Faraday rotation in bilayer and trilayer graphene in the quantum Hall regime

Optical Hall conductivity, as directly related to Faraday rotation, is theoretically studied for bilayer and trilayer graphene. In bilayer graphene, the trigonal warping of the band dispersion greatly affects the resonance structures in Faraday rotation not only in the low-energy region where small Dirac cones emerge, but also in the higher-energy parabolic bands as a sequence of satellite resonances. In ABA-stacked trilayer, the resonance spectrum is a superposition of effective monolayer and bilayer contributions with band gaps, while ABC trilayer exhibits a distinct spectrum peculiar to the cubic-dispersed bands with a strong trigonal warping, where the signals associated with low-energy Dirac cones should be directly observable owing to a large Lifshitz transition energy ($\sim 10\mathrm{meV}$).

Figure 1

(a) Lattice structure of bilayer graphene with A/B sublattice carbons indicated in blue/red. The unit cell consists of A${}_1$ and B${}_1$ carbon atoms in the top layer and A${}_2$ and B${}_2$ carbon atoms in the bottom. ${\gamma }_0$: the nearest-neighbor (A${}_1\leftrightarrow {\text{B}}_1,{\text{A}}_2\leftrightarrow {\text{B}}_2$) hopping within each layer, ${\gamma }_1$: the vertical (${\text{B}}_1\leftrightarrow {\text{A}}_2$) interlayer hopping, and ${\gamma }_3$: an oblique (${\text{A}}_1\leftrightarrow {\text{B}}_2$) interlayer hopping. Two layers are separated by $d=0.334$ nm. (b) Low-energy band dispersion of bilayer graphene with the trigonal warping effect with a Lifshitz transition where the topology of Fermi surface changes from four Dirac cones to one, trigonally warped parabolic dispersion.

Figure 2

For bilayer graphene QHE system without the trigonal warping effect ($v_3=0$), we show (a) the optical longitudinal conductivity ${\sigma }_{xx}({\epsilon }_F,\omega )$, and (b) optical Hall conductivity ${\sigma }_{xy}({\epsilon }_F,\omega )$, grey-scale plotted against the Fermi energy ${\epsilon }_F$ and the frequency $\omega$.

Figure 3

(a) Low-energy band structure and (b) Landau levels against magnetic field for bilayer graphene with the trigonal warping included. (c) The longitudinal ${\sigma }_{xx}({\epsilon }_F,\omega )$ and (d) Hall ${\sigma }_{xy}({\epsilon }_F,\omega )$ grey-scale plotted against the Fermi energy ${\epsilon }_F$ and the frequency $\omega$ for a magnetic field $B=1$ T [a vertical dashed line in (b)]. (e) A diagram indicating allowed resonances in ${\sigma }_{xy}$. Vertical dotted lines indicate the Landau levels. Note that “$-$1” ($n=-1$) and “1$-$” ($n=1$, $s=-1$)” are different.

Figure 4

Optical longitudinal (${\sigma }_{xx}$; dashed line) and Hall (${\sigma }_{xy}$; solid line) conductivities plotted against the frequency $\omega$ for a magnetic field $B=1$ T and the Fermi energy ${\epsilon }_F=-20$ meV.

Figure 5

Lattice structure of trilayer graphene with (a) ABA stacking and (b) ABC stacking.

Figure 6

(a) Low-energy band structure, and (b) Landau levels against magnetic field for ABA-stacked trilayer graphene. (c) Longitudinal ${\sigma }_{xx}({\epsilon }_F,\omega )$ and (d) Hall ${\sigma }_{xy}({\epsilon }_F,\omega )$ plotted against the Fermi energy ${\epsilon }_F$ and the frequency $\omega$ for a magnetic field $B=1$ T [a dashed line in (b)]. (e) A diagram indicating allowed resonances in ${\sigma }_{xy}$.

Figure 7

(a) Low-energy band structure and (b) Landau levels against magnetic field for ABC-stacked trilayer graphene. (c) Longitudinal ${\sigma }_{xx}({\epsilon }_F,\omega )$ and (d) Hall ${\sigma }_{xy}({\epsilon }_F,\omega )$ plotted against the Fermi energy ${\epsilon }_F$ and the frequency $\omega$ for a magnetic field $B=1$ T [a dashed line in (b)]. (e) A diagram indicating allowed resonances in ${\sigma }_{xy}$.