Gapless edge states in (C,O,H)-built molecular system with p-stacking and hydrogen bonds

Abstract

The gapless edge states have been found in a 2D molecular system built with light atoms: C,O,H. This prediction is done on the basis of combined density functional theory (DFT) and tight-binding calculations. The system does not exhibit any effect of the spin-orbit coupling (SOC), neither intrinsic nor Rashba type. The band structure and the edge states are tuned with a strength of the p-stacking and O...H interactions. The elementary cell of this noncovalent structure, does not have the 3D inversion or rotational symmetry. Instead, the system transforms via a superposition of two reflections: with respect to the xz and xy mirror planes, both containing the non-periodic direction. This superposition is equivalent to the inversion in the 2D subspace, in which the system is periodic. The energy gap obtained with the DFT method is 0.11 eV, and largely opens (above 1 eV) with the GW and hybrid-DFT approaches. The bands inversion is partial, i.e. the Bloch states are mixed, with the "swapping" and "non-swapping" atomic contributions.

Figure 1

The chemical structure of the studied phenalene derivative.

Figure 2

The geometries of five molecular wires, called accordingly to the stacking patterns: “on-top”, “rotated”, “rotated-shifted”, “flipped” and “flipped-shifted”, as well as their band structures, for three intermolecular stacking distances. The pressures are given above the corresponding panels. The Fermi level is at zero energy.

Figure 3

Band structures of a wire with the “rotated-shifted” molecular order, for a chosen intermolecular distance of 2.3 Å. The results were obtained with the DFT method and the GW approach for two electric-field polarizations: perpendicular and parallel to the axis of the π-stack. The calculated points are marked with circles and the band-lines are interpolated.

Figure 4

The geometric parameters of the 2D structure formed by a π-stacked wires with the “rotated-shifted” molecular order. The planar intermolecular hydrogen-bonds connect the columns. The band structure was obtained with the DFT method for d _π-stack = 2.58 Å and d _OH = 1.76 Å.

Figure 5

Two mirror planes in the 2D structure formed by the π-stacked wires with the “rotated-shifted” molecular order, which are repeated along the hydrogen intermolecular bonds.

Figure 6

The edge states in the 2D structure presented in Fig. 4. The thickness of the π-stacked layers was 50 elementary cells along the z-axis. The y-axis along O-H bonds was treated periodically. The intermolecular distances are given above the corresponding plots.

Figure 7

Visualization of the highest-energy occupied eigenvectors of the tight-binding Hamiltonian of the 2D structure presented in Fig. 4. The π-stacking and O-H distances were varied. The matrix elements involving the Wannier functions were included when $\langle W_m|H|W_n\rangle \ge 0.01\mathrm{eV}$.

Figure 8

The case with gapless edge states: d _π-stack = 2.3 Å and d _OH = 1.76 Å. Squares of the highest occupied Bloch state (HOBS) and lowest unoccupied Bloch state (LUBS), at three chosen k-points: k_A = 0.3(3)π, k_B = 0.3 π, k_C = 0.36(6)π, are plotted for two chosen 2D slices - just below the upper molecule (M1) and above the lower molecule (M2) in the elementary cell. All plots are done with the same scale for the atomic coefficients. As a reference, a scheleton of the upper molecule is displayed in all panels.