Multilayer graphenes with mixed stacking structure: Interplay of Bernal and rhombohedral stacking

We study the electronic structure of multilayer graphenes with a mixture of Bernal and rhombohedral stacking and propose a general scheme to understand the electronic band structure of an arbitrary configuration. The system can be viewed as a series of finite Bernal graphite sections connected by stacking faults. We find that the low-energy eigenstates are mostly localized in each Bernal section, and, thus, the whole spectrum is well approximated by a collection of the spectra of independent sections. The energy spectrum is categorized into linear, quadratic, and cubic bands corresponding to specific eigenstates of Bernal sections. The ensemble-averaged spectrum exhibits a number of characteristic discrete structures originating from finite Bernal sections or their combinations likely to appear in a random configuration. In the low-energy region, in particular, the spectrum is dominated by frequently appearing linear bands and quadratic bands with special band velocities or curvatures. In the higher-energy region, band edges frequently appear at some particular energies, giving optical absorption edges at the corresponding characteristic photon frequencies.

Figure 1

Lattice structures of (a) Bernal and (b) rhombohedral graphites. In each panel, the right figure is a top view, the middle is a schematic diagram of the lattice structure.

Figure 2

Example of Bernal-rhombohedral mixed graphene multilayers expressed by (5,7,4,1,3). Intralayer coupling by parameter ${\gamma }_0$ is shown as a vertical solid line, interlayer coupling by ${\gamma }_1$ is shown as a horizontal solid line.

Figure 3

(a) Decomposition of Bernal-rhombohedral mixed multilayers into incomplete Bernal sections. Dashed lines are connections between neighboring sections, which are to be switched off in the decomposition. (b) Decomposed incomplete Bernal section in the middle and (c) at one end of the whole system.

Figure 4

(Left) Energy bands of incomplete Bernal graphite with $N=8$. (Right) Some of wave functions from different categories at a small momentum along $p_x$ direction with $|p_x|\ll {\gamma }_1/v$. White and black circles represent positive and negative wave amplitudes, respectively, and the size is the absolute magnitude.

Figure 5

(a) Schematic of a stacking fault between the neighboring Bernal sections. (b) Structure of $(N_{i-1},2,N_{i+1})$.

Figure 6

Band structure and some wave functions of the graphene stack $(6,4,1,2)$. Dashed curves plotted for $p<0$ and $E>0$ represent the energy bands of the decomposed incomplete graphites, $N=6,4,1,$ and 2 ($N=6$ and 2 are the end-section type).

Figure 7

(a) Lists of the linear band velocity and the quadratic band mass in individual incomplete Bernal graphite sections from $N=1$ to 14. Solid and dashed bars represent the values of middle sections and end sections, respectively. (b) List of the same quantities for multilayer stacks of all possible configurations from a single layer to seven layers.

Figure 8

Averaged distribution of the velocity of low-energy linear bands in 100-layer Bernal-rhombohedral mixed graphite with the stacking-fault probability $p_s=0.1$, 0.2, and 0.4. Short horizontal bars attached to some major peaks are obtained by the approximation (see the text).

Figure 9

Averaged distribution of effective mass of low-energy quadratic bands similar to Fig. 8.

Figure 10

Averaged dynamical conductivity of 100-layer Bernal-rhombohedral mixed graphites with stacking-fault probability $p_s=0,0.2$, and 0.4. Histogram at the bottom is the averaged distribution of the band edges (band energies at zero momentum) in $p_s=0.2$.