Engineering Dirac cones and topological flat bands with organic molecules

We investigated polyhedral π-conjugated molecules with threefold rotation symmetry, which can be suitable building blocks for both Dirac cones and a topological flat-band system. The two dimensional network structures of such molecules can be characterized by intra- and intermolecular interactions. We constructed tight binding models of these structures by systematically changing the intra- $\left(t_0\right)$ and intermolecular $\left(t_1\right)$ transfer integrals as independent parameters. The degree of freedom for the interactions provides a topological flat band and a massless or massive Dirac cone. We analyzed the topological features and origins of the flat band and Dirac cone of a molecule-based triangular lattice.

Figure 1

Topological band structures of the model structures and polyhedral π-conjugated molecules. (a) Gapped Dirac cone of symmetry-broken honeycomb lattice. (b) Coexisting Dirac cone and flat band of kagome lattice. (c) and (d) Molecular structures and schematic images of crystal packings and band structures of polyhedral π-molecules $p$-TT (c) and NDI-$\mathrm{\Delta }$ (d). The three π-fragments that consist of molecules are blue.

Figure 2

Schematic images of molecular orbitals and signs of $t_0$. (a)–(c) The interaction can be categorized as Y-type molecules (a) and $\mathrm{\Delta }$-type molecules with odd (b) and even numbers (c) of nodes in π-fragments.

Figure 3

(a) Representative crystal packing of triangle lattice of Y-type molecules $o$-TT [52] and TTA [53]. The gray, white, red, and yellow spheres show carbon, hydrogen, oxygen, and sulfur atoms, respectively. (b) Schematic packing motif of polyhedral π-molecules. The left side shows the schematic image of $t_0$ and $t_1$ interactions between the π-fragments (gray circles with A, B, and C) of polyhedral molecules. The right side shows the hexagonal packing of intra- and intermolecular trimers of π-fragments (red triangles with α and blue triangles with β, respectively), and the vectors between them ($d$₁, $d$₂ and $d$₃).

Figure 4

Schematic image of an intermolecular MO overlaps. The sign of $t_1$ is determined by the overlap symmetry. (a) The case of $t_0<0$ with odd number of nodes in a trimer overlap. (b) The case of $t_0>0$ with even number of nodes.

Figure 5

Electronic band structures and Berry curvatures of polyhedral π-molecules in triangular lattices. (a)–(d) Calculated band structures with variable $t_0/t_1$ values, the signs of which are positive (a) and negative (b), and their spin Hall conductivity [(c) and (d)]. (e)–(h) 3D plots of the band structures for $t_0/t_1=+0.75$ [(e) and (g)] and −0.75 [(f) and (h)] with the color scale of Berry curvature without [(e) and (f)] and with SOC [(g) and (h)]. (i)–(l) Berry curvature of each band. The signs of $t_0$ and $t_1$ were set as $t_0<0$ and $t_1<0$ for $t_0/t_1>0$, and $t_0<0$ and $t_1>0$ for $t_0/t_1<0$.

Figure 6

Localized wavefunction that forms a topological flat band. The signs + and − represent the coefficients of $\pi \text{−}\mathrm{fragments}$. The coefficient of the π-fragment is zero when the MO is not shown.