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Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential
Optics Communications 283 (2010) 1795–1800
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Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential Yusuf Yakar a,*, Bekir Çakır b,*, Ayhan Özmen b a b
Physics Department, Faculty of Science and Letters, Aksaray University, Campus 68100 Aksaray, Turkey Physics Department, Faculty of Science, Selçuk University, Campus 42031 Konya, Turkey
a r t i c l e
i n f o
Article history: Received 4 September 2009 Received in revised form 26 October 2009 Accepted 11 December 2009
Keywords: Spherical quantum dot Binding energy Linear and nonlinear optical absorption coefficient Parabolic potential
a b s t r a c t In the effective mass approximation, we calculated the binding energy and wave function for the 1s-, 1p-, 1d- and 1f-states of a spherical quantum dot (QD) with parabolic potential by using a combination of quantum genetic algorithm (QGA) and Hartree–Fock–Roothaan (HFR) method. In addition, we also investigated the linear and the third-order nonlinear optical absorption coefficients as a function of the incident photon energy for the 1s–1p, 1p–1d and 1d–1f transitions. Our results are shown that the existence of impurity has great influence on optical absorption coefficients. Moreover, the optical absorption coefficients are strongly affected by the incident optical intensity, relaxation time, parabolic potential and dot radius. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Thanks to modern technology, semiconductor structures with three-dimensional confinement of electrons, called QDs, have been fabricated by using various techniques such as molecular beam epitaxy and etching [1]. These structures display an interesting behavior and play an important role in microelectronic and optoelectronic devices so that they can affect the electrical, optical and transport properties. Therefore, the electronic structure, the energy states, the binding energy, the relativistic effects, the optics properties and other physical properties of QDs with finite and infinite confinement potential have been studied in literature. Most of the theoretical studies in literature use variational approach [2– 13]. Other approximate methods such as perturbation [14], exact solution [15–18], QGA method [19–22], a combination of QGA and Hartree–Fock–Roothaan (HFR) [23,24] and others [25–27] have also been developed. When employing the confinement potential of infinite depth, a continuum-energy threshold does not exist and the excess electrons are always bound by the confinement potential, i.e., possess only discrete energy levels. Thus, binding and dissociation processes cannot be described. The problem can be solved if we consider the confinement potential of finite depth, which much * Corresponding authors. Tel.: +90 382 280 12 35; fax: +90 382 280 12 46 (Y. Yakar). E-mail addresses: [email protected] (Y. Yakar), [email protected] (B. Çakır). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.12.027
better describes the real structure of the QD, since the potential which confines the electrons always possesses the finite depth and range [28]. In literature, the dipole transition, the oscillator strength, the linear optical absorption coefficient and the photoionization cross-section of QDs have been investigated by various authors [9,10,29–31]. A parabolic potential is often considered to be a good representation of the potential in semiconductor structures. Recently, there has been a considerable interest in the linear and nonlinear optical absorption coefficients and refraction index change based on an optical (intersubband) transitions in parabolic QDs [32–40]. In 2009, Xie [41] carried out research on the linear and nonlinear optical properties of an exciton in a spherical parabolic QD by using the matrix diagonalization technique. Chen et al. [42], Yuan et al. [43] and Yao et al. [44] investigated these absorption coefficients for an asymmetric double triangular quantum well, for an off-center hydrogenic donor confined by a spherical QD with a parabolic potential and for a cylindrical QD system, respectively. All studies mentioned above are focused on the calculation of the optical transition between the lowest energy states, which is 1s–1p transitions for the ground (L = 0) and the first excited (L = 1) states of the QD. In a previous study [45], we computed the energies of the ground and excited states and the binding energy of a spherical QD with and without impurity. In addition, we also investigated the oscillator strength and, the linear and nonlinear optical absorption coefficients of the QD for the transitions between higher energy states. In the present study, we have extended our previous
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study to the spherical QD, GaAs/AlGaAs, with parabolic potential. We will calculate the binding energy of the 1s-, 1p-, 1d- and 1fstates. In addition, we will carried out a study on the linear and third-order nonlinear optical absorption coefficients for 1s–1p, 1p–1d and 1d–1f transitions of the spherical QD with and without impurity. 2. Theory In effective mass approximation, the Hamiltonian of a hydrogenic impurity in a spherical QD with parabolic potential confined by an finite spherical potential well, in atomic units, can be expressed as
