Dipole and quadrupole polarizabilities and oscillator strengths of spherical quantum dot

Chemical Physics 513 (2018) 213–220

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Dipole and quadrupole polarizabilities and oscillator strengths of spherical quantum dot ⁎

T

⁎

Yusuf Yakara, , Bekir Çakırb, , Ayhan Özmenb a b

Physics Department, Faculty of Arts and Sciences, Aksaray University Campus, 68100 Aksaray, Turkey Physics Department, Faculty of Sciences, Selcuk University Campus, 42031 Konya, Turkey

A R T I C LE I N FO

A B S T R A C T

Keywords: Static polarizability Quadrupole polarizability Dipole and quadrupole oscillator strengths Spherical quantum dot QGA and HFR method

In this study, the energy eigenvalues and eigenfunctions of the ground and excited states of a spherical quantum dot are calculated by using the Quantum Genetic algorithm (QGA) and Hartree-Fock Roothaan (HFR) method. Based on the calculated energies and wave functions, the static and dynamic dipole polarizabilities, the quadrupole polarizability, dipole and quadrupole oscillator strengths of spherical quantum dot are carried out as a function of dot size and the confining potential as perturbative. The results show that dot size and confining potential have a great influence on the polarizability and oscillator strength. It is found that the polarizability increases due to the spatial confinement effect in the strong confinement region. In the weak confinement region, the polarizability increases again until it reaches the saturation value. In addition, the peak positions of the dipole and quadrupole oscillator strengths shift toward smaller dot radii with the increases of the potential well depth.

1. Introduction In the past decade, the systems confined by different types of external potential have attracted much interest due to their potential device applications in semiconductor technology. Such systems are widely proposed to model a variety of problems in physics and chemistry. In the confined systems, the charge carriers (electrons and holes) have been restricted in one-, two- and three- dimensions by the external potential. When the motion of carriers is confined in all dimensions, such structures are referred to as zero-dimension nanostructures (or quantum dots, QDs). The spatial restriction causes drastic changes in their electronic and optical properties such as impurity energy, polarizability, localization of electrons, shell filling, photon absorption and ionization, etc. Therefore, in the last decade, a great number of theoretical studies on electronic structure and binding energies [1–9], optical properties [10–20], electric and magnetic field effects [21–27] and other properties [28–31] of QDs have been made employing various methods and different dot sizes. A simple but interesting example of a confined system is a hydrogen atom inside a spherical box as proposed by Michels et al. [32], who used an impenetrable cavity to simulate the effect of pressure on the static polarizability of hydrogen atom. The study of hydrogen atom at high pressures is one of the key problems in modern physics and astrophysical studies [33]. As is well known, external perturbations such as electric or magnetic field can provide much ⁎

valuable information about the confined systems. The application of an electric field gives rise to both a polarization of the carrier distribution and an energy shift (stark effect) in the quantum states. These effects lead to important changes in the energy spectra of the carriers. The polarizability describes the lowest-order the distortion of the electron cloud in the presence of an external uniform electric field. Therefore, a number of authors have been studying the dipole polarizability in QDs with different shape, size and the confinement potential using different methods. Dutt et al. [33] calculated 1 s static dipole polarizability (SDP) of the confined hydrogen atom by using the wave function obtained from the variational method. Montgomery and Sen [34] investigated the SDP and the dynamic dipole polarizability (DDP) for the ground state of hydrogenic impurity confined at the center of a spherical box with penetrable walls. Cohen et al. [35] used the variational-perturbation approach to obtain the SDP and DDP for the ground and few excited states of the confined hydrogen atom inside a spherical box with impenetrable walls. Das [36] solved the Scrödinger equation for hydrogenic system for two different plasma potential and investigated dipole and quadrupole polarizability of hydrogen like ions. Sen et al. [37] reported the variation of 1 s static dipole and quadrupole polarizability of hydrogen atom confined in a spherical box, with decreasing box radius and they found the same behavior for dipole and quadrupole polarizability. The first and second-order stark effects on various energy states, 1 s SDP and DDP in a spherical QD with infinite confining

Corresponding authors. E-mail addresses: [email protected] (Y. Yakar), [email protected] (B. Çakır).

https://doi.org/10.1016/j.chemphys.2018.07.049 Received 17 April 2018; Accepted 29 July 2018 Available online 31 July 2018 0301-0104/ © 2018 Elsevier B.V. All rights reserved.

