Investigation of magnetic field effects on binding energies in spherical quantum dot with finite confinement potential

Accepted Manuscript Research paper Investigation of Magnetic Field Effects on Binding Energies in Spherical Quantum Dot with Finite Confinement Potential Bekir Çakır, Yusuf Yakar, Ayhan Özmen PII: DOI: Reference:

S0009-2614(17)30670-X http://dx.doi.org/10.1016/j.cplett.2017.06.064 CPLETT 34926

To appear in:

Chemical Physics Letters

Received Date: Revised Date: Accepted Date:

31 May 2017 29 June 2017 30 June 2017

Please cite this article as: B. Çakır, Y. Yakar, A. Özmen, Investigation of Magnetic Field Effects on Binding Energies in Spherical Quantum Dot with Finite Confinement Potential, Chemical Physics Letters (2017), doi: http:// dx.doi.org/10.1016/j.cplett.2017.06.064

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Investigation of Magnetic Field Effects on Binding Energies in Spherical Quantum Dot with Finite Confinement Potential Bekir Çakır 1 1 2

Yusuf Yakar2,a

and

Ayhan Özmen1

Physics Department, Faculty of Science, Selcuk University, Campus 42075, Konya-Turkey

Physics Department, Faculty of Arts and Science, Aksaray University, Campus 68100, Aksaray-Turkey

Abstract: The magnetic effects on the energy states and binding energies of the ground and higher excited states of the spherical quantum dot are studied theoretically for various potential depths. Also, Zeeman transition energies in the case of

=0, 1 are carried out.

The results show that the energy states and binding energies in small dot radii are insensitive to the increase of magnetic field. In the case of negative m, in the strong confinement region, the binding energy increases as the confinement potential decreases. In the case of positive m, the binding energy decreases with the decrease of the confinement potential. Key words: Spherical quantum dot, magnetic field effect, binding energy, QGA and HFR method, Zeeman splitting

1. Introduction During the past few years, among low dimensional semiconductors, quantum dots in which charge carriers are confined in all three directions have attracted much interest due to their potential device applications in semiconductor technology. Spatial confinement of the motion of electron in spherical quantum dots (SQDs) leads to the formation of discrete energy levels, like real atoms. This limitation causes the drastic change and dramatically affects electronic structures and optical properties of QDs. The other effect causing a change in electronic and optical properties is the presence of impurity. Impurity dramatically alters performance and a

Corresponding author E-mail address: [email protected] (Y.Yakar) [email protected] (B.Çakır) Tel: +90 3822882163: fax: +90 3822882125

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physical properties of quantum devices [1]. Because of this, a great number of theoretical studies related to the electronic structure [2-8], binding energy [9-11], optical [12-22], and other properties [23-28] have been performed by many authors, employing various methods and different dot sizes. Recently, the investigation of energy states of QDs inside external perturbations is of great interest for a better understanding of their properties. The application of external perturbations such as electric and magnetic field reveals precious information about the impurity energy states. For example, an external magnetic field introduces an additional confinement potential and also modifies the symmetry of the impurity states and the nature of the wave functions leading to more complicated changes of the binding energy. Thus, in the presence of magnetic field, the electronic and optical properties of SQDs have been extensively studied by many researchers. Among of these studies, Xiao [29] calculated the binding energy for the ground state of a hydrogenic off-center impurity in a magnetic field within the effective mass approximation. Corella-Madueño et.al [30] studied the ground state and binding energy of a SQD with on-center and off-center impurity in a magnetic field. The magnetic field effects on the energy states and the binding energies of the ground and the first excited states of a parabolic quantum-well wires within a uniform magnetic field were carried out by An and Liu [31] using the finite-difference method. For a SQD with infinite potential barrier, the effect of the magnetic field on the electronic structure was investigated theoretically by Wu and Wan [32] employing the exact diagonalization and variational methods. Similarly, El Ghazi et al. [33] and Yeşilgul et al. [34] studied the effects of magnetic field on the binding energy by employing a variational approach. Some authors [35-38] calculated the effects of the magnetic field on the optical transitions such as refractive index and absorption coefficients in various dots with different sizes and shapes. Niculescu and Bejan [39] investigated theoretically the effect of external magnetic fields on the non-linear