( H¼
r2 2r þ c2p r2 þ VðrÞ for r < R r2 2r þ VðrÞ
ð1Þ
for r P R;
where cp ¼ hxp =2Ry is a dimensionless measure of the parabolic potential, xp corresponds to the harmonic oscillator frequency, Ry is the effective Rydberg energy unit. The term VðrÞ is the spherical confining potential well: V(r) = 0 for r < R and VðrÞ ¼ V 0 for r P R. The eigenstates of a system can be determined from the variational principle by minimizing the energy of the system corresponding to the trial wavefunction /i
Ei ¼ h/i jHj/i i=h/i j/i i:
ð2Þ
In the HFR approach, the wavefunction /i are written as linear combination of Slater-type orbitals (STOs), which are called basis sets. As a result our wave function is of the form
/i ¼ HðR r Þ/rR i i ¼ HðR r Þ
rr
r>R
r X rR ~ ~ crR ik vk fik ; r þ f1 HðR rÞg ik vk fik ; r ;
k¼1
k¼1
ð3Þ r
r>R
where HðxÞ is the Heviside step function, r (r ) are the size of the basis set used for the inner (outer) part of the wave function, rR (fr>R cr
vn‘m ðf; rh/Þ ¼ rn1 efr Y ‘m ðh; /Þ;
ð4Þ
where n; ‘; m are the quantum numbers of basis functions and Y ‘m ðh; /Þ is well-known complex spherical harmonics in Condon– Shortley phase convention. After substituting Eqs. (3) and (4) into Eq. (2), the kinetic, the Coulomb potential between the electron and impurity, the parabolic potential and the confining potential energy integrals can be obtained easily in terms of the expansion coefficients and the new integrals over STO for the spherical QD, see in Ref. [45]. Photoabsorption process may be defined as an optical (intersubband) transition in low-dimensional quantum mechanical systems. The photoabsorption occurs from a lower (i) state to an upper (f) state with absorbing a photon. The optical absorption computation is based on Fermi’s golden rule, and the total optical absorption coefficient is given [32,35,38,45]
aðx;IÞ ¼ a1 ðxÞ þ a3 ðx; IÞ 8 rffiffiffiffiffiffiffiffiffi<
9 >
4 = Mfi 2 qhC0 2IM fi qhC0 l h i ¼x h er e0 : Efi hx 2 þ ðhC0 Þ2 n e c E hx2 þ ðhC Þ2 i2 > ; r 0 0 fi
ð5Þ where the terms a1 ðxÞ and a3 ðx; IÞ denote the linear and the third-order nonlinear optical absorption coefficients, I is the
incident optical intensity, h x is the incident photon energy, l is the permeability of the system defined as l ¼ 1=e0 c2 , the q is the electron density in the QD, C0 ¼ 1=s is the relation rate, s is the pffiffiffiffi relaxation time of the 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. Efi ¼ Ef Ei denotes energy difference between lower and upper electronic states. Mfi ¼ h f jezjii is the dipole transition matrix element between lower and upper state, see in Ref. [45]. 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‘ ¼ 1, where ‘ is the angular momentum quantum number. As seen in Eq. (5), since the third-order nonlinear optical absorption coefficient a3 ðx; IÞ is negative and is proportional to the incident optical intensity I, the total absorption coefficient aðx; IÞ is decreases as I increases. Therefore, aðx; IÞ is reduced by one-half when I reach a critical value Ic, called the saturation intensity. 3. Computational method STOs are preferred in the quantum mechanical analysis of the electronic structure of a QD as they represent the correct behavior of the electronic wavefunctions. Therefore, we have chosen a linear combination of s(or p, d) STOs having different screening parameters for the orbitals (or p, d) type atomic orbital. To maintain the orthogonality of the orbitals the same set of screening parameters was used for all the one-electron spatial orbitals with the same angular momentum and seven basis sets ðr ¼ 7Þ were taken to calculate the expectation value of the energy. We combined the QGA r>R procedure and HFR method to determine the parameters crR and fr
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mAlGaAs ¼ 0:088m0 , eGaAs ¼ 13:18, eAlGaAs ¼ 12:24, V 0 ¼ 228 meV, nr = 1.3 and s ¼ 1:4 ps have been taken in this study. The effective Rydberg energy and Bohr radius corresponding to these material parameters are Ry ¼ 5:72 meV and a* ffi 100 Å.
functions have been obtained. The effective Rydberg energy 2 2 h =mr
5
5
Binding energy Eb(Hartree)
4
1d-Eb 1f-Eb
3
1s-Eb γp=0.2
4
Binding energy Eb
1s-Eb 1p-Eb
1s-Eb γp=0.4 3
1p-Eb γp=0.2 1p-Eb γp=0.4
2 1 0
γp=0.2
0
1
2
2
3
4
5
6
Rdot(a*)
1
0 0
1
2
3
4
5
6
Rdot(a*) Fig. 1. Binding energies of the ground and excited states of the spherical QD versus dot radius.
12
Z=0
1d-1f
1p-1d
10
1s-1p 8
Rdot=0.9a* τ=1.4ps γp=0.2
Total absorption coefficient α(106/m)
6 4 2 0 12
1d-1f
1p-1d
Z=1 10
1s-1p
8 I=20MW/m2
6
I=30MW/m2 I=50MW/m2
4
I=60MW/m2 I=70MW/m2
2 0 8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
Photon energy (R*y) Fig. 2. The total optical absorption coefficient a(x, I) of a QD with and without impurity versus the incident photon energy hx for five different values of the incident optical intensity I.
