Computation of the oscillator strength and absorption coefficients for the intersubband transitions of the spherical quantum dot

Optics Communications 282 (2009) 3999–4004

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Computation of the oscillator strength and absorption coefficients for the intersubband transitions of the spherical quantum dot Ayhan Özmen a, Yusuf Yakar b,*, Bekir Çakır a,*, Ülfet Atav a a b

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

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 27 May 2009 Accepted 22 June 2009

The electronic structure and optical properties of one-electron Quantum Dot (QD) with and without an on-center impurity were investigated by assuming a spherically symmetric confining potential of finite depth. The energy eigenvalues and the state functions of QD were calculated by using a combination of Quantum Genetic Algorithm (QGA) and Hartree–Fock Roothan (HFR) method. We have calculated the binding energy for the states 1s,1p,1d,1f, oscillator strengths, the linear and third-order nonlinear optical absorption coefficients as a function of the incident photon energy and incident optical intensity for the 1s–1p, 1p–1d and 1d–1f transitions. The existence of the impurity has great influence on the optical absorption spectra and the oscillator strengths. Also we found that the magnitudes of the total absorption coefficients of the spherical QD increase for transitions between higher states. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: Spherical quantum dot Binding energies Oscillator strength Linear and third-order nonlinear absorption coefficients QGA and HFR method

1. Introduction The semiconductor structures have found various application areas as electronic and electrooptical devices such as infrared photo detectors, light emitting diodes, single-electron transistors and quantum computers. Some authors have been focused on the electronic structure and some other physical properties of the zero-dimensional quantum dot, which is also called an artificial atom, because it exhibits the atomic properties like discrete energy levels and shell structures. Therefore, many authors have studied to the electronic structure, the ground and excited energy states, the binding energy, the relativistic effects etc. of the spherical QDs by using various methods such as perturbation [1], variational [2–5], exact solution [6,7], QGA [8–10] and a combination of QGA and HFR method [11,12]. In low-dimensional system, the physical properties of the spherical QD such as the dipole transition, the oscillator strength, the linear optical absorption coefficient and the photoionization cross section are studied by many authors using various computational techniques [13–18]. Besides, Wang and Guo [19] and Zhang et al. [20] investigated the linear and third-order nonlinear optical absorption coefficients in disk-like and one-dimensional parabolic QDs. In very recently, Xie [21–23] calculated the linear and thirdorder nonlinear optical absorption coefficients for a hydrogenic do-

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

(Y.

Yakar),

[email protected]

0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.06.043

nor in the spherical parabolic QD. Karabulut and Baskoutas [24] for a case of spherical QD with parabolic confining potential, Huang and Libin [25] for a parabolic QD with two-electrons, Chen et al. [26] for an asymmetric double triangular quantum well, Yuan [27] for an off-center hydrogenic donor confined by a spherical QD with a parabolic potential performed the linear and third-order nonlinear optical absorption coefficients. All studies mentioned above are focused on the calculation of the transitions between the lowest energy states, 1s–1p transition for the ground (L = 0) and the first excited (L = 1) states of the QD. The QGA method has been used investigation of electronic structure and other physical properties of QDs. The QGA method is a version of the Genetic Algorithm (GA) method. The QGA method is based on the principle of energy minimization just like in variational method. The conventional linear variational method used in atomic and molecular structure calculations assumes the nonlinear parameters known as a priori and determines only linear expansion coefficients, whereas in the QGA method both the linear and nonlinear parameters can be determined from the principle of energy minimization. We combined the QGA procedure and HFR method to determine the parameters cpk and fk in Eq. (5) to minimizing the total energy over Slater Type Orbitals (STOs). In the present paper, we calculated the ground and excited energy states and the binding energy of the spherical QD with and without impurity, assuming a spherical symmetric confining potential of finite depth. Moreover, we performed the optical dipole transition, the oscillator strength, the linear and third-order nonlinear optical absorption coefficients for the 1s–1p, 1p–1d and

