Electron-electron interactions in bilayer graphene quantum dots

A parabolic quantum dot (QD) as realized by biasing nanostructured gates on bilayer graphene is investigated in the presence of electron-electron interaction. The energy spectrum and the phase diagram reveal unexpected transitions as a function of a magnetic field. For example, in contrast to semiconductor QDs, we find a valley transition rather than only the usual singlet-triplet transition in the ground state of the interacting system. The origin of these features can be traced to the valley degree of freedom in bilayer graphene. These transitions have important consequences for cyclotron resonance experiments.

Figure 1

Schematic illustration of the potential profile for a parabolic bilayer graphene quantum dot.

Figure 2

Energy spectrum of a parabolic QD in BLG with $R=50$ nm and $U_b=150$ meV as a function of the magnetic field for (a) a single particle and (b) two noninteracting electrons. The lowest energy levels in (a) are labeled by the angular momentum $m$. The levels corresponding to the $K$ and $K^{\prime }$ valleys are, respectively, shown by the blue solid and red dashed curves. The black dotted curves are the LLs of a BLG sheet. The levels in (b) are labeled by $(M,\mathcal{T})$, where $M=m_1+m_2$ is the total angular momentum and $\mathcal{T}={\tau }_1+{\tau }_2$ is the total valley index. Levels having the same valley index are plotted using the same type of curve.

Figure 3

(a),(b) The same as Fig. 2 but in the presence of Coulomb interaction and for (a) $R=20$ nm and (b) $R=50$ nm. The levels are labeled by $(M,\mathcal{T},S)$ where $S$ indicates the total spin. The curves with the same total valley index $\mathcal{T}$ are shown with the same type of curve. (c) The average distance between the electrons as function of magnetic field for different single-particle basis functions. (d) The radius-magnetic field ($R-B$) phase diagram of the ground-state energy of a two-electron BLG QD.

Figure 4

The radial electron density for the two-electron QD of Fig. 3 for (a) $(M,\mathcal{T})\equiv (-2,0)$ and (b) $(M,\mathcal{T},S)\equiv (-3,2,1)$ and for $B=0$, $1.5$, 5 T. The upper insets show the total pair-correlation function for $B=0$ T and $B=5$ T. The lower insets show separate parts of the pair-correlation function for $B=0$. The black dot indicates the position of the first electron, which is pinned at ${\mathbf{r}}_1^{}=(0.4R,0)$.

Figure 5

The cyclotron transition energies for the QD of Fig. 3. The transition energies are labeled by their final states: $(M,\mathcal{T},S)$ for the triplet (or singlet) states and $(M,\mathcal{T})$ for the degenerate single-triplet states. Levels having the same total valley index are plotted using the same type of curve.