Estimating atomic positions from scanning tunneling microscopy images by means of the Maximum Entropy approach

S.D. Böhmig, M. Schmid and H. Störi
Institut für Allgemeine Physik, Technische Universität Wien
Wiedner Hauptstr. 8-10, A-1040 Wien, Austria


In this paper the Maximum Entropy approach is applied to restore and sharpen scanning tunneling microscopy (STM) images with atomic resolution. According to the STM theory of Tersoff and Hamann the process of data acquisition can be approximated by the convolution of a localized atomic density of states (i.e. narrow spots in the reconstruction) of the sample and a Gaussian which limits the resolution. In STM practice a good and robust estimation of the atomic core positions is necessary for different reasons, such as to be able to calculate the characteristics of the atomic lattice or to study non-periodicities.

In STM-literature deblurring is generally carried out by means of the linear Wiener filter. However, first, this technique tends to ``smooth out'' non-periodic structures in the image depending on the deblurring kernel. Second, the Wiener filter is essentially a low-pass filter thus constraining the amount of deblurring which should result in narrow and thus high- frequency peaks in the image. For the investigation of non-periodicities, such a filter is not suited, instead we have solved the inverse problem by the Maximum Entropy approach, i.e. by minimizing the mean square deviation between the measured and the reconstructed image using entropy as a regularization functional. The results are obtained without any assumption concerning the periodicity of the atomic arrangement. The width of the Gaussian kernel is derived from the Fourier spectrum of the image and the type of the ordered structure of the surface.

From the reconstruction the atomic core positions are extracted by detecting the local maxima and calculating their centres of mass. The accuracy of the method is quantified by comparing a periodic surface with different signal-to-noise ratios to an ideal lattice. The achieved accuracy of below 1 pixel amounts to a deviation between only 6% and 9% of the inter-atomic distance (approx. 11 pixels) depending on the quality of the measurements. In a further example, the method is used to measure the distances between rows of atoms and shifted rows, which are displaced along the row direction. Based on the atomic positions it can be shown that the row distances between the shifted rows and their neighbouring rows is the same as between two non-shifted rows, which has interesting implications for the interpretation of the shifted row reconstruction.

MaxEnt 94 Abstracts /