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Electronic properties of two interacting electrons in a quantum pseudodot under magnetic field: Perturbation theory and two parameters variational procedure
Superlattices and Microstructures 62 (2013) 166–174
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Electronic properties of two interacting electrons in a quantum pseudodot under magnetic field: Perturbation theory and two parameters variational procedure R. Khordad ⇑ Department of Physics, Yasouj University, Yasouj 75914-353, Iran
a r t i c l e
i n f o
Article history: Received 7 June 2013 Received in revised form 30 July 2013 Accepted 2 August 2013 Available online 12 August 2013 Keywords: Quantum pseudodot Variational method Magnetic field Interacting electron
a b s t r a c t In this paper, we have studied the ground state energy and wave function of a system of two interacting electrons in a two-dimensional (2D) quantum pseudodot under the influence of an external magnetic field. For this purpose, we have employed two different methods. First, we have used the perturbation theory and obtained analytic ground state energy of the system. Second, we have applied the variational procedure and considered a trial wave function with one and two variational parameters. We have found that against to a 2D quantum dot with parabolic potential, the results obtained from that first-order perturbation method is lower than the results with variational method. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The low-dimensional semiconductor structures and nanoparticles can be fabricated by the advanced growth techniques like chemical lithography, molecular-beam epitaxy, metal-organic chemical-vapor deposition (MOCVD), and arc discharge method [1–4]. Using the techniques, the growth of heterostructures consisting of alternate layers of two different semiconductors with controllable thickness has become possible. These heterostructures have stimulated new works in semiconductor physics over the past years. During the last decades, new semiconductor structures such as dots, antidots, wells, wellwires, pseudodot, and antiwells have been received a great attention both
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theoretically and experimentally by researchers. The structures may create new phenomena and they show great potential for device application in laser and optical modulation technology [5–12]. Nanostructure technologies allow the lateral confinement of electrons in one, two, and three dimensions in semiconductor structures called quantum wells, wires, and dots, respectively. Among the semiconductor structures, the quantum dots (QDs) with various shapes such as cubic, spherical, and cylindrical have received lots of attention by researchers during the last couple of decades as they play the role of key elements for potential applications [13–16]. In 1975, Esaki et al. were the first to present the concept of quantum wires and dots [17]. Quantum confinement of charge carriers in quantum dots leads to formation of discrete energy levels with energy spacings of a few meV or more, the enhancement of the density of states at specific energies and the drastic change of optical absorption spectra. Recently, the QDs are becoming more and more important because of their novel physical properties. In a typical QD, all electron are tightly bound, except for a few free electrons. It is worth mentioning that the QDs form the building units of larger structures. Hitherto, many works have been done both experimentally and theoretically by many authors on electronic and optical properties of QDs mainly because of their importance in the potential applications and development of optoelectronic devices [18–22]. For example, Sauvage and Boucand discussed nonlinear optical properties of quantum dots [23]. In the past few years, the physical properties in the semiconductor structures has drawn the attention of investigators due to their various novel quantum effects and superior electrical and optical properties [24–30]. In most of studies, the motion of electron and the confining potential are only considered. It means that the interaction between electrons is neglected. Investigation on electronic states (without the interaction between electrons) is simple and interesting, not only to understand how such levels differ from the bulk, but also in the fabrication and subsequent working of electronic and optical devices based on such systems. The first systematic study of the electronic states is an electron in a QD with a confinement potential. This problem is easily solved by the Schrödinger equation. In the typical QDs, the confined carriers substantially are under many real effects. Examples of the effects are electron-electron interaction, electron-hole interaction, magnetic and electric fields, temperature, and pressures. These effects play the important roles on electronic and optical properties of the QDs. Therefore, it is clearly very important to discuss these effects. One of the most effects in QDs is the electron–electron interaction. So far, several theoretical studies have been done on this effect in some of nanostructures such as two-dimensional QD [31–33]. There are several theoretical methods for studying the electron–electron interaction in semiconductor structures. Among the different theoretical methods, we can mention numerical diagonalization method [34], quantum Monte Carlo method [35], density functional method [36], and variational procedure [37]. Hitherto, several works have been performed on QDs with addition to the electron–electron interaction with and without magnetic field. For example, Filinov et al. [38] have studied a 2D quantum dot with considering the electron–electron interaction at zero magnetic field. Hirose and Wingreen have used the density functional method to study a QD under magnetic field [36]. In addition to the electron-electron interaction, there are many works on the electron-hole (exciton) interaction. It should be noted that excitonic effect also play an important role in the optical and