Momentum-selective optical absorption in triptycene molecular membrane

The optical properties of triptycene molecular membranes (TMMs) under the linearly and circularly polarized light irradiation have been theoretically studied. Since TMMs have the double-layered kagome lattice structures for their $\pi$ electrons, i.e., tiling of trigonal- and hexagonal-symmetric rings, the electronic band structures of TMMs have nonequivalent Dirac cones and perfect flat bands. By constructing the tight-binding model to describe the $\pi$-electronic states of TMMs, we have evaluated the optical absorption intensities and valley selective excitation of TMMs based on the Kubo formula. It is found that absorption intensities crucially depend on both the light polarization angle and the excitation position in momentum space, i.e., the momentum and valley selective optical excitation. The polarization dependence and optical selection rules are also clarified by using group theoretical analyses.

Figure 1

(a) Schematic structure of a triptycene molecule. The black and white carbons indicate ${sp}^2$ and ${sp}^3$ carbons, respectively. Each ${sp}^2$ carbon gives one $\pi$ electron. (b) Top view of the triptycene molecule together with the $\pi$ orbital at each ${sp}^2$ carbon. $\pi$ orbitals construct the triangle network drawn with green dashed lines. Schematic structure of (c) zigzag TMM and (d) armchair TMM. The yellow shaded rhombus indicates the unit cell of the TMMs. ${\mathit{a}}_{\mathit{1}}=(a,0)$, ${\mathit{a}}_{\mathit{2}}=(-a/2,\sqrt{3}a/2)$ are primitive vectors, where $a=8.92\AA$ for zigzag TMM and $a=22.17\AA$ for armchair TMM, respectively. The zigzag and armchair TMMs have 18 and $54\pi$-electronic sites in their unit cells, respectively. (e) First BZ of TMMs.

Figure 2

(a) Energy band structure of zigzag TMM together with the corresponding DOS on the basis of the tight-binding model. Black and cyan lines in the energy band structure indicate bonding and antibonding states between the upper and lower layers, respectively. Note that the optical transition between black and cyan subbands is prohibited. The representative optical transitions from the highest valence subband are named $T_1$, $T_2$, and $T_3$. (b) Left panel: Corresponding absorption spectrum of $T_1$, $T_2$, and $T_3$ optical transitions. The $k$ integration is performed within the whole first BZ. Right panel: Same for $T_1$, $T_2$, and $T_3$ optical transitions, but the $k$ integration is performed in the $1/3$ regions of the first BZ which separately contain $M_1$, $M_2$, and $M_3$ points. The angular dependence of absorption intensity at (c) $\mathrm{\Gamma }$, (d) $M_2$, and (e) $K_2$ points.

Figure 3

(a) Upper panel: Optical absorption spectrum of zigzag TMM under the circularly polarized light irradiation. The three dashed lines indicate the representative optical transition caused by $T_1$ at the $M$ point, $T_2$ at the $M$ point, and $T_2+T_3$ at the $K$ point, from the left. The integration is evaluated within the $1/6$ region of the first BZ containing the $K_2$ point. Lower panel: The difference of optical absorption between LCP and RCP. The dashed and solid lines correspond to the cases of $k$ integration for the $1/6$ regions of the BZ containing $K_1$ and $K_2$, respectively. Since TMM preserves both time-reversal and inversion symmetries, the $K_1$ and $K_2$ states polarize oppositely, i.e., no net valley polarization. (b) Momentum-space mapping of the absorption intensity difference between LCP and RCP, i.e., $\mathrm{\Delta }\alpha \left(\mathit{k}\right)$ for $T_1$ (upper panel) and $T_2$ (lower panel) transitions. (c) The distribution of the dipole vector in the momentum space for $T_1$ (upper panel) and $T_2$ (lower panel) transitions under the circularly polarized light irradiation. The green and orange arrows indicate the real and imaginary parts of the dipole vector, respectively.

Figure 4

(a) Energy band structure of armchair TMM together with DOS on the basis of the tight-binding model. The optical transitions from the highest valence subband to the lowest three conduction subbands are named $T_1$, $T_2$, and $T_3$. (b) Optical absorption spectrum of armchair TMM under linear (left panel) and circular (right panel) irradiation. For linear irradiation, $k$ integration is performed for the whole first BZ. There are three pronounced peaks at $\hslash \omega =2.76$, 2.84, and 2.94 eV. At the right side of the panel, the dominant optical transition and the high-symmetric points where the optical transitions occur are noted. For circular irradiation, $k$ integration is performed for the $1/6$ region of the first BZ containing $K_1$. (c)–(e) Momentum-space mapping of the absorption intensity difference between RCP and LCP, i.e., $\mathrm{\Delta }\alpha \left(\mathit{k}\right)$ for (c) $T_1$, (d) $T_2$, and (e) $T_3$ transitions.

Figure 5

(a) Schematic of the 2D kagome lattice. The yellow area represents the unit cell and the primitive vectors are ${\mathit{a}}_{\mathit{1}}=(a,0)$, ${\mathit{a}}_{\mathit{2}}=(-\frac{a}{2},\frac{\sqrt{3}a}{2})$. $-\gamma$ is the hopping parameter between the nearest-neighbor sites. (b) First BZ. (c) Energy band structure and DOS. The optical transition from ${\epsilon }_1$ to ${\epsilon }_2$ (${\epsilon }_3$) is defined as $T_1$ ($T_2$).

Figure 6

The real-space distribution of wave functions for the 2D Kagome lattice and corresponding angle dependence (bottom panel) of optical absorption at high-symmetric points: (a) $\mathrm{\Gamma }$ and (b) $M_2$. Black and white circles indicate the sign of the wave functions, and the radius of the circles indicates the amplitude of the wave functions. Red dots indicate the inversion center (IC) of the kagome lattice. (c) Same for the $K_2$ point. Here, the argument of complex wave function is indicated by green arrows. (d) Absorption intensity and polarization of the 2D kagome lattice, where $k$ integration for optical conductivity is performed (upper panel) for the whole first BZ (upper panel), for $1/3$ regions of the first BZ containing either the $M_1$, $M_2$ or $M_3$ point (middle panel), and for the $1/6$ region of the first BZ containing the $K_2$ point (lower panel).