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Fig. 1 shows the binding energies of the ground (1s-), the first (1p-), the second (1d-) and the third (1f-) excited states of the spherical QD with parabolic potential as a function of dot radius Rdot . The binding energy to the impurity of the electron is given by Eb ¼ E0 Eimp , where Eimp and E0 are the energies calculated for the case with and without the impurity. As seen from Fig. 1, as the dot radius decreases, the impurity binding energy reaches a critical peak value where the electron is most effectively confined by confining potential in the finite potential well and then it rapidly decreases with decreasing dot radius. The dot radius corresponding to this most effective confinement is Rdot ¼ 0:33a for 1s-, Rdot ¼ 0:55a for 1p-, Rdot ¼ 0:75a for 1d- and Rdot ¼ 0:93a for 1f-state, respectively. It is clearly seen from Fig. 1 that as dot radius decreases, the smaller ‘ states’ binding energies increase more quickly than those of the bigger ‘ states. As the confining electron is in a small ‘ state, most of time it distributes itself at the outer part of the impurity than the electron in big ‘ state. Because of this, the margin of the confining potential pushes at the small ‘ state first as the dot radius decreases. So its binding energy is influenced earlier than that of the big ‘ state. We have also displayed the binding energies of the ground and the first excited states of the QD into small frame in Fig. 1 as a function of the dot radius for two different values of cp : 0.2 and 0.4. When seen from this small figure, the effect of cp on the binding energy is unimportant when Rdot < 0.8a*, the case that corresponds to the strong spatial confinement, but this effect is important as the dot radius increases. That is, the greater cp is, the greater binding energy is. The reason is that the electron is closer to the impurity as cp increases. In addition, the binding energy for a given value of cp is bigger than the other case without cp , ðEb ðcp – 0Þ > Eb ðcp ¼ 0ÞÞ. Our results for the ground and
excited states agree well with the other results in literature [7,12,24,45,46]. We have presented in Fig. 2 the total optical absorption coefficient aðx; IÞ of the QD with and without impurity for 1s–1p, 1p–1d and 1d–1f transitions as a function of the incident photon energy h x for five different values of the incident optical intensity I. It can be seen from this figure that the maximum of absorption coefhx. ficient corresponds to the threshold photon energy, i.e., Efi This maximum value decreases with increasing incident optical intensity I, and the absorption is strongly bleached at sufficiently high incident optical intensities. Therefore, the nonlinear optical absorption coefficient a3 ðx; IÞ should be taken into account when the incident optical intensity I is comparatively strong, since it can reduce the total absorption coefficient. When the incident optical intensity I exceed a critical value Ic, which indicates saturation, the nonlinear term causes a collapse at the center of the total absorption peak splitting it into two peaks. On the other hand, it is clearly seen from the figure that this critical I value decreases when going to the transitions between higher energy states. The saturation, in Fig. 2, begins to occur at around I = 50 MW/m2 for 1s–1p, I = 48 MW/m2 for 1p–1d and I = 46 MW/m2 for 1d–1f transition of the QD with impurity. However, there is no a shift at the resonance peaks position with incident optical intensity. Understanding of the effects of impurity on the electronic states of semiconductor structures is important in semiconductor physics because their presence can dramatically alter the performance of quantum devices and optical and transport properties [47]. As seen from Fig. 2, the existence of the impurity causes a shift of the absorption peaks towards higher energies (blue shift) for all transitions. This blue shift is more enhanced in the transitions between
16
1d-1f
1p-1d
Z=0
12
12
Rdot=0.9a*
Z=0
1s-1p 8
10
4
8
0
6
1d-1f
-4
Total absorption coefficient α(106/m)
Absorption coefficient α(106/m)
1s-1p
1p-1d
1
α (ω) 3 α (ω,Ι) α(ω,Ι)
-8 -12 16
1d-1f
1p-1d 1s-1p
12
Z=1
8 4 0
Rdot=1.2a*
4 2 0 12
Rdot=0.9a*
Z=1
10 2
I=30MW/m τ=1.4ps γp=0.2 Rdot=1.2a*
8 6
2
I=70MW/m Rdot=0.9a*
-4 -8
4
τ=1.4ps γp=0.2
-12 8
9
2 0
10
11
12
13
Photon energy (R*y) Fig. 3. The linear (dashed), the third-order nonlinear (dotted) and the total (solid) optical absorption coefficients of a QD with and without impurity versus the incident photon energy hx .
4
5
6
7
8
9
10
11
12
13
Photon energy (R*y) Fig. 4. The total optical absorption coefficient a(x, I) for the 1s–1p, 1p–1d and 1d– 1f transitions of a QD with and without impurity versus the incident photon energy hx for two different values of the dot radius.
