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

Abstract

The epitaxial growth of molecular crystals at single-crystalline surfaces is often strongly related to the first monolayer at the substrate surface. The present work presents a theoretical approach to compare three-dimensional lattices of epitaxially grown crystals with two-dimensional lattices of the molecules formed within the first monolayer. Real-space and reciprocal-space representations are considered. Depending on the crystallographic orientation relative to the substrate surface, proper linear combinations of the lattice vectors of the three-dimensional unit cell result in a rhomboid in the xy plane, representing a two-dimensional projection. Mathematical expressions are derived which provide a relationship between the six lattice parameters of the three-dimensional case and the three parameters obtained for the two-dimensional surface unit cell. It is found that rotational symmetries of the monolayers are reflected by the epitaxial order. Positive and negative orientations of the crystallographic contact planes are correlated with the mirror symmetry of the surface unit cells, and the corresponding mathematical expressions are derived. The method is exemplarily applied to data obtained in previous grazing-incidence X-ray diffraction (GIXD) measurements with sample rotation on thin films of the conjugated molecules 3,4;9,10-perylenetetracarboxylic dianhydride (PTCDA), 6,13-pentacenequinone (P2O), 1,2;8,9-dibenzopentacene (trans-DBPen) and dicyanovinyl-quaterthiophene (DCV4T-Et2) grown by physical vapor deposition on Ag(111) and Cu(111) single crystals. This work introduces the possibility to study three-dimensional crystal growth nucleated by an ordered monolayer by combining two different experimental techniques, GIXD and low-energy electron diffraction, which has been implemented in the second part of this work.

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

“…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.…”

“…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).…”

“…The studies on PTCDA/Ag(111) and on P2O/Ag(111) have been published very recently (Simbrunner et al, 2021b). In the first part of our work (Simbrunner et al, 2022), we derived the mathematical framework to calculate the parameters of the two-dimensional lattice from the parameters of the associated three-dimensional lattice. In this work, we prove our results experimentally and compare these data with our recent data from LEED studies on the same molecules and substrates to correlate the properties of the multilayer with those of the monolayer.…”

“…For the area of the two-dimensional unit cell, the following relation is valid (Simbrunner et al, 2022):…”

“…The surface unit cell is spanned by two vectors, which are linear combinations of the three vectors of the threedimensional lattice. In the first part of our work (Simbrunner et al, 2022), a comprehensive mathematical framework has been developed to correlate the parameters of the corresponding three-and two-dimensional lattices. The knowledge of the orientation and parameters of the three-dimensional crystal lattice (or unit cell) allows the interpretation of the two-dimensional data by a direct comparison of the lattices.…”