electronic properties of nanostructures. In 1989, Takagahara studied theoretically the nonlinear optical properties of semiconductor microcrystallites, and showed that optical non-linearities are very large when one considers exciton effects [39]. In 1993, Chen discussed the third-order nonlinear optical susceptibility in silicon quantum wires, and also attained the enhancement of the nonlinear optical properties due to excitonic effects [40]. In 2002, Wang and Guo studied the excitonic effects on the third-order nonlinear optical properties in disk-like parabolic quantum dots [41]. Excitons dominate the optical properties of semiconductor nanostructures. For example, the sharp lines were observed in exciton photoluminescence from GaAs quantum wells [42–44]. Although many works performed on electronic and optical properties of semiconductor QDs, the electron-electron interaction effect in a quantum pseudodot has not been studied so far. For this purpose, in this paper, we intend to study this problem in details by using the perturbation theory and
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variational method with different parameters. The electronic properties, the perturbation theory and the variational procedure are briefly presented in Section 2. The results and discussion are presented in Section 3. Finally, the conclusions are given in Section 4. 2. Theory In the standard theoretical model of a quantum pseudodot, we include the following terms. First, we consider the motion of the electrons to be exactly two-dimensional (2D). Second, we take the confining potential to be 2D pseudoharmonic potential. This potential includes both harmonic quantum dot potential and antidot potential. The potential can be written as [6,12]
VðrÞ ¼ V 0
r r0 r0 r
2 ;
ð1Þ
where V0 is the chemical potential of the two-dimensional electron gas and r0 is the zero point of the pseudoharmonic potential. Third, we consider the interaction between electrons to be a pure Coulomb interaction. In absence of an external magnetic field, the Hamiltonian for two electrons interacting in a quantum pseudodot is
H¼
2 2 p21 r1 r0 p2 r2 r0 e2 þ V0 þ 2 þ V0 þ ; 2l r0 r1 2l r0 r2 jr2 r1 j
ð2Þ
where pi = (pix, piy) and ri = (xi, yi) are the 2D momentum operator and position of the ith electron, respectively. Also, l is the effective electronic mass and e is the electron’s charge. As we know, the Hamiltonian, Eq. (2) describes a system of two interacting electrons in a quantum pseudodot in absence of magnetic field. In the following, we intend to study the effect of an external magnetic field on the above system. Now, let us consider a system of two interacting electrons in a quantum pseudodot under uniform magnetic field along the z direction. The Hamiltonian will be written in the form
H¼
2 2 1 e 2 r1 r0 1 e 2 r2 r0 e2 p1 þ A1 þ V 0 þ p2 þ A2 þ V 0 þ : 2l c 2l c r0 r1 r0 r2 jr2 r1 j
ð3Þ
The vector potential A should be considered such that r A = B, where B is the applied uniform magnetic field along the z direction. The components of vector potential in the cylindrical coordinate system (for ith electron) are given by
Ai ¼
0;
Br i ;0 : 2
ð4Þ
It should be noted that even in the case of 2D quantum pseudodot in absence of magnetic field, Eq. (2), an exact analytic solution of the Hamiltonian is not available. Therefore, many researchers have applied the approximations or numerical calculations such as perturbation, diagonalization, and variational procedure. To use these methods, we should first solve the Schrödinger equation without a pure Coulomb interaction. To study a system of two independent electrons, we first investigate the single-electron states. In the following, we solve the Schrödinger equation for an electron in a 2D quantum pseudodot under an external magnetic field. 2.1. Single-electron states Using Eq. (2), the Schrödinger equation for an electron can be expressed as
! 2 2 h 1 @ @ 1 @2 hxc @w lx2c r 2 r r0 w i þ 2 r þ w þ V w ¼ Ew; 0 r @ u2 r0 r 2l r @r @r 2 @u 8 where xc = eB/lc is the cyclotron frequency.
ð5Þ
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169
Let us consider the wave function in the form
f ðrÞeimu wðr; uÞ ¼ pffiffiffiffiffiffiffi : 2p
ð6Þ
Substituting above wave function into the Schrödinger Eq. (5), we obtain the following equation for radial function [6,12] 2
d f ðrÞ dr
2
! 1 df ðrÞ b2 2 2 2 þ þ a 2 c r f ðrÞ ¼ 0; r dr r
ð7Þ
where the parameters a, b, and c are defined as
a2 ¼
2lðE þ 2V 0 Þ 2
h
b2 ¼ m2 þ
c2 ¼
2l V 0 2 h r20
2lV 0 r 20 h þ
2
ð8Þ
ð9Þ
;
e2 B2 2
meB ; hc
4h c2
ð10Þ
:
Using the following relation
f ðrÞ ¼ rjbj ecr
2 =2
LðrÞ;
ð11Þ
and applying n = cr2 instead of r, we shall obtain 2
n
d LðnÞ dn2
þ ðjbj n þ 1Þ
dLðnÞ þ nLðnÞ ¼ 0; dn
ð12Þ
where n = a2/4c (jbj + 1)/2 is an integer and LðnÞ ¼ Lbn ðnÞ is associated Laguerre polynomials. With respect to Eqs. 6, 11 and 12, the total wave function becomes
wn;m ðr; uÞ ¼ Arjbj ecr
2 =2
Fðn; jbj þ 1; cr2 Þeimu ;
1 where F(a, b; x) is the confluent hypergeometric function and A ¼ jbj! constant. Using n = a2/4c (jbj + 1)/2, we obtain energy spectrum as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jbj þ 1 8V mhxc En;m ðbÞ ¼ h n þ x2c þ 20 þ 2V 0 : 2 2 lr0
cjbjþ1 ðnþjbjÞ!
pn!
ð13Þ is the normalization
ð14Þ
It is to be noted that the quantum number m relates to the quantum number b [Eq. (9)]. Therefore, for this system, only two independent quantum numbers are required. Expressions (12) and (13) show the electron energy spectrum and wave functions in a quantum pseudodot system under the influence of external magnetic field. The ground-state energy and wave function correspond to n = 0 and m = 0
rffiffiffiffi w00 ðr; uÞ ¼ where X ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c cr2 =2 h 8V 1 ; E00 ¼ x2c þ 20 2V 0 ¼ hX 2V 0 : e 2 2 p lr 0
ð15Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 x2c þ 8V lr2 . 0
2.2. Perturbation method By ignoring the Coulomb interaction in Eq. (3), we would have a solution to the eigenvalue problem for the two-electron system, with eigenfunctions given by wn1 m1 ðr1 ; u1 Þwn2 m2 ðr2 ; u2 Þ and energy eigenvalues given by E ¼ En1 m1 þ En2 m2 .