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1d–1f transitions in a spherical QD with and without impurity. On the other hand, the linear and third-order nonlinear optical absorption coefficients for all transitions investigated as a function of the incident photon energy and dot radius.

is the kinetic energy,

2. Theory and formulations

is the coulomb energy between the electron and the impurity, and

hV i i ¼

Z

R

0

    Z 1 Z Z ð/irR Þ  r>R /ir>R ds; e r e r R ð9Þ

We consider a shallow hydrogenic impurity located at the center of a spherical QD confined by a finite spherical potential well with radius R. The Hamiltonian of this system can be written in the effective mass approximation as

H¼

r2 2m



Z

er r

þ VðrÞ;

ð1Þ

where Z is impurity charge, r is the distance between the electron to the impurity, m* is the effective mass of the electron and er denotes the real part of the relative dielectric constant of the medium. The confinement potential is assumed in a form of the spherical potential well, i.e. V(r) = 0 for r < R and V(r) = V0 for r > R. The time independent Schrödinger equation for this system is given by

Hwi ¼ Ei wi ;

ð2Þ

in which Ei is the energy and wi is the wave function for ith eigenstate of the system. The wavefunctions wi are orthonormalized, and the wavefunction and the probability flux should be continues at the interface between GaAs and AlGaAs (r = R), i.e

Z 0

R

ðwir
sþ

and

Z R

1

ðwr>R Þ wjr>R d i

s ¼ dij

r¼R

 1 dwir>R  ¼   mr>R dr 

ð3Þ

:

r¼R

Ei ¼ h/i jHj/i i=h/i j/i i:

ð4Þ

Within the HFR approach the wavefunction /i are written as lineal combinations of STOs, called basis set. We have used two different basis sets for inner and outer parts of the QD. As a result our wave function is of the form

/i ¼ HðR  rÞ/rR i i rr
r
k¼1

rr>R X

r>R

r>R cik vk ðfik ;~rÞ;

k¼1

ð5Þ r
r>R

where H(x) is the Heviside step function, r (r ) is the size of the basis set used for the inner (outer) part of the wave function, rR r>R cr
vnk ‘k mk ðfk ; rh/Þ ¼ rnk 1 efk r Y ‘k mk ðh; /Þ;

ð6Þ

where nk ; ‘k ; mk are the quantum numbers of basis functions and Y‘m(h,u) are well-known complex spherical harmonics in Condon– Shortley Phase convention. The energies of the system is given by

D E Ei ¼ hT i i þ hV i i þ V conf ; i where

hT i i ¼

Z 0

R

ð/ir
ð/ir>R Þ /ir>R ds;

ð10Þ

is the energy due to the confining potential. When the wave function given in Eq. (5) is substituted in Eqs. (8)–(10), these integrals are obtained in terms of the expansion coefficients and new integrals over STO’s. The integrals over STO’s have been extensively studied long before by many authors for the calculation of atomic system properties [28–30]. For QDs, the integrals given in Eqs. (8)–(10) can be easily evaluated by modifying the expressions for atomic systems by appropriate consideration of the boundaries. r>R and the screening constants The expansion coefficients crR , f are determined by using the QGA and HFR method from fr
Z R Z hM z ifi ¼ e ð/r
The eigenstates of the system can also be determined from the variational principle by minimizing the energy of the system corresponding to the trial wave function /i

¼ HðR  rÞ

R

1

R

1

 ð/r>R Þ r cos h/r>R ds : f i ð11Þ

  1 dwirR  ¼ wi r¼R   mr
  wr
Z D E V conf ¼ V0 i

r2 2mr
ð7Þ ! /r
Z

1 R

ð/r>R Þ  i

r2 2mr>R

! /r>R ds; i ð8Þ

The matrix element is important for the calculation of different optical properties of the system related to electronic transitions. In spherical QD dipole transitions are allowed only between states satisfying the selection rules D‘ = ±1, where ‘ is the angular momentum quantum number. The oscillator strength is a very important physical quantity in the study of the optical properties which are related to the electronic dipole-allowed transitions, and it is a dimensionless quantity. The oscillator strength Pfi is expressed as follows

