Aspects of anisotropic fractional quantum Hall effect in phosphorene

We have analyzed the effects of the anisotropic energy bands of phosphorene on magnetoroton branches for electrons and holes in the two Landau levels close to the band edges. We have found that the fractional quantum Hall effect gap in the lowest (highest) Landau level in the conduction (valence) band is slightly larger than that for conventional semiconductor systems and therefore the effect should be experimentally observable in phosphorene. We also found that the magnetoroton mode for both electrons and holes consists of two branches with two minima due to the anisotropy. Most importantly, in the long-wavelength limit a second mode with upward dispersion, well separated from the magnetoroton mode was found to appear, that is entirely a consequence of the anisotropy in the system. These novel features of the collective mode, unique to phosphorene, can be observed in resonant inelastic light-scattering experiments.

Figure 1

(a) Magnetic-field dependence of a few low-lying LLs in phosphorene, both in the conduction band (blue) and the valence (red) band. (b) Magnetic-field dependence of the $\nu =1/3$ gap in the conduction-band and the $\nu =2/3$ gap in the valence-band lowest LLs of phosphorene. The finite-size system considered for the FQHE gap has six electrons.

Figure 2

The energy spectrum of the six-electron FQHE system in phosphorene with $\nu =\frac{1}{3}$ in the lowest LL of the conduction band. The red and blue dashed lines show schematically the two branches of the magnetoroton mode. The green dashed line shows the second mode described in the text. Inset: the magnetoroton mode and the second mode for $N_e=7$.

Figure 3

The dispersion of the second mode of Fig. 2 for $N_e=6–8$.

Figure 4

The energy spectrum of the six-electron FQHE system with $\nu =\frac{7}{3}$ in the second LL of the conduction band for phosphorene. Inset: the same spectrum but for a conventional semiconductor system (such as GaAs).