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In first step, we consider the Coulomb interaction as a perturbation. To first order in perturbation theory, the energy correction of the ground state is given by
Z 2 Z e2 e2 2 2 w00 ðr 2 ; u Þ >¼ c dr1 dr2 ecðr1 þr2 Þ Eð1Þ ¼< w00 ðr 1 ; u1 Þ : 2 jr2 r1 j p jr2 r1 j
ð16Þ
Now, we calculate above integration by using the following identity [45] 1 Z 1 X e2 imðu u Þ dqe 1 2 J m ðqr1 ÞJm ðqr 2 Þ; ¼ jr2 r1 j m¼1 0
ð17Þ
where Jm(qr) is Bessel function with order m. First, we carry out integration over angular variables u1 and u2. Then, we obtain the following result by using straightforward calculations,
DEð1Þ ¼ e2
rffiffiffiffiffiffi
cp 2
ð18Þ
:
When the first-order correction of the energy is added to the unperturbed result, one can obtain
Eð1Þ ¼ hX 4V 0 þ DEð1Þ ¼ hX 4V 0 þ e2
rffiffiffiffiffiffi
cp 2
:
ð19Þ
2.3. Variational procedure with different parameters As we know, sometimes, three exist systems whose Hamiltonian are known but they cannot be solved exactly. One of the approximation methods that is suitable for solving such problems is the variational procedure, which is also called the Rayleigh-Ritz method [46]. This method is useful for determining upper bound values for the eigenenergies of a system whose Hamiltonian is known. It is particularly useful for obtaining the ground state and it is quite cumbersome to calculate the energy levels of the excited states. In this part, we use variational method to calculate the ground state energy of a 2D quantum pseudodot under an external magnetic field. We can perform this problem by choosing a trial wave function that depends on a number of parameters. According to the variational method, after choosing a trial wave function, we should calculate the expectation value of the Hamiltonian with respect to this trial wave function. Finally, we should minimize the trial energy with respect to all parameters. We first consider a normalized one-parameter trial wave function as
k2
/k ðr 1 ; r 2 Þ ¼
p
eðk
2
=2Þðr 21 þr 22 Þ
ð20Þ
;
where k is a positive variational parameter. To obtain the ground state energy, we should evaluate the following integral
Ek ¼
Z
dr1
Z
dr2 /k ðr 1 ; r2 ÞHðr 1 ; r 2 Þ/k ðr1 ; r 2 Þ;
ð21Þ
where the Hamiltonian is given in Eq. (3). After calculating the integral, we have 2
Ek ¼
h k2
l
þ
lX2 2k2
2
rffiffiffiffi
þe k
p 2
þ ð8:633024714Þð2k2 V 0 r 20 Þ 4V 0 :
ð22Þ
It is seen that the ground state energy depends on the parameter k. By differentiating with respect to k, we obtain 2
dEk 2h k lX2 ¼ 3 þ e2 dk l k
rffiffiffiffi
p 2
þ ð8:633024714Þð4kV 0 r 20 Þ ¼ 0:
ð23Þ
We search for the real positive root of Eq. (23). This root is called k0 that minimizes the energy [Eq. (22)].
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Now, we consider a normalized two-parameter trial wave function as
k 2 g 2 2 2 /k;g ðr 1 ; r2 Þ ¼ pffiffiffiffi eðk =2Þr1 pffiffiffiffi eðg =2Þr2 ;
p
ð24Þ
p
where k and g are positive variational parameters. To find the ground state energy, we should calculate the expectation value of the Hamiltonian, Eq. (3) by using the above wave function. After performing the integrals, we can obtain
Ek;g ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k2 lX2 h g2 lX2 h p 2 2 2 2 þ 2 þ þ þ ð8:633024714ÞðV r Þðk þ g Þ þ e k g 4V 0 : 0 0 2l 2l 4g2 4k k2 þ g2
ð25Þ
As previously, by differentiating with respect to parameters k and g, we can obtain two relations. Then, we should solve simultaneously these equations. After searching for the real roots of the equations, we can obtain the ground state energy. 3. Results and discussion In this paper, we have considered a GaAs quantum pseudodot with parameters l = 0.067,e = 13.18, 2 and r0 = 89.58 nm [12]. It is worth mentioning that we have inserted c = 1 and e2 ! 4epe in our calculations. In first step, we have divided Eq. (19) by ⁄X. Therefore, we have
ð1Þ ðnÞ ¼
rffiffiffiffi Eð1Þ 4V 0 DEð1Þ 4V 0 p : ¼1 þ ¼1 þn hX hX hX hX 2
ð26Þ
2 pffiffi where (1)(n) is a dimensionless energy and n ¼ 4epehcX is a dimensionless interaction parameter that gauges the strength of the Coulomb correlation relative to the confining potential and magnetic field. Table 1 shows the ground state energy = hX for given values of V0 (or n) at zero magnetic field. The results in this table obtained from two different methods: first-order perturbation theory and variational procedure with one and two parameters. It is seen from the table that the energies calculated from the one-parameter variational method are higher than the first-order perturbation theory for all V0 (or n). According to the results from the table, we can deduce an interesting point. For 2D quantum dot with the parabolic potential, at zero magnetic field, the ground state energy obtained from firstorder perturbation theory is too high [31]. But, for our system (2D quantum pseudodot), the obtained ground state energy by using the first-order perturbation theory is not too high. In this table, the fourth and fifth columns correspond to one-parameter and two-parameter variational method. It is obvious that the results obtained from variational method are higher than the perturbation theory. We also have found that the ground state energy decreases when V0 increases.
