ELECTRON ENERGETIC SPECTRA IN THE ELECTRIC FIELG OF THE CHARGED QUANTUM NANORING - BURDEYNYY, VOLODYMYR; KASIYANENKO , VASUL (Herald of Khmelnytskyi National University. Technical sciences, 2024)
Due to modern technology successes, the condensed matter physics is enriched with new objects of research, which allow not only to deepen the knowledge of relatively well-studied effects, but also predict very promissing possibilities of discovering phenomena appropriated exclusively to these objects, one of the examples of which are quantum rings (QRs). Concerning to the quantum ring conception as a rule one understands some closed curvilinear region which with surrounding matrix forms two-connected structure. In their composition, internal construction and other characteristic features, QRs are basically different from their surroundings. QRs consist of atoms, molecules or more complicated aggregates which in some sence are connected to each other more strongly than to the surrounding matrix. Quantum rings also are distinguished from other systems, similar in morphology, primarily by their characteristic sizes which by order of magnitude belong to the meso- or nanoscopic length scale Impressive advances in technology, such as X-ray lithography, selfassembling, molecular beam epitaxy, and atomic force microscopy have created opportunities to form quantum rings of various geometries with a number of charge carriers that achives from a few to tens or even hundreds (mesoscopic quantum rings) of electrons. As far as geometric forms there are reported about syntheses of QRs not only with circular geometry but also hexagonal, tetragonal and stadium form created on a semiconductor surface by applying the atomic transposition methods based on the technology of atom manipulation in a scanning tunneling or atomic force microscopy. A lot of results also point out on the possibility of controlled impact on the electron dynamics in a tunable periodic potential and promise the creation of two-dimensional artificial lattices on the surface of a semiconductor. The determining characteristic of any system, including charge carriers that are affected by QRs field, is their energy spectra. Optical, electrophysical and other properties, which provide pperspectives for the practical applications of both separated QRs and their complexes, depend on it. In the context of energy spectra, theoretical studies are mainly of a model nature and focus on the features of carrier confinement by model holding potentials, the influence of electric and magnetic fields or electromagnetic, particularly laser, radiation. However, despite of the large number of reports devoted to the energy spectra problem, some of its aspects remain at a long distance from complete on final results. This remark especially concerns the spectrum of carriers in the electrostatic field of charged QRs. A quantum ring , synthesized on the metal surface or, what can be more interesting due to enough promising practical applications, implanted in a semiconductor, is spacely limited by heterojunctions that are enable to accumulate some space charge controlled with special electrodes structure. The electric field of this charge by analogy with the fields of impurity centers, can modify the energy spectra causing appearance of energetic levels in the forbidden zone.The study of such energy states is the purpose of this paper. Here a thin uniformly charged ring immersed into a semiconductor matrix has been considered. By solving the Poisson equation, the electrostatic potential is found. It turns out that the potential can be expressed in terms of complete elliptic integrals of the first kind. Behavior of the potential in the vicinity of singular points allows to introduce and justify some approximate expressions for the electron potential energy. Due to accepting these approximations it became possible to separate the radial and transverse movements of carriers. It has been shown that the transverse movement in applied here basic approximation is described in terms of harmonic oscillator functions with corresponding energy values. If the ring is positively charged, a potential well for electrons appears. In the immediate vicinity of the ring the potential energy is approximated by a linear dependence on the radial coordinate. Therefore the corresponding dynamics are described by Airy functions, which made it possible to establish the dispersion equation in its explicit form and to find approximate solutions for eugen values of energy. It was also shown that obtained results are applicable and correct if quantum numbers are restricted by the ratio of Coulomb energy and minimum quantum mechanical energy associated with the localization of particles in the region commensurable with ring size.
Keywords: quantum nanoring; Poisson's equation; Schedinger's equation; quantum harmonic oscillator; dispersion equation; Airy functions