Quantum dots in AA-stacked bilayer graphene

While electrostatic confinement in single-layer graphene and AA-stacked bilayer graphene is precluded by Klein tunneling and the gapless energy spectrum, we theoretically show that a circular domain wall that separates domains of single-layer graphene and AA-stacked bilayer graphene can provide bound states. Solving the Dirac-Weyl equation in the presence of a global mass potential and a local electrostatic potential, we obtain the energy spectrum of these states and the corresponding radial probability densities. Depending on the mass potential profile, regular bound states can exist inside the quantum dot and topological bound states at the domain wall. Controlling the electrostatic potential inside the quantum dot enables the simultaneous presence of both types of states. We find that the number of nodes of the radial wave function of the regular bound states inside the quantum dot is equal to the radial quantum number. The energy spectra of the bound states display anticrossings, reflecting coupling of electron- and holelike states.

Figure 1

Low-energy spectrum of (a) SLG and (b) AA-BLG. Solid black and dashed red lines correspond to ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0$ and ${\mathrm{\Delta }}_1=-{\mathrm{\Delta }}_2=0.5{\gamma }_1$, respectively.

Figure 2

(a) and (b) Schematic pictures of the studied QDs. A circular DW with radius $R$ separates domains of SLG and AA-BLG. Mass potential profiles considered for layers 1 and 2 in the case of (c) and (e) QD1 and (d) and (f) QD2. Yellow and gray regions refer to SLG and AA-BLG.

Figure 3

Energy spectrum of the bound states in QD1 [see Fig. 2] as a function of the DW radius $R$ for $m=1$, ${\mathrm{\Delta }}_1=0.5{\gamma }_1$, $v_0^{<}=0.5{\gamma }_1$, $v_0^{>}=0$, and the mass potential profiles of (a) Fig. 2 and (b) Fig. 2. Dashed orange lines mark the energy gap $\pm 0.5{\gamma }_1$ outside the QD, and the dashed green line marks the top of the valence band $v_0^{<}-0.5{\gamma }_1$ inside the QD.

Figure 4

Band structure (a) inside and (b) outside QD1 with the parameters of Fig. 3. Note that the band structure is the same for the mass potential profiles of Figs. 2 and 2.

Figure 5

RPD for the labeled states in Fig. 3 (QD1). The dashed green line indicates the radius of the QD.

Figure 6

Spatial distributions of the local density of states for the bound states with energy $E_{n,1}^{\tau ,+1}\left(10l\right)$ in Fig. 3. The black dashed circles represent the QD with radius $R=10l$. The bottom and top rows refer to the $K$ and $K^{\prime }$ valleys, respectively.

Figure 7

Energy spectrum of the topological and bound states in QD2 [see Fig. 2] as a function of the DW radius $R$ for $m=1$, ${\mathrm{\Delta }}_1=0.5{\gamma }_1$, $v_0^{<}=0.5{\gamma }_1$, $v_0^{>}=0$, and the mass potential profile of (a) Fig. 2 and (b) Fig. 2. Dashed orange lines mark the energy gap $\pm 0.5{\gamma }_1$ outside the QD, and the dashed green line marks the top of the valence band $v_0^{<}-0.5{\gamma }_1$ inside the QD, as indicated in Fig. 4 for switched bands inside and outside the QD and the same $v_0$. In (c) the solid brown lines are the energy levels of the $K$ valley from (b), and the blue dashed lines are the energy levels of an AA-BLG nanodisk with ZZ1 edge for $m=1$.

Figure 8

RPD for the labeled states in Fig. 7 (QD2). The dashed green line indicates the radius of the QD.