spike, such as its center frequency, amplitude, area, and/or width, can be recorded and output to Output Device  770 . This information can be useful for identifying the root cause of the spike formation and thus can assist in the process of reducing or eliminating such root cause mechanism. 
     11. Detection and Measurement of PSD Bumps 
     Other phenomenon can give rise to PSD behavior that appears as a “bump” in the PSD that otherwise has the typical shape shown in  FIG.  3   . Such bumps generally occur at relative low frequencies. These bumps are distinguished from spikes by covering a relative wide range of frequencies, as opposed to the narrow frequency confines of a spike.  FIGS.  28 A and  28 B  show two examples of this so-called bump behavior in PSD, labeled as Bump Type I and Bump Type II. 
     Bump Type I ( FIG.  28 A ) is a large rise in the low-frequency PSD behavior above what would normally be considered the flat low-frequency regime characterized by PSD(0). Several mechanisms can give rise to this bump, such as the presence of photomask roughness that is then transferred to the wafer during a photolithography step. Uncompensated field distortions in the imaging tool used to capture the images being measured can also give rise to this kind of bump. Other mechanisms are possible as well. 
     Bump Type II ( FIG.  28 B ) occurs at low-to-mid frequencies such that the PSD behavior at frequencies higher and lower than the bump follows the expected behavior (as seen, for example, in  FIG.  3   ). When this type of PSD bump is found in the line-edge roughness PSD but not in the linewidth roughness PSD, the effect is sometimes called “wiggle” since it can be noticeable as a wiggle in the feature of the image. Such wiggle can be caused, for example, from stress or tension in the films used to make the features. Photolithography and subtractive etching of the film to form the features can relieve stress and allow the relaxed remaining film to wiggle. Other mechanism for causing wiggle are also possible. 
     Like white noise and spikes, bumps in the PSD are thought to arise through mechanisms separate from the stochastic mechanism that gave rise to the rest of the PSD. Thus, it is desirable to separate out the effects of the bump from the rest of the PSD. It is possible to use a procedure similar to spike detection and removal for bump detection and removal. However, this approach becomes problematic when the width of the bump is large due to the difficulty in defining a baseline PSD behavior over a large frequency range. While the larger frequency range of the bump means it is possible to distinguish bumps from spikes, it also means that different procedures for detecting and measuring bumps are likely required. 
     A separate technique of bump detection, measurement, and removal involves the use of a model for the bump. Like white noise and pink noise, the bump model adds directly to the typical PSD of the feature roughness. Thus, the bump model can be fit to the PSD simultaneously with the typical PSD model that does not include bump behavior. 
     A useful form for a bump model is given in Equation 10 below: 
       PSD bump ( f )= Ae   −(f-f     c     )     2     /2σ     w       2     (10)
 
     where A is the amplitude of the bump, f c  is the center frequency of the bump, and σ w  is the width of the bump. For a Type I bump ( FIG.  28 A ), the center frequency can be zero. Other models may also be used. Alternate parameterizations of the model can also be used, such as the area and center frequency of the bump. 
     The area of the bump above the baseline PSD, as determined for example from the best fit model, is a useful measure of the magnitude of the phenomenon that gave rise to the bump. For example, for the case of wiggle (a Bump Type II example,  FIG.  28 B ), the area represents the variance of the wiggle that adds to the variance caused by stochastic roughness. In other words, this approach for bump detection and measurement allows the total variance of the feature to be separated into a wiggle variance plus a stochastic roughness variance. 
     Referring to  FIG.  7   , the Information Handling System  750  can be modified to include the detection and/or removal of bumps using one of the exemplary methods described here. Information about each detected bump, such as its center frequency, amplitude, area, and/or width, can be recorded and output to Output Device  770 . This information can be useful for identifying the root cause of the bump formation and thus can assist in the process of reducing or eliminating such root cause mechanism. By subtracting the bump behavior from the total PSD, the remaining PSD can be characterized (using, for example, parameters such as PSD(0), correlation length, and roughness exponent) so that this remaining PSD reflects more accurately the mechanisms that gave rise to the PSD exclusive of the bump mechanism. 
     Referring now to  FIG.  30   , an example method  3000  to detect undesired spikes in a PSD dataset, and for removing spikes in a PSD dataset is illustrated. The method  3000  starts (block  3002 ) and generates, using an imaging device, a set of one or more images, each image of the set including an instance of a feature within a respective pattern structure, each image including measured linescan information corresponding to the pattern structure that includes noise (block  3004 ). Next the method proceeds to detect edges of the features within the pattern structure of each image of the set without filtering the images (block  3006 ) and generates a power spectral density (PSD) dataset representing feature geometry information corresponding to the edge detection measurements of the set of images (block  3008 ). If desired, an unbiased PSD data set can be generated from the biased PSD data set by subtracting SEM noise. Next, the method defines a threshold range and a threshold height (block  3010 ) and generates a baseline for a portion of the PSD dataset, by smoothly connecting a first PSD value of the portion of the PSD dataset to a second PSD value, wherein the first PSD value and the second PSD value are separated by the threshold range (block  3012 ), determines that a difference between a third PSD value of the portion of the PSD dataset and the baseline is greater than a threshold height (block  3014 ), and replaces the portion of the PSD dataset with the baseline for the portion of the PSD dataset (block  3016 ). Thereafter, the method ends (block  3018 ). 
     Referring now to  FIG.  31   , an example method  3100  to model bumps in a PSD dataset is illustrated. The method  3100  starts (block  3102 ) and generates, using an imaging device, a set of one or more images, each image of the set including an instance of a feature within a respective pattern structure, each image including measured linescan information corresponding to the pattern structure that includes noise (block  3104 ). Next the method proceeds to detect edges of the features within the pattern structure of each image of the set without filtering the images (block  3106 ) and generate a biased power spectral density (PSD) dataset representing feature geometry information corresponding to the edge detection measurements of the set of images (block  3108 ). If desired, an unbiased PSD data set can be generated from the biased PSD data set by subtracting SEM noise. A first bump is evaluated in the PSD dataset to create a bump model (block  3110 ); and fits a typical PSD model and the bump model to the PSD dataset to create a best fit model (block  3112 ). Thereafter, the method ends (block  3114 ). 
     The flowcharts of  FIG.  30    and  FIG.  31    include the steps that can be performed using the system  700  depicted in  FIG.  7   , including certain steps that can be carried out by the SEM  701  and certain other steps that can be carried out by the information handling system (IHS)  750  and its included processor  755  and storage  760 , both as described in detail herein. Instructions can be stored in storage  760  that, when executed by the processor, cause the processor to perform the methods disclosed herein and described by the flowcharts of  FIG.  30    and  FIG.  31   , in analogous fashion as other instructions stored in storage  760  that implement the inverse linescan model metrology tool  765  described herein. 
     12. Influence of Pixel Size and Magnification 
     With respect to the pixel size and magnification employed by SEM  701 ,  FIGS.  22 A and  22 B  show the biased and unbiased power spectral densities (PSDs), respectively, for a pattern of 16 nm lines and spaces for different magnifications and pixel sizes, assuming a white noise model. For a given number of frames of integration, changing the pixel size changes the electron dose per unit wafer area and the noise in the SEM image. Table 3 shows the measured 3σ linewidth roughness (LWR), as well as the other PSD parameters, for these different pixel size and magnification conditions. Under this range of conditions, the biased LWR varied by 0.63 nm (14%), while the unbiased LWR varied by only 0.07 nm (2%). The unbiased LWR is essentially unaffected by these metrology tool settings. Similar results are obtained for the measurement of LER and PPR. 
       FIGS.  22 A and  22 B  show power spectral densities as a function of pixel size and magnification. More particularly,  FIG.  22 A  shows the biased LWR PSD and  FIG.  22 B  shows the unbiased LWR PSD after noise has been measured and subtracted off. The SEM conditions for these results used a landing energy of 500 eV, 3 images per condition, and 16 nm resist lines and spaces. 
     TABLE 4 below shows the measured PSD parameters for the PSDs shown in  FIGS.  22 A and  22 B . 
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 Biased and unbiased 3σ LWR (nm) measurements 
               
