Algebraic approximations for transcendental equations with applications in nanophysics

Abstract

Using algebraic approximations of trigonometric or hyperbolic functions, a class of transcendental equations can be transformed in tractable, algebraic equations. Studying transcendental equations this way gives the eigenvalues of Sturm–Liouville problems associated to wave equation, mainly to Schroedinger equation; these algebraic approximations provide approximate analytical expressions for the energy of electrons and phonons in quantum wells, quantum dots (QDs) and quantum wires, in the frame of one-particle models of such systems. The advantage of this approach, compared to the numerical calculations, is that the final result preserves the functional dependence on the physical parameters of the problem. The errors of this method, situated between some few percentages and , are carefully analysed. Several applications, for quantum wells, QDs and quantum wires, are presented.

Keywords:

• nanostructures
• quantum mechanical calculation
• quantum dots
• quantum wells
• quantum wires

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Autoritatea Nationala pentru Cercetare Stiintifica [PN 09 37 01 02/ 2009], [grant number 04-4-1121-2015/2017]; HOPE Network [2013-3710_540130-LLP-1-2013-1-FR-ERASMUS-ENW].