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Y. Yakar et al. σ

potential were investigated as a function of dot radius using several formulas in a detailed study by our group [38]. The magnetic and confinemnt potential effects on the binding energy and the polarizability were investigated by Zounoubi et al. [39] for a shallow hydrogenic impurity placed at the center of rectangular and square quantum well wire, in which shallow impurity is called impurity requiring little energy- typically around the thermal energy or less- to ionize. In cylindrically confined hydrogen atom with on-center and off-center impurity, the 1 s SDP and DDP were computed by Ndenque et al. [40] using variational method. For excited states, Lumb et al. [41] solved the Schrödinger equation numerically to investigate the optical properties like oscillator strength and polarizability of the confined hydrogen atom with Gaussian Potential. In very recently, Ghosh et al. [42] investigated the various effects of spatially-varying effective mass, spatially-varying dielectric constant and anisotropy on the polarizability and dipole moment of doped QD. All of the studies mentioned above on calculation of the static and dynamic polarizability have been so far reported only for 1 s ground state of the confined hydrogenic impurity, except for [35,41]. To the best of our knowledge, there exists very little work concerning the SDP, DDP and SQP for the higher excited states of the spherical QD with finite confining potential. In the present study, our previous study [38] is extended to the spherical QD with finite confining potential. Since the potential confining electron always has the finite depth and range, the confinement potential of finite depth moreover much better describes the real QD. On the other hand, the application of the finite confinement potential allows us to determine the quantum ‘capacity’ of the QD [43]. The energies and the wave functions of the spherical QD are calculated by using QGA and HFR method, in which the wave function is constructed from Slater type basis functions. In addition, we have investigated the optical properties such as SDP, DDP, dipole and quadrupole oscillator strengths as a function of dot radius and confining potential. We have made our calculations specifically not only for the ground state but also for electronic excited states.

k=1 σ

+ (1−Θ(R−r ))

k=1

Pz(l) =

(l) Pxy =

2m∗ | 〈Mi(→l) f 〉rxy< R + 〈Mi(→l) f 〉rxy> R |2 (Ef −Ei ), ħ2

(1)

αz(l) = α‖(l) =

(6)

P (l)

∑ (E −zE )2 f

f >i

i

(7)

where Ef and Ei show final and initial energy of the states. On the other hand, if the applied radiation is polarized in the xy-plane, the orthogonal components of the polarizability along the x- and y-directions are written as [40]

(2a)

α⊥(l) = (α⊥(lx) + α⊥(ly) )/2

(8)

and

and in the case of infinite confining potential

α⊥(lx) = (2b)

P (l)

∑ (E −xE )2 f >i

f

i

(9)

A similar equation can be written for y-component when we replace x by y. The total polarizability is then written as [40,45]

where R is dot radius and V0 is the potential well depth, V0 > 0. The time-independent Schrödinger equation of the system can be written as

Hψnlm = Eψnlm ,

(5)

where l = 1,2 for dipole and quadrupole cases. Dipole transitions depend on dipole moment operator (α r) and they are allowed only between the states satisfying the selection rules Δl = ± 1 and ΔM = m−m 0 = 0 and ± 1. Quadrupole transitions are proportional to the quadrupole moment operator (α r2) and their values are only available for a small number of transition [45]. The quadrupole transitions satisfy the selection rule Δl = 0, ± 2 (the transition 0 → 0 forbidden) and ΔM = m−m 0 = 0 and ± 1. The polarizability uses the definition derived from the second order energy correction of the perturbation theory [46]. When a confined system in the state i is subjected to a time–independent electric field –static case- in the z-direction, the parallel component of the 2l -pole polarizability can be expressed in terms of oscillator strength as follows [41]

where m , K, εr and Z are the effective mass of electron, the electrical constant, the dielectric constant of the medium and impurity charge, respectively. The term VC(r) is the confining potential. In the case of finite confining potential, it has the form as follows:

