Automatic indexing of two-dimensional patterns in reciprocal space

Abstract

An indispensable part of the structure determination of crystalline two-dimensional (2D) materials and epitaxial thin films is the correct indexing of the acquired diffraction patterns. In our previous work, we described an effective algorithm to determine the 3D unit-cell parameters of complex systems comprising different orientations and polymorphs. In this work, we adapt the indexing method to 2D lattices in reciprocal space. Analyzing low-energy electron diffraction and Fourier-transformed scanning tunneling microscopy measurements, the method is exemplarily applied to thin films of conjugated molecules like 3,4:9,10-perylenetetracarboxylic dianhydride (PTCDA), 6,13-pentacenequinone (P2O), and vanadyl phthalocyanine (VOPc) grown by physical vapor deposition on Ag(111). In all cases unit cells (rhomboids) along with their sixfold rotationally or mirror symmetric counterparts are determined. The already known commensurate epitaxial relationship is reproduced for PTCDA on Ag(111), demonstrating the validity of our method. In the case of P2O/Ag(111) a point-on-line epitaxial condition is found. Our algorithm can be equally well applied to all kinds of 2D patterns in reciprocal space where a crystallographic indexing is required, e.g., electron diffraction data [such as transmission electron diffraction, selected area electron diffraction (SAED)] and fast Fourier transforms (FFTs) of scanning probe images. To demonstrate this aspect, we evaluate FFTs of scanning tunneling microscopy data for stacked VOPc/PTCDA heteroepitaxial layers on Ag(111) as well as SAED data of an epitaxial ${\mathrm{TiO}}_{2}/{\mathrm{LaAlO}}_{3}(100)$ heterostructure in cross section.