Linear response of twisted bilayer graphene: Continuum versus tight-binding models

We present a linear response calculation for twisted bilayer graphene. The calculation is performed for both the continuum and tight-binding models, with the aim of assessing the validity of the former. All qualitatively important features previously reported by us [Stauber et al., Phys. Rev. Lett. 120, 046801 (2018)] for the Drude matrix in the continuum model are also present in the tight-binding calculation, with increasing quantitative agreement for decreasing twist angle. These features include the chiral longitudinal magnetic moment associated with plasmonic modes, and the anomalous counterflow around the neutrality point, better interpreted as a paramagnetic response. We have addressed the differences between Drude and equilibrium response, and we showed that orbital paramagnetism is the equilibrium response to a parallel magnetic field over a substantial doping region around the neutrality point. Chirality also causes the equilibrium response to exhibit a nontrivial current structure associated with the nonvertical character of interlayer bonds in the tight-binding calculation.

Figure 1

$D_0$ (black), $D_1$ (red), and $D_{xy}$ (blue) entries of the Drude matrix for the tight-binding (solid lines) and continuum (dashed lines) models as functions of the chemical potential. Left panel: twist angle ${\theta }_{i=3}=9.4^{\circ }$. Right panel: twist angle ${\theta }_{i=6}=5.1^{\circ }$.

Figure 2

All entries of the tight-binding Drude matrix as functions of the chemical potential: $D_0$ (solid black), $D_1$ (solid red), $D_{xy}$ (solid blue), $D_i$ (dashed black), $D_2$ (dashed red), and $D_{xy}^{\prime }$ (dashed blue). Left panel: twist angle ${\theta }_{i=3}=9.4^{\circ }$. Right panel: twist angle ${\theta }_{i=6}=5.1^{\circ }$.

Figure 3

Total Drude weight as a function of the chemical potential for the tight-binding (solid line) and continuum models (dashed line). Left panel: twist angle ${\theta }_{i=3}=9.4^{\circ }$. Right panel: twist angle ${\theta }_{i=6}=5.1^{\circ }$.

Figure 4

Chiral Drude component as a function of the chemical potential for the tight-binding (solid line) and continuum models (dashed line). Left panel: twist angle ${\theta }_{i=3}=9.4^{\circ }$. Right panel: twist angle ${\theta }_{i=6}=5.1^{\circ }$.

Figure 5

Magnetic Drude component as a function of the chemical potential for the tight-binding (solid line) and continuum models (dashed line). Left panel: twist angle ${\theta }_{i=3}=9.4^{\circ }$. Right panel: twist angle ${\theta }_{i=6}=5.1^{\circ }$.

Figure 6

$D_0$ (black), $D_1$ (red) and $D_{xy}$ (blue) entries of the Drude matrix for the tight-binding (solid lines) and continuum (dashed lines) models as functions of the chemical potential. Left panel: twist angle ${\theta }_{i=1}=21.8^o$. Right panel: twist angle ${\theta }_{i=2}=13.2^o$.

Figure 7

Equilibrium magnetic response to a parallel magnetic field for the tight-binding (solid black line) and the continuum (dashed black line) models as functions of the chemical potential. The corresponding Drude responses of Fig. 5 are included for comparison (red lines). Left panel: twist angle ${\theta }_{i=3}=9.4^{\circ }$. Right panel: twist angle ${\theta }_{i=6}=5.1^{\circ }$.

Figure 8

Chiral components of the equilibrium response, ${\tilde{\chi }}_{xy}$ and ${\tilde{\chi }}_{xy}^{\prime }$, for the tight-binding calculation [see Eqs. (47) and (48)]. Left panel: twist angle ${\theta }_{i=3}=9.4^{\circ }$. Right panel: twist angle ${\theta }_{i=6}=5.1^{\circ }$.