Trembling motion of the wave packet in armchair graphene nanoribbons (AGNRs)

Abstract

A treatment of the trembling motion or Zitterbewegung (ZB) phenomena in the armchair graphene nanoribbons (AGNRs) by using long-wave approximation is presented theoretically. We are first interested to study the time dependence of the average values for the current density in AGNR. The longitudinal and transversal components of the current density operator are derived analytically in the Heisenberg representation. The wave packet in a Gaussian distribution is considered with half-width d and a carrier wave vector in the longitudinal orientation k[Formula: see text]. The average values of the current density which represent the current induced by the motion of the electrons along the nanoribbon are calculated numerically. The interference between two energy branches, or the corresponding upper and lower energy states, leads to the trembling motion in the armchair graphene nanoribbon, and hence, our results emphasized that the phenomena of ZB has an aperiodic and nontransient oscillation in the longitudinal and transversal direction, respectively. The average values for the current density for AGNRs are calculated with extremely large value of N and compared with infinite pristine graphene.

Theoretical Treatment of the Zitterbewegung phenomena in the multilayer armchair graphene nanoribbons (MLAGNRs)

“…(2) shows the structure diagram of AGNR armchair graphene nanoribbon with length L and width W . The width is represented by dashed lines on the y-axis of the ribbon, where it is given as a function of two carbon atoms per unit cell [21]:…”

“…They emphasized that the coupling between the bulk states and edge states has interesting features, and they also observed that the oscillation in position occurs along the direction of the wave packet propagation which is unavailable in the infinite graphene sheets and semiconductor nanowires. Majid and Savinskii [17,18] and Majid and Alla [19][20][21] analyzed in the detailed description of wave-packet evolution in a monolayer graphene, carbon nanotubes and graphene nanoribbon by using the Heisenberg representation, they investigated the influence of the different pseudospin polarizations on the wave-packet dynamics for the initial wave function and the phenomenon of ZBW. The time evolution of the wave packet in bilayer graphene is investigated theoretically by using the long wave approximation and the Heisenberg representation [22].…”