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optical process involving impurity states in a pyramidal QD with infinite potential depth. Mandal et al. [40] studied in detailed the effects of the electric and magnetic fields and the potential barrier on the optical absorption coefficients of a doped QD in presence and absence of noise. Using the compact density matrix approach Zhang et al. [41] investigated theoretically the effects of magnetic field on the optical properties in parabolic QD. In 2017, Zeeman splitting, Zeeman transition energies and nonlinear optical absorption coefficients of SQD with infinite potential barrier were performed in a detailed theoretical study by our group [42]. In all studies mentioned above, the authors have reported the magnetic field effects on the impurity energy states and the binding energies of the ground state (L=0) and the first excited state (L=1) of QDs with different size and shapes. Most of these studies have used the variational method and the trial wavefunction in their calculations. The theoretical investigation of the behaviors of the binding energies of the higher excited states with L=0, 1, 2 and 3 in SQD, GaAs/AlxGa1-xAs, in the presence of a magnetic field is the subject of this paper. We will be also investigated the effects of the confinement potential and the magnetic confinement on the binding energies of the excited states with positive and negative m cases. The spatial confinement, the magnetic field confinement, the confinement of the well-depth and the presence of impurity lead to considerable changes on the electronic energy states, binding energies and optical properties of QDs, which are of the potential applications in microelectronic and optoelectronic devices such as laser amplifiers, optical memories, single electron transistor, far-infrared photodetectors, light emitting diodes and high-speed electro optical modulators. To the best of our knowledge, for higher energy levels of QDs with finite potential depth, a detailed theoretical study on Zeeman transition energies and binding energies in the presence of external magnetic field has not been performed. Therefore, studies in this field are still important for both theoretical and practical applications of QDs.

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In the present study, we have extended our previous study [42] to the SQD with finite potential barrier. When the confinement potential with infinite depth in QDs is considered, a continuum energy threshold does not occur and excess electrons are always bound by the confinement potential. Thus, the dissociation processes cannot be defined. If the confinement potential with finite depth is taken into account, this problem can be solved. Since the potential confining the electrons always has the finite depth and range, it describes the real QD much better. In addition, the application of the finite confinement potential enables us to determine the quantum capacity of the dot, i.e., to give notice the number of electrons [43]. In this study, for a SQD, GaAs/AlxGa1-xAs, the wave functions and energy states including the linear Zeeman term have been obtained by employing a modified variational approach based on the Quantum Genetic Algorithm (QGA) and Hartree-Fock Roothaan (HFR) method, and the quadratic Zeeman term has been taken into account as a perturbation term. Both the perturbed and unperturbed energy levels and the binding energies have been calculated as a function of dot radius and magnetic field strength. 2. Theory In the effective mass approximation, the Hamiltonian of a confined system with finite potential depth in the presence of a magnetic field can be written as 2

H

where Z,

1  e  Ze 2 p  A   VC (r ) ,   c  4 r r 2m  

and

(1)

are the impurity charge, the effective mass of the electron and the

dielectric constant of material inside the QD, respectively. p is the linear momentum of the electron and

is the potential vector of the magnetic field. The term

is the finite

confinement potential defined by

.