0, r > R VC (r ) = ⎧ , ⎨ ⎩∞ , r ⩾ R

2m∗ | 〈Mi(→l) f 〉rz < R + 〈Mi(→l) f 〉rz > R |2 (Ef −Ei ). ħ2

Here, 〈Mi → f 〉 = 〈ψf |r (l) cos (l) θ| ψi 〉(l) is 2l -pole transition matrix element. On the other hand, if the applied radiation is polarized in the xy plane, 2l -pole oscillator strength can be expressed as follows

*

− V0, r > R VC (r ) = ⎧ , ⎨ ⎩ 0, r ⩾ R

(4)

(l)

In the effective mass approximation, the Hamiltonian of a hydrogenic impurity located in the center of a spherical QD can be written as

ħ2∇2 KZe 2 − + Vc (r ), 2m∗ εr r

∑ cpkr> R χk (ξkr > R, r )

in which Θ(x ) is the Heaviside step function, cpk and ξk show the expansion coefficients and the orbital exponents, respectively. In our calculations, we used the unnormalized complex STOs. The strength of optical transitions due to interaction of spherical QD with the applied radiation is commonly expressed in terms of a dimensionless quantity called oscillator strength. The oscillator strength is a physical quantity of practical importance in the study of the optical property. The oscillator strengths govern the intensity of the transitions observable in line spectra, and hence they are fundamental quantities in spectroscopy. This term determines the intensity of a specific spectral line in atomic spectrum [44] and also offers additional information on the fine structure. If the applied radiation is chosen in z-direction, 2lpole oscillator strengths for transitions from an initial state |i〉 to a final state |f 〉 is given by [21],

2. Theory and definitions

H=−

∑ cpkr< R χk (ξkr < R, r )

ψp (r ) = Θ(R−r )

α (l) = (α‖(l) + 2α⊥(l))/3

(10)

In the presence of an oscillating electric field, the other important quantity is the DDP, which describes the distortion of the electron and charge distribution of a system. The DDP is directly related to the parameters such as the Van der Waals constants, the frequency dependence of the refractive index, the dynamic dipole shielding factor, the Rayleigh scattering cross sections and the mean excitation energies

(3)

where nlm are the atomic quantum numbers. In HFR approach, the spatial part of the normalized one-electron wave function, ψ, can be expressed as linear combination of Slater type orbitals (STOs), χ, called basis functions, 214

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[47]. When the confined system is exposed to a radiation field with angular frequency ω , in the z-direction, the parallel component of the dynamic polarizability is written by [34]

αz(l) (ω) =

∑ f >i

Pz(l) (Ef −Ei )2−ω2

3. Results and discussion We have calculated the energies and the wave functions of 1s, 2p, 3d, 2s, 4f, 3p and 3s electronic states of a hydrogenic impurity located at the center of a spherical cavity with finite and infinite confining potential using QGA and HFR method. Also, the SDP and SQP for the 1s ground and higher excited states are computed as a function of dot radius and confining potential. The impurity energy states are labelled by the symbol Enℓ, in which n and ℓ are the principal quantum numbers and orbital angular momentum quantum numbers. The energy states are in m degeneracy just as a free–space hydrogen atom. We let the reduced mass and the dielectric constant be the same with a free-space hydrogen to make our results more conceivable, that is, the effective Bohr radius a∗ is equal to Bohr radius (0.529 Å) and the effective Rydberg energy R y∗ is equal to Rydberg energy (13.6 eV). In order to investigate the effects of the impurity charge Z and the confining potential VC on the energy states, the energies of 2 l and 3 l levels are displayed in Fig. 1 as a function of the dot radius at two different values of Z and VC . As it will be seen from these figures, for each n state, as the dot radius decreases, the energy of the state increases monotonically. In the case of finite confining potential, for R = 0.1 and Vc = −5eV , we have calculated the energy values as following: E1S = −0.5002 au, E2S = −0.1250 au, E3S = −0.0539 au, E2p = −0.1250 au, E3p = −0.0549 au, E3d = −0.0556 au, E4f = −0.0313 au. On the other hand, when dot radius is very large, for example, for R = 17, we have obtained the energy values: E1S = −0.6837 au, E2S = −0.3087 au, E3S = −0.2271 au, E2p = −0.3087 au, E3p = −0.2322 au, E3d = −0.2367 au, E4f = −0.2001 au. The results are in good agreement with the results obtained by Yang et al.[9], which are E2S = −0.1250 au and E2p = −0.1250 au for R = 0.1, and E2S = −0.3087 au and E2p = −0.3087 au for R = 17. In here, the values in Ref. [9] have been converted to atomic units (au) using the multiplier: 1/(27.2116). According to the results, it can be said that the energy of each n state approaches the corresponding energies of free hydrogen atom, that is,