               
                 as a function of pixel size and magnification. 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Pixel 0.8 nm 
                 Pixel 0.8 nm 
                 Pixel 0.5 nm 
                 Pixel 0.5 nm 
                 Pixel 0.37 nm 
               
               
                   
                 82kX 
                 164kX 
                 130kX 
                 264kX 
                 180kX 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Biased LWR 
                 5.10 
                 4.99 
                 4.67 
                 4.61 
                 4.47 
               
               
                 (3-sigma, nm) 
               
               
                 Unbiased LWR 
                 3.66 
                 3.65 
                 3.70 
                 3.67 
                 3.63 
               
               
                 (3-sigma, nm) 
               
               
                 Unbiased LWR 
                 15.95 
                 16.18 
                 17.2 
                 16.25 
                 16.35 
               
               
                 PSD(0) (nm 3 ) 
               
               
                 LWR Correlation 
                 5.08 
                 5.05 
                 5.31 
                 5.11 
                 5.38 
               
               
                 Length (nm) 
               
               
                   
               
            
           
         
       
     
     It has been found that the difference between biased and unbiased LWR is not constant, but varies with metrology tool settings, feature size, and process. Likewise, the ratio between biased and unbiased LWR varies with metrology tool settings, feature size, and process. TABLE 5 below shows the difference and ratio of biased to unbiased LWR for a variety of conditions. For these conditions, the ratio of biased to unbiased LWR varies from 1.09 to 1.66. The difference between biased and unbiased LWR varies from 0.32 nm to 2.19 nm in this particular example. 
     
       
         
           
               
             
               
                 TABLE 5 
               
             
            
               
                   
               
               
                 The relationship between biased and unbiased 
               
               
                 LWR for a variety of processes. 
               
            
           
           
               
               
               
            
               
                   
                 3σ LWR: 
                 3σ LWR (nm): 
               
               
                 Process 
                 Biased/Unbiased 
                 Biased − Unbiased 
               
               
                   
               
               
                 193i litho, 84 nm pitch, 500 V, 
                 1.20 
                 0.76 
               
               
                 512 rect pixels 
               
               
                 193i etch, 84 nm pitch, 800 V, 
                 1.14 
                 0.43 
               
               
                 512 rect pixels 
               
               
                 EUV litho, 32 nm pitch, 500 V, 
                 1.39 
                 1.44 
               
               
                 2048 0.8 nm pixels 
               
               
                 EUV litho, 32 nm pitch, 500 V, 
                 1.37 
                 1.34 
               
               
                 1024 0.8 nm pixels 
               
               
                 EUV litho, 32 nm pitch, 500 V, 
                 1.26 
                 0.97 
               
               
                 2048 0.5 nm pixels 
               
               
                 EUV litho, 32 nm pitch, 500 V, 
                 1.26 
                 0.94 
               
               
                 1024 0.5 nm pixels 
               
               
                 EUV litho, 32 nm pitch, 500 V, 
                 1.23 
                 0.84 
               
               
                 1024 0.37 nm pixels 
               
               
                 EUV litho, 36 nm pitch, 500 V, 
                 1.52 
                 1.86 
               
               
                 1024 0.8 nm pixels 
               
               
                 EUV litho, 32 nm pitch, 500 V, 
                 1.66 
                 2.19 
               
               
                 1024 rect pixels 
               
               
                 EUV etch, 32 nm pitch, 800 V, 
                 1.09 
                 0.32 
               
               
                 1024 rect pixels 
               
               
                   
               
            
           
         
       
     
