Electronic structure of stacking faults in rhombohedral graphite

The electronic structure of stacking faults and surfaces without and with an additional displaced layer is calculated for the case of rhombohedral (ABC) graphite. The full-potential local-orbital code and the generalized gradient approximation to density functional theory are used. All considered surfaces and interfaces induce surface/interface bands. All discovered surface and interface bands are restricted to the vicinity of the symmetry line $K-M$ in the two-dimensional Brillouin zone. There are groups of localized band pairs around $\pm 0$, $\pm 0.2$, and $\pm 0.6$ eV for one of the two considered types of stacking faults; $\pm 0$ and $\pm 0.5$ eV for the other type and for a displaced surface layer. At the $K$ point in the Brillouin zone, there is a one-to-one correspondence between these localized bands and the eigenvalues of those linear atomic clusters, which are produced by the perturbation of periodicity due to the displaced surface layer or due to the stacking faults. Some of the localized bands produce strong van Hove singularities in the local density of states near the surface or interface at energies up to several 0.1 eV. It is suggested to check these findings experimentally by appropriate spectroscopic methods. Undisturbed bulk (ABC) graphite is virtually a zero-gap semiconductor with a minute density of states at the Fermi energy. Both the surface and any of the considered stacking faults produce sharp peaks in the local density of states near the perturbation at energies of about 10 meV around the Fermi energy. This should provide a considerable contribution to the conductivity and its temperature dependence for samples with stacking faults or large surface-to-volume fraction.

Figure 1

The three basic layers in graphitic stacks presented in a hexagonal unit cell: A (blue), B (green), and C (red).

Figure 2

Band structure of bulk (ABC) graphite in the hexagonal unit cell shown in Fig. 1. The size of the symbols represents the related orbital weights.

Figure 3

DOS of bulk (ABC) graphite obtained by the linear tetrahedron method from $250\times 250\times 50$ $\mathbf{k}$ points in the hexagonal BZ.

Figure 4

Band structure of bulk (ABC) graphite projected onto a plane parallel to the layers. The discrete $k_z$ values given in the legend are in units $\overline{KH}$, e.g., the full red lines are the bands on the line $M\text{−}K\text{−}\Gamma$ in the 3D Brillouin zone, and the full black lines are bands on $L\text{−}H\text{−}A$. However, only 10% of the lines $M\text{−}K$ and $K-\Gamma$ in the ${\mathbf{k}}_{\parallel }$ space, centered around symmetry point $K$, are shown. The shaded region is quasicontinuously filled with projected bands.

Figure 5

Examples for the two types of SFs in highly symmetric supercell geometry. Left panel, type $\alpha$: (vacuum)${}_3$(ABC)${}_n$(BCA)${}_n$(vacuum)${}_3$ and right panel, type $\beta$: (vacuum)${}_3$C(ABC)${}_n$(BAC)${}_n$(vacuum)${}_3$. This figure is for $n=2$, but the numerical calculation was done for $n=4$ in either case. The plot shows the atoms in a plane parallel to the paper plane of Fig. 1. The yellow perpendicular bars indicate saturated $p_z$ bonds. The magenta stars denote inversion centers and the magenta horizontal line denotes a mirror plane. The vertical colored bars to the right of the figures indicate equivalent unit cells of the bulk lattice. The symmetry of $\alpha$ and $\beta$ slabs are $PB3M1$ and $PB6M2$, respectively.

Figure 6

Band structure of the $\alpha$-type slab (ABC)${}_4$(BCA)${}_4$ in the vicinity of the $K$ point. Orbital weights of $2p_z$ orbitals at some surface and interface atoms are indicated by the size of the related symbols. The lower panel shows a zoom to energies close to the Fermi level. For numbering of the lattice sites see Fig. 5. Shaded areas denote the projected bulk band structure.

Figure 7

Band structure of the $\alpha$-type slab (ABC)${}_4$(BCA)${}_4$ in the vicinity of the line $K\text{−}M$. Orbital weights of $2p_z$ orbitals at some interface atoms are indicated by the size of the related symbols. Only those weights are included, which amount to more than 50% of the maximum value. The symbol size is increased by a factor of 8/5 in comparison to Fig. 6. For numbering of the lattice sites see Fig. 5.

Figure 8

Local density of states (LDOS) for several sites close to the surface and interface of the $\alpha$-type slab (ABC)${}_4$(BCA)${}_4$. The data in the upper panel are obtained by the linear tetrahedron method with $500\times 500\times 1$ $\mathbf{k}$ points in the BZ. The zoomed version in the lower panel is calculated with a special grid of $200\times 200\times 1$ points located around the symmetry point $K$. The arrows mark the energies of the extrema in the localized bands at the $K$ point seen in the upper panel of Fig. 6.

Figure 9

Band structure of the $\beta$-type slab C(ABC)${}_4$(BAC)${}_4$ in the vicinity of the $K$ point. Orbital weights of $2p_z$ orbitals at some surface and interface atoms are indicated by the size of the related symbols. The lower panel shows a zoom to energies close to the Fermi level. For numbering of the lattice sites see Fig. 5. Shaded areas denote the projected bulk band structure.

Figure 10

Band structure of the $\beta$-type slab C(ABC)${}_4$(BAC)${}_4$ in the vicinity of the line $K\text{−}M$. Orbital weights of $2p_z$ orbitals at some interface atoms are indicated by the size of the related symbols. Only those weights are included which amount to more than 50% of the maximum value. The symbol size is increased by a factor of 8/5 in comparison to Fig. 9. For numbering of the lattice sites see Fig. 5.

Figure 11

Local density of states for several sites close to the surface and interface of the $\beta$-type slab C(ABC)${}_4$(BAC)${}_4$. The k grid is the same as in Fig. 8.

Figure 12

Example for a slab with displaced surface layers of type (CBC)(ABC)${}_n$(ABA). The structure is created by altering both surface layers of an (ABC)${}_{(n+2)}$ slab. Exactly speaking, the top layer, which used to be A in an (ABC)${}_{(n+2)}$ slab, was replaced by C, and the opposite surface was altered correspondingly from C to A to obtain a high symmetry. This figure is for $n=2$, but the numerical calculation was done for $n=6$.

Figure 13

Band structure of the slab (CBC)(ABC)${}_6$(ABA) with a displaced surface layer on either side in the vicinity of the $K$ point. Orbital weights of $2p_z$ orbitals at some surface and subsurface atoms are indicated by the size of the related symbols. The lower panel shows a zoom to energies close to the Fermi level. For numbering of the lattice sites see Fig. 12. Shaded areas denote the projected bulk band structure.

Figure 14

Local density of states for several sites close to the surface of the slab (CBC)(ABC)${}_6$(ABA) representing rhombohedral graphite with a displaced surface layer. The k grid is the same as in Fig. 8.