Tight-binding study of the magneto-optical properties of gapped graphene

We study the optical properties of gapped graphene in the presence of a magnetic field. We consider a model based on the Dirac equation, with a gap introduced via a mass term, for which analytical expressions for the diagonal and Hall optical conductivities can be derived. We discuss the effect of the mass term on electron-hole symmetry and $\pi$-${\pi }^{*}$ symmetry and its implications for the optical Hall conductivity. We compare these results with those obtained using a tight-binding model, in which the mass is modeled via a staggered potential and a magnetic field is included via a Peierls substitution. Considering antidot lattices as the source of the mass term, we focus on the limit where the mass term dominates the cyclotron energy. We find that a large gap quenches the effect of the magnetic field. The role of overlap between neighboring $\pi$ orbitals is investigated, and we find that the overlap has pronounced consequences for the optical Hall conductivity that are missed in the Dirac model.

Figure 1

Transitions contributing to the optical Hall conductivity in gapped graphene, in the case of zero temperature and chemical potential. The weights of each transition is indicated, with color indicating the sign. Note that for vanishing gap, $A_{\kappa }=B_{\kappa }$ and the Hall conductivity vanishes within each valley. This does not hold in the gapped case, where instead contributions from opposite valleys cancel.

Figure 2

Unit cell used in the tight-binding calculations. The colored circles indicate the carbon atoms included in the unit cell, with red and blue indicating different sublattices, A and B. The dashed rectangle indicates the fundamental unit cell, which is repeated in order to ensure translational symmetry. The lattice vectors of the enlarged unit cell are denoted ${\mathbf{A}}_x=L_x\widehat{\mathbf{x}}$ and ${\mathbf{A}}_y=L_y\widehat{\mathbf{y}}$. Also shown are the three nearest-neighbor vectors ${\delta }_n$.

Figure 3

Diagonal conductivity ${\sigma }_{xx}$ in units of the DC graphene conductivity ${\sigma }_0$, as a function of photon energy $\hslash \omega$. Results are shown for a magnetic field $B=78$ T. Thick lines indicate results from the TB model including overlap, while thin lines are without overlap. Dashed lines are DE results. Blue (red) coloring indicates the real (imaginary) part of the conductivity. Note the resonance around $\hslash \omega =4.4$ eV, which arises due to the van Hove singularity at the $M$ point of graphene, and is thus absent in the DE results. The inset offers a closer view of the low-energy oscillations, illustrating the excellent agreement between all three methods in this energy range. The vertical lines mark the transition energies predicted by the DE model.

Figure 4

Diagonal conductivity ${\sigma }_{xx}$ in units of the DC graphene conductivity ${\sigma }_0$, as a function of photon energy $\hslash \omega$. Results are shown for a magnetic field $B=78$ T. The chemical potential is set at the value of the mass term $\mu =\Delta$. Thick lines indicate results from the TB model including overlap, while thin lines are without overlap. Dashed lines are DE results. Blue (red) coloring indicates the real (imaginary) part of the conductivity. The inset offers a closer view of the low-energy oscillations, illustrating the excellent agreement between all three methods in this energy range. The vertical lines mark the transition energies predicted by the DE model.

Figure 5

Diagonal conductivity ${\sigma }_{xx}$ in units of the DC graphene conductivity ${\sigma }_0$, as a function of photon energy $\hslash \omega$. Results are shown for a magnetic field $B=78$ T and for four different values of the mass term, all calculated using the TB model with overlap. Full (dashed) lines indicate the real (imaginary) part of the conductivity. For comparison, the thin lines show the optical conductivity in the absence of a magnetic field. The vertical lines indicate the resonance energies predicted by the DE model.

Figure 6

DC Hall conductivity in gapped graphene as a function of the chemical potential. The thick, blue lines are results of the TB model including overlap, while thin, red lines are without overlap. Dashed, black lines are the DE results. Note that in contrast to the other figures, here the conductivity is given in units of $4e^2/h$. (a) DC Hall conductivity in the full range of chemical potentials considered, illustrating the discrepancies between the three models and the abrupt change in behavior near $\mu =(1+s)\left|t\right|$ for the TB model results. The inset shows the band structure of gapped graphene without magnetic field. (b) and (c) offer closer views of the regimes indicated by black rectangles in panel (a). Horizontal lines in (b) indicate ${\sigma }_{xy}^0=-(4e^2/h)(n+1/2)$, while those in (c) indicate ${\sigma }_{xy}^0=(2e^2/h)n$.

Figure 7

Optical Hall conductivity ${\sigma }_{xy}$ in units of the DC graphene conductivity ${\sigma }_0$, as a function of photon energy $\hslash \omega$. Results are shown for a magnetic field $B=78$ T. The chemical potential is set at the value of the mass term $\mu =\Delta$. Thick lines indicate results from the TB model including overlap, while thin lines are without overlap. Dashed lines are DE results. Blue (red) coloring indicates the real (imaginary) part of the conductivity. Vertical lines indicate the resonance energies predicted by the DE model. The inset illustrates the transitions contributing to the Hall conductivity. See text for more details.

Figure 8

Optical Hall conductivity ${\sigma }_{xy}$ in units of the DC graphene conductivity ${\sigma }_0$, as a function of photon energy $\hslash \omega$. Results are shown for a magnetic field $B=78$ T and for four different values of the mass term, all calculated using the TB model with overlap. The chemical potential is in each case fixed at the lowest Landau level $\mu =\Delta$. Full (dashed) lines indicate the real (imaginary) part of the conductivity. The inset shows a closer view of the first resonance. The results of the TB model without overlap are shown by thin, black lines, illustrating the slight discrepancies between the models in this regime. The DE results are practically identical to those of the TB model without overlap.

Figure 9

Optical Hall conductivity with a chemical potential fixed at midgap, calculated using the TB model with overlap. The vertical lines indicate the transition energies predicted by the DE model.