Removal of multiple-tip artifacts from scanning tunneling microscope images by crystallographic averaging
© Straton et al. 2015
Received: 9 September 2015
Accepted: 28 October 2015
Published: 14 November 2015
Crystallographic image processing (CIP) techniques may be utilized in scanning probe microscopy (SPM) to glean information that has been obscured by signals from multiple probe tips. This may be of particular importance for scanning tunneling microscopy (STM) and requires images from samples that are periodic in two dimensions (2D). The image-forming current for double-tips in STM is derived with a slight modification of the independent-orbital approximation (IOA) to allow for two or more tips. Our analysis clarifies why crystallographic averaging works well in removing the effects of a blunt STM tip (that consists of multiple mini-tips) from recorded 2D periodic images and also outlines the limitations of this image-processing technique for certain spatial separations of STM double-tips. Simulations of multiple mini-tip effects in STM images (that ignore electron interference effects) may be understood as modeling multiple mini-tip (or tip shape) effects in images that were recorded with other types of SPMs as long as the lateral sample feature sizes to be imaged are much larger than the effective scanning probe tip sizes.
Scanning probe microscopy (SPM) images are often degraded due to the effects of two (or more) protrusions on the probe tip (i.e. effective mini-tips on a blunt tip), as well as containing sample tilt errors, image bow and drift, and stepping errors that occur while scanning the tip in two dimensions (2D) over the sample surface. Averaging methods have long been used to remove scanning errors. There are also well-established techniques for straightening out keystone-shaped images that result from sample tilt and image drift, and for the removal of image bow by z-flattening using least-squares higher-order polynomials to model this distortion [1–3]. Removing multiple-tip artifacts from SPM images has, however, only recently been accomplished through the adoption of crystallographic image processing (CIP) techniques [4–7], which one may consider as being a kind of a crystallographic averaging in reciprocal (Fourier) space of the intensity of symmetry-related features in direct space.
The transmission electron crystallography community developed CIP to enable the extraction of structure factor amplitudes and phase angles from (parallel illumination) high-resolution phase contrast images of crystalline materials within the weak phase object approximation [8, 9]. It has also been used for the correction of these images for the effects of the phase contrast transfer function, two-fold astigmatism, sample tilt away from low-indexed zone axes, and beam tilt away from the optical axis of the microscope. The central ideas of this kind of 2D crystallographic symmetry averaging have also been applied to scanning transmission electron microscopy (STEM) in order to increase the signal-to-noise ratio of Z-contrast imaging .
In the context of SPM, CIP addresses multiple scanning probe tip imaging artifacts effectively. This is an application that is beyond its original conception by the electron crystallography community and also does not apply to Z-contrast STEM imaging.
Since one may define 2D image-based crystallography independent of the source of the 2D patterns as being concerned with categorizing, specifying, and quantifying 2D long-range ordered patterns , CIP is also a good term for procedures as applied to SPM images of 2D periodic objects.
This process consists in its simplest form in the application of a Fourier transform to the 2D digitized image (called Fourier analysis), detection of the most likely plane symmetry in reciprocal space, enforcement of this symmetry by averaging of the symmetry related Fourier coefficients to remove all kinds of degradations, and finally inverse-Fourier image reconstruction (called Fourier synthesis into direct space). Irregularities in the 2D periodic array that is to be imaged, e.g. 2D periodic motif vacancies, are “averaged out” by CIP. For representative results, one should therefore aim for a ratio of regularly repeating features to irregularities of at least 50 (or better 100) to one.
By means of CIP, one can also extract the prevailing point spread function1 of the SPM  and use it for the correction of subsequently recorded images . One may refer to this function loosely as the “effective scanning probe tip” as it represents the convolution of the effects of the actual tip shape with all kinds of scanning and signal processing irregularities.
