Topologically Protected Conduction State at Carbon Foam Surfaces: An Ab initio Study

We report results of ab initio electronic structure and quantum conductance calculations indicating the emergence of conduction at the surface of semiconducting carbon foams. The occurrence of new conduction states is intimately linked to the topology of the surface and not limited to foams of elemental carbon. Our interpretation based on rehybridization theory indicates that conduction in the foam derives from first- and second-neighbor interactions between $p_{\parallel }$ orbitals lying in the surface plane, which are related to $p_{\perp }$ orbitals of graphene. The topologically protected conducting state occurs on bare and hydrogen-terminated foam surfaces and is thus unrelated to dangling bonds. Our results for carbon foam indicate that the conductance behavior may be further significantly modified by surface patterning.

Figure 1

Geometry of carbon foam and thin foam slabs. (a) Structure of bulk foam and that of an individual foam cell in side and top view, allowing us to distinguish $s{p}^2$ and $s{p}^3$ sites. The dashed line shows the long cell axis. The perspective view in the right panel depicts a larger bulk segment. Structure of (b) an $s{p}^3$-terminated and (c) an $s{p}^2$-terminated thin foam slab. The tilted view used in left panels depicts the structure and primitive unit cells. The right panels in (b) and (c) are side views of the structure that better illustrate the type of termination and illustrate partial hydrogen coverage.

Figure 2

Electronic structure of thin foam slabs. DFT-based band structure of a thin, hydrogen-covered foam slab with (a) $s{p}^3$ and (b) $s{p}^2$ termination on both sides. (c) Charge distribution in the $s{p}^2$-terminated slab corresponding to states in the energy range $E_F-2\mathrm{eV}<E<E_F+2\mathrm{eV}$, indicated by shading in (b). (d) Conductance $G$ of $s{p}^3$- and $s{p}^2$-terminated slabs along the armchair direction, in units of the conduction quantum $G_0$.

Figure 3

Interpretation of electronic structure results for the $s{p}^2$-terminated slab (a)–(c) and bulk foam (d)–(f) using the LCAO technique. Band structure results considering only nearest-neighbor interaction [(a) and (d)], results of calculations that also include second-neighbor interaction between $p_{\parallel }$ states [(b) and (e)], and the corresponding densities of states [(c) and (f)]. (g) High-symmetry points in the slab and bulk Brillouin zones. (h) Rehybridized orbitals used in the calculation. $p_{\parallel }$ orbitals, shown in darker (green) shade, are responsible for conduction. (i) Tilted top view of the second neighbor-interaction between $p_{\parallel }$ states that dominate band dispersion near $E_F$. Connectivity diagram for the bulk (j) and thin foam slabs (k) that helps explain the semiconducting behavior of bulk and $s{p}^3$-terminated foam, and the origin of the conducting state at the $s{p}^2$-terminated surface.