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lower levels where the electron is more localized near the impurity. In Fig. 3, we have plotted the linear, the third-order nonlinear and the total optical absorption coefficients for transitions between higher electronic states of the QD with and without impurity as a function of the incident photon energy hx. The linear absorption coefficient a1 ðxÞ is positive, whereas the third-order nonlinear optical absorption coefficient a3 ðx; IÞ is negative. So the total optical absorption coefficient aðx; IÞ is significantly reduced by the nonlinear contribution. As seen in Fig. 3, the nonlinear term has a significant effect on the total absorption coefficient. Therefore, the contributions of both the linear and the third-order nonlinear absorption terms should be considered in calculation of the absorption spectrum of QDs, especially for those operating under high-incident optical intensity I. There are resonance peaks at a photon energy value of 8.51 Ry for 1s–1p, 10.40 Ry for 1p–1d and 11.47 Ry for 1d–1f transition in case without impurity, and 9.76 Ry for 1s–1p, 10.81 Ry for 1p–1d, 11.97 Ry for 1d–1f transition in case with impurity as expected, which correspond to the energy difference between the states considered. On the other hand, it is seen from this figure that the amplitudes of the optical absorption coefficient increase when going to the transitions between higher levels. The reason of this situation is the electronic dipolar transi-
Total absorption coefficient α(106/m)
8
1p-1d γp=0.8
1s-1p γp=2 1p-1d γp=2
1d-1f γp=0.8
1d-1f γp=2
1s-1p γp=0.8
7 6 5
tion matrix elements and the increment in energy interval of two different electronic states in which an optical transition occurs. Fig. 4 shows the total optical absorption coefficients of the spherical QD as a function of the photon energy h x for two different dot radius Rdot = 0.9a* and 1.2a* for 1s–1p, 1p–1d and 1d–1f transitions. As seen in Fig. 4, the effect of the dot radius is obvious. It is really seen that the total optical absorption coefficients of the QD with and without impurity for small dot radius are much stronger than that of the large dot radius QDs because the optical absorption spectrum depends on the electron density in QDs, i.e., it depends on the QD volume as 1/VQD, and so it changes with radius as 1/R3. Optical absorption coefficients obtained for the 1s– 1p case are qualitatively consistent with the results for literature [32,35,37,40]. We have displayed in Fig. 5 the total optical absorption coefficient aðx; IÞ for all transitions of the QD with impurity as a function of the photon energy hx for two different values of cp = 0.2 and 2. A very important feature of these figures is that as cp increases, absorption peaks move to the right side, that is, there is a blue shift. This shift is more enhanced in the transitions between lower states where the electron is located near a impurity. However, the shift decreases when going to the higher electronic states. Fig. 6 shows the total optical absorption coefficients aðx; IÞ of the QD with impurity for 1s–1p, 1p–1d and 1d–1f transitions as a function of the incident photon energy h x for six different values of the relaxation time s. One may see from Fig. 6 that the amplitudes of the total optical absorption spectra will increase significantly when the relaxation time s increases, and the bigger the absorption peak is, the sharper the peak intensity will be and then the absorption peaks will be significantly split up into two peaks, which are in consequence of the absorption at linear center that will be strongly bleached. The strong absorption saturation begins to occur at around s = 2.6 ps for 1s–1p, s = 2.3 ps for 1p–1d and s = 2 ps for 1d–1f transition.
2
I=30MW/m Rdot=1a*
4
τ=1.4ps
3
5. Conclusion
2 We investigated the binding energies of the ground and excited states of the spherical QD with parabolic potential. Moreover, we also calculated the linear a1 ðxÞ, the third-order nonlinear a3 ðx; IÞ and the total aðx; IÞ optical absorption coefficients for transitions between higher energy states. The results show that the binding energies increase as dot radius decreases reaching to a maximum value, and then rapidly decrease with decreasing dot radius. As for the optical absorption coefficients, the linear optical absorption
Z=1
1 0 7
8
9
10
11
Photon energy(R*y)
7 Z=1
6
Total absorption coefficient α(10 /m)
Fig. 5. The total optical absorption coefficient a(x, I) of a QD with impurity versus the incident photon energy hx for two different values of cp.
τ=0.4ps τ=0.6ps τ=1ps τ=1.8ps τ=2.3ps τ=2.6ps
6 5 4
1s-1p
1d-1f
1p-1d
2
I=20MW/m Rdot=1.2a* γp=0.2
3 2 1 0 6.0
6.5
7.0
7.5
8.0
8.5
Photon energy(R*y) Fig. 6. The total optical absorption coefficient a(x, I) of a QD with impurity versus the incident photon energy hx for six different values of the relaxation time s.
