The Ground State Electronic Properties of Two Electron Quantum Dot in External Magnetic and Electric Fields

The ground state energies of two interacting electrons in two dimensions are studied within the framework of shifted 1/N expansion. The effect of external uniform magnetic and electric have been studied. Energies of the relative part of the Hamiltonian of the system are calculated for both weak and strong field regimes. Our results show a very good agreement with those obtained by other computational methods like asymptotic integration (AIM) and exact diagonalizaion methods.


Introduction
Quantum dots QDs are generally refer to nanostructures made from semiconductor materials, in which carries are confined in all spatial dimensions.Modern semiconductor processing techniques allowed the artificial creation of quantum confinement shape of few electrons.Because of the similarity between atoms and quantum dots they are called artificial atoms.However, there are considerable differences in physical characterizations between them.The confining potential of nucleus is singular but that of quantum dots is nonsingular, which makes the parabolic confinement approximation possible.And artificial atoms, typical dimensions range from nanometers to few microns, are much larger than real atoms.Furthermore, there are mainly two energy scales in quantum dots; the Coulomb energy and the confinement potential energy.
The main technological motivation to investigate semiconductor quantum dots is that smaller structures should be faster, dissipate less heat, and quantum mechanical effects are so relevant in such structures that devices with fundamentally new properties can be obtained.Therefore, QDs have many applications in microelectronic and optoelectronic devices such as photodetectors, far-infrared (FIR) laser amplifiers, one electron transistors and high-speed electro-optical modulators (Mendoza, Vazquez, del Castillo-Mussot, & Spector, 2005;Li & Xia, 2007;Zhu & Gu, 1993;Dutta & Das, 1990;Yoffe, 2001;Ahn & Chuang, 1987).
Electric and magnetic fields influence have been applied to quantum dots using different approaches.Rezaei (Rezaei & Kish, 2013) studied the fields effect on the two dimensional quantum dot using direct matrix diagonalization method.Soylu (Soylu, 2012) investigated the influence of the fields on various quantum states using a asymptotic iteration method.The ground state energies of hydrogenic impurities in cylindrical quantum dots were studies using finite difference method (Wang, Wei, & Yi, 2010), complex absorbing potential (CAP) (Sahoo, Lin, & Ho 2008), and the finite difference methods have been applied to quantum dot under the influence of electric and magnetic fields within the effective mass approximation (Hong, Li-Xue, Xue, Chun-Yuan, & Liun-Jun, 2011).
The shifted 1/N-expansion, proposed by Sukhatme and Imbo (Sukhatme & Imbo 1983;Imbo, Pagnamenta, & Sukhatme, 1984), is a powerful tool to solve Schrödinger equation for spherical symmetric potentials (Dutta, Mukherji, & Varshni, 1986;Roy & Roychoudhury, 1987;Roychoudhury, & Varshni, 1988;Chatterjee, 1990), and also it was extended successfully to relativistic potentials (Roychoudhury & Varshni 1989;Mustafa & Sever, 1991).The method is simple, and it gives accurate results of energy eigenvalues calculations of the system without dealing with robust numerical calculations or trail wave functions.The shifted 1/N-expansion method has already been used to study various systems, such as two-dimensional magnetoexcitons (Quiroga, Camacho, & Gonzalez, 1995), shallow donor impurities (El-Said, 1994), two-electron spherical quantum dot (Pino & Villalba, 2001), two interacting electrons in two dimensional quantum dot with the presence of magnetic field (El-Said, 2000;Gomez & Romero, 2009).And recently, we have used the method to calculate energies and binding energies for quantum dot with Gaussian potential confinement (Al-Hayek & Sandouqa, 2015), the results show a very good agreement with other computational methods like asymptotic integration method (AIM) and exact diagonalization method.
In this approach, the calculations are carried out for states with arbitrary quantum numbers (the principal and magnetic quantum numbers n and m, respectively) using forth-order perturbation theory in the shifted expansion parameter 1/ k , where 2 k N m a = + − .N is the number of spatial dimensions and (a) is a suitable shift parameter which will be discussed later.
The rest of the this work is organized as follows.In section 2, we formulate the Hamiltonian of two electron quantum dot under the influence of external magnetic and electric field.In section 3, we present the shifted 1/N expansion method for arbitrary spherical potential V(r), and we apply the method on the potential of the problem.Numerical energy results and discussion are given in section 4. Conclusion is presented in section 5.