The symmetrizing is done in reciprocal space because of its computational efficiency. Since the Fourier coefficients were symmetrized, the CIP processed images are also symmetrized to the chosen 2D space group. The 2D space groups are also known as plane symmetry groups and combine 2D translations symmetries with 2D point symmetries, see “Appendix A”. We use the international (Hermann–Mauguin) notations for plane symmetry and 2D point symmetry groups  throughout the paper. When compared to CIP, conventional Fourier filtering  of 2D periodic images leads to translation averaging only. This means that the latter technique does not take advantage of the site symmetries in the plane groups (so that pure translation averaging will be up to 12 times less effective than CIP).
We note that both the unobscured image, Fig. 1a, and the obscured one, Fig. 1b, possess the same translation symmetry, which is that of the square 2D Bravais lattice. It was noted in Ref.  that subsequently recorded images from the same 2D periodic array that possess variations in the motif but possess the same translation symmetry are the hallmarks of blunt scanning probe tips. While obscured images have typically been discarded in the past, CIP presents an alternative to recover information from them. Figure 1c shows the inverse-Fourier image reconstruction after p4 symmetry enforcement in reciprocal space (following the guidelines in “Appendix B”) of the fully obscured portion of Fig. 1b. One sees a quite faithful reproduction (apart from a decrease in contrast) of the one-tip image, Fig. 1a as the 2D point symmetry of the motif is restored to group 4.
In the case of images that were recorded with multiple mini-tips, the whole plane symmetry enforcing procedure can, by virtue of the Fourier shift theorem , be thought of as aligning the 2D periodic motifs of all independent SPM images from the multiple mini-tips on top of each other, thus enhancing the signal-to-noise ratio significantly when done correctly. Within this context, CIP can be understood as a “sharpening up” of the effective scanning probe tip.
The present work shows in detail why CIP works and builds upon prior work [4–7] that shows how it is done at a practical level. In order to show in detail why CIP works, we will modify a common approach for simplifying the details of the problem, the independent-orbital approximation (IOA) to allow for the beating of signals from multiple mini-tips in STM. That is, we explore how “scanning tunneling probe tip surface structures” add both linearly and quantum mechanically to the recorded signal in convolution with the features of the “sample surface structure”.
Although the underlying physics of the IOA approach is specific to STM imaging, simulations of multiple-tip effects that ignore electron interference effects may be understood as modeling multiple mini-tip (or tip shape) effects in images that are recorded with other types of SPMs (where quantum mechanical interference effects can be safely ignored). It is well known that the nominal probe size is in STM imaging typically of the same (atomic or molecular) order of magnitude as the sample surface features that are to be imaged. For CIP to be applicable to images of 2D periodic arrays that were recorded with other types of SPMs (footnote 1), the effective probe size has to be much smaller than the lateral size of the features to be imaged. Although this requirement is trivial for any kind of meaningful imaging with SPMs (other than STMs, atomic or molecular resolution atomic force microscopes, and critical dimension SPMs3), it needs to be stated repeatedly as the literature abounds with conclusions that largely ignore it.
We first review the IOA, show how to modify it for two tips, and then trace back the resultant image to the salient details within its Fourier transform to show why CIP works. The changes wrought in the tunneling current by having two (or more) tips are outlined thereafter. The arrangements of multiple mini-tips in our analyses do not possess projected 3D point symmetries higher than 1, i.e. 360 degree rotations about arbitrary axes.
We begin with a treatment of double-tips since one may consider it a worst-case scenario of multiple tips, as will be illustrated later in the paper. We also examine the effect of double-tip height variations on the images and on the applicability of CIP.
In particular, we show that the 2D Fourier transform of the derived current resulting from two tips is comprised of the same Fourier coefficients as a single tip. The currents from the two tips differ in a phase term in reciprocal space  arising from the addition of complex numbers with different phases. These phase differences between two contributors may reduce the amplitudes (at a given reciprocal space point). CIP lessens this effect by averaging the Fourier coefficient amplitude and phase at such a point with amplitudes and phases at symmetry-related points.
We show the wide range of double-tip separations that are amenable to CIP. There are, however, certain double-tip separations for which some of these phases take prominent Fourier coefficients to zero, thereby obscuring the current map to the extent that even CIP cannot improve it.