(2)

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If the magnetic field B is applied in the z-direction

, the Hamiltonian in spherical

coordinates can be written as follows, in Hartree units, [1,42]

2 Z 1 1 H     Lz   2 r 2 sin 2 θ  VC (r ) , rr 2 8 2m

(3)

where the first and second terms are the electron kinetic energy and the coulomb potential energy, the third and fourth terms corresponding to the effect of magnetic field are paramagnetic and diamagnetic energies, respectively. Here,

is the z component of the

angular momentum operator of the electron. In this case the magnetic field preserves the azimuthal symmetry, i.e. m is still a “good” quantum number. The term

in Eq.(3) is a

dimensionless measure of the magnetic field strength and defined by effective Bohr radius. Since

,

is the

depends on both the effective mass and the electric

permittivity values, for a given value of the magnetic field intensity B, it may be very different from one semiconductor to another. The Hamiltonian given in Eq.(3) is difficult to resolve exactly because it involves two potentials [30]: One is the impurity attraction potential and the other is the magnetic potential. A perturbation method can be used to overcome this difficulty. In the solution of the Schrödinger equation, the normalized wave function can be written as a linear combination of Slater type orbitals (STOs),  , called basis functions, 



k 1

k 1

 p( 0) r   ( R  r ) c rpk R  k  kr  R , r  1  ( R  r )  c rpk R  k  kr  R , r , where (x) is the Heviside step function, k is the quantum numbers

(4)

of basis functions,

is the size of basis sets, c pk and  k are the expansion coefficients and the orbital exponents, respectively. In our calculations, we have employed the unnormalized complex STOs.

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For the SQD with finite potential barrier,

the unperturbed energy including the

paramagnetic term may be written as follows in terms of STOs

E0 







 c rpk R  k  kr  R , r  k 1



c



k 1

r R pk

 k 

r R k

2 Z 1    Lz  VC r   rr 2 2m



c k 1

 Z 1 ,r     Lz  VC r   rr 2 2m



2

rR pk



c k 1

 k  kr  R , r  r R pk

 k 

r R k

,r



,

(5)

and -the diamagnetic energy which is taken into account as perturbation energy is expressed as follows

E 



c k 1



rR pk

 k  kr  R , r    2 r 2 sin 2 



1 8







c k 1

1 c rpk R  k  kr  R , r   2 r 2 sin 2   8 k 1

rR pk



c k 1

 k  kr  R , r 

r R pk

 k  kr  R , r 

,

(6)

where, in the weak magnetic fields, the value of E 0 is very small when it is compared to E  . The external magnetic field removes the

degeneracy on energy levels. The splitting is

associated with what is called the orbital angular momentum quantum number (l), and the number of split levels in the presence of the magnetic field is 2l+1, that is, each level is split into 2l+1 terms. The paramagnetic term called linear Zeeman effect is proportional to the and it causes the splits of the energy levels. The diamagnetic term called Quadratic Zeeman effect is proportional to

and it causes a shift in the Zeeman energy levels. The electric

dipole transitions between Zeeman levels are referred to as Zeeman transitions. Spectral lines corresponding to these transitions show also polarization effects. Polarization has to do with the direction where the electromagnetic fields are vibrating. Since electric dipole operator has odd parity, the allowed transitions between levels must be corresponding to

. In this case, the transition

is called as π transition, in which electric dipole vector makes linear

oscillation along z-axis. In transitions

-

corresponding to Δm=+1(-1), the absorbed or

emitted light is circularly polarized and the dipole vector rotates at counter-clockwise

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(clockwise) direction in the xy plane about the z-axis. 3. Results and discussion We have calculated the Zeeman transition energies and the binding energies in the SQD, GaAs/AlxGa1-xAs, with finite potential depth under a uniform magnetic field. In calculations, the material parameters of GaAs for the well region and of AlxGa1-xAs for the barrier region are employed, where x is Aluminium concentration. The difference between the band gaps of





