Resonant longitudinal Zitterbewegung in zigzag graphene nanoribbons

The Zitterbewegung of a wave packet in a zigzag graphene nanoribbon is theoretically investigated. The coupling between edge states and bulk states results in intriguing properties. Apart from the oscillation in position perpendicular to the direction of motion, we also observe an oscillation along the direction of propagation which is not present in semiconductor nanowires or infinite graphene sheets. We also observe a resonance of its amplitude with respect to the central momentum of the wave packet. We show here that this longitudinal Zitterbewegung is caused by the interplay between bulk and edge states, which is a unique property of a zigzag nanoribbon.

Figure 1

Schematic of the initial wave packet at time $t=0$ in a zigzag graphene nanoribbon.

Figure 2

Variation of $k_n$ with $k_x$ near $K(k_x=0)$ point. The solid blue and the red dashed lines are respectively the real and imaginary solution for $k_n$ and correspond to the bulk and the edge states. The imaginary solution (edge state) appears for $k_{x}L\ge 1$, the limiting value ($k_{x}L=1$) being denoted by the vertical black dashed line.

Figure 3

Band structure of a ZGNR near the $K$ and $K^{\prime }$ points. The blue lines are due to the bulk state and the red lines are due to the edge states. The green region denotes the continuous spectrum for an infinite graphene sheet.

Figure 4

Norm of the wave packet at $t=0$ for three different values of $k_x^0$ with $d=L/4$. For $k_x^0=-8/L$ the bulk states are dominant. For $k_x^0=1/L$ both bulk and edge gives the same contribution and for $k_x^0=8/L$ the states are localized at the edges. The width of the ribbon is $L$ and the same scale is used for the length. The blue dashed line denotes $x=0$.

Figure 5

Expectation values for $x$ and $y$ for the complete wave function ($\Psi$) and for the positive and negative components ($+/-$), respectively. Here we use $k_x^0=1/L$ and $d=L/4$.

Figure 6

Zitterbewegung component of (a) $\langle x\left(t\right)\rangle$ and (b) $\langle y\left(t\right)\rangle$ for a wave packet with Gaussian width $d=L/4$. (c) and (d) show the variation of amplitude with the central momentum.

Figure 7

Variation of (a) $x_Z^{\text{max}}$ and (b) $\Gamma$ with $k_x^0$ for different $d$ (given in legend).