Microsc. Microanal. 21, 771–777, 2015 doi:10.1017/S1431927615000343
© MICROSCOPY SOCIETY OF AMERICA 2015
A Simple Metric for Determining Resolution in Optical, Ion, and Electron Microscope Images Alexandra E. Curtin,* Ryan Skinner, and Aric W. Sanders Boulder Laboratories, National Institute of Standards and Technology, Boulder, CO 80305, USA
Abstract: A resolution metric intended for resolution analysis of arbitrary spatially calibrated images is presented. By fitting a simple sigmoidal function to pixel intensities across slices of an image taken perpendicular to light–dark edges, the mean distance over which the light–dark transition occurs can be determined. A fixed multiple of this characteristic distance is then reported as the image resolution. The prefactor is determined by analysis of scanning transmission electron microscope high-angle annular dark field images of Si . This metric has been applied to optical, scanning electron microscope, and helium ion microscope images. This method provides quantitative feedback about image resolution, independent of the tool on which the data were collected. In addition, our analysis provides a nonarbitrary and self-consistent framework that any end user can utilize to evaluate the resolution of multiple microscopes from any vendor using the same metric. Key words: microscopy, helium ion microscopy, optical microscopy, Rayleigh criterion
I NTRODUCTION Although one of the most-quoted properties of a microscope, the term “resolution” can have multiple meanings and can be characterized by many different parameters. As they are often based on the specific physics of each microscopy method, these criteria are not easily generalized. In addition, many of the established metrics require precise knowledge of the subject geometry, which adds both complexity and additional overhead spent characterizing the sample. Owing to the myriad of resolution criteria used, comparison of the performance of different imaging devices tends to be subjective. At the very least, a degree of ambiguity is introduced when attempting to convert data from one metric to another. In this study, we developed a simple, reliable, and universal metric, which can be used to evaluate images quickly. Our approach is based on the fact that edges separating light and dark regions contain information about the resolving power of the imaging device. By performing our resolution analysis on the image itself, we have de-coupled our method from a particular tool or collection method, allowing the application of our resolution metric to be widely applicable. We will show that our analysis is quantitative, applicable to optical, transmission electron microscopy (TEM), and scanning electron/ ion microscope images, and produces results consistent with the Rayleigh criterion where appropriate. When a manufacturer of scientific equipments sells a microscope to a laboratory, the performance of that microscope is judged by its best attainable resolution. The resolution quoted by the manufacturer is measured by various means under ideal laboratory conditions and with ideal samples. Received October 1, 2014; accepted February 24, 2015 *Corresponding author. [email protected]
The analytical means used by vendors to determine resolution from these images vary greatly with many relying on contrastto-gradient methods. A simple method for translating this performance to images collected in imperfect laboratory conditions and from specimens with different scattering properties becomes a powerful tool for scientists trying to analyze their own data and independently evaluate the image resolutions claimed by multiple vendors. Our work presents a nonarbitrary, analytical platformindependent means of fitting light/dark ed, ges in microscope images from the micrometer down to the picometer scale. Various methods of fitting edges in an image to a function have been used in the past to determine image resolution (Ishitani & Sato, 2004). We have chosen to use a sigmoidal curve, or logistic function, to fit our light/dark edges. The sigmoidal curve is more simply applied than the error function, relying on only a single fit parameter to describe edge sharpness. The sigmoidal on its own reports the width of an edge, but we cannot know quantitatively whether this is an accurate estimate of image resolution. Our method extends past this, however, by tying the sigmoidal fit to the fit of a known sample. Scanning transmission electron microscopy high-angle annular dark field (STEM-HAADF) images of atomic dumbbells present a wealth of features with known spacings, but these are not easily fit with an odd function. Instead, we use a Gaussian fit of atomic dumbbells. Parameters from this fit, when analytically tied to the sigmoidal function, provide a prefactor to fix the sigmoidal fit parameter in the real world. It is important to note here that there are at least two methods including Fourier analysis for assessing the resolution of a STEM image. Our intent here is not to supplant these methods, but rather to use our STEM images of known dumbbell spac as a means of solving for the proportionality
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Alexandra E. Curtin et al.