GaAs and AlxGa1-xAs is E g x   1.155x  0.37 x 2 eV, the effective dielectric constant is

 x   13.18  3.12 x  , the effective electron mass is mx   0.0665  0.0835x m0 , m0 is the free-electron mass. 60% of the band gap difference has been used to determine the magnitude of the confining potential. We have employed the effective Hartree energy and the effective Bohr radius a0*  4 r  2 m*e 2 as the units of the energy and the length for an electron. In quantum mechanical analysis, the STOs are preferred to investigate the electronic structure of QD, because they represent correct behaviour of the electronic wave functions. We have chosen a linear combination of s(or p, d, f) STOs having different screening parameters for a s(or p, d, f ) type atomic orbital. In order to maintain the orthogonality of orbital, we used the same set of screening parameters for all the one-electron spatial orbital with the same angular momentum and employed seven basis sets to calculate the energy expectation value. QGA procedure and HFR method are used to determine the expansion coefficients and the screening parameters of the wavefunction minimizing the total energy, whose details are given in Ref. [9] In Fig.1, we show the diamagnetic energy (or perturbation energy) for the 1s-, 2p-, 3d- and 4f- levels with m=

as a function of magnetic field for three different values of R.

1s Z=1 x=0.3

1.2 0.9

1.2 m=1

0.9

0.6

R=3 R=2

0.0 0.5

1.0

0.6

m=1

R=2

R=3

0.3

0.0

Diamagnetic energy (au)

(a)

1.5

(b)

2p Z=1 x=0.3

1.5

0.3

m=1

R=1

R=1

2.0 0.0

0.5

1.0

0.0 1.5

2.0

1.5 1.2 0.9

1.5 3d Z=1 x=0.3

0.6

(c)

m=2

R=3

4f Z=1 x=0.3

0.0 1.0

1.5

m=1 m=3 m=2

R=3

m=1

R=2

m=2 m=1

m=3 m=2 m=1

R=1

0.5

1.2

m=3 m=2

m=1

R=2

0.3

0.0

m=2 m=1

(d)

R=1

2.0 0.0

0.5

1.0

1.5

Diamagnetic energy (au)

1.5

0.9 0.6 0.3 0.0

Diamagnetic energy (au)

Diamagnetic energy (au)

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2.0

Fig.1 Diamagnetic energy versus the magnetic field for three different values of R=1, 2 and 3: (a) for 1s level, (b) for 2p level with m=1, (c) for 3d levels with m=1,2 and (d) for 4f levels with m=1,2,3. In all energy states, as the magnetic field increases, the diamagnetic energy increases monotonically. As it will be seen in energy curves, while the diamagnetic energy is relatively insensitive to the increase of magnetic field in small dot radii, this energy becomes more sensitive in large dot radius. This is a consequence of the competition between the magnetic field effect and the spatial confinement effect. That is, as the spatial confinement is more dominant in small dot radii, the confinement effect of the magnetic field becomes more dominant in large dot radii. As seen in (c) and (d), for a given R value, when the value of the magnetic quantum number m increases, the diamagnetic energy increases monotonically.

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This should be expected because the wave functions with larger m in the xy plane have a larger spatial extent resulting in larger diamagnetic interaction energies. For the 1s ground state, our result is in good agreement with the result obtained with the exact diagonalization and variational methods [32].

Energy states (au)

2.5

(a)

Z=1

1s 2p w ith m=+1 2p w ith m=0 2p w ith m= -1

2.0 1.5 1.0

x=0.4 =0.5

0.5 0.0 -0.5 -1.0 0

2

4

6

8

10

12

14

Transition energy(au)

Dot radius R 2.5

(b)

2p-1s 2.0

x=0.4

E for M= +1 E for M=0 E for M= -1

1.5 1.0 0.5 0.0 0

2

4

6

8

10

12

14

Dot radius R Fig. 2 Perturbed 1s and 2p Zeeman energy states in (a) and Zeeman transition energies in (b) versus dot radius.

We present the perturbed 1s and 2p energy states in Fig.2 (a) and show the Zeeman transition energies in the case of

in Fig.2 (b). As seen in 2p level, the magnetic

field removes the m degeneracy on non-s levels and these levels are split into sublevels due to

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the Zeeman effect. In the strong (R<1) and the intermediate ( 1
, because of the diamagnetic term.

