Persistent currents and electronic properties of Mandelbrot quantum rings

Abstract

In this study, we investigate the persistent current, and electronic energy levels of Mandelbrot quantum rings. For this purpose, three types of Mandelbrot quantum rings are proposed. Furthermore, Mandelbrot equation is generalized by introducing parameter m, which makes Mandelbrot’s shape more symmetric by adding new branches to it, on the other hand, the iteration parameter M, controls geometrical deficiencies. We explain the procedure needed to form these structures, including a padding scheme, then we solve the resulting two-dimensional Schrodinger equation using the central finite difference method with uniform distribution of the mesh points. Thereafter, we obtain the persistent current in different situations including different Mandelbrot orders and quantum ring shapes. We show that the persistent current can have different shapes and intensities by changing the described geometrical parameters of Mandelbrot quantum rings. We explain this phenomenon by considering symmetries in the potential, and consequently the wavefunction.

Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration

“…Ring-shaped quantum dots play an important role in confining the electrons along a circular orbit. Remarkably, the effect of Mandelbrot fractality is explored by considering different types of Mandelbrot rings [6]. According to research by cognitive neuroscientists, computer-generated fractals can reduce stress in viewers in precisely the same way as fractals seen in nature [7].…”

“…One key idea is escape when figuring out if a point in the complex plane is contained in a Mandelbrot set. Thus, it is believed that accurately creating a high-level "Mandelbrot" is challenging and intricate [6]. One especially unique feature of fractals that is lacking in other systems is scaling invariance.…”