constant or prefactor our sigmoidal fit parameter. The sigmoidal fit can then be applied to lower magnification images. As a second check to our method, when we apply the sigmoidal fit to optical images, we can directly compare the fit to the Rayleigh criterion resolution limit for the particular microscope conditions. The Rayleigh criterion is one of the most well-known metrics for determining the minimum resolvable detail in diffraction-limited systems. When radiation from a point source diffracts through a circular aperture, the resulting pattern that falls on a detector is an Airy disk. When two such point sources are present, as when viewing a double star through a telescope, the Rayleigh criterion defines the points to be “just resolved” when the maximum of one Airy disk intersects the first minimum of the other. A second resolution condition commonly mentioned in tandem with the Rayleigh criterion is the Sparrow criterion. The Sparrow criterion states that the minimum resolvable distance is reached when the dip is no longer distinguishable. However, as the Sparrow criterion is vulnerable to noise in the acquired image, the Rayleigh criterion provides us with the most mathematically rigorous metric for a diffraction-limited system. For our purposes, we will compare the quality of our fit of light/dark transitions to the Rayleigh criterion for optical images of different magnifications. The Rayleigh criterion provides standard and sensible ways of defining resolution, but it is only applicable to specific diffraction-limited systems. In the fields of electron and ion microscopy, resolution of imaging systems is often limited by spherical and chromatic aberrations, as well as by astigmatism, before diffraction effects begin to degrade image quality. We will use this criterion later to check the effectiveness of our image analysis for images derived from our optical microscope. For scanning electron microscopy (SEM) images, there are thoroughly developed Fourier transform techniques for determining image sharpness, which could be extended to characterize the resolution (Postek & Vladár, 1998; Ishitani & Sato, 2002; Joy et al., 2010). This approach takes advantage of the fact that higher-frequency components of the image are maximized when the subject is in focus with minimal astigmatism. The highest spatial frequencies from the image correspond to the sharpest edges, and are presumably correlated with image resolution. Although we will not compare this method in this study, it is worth noting the effectiveness of Fourier methods for certain types of images. Where our analysis is easily applied to images containing straight light/ dark transitions, the well-known contrast transfer function and similar methods work well for SEM images containing high spatial frequency detail.
M ETHODS We considered image resolution based on the sharpness of image features with strong light/dark edge contrast. To quantify “sharpness” we fit a selected light/dark edge to a sigmoidal line profile. Every line profile crossing the edge is fit,
and the resulting curves were averaged. Our sigmoidal curve of choice, the logistic function, is defined as follows: a : (1) S ðx Þ ¼ c + 1 + Exp - xb+ x0 Here, S(x) describes the amplitude of the function, and a is a scaling factor. The parameters c and x0 provide x- and y-offsets, respectively. The logistic function is ideal for such a fit, as it applies a curve to the region of contrast transition but does not require a round feature or particle. The units are arbitrary as they are taken from the gray-scale range of the image. The relevant parameter for defining image resolution in the logistic function is b. We will show that b is proportional to the resolution of the image, such that resolution = m × b, where m will be the prefactor tying the width of the sigmoidal, b, to a quantitative statement about resolution. Without an empirically determined value for m—that is, without determining the relationship between the fit parameter b and the actual resolution of an image—we leave room for our edge fits to produce nonphysical results. Therefore, we turn to experimental methods to determine m, and then compare the calculated resolution of optical images to the Rayleigh criterion. STEM-HAADF imaging provides direct images of crystal structures for measuring known distances in an electron microscope. In particular, attempting to resolve columns of atoms from materials along different crystallographic directions, with different spacings, demonstrates where a microscope loses the ability to resolve the space between two points. In this study, we consider several such dumbbell systems and choose an image where the gap between dumbbell atoms was resolved. To quantify the gap, we fit the dumbbell (or any dumbbell in the image) to two Gaussian curves spaced some distance apart such that ðx - x1 Þ2 ðx - x2 Þ2 GðxÞ ¼ a Exp + Exp : (2) 2σ 2 2σ 2 Again the units of G(x) and a are arbitrary. The values x, x1, and x2 define the pixel position and centers of the two Gaussians, respectively. The relevant parameter is the standard deviation of each Gaussian, σ. The distance between the centers of the two curves should match the dumbbell spacing, where the width of each Gaussian should estimate the resolution of the image. We find that, as with the sigmoidal, the fit parameter, σ, in the exponential of the Gaussian overestimates how good our image resolution is. Although the quality of the fit is good, σ is not the correct metric for edge sharpness. It is more reasonable to consider the width of the Gaussian at the 1/e point. This is the point where G(x) = 1/e and x = √2σ. The full width at the 1/e point of the Gaussian is 2√2σ, and the resolution of the curve is √2σ. This factor of √2 is now incorporated into m along with the analytical relation between σ in the Gaussian curve and b in the sigmoidal curve. We can solve for the relation between the slopes of the two fit curves, equating one side of the Gaussian peak with
A Simple Metric for Determining Resolution
773
and extracted the resolution of each image. We compared these results with other methods, particularly the Rayleigh criterion for optical tools, and found that our method is robust enough to be applied across many length scales on a wide variety of images.
