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ZnS quantum dot
ARTICLE IN PRESS
Microelectronics Journal 37 (2006) 904–909 www.elsevier.com/locate/mejo
Quantum size dependent optical nutation in a core-shell CdSe/ZnS quantum dot Shaohua Gong, Duanzheng Yao, Xiaobo Feng, Hongliang Jiang Department of Physics, Wuhan University, Wuhan 430072, China Received 3 December 2005; accepted 29 January 2006 Available online 3 April 2006
Abstract The energy eigenvalues and eigenfunctions have been obtained for a core-shell CdSe/ZnS quantum dot structure under effective-mass approximation. The electric transition dipole moment is calculated for the 1s–2s electronic transition. The optical nutation signal of the transition of electrons has been calculated numerically based on optical Bloch equations. Particularly, we have investigated the quantum size, the core’s radius and the shell’s thickness, dependent optical nutation. It is shown from calculation results that the optical nutation signal is sensitive to the size and structure change. And the reasons for the variation of the Rabi frequency have been discussed based on the theory of the quantum size confined effect (QSCE). r 2006 Elsevier Ltd. All rights reserved. PACS: 78.55.Et; 78.67. Hc; 42.60. Rn Keywords: Core-shell QDs; Optical nutation; Quantum size effect; Electric transition dipole moment
1. Introduction The rapid progress of nano-science is a result of its rich technology applications. The low-dimensional semiconductor structures, such as quantum wells (QWs), quantum wires (QWRs) and quantum dots (QDs), have been applied in optoelectronic materials and devices for optical communication, optical computing and biological probe [1–4]. Their wonderful optical properties, especially the optical nonlinear properties, have attracted many scientists’ interest. Many experimental and theoretical investigations are carrying on hotly. The core-shell CdSe/ZnS QD is one of them and it is a kind of wide direct gap semiconductor QDs. Its absorption and photoluminescence spectra were studied by Azad Malik et al. [5]. Its conduction mechanism and electro-luminescence spectra were studied by Mhikmet and Talapin et al. [6]. Optical coherent transient effect (OCTE) arises in an active medium when it is irradiated by a coherent radiation source over a time period shorter than the medium’s Corresponding author. Tel.: +86 27 68764026.
E-mail address: [email protected] (D. Yao). 0026-2692/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2006.01.015
relaxation time. Among these different coherent transient phenomena, optical nutation is an important tool for the study of relaxation mechanisms and hot-carrier effects in semiconductor materials. For example, it’s an important approach to calculate the matrix elements of transition dipole and the lifetime of excited states [7–8]. Thanks to the remarkable growth of femtosecond laser technology, we can observe them in QWs, QWRs and QDs of semiconductor materials. The optical nutation has been investigated in bulk semiconductor using two-level model by Singh [9]. Pranay K. Sen has studied the effect in semiconductor QWs and QWRs, even those low-dimension structures were put into a strong magnetic field [10–11]. However, there are few researches about it in core-shell QDs. In this paper, we make an attempt to theoretically investigate the optical nutation phenomena of the 1s–2s electronic transition in the core-shell CdSe/ZnS QD adopting two-level model. Particularly, we have studied the quantum size dependent optical nutation. In addition, we keep the radius of the QD smaller than the exciton Bohr radius of the materials in the calculation, i.e., it is a strong confined system. The calculation results have been discussed.
