Electronic structure of twisted graphene flakes

We study the electronic structure of bilayer graphene flakes in which the constituent layers are mutually rotated by some angle $\theta$. The large system sizes involved (up to ${10}^5$ carbon atoms) necessitate the use of a tight-binding approach in conjunction with Lanczos diagonalization. We find that a single moiré spot is sufficient for the low-energy density of states to resemble closely that of the periodic analog of such flakes, the graphene twist bilayer, implying that the low-energy physics in this system is well described as that of a “moiré quantum well” trapping low-energy graphene electrons. Furthermore, a graphene twist flake consisting of a single moiré unit cell leads already to electron localization on the AA “moiré spot,” in agreement with this moiré quantum well picture. The electron density fluctuations induced by the moiré lattice in twist graphene flakes are significant, being an order of magnitude greater than those generated by the rippling of suspended graphene. Finally, we determine the electronic properties of such flakes in the presence of an external magnetic field, finding a “zero-mode” structure and Landau states that exhibit an electron current well described as a charge flow on a torus situated at the AA regions of the moiré lattice.

Figure 1

Graphene twist flake. Black and red dots denote the first and second layers, respectively. Green (black) circles indicate the AB (AA) regions of the flake. The moiré periodicity $D$ is the distance between the centers of two AA spots.

Figure 2

Distribution of interlayer bond distances for three types of bilayer graphene structures: AA stacked (red), AB stacked (black), and twist sheet with rotation angle of $\theta =1.{05}^{\circ }$ (blue).

Figure 3

(a)–(f) Band structures of several graphene systems obtained with DFT (red), tight-binding method using the parametrization from this work (blue), and tight-binding method using the parametrization by Tang et al. (Ref. 25) (green).

Figure 4

Interlayer hopping integrals ($pp\sigma$) for several bilayer systems using the parametrization from this work: AA stacked (red), AB stacked (green), and twist sheet with rotation angle of $\theta =1.{05}^{\circ }$ (black). For comparison, the hopping integrals of an AB-stacked bilayer system calculated with the parametrization by Tang et al. (Ref. 25) (green) is plotted. In the latter case, the integrals are too small by an order of magnitude and can not be used to correctly describe the interlayer interaction in graphene systems.

Figure 5

Inset plot shows (i) the density of states (DOS) for single-layer graphene (black line) and (ii) total DOS and partial DOS for single-layer flakes in which all states are weighted by their projection onto the inner region of the graphene flake for different values of $\Delta r$ [the green region of the flake indicated in panel (b)]. Note that as $\Delta r$ increases, the magnitude of the zero-mode peak in the DOS decreases. The main plot shows the ratio formed from the number of states in the energy range [$-0.1,0.1$] eV with each eigenvalue weighted by its projection onto the inner region to the total number of states in the energy range [$-0.1,0.1$] eV; full circles are for a single-layer flake, triangles for a nanoribbon with zigzag edge geometry, and squares for the graphene twist flake described in Sec. 4. In the case of the single-layer flake, the colors of the main and inset plots are identical for identical values of $\Delta r$.

Figure 6

DOS of bilayer flake ($N=49000$/layer, red) with rotation angles between $\theta =32.{20}^{\circ }$ and $0^{\circ }$. The Gaussian smearing used in generating the DOS is 15 meV. The DOS of the corresponding periodic systems is plotted in green. The outer region of $\Delta r=2\text{nm}$ is neglected and does not contribute to the flake DOS shown here.

Figure 7

Charge density obtained by integrating all states in the energy window of $[-0.4,0]$ eV, for rotation angles of $\theta =32.{20}^{\circ },5.{08}^{\circ },2.{13}^{\circ },0.{505}^{\circ }$. Electrons in the outer region (shaded gray) of $\Delta r=2\text{nm}$ are neglected.

Figure 8

(a) Full DOS ($\Delta r=0$) of a bilayer flake ($N=49000$/layer) with a rotation angle of $\theta =5.{08}^{\circ }$ (blue) and $32.{20}^{\circ }$ (red). For comparison, the DOS of a infinite sheet with a rotation angle of $\theta =5.{08}^{\circ }$ (green) is plotted. The inset shows a projection of the flake DOS onto the outer edge atoms. (b) Standard deviation of the edge DOS peak for three different flake systems.

Figure 9

(i)–(iii) Density of states (DOS) of a bilayer flake ($N=11000$/layer, $r=8.5\text{nm}=34a$) for rotation angles of $\theta =32.{20}^{\circ },5.{08}^{\circ },2.{13}^{\circ }$ (red) and a magnetic length of $l=6a$. The corresponding single-layer flake is plotted in green. (a)–(h) DOS of bilayer flake with $\theta =2.{13}^{\circ }$ (red) and magnetic lengths between $l=\infty$ and $l=2a$, projection onto AA (AB) regions are shown in green (blue).

Figure 10

Zoom of the DOS of a bilayer flake ($N=11000$/layer, $r=8.5\text{nm}=34a$, $\theta =2.{13}^{\circ }$, $l=4a$) (red), projection onto AA (AB) regions are shown in green (blue). Vertical lines mark the energies of the states whose current densities are plotted in Figs. 11, 12, and 13.

Figure 11

Current density of a Landau state with energy $E=-0.22\text{eV}$. (a) Intralayer current in layer 1; (b) intralayer current in layer 2; (c) interlayer current; (d) electron density (plotted for both layers superimposed). Note that the light shaded gray region corresponding to the edge region of the flake in panel (d) indicates that the electron density is not plotted in this region. Arrows indicate the direction and the color refers to the magnitude of the current. Interlayer currents are, to a very good approximation, perpendicular to the flake plane. Electron current is well described as a charge flow on the surface of a torus situated at the AA moiré spot (see text for details).

Figure 12

Current density of a Landau state with energy $E=-0.15\text{eV}$. (a) Intralayer current in layer 1; (b) intralayer current in layer 2; (c) interlayer current; (d) electron density (plotted for both layers superimposed). Note that the light shaded gray region corresponding to the edge region of the flake in panel (d) indicates that the electron density is not plotted in this region. Arrows indicate the direction and the color refers to the magnitude of the current. Interlayer currents are, to a very good approximation, perpendicular to the flake plane. Currents are flowing counterclockwise and “bounce off the edge,” leading to an overall clockwise rotation around the flake boundary.

Figure 13

Current density of a state at $E=-0.09\text{eV}$. (a) Intralayer current in layer 1; (b) intralayer current in layer 2; (c) interlayer current; (d) electron density (plotted for both layers superimposed). Note that the light shaded gray region corresponding to the edge region of the flake in panel (d) indicates that the electron density is not plotted in this region. Arrows indicate the direction and the color refers to the magnitude of the current. Interlayer currents are, to a very good approximation, perpendicular to the flake plane. Currents are flowing exclusively on AB regions of the flake showing a rather irregular behavior.