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Two interacting electrons in a Gaussian confining potential quantum dot Wenfang Xie Department of Physics, Guangzhou University, Guihua Gang East 1, Guangzhou 510405, People’s Republic of China Received 15 April 2002; received in revised form 22 November 2002; accepted 17 December 2002 by D. Van Dyck

Abstract Two interacting electrons in a Gaussian confining potential quantum dot are considered under the influence of a perpendicular homogeneous magnetic field. The energy levels of the low-lying states are calculated as a function of magnetic field. Calculations are made by using the method of few-body physics within the effective-mass approximation. A ground state behavior (singlet ! triplet state transitions) as a function of the strength of a magnetic field has been found in the weak confinement case as a two-electron quantum dot with parabolic confining potential. q 2003 Elsevier Ltd. All rights reserved. PACS: 73.20.Dx; 73.20.Mf; 73.40.Kp Keywords: A. Quantum dot; A. Semiconductors; D. Electron–electron interactions

Quantum dots (QDs) are structures in which the charge carriers (electrons or holes) are confined in all three dimensions [1,2]. Due to the singular nature (d-functionlike) of their density of states, they are interesting for optoelectronic device applications. From the fundamental physics point of view, they are like artificial atoms in which the number of electrons can be increased almost unlimited and in a controlled way. Recently, advances in nanofabrication technology have made it possible to manufacture QDs containing one, two, and more electrons, which are intensively investigated experimentally and theoretically. The experimental study of semiconductor QDs is expanding rapidly [3 – 6], and electron – electron interaction and correlation effects are shown to be of great importance [7 –9] in such systems. In the meantime, a large number of theoretical investigations of electronic structures and related magnetic and optical properties in QDs have been performed to explain the experimental observations. In experimentally realized QDs, the extension in the x – y plane is much larger than in the z direction. Assuming the z extension could be effectively considered zero, the electronic properties in these nanostructures have been E-mail address: [email protected] (W. Xie).

described successfully [1,10] within the model of the single-electron motion in the two-dimensional (2D) harmonic oscillator potential in the presence of a magnetic field. Based on a numerical solution of the Coulomb interaction between electrons, a complex ground state behavior (singlet ! triplet state transitions) as a function of a magnetic field has been predicted [8,11]. Remarkably, these ground state transitions for N ¼ 2 have been observed experimentally [12]. In recent years, various theoretical approaches have been employed to calculate the energy spectra of the two interacting electrons in QDs. In 1996, Zhu et al. [13] studied the quantum-size effects on the energy spectra of two electrons by using of the expansion in a power series. Peeters and Schweigert [14] calculated the energy levels of a disk containing two electrons as a function of an external magnetic field. Analytic expressions are given for the energy spectrum of the two interacting electrons in a harmonic oscillator potential under a perpendicular homogeneous magnetic field by Dineykhan and Nazmitdinov [15]. In 1998, Garcia-Castelan et al. [16,17] studied the energies for two interacting electrons in a harmonic QD using the WKB approximation. However, the parabolic potential possesses infinite depth and range. It is

0038-1098/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0038-1098(03)00335-1

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W. Xie / Solid State Communications 127 (2003) 401–405

inappropriate for a description of the experimentally measured charging of the QD by the finite number of excess electrons [18]. Some experimental results suggest that, the real confining potential is nonparabolic and possesses a well-like shape. When a QD is small (i.e. when its radius is comparable to the characteristic length of the variation of the lateral potential near the edge), a good approximation offers simple smooth potentials, such as a Gaussian well, VðrÞ ¼ 2V0 expð2r2 =L2 Þ: In 2000, Adamowski et al. [19] studied two electrons confined in QDs under an assumption of a Gaussian confining and its parabolic approximation. Recently, a numerical exact calculation for the energy spectra of two electrons in a finite height cylindrical QD was given by a coupled-channel method [20]. In the present paper, we propose a Gaussian confining potential to study the properties of the low-lying states of a two-electron system in a QD under a magnetic field using the method of few-body physics. This potential possesses the finite depth and range. Consider two electrons moving in a QD under the confined Gaussian potential, we make the generalization that the Hamiltonian in the system within the effective-mass approach when a magnetic field is applied perpendicular to the x – y plane is given by " # X 1 ! e! 2 e2 H¼ A p þ þVðr Þ þ i i i p 2me c e r12 i¼1;2 2 gp mB BSz

