Band-unfolding approach to moiré-induced band-gap opening and Fermi level velocity reduction in twisted bilayer graphene

We report on the energy spectrum of electrons in twisted bilayer graphene (tBLG) obtained by the band-unfolding method in the tight-binding model. We find the band-gap opening at particular points in the reciprocal space, that elucidates the drastic reduction of the Fermi-level velocity with the tiny twisted angles in tBLGs. We find that moiré pattern caused by the twist of the two graphene layers generates interactions among Dirac cones, otherwise absent, and the resultant cone-cone interactions peculiar to each point in the reciprocal space causes the energy gap and thus reduces the Fermi-level velocity.

Figure 1

Atomic structure and BZs of tBLG. (a) Atomic structure of tBLG with $\theta =2.{88}^{\circ }$ with blue arrows being the lattice vectors of the SC. (b) BZ of tBLG with $\theta =9.{43}^{\circ }$. Black dots represent either a $K_{\mathrm{SC}}$ or $K_{\mathrm{SC}}^{\prime }$ point of the SBZ, and the red and blue lines depict the PBZs of the two graphene monolayers twisted to each other. The BZs in the extended scheme for (c) $\theta =9.{43}^{\circ }$ and for (d) $\theta =5.{09}^{\circ }$ are shown.

Figure 2

Calculated energy-band structures of tBLG with (a) $\theta =9.{43}^{\circ }$ and (b) $\theta =2.{88}^{\circ }$ by the TB calculation. Enlarged band structures near $E_{\mathrm{F}}$ are shown in (c) and (d).

Figure 3

Fermi velocity of tBLG normalized to that of a monolayer as a function of the twisted angle $\theta$.

Figure 4

Contour plots of the unfolded bands, i.e., the spectral function of tBLG with (a) $\theta =32.{20}^{\circ }$, (b) $\theta =9.{43}^{\circ }$, (c) $\theta =5.{09}^{\circ }$, and (d) $\theta =3.{89}^{\circ }$. The energy bands in the SBZ are unfolded to the PBZ of a monolayer graphene shown in the red lines in Fig. 1. The energy bands of the AA-stacking bilayer are shown by white dotted lines.

Figure 5

Contour plots of the unfolded bands of tBLGs near the Fermi energy with (a) $\theta =9.{43}^{\circ }$, (b) $\theta =5.{09}^{\circ }$, (c) $\theta =3.{89}^{\circ }$, and (d) $\theta =2.{88}^{\circ }$. The energy bands in the SBZ are unfolded to the PBZ of a monolayer graphene shown in the red lines in Fig. 1. The energy bands of the tBLG having no interlayer interaction and of the AA-stacking bilayer are also shown by white dash-dotted lines and by white dotted lines, respectively.

Figure 6

Contour plots of the unfolded bands of tBLGs near the Fermi energy and total density of states with $\theta =5.{09}^{\circ }$. The energy band of the tBLG having no interlayer interaction and of the AA-stacking bilayer are also shown by white dash-dotted lines and by white dotted lines, respectively.

Figure 7

Contribution from each graphene monolayer to the spectral function of tBLG with $\theta =5.{09}^{\circ }$ unfolded to the red PBZ. The contribution from one monolayer corresponding to the red PBZ and the contribution from the other monolayer corresponding to the blue PBZ are shown in (a) and (b), respectively. The unfolded energy bands of the tBLG having no interlayer interaction and the energy bands of the AA-stacking BLG are shown by white dash-dotted and white dotted lines, respectively.

Figure 8

(a) Arrangements of the supercell Brillouin zone (SBZ) and the primitive BZs (PBZs) with several $K$ and $K^{\prime }$ points. Black dots depict $K_{\mathrm{SC}}$ and $K_{\mathrm{SC}}^{\prime }$ points at the corners of SBZs and the red and blue PBZs twisted to each other are shown by red and blue lines, respectively. The $K$ and $K^{\prime }$ points of the PBZs are labeled as $K_l(l=1$ to 6) as in the central figure. The enlargements near $K_4$ and $K_6$ are also shown. The symmetry lines discussed in the text are shown by black arrows. (b) Energy bands along the symmetry line depicted by the black arrow in the SBZ. (c),(d) The spectral functions along the same symmetry lines which are unfolded near (c) the region of the $K_4$ point and (d) the region of the $K_6$ point. (e) Dirac cones at $K_4$ and $K_6$ points in red and blue PBZs folded in the SBZ. The $K_e$ and $K_f$ points lie on the perpendicular bisector of each of two Dirac cones.

Figure 9

Variation of the energy gap points $K_e$ (blue squares) and $K_f$ (red triangles) as a function of the twisted angle in tBLG. The plotted points are the distances $\mathrm{\Delta }k$ of those points from the symmetry point $K_{\mathrm{SC}}^{\alpha }$ (see text). The side lengths of the supercell BZ (SBZ) are also shown by black circles.

Figure 10

Supercell Brillouin zone (SBZ) near the $K_4$ point in Fig. 8. The symmetry lines $KM$ and $K\mathrm{\Gamma }$ along which the unfolded bands are calculated in Fig. 7 are shown by red dashed lines.