Density functional investigation of rhombohedral stacks of graphene: Topological surface states, nonlinear dielectric response, and bulk limit

A comprehensive density-functional theory (DFT)-based investigation of rhombohedral (ABC)-type graphene stacks with finite and infinite layer numbers and zero or finite static electric fields applied perpendicular to the surface is presented. Electronic band structures and field-induced charge densities are critically compared with related literature data including tight-binding and DFT approaches as well as with our own results on (AB) stacks. It is found that the undoped AB bilayer has a tiny Fermi line consisting of one electron pocket around the K point and one hole pocket on the line K-$\Gamma$. In contrast to (AB) stacks, the breaking of translational symmetry by the surface of finite (ABC) stacks produces a gap in the bulklike states for slabs up to a yet unknown critical thickness $N^{\mathrm{semimet}}\gg 10$, while ideal (ABC) bulk ($\beta$ graphite) is a semimetal. Unlike in (AB) stacks, the ground state of (ABC) stacks is shown to be topologically nontrivial in the absence of an external electric field. Consequently, surface states crossing the Fermi level must unavoidably exist in the case of (ABC)-type stacking, which is not the case in (AB)-type stacks. These surface states in conjunction with the mentioned gap in the bulklike states have two major implications. First, electronic transport parallel to the slab is confined to a surface region up to the critical layer number $N^{\mathrm{semimet}}$. Related implications are expected for stacking domain walls and grain boundaries. Second, the electronic properties of (ABC) stacks are highly tunable by an external electric field. In particular, the dielectric response is found to be strongly nonlinear and can, e.g., be used to discriminate slabs with different layer numbers. Thus, (ABC) stacks rather than (AB) stacks with more than two layers should be of potential interest for applications relying on the tunability by an electric field.

Figure 1

Sketch of the considered stackings of graphene. Left panel: Graphene layer; central panel: (AB) stacking; right panel: (ABC) stacking. The (AB) stacking involves continuous lines of $p_z$ bonds for one kind of atoms and dangling bonds for the other atoms, while the (ABC) stacking is characterized by dimer-like $p_z$ bonds.

Figure 2

Band gap of AB bilayers vs external field: Comparison of our FPLO-slab results (bullets) with several supercell approaches and with experimental data ( + signs; Ref. 36). QE results (this work) are denoted by open triangles (up: without dipole corrections; down: with dipole corrections), literature data by squares (Ref. 11), diamonds (Ref. 37), and crosses (Ref. 13), respectively. Lines are intended to guide the eye. The insert is a blowup of the low-field region marked in the main figure. All curves show the minimum gap except that by Aoki and Amawashi,[13] where the gap at the K point is given.

Figure 3

DFT band structure for AB bilayer (a, e), ABC trilayer (b, f), ABCA tetralayer (c, g), and ABCAB pentalayer (d, h). Panels (a)–(d) refer to zero field and (e)–(h) to $E_{\mathrm{ext}}=0.08$ V/Å. The inserts show the subtleties around the Fermi level, which lies at zero energy (dashed lines). The fine but still limited k mesh yields a slightly shifted position of the Fermi level in panels (a) and (b). A higher resolution for the bilayer case is given in the following figures. In the case of the trilayer we note that the Fermi level should be situated exactly at the band crossing. The band structures are shown close to the symmetry point K (vertical lines) along the lines K $\to$ M and K $\to \Gamma$ as indicated below the figure. In the main figures, 20$\%$ of the line K-M and 10$\%$ of the line K-$\Gamma$ are shown, while the inserts are restricted to 5$\%$ of K-M and 2.5$\%$ of K-$\Gamma$. The energy scale on the left refers to the main figures; the energy scale on the right refers to the inserts.

Figure 4

Blow up of the band structure of the AB bilayer without external field around the K point. The left end of the plot lies $0.5\%$ of the distance K-$\Gamma$ from the K point toward the $\Gamma$ point, and the right end $0.5\%$ of the distance K-M toward the M point.

Figure 5

Density of states of the AB bilayer on the meV scale. The minor peaks originate from the finite resolution of the k mesh.

Figure 6

3D plot of the band structure of the AB bilayer without external field around the K point. The Fermi lines of the tiny electron (around K) and hole (on the line K-$\Gamma$) pockets are indicated. The data originate from the same calculation as those of Fig. 4. Note that the scaling of the two k axes differs by a factor of about 2.

Figure 7

Minimum band gap for AB bilayer and 3, 4, and 5-layer (ABC)-type graphene systems vs $E_{\mathrm{ext}}$.

Figure 8

Energy bands around the K point of a 10-layer (ABC) stack (black, thin lines) together with the (ABC) bulk bands on a part of the lines M-K-$\Gamma$ (red, thick lines) and L-H-A (blue, dashed lines) lying on the bottom and the top plane of the hexagonal Brillouin zone. Along K-M and H-L about 24$\%$ of the whole line are shown, and along K-$\Gamma$ and H-A about 12$\%$. The Fermi level of the bulk system is adjusted by 18 meV to get the correct balance between electron and hole pockets. For the sake of an easier comparison of slab and bulk calculations, the bulk calculation has been done in a hexagonal unit cell comprising six atoms, although a smaller rhombohedral cell with two atoms exists.

