Symmetry-directed electronic and optical properties in a two-dimensional square-lattice ZnPc-MOF

Abstract

The electronic structure of materials is fundamentally governed by their crystal symmetry. While most research on two-dimensional materials has focused on hexagonal lattices, such as graphene, hexagonal boron nitride, and transition metal dichalcogenides. This work explores a square-lattice system: the experimentally realized phthalocyanine-based metal-organic framework (ZnPc-MOF). Using group representation theory, we classify the electronic bands of ZnPc-MOF monolayer, AA- and AB-stacked bilayers, and twisted bilayers in terms of the irreducible representations (irreps) of their little groups. We find that bands in the AB-stacked bilayer remain two-fold degenerate along the $Y$ and $Y^{\prime}$ high-symmetry lines, as a consequence of the sole presence of two-dimensional irreps along these directions. We further derive optical transition selection rules to interpret the optical conductivity, revealing pronounced polarization-dependent optical responses. Additionally, we investigate the quasicrystalline electronic states in the 45$^{\circ}$ twisted bilayer (ZnPc-MOF quasicrystal) using the resonant coupling Hamiltonian. Compared to graphene quasicrystals, ZnPc-MOF quasicrystal exhibits weaker resonant coupling strengths, yet its quasicrystalline states lie closer to the Fermi energy, suggesting a greater contribution to low-energy electronic phenomena.

Figures (9)

• Figure 1: Structures of ZnPc-MOF (a) monolayer, (b)AA-stacked, (c) AB-stacked and (d) $36.87^{\circ}$ twisted bilayer $(m=2,n=1)$. Four unit cells are shown in (a)-(c), and one unit cell in (d). The middle point of each subfigure (Zn atom position) is as the origin when we construct the symmetry operation matrixes.
• Figure 2: Band structures of (a) monolayer, (b) AA-stacked, and (c) AB-stacked bilayer. Empty circles and solid red lines represent band structures obtained from DFT calculations and tight-binding model, respectively. The vdW correction is included in all DFT calculations for bilayer structures. The inset in (a) displays the BZ and high-symmetry points and lines.
• Figure 3: Irreps of electronic states at high-symmetry points and along high-symmetry lines for (a) monolayer, (b) AA-stacked, (c) AB-stacked, and (d) $36.78^\circ$ twisted bilayer. Irreps along high-symmetry lines are omitted in (d) due to the high density of closely spaced bands. The three bands of the monolayer in (a) are labeled $V_1$, $C_1$, and $C_2$. The six bands of the AA-stacked bilayer in (b) are labeled $V_{1\pm}$, $C_{1\pm}$, and $C_{2\pm}$, where $\pm$ denotes bonding/anti-bonding character. The six bands of the AB-stacked bilayer in (c) are labeled $V_1$, $V_2$, $C_1$, $C_2$, $C_3$, and $C_4$. In (a) and (c), shaded regions highlight $V_1 \rightarrow C_1$ interband transitions, with blue, red, and gray indicating transitions allowed by $x$-polarized, $y$-polarized, and both polarized lights, respectively. (e) and (f) show minimum transition energies for interband transitions $V_{1+} \rightarrow C_{i+}$ and $V_{1-} \rightarrow C_{i-}$ ($i = 1,2$) in the AA-stacked bilayer. (g) shows minimum and maximum transition energies for interband transition $C_{1+} \rightarrow C_{2+}$ in the AA-stacked bilayer. Blue, red, and gray arrows mark transitions excited by $x$-polarized, $y$-polarized, and both polarizations, respectively.
• Figure 4: Optical absorption spectra of charge neutral (a) monolayer, (b) AA-stacked, (c) AB-stacked, and (d) 36.87$^{\circ}$ twisted bilayer. The areas in shade for (b), (c) and (d) show the optical absorption spectra considering the relaxation. The vertical gray and green dashed lines show the starting energies of absorption steps and absorption peaks. The vertical blue dashed line in (b) shows the maximum transition energy for interband transition $C_{1+} \rightarrow C_{2+}$ as shown in Fig. \ref{['fig:band_irrep']} (g).
• Figure 5: Band structure comparison between rigid and relaxed structures for (a) AA-stacked, (b) AB-stacked, and (c) 36.87$^{\circ}$ twisted bilayer. Solid red and dashed green lines are the band structures of rigid and relaxed structures, respectively. The Fermi energy of the rigid structure is set to 0 eV.