Magnetic field dependence of energy levels in biased bilayer graphene quantum dots

Using the tight-binding approach, we study the influence of a perpendicular magnetic field on the energy levels of hexagonal, triangular, and circular bilayer graphene (BLG) quantum dots (QDs) with zigzag and armchair edges. We obtain the energy levels for AB (Bernal)-stacked BLG QDs in both the absence and the presence of a perpendicular electric field (i.e., biased BLG QDs). We find different regions in the spectrum of biased QDs with respect to the crossing point between the lowest-electron and -hole Landau levels of a biased BLG sheet. Those different regions correspond to electron states that are localized at the center, edge, or corner of the BLG QD. Quantum Hall corner states are found to be absent in circular BLG QDs. The spatial symmetry of the carrier density distribution is related to the symmetry of the confinement potential, the position of zigzag edges, and the presence or absence of interlayer inversion symmetry.

Figure 1

Sketches of (a),(b) armchair and (c),(d) zigzag BLG QDs with (a),(c) hexagonal, (b),(d) triangular, and (e),(f) circular geometries considered in this work. The first four geometries are characterized by the number of carbon hexagons $N$ at their edges. The circular dots (e) and (f) are characterized by their radius $R$. (a)–(e) QDs are defined by cutting the BLG lattice and (f) is defined by a size-dependent staggered potential $M_i$, where the atoms belonging to sublattices $A_b \left(A_u\right)$ and $B_b \left(B_u\right)$ have a mass-term potential given by $+M_0 \left(-M_0\right)$ and $-M_0 (+M_0$), respectively, and $M_i=0$ in the region inside the dot, represented in green. Bottom (top) atoms are given by black open (blue closed) symbols.

Figure 2

Energy levels (a) in the absence $\left(V=0\right)$ and (b) in the presence ($V=0.1$ eV) of a bias potential as a function of the magnetic flux $\left(\varphi /{\varphi }_0\right)$ through a hexagonal BLG QD with armchair edges and side length $L\approx 10.366$ nm ($N=25$ carbon hexagons in each side), as sketched in Fig. 1. The red dashed curves are the LLs of an infinite BLG sheet.

Figure 3

The probability density for the bottom $|{\mathrm{\Psi }}_b{|}^2$ and top $|{\mathrm{\Psi }}_u{|}^2$ graphene layers is shown for states labeled in Fig. 2 by the points 1 ($E\approx 0.20182$ eV), 2 ($E\approx 0.21093$ eV), 3 ($E\approx 0.1$ eV), 4 ($E\approx 0.07421$ eV), and 5 ($E\approx 0.01587$ eV), respectively, for the unbiased and biased cases and for $\varphi /{\varphi }_0=4.0\times {10}^{-3}$. Red (blue) colors stand for high (low) density.

Figure 4

(a),(b) The same as in Figs. 2 and 2, but now for the armchair triangular BLG QD with lateral size $L\approx 25.418$ nm (equivalent to $N=60$ carbon hexagons in each side), as sketched in Fig. 1. The red dashed curves are the LLs of an infinite BLG sheet. (c) An enlargement of the spectrum for low energies. (d) The probability density for the bottom $|{\mathrm{\Psi }}_b{|}^2$ and top $|{\mathrm{\Psi }}_u{|}^2$ layers is shown for states 1 and 2 denoted in (c) with energies $E_1=0.0345$ and $E_2=0.0368$ eV. Red (blue) colors stand for high (low) density.

Figure 5

(a) An enlargement of the energy spectrum around the crossing point ${\varphi }_C$ for a biased triangular BLG QD [see Fig. 4]. The red dashed curves are the two lowest LLs of an infinite sheet of BLG. (b) The probability density contribution from each layer (bottom $|{\mathrm{\Psi }}_b{|}^2$ and top $|{\mathrm{\Psi }}_u{|}^2$), as well as the total density $\left(|{\mathrm{\Psi }}_b{|}^2+{\left|{\mathrm{\Psi }}_u\right|}^2\right)$, are shown for states labeled by 1 ($E\approx 1.7039\times {10}^{-4}$ eV), 2 ($E\approx -1.7039\times {10}^{-4}$ eV), 3 ($E\approx 9.9892\times {10}^{-4}$ eV), 4 ($E\approx 1.3996\times {10}^{-3}$ eV), 5 ($E\approx 2.0607\times {10}^{-3}$ eV), 6 ($E\approx 2.6142\times {10}^{-3}$ eV), and 7 ($E\approx 6.9856\times {10}^{-3}$ eV) in the energy spectrum. Red (blue) colors stand for high (low) density.