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coefficient is not related to the incident optical intensity, whereas the incident optical intensity I have a great influence on the nonlinear optical absorption coefficient. In addition, calculated results have shown that the existence of an impurity has a great effect on the optical absorption spectra and it moves all absorption spectra to the right side, especially in transitions between lower states which an optical absorption occurs. When going to the transitions between higher states, the magnitudes of the linear, nonlinear and total absorption spectra increase. On the other hand, the value of cp , relaxation time s and dot radius have great influence on the optical absorption coefficients. To our knowledge, there has been no report of the absorption spectra which includes the transitions between higher states in spherical QD. With respect to the lack of such studies, we believe that our study makes an important contribution to the literature. The theoretical investigation of the optical absorption coefficient in a spherical QD will lead to a better understanding of the properties of QDs. Such theoretical studies may have profound consequences for practical application of the electrooptical devices, and optical absorption saturation also has extensive application in optical communication. Acknowledgement This work is partially supported by Selçuk University BAP office. References [1] M.A. Reed, J.N. Randall, R.J. Aggarwal, R.J. Matyi, T.M. Moore, A.E. Wetsel, Phys. Rev. Lett. 60 (1988) 535. [2] Y.P. Varshni, Superlattices Microstruct. 23 (1998) 145. [3] S.T. Perez-Merchancano, R. Franco, J. Silva-Valencia, Microelectron. J. 39 (2008) 383. [4] E. Kasapog˘lu, H. Sari, I. Sökmen, Physica B 390 (2007) 216. [5] Y.F. Huangfu, Z.W. Yan, Physica E 40 (2008) 2982. [6] E. Sadeghi, Physica E 41 (2009) 1319. [7] I.F.I. Mikhail, I.M.M. Ismail, Phys. Status Solidi B 244 (2007) 3647.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential Yusuf Yakar a,*, Bekir Çakır b,*, Ayhan Özmen b a b
Physics Department, Faculty of Science and Letters, Aksaray University, Campus 68100 Aksaray, Turkey Physics Department, Faculty of Science, Selçuk University, Campus 42031 Konya, Turkey
a r t i c l e
i n f o
Article history: Received 4 September 2009 Received in revised form 26 October 2009 Accepted 11 December 2009
Keywords: Spherical quantum dot Binding energy Linear and nonlinear optical absorption coefficient Parabolic potential
a b s t r a c t In the effective mass approximation, we calculated the binding energy and wave function for the 1s-, 1p-, 1d- and 1f-states of a spherical quantum dot (QD) with parabolic potential by using a combination of quantum genetic algorithm (QGA) and Hartree–Fock–Roothaan (HFR) method. In addition, we also investigated the linear and the third-order nonlinear optical absorption coefficients as a function of the incident photon energy for the 1s–1p, 1p–1d and 1d–1f transitions. Our results are shown that the existence of impurity has great influence on optical absorption coefficients. Moreover, the optical absorption coefficients are strongly affected by the incident optical intensity, relaxation time, parabolic potential and dot radius. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Thanks to modern technology, semiconductor structures with three-dimensional confinement of electrons, called QDs, have been fabricated by using various techniques such as molecular beam epitaxy and etching [1]. These structures display an interesting behavior and play an important role in microelectronic and optoelectronic devices so that they can affect the electrical, optical and transport properties. Therefore, the electronic structure, the energy states, the binding energy, the relativistic effects, the optics properties and other physical properties of QDs with finite and infinite confinement potential have been studied in literature. Most of the theoretical studies in literature use variational approach [2– 13]. Other approximate methods such as perturbation [14], exact solution [15–18], QGA method [19–22], a combination of QGA and Hartree–Fock–Roothaan (HFR) [23,24] and others [25–27] have also been developed. When employing the confinement potential of infinite depth, a continuum-energy threshold does not exist and the excess electrons are always bound by the confinement potential, i.e., possess only discrete energy levels. Thus, binding and dissociation processes cannot be described. The problem can be solved if we consider the confinement potential of finite depth, which much * Corresponding authors. Tel.: +90 382 280 12 35; fax: +90 382 280 12 46 (Y. Yakar). E-mail addresses: [email protected] (Y. Yakar), [email protected] (B. Çakır). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.12.027
better describes the real structure of the QD, since the potential which confines the electrons always possesses the finite depth and range [28]. In literature, the dipole transition, the oscillator strength, the linear optical absorption coefficient and the photoionization cross-section of QDs have been investigated by various authors [9,10,29–31]. A parabolic potential is often considered to be a good representation of the potential in semiconductor structures. Recently, there has been a considerable interest in the linear and nonlinear optical absorption coefficients and refraction index change based on an optical (intersubband) transitions in parabolic QDs [32–40]. In 2009, Xie [41] carried out research on the linear and nonlinear optical properties of an exciton in a spherical parabolic QD by using the matrix diagonalization technique. Chen et al. [42], Yuan et al. [43] and Yao et al. [44] investigated these absorption coefficients for an asymmetric double triangular quantum well, for an off-center hydrogenic donor confined by a spherical QD with a parabolic potential and for a cylindrical QD system, respectively. All studies mentioned above are focused on the calculation of the optical transition between the lowest energy states, which is 1s–1p transitions for the ground (L = 0) and the first excited (L = 1) states of the QD. In a previous study [45], we computed the energies of the ground and excited states and the binding energy of a spherical QD with and without impurity. In addition, we also investigated the oscillator strength and, the linear and nonlinear optical absorption coefficients of the QD for the transitions between higher energy states. In the present study, we have extended our previous