RESULTS
Figure 1. We can match the slopes for a logistic curve (solid, blue curve) for a light/dark step and a Gaussian curve (dashed, red curve) q for single bright feature using the analytical relation ffiffiffiffiffiffiffiffiffiffiffiffi ffi b ¼ σ=4 12 lnð2Þ. When we center the two curves over the same co-ordinate, x0, we can clearly see the agreement between their slopes over the region in the gray box from xL = −1/2mb + x0 to xR = 1/2mb + x0. The evaluation of S(x) at these bounds is given in the text. These relations allow us to use the Gaussian fits of atomic dumbbells to inform the choice of the prefactor m modifying the logistic function fit of larger edges.
the light/dark transition in the sigmoidal curve at the 1/2 maximum point. Figure 1 shows a Gaussian curve of width σ = 1. We found that a sigmoidal function describing the same light/dark transition on the right-hand side of the curve may be found by choosing the appropriate offset, x0, and using the following relation: σ b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi : (3) 4 12 lnð2Þ This expression allows us to relate the sharpness of a Gaussian fit to that of an equivalent sigmoidal model. As we know how to derive resolution from σ for STEM images, we can use the above relation and the factor of √2 and find m = 3.33. The boxes in Figure 1 mapping the overlapping curves to regions along the x and y axes are chosen to show the corrected resolution of an edge with this transition. The bottom box has a width m × b and is centered about x0. The green box on the right spans the y axis from S(xL = −1/2 mb + x0) to S(xR = 1/2 mb + x0). If we evaluate S(x) at these co-ordinates, we can find that a h qffiffiffiffiffiffiffiffiffiffiffiffiffii : SðxÞ ¼ c + (4) 1 + Exp ± 3 12 lnð2Þ This figure shows the agreement between the Gaussian fit and the sigmoidal curve, justifying our derivations of resolution from b via σ. Knowing m, we fit a selection of data from an optical microscope, an SEM, and a helium ion microscope (HIM)
In the following sections, certain commercial instruments are identified to foster understanding, both indirectly (via references) and directly, as in Figure 4. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. These tools were used to generate the data used to test our resolution analysis, and, as such, we make no claims about their ultimate capabilities or quality. Our laboratory is equipped with a 200 kV aberrationcorrected STEM that has been used to image dumbbell systems including GaN, Ge , and Si . Figure 2a shows the STEM-HAADF image of Si chosen for this analysis (image collected at NIST, Boulder Laboratories, courtesy of Toshihiro Aoki). By surveying the data, we found that it is in the Si images where the dumbbell is consistently resolved, with an atomic spacing of 136 pm (or a light/dark transition of 67 pm). Images from the same tool on GaN and Ge with smaller dumbbell spacings of 81.6 and 82.0 pm, respectively, did not have the dark region between the atoms reliably resolved. The atoms in Figure 2a are not wide enough to fit with the logistic function. However, the analytical relation between the sigmoidal and a Gaussian outlined in the previous section means that we can fit the complete dumbbell as two Gaussians of width 2√2 × σ spaced some distance apart. The width of the Gaussians accurately describes how quickly the edges of the dumbbell drop off, both at the edges of the dumbbell and at the dark contrast in the center. With this in mind, we calculated σ for the Si data and used it to show that 3.33 is a valid choice for m. Although we first checked by choosing a dumbbell at random, we formalized our method by extracting the data for half of the dumbbells in the image. From these 104 dumbbells, 90 could be fit. In keeping with the sigmoidal method, where we averaged the edge transitions over many line profiles, each dumbbell was composed of ten lines of data, generating ten Gaussian fits. The parameter σ was solved for each of these 90 dumbbells, and an average data profile and average fit curve were generated. The remaining 14 dumbbells failed the fit either because the signal:noise was poor or because the dumbbell may have been poorly centered in the selected ten lines of data. Figure 2b shows the average fit of the two Gaussians plotted over an average profile calculated from the 90 dumbbells. The average σ was 45 pm with a standard deviation in the data set of 11 pm. We note that if σ from the Gaussian fits were strictly accurate, then light/dark
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Figure 2. a: A scanning transmission electron microscopic image of Si atomic dumbbells (image collected at NIST, Boulder Laboratories, courtesy of Toshihiro Aoki). Half of the atomic dumbbells were pulled out for analysis. Of these 104 features, 90 dumbbells were able to be fit to two Gaussians of width σ and centered at the coordinated x1 and x2. In (b) an average profile for the 90 dumbbells is shown (blue, jagged line), is shown with an overlay of the Gaussian fit for the two peaks (red line).
