Emergence of bound states in ballistic magnetotransport of graphene antidots

An experimental method for detection of bound states around an antidot formed by a hole in a graphene sheet is proposed via measuring the ballistic two-terminal conductance. In particular, we consider the effect of bound states formed by a magnetic field on the two-terminal conductance and show that one can observe Breit-Wigner-like resonances in the conductance as a function of the Fermi level close to the energies of the bound states. In addition, we develop a numerical method utilizing a reduced computational effort compared to the existing numerical recursive Green's function methods.

Figure 1

Graphene strip including a hole and contacted by metallic leads. A magnetic field $B$ perpendicular to the sheet of graphene generates bound state $H$ localized to the hole and propagating edge states ${\xi }_1$ and ${\xi }_2$. The edge states are coupled to the bound states due to scattering on impurities in the graphene lattice. The couplings are modeled by the hopping terms ${\Gamma }_1$ and ${\Gamma }_2$.

Figure 2

Left panels: Graphene strip of width $W$ and length $L$. The hole in the strip has a diameter $2R$. Blue circles correspond to atomic vacancies in the lattice. The wave functions localized to the hole are also plotted in (a) and (b) at energies $E=0.619\hslash {\omega }_c$ and $E=0.606\hslash {\omega }_c$, respectively. Right panels: The conductance (solid black line) in units of ${\sigma }_0=2e^2/h$ as a function of the Fermi energy. Vertical dashed and solid red lines correspond to BSs localized to the hole, where the solid line labels the energy of the BS plotted in the left panels. Data in the figures were calculated at magnetic fields given by the magnetic length $l_B$ (see the numerical values in the subplots). The magnetic flux in the hole is $\varphi =\pi R^{2}B=0.817{\varphi }_0$ (${\varphi }_0=h/e$ is the flux quantum) and $\varphi =3.72{\varphi }_0$ for (a) and (b), respectively. The metallic contacts were modeled as heavily doped graphene leads shifted by a potential of $V=-0.27\gamma$ (where $\gamma$ is the coupling term between the nearest-neighbor sites of the lattice).

Figure 3

Energy eigenvalues of BSs (in units of $\hslash {\omega }_c$) localized to the hole in the graphene strip as a function of the magnetic flux of the hole (in units of ${\varphi }_0=h/e$), calculated for a hole with a diameter of $2R=18.9$ nm. At low magnetic fluxes the lines tend to the closest Landau level, whereas at higher magnetic fields they become parallel to each other.

Figure 4

Solid black line: The calculated conductivity (in units of ${\sigma }_0=2e^2/h$) of a graphene strip with zigzag edges between contacts, shown in the inset as a function of the aspect ratio $W/L$. The length of the strip was $L=20.4$ nm, and the calculations were performed at the energy of $E=0.3\times {10}^{-4}\gamma$. The metallic contacts were modeled as heavily doped graphene leads shifted by a potential of $V=-0.27\gamma$. (Here $\gamma$ is the coupling term between the nearest-neighbor sites of the lattice.) The theoretical value for the minimal conductivity of graphene is ${\sigma }_{\min }=2{\sigma }_0/\pi$ (dashed red line).

Figure 5

(a) Schematic representation of an infinite ribbon made of unit cells coupled to each other by nearest-neighbor couplings. The lattice constant is $a$ and the distance between the first and the $N\mathrm{th}$ unit cell is $L_N=(N-1)a$. (b) The UC of a zigzag-edged graphene ribbon is shown by a dashed rectangle. The translational vector is denoted by $\mathbf{a}$.