     13. Edge Detection Embodiments 
       FIG.  23    is a flowchart that depicts a representative overall process flow that the disclosed SEM edge detection system employs to detect edges of a pattern structure. For discussion purposes, the process described in the flowchart of  FIG.  23    is applied to sample  2400  of  FIG.  24 A . Sample  2400  is a pattern structure that may also be referred to as pattern structure  2400 . The flowchart of  FIG.  23    includes the steps carried out by inverse linescan model metrology tool  765  to determine the edges of the pattern structure. 
     Process flow commences at start block  2300  of  FIG.  23   . As seen in  FIG.  7   , an information handling system (IHS)  750  is coupled to SEM  701  to receive SEM linescan image information from SEM  701 . IHS  750  includes a processor  755  and storage  760  coupled thereto. Storage  760  may include volatile system memory and non-volatile permanent memory such as hard drives, solid state storage devices (SSDs) and the like that permanently store applications and other information. Storage  760  stores the inverse linescan model (ILM) metrology tool  765  disclosed herein and described by the flowchart of  FIG.  23   . SEM  701  includes a controller (not shown) that IHS  760  instructs to perform image acquisition on pattern structure  800  and that provides linescan information from SEM  701  to IHS  750 . 
     As per block  2305 , SEM  701  sends an SEM image of pattern structure  800  to IHS  750 , and in response, IHS  750  loads this SEM image into system memory within storage  760 . IHS  750  preprocesses the pattern structure image from the SEM  701 , as per block  2310 . For example, this preprocessing of the loaded SEM image may include adjusting grayscale values and subtracting out background tilts of intensity levels. Optionally, as per block  2315 , IHS  750  may perform filtering of the loaded image, although this is generally not preferred. 
     In the case of a pattern structure such as the vertical lines and spaces seen in the pattern structure  2400  of  FIG.  24 A , the inverse linescan metrology tool  765  averages vertically over the axis of symmetry to generate an average linescan, as per block  2320 . An average linescan may be a grayscale value as a function of horizontal position wherein all of the vertical pixels have been averaged together. This averages out much of the SEM noise contained in the SEM image and produces a linescan that is more representative of the physical processes that generate a linescan without noise.  FIG.  24 B  shows a single linescan at one Y-pixel position.  FIG.  24 C  shows the averaged linescan that is generated by averaging over all Y-pixels. 
     While the example shown here is for vertical lines and spaces, any pattern with an axis of symmetry can be so processed to produce an average linescan. For example, long lines, long spaces, or long isolated edges can be so processed whenever the length of the line is sufficient to allow adequate averaging. Contact holes or pillars, with circular or elliptical symmetry, can also be averaged in a radial direction to produce an average linescan. 
     As per block  2325 , tool  765  calibrates the inverse linescan model to the averaged linescan that was obtained in the manner described above. It is noted that the linescan model includes two kinds of parameters, namely 1) parameters that depend upon the materials and the properties of the SEM, and 2) parameters that depend on the geometry of the feature on the sample. Tool  765  can calibrate all of these parameters. Tool  765  finds the best fit of the model to the average linescan of  FIG.  24 C , as per block  2325 . The values of the best fit parameters of the model are then the calibrated values. 
     That calibrated model is applied to a single linescan as shown in  FIG.  24 B . The best fit of the model to the single linescan of  FIG.  24 B  is found, however, in this case tool  765  fixes all of the parameters that relate to the materials and SEM imaging tool. In this scenario, tool  765  varies only the parameters related to the geometry of the feature of the pattern structure in order to find the best fit of the calibrated model to a single linescan. 
     In a simplified scenario, the only parameters varied in block  2330  would be the positions of the edges of the feature. In one embodiment, it is assumed that the vertical dimension of the feature exhibits a predetermined thickness and that only the edge positions of the feature are varying. Next, the calibrated inverse linescan model is fit to every single horizontal cut through the 2D image of the feature, as per block  2330 . We take the top horizontal row of pixels, and then the next row of pixels that are one pixel down, and then the next horizontal row of pixels down, and so forth. An example of one such single linescan is shown in  FIG.  24 B . The resulting best fit edge positions are the detected edges. 
     After the edges of the feature are detected in the manner described above, tool  765  may detect that the sample was rotated slightly during image acquisition, resulting in parallel tilted lines (that is, lines that are not perfectly vertical). Such tilting or rotation may contribute to inaccuracy of the detected edges by changing the average linescan and thus the calibrated ILM. Image rotation can be detected by fitting all the edges in the image to a set of parallel lines and determining their slope compared to vertical. If the slope is sufficiently different from the vertical case, the rotation should be removed. One possible criterion would be to compare the pixel position of the best fit line at the top of the image to the pixel position of the best fit line at the bottom of the image. If these pixel positions differ by some threshold, such as two pixels, then the image rotation is considered to be sufficiently large that its removal is required. 
     If such tilting/rotation is detected, as per block  2335 , then the prior calibration is considered to be a first pass calibration and calibration is repeated. More particularly, if such tiling/rotation is detected, the rotation is subtracted out by shifting some rows of pixels to bring the edges into vertical alignment, as per block  2345 , and calculating a new average linescan. Calibration of the model is then repeated as per block  2350  and  2325 . Another fitting is performed as well, as per block  2330 . Ultimately, tool  765  outputs geometry feature information (such as edge positions) describing the geometry of the feature that corresponds to the linescan image information provided to tool  765 . 
     Like image rotation, the roughness of the features themselves contributes inaccuracies to the calibration of the ILM. Optionally, after a first pass edge detection, each row of pixels can be shifted to not only subtract out image rotation, but to subtract out the feature roughness as well. The final result after the shifting of each row of pixels is a vertical edge where the edge position varies by less than one pixel from a perfect vertical line. These shifted rows of pixels can then be averaged vertically to produce a more accurate average linescan for use in ILM calibration. 
     In actual practice, information handling system  760  may include an interface  757  coupled between processor  755  and an output device  770  such as a display, printer, or other device so that the user may observe the feature edges determined by metrology tool  765 . Interface  757  may be a graphics interface, a printer interface, network interface, or other hardware interface appropriate for the particular type of output device  770 . 
     14. Assessing the Quality of Devices Using Unbiased Roughness Measurements 
     The measurement of roughness of various pattern structures can be used to assess the quality of the devices being fabricated. For example, the yield and/or performance of a device might depend on the magnitude of the roughness of one or more patterns that make up that device, as well as the frequency content of the roughness of those patterns. 
     The use of roughness measurements to assess device quality can be as simple as defining a “specification” for the roughness: the measured roughness of a specific target pattern must not exceed a specified value. The specification is set based on its relationship to device yield and/or performance. When devices the “meet” the specification (have roughness that is at or below the specification for the target patterns being measured), it is known or assumed that the device will have acceptable yield and performance. 
     Alternately, the measurement of roughness can serve as an input into a model that predicts device yield and/or performance. Such models could be run in real time, or results could be precomputed and placed in a table for look-up as needed. 
