Quantum mechanical description of Electronic transitions in Cylindrical nanostructures, including pores in semiconductor
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Quantum mechanical description of Electronic transitions in Cylindrical nanostructures, including pores in semiconductors Yuri V. Vorobiev1, Pavel Horley2 and Jesus González-Hernández2 CINVESTAV-Querétaro, Libramiento Norponiente 2000, Fracc. Real de Juriquilla, CP 76230 Querétaro, QRO., México. 2 CIMAV Chihuahua/Monterrey, Avenida Miguel de Cervantes 120, CP 31109, Chihuahua, CHIH., México
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ABSTRACT Cylindrical nanostructures NS (namely, nanowires and pores) with rectangular or circular cross-section are analyzed using Mirror Boundary Conditions (MBC) in solution of the Schrödinger equation. The MBC are formulated as equivalence of the module of electron’s Ψfunction in an arbitrary point inside the NS and its images in NS’s walls treated as mirrors. Thus the two types of MBC – odd (OMBC) and even (EMBC) could be applied, when Ψ-functions in real point and its images are equated with the opposite or the same sign correspondingly. The first case gives zero value of Ψ-function at NS’s boundaries and therefore corresponds to the strong quantum confinement, whereas the second (non-zero Ψ-function at the boundary) gives a weak confinement. The analytical expressions for energy spectra of electron in a NS found for all cases examined contain no adjustable parameters, and show reasonable agreement with experimental data found in the literature. INTRODUCTION Nanostructures (NS) of different kind have been actively studied during the last two decades, both theoretically and experimentally; a special interest was focused on quasi-onedimensional NS such as nanowires, nanorods and elongated pores that can not only modify the main material’s parameters, but can introduce totally new characteristics like optical and electrical anisotropy, birefringence etc. All these elongated NS can be approximated as cylinders with different shape of cross-section. Theoretical treatment of NSs is based on the solution of the Schrödinger equation, usually within the effective mass approximation [1-4], although for small NS its application could be questioned. An important element of this description is the boundary conditions; the traditional impenetrable wall conditions (i) are not realistic, and (ii) in many cases could not be written in simple analytical form making the analysis quite difficult. Recently [5-7] we introduced a new mirror boundary conditions (MBC) assuming that electron in a NS is reflected by NS’s walls acting as mirrors; it is obvious that this assumption favors the effective mass approximation. Additional advantage of this approach is the possibility to treat pore as “inverted” NS – a void surrounded by semiconductor material – considering the “reflection” of particle wave function from the pore’s boundary. Thus, essentially the same solution of the Schrödinger equation (and the energy spectrum) will describe a pore and NS of the same geometry and size.
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In our approach, the boundary condition equalizes absolute values of the particle’s Ψfunction in an arbitrary point inside the NS and the corresponding image point
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