Figure 6

(a) A zoom of the magnetic energy spectrum for a hexagonal armchair BLG QD [Fig. 2] near the zero-energy level. The red dashed curves are the lowest zeroth LLs of an infinite sheet of BLG. (b) The total probability densities are shown for states labeled by 1 to 8 at $\varphi /{\varphi }_0=10.0\times {10}^{-3}$. Red (blue) colors stand for high (low) density.

Figure 7

The same as Fig. 6, but now for the armchair triangular BLG QD.

Figure 8

Energy levels (a) in the absence and (b) in the presence of a bias potential as a function of the magnetic flux $\left(\varphi /{\varphi }_0\right)$ for a hexagonal BLG QD with zigzag edges and $L\approx 10.084$ nm ($N=41$ carbon hexagons in each side), as sketched in Fig. 1. The red dashed curves are the LLs of an infinite sheet of BLG. The bias potential is $V=0.1$ eV.

Figure 9

(a) A zoom of the hexagonal zigzag QD energy spectrum [Fig. 8] near the zero-energy level. The red dashed curves are the lowest zeroth LLs of an infinite sheet of BLG. (b),(c) The total probability densities are shown for states labeled by 1–2, 3–4, 5–6, and 7–12 at $\varphi /{\varphi }_0=0.15\times {10}^{-3},0.22\times {10}^{-3},0.3\times {10}^{-3}$, and $10.0\times {10}^{-3}$, respectively. Larger red circles represent higher-density amplitudes.

Figure 10

The same as in Fig. 8, but for the zigzag triangular BLG QD with side length $L\approx 25.415$ nm ($N=103$ carbon hexagons along each side), as sketched in Fig. 1.

Figure 11

(a) A zoom of the triangular zigzag magnetic energy spectrum [Fig. 10] near the zero-energy level. The red dashed curves are the lowest zeroth LLs of an infinite sheet of BLG. (b) The total probability densities are shown for states labeled by the points 1 to 6 at $\varphi /{\varphi }_0=0.2\times {10}^{-3}$ and for points 7 to 12 at $\varphi /{\varphi }_0=10.0\times {10}^{-3}$. Red (blue) colors stand for high (low) density.

Figure 12

Energy levels as a function of the magnetic flux $\left(\varphi /{\varphi }_0\right)$ for a circular BLG QD with radius $R=10$ nm (a) in the absence and (b) in the presence of a bias potential, and (c) considering a mass potential profile surrounding an unbiased region, as sketched in Figs. 1 and 1, respectively. The red dashed curves are the LLs of an infinite sheet of BLG. The bias potential is $V=0.1$ eV. The inset in (b) shows a zoom of the low-energy oscillations at low magnetic fields. ${\varphi }_R=B\pi R^2$ is the magnetic flux threading the circular dot.

Figure 13

The probability amplitude for the bottom $\left(|{\mathrm{\Psi }}_b{|}^2\right)$ and top $\left(|{\mathrm{\Psi }}_u{|}^2\right)$ layers, as well as the total contribution ($|{\mathrm{\Psi }}_b{|}^2+{\left|{\mathrm{\Psi }}_u\right|}^2$) are shown for points 1 and 2 of the energy levels indicated in Fig. 12 at the magnetic flux $\left(\varphi /{\varphi }_0=0.5\times {10}^{-3}\right)$. The maximum amplitudes match with the six largest zigzag parts of the circular BLG QD with $R=10$ nm. Larger red circles represent higher-density amplitudes.