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study to the spherical QD, GaAs/AlGaAs, with parabolic potential. We will calculate the binding energy of the 1s-, 1p-, 1d- and 1fstates. In addition, we will carried out a study on the linear and third-order nonlinear optical absorption coefficients for 1s–1p, 1p–1d and 1d–1f transitions of the spherical QD with and without impurity. 2. Theory In effective mass approximation, the Hamiltonian of a hydrogenic impurity in a spherical QD with parabolic potential confined by an finite spherical potential well, in atomic units, can be expressed as
( H¼
r2 2r þ c2p r2 þ VðrÞ for r < R r2 2r þ VðrÞ
ð1Þ
for r P R;
where cp ¼ hxp =2Ry is a dimensionless measure of the parabolic potential, xp corresponds to the harmonic oscillator frequency, Ry is the effective Rydberg energy unit. The term VðrÞ is the spherical confining potential well: V(r) = 0 for r < R and VðrÞ ¼ V 0 for r P R. The eigenstates of a system can be determined from the variational principle by minimizing the energy of the system corresponding to the trial wavefunction /i
Ei ¼ h/i jHj/i i=h/i j/i i:
ð2Þ
In the HFR approach, the wavefunction /i are written as linear combination of Slater-type orbitals (STOs), which are called basis sets. As a result our wave function is of the form
/i ¼ HðR r Þ/r
rr
r>R
r X r
k¼1
k¼1
ð3Þ r
r>R
where HðxÞ is the Heviside step function, r (r ) are the size of the basis set used for the inner (outer) part of the wave function, r
vn‘m ðf; rh/Þ ¼ rn1 efr Y ‘m ðh; /Þ;
ð4Þ
where n; ‘; m are the quantum numbers of basis functions and Y ‘m ðh; /Þ is well-known complex spherical harmonics in Condon– Shortley phase convention. After substituting Eqs. (3) and (4) into Eq. (2), the kinetic, the Coulomb potential between the electron and impurity, the parabolic potential and the confining potential energy integrals can be obtained easily in terms of the expansion coefficients and the new integrals over STO for the spherical QD, see in Ref. [45]. Photoabsorption process may be defined as an optical (intersubband) transition in low-dimensional quantum mechanical systems. The photoabsorption occurs from a lower (i) state to an upper (f) state with absorbing a photon. The optical absorption computation is based on Fermi’s golden rule, and the total optical absorption coefficient is given [32,35,38,45]
aðx;IÞ ¼ a1 ðxÞ þ a3 ðx; IÞ 8 rffiffiffiffiffiffiffiffiffi<
9 >
4 = Mfi 2 qhC0 2IM fi qhC0 l h i ¼x h er e0 : Efi hx 2 þ ðhC0 Þ2 n e c E hx2 þ ðhC Þ2 i2 > ; r 0 0 fi
ð5Þ where the terms a1 ðxÞ and a3 ðx; IÞ denote the linear and the third-order nonlinear optical absorption coefficients, I is the
incident optical intensity, h x is the incident photon energy, l is the permeability of the system defined as l ¼ 1=e0 c2 , the q is the electron density in the QD, C0 ¼ 1=s is the relation rate, s is the pffiffiffiffi relaxation time of the 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. Efi ¼ Ef Ei denotes energy difference between lower and upper electronic states. Mfi ¼ h f jezjii is the dipole transition matrix element between lower and upper state, see in Ref. [45]. 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‘ ¼ 1, where ‘ is the angular momentum quantum number. As seen in Eq. (5), since the third-order nonlinear optical absorption coefficient a3 ðx; IÞ is negative and is proportional to the incident optical intensity I, the total absorption coefficient aðx; IÞ is decreases as I increases. Therefore, aðx; IÞ is reduced by one-half when I reach a critical value Ic, called the saturation intensity. 3. Computational method STOs are preferred in the quantum mechanical analysis of the electronic structure of a QD as they represent the correct behavior of the electronic wavefunctions. Therefore, we have chosen a linear combination of s(or p, d) STOs having different screening parameters for the orbitals (or p, d) type atomic orbital. To maintain the orthogonality of the orbitals the same set of screening parameters was used for all the one-electron spatial orbitals with the same angular momentum and seven basis sets ðr ¼ 7Þ were taken to calculate the expectation value of the energy. We combined the QGA r>R procedure and HFR method to determine the parameters cr
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mAlGaAs ¼ 0:088m0 , eGaAs ¼ 13:18, eAlGaAs ¼ 12:24, V 0 ¼ 228 meV, nr = 1.3 and s ¼ 1:4 ps have been taken in this study. The effective Rydberg energy and Bohr radius corresponding to these material parameters are Ry ¼ 5:72 meV and a* ffi 100 Å.
functions have been obtained. The effective Rydberg energy 2 2 h =mr
5
5
Binding energy Eb(Hartree)
4
1d-Eb 1f-Eb
3
1s-Eb γp=0.2
4
Binding energy Eb
1s-Eb 1p-Eb
1s-Eb γp=0.4 3
1p-Eb γp=0.2 1p-Eb γp=0.4
2 1 0
γp=0.2
0
1
2
2
3
4
5
6
Rdot(a*)
1
0 0
1
2
3
4
5
6
Rdot(a*) Fig. 1. Binding energies of the ground and excited states of the spherical QD versus dot radius.
12
Z=0
1d-1f
1p-1d
10
1s-1p 8
Rdot=0.9a* τ=1.4ps γp=0.2
Total absorption coefficient α(106/m)
6 4 2 0 12
1d-1f
1p-1d
Z=1 10
1s-1p
8 I=20MW/m2
6
I=30MW/m2 I=50MW/m2
4
I=60MW/m2 I=70MW/m2
2 0 8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
Photon energy (R*y) Fig. 2. The total optical absorption coefficient a(x, I) of a QD with and without impurity versus the incident photon energy hx for five different values of the incident optical intensity I.
