Perfect transmission of a Dirac particle in one-dimension double square barrier

Abstract

Dirac particles can undergo perfect transmission through a sufficiently high potential barrier in the Klein zone. Although the perfect Klein tunneling (often referred to as the Klein paradox) is similar to the non-relativistic resonant transmission which occurs only when the kinetic energy exceeds the barrier, the underlying mechanism is believed to be fundamentally distinct. In this work, we show that for the relativistic double-barrier model the perfect-transmission curve can pass continuously from the above-barrier zone to the Klein zone. Additionally, in the Klein zone, perfect transmission occurs even for subcritical barrier heights, supported by both bound-state analysis and wave-packet dynamics. These findings suggest a connection between perfect Klein tunneling and resonant transmission, and provide new insights into the physical nature of the Klein paradox.

Figures (5)

• Figure 1: Schematic illustration of the particle transmission through a single potential barrier. (a) $V_0 < 2mc^2$ or non-relativistic case. The green shaded region denotes the above-barrier zone where $E_k > V_0$ and resonance transmission with probability reaching 1 may occur. The gray shaded region is the normal-quantum-tunneling zone where $E_k < V_0$ and the transmitted wave is exponentially damped down. (b) $V_0 > 2mc^2$. The Klein zone ($0 < E_k < V_0 - 2mc^2$) appears at the bottom of the barrier, shown as the yellow shaded region.
• Figure 2: Potential barrier (red-solid line) and potential well (blue-dashed line), where $a$ and $d$ denote the barrier width and the distance between the two barriers, respectively.
• Figure 3: Perfect transmission curves for Dirac particles through a one-dimensional double potential barrier, each barrier of width $a =2.5\lambda$. (a) and (b) correspond to inter-barrier distances $d = 0\lambda$ and $d = 5\lambda$, respectively. (c) Close-up of a part in (b). The above-barrier zone lies above the black dash-dot-dot line $E_k =V_0$ and the Klein zone is below the black dash-dot-dot line $E_k=V_0-2mc^2$. The gray region between the two lines ($E_k =V_0$ and $E_k=V_0-2mc^2$) denotes the normal-tunneling zone in which the transmission probability decays exponentially as the barrier height and width increase. The blue-solid, red-dashed, and black-dotted curves represent perfect-transmission conditions. The intersections of the perfect-transmission curves with the horizontal axis ($E_k = 0$) correspond respectively to the zero-energy resonance ($V_0<0$) and the zero-momentum resonance ($V_0>2mc^2$).
• Figure 4: Eigenvalue spectrum of bound Dirac particles in a one-dimensional double potential well, each well of width $a = 2.5\lambda$. (a) and (b) correspond to inter-well distances $d = 0\lambda$ and $d = 1\lambda$, respectively. The intersections of the curve with the upper and lower axes correspond respectively to the half-bound and supercritical states, which are related to the zero-energy and zero-momentum resonances.
• Figure 5: Wave-packet dynamics of a Dirac particle impinging on a double barrier. The barrier heights in panels (a)–(d) are $V=0.4$, $V=0.8$, $V=2.55$, and $V=4$, respectively. The wave packet is represented by the red solid curve. Two separated potential barriers are enclosed by the gray dashed lines. In all panels, the same initial state is used, and the barriers are identical except for their heights. Panels (a)–(d), from top to bottom, show the time evolution of the wave packet in the above-barrier zone, the normal-quantum-tunneling zone, the Klein zone below the supercritical threshold, and the Klein zone above the supercritical threshold, respectively. The parameters used in the simulations are given in the main text.