# 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

### Abstract

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 / mas@mrao.cam.ac.uk