The independent-orbital approximation
Two scanning probe tips
If one were imaging using an atomic state with two lobes aligned parallel to the x-axis, one could follow the procedure Chen outlines  in which for a quantum mechanical p x tip state, say, one takes “derivatives of the sample wave function at the nucleus of the apex atom of the tip” with respect to x to get the tunneling matrix elements. This results in the current images from each sample atom being doubled, as pictured in his 1987 paper .
In many cases, however, an STM tip having a pair of mini-tips—due to manufacturing error, damage to the tip, or the originally atomically sharp tip having picked up some material from the sample or the surrounding—is likely to have them separated by a much larger distance than the lobes of an atomic orbital. Indeed the separation distance will likely be of the same order as the inter-atomic or inter-molecular spacings of the sample.
In such a case, we can treat such a doubled tip as two well-spaced s tips (keeping our s sample), for example, and rely upon the reciprocity principle:  by “interchanging the tip state and the sample state, the conductance distribution [and hence the image is] unchanged”. We saw above that a p x tip state imaging a real-space structure would result in a current image having each sample atom (or molecule) doubled. One would get a similar looking current image using a single tip on a lattice/structure one has cloned, after shifting the second lattice/structure’s origin along the x-axis by the distance between the lobes of the p x tip. With a double-tip whose spacing is significantly larger, the same principle applies. We will see, however, that tip separations on a scale matching the sample lattice constant give the new possibility that the two currents will beat against each other.
As the pair of s tips (on a blunt scanning probe tip) is scanned over the surface, each tip would encounter the largest charge density in the x direction at different positions of the scanning head holding the two tips. If the tip separation w were precisely (an integer times) the periodicity of the real-space lattice/structure, the conduction signal would simply be twice as large and the topographic image would be unchanged except for brightness from what a single tip would yield. If, on the other hand, the tips were separated by any other distance, the two tips would register different tunneling charge densities at each position of the scanning head, and the pair of conduction signals would beat against each other, altering the topographic image registered.
For our single-tip on a cloned lattice/structure, we still have atoms that are independent of each other so that they do not shift position when new neighbors are slipped into the interstices by the duplication and shift process. This is a reasonable assumption if the spacing between atoms is (much) larger than the atomic extent.
The transform (10) also reveals that suppression of Fourier components in the horizontal direction in reciprocal space by the phase terms Cos[n bu], seen in Fig. 3, is the cause of the significant change in the image registered by this model double STM tip in Fig. 2. In Fig. 3d, for π/4 − ε, this suppression becomes so severe that the character of the original image is entirely obscured for vanishing ε, see Fig. 2d.
For bu = 0.77, Fig. 2d, we are beyond the limit at which one might confidently use CIP without a priori knowledge and/or an unambiguous determination of the underlying translation symmetry. With our prior knowledge of the underlying plane symmetry of the sample 2D periodic array, and/or with our recently developed geometric Akaike information criterion (AIC) for the unambiguous identification of 2D Bravais lattices  (see “Appendix B”), we can direct the popular CIP program CRISP  to produce a reconstruction, Fig. 4d, much more faithful to the IOA p4mm wave functions, Fig. 2 a than that contained in the two-tip image, Fig. 2d.
Different heights for the two tips
We assumed a worst-case scenario in Eq. (12) in which the two tips were at precisely the same distance z above the surface structure. If one of the two tips is closer to the sample, its current will dominate the current from the higher tip, thereby exponentially reducing the obscuration of the image. In Sect. “Results”, above, we represented a double-tip by a single tip above a cloned lattice/structure, having shifted the clone one way along the x-axis and the original the other way by the same amount. In modeling two tips at different heights in such an approach, one could also raise the cloned lattice/structure higher than the original to yield the exponential dominance of the current from that original lattice/structure.
Tsukada, Kobayashi, and Ohnishi  found a reduction in interference with tip-elevation angle in their calculations using an antibonding H2 orbital model for a tip on graphite. By the time they reached a 0.26 rad elevation difference, the interference was much reduced.
