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

By quantitative low-energy electron diffraction (LEED) we investigate the extensively studied commensurate charge density wave (CDW) phase of trigonal tantalum disulphide $(1T-{\mathrm{TaS}}_2)$, which develops at low temperatures with a $(\sqrt{13}\times \sqrt{13})R13.9^{\circ }$ periodicity. A full-dynamical analysis of the energy dependence of diffraction spot intensities reveals the entire crystallographic surface structure, i.e., the detailed atomic positions within the outermost two trilayers consisting of 78 atoms as well as the CDW stacking. The analysis is based on an unusually large data set consisting of spectra for 128 inequivalent beams taken in the energy range 20–250 eV and an excellent fit quality expressed by a best-fit Pendry $R$ factor of $R=0.110$. The LEED intensity analysis reveals that the well-accepted model of star-of-David-shaped clusters of Ta atoms for the bulk structure also holds for the outermost two ${\mathrm{TaS}}_2$ trilayers. Specifically, in both layers the clusters of Ta atoms contract laterally by up to 0.25 Å and also slightly rotate within the superstructure cell, causing respective distortions as well as heavy bucklings (up to 0.23 Å) in the adjacent sulfur layers. Most importantly, our analysis finds that the CDWs of the first and second trilayers are vertically aligned, while there is a lateral shift of two units of the basic hexagonal lattice (6.71 Å) between the second and third trilayers. The results may contribute to a better understanding of the intricate electronic structure of the reference compound $1T-{\mathrm{TaS}}_2$ and guide the way to the analysis of complex structures in similar quantum materials.

Figure 1

(a), (b) LEED pattern of the $(\sqrt{13}\times \sqrt{13})R13.9^{\circ }$ phase for different energies. (c), (d) Comparison of raw experimental IV spectra for beam triples indicated in the LEED image (a) linked by a ${120}^{\circ }$ rotation. For beam labeling, see text.

Figure 2

(a) Ball model of a ${\mathrm{TaS}}_2$ trilayer with the two symmetry-equivalent unit meshes of the $(\sqrt{13}\times \sqrt{13})R13.9^{\circ }$ CDW phase (yellow, cyan) and that of the unreconstructed ($1\times 1$) trilayer (white). Also shown in light purple is the hexagonal Wigner-Seitz (WS) cell for the yellow unit cell. (b) Labeling of Ta and S sites within the trilayer WS cell. (c) Alternative CDW unit cell (light green) illustrating the 13 different registry positions, where the central Ta atoms of the next (higher) trilayer can sit. (d) Projection of the registry positions into the WS cell.

Figure 3

Selection of experimental and best-fit IV spectra with single-beam $R$ factors close to the overall $R$ factor $R=0.110$ of the analysis. For beam labeling, see text.

Figure 4

$R$ factor values calculated within the displayed energy increments of 20-eV width. Also shown is the overall $R$ factor level and that for energies above 100 eV. The color density of the columns corresponds to the statistical weight of the respective energy range.

Figure 5

Comparison of calculated spectra for beam triples linked by a ${120}^{\circ }$ rotation.

Figure 6

Top row: error curves for layer distances (left) and vibrational amplitudes (right). Below: error curves for vertical (left) and rotational lateral displacements (right) of exemplary triples of atoms (S4, Ta2b) within the first (middle row) and second trilayer (bottom row).

Figure 7

(a) Ball model of the best-fit structure in top view. The atomic displacements from the unreconstructed positions are five times enlarged for better visibility. Also shown are the $(\sqrt{13}\times \sqrt{13})R13.9^{\circ }$ unit mesh (yellow) and the star-of-David formation of Ta atoms (white). (b) Side view in direction of a superstructure unit vector (indicated by black arrows above) and the stacking sequence of layers (blue).

Figure 8

Schematic representation of the layer reconstruction within surface (left) and subsurface (“bulk”) trilayers (right) as derived from the LEED-IV analysis. Vertical displacements are color coded, size and direction of the lateral ones are indicated by solid arrows (lengths according to the scale bars). Dashed arrows correspond to the XRD analysis of Brouwer and Jellinek [23].

Figure 9

(a) Room-temperature STM survey image of the ${\mathrm{TaS}}_2$ sample taken 30 h after cleavage (${\mathrm{U}}_{\text{sample}}=-90$ mV; $I=1.3$ nA). (b), (c) Same as (a) but with atomic resolution. (${\mathrm{U}}_{\text{sample}}=+5$ mV; $I=3.4$ nA).

Figure 10

Comparison of spectra calculated for different second to third layer stacking (blue) with the experiment (red) for two selected beams. The displayed single-beam $R$ factors refer to the shown energy range 150–250 eV. Note that the fitting stage corresponds to the first line in the lower part of Table 2, i.e., the spectra shown for the favorite {7,8,11} stacking class (green boxes) are not identical to the best-fit spectra given in Fig. 3 and the Supplemental Material.