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

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.