transitions over length scales considerably shorter than the specifications of our microscope would be visible. The halfdistance of the dumbbell spacing for the Si in this image was 67 pm. The calculated resolution of this image was √2 × σ = 64 pm, close to the light/dark transition distance for the image and near the specification of the microscope. This supports our choice of the 1/e point for defining resolution from the Gaussian, and, consequently, the finding of m = 3.33. Although the data had a 24% standard deviation, we know that the negative side of this error is not realistic for the capabilities of the instrument. That is to say, our resolution on some days may be 24% higher than 64 pm, or closer to 80 pm, but we cannot resolve features on the negative side, as small as 50 pm (Xin et al., 2013).
We applied our averaged sigmoidal fit to a collection of optical, SEM, and HIM images. In each case, light/dark edges were selected over regions of interest large enough to provide many line profiles. Our SEM images were collected on a dual-beam focused ion beam (FIB)/SEM system on a Geller SEM magnification standard made of patterned antireflective chromium on quartz. In general, our sigmoidal analysis reported an edge resolution of roughly 2 nm. Variations around this value may have been due to focus quality of different images, room vibration, or other features within the region of interest (ROI) that caused the algorithm to locate the edge transition incorrectly. In our images, we were careful to set the pixel resolution below the optimal resolution quoted by the manufacturer. The value of 2 nm, then, may be considered an accurate representation of the microscope’s performance on the particular day of capture. The HIM images were collected on a graphite sample. Similar to the SEM images, we took care to make sure that the pixel size was
© MICROSCOPY SOCIETY OF AMERICA 2015
A Simple Metric for Determining Resolution in Optical, Ion, and Electron Microscope Images Alexandra E. Curtin,* Ryan Skinner, and Aric W. Sanders Boulder Laboratories, National Institute of Standards and Technology, Boulder, CO 80305, USA
Abstract: A resolution metric intended for resolution analysis of arbitrary spatially calibrated images is presented. By fitting a simple sigmoidal function to pixel intensities across slices of an image taken perpendicular to light–dark edges, the mean distance over which the light–dark transition occurs can be determined. A fixed multiple of this characteristic distance is then reported as the image resolution. The prefactor is determined by analysis of scanning transmission electron microscope high-angle annular dark field images of Si . This metric has been applied to optical, scanning electron microscope, and helium ion microscope images. This method provides quantitative feedback about image resolution, independent of the tool on which the data were collected. In addition, our analysis provides a nonarbitrary and self-consistent framework that any end user can utilize to evaluate the resolution of multiple microscopes from any vendor using the same metric. Key words: microscopy, helium ion microscopy, optical microscopy, Rayleigh criterion
I NTRODUCTION Although one of the most-quoted properties of a microscope, the term “resolution” can have multiple meanings and can be characterized by many different parameters. As they are often based on the specific physics of each microscopy method, these criteria are not easily generalized. In addition, many of the established metrics require precise knowledge of the subject geometry, which adds both complexity and additional overhead spent characterizing the sample. Owing to the myriad of resolution criteria used, comparison of the performance of different imaging devices tends to be subjective. At the very least, a degree of ambiguity is introduced when attempting to convert data from one metric to another. In this study, we developed a simple, reliable, and universal metric, which can be used to evaluate images quickly. Our approach is based on the fact that edges separating light and dark regions contain information about the resolving power of the imaging device. By performing our resolution analysis on the image itself, we have de-coupled our method from a particular tool or collection method, allowing the application of our resolution metric to be widely applicable. We will show that our analysis is quantitative, applicable to optical, transmission electron microscopy (TEM), and scanning electron/ ion microscope images, and produces results consistent with the Rayleigh criterion where appropriate. When a manufacturer of scientific equipments sells a microscope to a laboratory, the performance of that microscope is judged by its best attainable resolution. The resolution quoted by the manufacturer is measured by various means under ideal laboratory conditions and with ideal samples. Received October 1, 2014; accepted February 24, 2015 *Corresponding author. [email protected]
