Gate-tunable valley currents, nonlocal resistances, and valley accumulation in bilayer graphene nanostructures

Using the Büttiker-Landauer formulation of transport theory in the linear response regime, the valley currents and nonlocal resistances of bilayer graphene nanostructures with broken inversion symmetry are calculated. It is shown that broken inversion symmetry in bilayer graphene nanostructures leads to striking enhancement of the nonlocal four-terminal resistance and to valley currents several times stronger than the conventional electric current when the Fermi energy is in the spectral gap close to the energy of Dirac point. The scaling relation between local and nonlocal resistances is investigated as the gate voltage varies at zero Fermi energy and a power law is found to be satisfied. The valley velocity field and valley accumulation in four-terminal bilayer graphene nanostructures are evaluated in the presence of inversion symmetry breaking. The valley velocity and nonlocal resistance are found to scale differently with the applied gate voltage. The unit cell-averaged valley accumulation is found to exhibit a dipolar spatial distribution consistent with the accumulation arising from the valley currents. We define and calculate a valley capacitance that characterizes the valley accumulation response to voltages applied to the nanostructure's contacts.

Figure 1

Four-terminal BLG nanostructure with armchair edges. The bottom (top) layer is shown in black (blue). Each contact is composed of 40, semi-infinite 1D ideal leads (shown by red wavy lines) that are attached to both layers and connect the nanostructure to the reservoirs. The electric current flows through current contacts 1 and 2 while there is no net electric current entering or leaving the voltage contacts 3 and 4. In nonlocal resistance studies the potential difference is measured between contacts 3 and 4. Upper right inset: Two types of first Brillouin zone of bilayer graphene, hexagonal (solid) and rhombic (dotted). Lower right inset: Side view of four carbon atoms of a unit cell in bilayer graphene in AB stacking. The inversion symmetry point in the unit cell is the intersection of dotted lines.

Figure 2

(a) Calculated nonlocal resistance $R_{\text{NL}}$ [Eq. (7)] of the nanostructure of Fig. 1 in the linear response regime at zero temperature for different values of the gate voltage $V_g=0$ eV (green), $V_g=0.3$ eV (red), and $V_g=0.5$ eV (blue) as a function of Fermi energy $E_F$. (b) Normalized valley velocity $v_x^{\text{val}}/v_y$ of the BLG nanostructure for different values of the gate voltage as a function of Fermi energy when the net current flows between current contacts 1 and 2. (c) Comparison of the $x$ and $y$ components of the normalized valley velocity as a function of Fermi energy at $V_g=0.5$ eV. (d) Comparison of the $x$ and $y$ components of normalized valley velocity as a function of Fermi energy at $V_g=0.3$ eV.

Figure 3

(a) The normalized valley velocity $\left(v_x^{\text{val}}/v_y\right)$ (purple line) and nonlocal resistance $R_{\text{NL}}$ (green line) of the bilayer graphene nanostructure as a function of gate voltage at zero Fermi energy $E_{\mathrm{F}}=0$. (b) The scaling relation between the local and nonlocal resistance as the gate voltage varies from 0 to 0.2 eV. Red line is the power law $R_{\text{NL}}\sim R_{\mathrm{L}}^{\alpha }$ fitted to the simulation data. Upper left inset: The configuration of nonlocal resistance measurements. Lower right inset: The configuration of local resistance measurements.

Figure 4

The calculated total valley capacitance, valley velocity field, and unit cell-averaged valley capacitance of the bottom layer of a bilayer graphene nanostructure when the net electric current flows between current contact 1 and 2 at $V_g=0.15$ eV and $E_F=0$. (a) The total valley capacitance [Eq. (13)] is represented by blue (red) disks when the on-site valley accumulation is positive (negative). (b) The valley velocity field (black arrows) and average valley capacitance (red and blue disks), Eq. (14).