(11)

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7

Z2Ry∗

-0.1

a

2p 2s

Z=1 Vc = -5eV 2s

Z=2

2p

2p

Vc= -5eV

-0.3 2p

-0.4

2s

Vc= -10eV

-0.5

0 2 4 6 8 10 12 14 16 18 20 Dot radius R 0.0

b 3d

3p

3s

Z=1

-0.2 3d 3p

3s

Energy states(au)

Energy states(au)

2s

-0.6

0.0

-0.3

c

Z=1

-0.2

0 2 4 6 8 10 12 14 16 18 20 Dot radius R

-0.1

Z2Ry∗

En ≅ − 2 when R → 0 , and En ≅ − 2 −V0 when R→ very large [9]. n n On the other hand, in the case of infinite confining potential, we have calculated the energy values: for R = 1, E1S = 2.3742 au, E2S =

Energy states(au)

Energy states(au)

here for x- and y-components of the dynamic polarizability, similar equations can be written. As can be clearly seen in Eq. (11), when the frequency is close to the energy difference, Ef −Ei , the dynamic polarizability shows singularities for any electronic state f. This is particularly important in cavities where the symmetry breaking due to confinement leads to very close- nearly degenerate- atomic levels [46]. In Eqs. (7), (9) and (11), when l = 1, for 1s SDP and 1s DDP, the summation is over in all the bound and continuum states. The computation of the SDP and DDP using the sum-over-states method requires accurate evaluation of wave functions and energies of the system. The accurate evaluation of the integrals over all continuum states is very complex for the present numerical capability [34]. The major contribution to the 1s state dipole polarizability comes from overlapping of 1s ground state with the first few excited states [46]. Therefore, in our study, this sum was terminated after two cases including 1s → 2p and 1s → 3p dipole transitions. QGA procedure and HFR method are used to determine the expansion coefficients and the screening parameters of the wave function minimizing the total energy, the details of which are given in Ref. [48]. We have constructed one–electron wave functions from a linear combination of STOs. STOs represent the correct behavior of the real wave functions around an impurity. Having this correct behaviour is especially important in the regions very close to or far from the impurity. Therefore, 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 have used the same set of screening parameters for the wave function with the same angular momentum and employed seven basis functions to calculate the energy’s expectation value.

Vc = -5eV

-0.4

Z=2

-0.5 0 2 4 6 8 10 12 14 16 18 20 Dot radius R

d

Z=1

-0.1 3d

3p 3s

Vc= -5eV

-0.2 -0.3

3d 3p

3s

-0.4 -0.5

Vc= -10eV

0 2 4 6 8 10 12 14 16 18 20 Dot radius R

Fig. 1. Energies of n = 2 and 3 levels of spherical QD as a function of dot radius: for Z = 1 and 2 in (a, b) and for Vc = −5eV and −10 eV in (c, d).

215

Chemical Physics 513 (2018) 213–220

a

4 1s-Dz

3

Vc = -5eV Vc = -10eV

Z=1

1

Vc = infinity

0 2

4

6 8 10 Dot radius R

12

2p-Dz

b

80

Vc = -5eV

40

Vc = -10eV Vc = infinity

0 0

5

10 15 Dot radius R

20

3d-Dz 'M=0

400

c

300 200

Vc = -5eV

100

Vc = -10eV Vc =infinity

0 0

5

10

15

20

25

1.0

'M=+1

0.5

Z=1

30

Vc = -5eV Vc = -10eV Vc =infinity

2

4

6 8 10 Dot radius R

12

14

80 2p-Dx

60

e)

(l=1)

'M=+1

40

Vc = -5eV Vc = -10eV

20

Vc =infinity

Z=1

0 0

5

800 700 600 500 400 300 200 100 0

10 15 Dot radius R

3d-Dx

20

25

(l=1)

f)

'M=+1

Vc = -5eV Vc = -10eV

Z=1

0

Dot radius R

(l=1)

0.0

25

500 (l=1)

