Funnel-shaped electronic structure and enhanced thermoelectric performance in ultralight ${\mathrm{C}}_x{\left(\mathrm{BN}\right)}_{1\text{−}x}$ biphenylene networks

Carbon biphenylene has stimulated substantial research because of extraordinary properties introduced by the metallic character, e.g., the ultrahigh electron thermal transport. Here, inspired by the synthesis of carbon biphenylene [Fan et al., Science 372, 852 (2021)], we identify the stability of ${\mathrm{C}}_x{\left(\mathrm{BN}\right)}_{1\text{−}\mathrm{x}}$ biphenylene as $\text{CBN}$ and ${\mathrm{C}}_4\mathrm{BN}$ semiconductors with four-, six-, and eight-membered periodic rings of irregularly $s{p}^2$-hybridized atoms via structural searches. Unexpectedly, we confirm that $\text{CBN}$ biphenylene exhibits a peculiar funnel-shaped band structure, which is a direct consequence of the delocalization/localization of $\pi$ bonds formed by B-, N-, or C-$p_z$ electrons. The band structure greatly improves the thermoelectric performance by enhancing the power factor, although the lattice thermal conductivity is relatively large after including four-phonon scattering resistance because of low atomic masses. The similar behaviors are absent in ${\mathrm{C}}_4\mathrm{BN}$ biphenylene because of the localization of $\pi$ bonds formed by C-$p_z$ electrons, although with a stronger anharmonicity and thus a lower lattice thermal conductivity. The anomalous power factor can be explained by the constant $\tau$ approximation: the $p$-type doping controls the carrier group velocities and thus realizes the tunability of tensor ratio $K_1/K_0$. Our analysis suggests that the funnel-shaped electronic structure could be reproduced in two-dimensional semiconductor systems with the small electronegativity difference and the comparable stoichiometry. Our work realizes the thermoelectric improvement through controlling the shape of band structure, which provides insights for designing promising two-dimensional thermoelectric materials.

Figure 1

(a) Structural schemes of CBN and ${\mathrm{C}}_4\mathrm{BN}$ biphenylene with the electron localization functions. (b) Infrared response spectra as well as vibrational modes of the main peaks. (c) Lattice thermal conductivity versus temperature including three-phonon scattering as well as four-phonon scattering along the armchair (solid) and zigzag (dashed) directions. (d) Three- and four-phonon scattering rates of allowed scattering channels at 300 K.

Figure 2

Figure of $ZT$ along the armchair direction, the electrical conductivity $\sigma$, and the Seebeck coefficient $S$ along the armchair (solid line) and zigzag (dashed line) directions versus the chemical potential $E_f$ and temperature $T$ in (a) CBN and (b) ${\mathrm{C}}_4\mathrm{BN}$ biphenylene, together with the relative percentage ${\kappa }_L/{\kappa }_{\text{total}}$ in the insets.

Figure 3

HSE06 three-dimensional band structures near the Fermi level, projected band structures along the high-symmetry path of the Brillouin zone (red, blue, and green symbols represent the inserted C-, N-, and B-$p_z$ orbitals, respectively), electronic density of states, and spatial distributions of valence band edge states at the high-symmetry points in (a) CBN and (b) ${\mathrm{C}}_4\mathrm{BN}$ biphenylene networks.

Figure 4

Atomic orbital hybridization near the $E_f$ in (a) CBN and (b) ${\mathrm{C}}_4\mathrm{BN}$ biphenylene networks, where $p_x, p_y$, and $p_z$ orbitals of C, B, and N atoms and symmetry are labeled.