Landau levels and magnetic oscillations in gapped Dirac materials with intrinsic Rashba interaction

A new family of the low-buckled Dirac materials which includes silicene, germanene, etc. is expected to possess a more complicated sequence of Landau levels than in pristine graphene. Their energies depend, among other factors, on the strength of the intrinsic spin-orbit (SO) and Rashba SO couplings and can be tuned by an applied electric field $E_z$. We studied the influence of the intrinsic Rashba SO term on the energies of Landau levels using both analytical and numerical methods. The quantum magnetic oscillations of the density of states are also investigated. A specific feature of the oscillations is the presence of the beats with the frequency proportional to the field $E_z$. The frequency of the beats becomes also dependent on the carrier concentration when Rashba interaction is present allowing experimental determination of its strength.

Figure 1

Energies of Landau levels in ${\Delta }_{\text{SO}}$ units at the ${\mathbf{K}}_{\pm }$ points (upper and lower panels, respectively) as a function of $\Delta /{\Delta }_{\text{SO}}$ for ${\Delta }_R=0$ and $\hslash \omega =1.62{\Delta }_{\text{SO}}$. The solid (blue) and dashed (red) curves correspond to $\sigma =\uparrow ,\downarrow$, respectively.

Figure 2

Energies of Landau levels in ${\Delta }_{\text{SO}}$ units at the ${\mathbf{K}}_{\pm }$ points (upper and lower panels, respectively) as a function of $\Delta /{\Delta }_{\text{SO}}$ for ${\Delta }_R=150{\Delta }_{\text{SO}}$ and $\hslash \omega =1.62{\Delta }_{\text{SO}}$. The color marking corresponds to the not conserving spin quantum number: blue and red correspond to $\sigma =\uparrow ,\downarrow$, respectively, and their mixture reflects the superposition of the spins.

Figure 3

Dependence of the energy difference $\delta \epsilon ={\epsilon }_{n\xi }(R\ne 0,\Delta )-{\epsilon }_{n\xi }(R=0,\Delta )$ in $\text{meV}$ as a function of $\Delta$ in $\text{meV}$ for $n=1,10,{10}^2,{10}^3$ at the ${\mathbf{K}}_{-}$ point, $B=0.2\mathrm{T},{\Delta }_{\text{SO}}=4.2\mathrm{meV}$, and ${\Delta }_R=21\mathrm{meV}$. The Fermi velocity $v_F=5.5\times {10}^5\mathrm{m}/\mathrm{s}$.

Figure 4

Dependence of the relative energy shift $\delta E/E_{n\xi \sigma }^{+}$ at ${\mathbf{K}}_{-}$ point as a function of magnetic field $B$ for ${\Delta }_{\text{SO}}=10\text{meV}$ and ${\Delta }_R=50\text{meV}$. The long-dashed (red) line is for $n=0,n=1$—the dash-dotted (black) line, $n=10$—the solid (blue) line, and $n=100$—the short-dashed (green) line. All thick lines are plotted using the energy difference $\delta E={\epsilon }_{n\xi }(R\ne 0,\Delta =-\xi {\Delta }^{\mathrm{cros}})-{\epsilon }_{n\xi }(R=0,\Delta =-\xi {\Delta }^{\mathrm{cros}})$, which is computed using the numerical solution of the general Eq. (3.7) and the thin lines are calculated using the approximate Eq. (3.26).

Figure 5

Dependence of the relative energy shift $\delta E/E_{n\xi \sigma }^{+}$ at $K_{-}$ point as a function of magnetic field $B$ for ${\Delta }_{\text{SO}}=10\text{meV}$ and $n=50$. The dashed (red) line is for ${\Delta }_R=1\text{meV},{\Delta }_R=5\text{meV}$—the dash-dotted (black) line, and ${\Delta }_R=10\text{meV}$—solid (blue) line. All thick lines are plotted using the energy difference $\delta E={\epsilon }_{n\xi }(R\ne 0,\Delta =-\xi {\Delta }^{\mathrm{cros}})-{\epsilon }_{n\xi }(R=0,\Delta =-\xi {\Delta }^{\mathrm{cros}})$, which is computed using the numerical solution of the general Eq. (3.7) and the thin lines are calculated using the approximate Eq. (3.26).