Tunable Goos--Hänchen shifts and group delay time in single-barrier silicene

Abstract

We investigate the Goos--Hänchen (GH) shifts and group delay time of Dirac fermions traversing a rectangular electrostatic potential barrier in silicene. By analyzing their dependence on the incident angle, barrier height, barrier width, and incident energy, we demonstrate that the GH shifts exhibit pronounced oscillations arising from quantum interference within the barrier region. The amplitude and number of oscillation peaks increase with increasing energy, barrier width, and incidence angle, resulting in enhanced lateral beam displacement. Meanwhile, the group delay time exhibits resonant features associated with the formation of quasi-bound states, increasing with barrier width, energy, and incidence angle, while decreasing with increasing barrier height. These results clarify how barrier-induced quantum interference controls both the lateral and temporal dynamics of Dirac fermions in silicene, highlighting the potential role of electrostatic barriers in enabling tunable transport in two-dimensional Dirac materials.

Figures (8)

• Figure 1: Schematic illustration of the Goos–Hänchen (GH) shift and group delay time for Dirac fermions traversing a single electrostatic potential barrier in silicene. The upper-left panel shows a top view of the silicene honeycomb lattice with two inequivalent sublattices $A$ and $B$, while the lower-left inset presents a side view highlighting the intrinsic buckled structure of silicene. On the right, an incident Dirac wave packet impinges on a rectangular electrostatic potential barrier of height $V_0$ and width $L$ at an angle $\phi$. Upon transmission, the wave packet undergoes a lateral displacement along the interface (GH shift) and experiences a finite group delay time due to multiple reflections and phase accumulation inside the barrier.
• Figure 2: The GH shifts in transmissions $S_t/\lambda$ versus the incident angle $\phi$ (a,b), energy $E$ (c), and barrier height $V_0$ (d).
• Figure 3: The group delay time in transmission $\tau_{t}/\tau_{0}$ versus the incident angle $\phi$.
• Figure 4: The group delay time in transmission $\tau_{t}/\tau_{0}$ versus the barrier height $V_0$ for barrier width $d=40$ nm, different values of energy $E$ and incident angle $\phi$.
• Figure 5: The group delay time in transmission $\tau_{t}/\tau_{0}$ versus the barrier height $V_0$ for incident angle $\phi=10^\circ$, different values of energy $E$ and barrier width $d$.