Pfi ¼

2m h

2

 2 DEfi Mfi  ;

ð12Þ

where DEfi = Ef  Ei denotes difference of the energy between lower and upper states. The oscillator strength can offer additional information on the fine structure and selection rules of the optical absorption [31]. Photoabsorption process may be defined as an optical (intersubband) transition in low dimensional quantum mechanical systems. The photoabsorption occurs from a lower state to an upper state with absorbing a photon. The absorption computations for intersubband are based on Fermi’s golden rule, and the total optical absorption coefficient is given [19,20,25]

aðx;IÞ ¼ a1 ðxÞþ a3 ðx;IÞ

8 9  2  4 rffiffiffiffiffiffiffiffiffi> > < =     M q  h C 2I M q h C l h o o fi fi i ¼x ; h i  2> 2  2 2 er e0 > 2 : Efi hx þ ðhCo Þ nr eo c Efi hx þ ðhCo Þ ; ð13Þ

where the term a1(x) and a3(x, I) denote the linear and third-order nonlinear optical absorption coefficients, I is the incident optical intensity,  hx is the incident photon energy, l is the permeability of the system defined as l = 1/e0c2, the q is the electron density pffiffiffiffi in the QD, C0 is the relaxion ratio for states f and i, nr ¼ er represents the refraction index of the semiconductor, c is the speed of light in vacuum and e0 is the electrical permittivity of the vacuum, respectively.

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Since the third-order nonlinear optical absorption coefficient

24

a3(x, I) is negative and is proportional to the incident optical intensity I, the total absorption coefficient a(x,I) is decreases as I increases. Therefore, a(x,I) is reduced by one-half when I reaches a

21

4. Results and discussion In the production of quantum nanostructures GaAs and AlGaAs are extensively used for the well and barrier regions respectively. Also the physical parameters of these materials are well known. Therefore, we have used these material parameters in this study and assumed that the confining potential is a spherical well of finite depth. The atomic units (a.u.) have been used in the determination of electronic energies and wave functions, in which the electron charge e, the Planck constant  h, the electron mass m0 and 4pe0 are assumed as unit. The effective Rydberg energy 2 2 h =mr
eV) -2

Energy (10

STOs are preferred in the quantum mechanical analysis of the electronic structure of a QD as they represent correct behaviour of the electronic wavefunctions. Therefore, we have chosen a linear combination of s (or p, d) STOs having different screening parameters for a s (or p, d) type atomic orbital. To maintain the orthogonality of orbital the same set of screening parameters was used for all the one-electron spatial orbital with the same angular momentum, and took five basis sets (r = 5) to calculate the expectation value of the energy. The QGA method used in our calculation is based on three basic genetic operations: these are reproduction, crossover and mutation. The method starts up with an initial random population of possible solutions of the problem. A fitness value is assigned to each individual in the population. In the reproduction process the individuals of current population are copied to the next generation, according to their fitness values. Therefore, in performing reproduction process, a selection procedure is necessary to choose the individuals to be copied to the next generation. In the crossover operation two individuals randomly selected from the present are combined to obtain two new individuals of the next generation population. Another operation in QGA, the mutation process plays an important role in getting out of local minima and is implemented at lower probabilities than other operations. In this process, the genetic information is changed randomly. Here we have given only the outline of the procedure. The details of the QGA used in this study are given in our previous work [11].

15 12 9 6 3 0 0

1

2

3

4

5

6

*

Rdot(a ) Fig. 1. The ground and excited state energies of the QD with (with solid circles) and without (without solid circles) the impurity as a function of dot radius.

