Density of states as a probe of electrostatic confinement in graphene

We theoretically analyze the possibility to confine electrons in single-layer graphene with the help of metallic gates, via the evaluation of the density of states of such a gate-defined quantum dot in the presence of a ring-shaped metallic contact. The possibility to electrostatically confine electrons in a gate-defined “quantum dot” with finite-carrier density, surrounded by an undoped graphene sheet, strongly depends on the integrability of the electron dynamics in the quantum dot. With the present calculations, we can quantitatively compare confinement in dots with integrable and chaotic dynamics, and verify the prediction that the Berry phase associated with the pseudospin leads to partial confinement in situations where no confinement is expected according to the arguments relying on the classical dynamics only.

Figure 1

(a) Gate-defined graphene quantum dot (yellow) surrounded by an intrinsic graphene sheet and coupled to source and drain reservoirs (blue). (b) The setup considered here: A quantum dot (yellow) surrounded by an undoped graphene sheet and coupled to a ring-shaped metallic contact (blue). The distance between the dot's center and the contact is denoted $L$. The dot in (a) is disk shaped and has an integrable classical dynamics, for which the reflection angles at the dot's boundary are a constant of the motion. The dot in (b) is stadium shaped. In this case, classical trajectories sooner or later hit the dot's boundary arbitrarily close to perpendicular incidence.

Figure 2

Density of states for a circular quantum dot. Resonances are labeled according to their angular momentum $\left|m\right|$ ($R/L=0.2$).

Figure 3

Density of states for a stadium quantum dot. ($R/L=0.2,2a/R=\sqrt{3}$.)

Figure 4

Density of states for the first resonance, as well as resonance height $\mathcal{A}$ and width $\Gamma$ (insets) of the stadium dot for various values of the ratio $R/L$. ($2a/R=\sqrt{3}$.)

Figure 5

Density of states for a circular quantum dot in the presence of a flux tube. Resonances are labeled according to their kinematic angular momentum $\left|\mu \right|$. ($R/L=0.2$.)

Figure 6

Density of states for a stadium-shaped quantum dot in the presence of a $\pi$-flux tube. The flux is shifted from the center of the stadium in order to break inversion symmetry and obtain a truly chaotic structure; see also Ref. [19]. ($R/L=0.2,2a/R=\sqrt{3},d=2a/3$.)

Figure 7

Density of states for the first “resonance” and the corresponding resonance width (inset) of the stadium dot for various values of the ratio $R/L$. Both height and width of the feature saturate in the limit $R/L\to 0$. ($2a/R=\sqrt{3},d=2a/3$.)