Table 1 The ground state energy =hX of a 2D quantum pseudodot as a function of V0 (or dimensionless Coulomb coupling parameter n) at zero magnetic field. The results obtained from first-order perturbation theory (1)(n), one-parameter variational trial wave function (var. 1), and twoparameter variational trial wave function (var. 2). V0 (meV)
n
(1) (n)
(var. 1)
(var. 2)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
3.94 3.31 2.99 2.78 2.63 2.52 2.42 2.34 2.27 2.21
4.749564 3.472473 2.695042 2.116690 1.647081 1.246656 0.894453 0.577976 0.289149 0.022434
14.909176 13.632084 12.854654 12.276302 11.806693 11.406268 11.054065 10.737588 10.448761 10.182046
16.281255 15.234587 14.985621 14.462120 13.786492 13.516768 13.024865 12.895718 12.228963 11.925468
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R. Khordad / Superlattices and Microstructures 62 (2013) 166–174 Table 2 The same as Table 1, but for different magnetic field with V0 = 0.68346 meV. B (T)
n
(1)(n)
(var. 1)
(var. 2)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
2.43 2.41 2.35 2.26 2.16 2.06 1.96 1.87 1.78 1.71 1.64
0.949966 0.979342 1.057076 1.160100 1.267057 1.364495 1.447695 1.515936 1.570717 1.614156 1.648339
11.054065 11.101129 11.227350 11.399010 11.585019 11.760143 11.918940 12.057344 12.176229 12.277825 12.364662
14.002489 14.185879 14.52381 14.901895 15.189823 15.878458 16.099018 16.595784 16.978279 17.188425 17.786562
0.0115 0.0110 0.0105
λ
0.0100 0.0095 0.0090 0.0085 0.0080 0.0075 0.0
0.2
0.4
0.6
0.8
1.0
B (T) Fig. 1. Variational parameter (in one-parameter variational method) as a function of magnetic field for V0 = 0.68346 meV.
In Table 2, we have presented the ground state energy = hX for given values of magnetic field (or n) with V0 = 0.68346 meV. At similar to the 1, the results in this table obtained from two different methods: the first-order perturbation theory and variational procedure with one and two parameters. It is observed from the figure that the ground state energy (obtained from all procedures) increases when the magnetic field increases. The reason is as follow: when the magnetic field is applied to a system, an additional energy is added to the system. Therefore, the ground state energy is increases. As we see from the table, the energies calculated from the one-parameter variational method are higher than the first-order perturbation theory for all magnetic fields (or n). One can observe from the tables that the ground state energy by using the two-parameter variational method is higher than one-parameter variational and first-order perturbation. In Fig. 1, we have presented the variational parameter (in one-parameter variational method) as a function of magnetic field for V0 = 0.68346 meV. It is clear that the variational parameter increases when the magnetic field increases. It is obvious that by increasing the magnetic field, the parameter which minimizes the energy is increased. Fig. 2 shows the ground state energy as a function of magnetic field with r0 = 89.58 nm and V0 = 0.68346 meV. In this figure, we have presented our results obtained by the perturbation theory, variational method with one and two parameters. We observe from the figure that the ground state energy increases when the applied magnetic field increases.
R. Khordad / Superlattices and Microstructures 62 (2013) 166–174
173
32
First-order perturbation One-parameter variation Two-parameter variation
28 24
E (meV)
20 16 12 8 4 0
0.0
0.2
0.4
0.6
0.8
1.0
B (T) Fig. 2. The ground state energy as a function of magnetic field with r0 = 89.58 nm and V0 = 0.68346 meV.
5
First-order perturbation One-parameter variation Two-parameter variation
E (meV)
4
3
2
1
0 0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
V0 (meV) Fig. 3. The ground state energy as a function of V0 with B = 1 T and r0 = 89.58 nm.
In Fig. 3, we have displayed the ground state energy as a function of V0 with B = 1 T and r0 = 89.58 nm. This figure shows our results obtained by the perturbation theory, variational method with one and two parameters. By increasing V0, the potential is deeper and thereby the ground state energy decreases.
4. Conclusion In this work, we have studied a system of two interacting electrons in a quantum pseudodot under the influence of an external magnetic field. We tried to obtain electronic properties of ground state.