with impurity lies under the 1s energy state of the case without impurity about R > 2.6a* because of the impurity attractive coulomb potential. In the same way, the 1d and 1f energy states of the QD with impurity lie under the 1p and 1d energy state of the case without impurity about 3.8a* and 5a*, respectively. This is because that the electrons become less confined when the dot radius increases and their kinetic energies therefore decrease. For the case with impurity the contribution of the coulomb interaction becomes dominant for large dot radii. The effect of the attractive coulomb potential of the impurity on the total energy can be more clearly seen in Fig. 2, which the difference between the total energies of the cases with and without impurity is given. This difference can be regarded as the binding of the electron to the impurity. This binding energy increases with decreasing dot radius. At some critical radius it reaches to a maximum value and then it rapidly decreases with decreasing dot radius. For smaller dot radii the contribution of confinement becomes dominant resulting in a rapid decrease of the binding energy, eventually forcing the electron to be unbound. This results are in good agreement with the previous studies [2,6,18]. The oscillator strength is a useful parameter to obtain information on electronic structure and optical properties of a QD. The

1s 1p 1d 1f

50

40

-3

3. Computational technique

18

Binding energy (10 eV)

value, called the saturation intensity.

1s 1p 1d 1f

30

20

10

0 0

1

2

3

4

5

Rdot(a* ) Fig. 2. Binding energy of the spherical GaAs/AlGaAs QD as a function of the dot radius.

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In Fig. 4 we show the linear a1(x), third-order nonlinear a3(x, I) and total a(x,I) optical absorption coefficients for transitions between various states of the QD with and without impurity as a function of the incident photon energy h  x. As seen from Fig. 4, the linear absorption coefficient a1(x) is positive, whereas the third-order nonlinear optical absorption coefficient a3(x, I) is negative. So the total optical absorption coefficient a (x,I) is significantly reduced by the nonlinear contribution. Therefore, the contributions of both the linear and third-order nonlinear absorption terms should be considered in calculation of absorption spectrum of the QDs especially for those operating under high incident optical intensity I. Also we can see from Fig. 4 that the strengths of both the linear and nonlinear terms are larger for transitions between higher levels. The total optical absorption coefficients a(x,I) of the QD for 1s– 1p, 1p–1d and 1d–1f transitions are displayed as a function of the incident photon energy  hx for six different values of the incident optical intensity in Fig. 5. It can be seen from this figure that the maximum of absorption coefficient corresponds to the threshold hx. This maximum value decreases with photon energy, i.e. Efi   increasing incident optical intensity I, and the absorption is strongly bleached at sufficiently high-incident optical intensities. Therefore, the nonlinear optical absorption coefficient a3(x, I) should be taken into account when the incident optical intensity I is comparatively strong, which can reduce total absorption coefficient. When the incident optical intensity I exceeds a critical value Ic, the nonlinear term causes a collapse at the center of the total absorption peak splitting it into two peaks. It is clearly seen that this critical intensity Ic is smaller for transitions between higher states while the intensity of the total absorption spectra increases for the transitions between higher states. For 1s–1p, 1p–1d and 1d–1f transitions we display the total optical absorption spectra of the spherical QD as a function of the photon energy  hx at three different dot radii in Fig. 6. The total optical absorption coefficients for small dot radii are much stronger than that for large dot radii since the absorption spectrum depends on the QD volume. Also it is seen obviously from Fig. 6 that existence of the impurity causes a shift on absorption spectra

1.8 1.6

Oscilator strength

1.4 1.2 1.0 0.8 0.6 0.4

1s-1p

0.2

1p-1d

1d-1f

0.0 0

5

10

15

20

25

Rdot (a* ) Fig. 3. The ground and excited state energies of the QD with and without the impurity (line with and without solid circles respectively) as a function of dot radius.

oscillator strength involves two parameters, One of them is the dipole matrix element and the other is difference between energy states. The matrix element is very small although the energy differences are very high in small QDs. The oscillator strengths for various transitions are shown in Fig. 3. In the absence of an impurity the oscillator strength rapidly increases with dot radius and then reach to a limit value for all transitions. When there is an impurity, the rapid increase of the oscillator strengths for small dot radii is similar to that of without impurity. However, for the larger dot radii the oscillator strength goes through a maximum and then it decreases to a value corresponding to that of free hydrogen. The effect of the impurity on the oscillator strengths is larger for transitions between lower states and it decreases as the levels go up. For the 1s–1p transitions, our results are good in agreement with the previous studies [16,18].