     For example, stochastic effects that give rise to edge roughness of a pattern structure also give rise the catastrophic detects such as the merging of two edges that should remain separate. For a pattern of lines and spaces, two neighboring lines can bridge the space between them creating a merger across the space. If those two lines are current-carrying wires, the result is a short circuit. If the two edges of a single line merge the result is a break in the line. If that line is current-carrying wire, the result is an open circuit. Defects such as these are labeled “catastrophic” since the occurrence of just one could render the device inoperable. Thus, a yield model for roughness could take the output of a roughness measurement of a pattern structure and predict the probability of catastrophic defect, and thus predict a stochastic-limited yield. 
     The use of biased roughness measurements to assess the quality of a device is problematic since the biased measurement can overestimate or underestimate the true roughness of the pattern structures that were measured. This in turn could lead to underestimation or overestimation of the device impact of that roughness. 
     Further, the bias in the roughness measurements in not necessarily fixed. The bias in the measurements can vary from measurement to measurement due to variations in the measurement tool, or due to variations in the pattern structure that do not affect the true roughness of the pattern. 
     For these reasons, it is preferred that the assessment of device quality be based on unbiased roughness measurements, where random and/or systematic errors in the measurement of roughness are removed. 
     Further, some aspects of device quality are sensitive to the frequency content of the roughness. Low frequency roughness behaves like an error in the mean feature width, edge position, or center-line position of the feature. Mid-frequency roughness can produce, for example, scattering of electrons flowing through a metal wire, increasing its resistance. For an optical waveguide, the roughness frequency that matter most depends on the wavelength of the light passing through the waveguide. For some devices, roughness at frequencies higher than a certain cut-off may not have any affect at all on the performance of the device. 
     Thus, it is desirable to produce an unbiased estimation of the true PSD behavior of the roughness as well. Integrating the unbiased PSD over a certain frequency range will provide an estimate of the magnitude of the roughness only over that frequency range, ignoring other frequencies of the roughness. This roughness over a set frequency range can then be used as a specification, or as an input to a device quality model. 
     Note that the measured pattern structure or structures need not duplicate exactly the form of the pattern structure or structures that make up the device of interest. The only requirement is that the pattern structures that are measured produce roughness measures that are predictive of device quality. 
     15. Assessing the Quality of a Process or Material or Tool Using Unbiased Roughness Measurements 
     Roughness measurements of a pattern structure or structures can be used to assess the quality of the processes and/or process materials and/or process tools used to fabricate that pattern structure. For example, repeated measurements of roughness can be used to determine temporal variations in roughness or spatial variations in roughness using standard assessment techniques. 
     The assessment of temporal variations in a process parameter is commonly accomplished through either time-series analysis or statistical process control (SPC). Both techniques can identify behavior that deviates from historical trends, thus indicating a process variation that might need attention or an action such as a process adjustment. A drift or an abrupt change in the magnitude of the roughness or in its frequency components could indicate a problem with the process, a problem with a material, or a problem with a tool used in the fabrication of the measure pattern structure. 
     Spatial analysis of roughness can be used to indicate a systematic spatial signature present in the fabrication process. For example, a variation across the substrate (such as a wafer) might indicate a problem with the uniformity of the etch process even though other metrics (such as a measurement of the average dimensions of the pattern structures) do not show a similar spatial signature or uniformity problem. A variation across the exposure field of a lithography tool might indicate a problem that is similarly unnoticed by other measurements. 
     Changes in a process material, such as a photoresist used in a lithography process, can result in a change in the measured pattern roughness, including its frequency behavior as exhibited by the power spectral density or other equivalent measure. 
     The use of roughness measurements to assess the quality of a process, process material, or process tool can be significantly degraded when the roughness measurements are biased by noise and/or systematic errors in the measurement to produce a biased roughness measurement. 
     Temporal and spatial variations in roughness include variations in the frequency components of the roughness. For example, a spatial variation in the correlation length (determined, for example, by measurement and analysis of the PSD) might indicate a variation in a temperature dependent process such as diffusion, which in turn might indicate a temperature uniformity problem during a baking step of the fabrication process. 
     Biased measurement can be higher or lower than the true value, depending on the source of bias and whether image filtering was used before edge detection, for example. Further, the bias in the roughness measurements in not necessarily fixed. The bias in the measurements can vary from measurement to measurement due to variations in the measurement tool, or due to variations in the pattern structure that do not affect the true roughness of the pattern. 
     As a result, the use of unbiased measurements for the assessment of the quality of a process, process material, or process tool is highly desirable. 
     Note that the measured pattern structure or structures need not duplicate exactly the form of the pattern structure or structures that make up the device or devices being fabricated by the process of interest. The only requirement is that the pattern structures that are measured produce roughness measures that are predictive of process, process material, or process tool quality. 
     16. Assessing the Quality of a Metrology Tool and Process Using Unbiased Roughness Measurements 
     The unbiased measurement of roughness necessarily entails the determination of the measurement bias, whether that bias is caused by random errors such as image noise and edge detection noise, or systematic errors such as distortion. As a result, the measurement bias is an output of an unbiased measurement of roughness. The determination of measurement bias can be used to assess the quality of the tool and process used to measure roughness. 
     For a given measurement tool, measurement process, and pattern structure to be measured, the roughness measurement bias should be a fixed quantity. Further, the edge detection metrology noise as well as the systematic errors such as measurement distortion should individually be constant. Thus, changes in these quantities could be an indication of a change in the ability to measure these quantities. By tracking the detected roughness measurement errors over time it is possible to assess changes in the measurement and to assess the quality of the measurement process and/or the measurement tool. 
     17. Controlling a Fabrication Process Using Unbiased Roughness Measurements 
     The above sections describe how unbiased measurements of roughness of a pattern structure can be used to assess the quality of the device that incorporates the pattern structure. Further, the above sections describe how unbiased measurements of roughness of a pattern structure can be used to assess the quality of the process, process materials, and process tools used to make the pattern structure. Further, the above sections describe how unbiased measurements of roughness of a pattern structure can be used to assess the quality of the of the metrology used to measure said roughness. 
     Once the quality of the devices, processes, materials, process tools, and measurement tools have been assessed, that assessment can be used to control the fabrication of the devices. Process control is a well-known application of measurement results including feedback control, feed-forward control, and advanced process control (APC) such as run-to-run control. 
     Feedback control uses a measurement result (or many measurement results) to determine a change in the fabrication process (such as a change in a setting, a change in a material, or a change in a tool) that would have produced a better result had those changes been implemented prior to the fabrication of the measured pattern structure. These changes are then implemented for the fabrication of future pattern structures under the assumption that they will affect the desire correction for the future results. 
     Feedforward control uses a measurement result (or many measurement results) to determine a change in a subsequent process step that could compensate for the errors measured in the current process step. 
