Computing of the Ground State Energy of a Hydrogen Like Impurity in a Spherical Quantum Dot using QPSO Algorithm

Computing of the Ground State Energy of a Hydrogen Like Impurity in a Spherical Quantum Dot using QPSO Algorithm

Mohsen Samet Omran (Department of Physics, University of Semnan, Semnan, Iran)
Copyright: © 2016 |Pages: 14
DOI: 10.4018/IJEOE.2016100103
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In this work, quantum particle swarm optimization (QPSO1) algorithm method is applied to the problem of impurity at the center of a spherical quantum dot for infinite confining potential case. For this purpose, a trial variational wave function is considered for ground state, and then energy values are calculated as a function of the radius of a spherical quantum dot. Also, the evolution of the energy eigenvalue for different dot radii and different optimized parameter is determined. The energy converges remarkably fast, after a few numbers of iteration. In comparison with the two other available methods, standard variational procedure and genetic algorithm method (GA), the results coming out from QPSO algorithm are in more satisfactory agreement with the real values.
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1. Introduction

The recent developments in the fabrication technology have given an opportunity to confine the electrons in two-, one- and zero-dimensional semiconductor structures (Banyai, Koch, 1993; Bimberg, Grundmann, Ledentsov, 1999). Semiconductor quantum nanostructures (quantum wells, wires or dots) have found various application areas especially as electronic devices such as single electron transistor, quantum well and quantum dot infrared photo detector (QWIP and QDIP) (Levine, 1993; Ryzhii, 1996; Lee, Hirakawa, and Shimada, 2000). Therefore, these structures have been intensively studied both theoretically and experimentally in condensed matter physics (Lee, Lee, Shin, Yu, Kim, Ihm, 1997; Jovanovic, Leburton, 1994). The studies performed so far to determine the electronic properties of semiconductor quantum nanostructures are based generally either on variational or on numerical procedures (Bellessa, Combescot, 1999; Varshni, 1999; Bose and Sarkar, 2000). Many analytical and numerical studies on energy levels and other physical properties of quantum dots (QDs) have been reported (Sinha, 2000). Over the past several decades, population-based random optimization techniques, such as evolutionary algorithm and swarm intelligence optimization, have been widely employed to solve global optimization (GO) problems and minimization problems for the quantum mechanical systems (Chaudhury, Bhattacharyya, 1998). Four well-known paradigms for evolutionary algorithms are genetic algorithms (GA) (Goldberg, 1989), evolutionary programming (EP) (Fogel, 1994), evolution strategies (ES) (Rechenberg, 1994) and genetic programming (GP) (Koza, 1992). These methods are motivated by natural evolution.

The particle swarm optimization (PSO) method is a member of a wider class of swarm intelligence methods used for solving GO problems. Particle Swarm Optimization (PSO), motivated by the collective behaviors of bird and other social organisms, is a novel evolutionary optimization strategy introduced by J. Kennedy and R. Eberhart in 1995 (Kennedy, Eberhart, 1995), which relies on the exchange of information between individuals. Each particle flies in search space with a velocity, which is dynamically adjusted according to its own flying experience and its companions’ flying experience. PSO’s performance is comparable to traditional optimization algorithms such as simulated annealing (SA) and the genetic algorithm (GA) (Angeline, 1998; Eberhart, Shi, 1998). Since its origin in 1995, many revised versions of PSO have been proposed to improve the performance of the algorithm. In 1998, Shi and Eberhart introduced inertia weight W into evolution equation to accelerate the convergence speed (Shi, Eberhart, 1998). In 1999, Clerc employed Constriction Factor K to guarantee convergence of the algorithm and release the limitation of velocity (Clerc, 1999). Clerc in 2002 did trajectory analysis of PSO (Clerc, Kennedy, 2002). As far as the PSO itself concerned, however, it is not a global optimization algorithm, as has been demonstrated by Van den Bergh (Bergh, 2002; Nguyen and Kachitvichyanukul, 2010; Lourenço, and Pereira, 2011).

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