Interplay of Aharonov-Bohm and Berry phases in gate-defined graphene quantum dots

We study the influence of a magnetic flux tube on the possibility to electrostatically confine electrons in a graphene quantum dot. Without a magnetic flux tube, the graphene pseudospin is responsible for a quantization of the total angular momentum to half-integer values. On the other hand, with a flux tube containing half a flux quantum, the Aharonov-Bohm phase and Berry phase precisely cancel, and we find a state at zero angular momentum that cannot be confined electrostatically. In this case, true bound states only exist in regular geometries for which states without zero-angular-momentum component exist, while nonintegrable geometries lack confinement. We support these arguments with a calculation of the two-terminal conductance of a gate-defined graphene quantum dot, which shows resonances for a disk-shaped geometry and for a stadium-shaped geometry without flux tube, but no resonances for a stadium-shaped quantum dot with a $\pi$-flux tube.

Figure 1

The geometry under consideration: A quantum dot (here with circular shape), consisting of a region of graphene with a nonzero spatially uniform carrier density surrounded by a carrier-free graphene layer, which is coupled to leads in a two-terminal geometry. The total system has a rectangular shape of dimension $L\times W$; the size of the quantum dot is denoted $R$. In this paper we consider the effect of a magnetic flux tube through the quantum dot (indicated in red). If the flux tube carries half a flux quantum, the Aharonov-Bohm phase precisely cancels the Berry phase that is accumulated in a cyclic orbit inside the quantum dot.

Figure 2

Two-terminal conductance of a graphene sheet containing a disk-shaped quantum dot without (top) and with (bottom) $\pi$-flux tube. Model parameters are $R/L=0.2$ and $W/L=6$. Without flux tube, resonances have definite angular momentum, with quantum number $\left|m\right|$ indicated at each resonance (data taken from Ref. 14). Without flux tube, resonances are labeled by the kinematic angular momentum quantum number $\left|\mu \right|$. No resonance is found for $\mu =0$.

Figure 3

First two resonances for a disk-shaped quantum dot with $\pi$-flux tube, for different coupling strengths to the leads. Calculations are performed for $W/L=8$ and various $R/L$, as indicated in the figure. The second resonance is shown enlarged in the inset.

Figure 4

Two-terminal conductance of a graphene sheet containing a stadium-shaped quantum dot without (top) and with (bottom) a $\pi$ flux. Parameters for the calculation are $R/L=0.2$, $W/L=12$, $a/R=\sqrt{3}/2$, $d=2a/3$. Without flux tube, the calculation for the conductance was done with the method of Ref. 17.

Figure 5

Behavior of the first three quasiresonances of the stadium-shaped quantum dot without (top) and with (bottom) $\pi$ flux upon changing the coupling to the leads $R/L$. The other parameters are the same as in Fig. 4.