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Fig. 1 shows the binding energies of the ground (1s-), the first (1p-), the second (1d-) and the third (1f-) excited states of the spherical QD with parabolic potential as a function of dot radius Rdot . The binding energy to the impurity of the electron is given by Eb ¼ E0 Eimp , where Eimp and E0 are the energies calculated for the case with and without the impurity. As seen from Fig. 1, as the dot radius decreases, the impurity binding energy reaches a critical peak value where the electron is most effectively confined by confining potential in the finite potential well and then it rapidly decreases with decreasing dot radius. The dot radius corresponding to this most effective confinement is Rdot ¼ 0:33a for 1s-, Rdot ¼ 0:55a for 1p-, Rdot ¼ 0:75a for 1d- and Rdot ¼ 0:93a for 1f-state, respectively. It is clearly seen from Fig. 1 that as dot radius decreases, the smaller ‘ states’ binding energies increase more quickly than those of the bigger ‘ states. As the confining electron is in a small ‘ state, most of time it distributes itself at the outer part of the impurity than the electron in big ‘ state. Because of this, the margin of the confining potential pushes at the small ‘ state first as the dot radius decreases. So its binding energy is influenced earlier than that of the big ‘ state. We have also displayed the binding energies of the ground and the first excited states of the QD into small frame in Fig. 1 as a function of the dot radius for two different values of cp : 0.2 and 0.4. When seen from this small figure, the effect of cp on the binding energy is unimportant when Rdot < 0.8a*, the case that corresponds to the strong spatial confinement, but this effect is important as the dot radius increases. That is, the greater cp is, the greater binding energy is. The reason is that the electron is closer to the impurity as cp increases. In addition, the binding energy for a given value of cp is bigger than the other case without cp , ðEb ðcp – 0Þ > Eb ðcp ¼ 0ÞÞ. Our results for the ground and
excited states agree well with the other results in literature [7,12,24,45,46]. We have presented in Fig. 2 the total optical absorption coefficient aðx; IÞ of the QD with and without impurity for 1s–1p, 1p–1d and 1d–1f transitions as a function of the incident photon energy h x for five different values of the incident optical intensity I. It can be seen from this figure that the maximum of absorption coefhx. ficient corresponds to the threshold photon energy, i.e., Efi This maximum value decreases with increasing incident optical intensity I, and the absorption is strongly bleached at sufficiently high incident optical intensities. Therefore, the nonlinear optical absorption coefficient a3 ðx; IÞ should be taken into account when the incident optical intensity I is comparatively strong, since it can reduce the total absorption coefficient. When the incident optical intensity I exceed a critical value Ic, which indicates saturation, the nonlinear term causes a collapse at the center of the total absorption peak splitting it into two peaks. On the other hand, it is clearly seen from the figure that this critical I value decreases when going to the transitions between higher energy states. The saturation, in Fig. 2, begins to occur at around I = 50 MW/m2 for 1s–1p, I = 48 MW/m2 for 1p–1d and I = 46 MW/m2 for 1d–1f transition of the QD with impurity. However, there is no a shift at the resonance peaks position with incident optical intensity. Understanding of the effects of impurity on the electronic states of semiconductor structures is important in semiconductor physics because their presence can dramatically alter the performance of quantum devices and optical and transport properties [47]. As seen from Fig. 2, the existence of the impurity causes a shift of the absorption peaks towards higher energies (blue shift) for all transitions. This blue shift is more enhanced in the transitions between
16
1d-1f
1p-1d
Z=0
12
12
Rdot=0.9a*
Z=0
1s-1p 8
10
4
8
0
6
1d-1f
-4
Total absorption coefficient α(106/m)
Absorption coefficient α(106/m)
1s-1p
1p-1d
1
α (ω) 3 α (ω,Ι) α(ω,Ι)
-8 -12 16
1d-1f
1p-1d 1s-1p
12
Z=1
8 4 0
Rdot=1.2a*
4 2 0 12
Rdot=0.9a*
Z=1
10 2
I=30MW/m τ=1.4ps γp=0.2 Rdot=1.2a*
8 6
2
I=70MW/m Rdot=0.9a*
-4 -8
4
τ=1.4ps γp=0.2
-12 8
9
2 0
10
11
12
13
Photon energy (R*y) Fig. 3. The linear (dashed), the third-order nonlinear (dotted) and the total (solid) optical absorption coefficients of a QD with and without impurity versus the incident photon energy hx .
4
5
6
7
8
9
10
11
12
13
Photon energy (R*y) Fig. 4. The total optical absorption coefficient a(x, I) for the 1s–1p, 1p–1d and 1d– 1f transitions of a QD with and without impurity versus the incident photon energy hx for two different values of the dot radius.
