Automatic indexing of two-dimensional patterns in reciprocal space

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