The analytical means used by vendors to determine resolution from these images vary greatly with many relying on contrastto-gradient methods. A simple method for translating this performance to images collected in imperfect laboratory conditions and from specimens with different scattering properties becomes a powerful tool for scientists trying to analyze their own data and independently evaluate the image resolutions claimed by multiple vendors. Our work presents a nonarbitrary, analytical platformindependent means of fitting light/dark ed, ges in microscope images from the micrometer down to the picometer scale. Various methods of fitting edges in an image to a function have been used in the past to determine image resolution (Ishitani & Sato, 2004). We have chosen to use a sigmoidal curve, or logistic function, to fit our light/dark edges. The sigmoidal curve is more simply applied than the error function, relying on only a single fit parameter to describe edge sharpness. The sigmoidal on its own reports the width of an edge, but we cannot know quantitatively whether this is an accurate estimate of image resolution. Our method extends past this, however, by tying the sigmoidal fit to the fit of a known sample. Scanning transmission electron microscopy high-angle annular dark field (STEM-HAADF) images of atomic dumbbells present a wealth of features with known spacings, but these are not easily fit with an odd function. Instead, we use a Gaussian fit of atomic dumbbells. Parameters from this fit, when analytically tied to the sigmoidal function, provide a prefactor to fix the sigmoidal fit parameter in the real world. It is important to note here that there are at least two methods including Fourier analysis for assessing the resolution of a STEM image. Our intent here is not to supplant these methods, but rather to use our STEM images of known dumbbell spac as a means of solving for the proportionality
772
Alexandra E. Curtin et al.
constant or prefactor our sigmoidal fit parameter. The sigmoidal fit can then be applied to lower magnification images. As a second check to our method, when we apply the sigmoidal fit to optical images, we can directly compare the fit to the Rayleigh criterion resolution limit for the particular microscope conditions. The Rayleigh criterion is one of the most well-known metrics for determining the minimum resolvable detail in diffraction-limited systems. When radiation from a point source diffracts through a circular aperture, the resulting pattern that falls on a detector is an Airy disk. When two such point sources are present, as when viewing a double star through a telescope, the Rayleigh criterion defines the points to be “just resolved” when the maximum of one Airy disk intersects the first minimum of the other. A second resolution condition commonly mentioned in tandem with the Rayleigh criterion is the Sparrow criterion. The Sparrow criterion states that the minimum resolvable distance is reached when the dip is no longer distinguishable. However, as the Sparrow criterion is vulnerable to noise in the acquired image, the Rayleigh criterion provides us with the most mathematically rigorous metric for a diffraction-limited system. For our purposes, we will compare the quality of our fit of light/dark transitions to the Rayleigh criterion for optical images of different magnifications. The Rayleigh criterion provides standard and sensible ways of defining resolution, but it is only applicable to specific diffraction-limited systems. In the fields of electron and ion microscopy, resolution of imaging systems is often limited by spherical and chromatic aberrations, as well as by astigmatism, before diffraction effects begin to degrade image quality. We will use this criterion later to check the effectiveness of our image analysis for images derived from our optical microscope. For scanning electron microscopy (SEM) images, there are thoroughly developed Fourier transform techniques for determining image sharpness, which could be extended to characterize the resolution (Postek & Vladár, 1998; Ishitani & Sato, 2002; Joy et al., 2010). This approach takes advantage of the fact that higher-frequency components of the image are maximized when the subject is in focus with minimal astigmatism. The highest spatial frequencies from the image correspond to the sharpest edges, and are presumably correlated with image resolution. Although we will not compare this method in this study, it is worth noting the effectiveness of Fourier methods for certain types of images. Where our analysis is easily applied to images containing straight light/ dark transitions, the well-known contrast transfer function and similar methods work well for SEM images containing high spatial frequency detail.
