Electronic structure of turbostratic graphene

We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i.e., two layers rotated by some angle $\theta$. It is shown that, due to the weak interaction between graphene layers, many features of this system can be understood by interference conditions between the quantum states of the two layers, mathematically expressed as Diophantine problems. Based on this general analysis we demonstrate that while the Dirac cones from each layer are always effectively degenerate, the Fermi velocity $v_F$ of the Dirac cones decreases as $\theta \to 0°$; the form we derive for $v_F\left(\theta \right)$ agrees with that found via a continuum approximation in [J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007)]. From tight-binding calculations for structures with $1.47°\le \theta <30°$ we find agreement with this formula for $\theta \gtrsim 5°$. In contrast, for $\theta \lesssim 5°$ this formula breaks down and the Dirac bands become strongly warped as the limit $\theta \to 0$ is approached. For an ideal system of twisted layers the limit as $\theta \to 0°$ is singular as for $\theta >0$ the Dirac point is fourfold degenerate, while at $\theta =0$ one has the twofold degeneracy of the $AB$ stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined globally, in contrast to the “nearsightedness” [W. Kohn, Phys. Rev. Lett. 76, 3168 (1996)] of electronic structure generally found in condensed matter.

Figure 1

(Color online) Illustration of the commensuration cell for the case of a misorientation angle of $\theta =21.78°$, generated by a $\left(p,q\right)$ pair of (1,3) (lattice vectors ${\mathbf{t}}_1$ and ${\mathbf{t}}_2$). Shown also are the unit cells of the unrotated graphene layer (vectors ${\mathbf{a}}_1$, ${\mathbf{a}}_2$) and rotated graphene layer (vectors $\mathbf{R}{\mathbf{a}}_1$, $\mathbf{R}{\mathbf{a}}_2$). For explanations of other symbols refer to Sec. 2.

Figure 2

(Color online) Shown is the number of C atoms, $N_C$, in the commensuration cell as a function of the relative orientation of the two graphene layers, for $N_C<7000$. Inset displays the moiré pattern for the cell indicated number 4. Band structures of twist bilayers corresponding to the points labeled 1–4 are displayed in panels 1–4 of Fig. 8. The dashed line corresponds the lower bound $N_C={\left({\text{sin}}^2\theta /2\right)}^{-1}$; for commensuration cells that fall on this line the moiré periodicity is equal to the commensuration periodicity, see Sec. 2 for details.

Figure 3

(Color online) Brillouin zones of the unrotated (U) and rotated (R) graphene layers, as well as the Brillouin zone of the bilayer supercell (B) for the case $\left(p,q\right)=\left(1,5\right)$, corresponding to $\theta =13.17°$. In this Figure ${\mathbf{b}}_1$, ${\mathbf{b}}_2$ are the reciprocal lattice vectors of the unrotated graphene layer, $\mathbf{R}{\mathbf{b}}_1$, $\mathbf{R}{\mathbf{b}}_2$ the reciprocal lattice vectors of the rotated layer, and ${\mathbf{g}}_1$, ${\mathbf{g}}_2$ the reciprocal lattice vectors of the bilayer supercell. Special $K$ points of various Brillouin zones indicated by subscript U, R, and B. The separations of special $K$ points indicated are $\Delta K=\left|{\mathbf{K}}_R-{\mathbf{K}}_U\right|=\left|{\mathbf{K}}_R^{\ast }-{\mathbf{K}}_U^{\ast }\right|$, $\Delta K_1=\left|{\mathbf{K}}_U^{\ast }-{\mathbf{K}}_R\right|$, and $\Delta K_2=\left|{\mathbf{K}}_U-{\mathbf{K}}_R^{\ast }\right|$.