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R. Khordad / Superlattices and Microstructures 62 (2013) 166–174
For this purpose, we have calculated the ground state energy and wave function by using various methods. We have applied the first-order perturbation and variational method with one and two parameters. According to the results, we have found that the ground state energy obtained from first-order perturbation has lowest values. The results with variational method are higher than the perturbation theory. We also deduced an interesting point. Against 2D quantum dot, the results with perturbation theory are lower than variational method. According to the results obtained from the present work, it is deduced that the magnetic field and V0 play important roles in the ground state energy a system of two interacting electrons in a quantum pseudodot. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Electronic properties of two interacting electrons in a quantum pseudodot under magnetic field: Perturbation theory and two parameters variational procedure R. Khordad ⇑ Department of Physics, Yasouj University, Yasouj 75914-353, Iran
a r t i c l e
i n f o
Article history: Received 7 June 2013 Received in revised form 30 July 2013 Accepted 2 August 2013 Available online 12 August 2013 Keywords: Quantum pseudodot Variational method Magnetic field Interacting electron
a b s t r a c t In this paper, we have studied the ground state energy and wave function of a system of two interacting electrons in a two-dimensional (2D) quantum pseudodot under the influence of an external magnetic field. For this purpose, we have employed two different methods. First, we have used the perturbation theory and obtained analytic ground state energy of the system. Second, we have applied the variational procedure and considered a trial wave function with one and two variational parameters. We have found that against to a 2D quantum dot with parabolic potential, the results obtained from that first-order perturbation method is lower than the results with variational method. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The low-dimensional semiconductor structures and nanoparticles can be fabricated by the advanced growth techniques like chemical lithography, molecular-beam epitaxy, metal-organic chemical-vapor deposition (MOCVD), and arc discharge method [1–4]. Using the techniques, the growth of heterostructures consisting of alternate layers of two different semiconductors with controllable thickness has become possible. These heterostructures have stimulated new works in semiconductor physics over the past years. During the last decades, new semiconductor structures such as dots, antidots, wells, wellwires, pseudodot, and antiwells have been received a great attention both
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theoretically and experimentally by researchers. The structures may create new phenomena and they show great potential for device application in laser and optical modulation technology [5–12]. Nanostructure technologies allow the lateral confinement of electrons in one, two, and three dimensions in semiconductor structures called quantum wells, wires, and dots, respectively. Among the semiconductor structures, the quantum dots (QDs) with various shapes such as cubic, spherical, and cylindrical have received lots of attention by researchers during the last couple of decades as they play the role of key elements for potential applications [13–16]. In 1975, Esaki et al. were the first to present the concept of quantum wires and dots [17]. Quantum confinement of charge carriers in quantum dots leads to formation of discrete energy levels with energy spacings of a few meV or more, the enhancement of the density of states at specific energies and the drastic change of optical absorption spectra. Recently, the QDs are becoming more and more important because of their novel physical properties. In a typical QD, all electron are tightly bound, except for a few free electrons. It is worth mentioning that the QDs form the building units of larger structures. Hitherto, many works have been done both experimentally and theoretically by many authors on electronic and optical properties of QDs mainly because of their importance in the potential applications and development of optoelectronic devices [18–22]. For example, Sauvage and Boucand discussed nonlinear optical properties of quantum dots [23]. In the past few years, the physical properties in the semiconductor structures has drawn the attention of investigators due to their various novel quantum effects and superior electrical and optical properties [24–30]. In most of studies, the motion of electron and the confining potential are only considered. It means that the interaction between electrons is neglected. Investigation on electronic states (without the interaction between electrons) is simple and interesting, not only to understand how such levels differ from the bulk, but also in the fabrication and subsequent working of electronic and optical devices based on such systems. The first systematic study of the electronic states is an electron in a QD with a confinement potential. This problem is easily solved by the Schrödinger equation. In the typical QDs, the confined carriers substantially are under many real effects. Examples of the effects are electron-electron interaction, electron-hole interaction, magnetic and electric fields, temperature, and pressures. These effects play the important roles on electronic and optical properties of the QDs. Therefore, it is clearly very important to discuss these effects. One of the most effects in QDs is the electron–electron interaction. So far, several theoretical studies have been done on this effect in some of nanostructures such as two-dimensional QD [31–33]. There are several theoretical methods for studying the electron–electron interaction in semiconductor structures. Among the different theoretical methods, we can mention numerical diagonalization method [34], quantum Monte Carlo method [35], density functional method [36], and variational procedure [37]. Hitherto, several works have been performed on QDs with addition to the electron–electron interaction with and without magnetic field. For example, Filinov et al. [38] have studied a 2D quantum dot with considering the electron–electron interaction at zero magnetic field. Hirose