Time-dependent transport in graphene nanoribbons

We theoretically investigate the time-dependent ballistic transport in metallic graphene nanoribbons of width $W$ after the sudden switching of a bias voltage. The potential drop is linear across a central part of length $L$ where the current is calculated. During the early transient time the current does not grow linearly in time but remarkably reaches a temporary plateau. Such behavior allows us to define a transient conductivity, the value of which coincides with the minimal conductivity of two-dimensional graphene. At time $L/v_F$ ($v_F$ being the Fermi velocity) a crossover takes place: the current changes abruptly and saturates at its final steady-state value (second plateau). We show that the two plateaus develop with damped oscillations of totally different nature and demonstrate that the occurrence of the first plateau is independent of the boundary conditions. The transition from quasi-one-dimensional to bulk behavior $\left(W\to \infty \right)$ is also analyzed.

Figure 1

(Color online) Schematic representation of the system. The nanoribbon has semi-infinite L and R graphenic reservoirs. The linear drop of the bias voltage in region C is also shown.

Figure 2

(Color online) Ladder model for fixed transverse momentum $k_y$. The bias profile is the same as in Fig. 1.

Figure 3

(Color online) Total current $I\left(t\right)$ with geometric parameters $L\approx 25\text{nm}$, $W\approx 106\text{nm}$, $N_r=2000$, and applied voltage $V=8\times {10}^{-4}\text{Volt}$. Current is in units of $V{\sigma }_0$, where ${\sigma }_0=e^2/h$ is the quantum of conductivity and time in units of $t_v=2.5\times {10}^{-16}\text{s}$. The values of the dc conductivity (dashed line) ${\sigma }_{\text{dc}}/{\sigma }_0=4$ (Ref. 32) and of the minimal conductivity (dotted line) $\left({\sigma }_{\text{min}}\times W/L\right)/{\sigma }_0=\pi /2\times W/L$ are also shown.

Figure 4

(Color online) Current $I\left(t\right)$ for different widths $W_1=106\text{nm}$, $W_2=27\text{nm}$, $W_3=13\text{nm}$. The rest of parameters are as in Fig. 3.

Figure 5

(Color online) Current $I\left(t\right)$ for three different armchair open boundary nanoribbons. The parameters are $L=5.5\text{nm}$, $W_1=3.6\text{nm}$, $W_2=2.1\text{nm}$, $W_3=0.6\text{nm}$ and the applied bias is $V=0.03\text{Volt}$. The leads have $N_r=9$ cells. The values of the dc conductivity (dashed line) ${\sigma }_{\text{dc}}/{\sigma }_0=2$ (Refs. 10, 32) and of the minimal conductivity (dotted line) $\left({\sigma }_{\text{min}}\times W/L\right)/{\sigma }_0=\pi /2\times W/L$ are also shown.