ð1Þ

radius ap ¼ e aB =mpe as a unit of length, where Ry is the atomic Rydberg, aB is the atomic Bohr radius. The Hamiltonian has cylindrical symmetry with respect to the QD axis, i.e. z-axis; which implies that the z-component of the total orbital angular momentum, Lz ; is a conserved quantity, i.e. a good quantum number. The total spin of two electrons, i.e. S; and its projection along the z-axis; Sz ; are conserved quantities. Hence, the eigenstates of the two electrons in QDs can be classified according to the total orbital angular momentum and the total spin momentum of the electrons along the z direction, i.e. after solving our Hamiltonian, a series of energy levels which we indicate by the quantum numbers ðLz ; Sz Þ: To obtain the eigenfunction and the eigenenergy associated with the two electrons in a Gaussian potential QD, we diagonalized H: As we know, the two electrons obey Fermi – Dirac statistics which means that the electronic part of the total wavefunction must be antisymmetric, i.e. when S ¼ 0 the spatial part of the electronic wavefunction must be symmetric and when S ¼ 1 the spatial part of the electronic wavefunction must be anti-symmetric. Thus, S can be used as a quantum number which indicates the parity of the state. Under the center-of-mass frame, we introduce the center! ! of-mass coordinate j ¼ ð r 1 þ r 2 Þ=2 and the relative ! ! ! coordinate h ¼ r 12 ¼ r 1 2 r 2 and the Hamiltonian equation (4) can be rewritten as H ¼ H0 þ Vr þ Vg

ð5Þ

with

with Vðri Þ ¼ 2V0 expð2ri2 =2R2 Þ !

ð2Þ

!

where r i ðp i Þ is the position vector (the momentum vector) of the ith electron originating from the center of the dot; mpe ! ! is the effective mass of an electron; r12 ¼ l r 1 2 r 2 l is the p electron – electron separation; g is the effective Lande factor; mB is the Bohr magneton; Sz is the z-component of the total spin, e is the effective dielectric constant of the QD, V0 is the height of the potential well and V0 . 0; and R is the range of the confinement potential, which corresponds to a radius of the QD. For r=R p 1; Gaussian potential (2) can be approximated by the parabolic potential Vðri Þ ¼ 2V0 þ g2 ri2

ð3Þ

where!g2 ¼ V0 =2R2 : With the symmetric gauge for magnetic field A ¼ ðB=2Þ ð2y; x; 0Þ; the Hamiltonian then reads " # X p2i 1 p e2 2 2 m H¼ þ ð v =2Þ r þ Vðr Þ þ e i c i p 2 2me e r12 i¼1;2 þ

1 v L 2 gp mB BSz 2 c z

ð4Þ

where vc ¼ eB=ðmpe cÞ is the cyclotron frequency and Lz is the total orbital angular momentum along in the z-direction: Throughout the present paper, we use the donor Rydberg Ryp ¼ mpe Ry=e 2 as a unit of energy, and the donor Bohr

H0 ¼

Vr ¼

P2j P2h 1 1 1 þ Mðvc =2Þ2 j2 þ þ mðvc =2Þ2 h2 þ 2M 2m 2 2 2 vc Lz 2 gp mB BSz

ð6Þ

e2 eh

ð7Þ

Vg ¼ 2V0 expð2r12 =2R2 Þ 2 V0 expð2r22 =2R2 Þ

ð8Þ

where M ¼ 2mpe ; and m ¼ mpe =2: To obtain the eigenfunction and the eigenenergy, we diagonalize H in a model space spanned by the translational invariant 2D harmonic product bases F½K ¼ ~ fvn l ðh~Þfvn l ðj~ÞLx }; where xS ¼ ½zð1Þzð2ÞS ; zðiÞ is the A{½ S 1 1 2 2 spin state of the ith electron and the spins of two electrons v are coupled to S; fnl is a 2D harmonic oscillator state with frequency v (v is considered as an adjustable variational parameter), an energy ð2n þ lll þ 1Þ"v; and A~ is the antisymmetrizer. ½K denotes the whole set of quantum numbers ðn1 ; l1 ; n2 ; l2 Þ in brevity, l1 þ l2 ¼ L is the total angular momentum. The angular momentum L ¼ odd if the spin S ¼ 1 and L ¼ even if S ¼ 0 such that the wave function is antisymmetrized. The matrix elements of H are

W. Xie / Solid State Communications 127 (2003) 401–405

then given by the following expressions: F½K lH0 lF½K 0 ¼ ½2ðn1 þ n2 Þ þ ll1 l þ ll2 l þ 2"v þ

1 2

vc Lz

2 gp mB BSz d½K;½K 0

F½K lVr lF½K 0 ¼ UnI 1 n01 dl1 ;l01 dn2 ;n02 dl2 ;l02 D E F½K lVg lF½K 0 ¼

X

ð9Þ ð10Þ

B½K½K 00 B½K 0 ½K 000 ðUnII001 n0001 dn002 ;n0002

½K 00 ½K 000

þ UnII002 n0002 dn001 ;n0001 Þdl1 ;l01 dl2 ;l02

ð11Þ

with I 0 ¼ Unn

ð1 0

Rnl ðrÞ

II 0 ¼ 2V0 Unn

B½K½K 0 ¼

ð

ð1 0

e2 R 0 ðrÞr dr er n l

Rnl ðrÞexpð2r 2 =2R2 ÞRn0 l ðrÞr dr

F½K ðh~; j~ÞF½K 0 ðh~0 ; j~0 Þdh~ dj~

ð12Þ ð13Þ ð14Þ

where Rnl ðrÞ is the radial part of 2D harmonic oscillator function, B½K½K 0 is the transformation bracket of 2D harmonic product states with two different sets of co-