and Wingreen have used the density functional method to study a QD under magnetic field [36]. In addition to the electron-electron interaction, there are many works on the electron-hole (exciton) interaction. It should be noted that excitonic effect also play an important role in the optical and electronic properties of nanostructures. In 1989, Takagahara studied theoretically the nonlinear optical properties of semiconductor microcrystallites, and showed that optical non-linearities are very large when one considers exciton effects [39]. In 1993, Chen discussed the third-order nonlinear optical susceptibility in silicon quantum wires, and also attained the enhancement of the nonlinear optical properties due to excitonic effects [40]. In 2002, Wang and Guo studied the excitonic effects on the third-order nonlinear optical properties in disk-like parabolic quantum dots [41]. Excitons dominate the optical properties of semiconductor nanostructures. For example, the sharp lines were observed in exciton photoluminescence from GaAs quantum wells [42–44]. Although many works performed on electronic and optical properties of semiconductor QDs, the electron-electron interaction effect in a quantum pseudodot has not been studied so far. For this purpose, in this paper, we intend to study this problem in details by using the perturbation theory and
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variational method with different parameters. The electronic properties, the perturbation theory and the variational procedure are briefly presented in Section 2. The results and discussion are presented in Section 3. Finally, the conclusions are given in Section 4. 2. Theory In the standard theoretical model of a quantum pseudodot, we include the following terms. First, we consider the motion of the electrons to be exactly two-dimensional (2D). Second, we take the confining potential to be 2D pseudoharmonic potential. This potential includes both harmonic quantum dot potential and antidot potential. The potential can be written as [6,12]
VðrÞ ¼ V 0
r r0 r0 r
2 ;
ð1Þ
where V0 is the chemical potential of the two-dimensional electron gas and r0 is the zero point of the pseudoharmonic potential. Third, we consider the interaction between electrons to be a pure Coulomb interaction. In absence of an external magnetic field, the Hamiltonian for two electrons interacting in a quantum pseudodot is
H¼
2 2 p21 r1 r0 p2 r2 r0 e2 þ V0 þ 2 þ V0 þ ; 2l r0 r1 2l r0 r2 jr2 r1 j
ð2Þ
where pi = (pix, piy) and ri = (xi, yi) are the 2D momentum operator and position of the ith electron, respectively. Also, l is the effective electronic mass and e is the electron’s charge. As we know, the Hamiltonian, Eq. (2) describes a system of two interacting electrons in a quantum pseudodot in absence of magnetic field. In the following, we intend to study the effect of an external magnetic field on the above system. Now, let us consider a system of two interacting electrons in a quantum pseudodot under uniform magnetic field along the z direction. The Hamiltonian will be written in the form
H¼
2 2 1 e 2 r1 r0 1 e 2 r2 r0 e2 p1 þ A1 þ V 0 þ p2 þ A2 þ V 0 þ : 2l c 2l c r0 r1 r0 r2 jr2 r1 j
ð3Þ
The vector potential A should be considered such that r A = B, where B is the applied uniform magnetic field along the z direction. The components of vector potential in the cylindrical coordinate system (for ith electron) are given by
Ai ¼
0;
Br i ;0 : 2
ð4Þ
It should be noted that even in the case of 2D quantum pseudodot in absence of magnetic field, Eq. (2), an exact analytic solution of the Hamiltonian is not available. Therefore, many researchers have applied the approximations or numerical calculations such as perturbation, diagonalization, and variational procedure. To use these methods, we should first solve the Schrödinger equation without a pure Coulomb interaction. To study a system of two independent electrons, we first investigate the single-electron states. In the following, we solve the Schrödinger equation for an electron in a 2D quantum pseudodot under an external magnetic field. 2.1. Single-electron states Using Eq. (2), the Schrödinger equation for an electron can be expressed as
! 2 2 h 1 @ @ 1 @2 hxc @w lx2c r 2 r r0 w i þ 2 r þ w þ V w ¼ Ew; 0 r @ u2 r0 r 2l r @r @r 2 @u 8 where xc = eB/lc is the cyclotron frequency.
ð5Þ
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169
Let us consider the wave function in the form
f ðrÞeimu wðr; uÞ ¼ pffiffiffiffiffiffiffi : 2p
ð6Þ
Substituting above wave function into the Schrödinger Eq. (5), we obtain the following equation for radial function [6,12] 2
d f ðrÞ dr
2
! 1 df ðrÞ b2 2 2 2 þ þ a 2 c r f ðrÞ ¼ 0; r dr r
ð7Þ
where the parameters a, b, and c are defined as
a2 ¼
2lðE þ 2V 0 Þ 2
h
b2 ¼ m2 þ
c2 ¼
2l V 0 2 h r20
2lV 0 r 20 h þ
2
ð8Þ
ð9Þ
;
e2 B2 2
meB ; hc
4h c2
ð10Þ
:
Using the following relation
f ðrÞ ¼ rjbj ecr
2 =2
LðrÞ;
ð11Þ
and applying n = cr2 instead of r, we shall obtain 2
n
d LðnÞ dn2
þ ðjbj n þ 1Þ
dLðnÞ þ nLðnÞ ¼ 0; dn
ð12Þ
where n = a2/4c (jbj + 1)/2 is an integer and LðnÞ ¼ Lbn ðnÞ is associated Laguerre polynomials. With respect to Eqs. 6, 11 and 12, the total wave function becomes
wn;m ðr; uÞ ¼ Arjbj ecr
2 =2
Fðn; jbj þ 1; cr2 Þeimu ;
1 where F(a, b; x) is the confluent hypergeometric function and A ¼ jbj! constant. Using n = a2/4c (jbj + 1)/2, we obtain energy spectrum as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jbj þ 1 8V mhxc En;m ðbÞ ¼ h n þ x2c þ 20 þ 2V 0 : 2 2 lr0
cjbjþ1 ðnþjbjÞ!
pn!
ð13Þ is the normalization
ð14Þ
It is to be noted that the quantum number m relates to the quantum number b [Eq. (9)]. Therefore, for this system, only two independent quantum numbers are required. Expressions (12) and (13) show the electron energy spectrum and wave functions in a quantum pseudodot system under the influence of external magnetic field. The ground-state energy and wave function correspond to n = 0 and m = 0
rffiffiffiffi w00 ðr; uÞ ¼ where X ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c cr2 =2 h 8V 1 ; E00 ¼ x2c þ 20 2V 0 ¼ hX 2V 0 : e 2 2 p lr 0
ð15Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 x2c þ 8V lr2 . 0
2.2. Perturbation method By ignoring the Coulomb interaction in Eq. (3), we would have a solution to the eigenvalue problem for the two-electron system, with eigenfunctions given by wn1 m1 ðr1 ; u1 Þwn2 m2 ðr2 ; u2 Þ and energy eigenvalues given by E ¼ En1 m1 þ En2 m2 .