1s-Dx

1.5

0

(l=1)

'M=0

120

d)

2.0

14

200 160

2.5

Static polarizability (au)

Static polarizability (au)

(l=1)

'M=0

2

0

Static polarizability (au)

Static polarizability (au)

5

Static polarizability (au)

Static polarizability (au)

Y. Yakar et al.

5

10 15 20 Dot radius R

Vc =infinity

25

30

Fig. 2. Parallel components in (a, b, c) and orthogonal components α⊥(lx= 1) in (d, e, f) of static dipole polarizabilities for 1s, 2p and 3d levels as a function of dot radius for three different values of the confining potential VC = −5 eV, −10 eV and infinity.

αz(l = 1)

itself at the outer part of the impurity. The margin of the confinement potential first pushes the small l state. Thus, the energy of electron is affected earlier than the big l state. In (a, b), it is seen that as the impurity charge increases, the non-degenerate range decreases. This is because the electron cloud is pulled toward the impurity, that is, Coulomb interaction energy increases. On the other hand, in (c, d), as the depth of the confinement potential increases, the non-degenerate range does not change. But, the interval between energy levels increases, that is, the splitting increases with increasing VC . This is due to the fact that electron wave functions remain inside the well with the increase of the well depth. For 1s → np, 2p → 3d and 3d → 4f dipole matrix transitions, in the case of ΔM = 0 , the parallel component αz(l = 1) of the SDP is displayed in Fig. 2(a–c), and the orthogonal component α⊥(lx= 1) is illustrated in Fig. 2(d–f) in the cases of ΔM = +1 as a function of dot radius. In these figures, we have investigated the effects of potential well depth and dot radius on the polarizability. The dominant contribution to the 1 s polarizability comes from the 1s → 2p transition, which makes this contribution to the polarizability large, and the 1 s polarizability is not significantly modified by the appearance of new p states in additional shells [46]. Similarly, it can be said that the major contribution to the

16.6051 au, E3S = 41.0263 au, E2p = 8.2258 au, E3p = 27.6937 au, E3d = 14.9774 au, E4f = 22.9105 au. For R = 10, the energies are E1S = −0.5 au, E2S = −0.1107 au, E3S = 0.0879 au, E2p = −0.1189 au, E3p = 0.0504 au, E3d = −7.0806e−3 au, E4f = 0.0891 au. In the case of infinite confining potential, our results are in good agreement with the obtained results [5], which are E1S = 2.3739 au, E2S = 16.5703 au, E3S = 40.8637 au, E2p = 8.2231 au, E3p = 27.4740 au, E3d = 14.9674 au, E4f = 22.8958 au for R = 1 and E1S = −0.5 au, E2S = −0.1128 au, E3S = 0.0886 au, E2p = −0.1189 au, E3p = 0.0492 au, E3d = −0.0071 au, E4f = 0.0882 au for R = 10. From these results, in the case of infinite confining potential, as the dot radius is very large, it can be said that the energy of each n state approaches the corresponding energy of free hydrogen atom. In Fig. 1, for each n state, the impurity energies are in l degeneracy as the dot radius is at these two extreme situations, but the l degeneracy disappears when dot radius is between two extreme situations. The presence of the confining potential splits the angular momentum sublevel degeneracy. For a given R and VC , for a fixed n, the sublevel with the largest l has the lowest energy. As seen in energy curves, the energy of the state with small l increases more than the state with big l. The reason for this behavior is that when the confined electron is in small l state, it mostly distributes 216

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a

D z(l=1)Z

R=5

1s

'M=0

2p 3d

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Dynamic polarizability (au)

