Semimetallic square-octagon two-dimensional polymer with high mobility

The electronic properties of $\pi$-conjugated two-dimensional (2D) polymers near the Fermi level are determined by structural topology and chemical composition. Thus tight-binding (TB) calculations of the corresponding fundamental network can be used to explore the parameter space to find configurations with intriguing properties before designing the atomistic 2D polymer network. The vertex-transitive fes lattice, which is also called a square-octagon, 4-8, or 4.8${}^2$ lattice, is rich in interesting topological features including Dirac points and flat bands. Herein, we study its electronic and topological properties within the TB framework using representative parameters for chemical systems. Secondly, we demonstrate that the rational implementation of band structure features obtained from TB calculations in 2D polymers is feasible with a family of 2D polymers possessing fes structure. A one-to-one band structure correspondence between the fundamental network and 2D polymers is found. Moreover, changing the relative length of linkers connecting the triangulene units in the 2D polymers reflects tuning of hopping parameters in the TB model. These perturbations allow sizable local band gaps to open at various positions in the Brillouin zone. From analysis of the Berry curvature flux, none of the polymers exhibits a large topologically nontrivial band gap. However, we find a particular configuration of semimetallic characteristics with separate electron and hole pockets, which possess very low effective masses both for electrons (as small as $m_{\mathrm{e}}^{*}=0.05$) and for holes (as small as $m_{\mathrm{h}}^{*}=0.01$).

Figure 1

(a) The fes network with squares and octagons. Exemplary first- and second-neighbor hoppings within the square, $t_1^s$ and $t_2^{ss}$, as well as between squares, $t_1^o$ and $t_2^{so}$, are shown. Arrows indicate which sites are involved in the interaction. (b) The corresponding fes band structure with only first-neighbor hoppings ($t_1^o=t_1^s$). (c)–(e) Band structure details near the originally triply degenerate points at $\mathit{M}$ (red box) and $\mathrm{\Gamma }$ (blue box) when other interactions are added to $t_1^o=t_1^s$, i.e., (c) first-neighbor hoppings and SOC, (d) under consideration of first- and second-neighbor hoppings, and (e) first- and second-neighbor hoppings and SOC. Yellow boxes show small band gap openings due to SOC. Band structures are obtained from TB; for details on hopping parameters, see SM Sec. S1 [39].

Figure 2

The fes band structures and corresponding Chern numbers with different hopping parameters for intrasquare and intersquare interactions. The strength of SOC considered is $\lambda =2\times {10}^{-5}\mathrm{eV}$: (a) $t_1^s=t_1^o=-3.438$ eV, $t_2^{ss}=t_2^{so}=-0.126$ eV, (b) $t_1^s=-3.438$ eV, $t_1^o=-4.599$ eV, $t_2^{ss}=-0.126$ eV, $t_2^{so}=-0.185$ eV, and (c) $t_1^s=-3.438$ eV, $t_1^o=-2.562$ eV, $t_2^{ss}=-0.126$ eV, $t_2^{so}=-0.086$ eV. Dash-dotted, dashed, and dotted lines indicate the Fermi level for $1/4$, $1/2$, and $3/4$ fillings, respectively.

Figure 3

(a) Atomistic structure of a hypothetical 2D polymer with the underlying fes net topology. Light blue square and orange areas indicate the position of the square and octagon of the fes net, respectively. Triangulene connectors (red circle) are put on vertex positions of the fundamental fes network, while two types of linkers are put on the square and octagon edges. The relative interaction strength of connectors (effectively corresponding to relative strength of hopping parameters in the TB picture) can be controlled by leaving one linker unchanged and changing the length of the other linker. Here, the number of ${\mathrm{C}}_2$ units in the linear linker (blue circle) is changed to control its length. Hydrogen, carbon, and oxygen atoms are shown in white, gray, and red, respectively. (b)–(e) Band structures of the hypothetical 2D polymers with different number of ${\mathrm{C}}_2$ units as linkers connecting the squares from first principles. The black dashed line indicates the Fermi level. For (b), the electronic effective mass is $m_{\mathrm{e}}^{*}=0.01$ (blue band, $\mathrm{\Gamma }$ point) and for holes the electronic effective mass is $m_{\mathrm{h}}^{*}=0.05$ (purple band, $\mathit{M}$ point).

Figure 4

Extracted TB band structure and Berry flux ${\mathrm{\Omega }}_n\left(\mathbf{k}\right)$ around $\mathit{M}$ of the hypothetical 2D polymer with carbonyl bridge group. (a) Without a linear linker between triangulene units [cf. Fig. 3] and (b) with two ${\mathrm{C}}_2$ units as linear linker [cf. Fig. 3]. The colored frames of the Berry curvature plots indicate the corresponding band. In the structure without a linear linker, the lower two bands (green and blue) form a Dirac cone at $\mathit{M}$ and consequently have an enhanced Berry curvature flux in the vicinity of that point. Albeit very close in energy, the third band (purple) exhibits almost no Berry curvature flux. By inserting the linker (b), the Berry curvature flux is transported to higher-lying bands in the spectrum, bringing it closer to the Fermi level. See Table S6 in the SM [39] for TB parameters.