Computer Algebra in Nanosciences: Modeling Electronic States in Quantum Dots
- 638 Downloads
In the present paper we discuss single-electron states in a quantum dot by solving the Schrödinger equation taking into account spatial constraints, in which the confinement is modeled by a spherical potential wall (particle-in-a-sphere model). After the separation of variables we obtain second order ordinary differential equations, so that automatic methods for finding a closed-form solution are needed. We present a symbolic algorithm implemented in Maple based on the method of indeterminate coefficients, which reduces the obtained equations to the well-known differential equations. The latter can be solved in terms of hypergeometric or Bessel functions. The usage of indeterminate coefficients allows one to obtain the solution of the problem equations in terms of control parameters, which can then be choosen according to the purposes of a nanotechological process.
KeywordsComputer Algebra Symbolic Computation Spatial Constraint Radial Equation Symbolic Algorithm
Unable to display preview. Download preview PDF.
- 1.Bronstein, M., Lafaille, S.: Solutions of linear ordinary differential equations in terms of special functions. In: Proceedings of the 2002 international symposium on Symbolic and algebraic computation, Lille, France, July 07-10, pp. 23–28 (2002)Google Scholar
- 3.Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. McGraw-Hill Book Company, Inc., New York (1953)Google Scholar
- 4.Gonis, A.: Theory, Modeling, and Computation in Materials Science, LLNL, Livermore, CA (1993)Google Scholar
- 7.Theory and modeling in nanoscience. Report of the May 10-11, 2002 Workshop, DOE U.S. LBNL-50954 (2002)Google Scholar
- 8.Theory, simulation, and modeling in nanoscience, LLNL Nanoscience Home Page, http://www.llnl.gov/nanoscience
- 9.Yoffe, A.D.: Low-dimensional systems: quantum size effects and electronic properties of semiconductor microcristallities (zero-dimensional systems) and some quasi-two-dimensional systems. Adv. Physics 51 (2002)Google Scholar
- 10.Yoffe, A.D.: Semiconductor quantum dots and related systems: electronic, optical, luminescence and related properties of lowdimensional systems. Adv. Physics 50 (2001)Google Scholar