Eur. Phys. J. B 74, 415–418 (2010) DOI: 10.1140/epjb/e2010-00080-y Regular Article T HE EUROPEAN P HYSICAL JOURNAL B A quasi-analytical approach to study energy levels of a two-electron quantum dot H. Hassanabadi a Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316 Shahrood, Iran Received 21 October 2009 / Received in final form 18 January 2010 Published online 9 March 2010 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2010 Abstract. Our approach is quasi-analytical. We consider the Hamiltonian of a two-electron quantum dot composed of quadratic plus Coulomb terms as well as a term related to the interaction with the external magnetic field. To avoid the complexity, the Taylor expansion of the effective potential is introduced into the problem and thereby a solution is found for the eigenvalues of the corresponding two-body Schr¨odinger equation in terms of the Wigner parameter. We have finally made a comparison with some other theoretical results. 1 Introduction Quantum dots are increasingly being used in differ- ent areas of science and technology including quan- tum information, optoelectronic devices, nanotechnology, etc. [1,2]. Among different quantum dots, two-electron- quantum dots are approximately the simplest basis to study the electron-electron interaction [3]. Just like all other branches of science, both experimentalist and theo- rist are working on the problem with their own strate- gies. Some of the former have measured for example the excitation spectrum of such systems with finite bias spectroscopy on vertical few-electron quantum dots [4,5]. In the second category, both numerical and approxi- mate methods have been suggested [6,7]. The special case of interest for us in the present work is however the question of the operating confining potential. Dineykhan et al. analyzed the two-electron-quantum dot by con- sidering a harmonic term and under the influence of a perpendicular homogeneous magnetic field [8]. Taut have presented some particular analytical solutions of the two- dimensional Schr¨ odinger equation corresponding to a two- electron-system interacting under Coulomb potentials and an oscillator potential under the influence of a homoge- neous magnetic field [9]. Ghoes and Samanta have stud- ied such systems system using the ansatz method in the presence of oscillating, linear and Coulomb terms while within their work spin interaction and magnetic field are not included [10]. Zhu et al. used the technique of expan- sion in power series to study the quantum-size effects on the energy spectra of the system [11]. In relatively simi- lar studies, Low-lying energy levels of a two-dimensional two-electron system have been studied for various confin- ing potentials in the present of a perpendicular magnetic field [12–14]. Also, the analytical study of the energy spec- a e-mail: h.hasanabadi@shahroodut.ac.ir trum of two interacting electrons in a parabolic confining potential has been carried for particular values of the mag- netic filed [15,16]. These, however, are not the whole story. Many other methods including the WKB approximation [17–19], shifted 1/N expansion method [20] are also present in this annals just like any similar few body problem. In the present study, a two-electron-quantum dot in the presences of an external magnetic field is considered, the corresponding energy eigenvalues of the system are obtained by considering a quadratic confining term as well as a Coulomb interaction between the two electrons. A comparison of our work with some other results (WKB, WKB-DP method and Ref. [21]) is given in Table 1. 2 Theory of the problem Here, for the two-electron-quantum dot, we consider the potential a combination of quadratic and Coulomb terms. The corresponding Hamiltonian in the presence of an ex- ternal magnetic field therefore reads H = 1 2m ∗ e 2  i=1  -→ p i - e c -→ A  2 + V ( -→ r 1 , -→ r 2 )+ H spin (1) with V ( -→ r 1 , -→ r 2 )= 2  i=1 1 2 m ∗ e ω 2 0 -→ r 2 i + e 2 k| -→ r 1 - -→ r 2 | (2) and H spin = g ∗ μ B  -→ S 1 + -→ S 2  · -→ B (3) where H spin ,μ B = e 2m ∗ e ,m ∗ e ,g ∗ , and κ represent the Zeeman energy, the Bohr magneton, the effective mass,