Magneto-optical conductivity of silicene and other buckled honeycomb lattices

The magneto-optical longitudinal, transverse Hall, and circularly polarized responses of silicene and other materials described by a Kane-Mele Hamiltonian are calculated. Particular attention is paid to the effects of an external electric field and finite charge doping. The energy of interband transitions can be tuned by varying the electric field. The onset frequency of the absorptive peaks moves differently between the topological insulator and band insulator regimes. This may be used to verify experimentally the existence of the two insulating phases as well as provide a measure of the spin-orbit band gap. The zeroth Landau level splits between four spin and valley distinct energies resulting in valley-spin-polarized levels in the density of states. With charge doping, transitions between these levels allow for a spin- and valley-polarized response in the conductivity whereby charge carriers of specific spin and valley index can be isolated by tuning the incident photon frequency. Increasing the chemical potential is shown to redistribute spectral weight from interband transitions to a strong low-energy intraband response. For large chemical potential, this intraband feature is associated with the semiclassical cyclotron resonance frequency which is shown to linearly increase with magnetic field.

Figure 1

Upper: The crystal structure of silicene is based on the 2D honeycomb lattice. Due to the larger ionic size of silicon atoms, the silicene lattice is buckled with $A$ and $B$ sublattices sitting in vertically separated parallel planes. Lower: Side view of the vertical buckling.

Figure 2

Schematic representation of the band structure evolution at the $K$ point as the perpendicular electric field strength is increased. The dashed blue curves represent spin-up bands and the solid red curves represent spin-down bands. A finite electric field spin splits the bands and the system transitions from a TI to a BI as ${\Delta }_z$ becomes greater than ${\Delta }_{\mathrm{so}}$. When ${\Delta }_z={\Delta }_{\mathrm{so}}$, the lowest band gap closes and the system is referred to as a VSPM.

Figure 3

Schematic representation of the Landau level formation in the TI regime about the $K$ and $K^{\prime }$ points (upper and lower panels, respectively). The blue bands shown to the left are spin up while the red bands seen in the middle are spin down. These bands are associated with no magnetic field and are separated for clarity. The Landau levels are shown to the right and are not spin degenerate at a given valley. All but the $n=0$ levels have an equal-energy counterpart of opposite spin label at the other valley. The Fermi energy at charge neutrality (i.e., chemical potential $\mu =0$) is given by the dashed green line. The two lowest energy interband transitions are given by the black arrows.

Figure 4

Schematic representation of the Landau levels in the TI (upper panel) and BI (lower panel) regimes about the $K$ and $K^{\prime }$ points. Again, blue and red signify spin up and down, respectively. While the relative position of the $n\ne 0$ levels remain the same, the $n=0$ level undergoes an important transition due to the band inversion. That is, the $n=0$ spin-up level at the $K$ point switches between positive and negative energy while the $n=0$ spin-down level switches between negative and positive energy at the $K^{\prime }$ point.

Figure 5

Landau level energies as a function of magnetic field for ${\Delta }_z=0.5{\Delta }_{\mathrm{so}}$. All $n\ne 0$ levels scale as $\sqrt{B}$ while the location of the $n=0$ levels are fixed by the strength of ${\Delta }_{\mathrm{so}}$ and ${\Delta }_z$. The left panel shows low magnetic field results while the right panel illustrates the evolution over a large magnetic field range (note the factor of ${10}^3$).

Figure 6

Upper: Total electronic density of states for silicene in a magnetic field in units of[45] $eB/\left(2\pi \hslash c{\Delta }_{\mathrm{so}}\right)$. Lower: Spin-dependent density of states in the TI (left) and BI (right) regimes for ${\Delta }_z=0.75{\Delta }_{\mathrm{so}}$ and $1.25{\Delta }_{\mathrm{so}}$, respectively. Dashed blue curves correspond to spin up and the solid red curves correspond to spin down. The levels associated with $n=0$ are spin- and valley-polarized while all higher features are spin-valley degenerate and of double intensity.

Figure 7

Real part of the zero-temperature longitudinal magneto-optical conductivity of silicene for varying electric field strength in a magnetic field of strength $B/{\Delta }_{\mathrm{so}}^2=65.7$ G/meV${}^2$ with a scattering rate of $\eta =0.05{\Delta }_{\mathrm{so}}$ and $\mu =0$. The behavior of the interband responses is a signature of the two insulating regimes. The results are vertically offset by 15 units.

Figure 8

Real part of the zero-temperature longitudinal conductivity of silicene (${\Delta }_z=0$) for varying chemical potential in a magnetic field of strength $B/{\Delta }_{\mathrm{so}}^2=65.7$ G/meV${}^2$ with a scattering rate of $\eta =0.05{\Delta }_{\mathrm{so}}$. With increased $\mu$, the spectral weight of interband transitions is redistributed to a single intraband feature. The two red arrows mark the spin polarization of the two lowest features of the dashed red curve. These results correspond to the Kane-Mele model for a 2D quantum spin Hall insulator.

Figure 9

Real part of the zero-temperature longitudinal conductivity of silicene for varying chemical potential for $B/{\Delta }_{\mathrm{so}}^2=65.7$ G/meV${}^2$ and $\eta =0.05{\Delta }_{\mathrm{so}}$ in the (a) TI, (b) VSPM, and (c) BI regimes. Four spin- and valley-polarized responses can be generated. The results are vertically offset by (a) 10 units, (b) 16 units, and (c) 9 units.

Figure 10

Schematic of the allowed transitions between Landau levels at both valleys in the TI regime for various values of chemical potential. Dashed blue lines and circles represent spin-up levels, solid red lines and circles represent spin-down levels, and dash-dotted purple lines represent spin-degenerate levels. The various chemical potential values used in Fig. 9a are marked by the solid cyan line.

Figure 11

Imaginary part of the zero-temperature transverse Hall conductivity of silicene for varying chemical potential for $B/{\Delta }_{\mathrm{so}}^2=65.7$ G/meV${}^2$, $\eta =0.05{\Delta }_{\mathrm{so}}$, and ${\Delta }_z=0.75{\Delta }_{\mathrm{so}}$. These results correspond to those in Fig. 9a and are vertically offset by 10 units.

Figure 12

The absorptive part of the conductivity response to (a) right-handed and (b) left-handed circularly polarized light. In the response to right-handed polarization, the quartet of peaks that meet the criteria for a finite transverse Hall response are of double weight; they are absent in the response to left-handed polarization. All higher features look identical to the longitudinal response.

Figure 13

(a) The onset of the strong intraband absorption peak for various magnetic fields. (b) A comparison of the actual cyclotron resonance frequency (red circles) found by the onset frequencies in (a) with Eq. (19) (solid black line). The parameters used are $\mu /{\Delta }_{\mathrm{so}}=100$, ${\Delta }_z=0$, and $\eta /{\Delta }_{\mathrm{so}}=0.05$.