Narrow Results

In close analogy with optical Fabry–Pérot (FP) interferometer, we rederive the transmission and reflection coefficients of tunneling through a rectangular double barrier (RDB). Based on the same analogy, we also get an analytic finesse formula for its filtering capability of matter waves, and with this formula, we reproduce the RDB transmission rate in exactly the same form as that of FP interferometer. Compared with the numerical results obtained from the original finesse definition, we find the formula works well. Next, we turn to the elusive time issue in tunneling, and show that the "generalized Hartman effect" can be regarded as an artifact of the opaque limit βl → ∞. In the thin barrier approximation, phase (or dwell) time does depend on the free inter-barrier distance d asymptotically. Further, the analysis of transmission rate in the neighborhood of resonance shows that, phase (or dwell) time could be a good estimate of the resonance lifetime. The numerical results from the uncertainty principle support this statement. This fact can be viewed as a support to the idea that, phase (or dwell) time is a measure of lifetime of energy stored beneath the barrier. To confirm this result, we shrink RDB to a double Dirac δ-barrier. The landscape of the phase (or dwell) time in k and d axes fits excellently well with the lifetime estimates near the resonance. As a supplementary check, we also apply phase (or dwell) time formula to the rectangular well, where no obstacle exists to the propagation of particle. However, due to the self-interference induced by the common cavity-like structure, phase (or dwell) time calculation leads to a counterintuitive "slowing down" effect, which can be explained appropriately by the lifetime assumptions.

The present paper investigates that the tunneling time for bilayer graphene potential barrier with monolayer graphene leads to all range of energy. Numerical results reveal that parameters such as the incident energy and angle plays a significant role in inducing of the Hartman effect. In contrast to single-layer graphene, in the bilayer graphene, due to the chirality of quasi-particles induction of Klein and Hartman effects occur in the normal incidence case. Moreover, it is demonstrated that even for energy levels above barrier, the Hartman effect is present.

We theoretically study the spin-dependent group delay time through ferromagnetic bilayer graphene superlattice in the absence and presence of the bandgap. It is found that the group delay time depends on the spin degree of freedom and exhibits an oscillatory behavior with respect to the Fermi energy and barrier width. Furthermore, in the absence of the bandgap, the superluminal or Hartman effect exists only for the normal angle of incidence. Moreover, when bandgap value is large enough $(\mathrm{\Delta }\ge 60\mathrm{\text{meV}})$, the Hartman effect can be observed for all angles of incidence. These results are contrary to the observed behavior for monolayer graphene superlattice.

This paper investigates the problem of a relativistic Dirac half-integer spin free particle tunneling through a rectangular quantum-mechanical barrier. If the energy difference between the barrier and the particle is positive, and the barrier width is large enough, there is proof that the tunneling may be superluminal. For first spinor components of particle and antiparticle states, the tunneling is always superluminal regardless the barrier width. Conversely, the second spinor components of particle and antiparticle states may be either subluminal or superluminal depending on the barrier width. These results derive from studying the tunneling time in terms of phase time. For the first spinor components of particle and antiparticle states, it is always negative while for the second spinor components of particle and antiparticle states, it is always positive, whatever the height and width of the barrier. In total, the tunneling time always remains positive for particle states while it becomes negative for antiparticle ones. Furthermore, the phase time tends to zero, increasing the potential barrier both for particle and antiparticle states. This agrees with the interpretation of quantum tunneling that the Heisenberg uncertainty principle provides. This study’s results are innovative with respect to those available in the literature. Moreover, they show that the superluminal behaviour of particles occurs in those processes with high-energy confinement.