Transmission through Cantor structured Dirac comb potential

Abstract

In this study, we introduce the Cantor-structured Dirac comb potential, referred to as the Cantor Dirac comb (CDC-$ρ_{N}$) potential system, and investigate non-relativistic quantum tunneling through this novel potential configuration. This system is engineered by positioning delta potentials at the boundaries of each rectangular potential segment of Cantor potential. This study is the first to investigate quantum tunneling through a fractal geometric Dirac comb potential. This potential system exemplifies a particular instance of the super periodic potential (SPP), a broader class of potentials that generalize locally periodic potentials. Utilizing the theoretical framework of SPP, we derived a closed-form expression for the transmission probability for this potential architecture. We report various transmission characteristics, including the appearance of band-like features and the scaling behavior of the reflection coefficient with wave vector $k$, which is governed by a scaling function expressed as a finite product of the Laue function. A particularly striking feature of the system is the occurrence of sharp transmission resonances, which may prove useful in applications such as highly sharp transmission filters.

Figures (15)

• Figure 1: This figure illustrates the formation of the GC potential (depicted in a white background) and the CDC-$\rho_{2}$ system (depicted in a light gray background). It is important to note that stage $S=0$ marks the initial stage of the GC potential, while stage $S=1$ represents the initial stage of the CDC-$\rho_{2}$ system. The GC potential is constructed through an iterative process where a fraction $\rho^{-1}$ of the length of the potential segment at each stage is removed. The CDC-$\rho_{2}$ system is created by positioning delta potentials at the boundaries of each potential segment of the GC potential at every stage. Moreover, the construction of the CDC-$\rho_{2}$ system at each stage can also be seen as a super periodic repetition of delta potentials in a defined manner. The super-periodic distances, denoted by $r_{q}$, are characterized by the relationship $r_{q}=f(L, \rho)$.
• Figure 2: This figure shows the formation of CDC-$\rho_{3}$ and CDC-$\rho_{4}$ systems. The opaque regions represent the rectangular potential segment and the whole system represents the low lacunarity polyadic Cantor potential for $N=3$ and $N=4$, where $N$ is the count of rectangular segments at the first stage (the second stage in the context of CDC-$\rho_{N}$ system. Delta potentials placed at segment boundaries create the polyadic Cantor-structured Dirac comb potential, as highlighted by the black vertical lines, thereby representing the CDC-$\rho_{3}$ and CDC-$\rho_{4}$ systems.
• Figure 3: Plots showing the transmission profile of the super periodic delta potential of order $S=1$, $2$, $3$ and $4$ with the super periodic distances $(r_{1}, r_{2}, r_{3}, r_{4}) = (1.15, 2.5, 4.5, 10)$. The super periodic count of delta potential for each order is mentioned in the plot legend of the figure. Here, the potential parameter is set to $V=15$. The grid lines labeled with the numbers $1$, $2$, $3$, and $4$ indicate the transmission resonance energy points $k^{\star}$ with values $2.5835090$, $5.1747801$, $2.2032074$, and $4.4183635$ at the $k$-axis respectively. Transmission resonance points labeled as $1$ and $2$ arise due to the SPP of order $1$ and serve as transmission resonance points for subsequent SPPs. Similarly, resonance points labeled as $3$ and $4$ correspond to the SPP of order $2$, and they are resonance points for higher-order SPPs but not for the order $1$. A zoomed view of these transmission resonances is provided in Fig. \ref{['zoomresonance']}.
• Figure 4: A magnified view of the transmission resonance points of Fig. \ref{['transmission_resonance_01']}, labelled as $1$, $2$, $3$, and $4$, is provided.
• Figure 5: The transmission profile (blue curve) of the super periodic delta potential and the argument of the Chebyshev polynomial $\Gamma_{S}$ (red curve) are shown. The potential parameter is set at $V=5$, with super periodic distances of $r_{1}=2$, $r_{2}=3.5$, $r_{3}=6.5$, and $r_{4}=13$. In (a), we have the super periodic delta potential of order $S=1$ characterized by $(N_{1}, r_{1})=(4,2)$. In (b), the order is $S=2$ with $(N_{1}, N_{2}, r_{1}, r_{2})=(2,4,2,3.5)$. In (c), the order is $S=3$, defined by $(N_{1}, N_{2}, N_{3}, r_{1}, r_{2}, r_{3})=(2,2,4,2,3.5,6.5)$. Lastly, in (d), the order is $S=4$ with $(N_{1}, N_{2}, N_{3}, N_{4}, r_{1}, r_{2}, r_{3}, r_{4})=(2,2,2,4,2,3.5,6.5,13)$. Across all these cases, it is clear that the transmission probability nearly vanishes when $\vert\Gamma_{1}\vert>1$, $\vert\Gamma_{2}\vert>1$, $\vert\Gamma_{3}\vert>1$, and $\vert\Gamma_{4}\vert>1$.