     Advanced process control (APC) or run-to-run control uses feedforward or feedback loops very quickly so that changes (either forward or backward) can be implemented with very little delay, reducing the amount of product that is fabricated with the process exhibiting the measured error. 
     Unbiased roughness measurements enhance the efficacy of each of these control approaches. While the uses of biased roughness in feedforward, feedback, and APC control loops is possible, the results are often less than desired (and sometimes worse than no control at all) due to the biases in the measurements. Further, since the bias in the measurements can change, it is possible that a feedback loop, for example, could cause process changes in response to a change in measurement noise rather than a change in the actual pattern structure, thus making the process more unstable rather than more stable. 
     While the above embodiments describe the use of unbiased roughness measurements of a pattern structure or structures to assess and improve the quality of a fabrication process, the same approaches can be used to reduce the cost of manufacturing while keeping the quality of the fabrication process constant. For example, process quality assessment might determine that a process using a shorter etch time produces equivalent unbiased roughness measurement results as a process with longer etch times. Since the shorter etch time results in higher process throughput, the cost per device is reduced without a reduction in process quality. Biased roughness measurement may not provide this same result if the bias in the measurement changes with etch time. 
     As a second example, the results shown in  FIG.  21    show that reducing the number of frames of integration in a CD-SEM have a significant impact on biased roughness measurements for all cases, but has very little impact on unbiased roughness measurement at 8 frames or greater. The throughput of a SEM metrology tool is roughly inversely proportional to the number of frames of integration used, and so the cost of a measurement is roughly proportional to the number of frames of integration used in the measurement. By reducing the number of frames of integration, say, from 32 to 16, a significant reduction in metrology cost (close to a factor of 2) is achieved without loss of metrology precision or accuracy, but only if unbiased roughness measurements are used. 
     While the embodiments described above make reference to the measurement of structures found on semiconductor wafers, as used in the manufacture of semiconductor devices, the invention is not limited to these applications. The invention can be usefully employed to measure the roughness of feature edges found on flat panel displays, microelectromechanical systems, microfluidic systems, optical waveguides, photonic devices, and other electronic, optical, or mechanical devices. Further, the invention can be used to measure the feature edge characteristics of naturally occurring structures such as crystals or minerals, or manmade structures such as nanoparticles or other nanostructures. Further, the invention can be used to measure the feature edge characteristics of biological samples as well. 
     While the embodiments described above make reference to measurements using a scanning electron microscope, the invention is not limited to that imaging tool. Other imaging tools, such as optical microscopes, stimulated emission and depletion (STED) microscopes, x-ray microscopes, transmission electron microscopes (TEM), focused ion beam microscopes, and helium ion microscopes, can also be used. Other forms of microscopes, such as scanning probe microscopes (atomic force microscopes (AFM) and scanning near-field optical microscopes (SNOM), for example) can be used as well. 
     While the embodiments described above make reference to top-down images of nominally planar pattern structures to measure edge roughness, the invention is not limited to such pattern structure geometries. Three-dimensional structures, non-flat structures, curved surfaces, or tilted structures can be measured using this invention. Besides edge roughness, surface roughness can be measured and analyzed using similar techniques as described in this invention. 
     While the embodiments described above make reference to the measurement of roughness, the invention can be used to make other measurements as well. For example, highly accurate determination of pattern structure edges can be used in the measurement of feature width, feature placement, edge placement, and other similar measures. Contours of measured features can be used for many purposes, such as modeling or controlling the performance of the measured device. By collecting and statistically averaging the measurement of many samples, even greater accuracy (lower uncertainty) can be obtained. 
     17. Method/Apparatus for Determining Unbiased Local Critical Dimension Uniformity (LCDU), Local Pattern Placement Error (LPPE), and Local Edge Placement Error (LEPE) from Measurement of Biased Values. 
     Roughness metrics, such as linewidth roughness, line-edge roughness, and pattern placement roughness, are not the only stochastic metrics that can be biased by image noise. Image noise that leads to edge detection noise can also bias other metrics, such as the local critical dimension uniformity (LCDU), local pattern placement error (LPPE), and local edge placement error (LEPE), for example. 
     Critical dimension uniformity (CDU) is generally expressed as the standard deviation of the critical dimension (or sometimes, three times the standard deviation), based on some sampling plan for choosing which features to measure. For example, a sampling of nominally identical features spaced across a wafer will allow for the measurement of the across-wafer CDU. A sampling of features across the exposure field of a lithography tool will result in across-field CDU. Many other samplings may be possible. These versions of CDU may be referred to as global CDU (GCDU) due to the relatively large distances between the sampled features. Given large enough sample sizes the GCDU may not be influenced by stochastic effects. 
     GCDU may be caused by longer-range variations in the lithography process, such as across-wafer variation in film thicknesses or properties, across-wafer variation in the resist development process, or across-wafer variation of temperature during a resist baking step, among other factors. Across-lithography-field variations can be caused by dose or focus variations in the lithography tool or aberrations that vary across the lithography tool&#39;s imaging field. These causes of variation are not stochastic—that is, they are not caused by the fundamental randomness that exists at the molecular scale—but instead may be caused by non-stochastic errors in the patterning process. 
     When measuring the CDU for nominally identical features that are very close to each other, such as the features found in one field-of-view (FOV) of a SEM image, these features may be influenced by the longer-range effects that influence GCDU uniformly, so that every feature within the SEM FOV may be influenced approximately the same by these global effects. A typical SEM FOV can be between a few hundred nanometers on a side to a few tens of microns on a side, with about one micron on a side very common. Variations of CD (for features that are nominally identical) over distances found in one SEM image FOV or shorter may be referred to as local CDU (LCDU). Thus, LCDU may be measured as the standard deviation (sometimes multiplied by three) of the features found in one SEM FOV. The standard deviation represents the variation of data about a mean. For LCDU, the mean being used is the mean CD for that single SEM image. 
     Multiple SEM images of nominally the same types of features can be used to measure both GCDU and LCDU. For each SEM image, both a mean and a standard deviation of the CDs of the features within that image are calculated. The standard deviation of the means of each image may be the GCDU (the image-to-image variation of the image mean CD). The mean of the standard deviations of each image may be the LCDU. In some instances, the mean of the variances of each image may be calculated, and the square root of this mean variance may be used as the LCDU. 
     SEM image noise may result in edge detection noise, and this in turn may result in measurement uncertainty in the measurement of CD from that SEM image. For example, the mean CD of a line feature may be determined by measuring the width of the line at N points along its length, then finding the average of those N width measurements. For example, for a vertically oriented line, the edges may be detected and the feature width along each row of pixels in the SEM image may be measured. Alternatively, measurements of every other row of pixels, or some other sampling along the length of the line, may be measured. If σ noise  is the uncertainty in measuring the width at one point along the line (caused by edge detection noise), and each measurement of the N points along the line are statistically independent, the standard error of the mean CD (the uncertainty in the mean CD caused by edge detection noise) may be represented as follows: 
     
