Bound states and magnetic field induced valley splitting in gate-tunable graphene quantum dots

The magnetic field dependence of energy levels in gapped single-layer and bilayer graphene quantum dots (QDs) defined by electrostatic gates is studied analytically in terms of the Dirac equation. Due to the absence of sharp edges in these types of QDs, the valley degree of freedom is a good quantum number. We show that its degeneracy is efficiently and controllably broken by a magnetic field applied perpendicular to the graphene plane. This opens up a feasible route to create well-defined and well-controlled spin and valley qubits in graphene QDs. We also point out the similarities and differences in the spectrum between single-layer and bilayer graphene quantum dots. Striking in the case of bilayer graphene is the anomalous bulk Landau level (LL) that crosses the gap, which results in crossings of QD states with this bulk LL at large magnetic fields in stark contrast to the single-layer case where this LL is absent. The tunability of the gap in the bilayer case allows us to observe different regimes of level spacings directly related to the formation of a pronounced “sombrero” in the bulk band structure. We discuss the applicability of such QDs to control and measure the valley isospin and their potential use for hosting and controlling spin qubits.

Figure 1

(Color online) QD in single-layer graphene with a constant mass term $\Delta$. An electrostatic potential with height $U_0$ gives rise to bound states (dashed line) in the conduction band (c) defining a QD of radius $R$. Note that the confining potential $U\left(r\right)$ is repulsive for holes in the valence band (v).

Figure 2

(Color online) Bound-state levels as a function of the QD radius $R$ with $U_0=\Delta$ and for $j=1/2$ at zero magnetic field. Full lines correspond to $\tau =+1$; dashed lines correspond to $\tau =-1$.

Figure 3

(Color online) Numerical evaluation of characteristic Eqs. (11, 12) as a function of $R/l_B$, with $l_B={\left(\hslash /eB\right)}^{1/2}$ as the magnetic length and $R$ as the QD radius. We use $\Delta =10\delta$ and $U_0=\Delta$. (a) The parameter regime of small $B$ fields where we observe a breaking of the level degeneracy, shown for $(j,\tau )=(\frac{1}{2},\pm 1),(-\frac{1}{2},\pm 1)$, and $(\pm \frac{3}{2},\pm 1)$. The full lines are for $\tau =1$ and the dashed lines are for $\tau =-1$ corresponding to the two valleys of graphene. (b) Same parameters as in (a), but for larger magnetic fields. The energy levels converge to the bulk Landau levels with increasing $R/l_B$. Included are levels for $j=\pm \frac{1}{2},\pm \frac{3}{2},\pm \frac{5}{2}$.

Figure 4

QD in bilayer graphene: A back gate and dopants on top of the bilayer control the voltage $V$ between the layers—leading to a controllable gap opening—as well as the Fermi energy (band filling). An additional top gate allows to induce a spatially inhomogeneous electrostatic potential $U\left(r\right)$ analogous to the single-layer model, which leads to bound states in the conduction band (or valence band) of the bilayer. Another possibility is to use a split top gate (instead of a combination of top gate and dopants) to achieve a similar confinement.

Figure 5

(Color online) Energy levels in a relatively small bilayer QD (radius $R=25\text{nm}$) as a function of the magnetic field. The other parameters are as follows: $t_{\perp }=0.4\text{eV}=15.19\hslash v/R$, $V=1.9\hslash v/R$, $U_0=1.52\hslash v/R$, and $s=1$ (i.e., positive $B$ field). The solid and dashed lines are for different valleys.

Figure 6

(Color online) Energy levels in a relatively small bilayer QD with the same parameters as in Fig. 5 except that the bias field $V$ is about three times larger: $V=6\hslash v/R$.

Figure 7

(Color online) (a) Merging of bilayer QD levels to the bulk LLs as function of magnetic field for a relatively large bilayer QD with $R=67.48\text{nm}$ and $t_{\perp }=0.4\text{eV}=41\hslash v/R$, $U_0=3.5\hslash v/R$, $V=5.13\hslash v/R$ for $m=0,\pm 1$ and $s=1$ (i.e., positive $B$ field). Full lines are for $\tau =+1$ and dashed lines are for $\tau =-1$. (b) Bulk LLs which are approached almost perfectly at high fields in this parameter regime. Note that the $n=0$ LL (see Sec. 3B) crosses the gap with a negative slope, whereas the other LLs $\left(n=1,2,3\right)$ have positive slope (full lines and dashed lines distinguish the two valleys). There is also a flat LL $\left(n=-1\right)$ at $V/2$ similar to the single-layer case.

Figure 8

(Color online) Effective bandwidth of the bilayer QD defined by the area between the lines $E_{>}$ and $E_{<}$ (dashed lines) as a function of magnetic field for $V=17.1\hslash v/R$ and $U_0=7\hslash v/R$, $s=1$, $t_{\perp }=41\hslash v/R$, and for valley $\tau =-1$. The dotted line displays the $n=0$ bulk LL (in valley $\tau =-1$) that crosses the QD bandwidth corresponding to the band gap at zero magnetic field (full lines). This bulk LL presents an escape channel for the bilayer QD if bound states cross this bulk LL. Note that no bulk states overlap with the effective bandwidth (from the same valley).