600 400 200 0 -200 -400 -600 -800 -1000 -1200

weak spatial confinement region (R ≥ 3), the external field on electron prevails over the spatial confinement, and the polarizability is more sensitive to the applied field until a certain dot radius. In the weak confinement region, the electron moves away from the impurity due to the influence of the external field, and so the polarizability increases. This situation continues until a certain dot radius. At this point the polarizability reaches a saturation value. On the other hand, we have seen the reduction of polarizability as the depth of potential well increases. This is because the electron wave function is more localized inside the QD and the polarizability reduces. As seen in (b) and (c), the 2p and 3d polarizability is maximum in the strong and intermediate regions. This is because 2p and 3d orbitals are farther away from the impurity according to 1s orbital. The electron wave function escapes out of the dot toward the material barrier, but the electron cannot go out of the well. Here the energy of the system remains constant until a certain dot radius, as seen in energy curves in Fig. 1, and so the polarizability of the system is also maximum. In (c), the dot radius increases, the polarizability starts to reduce and reaches a minimum value, and then increases again, like in 1s polarizability. It is seen that the 2p and 3d polarizabilities are larger than that of 1s, as expected. In (b) and (c), it is worth to note that, in the case of infinite confining potential, the polarizability is larger than finite case due to the wave function remaining inside the well. When we compare (a, b, c) to (d, e, f), similar behaviours are obtained for the orthogonal component of the SDP. However, it is worth to note that the magnitudes of the polarizability in the case of (a, b, c) are larger than those of (d, e, f). This is originated from the interaction between the applied field and the dipole operator. For the 1s static polarizability, similar behaviours have been reported by Zounoubi et al. [39] and Lumb [41] for the quantum well wire and the confined hydrogen atom with depth of Gaussian potential. Fig. 3 shows the parallel αz(l = 1) (ω) and orthogonal α⊥(lx= 1) (ω) components of the 1s, 2p and 3d DDP as a function of photon energy at R = 5 and 8, the the frequency step interval has been taken 0.01 au. The jumps (singularities) appearing on the figures define the frequencies corresponding the energy difference in Eq. (11). It is important to emphasize that the peak positions and the amplitudes of the DDP change continuously according to dot radius. As can be seen from Eq. (11), the DDP increases with the increase of ω up to a singularity, which is expressed at ωfi = Ef −Ei , and the presence of this pole leads to the sign inversion of the polarizability. When we compare (a, b) to (c, d), 300 200 100 0 -100 -200 -300 -400 -500 -600 -700

Dynamic polarizability (au)

Dynamic polarizability (au)

2p and 3d polarizabilities comes from the 2p → 3d and 3d → 4f transitions. In all figures, we have noticed three effects on the polarizability: the dot size effect, the confinement potential effect and the effect of the field direction. First two effects tend to reduce the spatial extension of the wave function and they change the polarizability of the system. The third effect has a change on the magnitude of the polarizability, as seen in (d, e, f). In the case of finite confining potential, the static polarizability largely depends on dot radius R. As will be seen in (a), when R > 7, the 1s polarizability is somewhat reduced. After reaching a constant value, it does not change anymore. For the 1 s-αz(l = 1) , for R = 7, we have calculated the polarizability values 4.32 au in the finite case and 4.35 au in the infinite case. Our results agree with the results reported in Ref. [33], in which they are calculated 4.34 au for 1s exact polarizability at R = 7. On the other hand, as the dot radius decreases, the polarizability starts to decrease and continues to decline until reaching to a certain minimum value. After this point, in the case of infinite confining potential, the polarizability goes to zero, whereas it increases rapidly again in the case of finite confining potential and reaches a maximum value in the strong confinement region, R < 1. The effect of the spatial confinement in very small dot radii is more dominant than the confinement potential. In the strong confinement region, as the dot radius decreases, the electron wavefunction becomes very localized, and so the polarizability decreases. If the dot radius is further reduced (or dot radius becomes very small), the wave function escapes out of the dot toward the material barrier, and the polarizability starts to increase rapidly again and becomes maximum. Also, the electron energy becomes maximum according to the Heisenberg principle. In the strong confinement region, R ≤ 1, since the electronic orbital is very localized, the charge distribution becomes less sensitive to the applied field. Therefore, in this region, the effect of the spatial confinement is more dominant than the effect of the applied field on the polarizability. In the intermediate confinement region (1 < R < 3), the effect of spatial confinement on electron mixes with the effect of the external field, the polarizability begins to become sensitive to the external field, and the polarizability starts to increase. As seen in polarizability curves, 1s polarizability reaches a minimum value at the critical dot radius, R ∼ 1.9. It can be said that this critical dot radius is the point where the electron is bound to the impurity, in which the total energy of the electron becomes negative. That is, the electron bonds to the impurity and so the 1s polarizability becomes minimum. In the