       
         
           
             
               S 
               ⁢ 
               
                 E 
                 ⁡ 
                 ( 
                 
                   mean 
                   ⁢ 
                       
                   CD 
                 
                 ) 
               
             
             = 
             
               
                 σ 
                 noise 
               
               
                 N 
               
             
           
         
       
     
     In some embodiments, if the N measurement points along the line represent different rows of pixels, and no filtering or other image processing is applied, then each of the N measurements may be statistically independent and the above formula can be used. Other specific definitions of mean CD determined from one feature on one SEM image can also be used, with different specific influence of edge detection noise on the standard error of the mean CD. The proper value of N to use in the above equation will depend on the specifics of the measurement of mean CD. 
     Uncertainty in the measurement of one mean CD may bias the measurement of LCDU. If LCDU true  is the true value of the LCDU (without edge detection measurement noise) and LCDU meas  is the measured value of the LCDU (including measurement noise), the bias adds to the true value to get the measured value represented as follows: 
       LCDU meas   2 =LCDU true   2   +SE (mean  CD ) 2    
     It may be desirable to know the true value of the LCDU. This may be obtained by subtracting the measurement uncertainty contribution to the measured LCDU using the above equation. As described above, PSD analysis for LWR measurement may be used to determine the measurement noise floor of the PSD, which in turn results in a measurement of σ noise , the same quantity used in unbiasing LWR. 
     The magnitude of the LCDU bias may be a function of N, the number of measurement points used to determine the mean CD of a single feature. In turn, N is a function of the length of the line being measured. For long lines (large N), the bias in LCDU measurement may be quite small. But for short lines (small N), the bias can be significant. While the above discussion refers to lines, the same principles apply to the measurement of spaces as well. 
     In general, short lines have larger LCDU bias than long lines due to the smaller value of N. But under similar SEM measurement conditions, short lines and long lines have about the same σ noise . Further, σ noise  may be more easily determined from longer lines since the analysis of the PSD of long lines may provide a more accurate assessment of a noise floor than the analysis of the PSD of shorter lines. In some embodiments, σ noise  is measured for long lines (or spaces) and applied to unbiasing the LCDU of shorter lines (or spaces). 
     When a line is so short that its length is similar to its width, the feature is often called a pillar. In some embodiments, “similar” in this instance may refer to a ratio of width/height of the feature between 0.33 and 3. Likewise, when a space is so short that its length is similar to its width, that feature is often called a contact hole (or via). Holes and pillars may have the greatest amount of LCDU bias since they represent the “shortest” features that may be measured. Often holes and pillars are round in shape, and edge detection may be performed as a function of polar angle about the center of the hole or pillar instead of as a series of horizontal measurement slices spaced apart in the vertical direction (rows of pixels, for example) as is done for lines and spaces. The same principles as discussed above for lines and spaces still apply to holes and pillars in terms of LCDU bias and the method for unbiasing the LCDU. 
     For holes and pillars, the determination of σ noise  may be more complicated if the feature edges are detected as a function of the polar angle instead of along horizontal rows of pixels. In such a case, edge detection at any given angle generally may involve interpolation between pixels. For such a case, neighboring points along the feature edge may not have statistically independent edge detection noise. Interpolation between pixels is an averaging of neighboring pixels, which is similar to filtering. The PSD from the feature edges of a hole or pillar detected in this way may exhibit a pink noise floor (that is, a noise floor that is not flat) rather than a white noise floor (that is, a noise floor that is flat). 
     For the case of a hole or pillar with a pink noise floor, the noise floor may be determined using a pink noise model and the σ noise  level can be determined. For such a case, the number of statistically independent edge detection points (the value of N used to determine the standard error of the feature&#39;s mean CD) may be less than half the number of edge points around the circumference of the feature. In this case, an effective number of edge points may be used to unbias the hole or pillar LCDU. 
     While the previous discussion describes a method for unbiasing the measurement of LCDU, the same approach may be used for other metrics, such as the local pattern placement error (LPPE) and the local edge placement error (LEPE). 
     As described above, unbiased PSD for very long lines and spaces can be used to predict the unbiased LCDU, LPPE, and LEPE for shorter lines and spaces. These predicted values of LCDU, LPPE, or LEPE can be used to validate or supplement the unbiasing procedure for measured LCDU, LPPE, or LEPE. 
     One type of double patterning can be used to make holes or pillars by crossing lines and spaces from different patterning steps at an angle with respect to each other (for example, but not limited to, perpendicular lines and spaces). The intersection of the lines or the spaces produces either pillars or holes depending on the details of the process. For example, crossed spaces can be used to make holes. 
     For such a double patterning process, measuring the unbiased PSDs of the lines and spaces of the two patterning steps can be used to predict the unbiased LCDU, LPPE, and LEPE of the holes or pillars that result. For example, the first patterning step may produce a feature width of w 1  and the second patterning step may produce a feature of width w 2 . The width of the second patterning step feature may be used as the length of the first patterning step feature for the purpose of predicting LCDU of the resulting hole or pillar feature in a direction perpendicular to the line direction of the first patterning step. Likewise, the width of the first patterning step feature may be used as the length of the second patterning step feature for the purpose of predicting LCDU of the resulting hole or pillar feature in a direction perpendicular to the line direction of the second patterning step. 
     Turning now to  FIGS.  32  and  33   , the flowcharts of  FIG.  32    and  FIG.  33    include the steps that can be performed using the system  700  depicted in  FIG.  7   , including certain steps that can be carried out by the SEM  701  and certain other steps that can be carried out by the information handling system (IHS)  750  and its included processor  755  and storage  760 , both as described in detail herein. Instructions can be stored in storage  760  that, when executed by the processor, cause the processor to perform the methods disclosed herein and described by the flowcharts of  FIG.  32    and  FIG.  33   , in analogous fashion as other instructions stored in storage  760  that implement the inverse linescan model metrology tool  765  described herein. 
       FIG.  32    is a flowchart that depicts a representative process flow to determine unbiased parameters from biased parameters and measurement of edge detection noise. 
     The method  3200  starts (block  3202 ) and determines, by a processor  755 , a measurement of edge detection noise. The method proceeds to receive a measurement of a biased parameter (LCDU, LPPE, LEPE, etc.) including measurement noise (block  3204 ). Based on the measurement of edge detection noise and a number of measurement points, the method determines a contribution of edge detection noise to the biased parameter (block  3206 ). This contribution of noise to one feature parameter is generally the edge detection noise divided by the square root of the number of independent edge measurement points that contribute to the one feature parameter. The method may determine an unbiased parameter by subtracting the contribution of noise from the biased parameter including the measurement noise (block  3208 ). The method may proceed to output the unbiased parameter (block  3210 ). Thereafter, the method ends (block  3210 ). 