Figure 9

(ABC) bulk bands in the vicinity of the line K-H projected onto the M-K-$\Gamma$ plane in the Brillouin zone. Five discrete $k_z$ values were adopted as shown in the legend. Along K-M and H-L about 24$\%$ of the whole line are shown, and along K-$\Gamma$ and H-A about 12$\%$. The Fermi level is adjusted by 18 meV to get the correct balance between electron and hole pockets. The blue and the red curves (from the top and the bottom plane of the hexagonal Brillouin zone, respectively) with the parabola-like maxima/minima are twofold degenerate in addition to spin degeneracy.

Figure 10

Brillouin zone of (ABC) graphene. (a) Three Brillouin zones of the corresponding real-space conventional six-atom hexagonal cell are shown together with the rhombohedral trigonal (RTG) BZ. High-symmetry points of the hexagonal cell are denoted with white labels and the points X and T of the RTG cell are denoted with black labels. Note that the RTG and the hexagonal L point are identical. The RTG X-point falls onto the hexagonal M point of the translated hexagonal cells, which means that bands seen at X in the rhombohedral band structure will be folded onto M in the hexagonal setup, which contains three times as many atoms as the RTG setup. In the same way the RTG T point is folded onto the hexagonal A point. (b) The path through the RTG BZ as used in the rhombohedral band structure of Fig. 11 (upper panel) is shown. Uppercase labels denote rhombohedral high-symmetry points, and lowercase labels denote symmetry points in the hexagonal BZ. Note that the paths X-U-k-$\Gamma$ and L-h-W-T form straight lines when continued into adjacent BZs. The segment T-$\Gamma$ does not cut the BZ boundary in any high-symmetry point and was added to compare with the band structure published in Ref. 24.

Figure 11

Band structure calculated in the rhombohedral two-atom setup. Upper panel: Band structure along the lines defined in Fig. 10b; lower panel: band structure along lines of the three hexagonal BZs shown in Fig. 10a. Black lines denote bands of the central BZ, green and red lines denote bands of the upper and lower BZ, respectively. This composed band structure is equivalent to a back-folded band structure obtained in a six-atom hexagonal setup using the first hexagonal BZ only.

Figure 12

Planar average of the induced electron density for an AB bilayer compared with six-layer (AB) and (ABC) stacks as a function of $z$ (coordinate perpendicular to the slab). The external field $E_{\mathrm{ext}}=0.1$ V/Å points from the left to the right. The dashed vertical lines indicate the positions of the carbon planes.

Figure 13

Partition of the total dipole moment $p_z$ (given in atomic units e$·$a${}_0$ = 8.478$\times {10}^{-30}$ As$·$m) of a bilayer into inter- and intralayer contributions. The total dipole moment of a single layer, which is only due to intralayer polarization, is given for comparison. Lines guide the eye.

Figure 14

Chemical electron transfer $N_{\mathrm{chem}}$ (upper panels) and physical electron transfer $\Delta N_{\mathrm{phys}}$ (lower panels) in five-layer graphene slabs with (ABC) stacking (left panels) and (AB) stacking (right panels). The excess charge is proportional to the line width of the circles. For better graphical presentation, the physical transfer is enlarged by a factor of 10 compared with the chemical transfer. Red (light) and blue (dark) stands for positive and negative values, respectively. Stacking direction is from top to bottom. The electric field of a strength $0.1$ V/Å points downward (external force on the electrons upward). The signs in the notation of A and B layer atoms refer to their location above or below the central layer. The numbers denote the position of the projection of the atomic position in the 2D unit cell. The green boxes mark the repetition unit in the slabs.

Figure 15

LDOS for the $p_z$ states of AB-bilayer atoms for vanishing and for finite external field, $E_{\mathrm{ext}}=0.1$ V/Å. The panels are arranged according to the atomic positions in the notation used in Fig. 14. Despite a dense mesh of points in $\mathbf{k}$ space with $600\times 600$ points in the 2D Brillouin zone and a linear interpolation of the energies and matrix elements within triangles, the visible numerical fluctuations could not be avoided. The Fermi level is at zero energy, and the gap at the finite field amounts to 0.074 eV. The energy grid spacing is 0.002 eV.

Figure 16

LDOS for the $p_z$ states of a 5-layer (ABC) slab for vanishing and for finite external field, $E_{\mathrm{ext}}=0.1$ V/Å. The panels are arranged in analogy to the atomic pattern in Fig. 14. The Fermi level is at zero energy, and the gap at the finite field amounts to 0.072 eV. The energy grid spacing is 0.002 eV.

Figure 17

Averaged nonlinear dielectric constant for the AB bilayer and for 3-, 4-, and 5-layer (ABC)-type graphene systems vs $E_{\mathrm{ext}}$.