Carbon Kagome Lattice and Orbital-Frustration-Induced Metal-Insulator Transition for Optoelectronics

A three-dimensional elemental carbon kagome lattice, made of only fourfold-coordinated carbon atoms, is proposed based on first-principles calculations. Despite the existence of 60° bond angles in the triangle rings, widely perceived to be energetically unfavorable, the carbon kagome lattice is found to display exceptional stability comparable to that of ${\mathrm{C}}_{60}$. The system allows us to study the effects of triangular frustration on the electronic properties of realistic solids, and it demonstrates a metal-insulator transition from that of graphene to a direct gap semiconductor in the visible blue region. By minimizing $s\text{−}p$ orbital hybridization, which is an intrinsic property of carbon, not only the band edge states become nearly purely frustrated $p$ states, but also the band structure is qualitatively different from any known bulk elemental semiconductors. For example, the optical properties are similar to those of direct-gap semiconductors GaN and ZnO, whereas the effective masses are comparable to or smaller than those of Si.

Figure 1

Orthogonal $p$ orbitals on a kagome lattice. (a),(b) In-plane $p_1$ and $p_2$ and (c) out-of-plane $p_3$ orbitals. (d)–(f) Three states derived from the $p_2$ orbital by applying symmetry operations, corresponding to the wave functions ${\psi }_1$, ${\psi }_2$, and ${\psi }_3$, respectively. I, II, and III are three lattice sites of the primitive cells (dotted rhombic lines). Yellow and blue ellipsoids represent negative and positive lobes of $p$ orbitals, respectively.

Figure 2

Crystal structures. (a) CKL. The unit cell consists of six C atoms in the form of two linked triangles. Each pair of (same color) atoms forms a zigzag chain in the vertical direction. (b) A schematic kagome lattice for the CKL, where G-1, G-2, and G-3 form three interlocked graphene lattices. Note that here each zigzag chain in the real structure is condensed to a lattice point. (c),(d) Same for the IGN with same legends. Notice the disappearance of G-3 in the IGN.

Figure 3

Electronic and optical properties. (a) Band structure of CKL. (b) Effective masses of CKL, ZnO, GaN, and Si calculated using the density-functional theory and Perdew-Burke-Ernzerhof method. For CKL, ZnO, and GaN, $m_e^{//}$ and $m_e^{\perp }$ are the in-plane and $c$-axis effective masses of electrons, $m_{hh}^{//}$ and $m_{lh}^{//}$ are the in-plane effective masses of heavy and light holes, whereas $m_h^{\perp }$ is the $c$-axis effective mass of heavy hole. For Si, $m_e^{//}$ and $m_e^{\perp }$ are the transverse and longitudinal masses of electrons, $m_{hh}^{//}$ and $m_{lh}^{//}$ are the effective masses of heavy and light holes along [001], whereas $m_h^{\perp }$ is the effective mass of heavy hole along [111] [40]. (c) Imaginary part of dielectric function ${ϵ}_2$ as a function of $E-E_{\text{gap}}$ for CKL, ZnO, GaN, and diamond, where $E_{\text{gap}}$ is the band gap. (d) Partial density of states (PDOS) of CKL, in which red and blue lines represent $s$ and $p$ states, respectively.

Figure 4

Band structure evolution with increased orbital frustration. (a) Zigzag carbon chain, (b) IGN, (c)–(e) intermediate structures, and (f) CKL. Colored bands in (a) split into three (red, blue, and green) bands in (b)–(f). (g) Wave functions for ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_{11}$, and ${\mathrm{\Gamma }}_{12}/{\mathrm{\Gamma }}_{13}$ states, respectively. Note the correspondences: ${\mathrm{\Gamma }}_{11}$, Fig. 1; ${\mathrm{\Gamma }}_{12}$, Fig. 1; ${\mathrm{\Gamma }}_{13}$, Fig. 1. Also, having two atoms at each kagome lattice site, the ${\mathrm{\Gamma }}_{11}$ state here appears “fatter” than the one in Fig. 1. The banana-shaped contours for ${\mathrm{\Gamma }}_{12}/{\mathrm{\Gamma }}_{13}$ are a result of in-phase addition of the schematic wave functions in Figs. 1 and 1. (h) Wave functions for ${\mathrm{\Gamma }}_2$, ${\mathrm{\Gamma }}_{21}$ [at $E=13.9\mathrm{eV}$ in (f)], and ${\mathrm{\Gamma }}_{22}/{\mathrm{\Gamma }}_{23}$.