15 Z=0

1p-1d

10

1d-1f

1s-1p

0

9

Absorption coefficient α (10 /m)

5

-5 -10 15 Z=1

10

1d-1f

1p-1d 1s-1p

5 0 -5

Lineer Nonlineer Total

-10 6. 5

7 .0

7. 5

8 .0

8. 5

9 .0

9. 5

10. 0

10. 5

11. 0

Photon energy (R y* ) Fig. 4. The linear (dashed), third-order nonlinear (dashed–dotted) and total optical (solid) absorption coefficients of the QD with and without impurity versus the incident photon energy  hx for I = 10 MW/m2 and R = 1a*.

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A. Özmen et al. / Optics Communications 282 (2009) 3999–4004

10 1d-1f

Z=0 1p-1d

8 1s-1p

9

Total absorption coefficient α (10 /m)

6 4 2 0

1d-1f

Z=1

8

1p-1d 2

1s-1p

I=20 MW/m 2 I=30 MW/m 2 I=38 MW/m 2 I=45 MW/m 2 I=55 MW/m 2 I=65 MW/m

6

4

2

0 6.5

7.0

7.5

8.0

8.5

9.0

9.5

10. 0

10.5

11.0

Photon energy (R*y) Fig. 5. The total optical absorption coefficients of the QD with and without impurity versus the incident photon energy  hx for six different values of the incident optical intensity I and R = 1a*.

14

R=0.9 a * R=1 a

9

Total absorption coefficient α (10 /m)

12

R=1.2 a

10

1d-1f

*

Z=0 1d-1f

*

1s-1p 1p-1d

8

1s-1p

6 4

1p-1d

1d-1f

1p-1d 1s-1p

2 0 14

1d-1f Z=1

1p-1d

12

1d-1f 1s-1p

10

1p-1d

8

1s-1p 1d-1f

6

1p-1d 1s-1p

4 2 0 5

6

7

8

9

10

11

12

13

*

Photon eneregy(R y) Fig. 6. The total optical absorption coefficients of 1s–1p, 1p–1d and 1d–1f transitions in the QD with and without impurity versus the incident photon energy  hx for three dot radii and I = 10 MW/m2.

towards higher energies (blue shift) for all transitions and dot radii. This blue shift is more enhanced in the transitions between lower states where the electron is more localized near the impurity. On the other hand, the intensity of the total absorption spectra increases for the transitions between higher levels. This is because that the electronic dipolar transition matrix elements are very high for the transitions between higher levels. Optical absorption coefficients obtained for the 1s–1p case are qualitatively consistent

with the results for the spherical QD with parabolic confinement potential in Refs. [19,20,22,23,25]. 5. Conclusion We calculated the energies and wave functions of the spherical QD with and without impurity for the ground and the excited states. The binding energies of the electron to the impurity were

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A. Özmen et al. / Optics Communications 282 (2009) 3999–4004

also calculated. Moreover, we computed the oscillator strength Pfi, the linear a1(x), the third-order nonlinear a3(x, I) and total a (x,I) optical absorption coefficients for GaAs/AlGaAs spherical QD. For 1s–1p transition these physical properties were investigated for various structures in the literature, in this study we have also considered the transitions between higher levels. The results show that the binding energies increase as the dot radius decease reaching to a maximum value as dot radius decreases and then rapidly decrease with decreasing dot radius. The impurity has great influence on the oscillator strength. As for the optical absorption coefficients, the linear optical absorption coefficient is not related to the incident optical intensity, whereas the incident optical intensity I have great influence on the nonlinear optical absorption coefficient. Moreover, when the incident optical intensity I increases the total optical absorption coefficient decreases. Also it has been observed that existence of the impurity causes a blue shift on the absorption spectra for all the transitions, especially for 1s–1p. The magnitudes of the linear, nonlinear and total absorption coefficients increase for transitions between higher levels. The theoretical studies may have profound consequences about practical application of the electrooptical devices, and the optical absorption saturation also has extensive application in the optical communication.

Acknowledgement This work is partially supported by Selçuk University BAP office.

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