Detecting and Correcting Piezoelectric-tube Actuator Drift Induced Distortion in Atomic-Resolution Scanning Tunneling Microscope Images from Crystal Surfaces

Since its development in 1981, Scanning Tunneling Microscopy (STM) has become a premier technology in imaging at atomic resolution across disciplines. The “Scanning” in STM refers typically to the movement of the sample relative to the probing tip, as controlled by piezoelectric-tube actuators. These actuators allow the sample to be moved at sub-nanometer precision, but over several repeated scans are susceptible to drift, especially when operating in open-loop mode. This drift leads to directly detectable distortions in the image [1], which inevitably decrease the fidelity and by extension the science that can be done with the image. To make matters worse, as the process to correct or account for this drift can be complex, many publications simply forgo the exercise entirely, leading to some “untenable conclusions” [1] being drawn from compromised data. These errors can often be detected by observing the crystallographic symmetries and pseudo-symmetries present in potentially compromised images and comparing them with the symmetries known to be present in the sample. For example, Highly Oriented Pyrolytic Graphite (HOPG) is often used as a calibration sample for STM and has well-documented symmetries. The breaking of these symmetries can be objectively quantified [2, 3], which clearly demonstrates the effect of actuator drift. This drift can be modeled in several ways, such as the transformation of the scanning area from its intended square or rectangle to a skewed parallelogram [1]. As a result, key morphological features may be largely preserved, and many qualitative observations can still be made. This is sufficient for many applications. But when distances and angles between atomic rows are of particular concern, those measurements cannot be directly recovered from a drift distorted image to much precision [1]. In detailing and mathematically formalizing these distortions, Henriksen and Stipp [1] provided a pathway for correcting the aberrations to achieve greater image fidelity in post processing — thus prospectively allowing for more accurate and useful quantitative data collection. The drift is formulated as a linear combination of a displacement vector of a particular point on the image (with a fixed origin) and a scaling factor. Through linear algebra, the aberrant transformation can be divined, and its reversal enacted. Simultaneously, there have been recent advances in information-theoretic classification and quantification methods for two-dimensional (2D) crystallographic symmetries [2, 3]. Starting from a digital image, these methods are not reliant on any particular imaging or microscopy technology. They allow for objective classifications into 2D Bravais lattice types, projected Laue classes, and plane symmetry groups, as well as the associated site symmetries. They are objective in that the classifications are made based on a geometric Akaike Information Criteria [2]. That is, the symmetries assigned to an image follow deterministically from the algorithm, instead of being approximated by human eyes or a computer program that uses arbitrarily chosen thresholds. To accommodate noise from a variety of sources including but not limited to minor defects in the surface itself and minor imaging errors in the machinery, the noise of the image is considered to be a singular aggregate term and more or less Gaussian in nature. When no singular noise source dominates, this assumption is justified by versions of the central limit theorem [4]. Figure 1 shows a STM image of HOPG that was recorded at the “Nano Transport System” of the Center for Nanophase Materials Science of the Oak Ridge National Laboratory [5]. Selected ratios of squared residuals for both minimal supergroups and maximal subgroups of plane symmetry groups and projected Laue classes are part of this figure. These plane and 2D point symmetry groups form hierarchical trees, with more complex symmetries subsuming simpler ones [2]. If the ratio of the residuals corresponding to each of these groups is sufficiently small, one can “climb” the tree and identify increasingly complex symmetries. These ratios assign plane symmetry group c1m1 and projected Laue class 2mm to this STM image. A theoretically perfect atomic resolution image of a perfectly crystalline HOPG sample surface would feature plane symmetry group p3m1 exactly. The experimental counterpart of that nonexistent image would also feature that plane symmetry in the absence of drift for a real-world HOPG sample, as one would on information theoretic grounds [2-4] be allowed to ascend from c1m1 to p3m1, whereby both symmetries would be broken by the same amount of generalized noise. (Note that drift is a major systematic distortion that exists in addition to a variety of minor distortions that are considered to be of a random nature.) The three-fold rotation site symmetries in Fig. 1 are more severely broken by drift in the piezo-electric tube actuators than the mirror lines. (With the same amount of drift, higher plane symmetries are more severely broken than lower symmetries.) Fortunately, these breakages can be corrected [1], restoring the symmetries to their predicted state. After making corrections to account for the drift in the sample image, the improvements in the detected symmetries can be objectively [2, 3] measured to demonstrate the success of the correction. The goal of this work is to implement the method put forth in [1] in software and evaluate its effectiveness using the aforementioned objective analyses [2-4]. This entails numerically calculating the drift and scaling factors from a trio of points on the image, taken from the Fourier periodicities, and then applying the inverse of the drift transformation. Putting numerical accuracy to such methodology poses the opportunity to evaluate the effectiveness of this method, but also to serve as a baseline for more potentially nuanced and mathematically complex methodologies that may arise in the future. Atomic resolution STM image of HOPG (from the on-line supporting material to [5]), the central area of its discrete Fourier transform, and a selection of ratios of squared residuals (from left to right). In the selected circular region of the STM image, the symmetry climbs the hierarchy tree of the plane symmetry groups freely from p1 to c1m1 because the discrete Fourier transform features discrete Bragg peaks and the c1m1 symmetrized model of the image features the lowest squared residual of the complex-valued Fourier coefficients for all higher symmetric plane symmetry groups. The c2mm and p3m1 symmetrized models fail to achieve sufficiently low residual ratios with respect to the c1m1 model and thus further refinements of the geometric model of the HOPG surface are not allowed on information theoretic grounds. Similarly, the least broken projected Laue class is 2mm, from which no ascent to either 4mm or 6mm is allowed by the ratios of the squared residuals of the Fourier coefficient amplitudes.