Electric-field-tunable band gap in commensurate twisted bilayer graphene

Bernal bilayer graphene exhibits a band gap that is tunable through the infrared with an electric field. We show that sublattice odd commensurate twisted bilayer graphene (C-TBG) exhibits a band gap that is tunable through the terahertz with an electric field. We show that from the perspective of terahertz optics the sublattice odd and even forms of C-TBG are “inflated” versions of Bernal and AA-stacked bilayer graphene, respectively, with energy scales reduced by a factor of 110 for the $21.{79}^{\circ }$ commensurate unit cell. This lower energy scale is accompanied by a correspondingly smaller gate voltage, which means that the strong-field regime is more easily accessible than in the Bernal case. Finally, we show that the interlayer coherence energy is a directly accessible experimental quantity through the position of a power-law divergence in the optical conductivity.

Figure 1

Real-space crystal structure and momentum-space band structure of the $21.79{}^{\circ }$ twisted bilayer graphene for various interlayer shift vectors. The band structure near the Fermi energy is qualitatively the same AB- and AA-stacked graphene bilayers but with an energy scale that is two orders of magnitude smaller. The structures vary from a gapless structure (SE-odd) with a quadratic band touching at zero shift to a gapped structure with linear band crossings above and below the Fermi energy (SE-even) at $\vec{t}/3$ shift, where $\vec{t}=(a_{M,1}+a_{M,2})/7$. Here the dashed gray line indicates the interlayer coherence scale, $V_0=3$ meV, a scale which is two orders of magnitude smaller than the scale in AB and AA bilayers. This small energy scale makes the strong-field limit much easier to obtain: to realize layer potential energies of $\pm \mathcal{E}=\pm 2V_0$ requires an electric field strength of just 0.0358 V/nm. In the strong-field limit, these materials exhibit a band inversion [colored by $c=\frac{1}{2}(1+|\langle 3_{\text{odd}}\left(0\right)|\psi \left(\mathcal{E}\right)\rangle {|}^2-|\langle 2_{\text{odd}}\left(0\right)|\psi \left(\mathcal{E}\right)\rangle {|}^2)$]. This inversion can be understood as the layered Dirac cones being separated in the strong-field limit, and then inverting when the interlayer coherence dominates the electric potential. (columns) Structures as a function of shift, (row 1) crystal structures, (row 2) band structures, and (row 3) band structures in an electric field.

Figure 2

Optical conductivity can be used to determine the coupling of the offset Dirac cones as a function of twist. The gapped systems exhibit a power-law divergence at the band edge, the gapless system has a finite dc conductivity, and all systems asymptote to twice the optical conductivity of the monolayer in the high-frequency limit. Band structures of the systems and interband transitions contributing to the optical conductivity are shown in the figure insets. Vertical dashed lines correspond to positions where the optical conductivity diverges. (a) the peak onsets at $V_0$ and the location is weakly dependent on scattering and temperature, and universal behavior onsets at $2V_0$; (b) the peak is at $V_0\sqrt{\epsilon /(1+\epsilon )}$, where $\epsilon =4{\mathcal{E}}^2/V_0^2$ and, universal behavior onsets near $2\mathcal{E}$; (c) the peak is at $2V_{0}sin[(\phi -\theta )/2]$, where $\phi =\pi /3$ and $\theta =38.{21}^{\circ }$ is the twist angle, and universal behavior onsets at $2V_0$; (d) the divergent peak is at the same location as in (c), but the onset of universal behavior is near $2\mathcal{E}$.