The Hamiltonian
The Hamiltonian of two electrons confined in a parabolic quantum dot under the effect of external electric and magnetic fields can be separated into center of mass H cm and relative motion H rm as follows: A Where M cm = 2m * is the total mass, and q = 2e is the total charge and ω 0 is the confinement potential.m * is the effective mass of each electron, ε and c are the dielectric constant of the dot material and the speed of light, respectively.We introduce the relative coordinate of the two electrons r = r1 -r2 and momentum P = p1 + p2, with the reduced mass µ = m * /2.
The magnetic field effect appears in the effective frequency Ω = (ω 0 2 + ω c 2 /4) 1/2 in the Hamiltonian, where ω c = eB/m * c is the cyclotron frequency.The magnetic field B is assumed to be uniform and perpendicular to the dot plane along the z axis.The vector potential is chosen to be in the symmetric gauge as A = (1/2) B × r.The eigenvalue of the angular momentum operator L z is mħ, = 0, ±1, ±2 ….is the azimuthal quantum number.We assume the term qF.r in Equation ( 3) to be F r, Equation (3) can be written as: The main task is to solve the Hamiltonian of the relative motion (Equation ( 3)).Using the substitution  is the characteristic length in the x-y plane, a * and R * are the effective Bohr radius and effective Redberg, respectively.λ represents the ratio of the electron Coulomb interaction to the harmonic confinement, any change in the Coulomb strength or harmonic confinement results in a change in the coupling constant λ.
Equation ( 6) related to the relative motion of the two electron quantum dot under the influence of both magnetic and electric fields confined by the potential ( )

1/N Expansion Formalism
Following the work of Imbo (Imbo, Pagnamenta, & Sukhatme, 1984) related to shifted 1/N expansion method, we formulating the radial Schrödinger equation for an arbitrary spherical potential V(r) as where . N is the number of spatial dimensions.
In terms of the shifted variable k k a = − (a is a shifted parameter), we rewrite Equation (3) as: where Q is a constant which rescales the potential (in large k limit) and it will be determined below.The energy eigenvalues are given by an expansion in powers of 1 / k .
The shifted 1/N expansion method consists in solving equation ( 4) systematically in terms of the expansion parameter1/ k .The leading contribution to the energy comes from the effective potential ( ) ( ) ( ) V r is assumed to be well behaved so that ( ) eff V r has a minimum at 0 r r = and there are well-defined bound states.Q is then determined from the following equation ( ) It is convenient to shift the origin of coordinates to 0 r r = by defining ( ) The energy eigenvalues are given by an expansion in powers of 1/ k where 2 k N m a = + − , N being the number of spatial dimensions and the shifted parameter a (see Appendix).
The shift parameter is fixed from the requirement that the term 1 E vanishes.Therefore, ( ) ω is the anharmonic frequency parameter given in (Appendix, Equation A7).For any specific choice of n, m and N, the constant Q should be such as to make Eqs.( 7) and ( 8) identical.This means using Eqs.( 10), (Equation A4), and ( 13), an explicit equation for determining the root 0 r , Having determined 0 r , all the energy eigenvalues can be computed.

Numerical Results Discussions
The calculated energies of two interacting electrons in a two-dimensional quantum dot using 1/N expansion are presented in Tables 1-3.Table 1 lists energy calculations for two interacting electrons in two dimensional quantum dot under the influence of magnetic field only (λ = 1, F = 0).Energies are obtained for ground state with different values of magnetic quantum number m.The energy eigenvalues increase as the magnitude of the orbital quantum number m increases.For comparison of energy accuracy of our results, we list the energies obtained by exact method (García-Castelán, Choe, & Lee, 1998) and asymptotic iteration method AIM (Soylu, 2012).Our results show a very good agreement.
Table 2 displays energy eigenvalues using magnetic field only with higher strength (λ = 10 and F = 0).The effect of enhancing the interaction term, λ, on energies is clearly seen by comparing the results in Table 2 with Table 1 for the same state.This increase in energy is due to increase of the Coulomb interaction energy between the two electrons comparable to the parabolic confinement one.The results show also a very good accuracy with diagonalization and AIM methods (García-Castelán, Choe, & Lee, 1998;Soylu, 2012).
Table 3 shows the calculated energy eigenvalues for the two-electron quantum dot system under the effect of both electric and magnetic field for λ = 1 and different electric field strengths (F = 0, F =10 and F = 100).The dependence of energies on the electric field strength is seen.As the electric field strength increases, the calculated energy eigenvalues increases.Our calculated energy results are compared with those obtained using AIM method.The data shows a very good accuracy of the shifted 1/N expansion method (Soylu, 2012).

Conclusion
In summary, we have calculated the energy eigevalues of two electron quantum dot in two dimensions under the influence of both magnetic and electric fields within the frame work of the shifted 1/N expansion.The effect of both electric and magnetic fields on the ground state energies of the system have been found for different magnetic quantum number m.The solution of the relative part of the Hamiltonian is made for different degrees of the Coulomb to confinement ratios (λ) and electric field strength (F).
The simplicity and efficiency of using the shifted 1/N expansion is shown in our work.Hence, no need to deal with robust numerical calculations or trial wave functions.Our results have a very good agreement with other different numerical methods.Finally, we think that the shifted 1/N expansion can be applied effectively to other kinds of potential confinements in spherical quantum dots.
Where ( ) P x is the perturbation term given by ( ) ( ) ( ) We can compare Eqs.(A1) and (A3) term by term to define all the anharmonic parameters in terms of k , Q, 0 r and potential derivatives.
Proceeding in a straightforward way we obtain the following identifications:

For
) and for any value of m the energy ( ) , E n m is given by an expansion in powers of 1 k

Table 1 .
Calculation results for the ground state energies of two-electron quantum dot with λ = 1 and F = 0.