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Electronic structure of nanorod of strongly prolate ellipsoidal shape

A charged particle (electron or hole) confined in nanorod of strongly prolate ellipsoidal shape is considered. The effective-mass Schrödinger equation is solved in prolate spheroidal coordinates and asymptotically exact expressions for the energy spectrum and wavefunctions are derived. It is shown that the treatment of the confinement energies is incorrect if the actual shape of ellipsoidal nanorod is not taken into account. In particular, developed earlier the approach based on the consideration of the problem in cylindrical coordinates with the use of parabolic potential leads to the incorrect expression for the energy spectrum. Rule of correlation between the states corresponding to spherical quantum dot and nanorod of strongly prolate ellipsoidal shape is suggested and an appropriate energy correlation diagram is constructed. The correlation diagram is in complete qualitative and quantitative agreement with the energy diagram obtained by numerical solution of the Schrödinger equation in spheroidal coordinates.