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In first step, we consider the Coulomb interaction as a perturbation. To first order in perturbation theory, the energy correction of the ground state is given by
Z 2 Z e2 e2 2 2 w00 ðr 2 ; u Þ >¼ c dr1 dr2 ecðr1 þr2 Þ Eð1Þ ¼< w00 ðr 1 ; u1 Þ : 2 jr2 r1 j p jr2 r1 j
ð16Þ
Now, we calculate above integration by using the following identity [45] 1 Z 1 X e2 imðu u Þ dqe 1 2 J m ðqr1 ÞJm ðqr 2 Þ; ¼ jr2 r1 j m¼1 0
ð17Þ
where Jm(qr) is Bessel function with order m. First, we carry out integration over angular variables u1 and u2. Then, we obtain the following result by using straightforward calculations,
DEð1Þ ¼ e2
rffiffiffiffiffiffi
cp 2
ð18Þ
:
When the first-order correction of the energy is added to the unperturbed result, one can obtain
Eð1Þ ¼ hX 4V 0 þ DEð1Þ ¼ hX 4V 0 þ e2
rffiffiffiffiffiffi
cp 2
:
ð19Þ
2.3. Variational procedure with different parameters As we know, sometimes, three exist systems whose Hamiltonian are known but they cannot be solved exactly. One of the approximation methods that is suitable for solving such problems is the variational procedure, which is also called the Rayleigh-Ritz method [46]. This method is useful for determining upper bound values for the eigenenergies of a system whose Hamiltonian is known. It is particularly useful for obtaining the ground state and it is quite cumbersome to calculate the energy levels of the excited states. In this part, we use variational method to calculate the ground state energy of a 2D quantum pseudodot under an external magnetic field. We can perform this problem by choosing a trial wave function that depends on a number of parameters. According to the variational method, after choosing a trial wave function, we should calculate the expectation value of the Hamiltonian with respect to this trial wave function. Finally, we should minimize the trial energy with respect to all parameters. We first consider a normalized one-parameter trial wave function as
k2
/k ðr 1 ; r 2 Þ ¼
p
eðk
2
=2Þðr 21 þr 22 Þ
ð20Þ
;
where k is a positive variational parameter. To obtain the ground state energy, we should evaluate the following integral
Ek ¼
Z
dr1
Z
dr2 /k ðr 1 ; r2 ÞHðr 1 ; r 2 Þ/k ðr1 ; r 2 Þ;
ð21Þ
where the Hamiltonian is given in Eq. (3). After calculating the integral, we have 2
Ek ¼
h k2
l
þ
lX2 2k2
2
rffiffiffiffi
þe k
p 2
þ ð8:633024714Þð2k2 V 0 r 20 Þ 4V 0 :
ð22Þ
It is seen that the ground state energy depends on the parameter k. By differentiating with respect to k, we obtain 2
dEk 2h k lX2 ¼ 3 þ e2 dk l k
rffiffiffiffi
p 2
þ ð8:633024714Þð4kV 0 r 20 Þ ¼ 0:
ð23Þ
We search for the real positive root of Eq. (23). This root is called k0 that minimizes the energy [Eq. (22)].
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Now, we consider a normalized two-parameter trial wave function as
k 2 g 2 2 2 /k;g ðr 1 ; r2 Þ ¼ pffiffiffiffi eðk =2Þr1 pffiffiffiffi eðg =2Þr2 ;
p
ð24Þ
p
where k and g are positive variational parameters. To find the ground state energy, we should calculate the expectation value of the Hamiltonian, Eq. (3) by using the above wave function. After performing the integrals, we can obtain
Ek;g ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k2 lX2 h g2 lX2 h p 2 2 2 2 þ 2 þ þ þ ð8:633024714ÞðV r Þðk þ g Þ þ e k g 4V 0 : 0 0 2l 2l 4g2 4k k2 þ g2
ð25Þ
As previously, by differentiating with respect to parameters k and g, we can obtain two relations. Then, we should solve simultaneously these equations. After searching for the real roots of the equations, we can obtain the ground state energy. 3. Results and discussion In this paper, we have considered a GaAs quantum pseudodot with parameters l = 0.067,e = 13.18, 2 and r0 = 89.58 nm [12]. It is worth mentioning that we have inserted c = 1 and e2 ! 4epe in our calculations. In first step, we have divided Eq. (19) by ⁄X. Therefore, we have
ð1Þ ðnÞ ¼
rffiffiffiffi Eð1Þ 4V 0 DEð1Þ 4V 0 p : ¼1 þ ¼1 þn hX hX hX hX 2
ð26Þ
2 pffiffi where (1)(n) is a dimensionless energy and n ¼ 4epehcX is a dimensionless interaction parameter that gauges the strength of the Coulomb correlation relative to the confining potential and magnetic field. Table 1 shows the ground state energy = hX for given values of V0 (or n) at zero magnetic field. The results in this table obtained from two different methods: first-order perturbation theory and variational procedure with one and two parameters. It is seen from the table that the energies calculated from the one-parameter variational method are higher than the first-order perturbation theory for all V0 (or n). According to the results from the table, we can deduce an interesting point. For 2D quantum dot with the parabolic potential, at zero magnetic field, the ground state energy obtained from firstorder perturbation theory is too high [31]. But, for our system (2D quantum pseudodot), the obtained ground state energy by using the first-order perturbation theory is not too high. In this table, the fourth and fifth columns correspond to one-parameter and two-parameter variational method. It is obvious that the results obtained from variational method are higher than the perturbation theory. We also have found that the ground state energy decreases when V0 increases.