In π polarization

transitions (Δm=0), since the magnetic quantum numbers are zero, the π polarization transition energies increase gradually as dot radius increases. Here since the paramagnetic term is zero, the small change is originated from the diamagnetic term. On the other hand, in large dot radii,

and

polarization transition energies change more quickly due to the

contributions coming from paramagnetic and diamagnetic energies. For

and

polarization transitions, while the contributions coming from the paramagnetic term are –γ and γ, the contributions coming from the diamagnetic term are of the same sign, that is, positive, and so the variation of

transitions according to dot radius is more stronger. Also,

the magnitudes of these transitions depend on the magnetic field strength. The diamagnetic energy (or quadratic Zeeman effect) is prominent in large magnetic fields and dot radii, that is, it behaves like a power law. Finally, it is worth noting that frequency of the light emitted in the Zeeman transitions can change by adjusting applied magnetic field strength. Similar behaviors are seen in Zeeman transitions between the perturbed 3d and 2p levels.

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Binding energy(au)

5 (a)

x=0.2 x=0.3 x=0.4

4 3

1s =0.5

2 1 0 0

1

2

3

4

5

Dot radius R

2.5

2p =0.5

(b)

2.0

(bi)

x=0.2, m=1 x=0.3, m=1 x=0.4, m=1

1.5

2.5 2.0

x=0.2, m=-1 x=0.3, m=-1 x=0.4, m=-1

1.0

1.5 1.0

0.5

0.5

0.0

0.0 0

1

2

3

4 0

1

Dot radius R

Binding energy (au)

Binding energy (au)

3.0 2p =0.5

3.0 2.5 2.0 1.0

4

3.0 2.5 2.0 1.5 1.0 0.5 0.0

(ci)

3d =0.5

x=0.2, m=-2 x=0.3, m=-2 x=0.4, m=-2

x=0.2, m=2 x=0.3, m=2 x=0.4, m=2

1.5

3

Dot radius R

(c)

3d =0.5

2

0.5 0.0 0

1

2 Dot radius R

3

4 0

1

2 Dot radius R

3

4

Binding energy (au)

Binding energy (au)

3.0

12

4f =0.5

2.5 2.0

(d)

1.0

2.5

x=0.2, m=-3 x=0.3, m=-3 x=0.4, m=-3

x=0.2, m=3 x=0.3, m=3 x=0.4, m=3

1.5

3.0

(di)

4f =0.5

2.0 1.5 1.0

0.5

0.5

0.0

0.0 0

1

2 Dot radius R

3

4 0

1

2

3

Binding energy (au)

Binding energy (au)

3.0

4

Dot radius R

Fig. 3 Binding energies of the perturbed 1s, 2p, 3d and 4f levels versus dot radius for three values of x=0.2, 0.3 and 0.4: (a) for 1s level, (b) and (bi) for 2p levels with m= 1, (c) and (ci) for 3d levels with m= 2, (d) and (di) for 4f levels with m= 3. Binding energy of impurity is defined as the energy of the system without the impurity minus the energy of the system with the impurity, that is,

.

The binding energy defined in this way is a positive quantity. The curves in Fig.3 display the variation of the binding energies of the perturbed 1s, 2p, 3d and 4f levels with m=0,

as a function of dot radius for three different values of x. For a given level,

as the dot radius increases, the binding energy first increases until it reaches a maximum value at a certain dot radius, which is the critical dot radius (Rc), and then decreases monotonically from a larger value. When the dot radius increases further to large values, the binding energy approaches its bulk value. In small R regions, since the electron is more strongly localized inside the QD, the Coulomb interaction between the electron and impurity is significantly raised and so the binding energy increases. As it will be seen in Fig.3 (b), (c) and (d), in the case of the positive m, the increase of the potential well-depth enhances the binding energy, and also shifts the maximum value of the binding energy to lower Rc values of dot radius. This indicates that the increase of potential depth results in more stability of the system at lower dot radii. While m is negative, this case is reversed. As seen in (bi), (ci) and (di), we

