Cloaking resonant scatterers and tuning electron flow in graphene

We consider resonant scatterers with large scattering cross sections in graphene that are produced by a gated disk or a vacancy, and show that a gated ring can be engineered to produce an efficient electron cloak. We also demonstrate that this same scheme can be applied to tune the direction of electron flow. Our analysis is based on a partial-wave expansion of the electronic wave functions in the continuum approximation, described by the Dirac equation. Using a symmetrized version of the massless Dirac equation, we derive a general condition for the cloaking of a scatterer by a potential with radial symmetry. We also perform tight-binding calculations to show that our findings are robust against the presence of disorder in the gate potential.

Figure 1

Sketch of the problem, showing the incoming electron, the impurity, and the cloak.

Figure 2

(a) Phase shifts ${\delta }_0$ (black), ${\delta }_1$ (red) and ${\delta }_2$ (blue), and (b) transport cross section ${\sigma }_T$ (six first partial waves are considered) as a function of the electrostatic potential $V$ for $E=0.02$ eV and $R=10$ nm.

Figure 3

Transport cross section for a vacancy of radius $R_1=1\text{nm}$ as a function $E$ considering the six first partial waves (solid) and a single partial wave ${\delta }_0$ (dashed).

Figure 4

(a) Cloaking efficiency as a function of $V_2$ and $R_2$ for $E=0.02\text{eV}$, $R_1=10\text{nm}$, and $V_1=-114\text{meV}$. (b) Cloaking efficiency for the same parameters of (a) and $R_2=20$ nm.

Figure 5

(a) Cloaking efficiency as a function of $V_2$ and $E$ for $R_1=10\text{nm}$, $V_1=-114\text{meV}$, and $R_2=20\text{nm}$. (b) Cloaking efficiency as a function of $E$ for the same parameters of (a) and $V_2=286$ meV.

Figure 6

Differential cross section as a function of $V_2$ and the scattering angle $\theta$ for $E=0.02\text{eV}$, $R_1=10\text{nm}$, $V_1=-114\text{meV}$, and $R_2=20\text{nm}$.

Figure 7

(a) Cloaking efficiency as a function of $V_2$ and $R_2$ for $E=2\text{meV}$ and a vacancy of radius $R_1=1\text{nm}$; (b) cloaking efficiency for the same parameters of (a), $E=1\text{meV}$ and $R_2=20$ nm.

Figure 8

(a) Cloaking efficiency as a function of $V_2$ and $E$ for a vacancy of radius $R_1=10\text{nm}$ and $R_2=20\text{nm}$. (b) Cloaking efficiency as a function of $E$ for the same parameters of (a) and $V_2=87$ meV.

Figure 9

Density plot of $|{\tilde{K}}_m|$, with $R=2.8$ Å, and $v_0=\pi /2$, as function of $k$ and $R_0$. Top panel: $|{\tilde{K}}_0|$; bottom panel: $|{\tilde{K}}_1|$.

Figure 10

Density plot of $|{\tilde{K}}_m|$, with $R=2.8$ Å, and $k=0.034$ 1/Å  (corresponding to $V_g=50$ V in a graphene FET on ${\mathrm{SiO}}_2$), as function of $v_0$ and $R_0$. Top panel: $|{\tilde{K}}_0|$; bottom panel: $|{\tilde{K}}_1|$.

Figure 11

Logarithm of the local density of states for $E=0.00\text{eV}$. Panel (a): $V_1=0.45t{V}_2=0$. Panels (b) and (c): $V_1=0.45t{V}_2=0.23t$. The sites with $V_i=V_1$ are black while the sites with $V_i=V_2$ are green.