Diverse magnetic quantization in bilayer silicene

The generalized tight-binding model is developed to investigate the rich and unique electronic properties of $AB$-bt (bottom-top) bilayer silicene under uniform perpendicular electric and magnetic fields. The first pair of conduction and valence bands, with an observable energy gap, displays unusual energy dispersions. Each group of conduction/valence Landau levels (LLs) is further classified into four subgroups, i.e., the sublattice- and spin-dominated LL subgroups. The magnetic-field-dependent LL energy spectra exhibit irregular behavior corresponding to the critical points of the band structure. Moreover, the electric field can induce many LL anticrossings. The main features of the LLs are uncovered with many van Hove singularities in the density-of-states and nonuniform delta-function-like peaks in the magnetoabsorption spectra. The feature-rich magnetic quantization directly reflects the geometric symmetries, intralayer and interlayer atomic interactions, spin-orbital couplings, and field effects. The results of this work can be applied to novel designs of Si-based nanoelectronics and nanodevices with enhanced mobilities.

Figure 1

The top view (a) and side view (b) of the atomic structure for bilayer silicene with intra- and interlayer atomic interactions. The first Brillouin zone along the high-symmetry points is illustrated in (c); the highly symmetric $\mathbf{K}$ (${\mathbf{K}}^{\prime }$) and $\mathrm{\Gamma }$ points and an extreme one, $\mathbf{T}(k_x=0.1\times \sqrt{3},k_y=0.1)$ (1/$\text{Å}$), are presented. The enlarged rectangular-shaped unit cell under a uniform perpendicular magnetic field is shown in (d).

Figure 2

The first pair of conduction and valence energy bands (a) at $E_z=0$, the state probabilities for the conduction band (b),(c) and the valence band (d),(e). Also shown in the inset of (a) are $E_z$-split energy bands.

Figure 3

At $B_z=20$ T, the low-lying conduction LL energies (a); the spatial amplitude distributions for the $B_{\uparrow }^2$- and $B_{\downarrow }^1$-dominated LLs (b) and the $B_{\uparrow }^1$- and $B_{\downarrow }^2$-dominated LLs (c). The numbers in the abscissa represent the quantum numbers of those Landau levels. They are determined by the number of zero points of the spatial distributions on the dominating sublattices.

Figure 4

At $B_z=20$ T, the low-energy valence LLs (a); the spatial wave functions for the (b) $B_{\uparrow }^2$-, (c) $B_{\downarrow }^1$-, (d) $B_{\uparrow }^1$-, and (e) $B_{\downarrow }^2$-dominated LLs.

Figure 5

The $B_z$-dependent LL energy spectrum: the blue, red, purple, and green curves are the different subgroups of conduction (a) and valence (b),(c) LLs based on the dominant $B_{\uparrow }^1$, $B_{\downarrow }^1$, $B_{\uparrow }^2$, and $B_{\downarrow }^2$ sublattices, respectively.

Figure 6

The density of states of the conduction (a) and valence (b) LLs under a magnetic field of $B_z=20$ T.

Figure 7

The $E_z$-dependent conduction-LL energy spectrum (a) at $B_z=20$ T; the ($n_{\downarrow 1}^c=1$ and $n_{\uparrow 2}^c=1$) and ($n_{\downarrow 1}^c=1$ and $n_{\uparrow 2}^c=2$) anticrossings (b), corresponding to the evolutions of wave functions in (c) and (e), respectively.

Figure 8

The $E_z$-dependent valence-LL energy spectrum (a) at $B_z=20$ T; the ($n_{\downarrow 2}^v=1$ and $n_{\uparrow 1}^v=1$) and ($n_{\downarrow 2}^v=1$ and $n_{\uparrow 1}^v=2$) anticrossings (b), corresponding to the evolutions of wave functions in (c) and (e), respectively.

Figure 9

The optical conductivities of $AB$-bt bilayer silicene under (a) zero magnetic field and (b) $B_z=40$ T.