       FIG.  33    is a flowchart that depicts a representative process flow to predict unbiased parameter for short features from measurement of unbiased power spectral density for long features. The method begins at block  3302  and receives an image of a semiconductor device. The method proceeds to determine, based on the image, one or more measurements of unbiased power spectral density data for a first feature (e.g., a long line or space) included in the image (block  3304 ). The first feature may be associated with a first property (e.g., a shape, a size, etc.), for example, a line with a long length. The method may predict, based on the one or more measurements of unbiased power spectral density data for the first feature, an unbiased parameter for a second feature (e.g., a hole or a pillar, or a short line or space) associated with a second property (e.g., a shape, a size, etc.), for example, a line that is shorter than the first feature (block  3306 ). The method may output the predicted unbiased parameter for the second feature (block  3308 ). 
     An unbiased parameter from a shorter feature can be predicted from an unbiased PSD associated with a longer feature as follows. The PSD may represent variance of a specific parameter (such as linewidth, edge position, or feature center position) per unit frequency, where frequency is one over a length along the feature (that is, perpendicular to the edge). High frequencies may represent short length scales, whereas low frequencies may represent long length scales. The area under the PSD is the total variance of the parameter. Thus, three multiplied by the square root of the area under the linewidth PSD may be a measure of the LWR for that feature. Using this PSD from a longer feature, the LWR of a shorter feature can be predicted by integrating the PSD from the highest frequency down to a frequency given as one over the length of the shorter feature. 
     Further, predictions of other unbiased parameters for this shorter feature can also be made using the so-called “conservation of roughness” principle: 
       LWR long feature   2 =LWR short feature   2 +LCDU short feature   2    
     The unbiased LWR of the long feature can be determined directly by measurements of that long feature as described above. Further, the unbiased PSD of this long feature can also be determined and used to predict the LWR of the short feature, as described above. Finally, the predicted unbiased LWR of the shorter feature can be subtracted (in quadrature) from the unbiased LWR of the long feature to determine the unbiased LCDU of the short feature using the conservation of roughness equation. 
     While the previous example relates unbiased LWR for a long feature to the prediction of unbiased LWR and unbiased LCDU for a short feature, other unbiased parameters can be predicted as well. Using measurement of pattern placement roughness (PPR) of a long feature, the unbiased LPPE of a short feature can be predicted in the same way. Likewise, using measurement of LER of a long feature, the unbiased LEPE of a short feature can be predicted. Further, as described above, crossed lines or spaces are sometimes used to make holes or pillars in a multiple dpatterning process. The approach used here to predict the LWR or LER or PPR of shorter lines or spaces can be used to predict the LCDU, LPPE, and LEPE of holes or pillars from these crossed lines or spaces. 
     In some embodiments, the system  700  may be used to control a lithography tool as described further herein. The unbiased parameter may be accounted for to change one or more operating parameters or a process window associated with controlling a lithography tool to manufacture a semiconductor device. As described previously, unbiased parameters can be used for process monitoring, statistical process control, and also in feedforward, feedback, and APC control loops for process control. As an example, the measurement of LCDU, LPPE, or LEPE can be combined with the measurement of global CDU and overlay when making a decision to rework a wafer or lot. In such a case, the use of biased LCDU, LPPE, or LEPE could result in poor decisions: reworking a wafer that does not need to be, or passing a wafer that should be reworked. 
     Consistent with the above disclosure, the examples of systems and methods enumerated in the following clauses are specifically contemplated and are intended as a non-limiting set of examples. 
     Clause 1. A method, comprising: 
     determining, by a processor, a measurement of edge detection noise; 
     receiving a measurement of a biased parameter including measurement noise; 
     based on the measurement of edge detection noise and a number of measurement points, determining a contribution of edge detection noise to the biased parameter; 
     determining an unbiased parameter by subtracting the contribution of noise from the biased parameter including the measurement noise; and 
     outputting the unbiased parameter. 
     Clause 2. The method of any clause herein, wherein the number of measurement points represent rows of pixels in a selected portion of a scanning electron microscope image. 
     Clause 3. The method of any clause herein, wherein determining the measurement of edge detection noise comprises determining a measurement of noise floor of a power spectral density dataset. 
     Clause 4. The method of any clause herein, wherein: 
     the biased parameter comprises a biased local critical dimension uniformity (LCDU), a biased local pattern placement error (LPPE), or a biased local edge placement error (LEPE), and 
     the unbiased parameter comprises an unbiased LCDU, an unbiased LPPE, or an unbiased LEPE. 
     Clause 5. The method of any clause herein, wherein the measurement points comprise a line or the measurement points comprise a space. 
     Clause 6. The method of any clause herein, wherein: 
     the number of measurement points represent a first line or first shape, and 
     the measurement of edge detection noise is determined using a power spectral density analysis of a second line or second shape having a length or size greater than the first line or first shape. 
     Clause 7. The method of any clause herein, wherein the measurement points represent a hole or pillar comprising a shape having a length similar to its width, and the determining the measurement of edge detection noise further comprises detecting feature edges as a function of a polar angle. 
     Clause 8. A method, comprising: 
     receiving an image of a semiconductor device; 
     determining, based on the image, one or more measurements of unbiased power spectral density data for a first feature included in the image, wherein the first feature is associated with a first property; 
     predicting, based on the one or more measurements of unbiased power spectral density data for the first feature, an unbiased parameter for a second feature associated with a second property, wherein the first property is associated with a value greater than the second property; and 
     outputting the predicted unbiased parameter for the second feature. 
     Clause 9. The method of any clause herein, wherein the first and second properties comprise a length or a size. 
     Clause 10. The method of any clause herein, wherein the first and second features comprise a line or a space. 
     Clause 11. The method of any clause herein, wherein: 
     the unbiased parameter comprises an unbiased LCDU, an unbiased LPPE, or an unbiased LEPE. 
     Clause 12. The method of any clause herein, wherein determining the one or more measurements of unbiased power spectral density data for the first feature having the first property further comprises: 
     determining a measurement of noise floor. 
     Clause 13. An apparatus, comprising: 
     a memory device storing instructions; and 
     a processing device communicatively coupled to the memory device, wherein the processing device executes the instructions to:
         determine a measurement of edge detection noise;   based on the measurement of edge detection noise and a number of measurement points, determine the contribution of edge detection noise to a biased parameter;   receive a measurement of a biased parameter including measurement noise;   determine an unbiased parameter by subtracting the contribution of noise from the biased parameter including the measurement noise; and   output the unbiased parameter.       