1400 1200 1000 800 600 400 200 0 -200 -400 -600

c

Dynamic polarizability (au)

b

D

Z

(l=1) z

R=8 'M=0

R=5

1s

'M=+1

2p 3d

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Photon energy hQ 3000 2500 2000 1500 1000 500 0 -500 -1000 -1500

D x(l=1)Z

Photon energy hQ

200 100 0 -100 1s -200 -300 0.2 0.3 0.4 0.5 0.6

1s 2p 3d

d

D x(l=1)Z

R=8 'M=+1

1s 2p 3d

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Photon energy hQ

Photon energy hQ

Fig. 3. Parallel components αz(l = 1) (ω) in (a, b) and orthogonal components α⊥(lx= 1) (ω) in (d, e) of dynamic dipole polarizabilities for 1 s, 2p and 3d levels as a function of photon energy hυ atVc = −5eV . 217

Chemical Physics 513 (2018) 213–220

5

of this behaviour is that, after a certain well depth, the electron wave function is pushed completely into the well and the polarizability reaches the maximum saturation point and then remains constant. In (a), for 1s-αz(l = 1) at R = 1 and 2, when the depth of VC decreases, the 1spolarizability increases monotonically and reaches a maximum value. This is because the effect of the spatial confinement is stronger than the well effect. As a result of this, the wave function escapes out of the dot toward the potential barrier, and so the value of 1s polarization increases. However, in the weak confinement regions, for example R = 5 and 8, the effect of external field starts to become dominant, the situation is reversed. In these regions, as the dot radius increases, 1s polarizability increases until a certain value of the VC and then remains constant. In (b), for 2p-αz(l = 1) , when the VC becomes large enough, the polarizability goes to constant values due to reaching saturation, like in (a). As the VC decreases, the polarizability starts to increase in all dot radii. As seen in polarizability curves, the smaller dot radius’ polarizability starts to increase earlier than the bigger dot radius. In order to see how the dot radius and the confining potential affect the quadrupole polarizability, in Fig. 5, we show the 1 s- αz(l = 2) and 2pαz(l = 2) as a function of dot radius in (a, b) and as a function of the VC in (c, d). Here we have considered the 1s → 3d and 2p → 4f quadrupole transition matrix elements in the case of Δl = +2 and ΔM = 0 . As seen in (a, b), in small dot radii, the SQP decreases with the increase in dot radius and reaches a minimum value at critical dot radius. Afterwards, in the weak confinement regions, as the dot radius increases, the SQP increases rapidly due to the field effect and reaches a maximum value and then decreases monotonically with the increase of dot radius. As seen in (c, d), the behaviors of the SQP resemble the behaviors of the SDP as the VC increases as negative. When we compare (c,d) in Fig. 5 to (a, b) in Fig. 4, in small values of VC, there are small discrepancies l = 1) between SDP and SQP. The reason is the variation of 〈Mi(→ f 〉z and

a 4 (l=1)

1s-D z

3

R=1

'L=1 'M=0

2

R=2 R=5 R=8

1 0 0

-5

-10

-15

-20

-25

Static polarizability (au)

Vc 140 120 100 80 60 40 20 0

b

(l=1)

2p-D z

R=1 R=2

'L=1 'M=0

0

-5

-10

R=5 R=8

-15

-20

-25

Vc Fig. 4. Parallel components αz(l = 1) of static dipole polarizabilities for 1 s and 2p levels as a function of confining potential Vc at R = 1, 2, 5 and 8.

l = 2) 〈Mi(→ f 〉z matrix elements. There is no previous data for comparison with our calculations for the SQP. To show the effect of the dot radius and the confining potential on the oscillator strength, in Fig. 6, we show the dipole Pz(l = 1) and the quadrupole Pz(l = 2) oscillator strengths as a function of dot radius. The effects of the confining potential and dot radius are clearly seen on the dipole and quadrupole oscillator strengths. As seen from the oscillator curves, in the case of finite confining potential, both dipole and

Quadrupole polarizability (au)

Quadrupole polarizability (au)

Quadrupole polarizability (au)

the amplitudes of the DDP in the case of ΔM = +1 are smaller than those of ΔM = 0 , like in SDP. In Fig. 4, we display the parallel components αz(l = 1) of the 1s and 2p SDP as a function of the confining potential Vc for four different values of R. In (a), it is clearly seen that 1s SDP does not change in great values of the VC , that is, the effect of the VC is very weak on the SDP. The reason

