Fractal butterflies in buckled graphenelike materials

We study theoretically the properties of buckled graphenelike materials, such as silicene and germanene, in a strong perpendicular magnetic field and a periodic potential. We analyze how the spin-orbit interaction and the perpendicular electric field influence the energy spectra of these systems. When the magnetic flux through a unit cell of the periodic potential measured in the magnetic flux quantum is a rational number, $\alpha =p/q$, then in each Landau level the energy spectra have a band structure, which is characterized by the corresponding gaps. We study the dependence of those gaps on the parameters of the buckled graphenelike materials. Although some gaps have weak dependence on the magnitude of the spin-orbit coupling and the external electric field, there are gaps that show strong nonmonotonic dependence on these parameters. For $\alpha =1/2$, the spin-orbit interaction also opens up a gap at one of the Landau levels. The magnitude of the gap increases with spin-orbit coupling and decreases with the applied electric field.

Figure 1

Energy spectrum (Hofstadter butterfly) of the germanene monolayer as a function of the parameter $\alpha$, which is the inverse magnetic flux through the unit cell in units of the flux quantum. The external electric field is $E_z^{}=50$ mV/Å. The amplitude of the periodic potential is $V_0^{}=40$ meV and its period is $a_0^{}=20$ nm.

Figure 2

Energy spectra of the germanene monolayer for $\alpha =1/3$. The external electric field is $E_z^{}=10$ mV/Å. The numbers near the lines are the amplitudes of the periodic potential, $V_0^{}$. The period of the potential is $a_0^{}=20$ nm. Two gaps ${\mathrm{\Delta }}_1^{}$ and ${\mathrm{\Delta }}_2^{}$ are marked in the figure.

Figure 3

Color online) Energy spectra of buckled graphenelike materials for $\alpha =1/3$. The external electric field is $E_z^{}=100$ mV/Å  (black lines) and $E_z^{}=0$ (red/medium gray lines). The numbers near the lines are the SO interaction constants, ${\lambda }_{\mathrm{SO}}^{}$. The period of the potential is $a_0^{}=20$ nm and its amplitude is $V_0^{}=20$ meV.

Figure 4

Gaps ${\mathrm{\Delta }}_1^{}$ and ${\mathrm{\Delta }}_2^{}$, defined in Fig. 2, as functions of the SO interaction, ${\lambda }_{\mathrm{SO}}^{}$. The period of the potential is 20 nm. (a) The electric field is zero and the amplitude of periodic potential is 10 meV. (b) The electric field is 100 mV/Å  and the amplitude of the periodic potential is 10 meV. (c) The electric field is 100 mV/Å  and the amplitude of the periodic potential is 20 meV.

Figure 5

Energy spectrum of the germanene monolayer as a function of the effective wave vector ${\kappa }_x^{}$ for various values of ${\kappa }_y^{}$. The external electric field is zero. The amplitude of periodic potential is 40 meV and its period is 20 nm.

Figure 6

Energy gap $\mathrm{\Delta }$ between two bands in $L{L}_{0,2}^{}$. (a) The period of the potential is 20 nm and its amplitude is 40 meV. The SO coupling is ${\lambda }_{\mathrm{SO}}^{}=43$ meV. (b) The electric field is 10 mV/Å. The period of the potential is 20 nm and its amplitude is 40 meV.