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have observed a successive displacement of the curve peak position in the case of negative m. This case in the binding energies can be expressed by classical approximation as follows: While the spatial confinement is more dominant in low energy orbitals, the confinement of the magnetic field becomes more dominant in higher energy orbitals. When a static magnetic field is applied on the confined system, the rotating speed of the electron is different in the positive and negative values of m. While the speed of the electron in the first case increases, it decreases in the second case. Because of the conservation of the angular momentum, contrary to the speed of the electron, the average radius of the electron shrinks or grows. Since the binding energy is related to Ze/r, in small R, the contribution coming from the diamagnetic term to the binding energy will be different in positive and negative m cases. In the spatial confinement, all levels are pushed into the dot with the increase of x, and so the average radius of the electron reduces. On the other hand, in the case of positive m, while the confinement effect of the magnetic field increases the average radius of the electron, this effect in the case of negative m reduces the average radius of the electron. As a result, since the spatial confinement becomes more dominant on the s-orbitals, the binding energy increases with the increase of x and so the Rc decreases. In the case of positive m, while the contributions coming from the spatial and magnetic confinements to the binding energy are of the same sign, they are of different sign in the case of negative m. Contrary to s- and porbitals, since d- and f-orbitals are of higher energy, the magnetic confinement is more important there. In the case of positive m, the energy increases and R c decreases as the x increases. However, this case is reversed at the energy states with negative m, like the binding energy of p-orbital. In addition, it is worth pointing out that the peak positions of the binding energies decrease while going to the higher excited states, as expected. In order to investigate the influence of magnetic field on the binding energy, in Fig. 4, we plot the variation of the binding energies of the ground and higher excited states with positive

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and negative m as a function of the dot radius for three values of the magnetic field

0.2,

0.5 and 2. As seen in Fig.4, in small dot radii R<1, the binding energy is relatively insensitive to the magnetic field due to the strong spatial confinement and is identical to the zero magnetic field case. As the dot radius increases, the binding energy increases until it reaches a maximum value and it decreases again monotonically. In this region, the main contribution to the binding energy is the spatial confinement energy of the electron. In the intermediate confinement region, the binding energy curves tend to deviate from each other due to the diamagnetic term and continue to decrease with the increase of dot radius. After reaching a certain minimum value, the binding energies begin to increase again while the dot radius is further increased in the weak confinement region, R>3. The physical reason of this is that the increase of the magnetic field in large dot radii shrinks the electron wave function and decreases the cyclotron radius for the electron. Thus, the magnetic confinement compels the electron to move “closer” to the core of impurity, and so the average distance between the electron and impurity decreases and the binding energy increases again. In Fig.4, an important feature is that, as seen in the binding energy curves, the peak position of the binding energy decreases and shifts to larger dot radius while going up excited states. For a fixed l level, in the strong confinement region, the other important feature is that the peak position of the binding energy is bigger in the case of negative m than that of positive m. The reason for this is explained above.

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3 2

(a)

(ai)

4

1s x=0.3

1s x=0.3

3 2

1

1

0

0 0

1

2

4 0

3

1

2

3

4

Dot radius R

Dot radius R

3.0

2.5

(b)

2.0

2p m=1 x=0.3

1.5

=0.2 =0.5 =2

1.0

(bi)

2.5 2.0

2p m=-1 x=0.3

1.5 1.0

0.5

0.5

0.0

0.0 0

1

2 3 Dot radius R

4

0

1

2 3 Dot radius R

4

3.0

Binding energy (au)

Binding energy(au)

Binding energy (au)

3.0

3.0 =0.2 =0.5 2

2.5 2.0 1.5 1.0

=0.2 =0.5 =2

(c) 3d m=2 x=0.3

(ci)

2.5 2.0

3d m=-2 x=0.3

1.5 1.0

0.5

0.5

0.0

0.0 0

1

2 Dot radius R

3

4 0

1

2 Dot radius R

3

4

Binding energy (au)

Binding energy (au)

=0.2 =0.5 g=2

4

Binding energy (au)

5

5

16

3.0 (di)

(d)

