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Application of qualitative methods for calculation of ideal quantum dots spectrum

Keywords:

A.M. Mandel – Ph.D. (Phys.-Math.), Professor, Department «Physics», Moscow State University of Technology STANKIN. E-mail: arkadimandel@mail.ru V.B. Oshurko – Dr.Sc. (Phys.-Math.), Professor, Head of Department «Physic», Moscow State University of Technology STANKIN. E-mail: vbo08@yandex.ru G.I. Solomakho – Ph.D. (Phys.-Math.), Professor, Department «Physics», Moscow State University of Technology STANKIN. E-mail: solgeo@mail.ru K.G. Solomakho – Post-graduate Student, Moscow State University of Technology STANKIN. E-mail: kirgeosol@gmail.com V.S. Veretin – Ph.D. (Phys.-Math.), Associate Professor, Departament «Chemistry and Physics », Plekhanov Russian University of Economics. E-mail: ilavvsse@mail.ru


Ideal quantum dot (IQD) could bind only one electronic state. Qualitative methods of analysis are best suited for such objects. The main objective of the work was to calculate the spectrum and wave functions of the electrons in close to IQD. In this case simplest mathematical methods were used. The main idea of this article is that there is no necessity to solve the Schrödinger equation within the cluster exactly, it suffices to consider a two-term asymptotic behavior in the vicinity of the cluster center. This approach allows us to obtain clear analytical results, suitable for engineering applications, with high degree accuracy. Considered cluster model gradually became more complicated. Proposed method has been tested first on a simple model of a spherical QD within homogeneous matrix. Then we have considered the case of a cylindrically symmetric QD with radius much larger than its thickness (so called “pill structure). In this case, the spectrum is defined both by the electron’s motion along the axis of the cluster and by radial motion. The first type of movement forms critical conditions for the occurrence level, and the second creates a fine structure. Accordingly, for the solution of the Schrödinger equation for the axis QD the two-term asymptotic behavior has been used, and the solution to the radial variable has been given by trigonometric functions. The next step of model complexity was cylindrically symmetric QDs on the surface of the matrix. Since surface effects additionally bound electron thus the critical thickness of the cluster required for the appearance of the first bound level decreases. In conclusion, the external magnetic field is introduced, splitting bound level into a doublet. Its components are associated with the orientation of the electron spin with respect to the external magnetic-field (normal Zeeman effect). The ratio between the Zeeman splitting and the absolute value of the energy level is essential for spintronics. Energy level is determined by the properties of used semiconductor materials and the geometrical dimensions of QD, the energy of the Zeeman splitting – g-factor and magnetic field induction. All these factors could be varied within rather wide limits, thus creating conditions for controlling spin states of electrons localized on QD.
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