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 TiO 2 /LaAlO 3 (100) heterostructure in cross section.

Correlation between two- and three-dimensional crystallographic lattices for epitaxial analysis. II. Experimental results

“…All errors given in the result tables are exclusively numerical errors (standard deviations) obtained from our algorithm (Simbrunner et al, 2021b). They do by no means reflect the absolute uncertainties, which are larger due to the occurrence of systematic and unsystematic errors that are not known precisely.…”

“…In this work, we will check our theoretically derived results with experimental data by indexing only x and y components of the scattering vector (q x , q y ) from GIXD experiments. Mathematical expressions were derived for an indexing method (Simbrunner et al, 2021b(Simbrunner et al, , 2022), which will be described and proposed for this purpose. Afterwards, we will also compare these findings with the results of recent LEED experiments on the same molecules to compare the crystallographic properties of the monolayer and multilayer.…”

“…where a a and b a are the lattice vectors of the adsorbate (molecular contact layer), a s and b s are the lattice vectors of the substrate, and A a and A s are the associated matrices. As A À1 s ¼ ð1=2ÞA Ã s (see Simbrunner et al, 2021b), the following relation can be deduced:…”

“…The mathematical basis for our indexing procedure was derived in our theoretical work (Simbrunner et al, 2022). A more detailed treatise was published recently (Simbrunner et al, 2021b). Please note that all vectors throughout Section 2 are entirely two-dimensional.…”

Correlation between two- and three-dimensional crystallographic lattices for epitaxial analysis. I. Theory

“…If a transformation matrix N exists so that m equals N(h 1 , h 2 ) T or N(k 1 , k 2 ) T , a vector of a superlattice is obtained. The reduced rhomboid is obtained by choosing the two shortest vectors which are not collinear and whose scalar products with all reciprocal vectors yield integers (Simbrunner et al, 2021b). This reduced rhomboid is equivalent to the Buerger cell in the three-dimensional case (Buerger, 1957).…”