Thus we see that the double-tip case is indeed some kind of a worst case. Additional tips provide nonzero contributions to the reciprocal space amplitudes at spatial frequencies that would otherwise be completely suppressed. This facilitates the application of CIP to bring out even more underlying information in the “sample”. So we expect that crystallographic averaging would work well in removing the effects of a blunt STM tip, consisting of multiple mini-tips.
Summary and conclusions
CIP may often be used to remove multiple-tip artifacts from SPM images. Alternatively, one can think of the application of CIP as being analog to the “sharpening up” of a blunt tip to enhance the signal-to-noise level.
We have modified the independent-orbital approximation (IOA) to account for the beating of signals from two tips. Tracing back the resultant image to the salient details within its Fourier transform shows why CIP is effective. The tunneling currents from the two tips differ in a phase term in reciprocal space that may reduce the Fourier amplitudes (and hence, the real-space modulation) at a given reciprocal space point. We show that CIP lessens this effect by averaging the amplitude and phase at such a point with amplitudes and phases at symmetry-related points.
We have also shown that the existence of more than two tips at random separations will tend to ameliorate pair-wise destructive beating of signals at a given reciprocal space point, providing additional amplitude at that Fourier point to restore some real-space modulation. Finally, we have recovered textbook knowledge that tip height variations will ameliorate image degradations because of the exponential falloff of the signal with the tip-surface distance.
In particular, we have shown that the 2D Fourier transform of the derived tunneling current resulting from two tips is comprised of the same Fourier coefficients as a single tip. We show the wide range of double-tip separations that are amenable to CIP. There are, however, certain double-tip separations for which some of these phases take prominent Fourier coefficients to zero, thereby obscuring the current map to the extent that even CIP cannot improve it.
Reference  demonstrates, for example, the application of CIP to two 2D periodic images (that were recorded from the same commercial calibration sample with the same atomic force microscope) under (i) standard and (ii) non-standard imaging conditions, i.e. an open feed back loop. That calibration sample was designed to possess plane symmetry p4mm and its lateral 2D periodic feature size were one order of magnitude larger than the nominal probe sizes. (The horizontal sample feature size was approximately a tenth of the nominal probe sizes.) The effective scanning probe tips were de-convoluted from these images and the one that corresponded to the standard imaging conditions was less than half of the size of its non-standard imaging conditions counterpart.
One duplicate of the p4 image was pasted on top of the p4 image and then shifted 3 pixels to the right and 15 pixels down, out of 550 pixels and a second duplicate was shifted up 9 pixels and right 26 pixels. The three layers were then combined using Photoshop’s overlay blend mode, the formulas for which are given at http://www.stackoverflow.com/questions/5825149/overlay-blend-mode-formula, with the opacity of the duplicate layers set at 70 and 30 %, respectively.
Critical dimension SPMs were developed specifically for the assessment of narrow and deep trenches as well as steep and high walls either as transients in the building-up of integrated circuits or in micro- and nano-electromechanical systems.
JS crafted the mathematical structure of the paper, generated the “hypothetical images” that model the obscurations CIP is capable of removing, performed some of the image processing, and drafted the whole paper. BM performed some of the image processing and helped edit the paper. TB contributed to the development of our method for the unambiguous detection of the underlying Bravais lattice of a 2D periodic SPM image, provided Fig. A1, and helped edit the paper. PM helped write and edit the paper, drafted the appendices, performed some of the image processing, and provided overall guidance to the whole project since applying CIP to SPM images was his basic idea (for which he also secured a patent for his employer). BM and TB each prepared Master of Science theses on the application of CIP to SPM images of 2D periodic arrays. All authors read and approved the final manuscript.
This research was supported by awards from Portland State University’s Venture Development Fund and the Faculty Enhancement program. A grant from Portland State University’s Internationalization Council is also acknowledged. JS would like to thank C. Julian Chen for helpful comments on charge density distributions.
PM secured for Portland State University a patent for applying CIP to SPM images. He is also a Deputy Editor-in-Chief of Advanced Structural and Chemical Imaging.
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