ARTICLE IN PRESS S. Gong et al. / Microelectronics Journal 37 (2006) 904–909
2. Model and theory The model used in calculation and analyses is an isolated CdSe/ZnS core-shell quantum dot with inner radius R1 and outer radius R2, shown in Fig. 1. Suppose the core-shell quantum dot forms a shell well and a centric barrier for the two kinds of materials have different potentials. The potential of the core is chosen to be referent zero point of energy, and the band-gap of ZnS is wider than that of CdSe, thus V C 40 [12]. On the premise of the effective mass approximation, the steady Schro¨dinger equation for electron can be written as
_2 q 2 q 1 q q r sin y þ qr sin y qy qy 2mi r2 qr 2 1 q þ 2 þ V i ðrÞ Fnlm ðrÞ ¼ EFnlm ðrÞ sin y qj2
905
relative to the position in the model and expressed as follows: ( m1 ; rpR1 ; (2) mi ¼ m2 ; R1 orpR2 ; 8 > < 0; V i ðrÞ ¼ V c ; > : 1;
0orpR1 ; R1 orpR2 ; r4R2 :
(3)
Due to the fact that the mass and the potential are symmetric spherically, the separation of radial and angular coordinates leads to Fnlm ðr; y; fÞ ¼ Rnl ðrÞY lm ðy; fÞ,
ð1Þ
where _ ¼ h=2p, h is Planck’s constant; E the energy eigenvalue; and Fnlm ðrÞ the responding eigenfunction. n is the principal quantum number, and l and m are the angular momentum quantum numbers. mi is the effective mass of electron in the ith region, and V i ðrÞ the potential. They are
(4)
where Rnl ðrÞ is the radial wave function, and Y lm ðy; fÞ is the spherical harmonics. For the spherical potential consists of three parts, the radial eigenfunction Rnl ðrÞ consists of three parts too, according to the position of the electron in the model. Consequently, to solve the Eq. (1), two cases need to be distinguished. In the region where E4V c , the solution of radial wave function Rnl ðrÞ is a linear combination of spherical Bessel function j l ðxÞ and spherical Neumann function nl ðxÞ [12,13]. 8 > < A1 j l ðknl;1 rÞ þ B1 nl ðknl;1 rÞ; rpR1 ; Rnl ðrÞ ¼ A2 j l ðknl;2 rÞ þ B2 nl ðknl;2 rÞ; R1 orpR2 ; (5) > : 0; r4R2 ; with knl;1 ¼ knl;2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mn1 E=_2 ;
(6)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2mn2 ðE V c ÞE=_2 ;
(7)
A1, A2, B1 and B2 are normalized constants. The boundary conditions [14,15] for the function are Rnl;1 ðR1 Þ ¼ Rnl;2 ðR1 Þ,
(8)
1 dRnl;1 1 dRnl;2 ¼ . m1 dr r¼R1 m2 dr r¼R2
(9)
For the wave function is limited when r ! 0, there is B1 ¼ 0. At the same time, the wave function has to vanish rapidly when r ! 1, namely Rnl;2 ðR2 Þ ¼ 0.
(10)
And the wave function must satisfy the normalization condition Z r1 Z r2 r2 Rnnl;1 Rnl;1 dr þ r2 Rnnl;2 Rnl;2 dr ¼ 1 (11) 0
Fig. 1. The core-shell CdSe/ZnS QD model and its schematic diagram of potential.
r1
The four coefficients A1, A2, B2 and E can be got by solving Eqs. (8)–(11), then the wave function and eigenenergy are obtained.
