Dirac wave transmission in Lévy-disordered systems

We investigate the propagation of electronic waves described by the Dirac equation subject to a Lévy-type disorder distribution. Our numerical calculations, based on the transfer matrix method, in a system with a distribution of potential barriers show that it presents a phase transition from anomalous to standard to anomalous localization as the incidence energy increases. In contrast, electronic waves described by the Schrödinger equation do not present such transitions. Moreover, we obtain the phase diagram delimiting anomalous and standard localization regimes, in the form of an incidence angle versus incidence energy diagram, and argue that transitions can also be characterized by the behavior of the dispersion of the transmission. We attribute this transition to an abrupt reduction in the transmittance of the system when the incidence angle is higher than a critical value which induces a decrease in the transmission fluctuations.

Figure 1

The propagation of Dirac electronic wave along the sequence of barriers and wells with thicknesses $w_j$ and $w_{j+1}$ following a Lévy-type distribution.

Figure 2

Average transmission probability $〈T〉$ as a function of system length $L$ for Dirac heterostructures with Lévy-type distribution of potential barriers with $V=50$ meV. Incidence angle increases from top $\theta =0$ to bottom $\pi /2$, and incidence energies are (a) $E=20$ meV and (b) $E=30$ meV.

Figure 3

Panels (a)–(d), show $〈-lnT〉$ as a function of length for Dirac heterostructures with Lévy-type distribution of potential barriers with $V=50$ meV. Incidence angle increases from bottom $\theta =0$ to top $\pi /2$, and incidence energies are $E=20,25,26$, and 30 meV. Panel (e) shows the phase diagram in terms of incidence angle $\theta$ and incidence energy $E$, dividing anomalous (AL) and “standard” localization (SL) regimes.

Figure 4

Average transmission [(a) and (c)] and $〈-lnT〉$ [(b) and (d)] as a function of system length for Dirac heterostructures with Lévy-type barrier distribution for $\alpha =0.5$. In (a) and (b) we have $E=25$ meV ($E<V$). The dashed lines are fitted using Eqs. (3) and (4). In (c) and (d) we have $E=60$ meV ($E>V$) and $\theta =\pi /5$. The dashed lines are fitted using Eqs. (3) and (2), whereas the straight line is fitted with Eq. (1).

Figure 5

Average transmission [(a) and (c)] and $〈-lnT〉$ [(b) and (d)] as a function of $L$ for Dirac heterostructures with Lévy-type barrier distribution. In (a) and (b) $\theta =2\pi /5$, while in (c) and (d) $\alpha =0.5$. The dashed lines are fitted using Eqs. (3) and (2) for top and bottom panels, respectively. Incidence energy is fixed $E=60$ meV.

Figure 6

Average transmission and its standard deviation as a function of incidence energy for a system length $L=5\mu \mathrm{m}$. In (a) and (b) we have $\alpha =0.25$, while in (c) and (d) $\alpha =0.50$. The incidence angle takes values $\theta =\pi /5,\pi /3$, and $2\pi /5$.