The energy levels and wave functions of single-electron states in nano sized quantum rings
UDC
530.145EDN
WDKVXPAbstract
The electronic states of nanoscale quantum rings containing one electron are studied. Compiled by the Schrodinger equation in the two-dimensional space, for a electron system in the ekhternal potentialfield, describing nanoscale quantum ring heterolayer semiconductor structure. As a potential describing quantum ring are considered potential Volcano and Hill. For the Hill potential theexact solutions of the two-dimensional Schrödinger equation, the study of which is usually carried out by numerical methods, are obtained. A model potential (potential of the "6-2"), describes a unified way as a quantum ring, and a quantum dot in the presence of an external homogeneous constant magnetic field. Analytical methods to obtain new exact solutions of the Schrödinger equation in the class of Heun's functions for the potential "6-2" and the capacity of the Hill. It is shown that the solution of the Schrödinger equation for a single electron in the potential of the "6-2" are expressed in elementary functions only in the case of certain relations between the parameters of the potential well, in other cases they are expressed in terms of Heun's functions. Found energy levels and electron wave functions in the ground state, and some low-lying states for the potential "6-2" for small values of the orbital angular momentum for the values of the parameters of the potential, when the solution of the Schrödinger equation is expressed in terms of elementary functions. A variational method for studying electron states for arbitrary values of the potential parameters, using the results as a starting solution. It is shown that this method allows to obtain the energy levels and wave functions of the ground state of an electron with sufficient accuracy for practical applications.
Keywords:
quantum rings, Hill potential, model potential, Schrodinger equation, Heun's functionsFunding information
Работа выполнена при поддержке РФФИ и Министерства образования и науки Краснодарского края (13-01-96525).
References
- Ihn T., Fuhrer A., Sigrist M. et al. Quantum mechanics of quantum rings // Adv. in Solid State Physics. 2003. Vol. 43. P. 139-154.
- Son S.B., Miao Q., Shin J.-Y. et al. Ring and Volcano structures formed by a metal Dipyrromethene complex // Bulletin of the Korean Chemical Society. 2014. Vol. 35. Iss. 6. P. 1727-1731.
- Васильевский И.С., Виниченко А.Н., Еремин И.С. и др. Особенности формирования ансамблей квантовых колец GaAs/AlGaAs и InGaAs/AlGaAs методом капельной эпитаксии // Вестник национального исследовательского ядерного университета "МИФИ", 2013, Т. 2, № 3, С. 267-272. [Vasil'evskii I.S., Vinichenko A.N., Eremin I.S. et al. Specific features of the formation of GaAs/AlGaAs and InGaAs/AlGaAs quantum ring ensembles by droplet epitaxy. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI" [Bulletin of the National Research Nuclear University "MEPI"], 2013, vol. 2, no. 3, pp. 267-272. (In Russain)]
- Маргулис В.А., Миронов В.А. Магнитный момент кольца Волкано // Физика твёрдого тела. 2008. Т. 50. Вып. 1. С. 148-153. [Margulis V.A., Mironov V.A. Magnitnyy moment kol'tsa Volkano [Magnetic moment of the Volcano ring]. Physics of the Solid State, 2008, vol. 50. no. 1, pp. 148-153.]
- Simonin J., Proetto C.R., Barticevic Z. et al. Single-Particle electronic spectra of quantum rings: A comparative study // Phys. Rev. B. 2004. Vol. 70. P. 205305-1-205305-8.
- Chakraborty T., Pietilainen P. Electron-Electron interaction and the persistent current in a quantum ring // Phys. Rev. B. 1994. Vol. 50. no. 12. P. 8460-8468.
- Ronveaux A. The Heun's Differential Equation, Oxford University Press, Oxford.1995. 380 p.
- Славянов С.Ю., Лай В. Специальные функции: единая теория, основанная на анализе особенностей. СПб.: Невский диалект, 2002. 312 с. [Slavyanov S. Yu., Lay W. Special Functions: A Unified Theory Based on Singularities. Oxford University Press, 2000, 312 p.]
- Зайцев В.Ф., Полянин А.Д. Справочник по обыкновенным дифференциальным уравнениям. М.: Физматлит, 2001. 576 с. [Zaitsev V.F., Polyanin A.D.Handbook of Exact Solutions for Ordinary Differential Equations. Chapman&Hall/CRC, London, 2003, 802 p.]
- Ландау Л.Д., Лифшиц Е.М. Квантовая механика (нерелятивистская теория). М.: Физматлит, 2004. 797 с. [Landau L.D., Lifshitz E.M. Quantum Mechanics. Non-relativistic theory. Pergamon Press, Oxford-New York, 1977, 677 p.]
- Сансоне Дж. Обыкновенные дифференциальные уравнения. Том 1. М.: ИЛ, 1953. 346 с. [Sansone G. Equazioni Differentiale Nei Campo Reale. Parte Primo/Bologna, 1948, 346 p. (In Italian)]
- Третяк Д.Н., Тумаев Е.Н. Квантовая тория низкоразмерных систем. Краснодар: Изд-во Кубанского университета, 2015. 212 с. [Tretyak D.N., Tumayev E.N. Kvantovaya teoriya nizkorazmernyh sistem [Quantum theory of low-dimensional systems]. Krasnodar, Kuban State Univesity, 2015, 212 p. (In Russian)]
Downloads
Downloads
Dates
Submitted
Accepted
Published
How to Cite
License
Copyright (c) 2016 Тумаев Е.Н.
This work is licensed under a Creative Commons Attribution 4.0 International License.