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lower levels where the electron is more localized near the impurity. In Fig. 3, we have plotted the linear, the third-order nonlinear and the total optical absorption coefficients for transitions between higher electronic states of the QD with and without impurity as a function of the incident photon energy hx. The linear absorption coefficient a1 ðxÞ is positive, whereas the third-order nonlinear optical absorption coefficient a3 ðx; IÞ is negative. So the total optical absorption coefficient aðx; IÞ is significantly reduced by the nonlinear contribution. As seen in Fig. 3, the nonlinear term has a significant effect on the total absorption coefficient. Therefore, the contributions of both the linear and the third-order nonlinear absorption terms should be considered in calculation of the absorption spectrum of QDs, especially for those operating under high-incident optical intensity I. There are resonance peaks at a photon energy value of 8.51 Ry for 1s–1p, 10.40 Ry for 1p–1d and 11.47 Ry for 1d–1f transition in case without impurity, and 9.76 Ry for 1s–1p, 10.81 Ry for 1p–1d, 11.97 Ry for 1d–1f transition in case with impurity as expected, which correspond to the energy difference between the states considered. On the other hand, it is seen from this figure that the amplitudes of the optical absorption coefficient increase when going to the transitions between higher levels. The reason of this situation is the electronic dipolar transi-
Total absorption coefficient α(106/m)
8
1p-1d γp=0.8
1s-1p γp=2 1p-1d γp=2
1d-1f γp=0.8
1d-1f γp=2
1s-1p γp=0.8
7 6 5
tion matrix elements and the increment in energy interval of two different electronic states in which an optical transition occurs. Fig. 4 shows the total optical absorption coefficients of the spherical QD as a function of the photon energy h x for two different dot radius Rdot = 0.9a* and 1.2a* for 1s–1p, 1p–1d and 1d–1f transitions. As seen in Fig. 4, the effect of the dot radius is obvious. It is really seen that the total optical absorption coefficients of the QD with and without impurity for small dot radius are much stronger than that of the large dot radius QDs because the optical absorption spectrum depends on the electron density in QDs, i.e., it depends on the QD volume as 1/VQD, and so it changes with radius as 1/R3. Optical absorption coefficients obtained for the 1s– 1p case are qualitatively consistent with the results for literature [32,35,37,40]. We have displayed in Fig. 5 the total optical absorption coefficient aðx; IÞ for all transitions of the QD with impurity as a function of the photon energy hx for two different values of cp = 0.2 and 2. A very important feature of these figures is that as cp increases, absorption peaks move to the right side, that is, there is a blue shift. This shift is more enhanced in the transitions between lower states where the electron is located near a impurity. However, the shift decreases when going to the higher electronic states. Fig. 6 shows the total optical absorption coefficients aðx; IÞ of the QD with impurity for 1s–1p, 1p–1d and 1d–1f transitions as a function of the incident photon energy h x for six different values of the relaxation time s. One may see from Fig. 6 that the amplitudes of the total optical absorption spectra will increase significantly when the relaxation time s increases, and the bigger the absorption peak is, the sharper the peak intensity will be and then the absorption peaks will be significantly split up into two peaks, which are in consequence of the absorption at linear center that will be strongly bleached. The strong absorption saturation begins to occur at around s = 2.6 ps for 1s–1p, s = 2.3 ps for 1p–1d and s = 2 ps for 1d–1f transition.
2
I=30MW/m Rdot=1a*
4
τ=1.4ps
3
5. Conclusion
2 We investigated the binding energies of the ground and excited states of the spherical QD with parabolic potential. Moreover, we also calculated the linear a1 ðxÞ, the third-order nonlinear a3 ðx; IÞ and the total aðx; IÞ optical absorption coefficients for transitions between higher energy states. The results show that the binding energies increase as dot radius decreases reaching to a maximum value, and then rapidly decrease with decreasing dot radius. As for the optical absorption coefficients, the linear optical absorption
Z=1
1 0 7
8
9
10
11
Photon energy(R*y)
7 Z=1
6
Total absorption coefficient α(10 /m)
Fig. 5. The total optical absorption coefficient a(x, I) of a QD with impurity versus the incident photon energy hx for two different values of cp.
τ=0.4ps τ=0.6ps τ=1ps τ=1.8ps τ=2.3ps τ=2.6ps
6 5 4
1s-1p
1d-1f
1p-1d
2
I=20MW/m Rdot=1.2a* γp=0.2
3 2 1 0 6.0
6.5
7.0
7.5
8.0
8.5
Photon energy(R*y) Fig. 6. The total optical absorption coefficient a(x, I) of a QD with impurity versus the incident photon energy hx for six different values of the relaxation time s.
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coefficient is not related to the incident optical intensity, whereas the incident optical intensity I have a great influence on the nonlinear optical absorption coefficient. In addition, calculated results have shown that the existence of an impurity has a great effect on the optical absorption spectra and it moves all absorption spectra to the right side, especially in transitions between lower states which an optical absorption occurs. When going to the transitions between higher states, the magnitudes of the linear, nonlinear and total absorption spectra increase. On the other hand, the value of cp , relaxation time s and dot radius have great influence on the optical absorption coefficients. To our knowledge, there has been no report of the absorption spectra which includes the transitions between higher states in spherical QD. With respect to the lack of such studies, we believe that our study makes an important contribution to the literature. The theoretical investigation of the optical absorption coefficient in a spherical QD will lead to a better understanding of the properties of QDs. Such theoretical studies may have profound consequences for practical application of the electrooptical devices, and optical absorption saturation also has extensive application in optical communication. Acknowledgement This work is partially supported by Selçuk University BAP office. References [1] M.A. Reed, J.N. Randall, R.J. Aggarwal, R.J. Matyi, T.M. Moore, A.E. Wetsel, Phys. Rev. Lett. 60 (1988) 535. [2] Y.P. Varshni, Superlattices Microstruct. 23 (1998) 145. [3] S.T. Perez-Merchancano, R. Franco, J. Silva-Valencia, Microelectron. J. 39 (2008) 383. [4] E. Kasapog˘lu, H. Sari, I. Sökmen, Physica B 390 (2007) 216. [5] Y.F. Huangfu, Z.W. Yan, Physica E 40 (2008) 2982. [6] E. Sadeghi, Physica E 41 (2009) 1319. [7] I.F.I. Mikhail, I.M.M. Ismail, Phys. Status Solidi B 244 (2007) 3647.
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