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By quantitative low-energy electron diffraction (LEED) we investigate theextensively studied commensurate charge density wave (CDW) phase of trigonaltantalum disulphide (1T-TaS$_2$), which develops at low temperatures with a$\left(\sqrt{13}\times\sqrt{13}\right)$R13.9{\deg} periodicity. Afull-dynamical analysis of the energy dependence of diffraction spotintensities reveals the entire crystallographic surface structure, i.e. thedetailed atomic positions within the outermost two trilayers consisting of 78atoms as well as the CDW stacking. The analysis is based on an unusually largedata set consisting of spectra for 128 inequivalent beams taken in the energyrange 20-250 eV and an excellent fit quality expressed by a bestfit PendryR-factor of R=0.110. The LEED intensity analysis reveals that the well-acceptedmodel of star-of-David-shaped clusters of Ta atoms for the bulk structure alsoholds for the outermost two TaS$_2$ trilayers. Specifically, in both layers theclusters of Ta atoms contract laterally by up to 0.25 $\r{A}$ and also slightlyrotate within the superstructure cell, causing respective distortions as wellas heavy bucklings (up to 0.23 $\r{A}$) in the adjacent sulphur layers. Mostimportantly, our analysis finds that the CDWs of the 1$^{\text{st}}$ and2$^{\text{nd}}$ trilayer are vertically aligned, while there is a lateral shiftof two units of the basic hexagonal lattice (6.71 $\r{A}$) between the2$^{\text{nd}}$ and 3$^{\text{rd}}$ trilayer. The results may contribute to abetter understanding of the intricate electronic structure of the referencecompound 1T-TaS$_2$ and guide the way to the analysis of complex structures insimilar quantum materials.

Surface structure and stacking of the commensurate $\left(\sqrt{13}\times\sqrt{13}\right)$R13.9{\deg} charge density wave phase of 1T-TaS$_2$(0001)