Figure 4

(Color online) Example of the relationship between ${\mathbf{k}}_2-{\mathbf{k}}_1$ and the shift of the lattice of solution vectors of the equation ${\mathbf{G}}_1=\mathbf{R}{\mathbf{G}}_2+{\mathbf{k}}_2-{\mathbf{k}}_1$. Shown in the middle panel are those $\mathbf{k}$ points that fold back to the $\Gamma$ point of the commensuration Brillouin zone, for the case $\left(p,q\right)=\left(11,31\right)$. The left and right panels display the reciprocal lattices ${\mathbf{G}}_1$ and $\mathbf{R}{\mathbf{G}}_2+{\mathbf{k}}_2-{\mathbf{k}}_1$ along with the solution vectors for the two different cases of ${\mathbf{k}}_2-{\mathbf{k}}_1$ indicated in the central panel. See Sec. 3E for details.

Figure 5

(Color online) Shown is the shift term $\left|\Delta \tilde{\mathbf{G}}\right|$ corresponding to the separations ${\mathbf{k}}_2-{\mathbf{k}}_1={\mathbf{K}}_U^{\ast }-{\mathbf{K}}_R$ for $p=1,2,17$ $\left(\delta =3\right)$, and ${\mathbf{k}}_2-{\mathbf{k}}_1={\mathbf{K}}_U-{\mathbf{K}}_R$ for $p=3\left(\delta =1\right)$. Note that this shift diverges in all cases as $\theta \to 0$.

Figure 6

(Color online) Tight-binding calculation of the Dirac point splitting in (i) AB bilayer, indicated by light shaded points and (ii) $\theta =30°\pm 8.21$ twist bilayers, indicated by dark shaded open squares $\left(\theta =38.21°\right)$ and diamond symbols $\left(\theta =21.79°\right)$. The ab initio values for the AB and twist bilayer splitting are given by the horizontal dot-dashed lines. Full/dashed lines are calculations with the environment dependence of the hopping matrix elements switched off/on.

Figure 7

(Color online) Convergence of two nearly degenerate eigenvalues on the lower branch of the bilayer Dirac cone. The rotation angle of the bilayer is $\theta =3.48°$ (corresponding to $\left(p,q\right)=\left(1,19\right)$—see Sec. 2). Eigenvalues are calculated at $\mathbf{k}=\mathbf{K}+\frac{2}{10}\left(\mathbf{\Gamma }-\mathbf{K}\right)$ with $\mathbf{K}$ and $\mathbf{\Gamma }$ the $\mathbf{k}$ vectors of the $K$ and $\Gamma$ points. Inset shows ${\text{log}}_{10}\left|⟨{\varphi }_{i_1{\mathbf{k}}_1}^{\left(1\right)}|\overline{\mathbf{V}}|{\varphi }_{i_2{\mathbf{k}}_2}^{\left(2\right)}⟩\right|$ plotted for ${\mathbf{k}}_2-{\mathbf{k}}_{\Gamma }=n{\mathbf{g}}_1$.

Figure 8

(Color online) Tight-binding band structures for rotation angles of $\theta =21.79°\left(1,3\right)$, $9.43°\left(1,7\right)$, $2.65°\left(1,25\right)$, $1.47°\left(1,45\right)$, displayed, respectively, clockwise from top left. Shown is the band structure generated by direct tight-binding calculation (wide black lines) and that generated with SLG basis approach, indicated by light shaded (green) lines. For comparison in each panel the folded back band structure of single layer graphene is shown (dashed lines). The numbering of the panels corresponds to that of Fig. 3.

Figure 9

(Color online) Splitting of bands at the Dirac point in misoriented graphene layers. Shown are tight-binding binding calculations for supercells generated with $p=1$ and $q$ an odd integer. First principles calculations for supercells generated by $\left(p,q\right)$ pairs of (1,3), (1,2), (1,5), (2,3), (1,7) in order of increasing $N$ are taken from Ref. 11.

Figure 10

(Color online) Relation between misorientation angle of the bilayer and Fermi velocity damping; dashed (black) line in the main panel is the best fit to Eq. (54) for data points with $\theta >5°$.