5 a

(l=2) 1s -D z

4

Vc=-5eV

3

Vc=-10eV

2

Vc=infinity

'L=2 'M=0

1 0 0

5

1600 1400 1200 1000 800 600 400 200 0

10 15 20 Dot radius R

(l=2) 2p-D z

Vc=-10eV

0

5

Fig. 5. Parallel component

αz(l = 2)

30

b

Vc=-5eV Vc=infinity

25

'L=2 'M=0

10 15 20 Dot radius R

25

30

Quadrupole polarizability (au)

Static polarizability (au)

Y. Yakar et al.

5

c

4 3

(l=2)

1s -D z

2

R=1 R=2

'L=2 'M=0

1

R=5 R=8

0 0

-5

-10

-15

-20

-25

Vc 750

d

600

(l=2)

2p-D z 'L=2 'M=0

450 300

R=1 R=2 R=5

150

R=8

0 0

-5

-10

-15

-20

-25

Vc

of static quadrupole polarizability in the case of ΔL = +2 as a function of R in (a–b) and Vc in (c–d). 218

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1.2

3.0 (a)

(au)

V c =-5ev

'M=0

V c =-10ev

(l=2)

1s-2p

V c =inf inity

0.6

Pz

(au) (l=1)

Pz

0.8

(d)

2.5

1.0

0.4

'M=0

1.5

V c=-5eV

1.0

V c=-10eV

0.5

V c=infinity

0.0

0.2 0

5

10

15

20

25

0

30

5

10 15 20 Dot radius R

Dot radius R

30

(au) (l=2)

V c =-10ev

2.0

V c =inf inity

1.5 1.0

2p-3d

0.5

'M=0

V c=-5eV

80

V c =-5ev

Pz

(au)

(e)

(b)

2.5 (l=1)

25

100

3.0

Pz

1s-3d

2.0

60 40

2p-4f

V c=-10eV

'M=0

V c=infinity

20 0

0.0 0

5

10

15

20

25

0

30

5

10

15

20

25

30

Dot radius R

Dot radius R

25 (c)

Pz

(l=1)

(au)

20 3d-4f 'M=0

V c =-5ev

15

V c =-10ev V c =inf inity

10 5 0 0

5

10

15

20

25

30

Dot radius R Fig. 6. Dipole oscillator strengths Pz(l = 1) in (a, b, c) for the transitions 1s-2p, 2p-3d and 3d-4f and the quadrupole oscillator strengths Pz(l = 2) in (d, e) for the transitions 1s-3d and 2p-4f as a function of dot radius for three different values of the confining potential VC = −5 eV, −10 eV and infinity.

others, and, as a consequence, the polarizability increases until a certain dot radius. 1s polarizability becomes minimum when the electron is bound to the impurity. In addition, the polarizability decreases when the potential well depth increases. On the other hand, it has been found that the dipole and quadrupole oscillator strengths strongly depend on the dot radius and the limiting potential. Theoretical investigation of the dipole and quadrupole polarizability 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.

quadrupole oscillator strengths exhibit similar behaviors. That is, as the dot radius increases, first, the oscillator strength decreases until a certain dot radius. After this point, it starts to increase until reaching a maximum value and then they decrease again monotonically as the dot radius is further increased. On the other hand, the VC has the influence on the oscillator strengths. As seen in (a, b, c), as the well depth increases, the peak positions of the oscillator strengths shift toward smaller dot radius.

4. Conclusion We have calculated the eigenenergies and the wave functions of the hydrogenic impurity confined at the center of a spherical box with finite depth. By using the calculated energies and the wave functions, we have investigated the parallel and orthogonal components of the dipole and quadrupole polarizability as perturbative. In addition, the dipole and quadrupole oscillator strengths are calculated as a function of dot radius and the confining potential. It is seen that the dot size, the confinement potential and impurity charge have a great effect on the electronic impurity energies. For the polarizability, the results present that, in the strong confinement region, the electron spatial confinement is dominant on the polarizability and prevails over the confinement of external field and the confinement of potential well. They compute with each other in the medium confinement region. As in the weak confinement region, the field confinement is more dominant than the

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