403

ordinates, which allows us to reduce the otherwise multiintegral into single-integral. Nonvanishing B½K½K 0 occurs only when both the states F½K ðh~; j~Þ and F½K 0 ðh~0 ; j~0 Þ have exactly the same eigenenergy and eigenangular momentum. Analytical expression for B½K½K 0 has already been derived in Ref. [21]. The set of canonical coordinates ðh~0 ; j~0 Þ are defined by h~0 ¼ ~r1 ; j~0 ¼ 2~r2 : The dimension of the model space is constrained by 0 # N ¼ 2ðn1 þ n2 Þ þ ll1 l þ ll2 l # 24: If N is increased by 2, the ratio of the difference in energy is less than 0.001%. Our numerical computation is carried out for one of the typical semiconducting materials, GaAs, as an example with the material parameters shown in the following: e ¼ 12:4; and mpe ¼ 0:067me ; where me is the single electron bare mass. The potential-well depth is taken to be V0 ¼ 50Ryp corresponds to the GaAs QDs, for which Ryp ¼ 6 meV and ap ¼ 10 nm:: We first took the magnetic field B ¼ 2T and plotted in Fig. 1 the energy spectrum of two interacting electrons in the Gaussian (solid curves) and parabolic (dashed curves) potentials as a function of the dot radius R which corresponds to the QDs with the strong and intermediate confinement, i.e. R # 3ap : The singlet and triplet states of the two interacting electrons with the total angular momentum L ¼ 0 (S states) and L ¼ 1 (P states) are denoted by 1 S; 3 S and 1 P; 3 P; respectively. The Fig. 1 shows that the qualitative properties of energy levels for the Gaussian potential and the parabolic potential are similar.

Fig. 1. The energy levels of two electrons confined in Gaussian (solid curves) and parabolic (dashed curves) potential as a function of the dot radius R with V0 ¼ 50Ryp and B ¼ 2 T: The levels are labeled b6y quantum numbers ðL; SÞ: Energy is expressed in donor Rydbergs Ryp and length in donor Bohr radii ap :

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W. Xie / Solid State Communications 127 (2003) 401–405

However, the quantitative differences are also obvious. In the strong confinement case, i.e. R # ap ; the energies in the Gaussian potential are obviously lower than those in the parabolic potential. When R=ap is larger, the calculation values both the Gaussian potential and the parabolic potential are reaching consistent, i.e. only for a larger QD the parabolic can be regarded as a fairly good approximation of the nonparabolic Gaussian potential. It is obvious that the

energy differences of the ground state are larger those of the excited states. Further, we took the dot radius R ¼ 1:0ap (i.e. strong confinement) and R ¼ 5:0ap (weak confinement) and plotted in Fig. 2 the energy spectra of two interacting electrons in the Gaussian potential QD as a function of the magnetic field strength for four levels, respectively. Note that as a function of the magnetic field, in the R ¼ 1:0ap

Fig. 2. The energy levels of two electrons confined in a Gaussian potential quantum dot as a function of the magnetic field strength B for the R ¼ 1:0ap (a) and R ¼ 5:0ap (b). The others are the same as in Fig. 1.

W. Xie / Solid State Communications 127 (2003) 401–405

case, there no occurs the ground state transition. This is obviously different that in a parabolic potential QD. Obviously, in Fig. 2b, it is readily seen that a ground state transition (singlet ! triplet state transition) occurs at B ¼ 9:2 T This is qualitatively the same as those for a twoelectron parabolic QD. The ground state transition is a result of the competition between the single-electron confinement energy and the many body electron – electron interactions [11]. Such magnetic filed induced angular momentum (and spin) transitions have been observed experimentally [12], for dots with parabolic-like confinement. The two-electron dot system with parabolic potential has been studied in great detail by Merkt et al. [22]. In conclusion, we have demonstrated a procedure to numerically solve the two interacting electrons with a Gaussian potential QD problem. We find that a ground state transition (singlet ! triplet state transition) occurs in the weak confinement case. However, in the strong confinement case, the transition does not occur as a two-electron QD system with a parabolic confining potential. The present results are useful to understand the optical and magnetic properties of QD material. The ground state transition predict a possibility to observe phenomena related to electron – electron interaction in QDs.

Acknowledgements This work is financially supported by the National Natural Science Foundation of China under Grant No. 10275014.

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