Aharonov-Bohm oscillations in phosphorene quantum rings

The Aharonov-Bohm (AB) effect in square phosphorene quantum rings, with armchair and zigzag edges, is investigated using the tight-binding method. The energy spectra and wave functions of such rings, obtained as a function of the magnetic flux $\mathrm{\Phi }$ threading the ring, are strongly influenced by the ring width $W$, an in-plane electric field $E_p$, and a side-gating potential $V_g$. Compared to a square dot, the ring shows an enhanced confinement due to its inner edges and an interedge coupling along the zigzag direction, both of which strongly affect the energy spectrum and the wave functions. The energy spectrum that is gapped consists of a regular part, of conduction (valence) band states, that shows the usual AB oscillations in the higher- (lower-) energy region, and of edge states, in the gap, that exhibit no AB oscillations. As the width $W$ decreases, the AB oscillations become more distinct and regular and their period is close to ${\mathrm{\Phi }}_0/2$, where the flux quantum ${\mathrm{\Phi }}_0=h/e$ is the period of an ideal circular ring ($W\to 0$). Both the electric field $E_p$ and the side-gating potential $V_g$ reduce the amplitude of the AB oscillations. The amplitude can be effectively tuned by $E_p$ or $V_g$ and exhibits an anisotropic behavior for different field directions or side-gating configurations.

Figure 1

Schematic view of (a) a SPQD and (b) a SPQR, with armchair edges along the $x$ direction and zigzag edges along the $y$ direction, in the presence of an external perpendicular magnetic field $B$. $L$ is the side length of the SPQD, and $L_{\text{in}}$ ($L_{\text{out}}$) is the length of the inner (outer) side of the SPQR, so the ring width is given by $W=(L_{\text{out}}-L_{\text{in}})/2$. The entire SPQR may be subject to an in-plane electric field, $F_x$ or $F_y$, or have different side-gating potentials $\pm V_g/2$ applied to its arms (indicated by the yellow regions). (c) Schematic lattice structure of phosphorene (top view). The unit cell (rectangle) consists of four inequivalent phosphorus atoms with two of them labeled by dark-gray solid circles and the other two labeled by light-gray solid circles. The symbols $t_1–t_5$ denote five hopping links between different phosphorus atoms.

Figure 2

(a)–(d) Energy spectrum of a SPQD, with side length $L=8$ nm, as a function of the magnetic flux at zero electric field and side-gating potential. (e)–(h) As in (a)–(d) for a SPQR, with an outer (inner) side length $L_{\text{out}}=8$ nm ($L_{\text{in}}=4$ nm). Both the dot and the ring have armchair edges along the $x$ direction and zigzag edges along the $y$ direction. The red rectangles shown in (a) and (e) are enlarged next to them, as indicated, and that in (g) is enlarged in the inset.

Figure 3

Probability densities, at zero magnetic flux ($\mathrm{\Phi }=0$), of the lowest-energy states marked by the black dots in Fig. 2: ($\alpha$) and ($\beta$) correspond to Figs. 2 and 2, respectively, ($\gamma$) and ($\delta$) to Figs. 2 and 2, respectively.

Figure 4

Probability densities of the energy states shown in Fig. 2 with energy $E$ and magnetic flux $\mathrm{\Phi }$, as indicated.

Figure 5

Energy spectrum of a SPQR, with armchair and zigzag edges, as a function of the magnetic flux. The ring width is $W=2$ nm in the upper panels and $W=1$ nm in the lower ones. The outer sides are $L_{\text{out}}=8$ nm long for both rings. The marking of the panels is the same as in Fig. 2.

Figure 6

Energy spectrum [(a)–(c)] and wave functions [($\alpha$)–($\gamma$)] of a SPQR, with outer (inner) side length $L_{\text{out}}=8$ nm ($L_{\text{in}}=6$ nm) and with armchair and zigzag edges, as a function of the magnetic flux, in the absence and presence of an in-plane electric field $F_x$ or an asymmetric side-gating potential $V_y$. (a) and ($\alpha$) are for $F_x=0$ and $V_y=0$, (b) and ($\beta$) for $F_x=0.05$ V/nm and $V_y=0$, and (c) and ($\gamma$) for $F_x=0$ and $V_y=0.25$ V. The energy levels indicated by the green curves have the same state index of 506 in the whole energy spectrum, as indicated in (a)–(c).

Figure 7

Schematic plot of a model square ring in the presence of an electric field $F_x$ or side-gating potentials $\pm V_g/2$. Here, $C_1,C_2$ denote two particle's traveling paths that are separated by two black dots, as indicated.

Figure 8

Energy spectrum ($E$) and AB-oscillation amplitude ($\mathrm{\Delta }E$) of a SPQR, with outer (inner) side length $L_{\text{out}}=8$ nm ($L_{\text{in}}=6$ nm) and with armchair and zigzag edges, as a function of the field $F_x$ along the armchair direction (left panels) or of the field $F_y$ along the zigzag direction (right panels). Only results for two energy levels are shown for the amplitude and their state indices are indicated in the bottom panels. The vertical thin black dashed lines indicate the positions of the amplitude peaks.

Figure 9

(a) and (b) Energy of a few lowest edge states, vs state index, in a SPQR of inner (outer) side length $L_{\text{in}}=4$ nm ($L_{\text{out}}=8$ nm), with armchair and zigzag edges, at $\mathrm{\Phi }=0$, for (a) $F_x=0$ and (b) $F_x=0.05$ V/nm. (c) and (d) show the probability densities of the two lowest edge states in (a) while (e) and (f) show those of the two lowest states in (b).