2.5 2.0

2.0

4f m=3 x=0.3

1.5 1.0

2.5

4f m=-3 x=0.3

1.5 1.0

0.5

0.5

0.0

0.0 0

1

2

3

4 0

1

2

3

Binding energy (au)

Binding energy (au)

3.0

4

Dot radius R Dot radius R Fig. 4 Binding energies of the perturbed 1s, 2p, 3d and 4f levels versus dot radius for three values of

=0.2, 0.5 and 2: (a) for 1s level, (b) and (bi) for 2p levels with m= 1, (c) and (ci)

for 3d levels with m= 2, (d) and (di) for 4f levels with m= 3.

Conclusions In this study, within the effective mass approximation, we have calculated the energy states and the wave functions of a spherical quantum dot with finite potential depth inside a uniform magnetic field, employing a modified variational approach method. In addition, we have investigated the perturbed binding energy of the ground and higher excited states. In calculations, the diamagnetic energy has been considered as perturbation term. The results show that, in the intermediate and weak confinement regions, the diamagnetic energy in all states increases with the increase of magnetic field. In R<1, the electron spatial confinement prevails over the magnetic field confinement and Zeeman splitting breaking the degeneracy on the magnetic quantum number m disappears in this region. On the other hand, it is seen that, in weak confinement region,

(

polarization transition energy shifts toward higher

(lower) energy as the magnetic field increases. Thus, the frequency of the emitted light in Zeeman transitions can be changed by setting the strength of applied magnetic field.

In the

weak spatial confinement regions, the increase of magnetic field increases the binding energy.

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It is found that, the binding energy is relatively insensitive to the magnetic field in the strong spatial confinement region. However, in the weak spatial confinement region, for a given dot radius, the binding energies increase with the increase of magnetic field due to the perturbation energy. In addition, the peak position of the binding energy changes according to whether m is positive or negative. In the strong spatial confinement region R

in the case

of negative m, the binding energy increases as the confinement potential decreases. Whereas in the same region, in the case of positive m, the binding energy decreases with the decrease of the confinement potential. In addition, the effect of the confinement potential (or the welldepth) shifts the maximum binding energies to lower values of dot radius. Theoretical investigation of the effect of the magnetic field on energy states of spherical QD will lead to a better understanding of the properties of low dimensional structures. Such theoretical studies may have profound consequences about practical applications of the electrooptical devices, and the results of this study will contribute to the research on related subjects. Acknowledgements This work has been supported by Research Fund of Aksaray University- Project Number: 2017-028. And also, it has been partially supported by Selçuk University BAP office. Referances [1] Z. Xiao, J. Zhu, F. He, J. Appl. Phys. 79 (1999) 9181 [2] H. E. Montgomery Jr, I. P. Vilademir, Phys. Lett A 377(2013) 2880. [3] N. Aquino, J. Garza, A. Flores-Riveros, J. F. Rivas-Silva, K. D. Sen, J. Chem. Phys. 124 (2006) 054311 [4] Y. Yakar, B. Çakır, A. Özmen, Int. J. Quant. Chem. 111 (2011) 4139; Y. Yakar, B. Çakır, A. Özmen, Philosophical Magazines 95 (2015) 311.

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3.0 2.5 2.0

x=0.2, m=-2 x=0.3, m=-2 x=0.4, m=-2

x=0.2, m=2 x=0.3, m=2 x=0.4, m=2

1.5 1.0

3.0 2.5 2.0 1.5 1.0 0.5 0.0

(ci)

3d =0.5

(c)

3d =0.5

0.5 0.0 0

1

2 Dot radius R

3

4 0

1

2 Dot radius R

3

4

Binding energy (au)

Binding energy (au)

22

23 Highlights

Zeeman transitions in the spherical quantum dot with finite potential depth are calculated. The effects of the magnetic field on binding energy are investigated. Paramagnetic and diamagnetic energies are carried out as a function of dot radius and magnetic field. The unperturbed wavefunctions and energy states are obtained from QGA and HFR method.