     Clause 14. The apparatus of any clause herein, wherein the number of measurement points represents the rows of pixels in used portion of a scanning electron microscope image. 
     Clause 15. The apparatus of any clause herein, wherein determining the measurement of edge detection noise comprises determining a measurement of noise floor of a power spectral density dataset. 
     Clause 16. The apparatus of any clause herein, wherein: 
     the biased parameter comprises a biased local critical dimension uniformity (LCDU), a biased local pattern placement error (LPPE), or a biased local edge placement error (LEPE), and 
     the unbiased parameter comprises an unbiased LCDU, an unbiased LPPE, or an unbiased LEPE. 
     Clause 17. The apparatus of any clause herein, wherein the measurement points comprise a line or the measurement points comprise a space. 
     Clause 18. The apparatus of any clause herein, wherein: 
     the number of measurement points represent a first line or first shape, and 
     the measurement of edge detection noise is determined using a power spectral density analysis of a second line or second shape having a length or size greater than the first line or first shape. 
     Clause 19. The apparatus of any clause herein, wherein the measurement points represent a hole or pillar comprising a shape having a length similar to its width, and the determining the measurement of edge detection noise further comprises detecting feature edges as a function of a polar angle. 
     Clause 20. The apparatus of any clause herein, wherein determining the one or more measurements of unbiased power spectral density data for the first feature having the first property further comprises: 
     determining a measurement of noise floor. 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
     Note that not all of the activities described above in the general description or the examples are required, that a portion of a specific activity may not be required, and that one or more further activities can be performed in addition to those described. Still further, the order in which activities are listed are not necessarily the order in which they are performed. 
     It can be advantageous to set forth definitions of certain words and phrases used throughout this patent document. The term “communicate,” as well as derivatives thereof, encompasses both direct and indirect communication. The terms “include” and “comprise,” as well as derivatives thereof, mean inclusion without limitation. The term “or” is inclusive, meaning and/or. The phrase “associated with,” as well as derivatives thereof, can mean to include, be included within, interconnect with, contain, be contained within, connect to or with, couple to or with, be communicable with, cooperate with, interleave, juxtapose, be proximate to, be bound to or with, have, have a property of, have a relationship to or with, or the like. The phrase “at least one of,” when used with a list of items, means that different combinations of one or more of the listed items can be used, and only one item in the list can be needed. For example, “at least one of: A, B, and C” includes any of the following combinations: A, B, C, A and B, A and C, B and C, and A and B and C. 
     The description in the present application should not be read as implying that any particular element, step, or function is an essential or critical element that must be included in the claim scope. The scope of patented subject matter is defined only by the allowed claims. Moreover, none of the claims invokes 35 U.S.C. § 112(f) with respect to any of the appended claims or claim elements unless the exact words “means for” or “step for” are explicitly used in the particular claim, followed by a participle phrase identifying a function. Use of terms such as (but not limited to) “mechanism,” “module,” “device,” “unit,” “component,” “element,” “member,” “apparatus,” “machine,” “system,” “processor,” or “controller” within a claim is understood and intended to refer to structures known to those skilled in the relevant art, as further modified or enhanced by the features of the claims themselves, and is not intended to invoke 35 U.S.C. § 112(f). 
     Benefits, other advantages, and solutions to problems have been described above with regard to specific embodiments. However, the benefits, advantages, solutions to problems, and any feature(s) that can cause any benefit, advantage, or solution to occur or become more pronounced are not to be construed as a critical, required, or essential feature of any or all the claims. 
     After reading the specification, skilled artisans will appreciate that certain features are, for clarity, described herein in the context of separate embodiments, can also be provided in combination in a single embodiment. Conversely, various features that are, for brevity, described in the context of a single embodiment, can also be provided separately or in any subcombination. Further, references to values stated in ranges include each and every value within that range. 
     The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.