M ETHODS We considered image resolution based on the sharpness of image features with strong light/dark edge contrast. To quantify “sharpness” we fit a selected light/dark edge to a sigmoidal line profile. Every line profile crossing the edge is fit,
and the resulting curves were averaged. Our sigmoidal curve of choice, the logistic function, is defined as follows: a : (1) S ðx Þ ¼ c + 1 + Exp - xb+ x0 Here, S(x) describes the amplitude of the function, and a is a scaling factor. The parameters c and x0 provide x- and y-offsets, respectively. The logistic function is ideal for such a fit, as it applies a curve to the region of contrast transition but does not require a round feature or particle. The units are arbitrary as they are taken from the gray-scale range of the image. The relevant parameter for defining image resolution in the logistic function is b. We will show that b is proportional to the resolution of the image, such that resolution = m × b, where m will be the prefactor tying the width of the sigmoidal, b, to a quantitative statement about resolution. Without an empirically determined value for m—that is, without determining the relationship between the fit parameter b and the actual resolution of an image—we leave room for our edge fits to produce nonphysical results. Therefore, we turn to experimental methods to determine m, and then compare the calculated resolution of optical images to the Rayleigh criterion. STEM-HAADF imaging provides direct images of crystal structures for measuring known distances in an electron microscope. In particular, attempting to resolve columns of atoms from materials along different crystallographic directions, with different spacings, demonstrates where a microscope loses the ability to resolve the space between two points. In this study, we consider several such dumbbell systems and choose an image where the gap between dumbbell atoms was resolved. To quantify the gap, we fit the dumbbell (or any dumbbell in the image) to two Gaussian curves spaced some distance apart such that ðx - x1 Þ2 ðx - x2 Þ2 GðxÞ ¼ a Exp + Exp : (2) 2σ 2 2σ 2 Again the units of G(x) and a are arbitrary. The values x, x1, and x2 define the pixel position and centers of the two Gaussians, respectively. The relevant parameter is the standard deviation of each Gaussian, σ. The distance between the centers of the two curves should match the dumbbell spacing, where the width of each Gaussian should estimate the resolution of the image. We find that, as with the sigmoidal, the fit parameter, σ, in the exponential of the Gaussian overestimates how good our image resolution is. Although the quality of the fit is good, σ is not the correct metric for edge sharpness. It is more reasonable to consider the width of the Gaussian at the 1/e point. This is the point where G(x) = 1/e and x = √2σ. The full width at the 1/e point of the Gaussian is 2√2σ, and the resolution of the curve is √2σ. This factor of √2 is now incorporated into m along with the analytical relation between σ in the Gaussian curve and b in the sigmoidal curve. We can solve for the relation between the slopes of the two fit curves, equating one side of the Gaussian peak with
A Simple Metric for Determining Resolution
773
and extracted the resolution of each image. We compared these results with other methods, particularly the Rayleigh criterion for optical tools, and found that our method is robust enough to be applied across many length scales on a wide variety of images.
RESULTS
Figure 1. We can match the slopes for a logistic curve (solid, blue curve) for a light/dark step and a Gaussian curve (dashed, red curve) q for single bright feature using the analytical relation ffiffiffiffiffiffiffiffiffiffiffiffi ffi b ¼ σ=4 12 lnð2Þ. When we center the two curves over the same co-ordinate, x0, we can clearly see the agreement between their slopes over the region in the gray box from xL = −1/2mb + x0 to xR = 1/2mb + x0. The evaluation of S(x) at these bounds is given in the text. These relations allow us to use the Gaussian fits of atomic dumbbells to inform the choice of the prefactor m modifying the logistic function fit of larger edges.
