Fano resonances in the conductance of graphene nanoribbons with side gates

The control of side gates on the quantum electron transport in narrow graphene ribbons of different widths and edge types (armchair and zigzag) is investigated. The conductance exhibits Fano resonances with varying side gate potential. Resonant and antiresonant peaks in the conductance can be associated with the eigenstates of a closed system, and these peaks can be accurately fitted with a Fano line shape. The local density of states (LDOS) and the electron current show a specific behavior at these resonances, which depends on the ribbon edge type. In zigzag ribbons, transport is dominated by intervalley scattering, which is reflected in the transmission functions of individual modes. The side gates induce $p-n$ interfaces near the edges at which the LDOS exhibits peaks. Near the resonance points, the electron current flows uniformly through the constriction, while near the antiresonances it creates vortices. In the armchair ribbons the LDOS spreads in areas of high potential, with current flowing near the edges.

Figure 1

(a) A graphene nanoribbon system. The colormap represents the change in the electron on-site energy caused by the two side gates SG1 and SG2. (b) and (c) Zoom of the edge of the ribbon showing the zigzag and the armchair edge. The width of the ribbon $\left(W_y\right)$ is defined by a number $n_y$ of atom lines (horizontal dashed blue lines) in the $y$ direction. Four ribbon types are considered: the zigzag ($n_y=46,W_y=9.66$ nm), the antizigzag ($n_y=47,W_y=9.87$ nm), the semiconducting armchair ($n_y=82,W_y=9.96$ nm), and the metallic armchair ($n_y=80,W_y=9.72$ nm). All of these have the length $\left(W_x\right)$ of 45 nm.

Figure 2

The dispersion relations for the considered four types of ribbons: the zigzag (a), the antizigzag (b), the semiconducting armchair (c), and the metallic armchair (d) ribbon. The thick black lines show results for no side gate potential $\left(U_c=0\right)$, while the thick blue lines show results for $U_c=0.2$ eV. We mark the incoming modes at $E_F=0.3$ eV with colored dots, and in (a) they are labeled by the decreasing value of their $k_x$ vector. The encircled area in (a) is shown enlarged and emphasizes the deformation of the first conduction band and the opening of new modes $n_1$ and $n_2$.

Figure 3

The conductance (thick black lines) as a function of the side gate potential at $E_F=0.3$ eV for the zigzag (a), the antizigzag (b), the semiconducting armchair (c), and the metallic armchair (d) ribbon. The colored curves show the transmission probabilities of individual modes. They are vertically displaced by $\Delta T=3$ for clarity. The legend in (a) indicates which line color corresponds to which incoming mode [the modes are numbered in Fig. 2]. The gray areas in (a) and (b) mark the range of $U_c$ values for which modes from both conduction and valence bands coexist in the constriction region. This is illustrated schematically in the insets of (b), which show how the dispersion in the constriction looks like for different values of the potential. The band bending is exaggerated for better view.

Figure 4

The conductance as a function of the Fermi energy and the side gate potential $U_c$ for zigzag (a), antizigzag (b), semiconducting armchair (c), and metallic armchair (d) ribbon. The blue (green) lines mark the minimal (maximal) energies of the electron (hole) bands. The white boxes highlight the areas with resonance peaks where a comparison with the eigenstates is made.

Figure 5

Conductance of the zigzag ribbon $(n_y=46)$ for different misalignments between the side gates: (a) $\Delta x=0$, (b) $\Delta x=l_c$/2, (c) $\Delta x=l_c$, and (d) $\Delta x=3l_c/2$, where $l_c$ is the graphene lattice constant.

Figure 6

Conductance (thick black lines) through the zigzag ribbon as a function of the side gate potential for $\Delta x=0.5l_c$ (a) and $\Delta x=l_c$ (b). The Fermi energy is 0.3 eV. Similarly to Fig. 3, transmissions of individual modes are displaced by $\Delta T=2$.

Figure 7

Comparison between the conductance of an open system and the eigenenergies (black lines) of a closed system (a ribbon detached from the leads) for the zigzag (a), the antizigzag (b), the semiconducting armchair (c), and the metallic armchair (d) ribbon. $E$ and $U_c$ run over the ranges bounded by the white rectangles in Fig. 4.

Figure 8

The probability density of the eigenstates ${\Psi }_i$ ($i=1$, 2, 3, 4) from Fig. 7. The white lines are equipotential lines for which $U(x,y)$ is equal to the eigenstate energy.

Figure 9

Comparison between the conductance of an open system and the eigenenergies (black lines) of a closed system (with next-nearest-neighbor interaction included, $t^{\prime }=0.1t$) for the zigzag (a), the antizigzag (b), the semiconducting armchair (c), and the metallic armchair (d) ribbon.

Figure 10

Fano resonances: comparison between the numerical results obtained by kwant (dots) and lines fitted by the normalized expression for the Fano resonance. (a) The sum of transmissions of the second and the third mode for the zigzag ribbon $U_c=0.3$ eV. (b) The transmission of the first mode for the zigzag ribbon $U_c=0.3$ eV. (c) The transmission of the first mode for the semiconducting armchair ribbon $U_c=0.45$ eV. (d) The sum of transmissions of the second and the third mode for the metallic armchair ribbon $U_c=0.9$ eV.

Figure 11

LDOS at $E=300$ meV for the zigzag ribbon at $U_c=737$ meV (a), the antizigzag ribbon at $U_c=750$ meV (b), the metallic armchair ribbon at $U_c=1084$ meV (c), and the semiconducting armchair ribbon at $U_c=1276$ meV (d). The white lines are equipotential lines for which $U(x,y)=E$.

Figure 12

The relative current density at $E=300$ meV for the same parameters as in Fig. 11.

Figure 13

The current in the zigzag ribbon at $E=0.3$ eV for (a) $U_c=U_{\min }-\Delta u$ before the zero-conductance dip, and (b) $U_c=U_{\min }+\Delta u$ after the zero-conductance dip ($U_{\min }=374.82$ meV, $\Delta u=0.15$ meV). For clarity, results are averaged and presented on a square mesh.

Figure 14

Conductance through the zigzag ribbon as a function of the side gate potential for different number $\left(n\right)$ of randomly removed carbon atoms. The Fermi energy is $E=300$ meV. For clarity, results for different $n$'s are moved vertically by $\Delta T=2$.