Table 1 The ground state energy =hX of a 2D quantum pseudodot as a function of V0 (or dimensionless Coulomb coupling parameter n) at zero magnetic field. The results obtained from first-order perturbation theory (1)(n), one-parameter variational trial wave function (var. 1), and twoparameter variational trial wave function (var. 2). V0 (meV)
n
(1) (n)
(var. 1)
(var. 2)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
3.94 3.31 2.99 2.78 2.63 2.52 2.42 2.34 2.27 2.21
4.749564 3.472473 2.695042 2.116690 1.647081 1.246656 0.894453 0.577976 0.289149 0.022434
14.909176 13.632084 12.854654 12.276302 11.806693 11.406268 11.054065 10.737588 10.448761 10.182046
16.281255 15.234587 14.985621 14.462120 13.786492 13.516768 13.024865 12.895718 12.228963 11.925468
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R. Khordad / Superlattices and Microstructures 62 (2013) 166–174 Table 2 The same as Table 1, but for different magnetic field with V0 = 0.68346 meV. B (T)
n
(1)(n)
(var. 1)
(var. 2)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
2.43 2.41 2.35 2.26 2.16 2.06 1.96 1.87 1.78 1.71 1.64
0.949966 0.979342 1.057076 1.160100 1.267057 1.364495 1.447695 1.515936 1.570717 1.614156 1.648339
11.054065 11.101129 11.227350 11.399010 11.585019 11.760143 11.918940 12.057344 12.176229 12.277825 12.364662
14.002489 14.185879 14.52381 14.901895 15.189823 15.878458 16.099018 16.595784 16.978279 17.188425 17.786562
0.0115 0.0110 0.0105
λ
0.0100 0.0095 0.0090 0.0085 0.0080 0.0075 0.0
0.2
0.4
0.6
0.8
1.0
B (T) Fig. 1. Variational parameter (in one-parameter variational method) as a function of magnetic field for V0 = 0.68346 meV.
In Table 2, we have presented the ground state energy = hX for given values of magnetic field (or n) with V0 = 0.68346 meV. At similar to the 1, the results in this table obtained from two different methods: the first-order perturbation theory and variational procedure with one and two parameters. It is observed from the figure that the ground state energy (obtained from all procedures) increases when the magnetic field increases. The reason is as follow: when the magnetic field is applied to a system, an additional energy is added to the system. Therefore, the ground state energy is increases. As we see from the table, the energies calculated from the one-parameter variational method are higher than the first-order perturbation theory for all magnetic fields (or n). One can observe from the tables that the ground state energy by using the two-parameter variational method is higher than one-parameter variational and first-order perturbation. In Fig. 1, we have presented the variational parameter (in one-parameter variational method) as a function of magnetic field for V0 = 0.68346 meV. It is clear that the variational parameter increases when the magnetic field increases. It is obvious that by increasing the magnetic field, the parameter which minimizes the energy is increased. Fig. 2 shows the ground state energy as a function of magnetic field with r0 = 89.58 nm and V0 = 0.68346 meV. In this figure, we have presented our results obtained by the perturbation theory, variational method with one and two parameters. We observe from the figure that the ground state energy increases when the applied magnetic field increases.
R. Khordad / Superlattices and Microstructures 62 (2013) 166–174
173
32
First-order perturbation One-parameter variation Two-parameter variation
28 24
E (meV)
20 16 12 8 4 0
0.0
0.2
0.4
0.6
0.8
1.0
B (T) Fig. 2. The ground state energy as a function of magnetic field with r0 = 89.58 nm and V0 = 0.68346 meV.
5
First-order perturbation One-parameter variation Two-parameter variation
E (meV)
4
3
2
1
0 0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
V0 (meV) Fig. 3. The ground state energy as a function of V0 with B = 1 T and r0 = 89.58 nm.
In Fig. 3, we have displayed the ground state energy as a function of V0 with B = 1 T and r0 = 89.58 nm. This figure shows our results obtained by the perturbation theory, variational method with one and two parameters. By increasing V0, the potential is deeper and thereby the ground state energy decreases.
4. Conclusion In this work, we have studied a system of two interacting electrons in a quantum pseudodot under the influence of an external magnetic field. We tried to obtain electronic properties of ground state.
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For this purpose, we have calculated the ground state energy and wave function by using various methods. We have applied the first-order perturbation and variational method with one and two parameters. According to the results, we have found that the ground state energy obtained from first-order perturbation has lowest values. The results with variational method are higher than the perturbation theory. We also deduced an interesting point. Against 2D quantum dot, the results with perturbation theory are lower than variational method. According to the results obtained from the present work, it is deduced that the magnetic field and V0 play important roles in the ground state energy a system of two interacting electrons in a quantum pseudodot. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
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