the light/dark transition in the sigmoidal curve at the 1/2 maximum point. Figure 1 shows a Gaussian curve of width σ = 1. We found that a sigmoidal function describing the same light/dark transition on the right-hand side of the curve may be found by choosing the appropriate offset, x0, and using the following relation: σ b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi : (3) 4 12 lnð2Þ This expression allows us to relate the sharpness of a Gaussian fit to that of an equivalent sigmoidal model. As we know how to derive resolution from σ for STEM images, we can use the above relation and the factor of √2 and find m = 3.33. The boxes in Figure 1 mapping the overlapping curves to regions along the x and y axes are chosen to show the corrected resolution of an edge with this transition. The bottom box has a width m × b and is centered about x0. The green box on the right spans the y axis from S(xL = −1/2 mb + x0) to S(xR = 1/2 mb + x0). If we evaluate S(x) at these co-ordinates, we can find that a h qffiffiffiffiffiffiffiffiffiffiffiffiffii : SðxÞ ¼ c + (4) 1 + Exp ± 3 12 lnð2Þ This figure shows the agreement between the Gaussian fit and the sigmoidal curve, justifying our derivations of resolution from b via σ. Knowing m, we fit a selection of data from an optical microscope, an SEM, and a helium ion microscope (HIM)
In the following sections, certain commercial instruments are identified to foster understanding, both indirectly (via references) and directly, as in Figure 4. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. These tools were used to generate the data used to test our resolution analysis, and, as such, we make no claims about their ultimate capabilities or quality. Our laboratory is equipped with a 200 kV aberrationcorrected STEM that has been used to image dumbbell systems including GaN, Ge , and Si . Figure 2a shows the STEM-HAADF image of Si chosen for this analysis (image collected at NIST, Boulder Laboratories, courtesy of Toshihiro Aoki). By surveying the data, we found that it is in the Si images where the dumbbell is consistently resolved, with an atomic spacing of 136 pm (or a light/dark transition of 67 pm). Images from the same tool on GaN and Ge with smaller dumbbell spacings of 81.6 and 82.0 pm, respectively, did not have the dark region between the atoms reliably resolved. The atoms in Figure 2a are not wide enough to fit with the logistic function. However, the analytical relation between the sigmoidal and a Gaussian outlined in the previous section means that we can fit the complete dumbbell as two Gaussians of width 2√2 × σ spaced some distance apart. The width of the Gaussians accurately describes how quickly the edges of the dumbbell drop off, both at the edges of the dumbbell and at the dark contrast in the center. With this in mind, we calculated σ for the Si data and used it to show that 3.33 is a valid choice for m. Although we first checked by choosing a dumbbell at random, we formalized our method by extracting the data for half of the dumbbells in the image. From these 104 dumbbells, 90 could be fit. In keeping with the sigmoidal method, where we averaged the edge transitions over many line profiles, each dumbbell was composed of ten lines of data, generating ten Gaussian fits. The parameter σ was solved for each of these 90 dumbbells, and an average data profile and average fit curve were generated. The remaining 14 dumbbells failed the fit either because the signal:noise was poor or because the dumbbell may have been poorly centered in the selected ten lines of data. Figure 2b shows the average fit of the two Gaussians plotted over an average profile calculated from the 90 dumbbells. The average σ was 45 pm with a standard deviation in the data set of 11 pm. We note that if σ from the Gaussian fits were strictly accurate, then light/dark
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Alexandra E. Curtin et al.
Figure 2. a: A scanning transmission electron microscopic image of Si atomic dumbbells (image collected at NIST, Boulder Laboratories, courtesy of Toshihiro Aoki). Half of the atomic dumbbells were pulled out for analysis. Of these 104 features, 90 dumbbells were able to be fit to two Gaussians of width σ and centered at the coordinated x1 and x2. In (b) an average profile for the 90 dumbbells is shown (blue, jagged line), is shown with an overlay of the Gaussian fit for the two peaks (red line).
transitions over length scales considerably shorter than the specifications of our microscope would be visible. The halfdistance of the dumbbell spacing for the Si in this image was 67 pm. The calculated resolution of this image was √2 × σ = 64 pm, close to the light/dark transition distance for the image and near the specification of the microscope. This supports our choice of the 1/e point for defining resolution from the Gaussian, and, consequently, the finding of m = 3.33. Although the data had a 24% standard deviation, we know that the negative side of this error is not realistic for the capabilities of the instrument. That is to say, our resolution on some days may be 24% higher than 64 pm, or closer to 80 pm, but we cannot resolve features on the negative side, as small as 50 pm (Xin et al., 2013).
We applied our averaged sigmoidal fit to a collection of optical, SEM, and HIM images. In each case, light/dark edges were selected over regions of interest large enough to provide many line profiles. Our SEM images were collected on a dual-beam focused ion beam (FIB)/SEM system on a Geller SEM magnification standard made of patterned antireflective chromium on quartz. In general, our sigmoidal analysis reported an edge resolution of roughly 2 nm. Variations around this value may have been due to focus quality of different images, room vibration, or other features within the region of interest (ROI) that caused the algorithm to locate the edge transition incorrectly. In our images, we were careful to set the pixel resolution below the optimal resolution quoted by the manufacturer. The value of 2 nm, then, may be considered an accurate representation of the microscope’s performance on the particular day of capture. The HIM images were collected on a graphite sample. Similar to the SEM images, we took care to make sure that the pixel size was
