Patent Publication Number: US-11664188-B2

Title: Edge detection system

Description:
CROSS REFERENCE TO RELATED PATENT APPLICATIONS 
     This patent application is a continuation of U.S. application Ser. No. 16/716,131, filed Dec. 16, 2019, titled “Edge Detection System”, 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.” Both applications are incorporated by reference herein as if reproduced in full below. 
    
    
     BACKGROUND 
     The disclosures herein relate generally to edge detection of pattern structures, and more particularly, to edge detection 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. 
     BRIEF SUMMARY 
     In one embodiment, an edge detecting system is disclosed that includes an imaging device that images a pattern structure which includes a predetermined feature to form a first image. The imaging device images the pattern structure to generate measured linescan information that includes image noise. The edge detection system also includes a processor that is coupled to the imaging device. The processor receives the measured linescan information including image noise from the imaging device. In one embodiment, the processor is configured to apply the measured linescan information to an inverse linescan model that relates measured linescan information to feature geometry information. The processor is further configured to determine, from the inverse linescan model, feature geometry information that describes feature edge positions of the predetermined feature of the pattern structure that corresponds to the measured linescan information. In one embodiment, the processor is further configured to form a second image of the pattern structure, the second image including a representation of the feature geometry information. In one embodiment, an output device is coupled to the processor to receive the second image from the processor. 
     In another embodiment, an edge detection system is disclosed that includes a scanning electron microscope (SEM) that images a pattern structure which includes a predetermined feature to form a first image. The imaging device is configured to scan the pattern structure to generate measured linescan information that includes image noise. In one embodiment, the edge detection system includes a processor, coupled to the scanning electron microscope (SEM), that receives the measured linescan information including image noise from the SEM. In one embodiment, the processor is configured to average the measured linescan information along an axis of symmetry of the feature to provide an averaged linescan, the processor being also configured to calibrate an inverse linescan model to the averaged linescan to form a calibrated inverse linescan model that relates measured linescan information to feature geometry information. The processor is further configured to apply the measured linescan information to the calibrated inverse linescan model. The processor is still further configured to fit the calibrated inverse linescan model to each horizontal scan across the pattern structure to determine feature edge positions corresponding to each horizontal scan. 
     In one embodiment, a method is disclosed that includes forming a first image, by an imaging device, of a pattern structure exhibiting a predetermined feature. The imaging device images the pattern structure to generate measured linescan information that includes image noise. The method also includes applying the measured linescan information to an inverse linescan model that relates measured linescan information to feature geometry information. The method further incudes determining, from the inverse linescan model, feature geometry information that describes feature edge positions of the predetermined feature of the pattern structure that corresponds to the measured linescan information. The method may also include displaying, via an output device, a second image that depicts the feature geometry information. In one embodiment, the imaging device may be a scanning electron microscope (SEM). The method may also include averaging the measured linescan information over an axis of symmetry of the predetermined feature to provide an averaged linescan. The method may still further include detecting at least one of tilt and rotation in the feature or features after the feature edge positions are determined, and in response to such detection, repeating the calibrating of the inverse linescan model to the averaged linescan if needed. 
     In another embodiment, a method is disclosed that includes scanning, by a scanning electron microscope (SEM), a pattern structure exhibiting a predetermined feature to form a first image of the pattern structure, wherein the SEM performs a plurality of horizontal scans across the pattern structure at different Y positions to generate measured linescan information that includes image noise. The method also includes averaging the measured linescan information along an axis of symmetry of the feature to provide an averaged linescan. The method further includes calibrating an inverse linescan model to the averaged linescan to form a calibrated inverse linescan model that relates measured linescan information to feature geometry information. The method still further includes applying the measured linescan information to the calibrated inverse linescan model. The method also includes fitting the calibrated inverse linescan model to each horizontal scan across the pattern structure to determine feature edge positions corresponding to each horizontal scan. The method may still further include detecting at least one of tilt and rotation in the feature or features after the feature edge positions are determined, and in response to such detection, repeating the calibrating of the inverse linescan model to the averaged linescan if needed. The method may also include displaying, by an output device, a second image that depicts the feature geometry information. The method may further include averaging the measured linescan information vertically over an axis of symmetry of the predetermined feature to provide an averaged linescan. 
    
    
     
       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    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. 
     
    
    
     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. A goal of this invention is 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. 
     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 detect edges in very noisy SEM images without the use of image filtering. 
     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 are believed to provide the first 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 below. 
     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. 
     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 true roughness adds to the measurement noise 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. 
     2. The Frequency Dependence of Line-Edge Roughness (LER), LineWidth 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 σ 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 LIξ. 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 
                     
                   
                   &gt;&gt; 
                   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. 
     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  701  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 could 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 
                   
                   , 
                   
                     
                       
                         SE 
                         ⁡ 
                         ( 
                         x 
                         ) 
                       
                       
                         SE 
                         ⁡ 
                         ( 
                         
                           - 
                           ∞ 
                         
                         ) 
                       
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           α 
                           f 
                         
                         ⁢ 
                         
                           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 secondaries 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 
                     
                     &gt; 
                     0 
                   
                   , 
                   
                     
                       
                         SE 
                         ⁡ 
                         ( 
                         x 
                         ) 
                       
                       
                         SE 
                         ⁡ 
                         ( 
                         ∞ 
                         ) 
                       
                     
                     = 
                     
                       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. 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 ora 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  3 σ 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 that pixel, relative noise levels 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 8.
 
σ biased   2 =σ unbiased   2 +σ noise   2   EQUATION 8
 
     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 8 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. Since an unbiased estimate of the feature roughness is desired, the measured roughness is corrected by subtracting an estimate of the noise term. 
     While several approaches for estimating the SEM 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. This smoothing makes edge detection significantly more robust when image noise is high.  FIG.  17    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.  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 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. As will be described next, the use of image filtering makes measurement and subtraction of image noise impossible.  FIG.  18    shows power spectral densities averaged from 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. 
     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.  4   . Correlations reduce high-frequency roughness so that the roughness becomes very small over very small length scales. SEM image noise, on the other hand, can be reasonably assumed to be white noise, so that the noise PSD is flat. Other models of the SEM image noise are also possible, for example using pixel-to-pixel correlation to describe the noise. 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), white SEM 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 image 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. It is common for SEM images to have noise behavior that is uncorrelated with its neighboring pixels in the Y (non-scan) direction, but slightly correlated with neighboring pixels in the X (scan) direction. If a correlation model is assumed or measured (for example, an exponential autocorrelation of noise in the X direction), a suitable noise expression for the PSD can be used to replace EQUATION 9. 
       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. 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   . 
     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. 
     9. Sensitivity to Metrology Tool Settings 
     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 may 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 frames of integration. The cases of 6 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. 
       FIG.  21    shows the biased and unbiased values of the 3σ linewidth roughness measured as a function of the number of frames of integration. 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 pixel noise model, such as an exponential autocorrelation noise model, can produce better noise subtraction especially for the low frames of integration. While the results shown are for LWR, similar results are obtained for the measurement of line edge roughness (LER) and pattern placement roughness (PPR). 
     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. 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 3 below shows the measured PSD parameters for the PSDs shown in  FIGS.  22 A and  22 B . 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 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 (3- 
                 5.10 
                 4.99 
                 4.67 
                 4.61 
                 4.47 
               
               
                 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 4 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 4 
               
             
            
               
                   
               
               
                 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 
                   
                   
               
               
                   
               
            
           
         
       
     
       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 . 
     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. 
     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. 
     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.