Zero-energy states in graphene quantum dots and rings

We present exact analytical zero-energy solutions for a class of smooth-decaying potentials, showing that the full confinement of charge carriers in electrostatic potentials in graphene quantum dots and rings is indeed possible without recourse to magnetic fields. These exact solutions allow us to draw conclusions on the general requirements for the potential to support fully confined states, including a critical value of the potential strength and spatial extent.

Figure 1

Radial wave-function components for the first two states ($N=0,1$) with angular momentum $m=1$ for the Lorentzian dot ($k=0$) and the first ring ($k=1$): (a) $k=0$, $N=0$; (b) $k=0$, $N=1$; (c) $k=1$, $N=0$; and (d) $k=1$, $N=1$. Solid (dotted) lines correspond to components ${\chi }_A$ ($i{\chi }_B$). Insets: shape of the probability density for each state.

Figure 2

Schematic representation of a proposed experimental setup in which a charged STM tip with characteristic radius $R_{\mathrm{tip}}$ produces confined states in the graphene sheet either by varying the charge on the tip or by fixing the tip voltage $V_{\mathrm{tip}}$ and moving the tip vertically with respect to the flake. The presence of bound states is exhibited by peaks in the conductivity whenever new states appear.

Figure 3

Comparison of a repulsive electric potential created by a charged STM tip (blue dotted line) to the solvable Lorentzian potential (red solid line) as a function of scaled radial distance. The potential profiles clearly are very similar. Chosen values of $h_1$ and $h_2$ here are arbitrary; the profiles resemble that shown here for any values so long as $h_2>h_1$. $U_0=V_0\hslash v_{\mathrm{F}}$ is the maximum value of the potential at the origin.