Electronic properties of bilayer phosphorene quantum dots in the presence of perpendicular electric and magnetic fields

Using the tight-binding approach, we investigate the electronic properties of bilayer phosphorene (BLP) quantum dots (QDs) in the presence of perpendicular electric and magnetic fields. Since BLP consists of two coupled phosphorene layers, it is of interest to examine the layer-dependent electronic properties of BLP QDs, such as the electronic distributions over the two layers and the so-produced layer-polarization features, and to see how these properties are affected by the magnetic field and the bias potential. We find that in the absence of a bias potential only edge states are layer polarized while the bulk states are not, and the layer-polarization degree (LPD) of the unbiased edge states increases with increasing magnetic field. However, in the presence of a bias potential both the edge and bulk states are layer polarized, and the LPD of the bulk (edge) states depends strongly (weakly) on the interplay of the bias potential and the interlayer coupling. At high magnetic fields, applying a bias potential renders the bulk electrons in a BLP QD to be mainly distributed over the top or bottom layer, resulting in layer-polarized bulk Landau levels (LLs). In the presence of a large bias potential that can drive a semiconductor-to-semimetal transition in BLP, these bulk LLs exhibit different magnetic-field dependences, i.e., the zeroth LLs exhibit a linearlike dependence on the magnetic field while the other LLs exhibit a square-root-like dependence.

Figure 1

(a) Top and (b) side views of the lattice structure of AB-stacked BLP. The upper and lower phosphorene layers, with interlayer separation $d$, are depicted by red and blue phosphorus atoms, respectively. The armchair (zigzag) edges are assumed to be along the $x$ ($y$) direction. External electric and magnetic fields, denoted by $F$ and $B$, are applied perpendicular to the phosphorene layers (i.e., along the $z$ direction). The black rectangle represents the unit cell of BLP with $a$ and $b$ being the lattice constants. The symbols $t_1$ and $t_2$ are the intralayer hopping parameters and $t_3$ is the interlayer hopping parameter. (c) Schematic plot of the first Brillouin zone (the red rectangle) with four high-symmetry points $\mathrm{\Gamma }, X, Y$, and $S$ as indicated by black dots.

Figure 2

Band structures of infinite BLP (MLP) subjected to different bias potential energies $V$ as indicated in (a)–(c) [(e)–(g)] and the band energies at the $\mathrm{\Gamma }$ point as a function of $V$ shown in (d) [(h)]. The red dashed line denotes the zero-energy reference.

Figure 3

(a) [(b)] Energy levels of an unbiased BLP QD, with dot size $L=3.5$ (7.5) nm, as a function of the magnetic flux $\mathrm{\Phi }$, where the colorbar shows the DOS values at these levels. (c)–(e) Probability densities of the electronic states denoted by points 1–3 shown in (a). (f)–(h) Probability densities of the electronic states denoted by points 4–9 shown in (a) and (b). The size of the blue and red dots shown in (c)–(h) represents the amplitude of the electronic probability density.

Figure 4

(a) Energy levels of an unbiased BLP QD with dot size $L=3.5$ nm, as a function of the magnetic flux $\mathrm{\Phi }$, where the colorbar shows the LPD ($\eta$) values at these levels. The inset shown in (a) shows a zoom of the edge levels around zero energy. (b)–(g) Layer-resolved probability densities of the electronic states indicated by points 1–3 shown in (a). $|{\mathrm{\Psi }}_j^t{|}^2$ and $|{\mathrm{\Psi }}_j^b{|}^2$ ($j=1,2,3$) denote the top-layer and bottom-layer electronic distributions, respectively. The size of the blue dots shown in (b)–(g) represents the amplitude of the electronic probability density. (h) Zero-field LPD ($\eta$) values of the edge and bulk states for different QD sizes $L$ as indicated. There are many points around $E=0$ due to the highly quasidegenerate edge states around this zero energy.

Figure 5

(a) [(b)] LPD-projected energy levels of a BLP QD with dot size $L=3.5$ (7.5) nm, at zero magnetic field ($\mathrm{\Phi }=0$), as a function of the bias potential energy $V$. (c) Results of a MLP QD with dot size $L=7.5$ nm shown for comparative purposes. (d) Layer-resolved probability densities $|{\mathrm{\Psi }}^t{|}^2$ and $|{\mathrm{\Psi }}^b{|}^2$ of the electronic states denoted by points 1–6 shown in (b). Here, the black rectangle shown in (b) is enlarged as an inset, and the vertical dashed lines shown in (a) and (b) mark the position at which the bias potential $V$ is equal to the interlayer coupling energy $t_3$.

Figure 6

DOS-projected energy levels of a BLP QD with dot size $L=5.5$ nm, as a function of the magnetic flux $\mathrm{\Phi }$, for different bias potential energies $V$: (a) $V=0$ eV, (b) $V=0.5$, and (c) $V=2$ eV. (d) Layer-resolved probability densities $|{\mathrm{\Psi }}^t{|}^2$ and $|{\mathrm{\Psi }}^b{|}^2$ of the electronic states denoted by points 1–8 shown in (a) and (b). The size of the blue dots shown in (d) represents the amplitude of the electronic probability density.