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In the region where EoV c , the radial wave function will keep the same as the above where rpR1 and r4R2 , but in the range of R1 orpR2 , the radial wave function is a linear combination of two Hankel functions hðþÞ and hðÞ l l . Rnl ðrÞ 8 0 A1 j l ðknl;1 rÞ þ B01 nl ðknl;1 rÞ; > > < ðþÞ ðÞ ¼ A02 hl ðiknl;2 rÞ þ B02 hl ðiknl;2 rÞ; > > : 0;
rpR1 ; R1 orpR2 ;
ð12Þ
r4R2 ;
with knl;2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mn2 ðV c E ÞE=_2 ;
(13) A01 ,
B01
A02 ,
B02
¼ 0, and and E can be It is obvious that got by solving Eqs. (8)–(11). Of course, we can restrict the calculations to 1s and 2s states. For l ¼ m ¼ 0 the spherical harmonics pffiffiffiffiffiffi Y0,0 (y, j) yields only a constant factor of 1= 4p. And these calculations can be performed for not only electrons but also holes. We consider a single photon transition process in a twolevel system. Due to the fact that the radius of the semiconductor QD is kept smaller than its Bohr radius, the kinetic energy of those charge carriers is bigger a lot than the Coulomb interaction energy between electron and hole. The energy of electrons and holes is quantized respectively. (the exciton Bohr radius is 5.6 nm in bulk CdSe materials.) Therefore the Coulomb interaction term can be neglected, and we can use the particle-in-the-box-model (PBM) to analyze. The Hamiltonian for this system irradiated with a femtosecond laser pulse can be written as H ¼ H M þ H I þ H R, where H
M
¼
"
Ea 0
# 0 , Eb "
H I ¼ m E 0 ¼
R
½H ; r ¼ i_
0
mab E 0
mba E 0
0
raa ga
rab gab
rba gab
rbb gb
# ,
! ,
where HM is the unperturbed Hamiltonian, Ea and Eb are the eigen-energies of a and b levels, respectively; HI is the interaction Hamiltonian between the light field and the QD; E0 is the electric component of the polarized light field, of transition, and ! 0 mab m¼ mba 0 is the electric dipole moment and mab ¼ hFa j erjFb i.
(14)
HR is the relaxation Hamiltonian; ga and gb are longitudinal relaxation velocities of a and b energy levels; gab is the transverse relaxation velocity. The Liouville equation for the density matrix is qr qr i_ ¼ H M þ H I þ r þ i_ qt qt damping Then, the optical Bloch equation can be derived using rotating-wave approximation and slowly changing approximation of amplitude. u u_ ¼ Dv gab u ¼ Dv , (15a) T2 v_ ¼ Du þ ww gab v ¼ Du þ ww
w_ ¼ wv g½w w0 ¼ wv
v , T2
w w0 , T1
(15b)
(15c)
where u, v and w are the three parameters of Bloch vector M ðu; v; wÞ. w is the Rabi frequency, in the case of nearresonant excitation, m E0 w ¼ ab , (16) _ where g ¼ 1=T 1 , gab ¼ 1=T 2 ; T1 and T2 are the longitudinal and transversal relaxation time of energy level respectively. D is the deviation of frequency between the irradiating light field frequency and the resonant frequency of the material, usually it is zero in the special case of a near-resonant excitation by a polarized quasi-monochromatic field. We can solve these three Eqs. (15a–c); the analytic solution as following: 0 wD 1 0 1 ð1 cos gtÞ g2 u B C w B C B C (17) g sin gt @ v A ¼ egt w0 B C, @ A w2 w 1 g2 ð1 cos gtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g ¼ w2 þ D2 is the rotational velocity of processional motion. We can calculate the induced effective polarization in the QD as Z 1 m w gt0 pffiffiffi e Peff ¼ mab v dvz ¼ ab v r p Z11 sin gt0 dvz , ðN 0a N 0b Þ expðv2z =v2r Þ ð18Þ g 1 where vz is the velocity component of atoms’ thermal motion at the direction of the laser pulse. vr is the rootmean-square value of the velocity of atoms’ thermal motion, and N 0a N 0b is the difference of particle population in unit volume between two energy levels in initial state. The numerical solution of the induced polarization can obtain by solving Eqs. (17), (18). The transient nutation
ARTICLE IN PRESS S. Gong et al. / Microelectronics Journal 37 (2006) 904–909
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signal can be calculate as 1 I S ¼ E 0 oLPeff ðt tÞ (19) 2 where o is the frequency of the incident laser pulse. L is the thickness of the sample and t is the time that the laser pulse takes to cross the sample. 3. Results and discussion Suppose the QD is interacted with a 150 femtosecond laser pulse under a near-resonant excitation, whose electric field amplitude E 0 ¼ 5 106 V=m, which is under weak excitation regime and low temperature [16], and N 0b ¼ 0. The parameters used in the calculation are as follows: mne;CdSe ¼ 0:13m0 , mne;ZnS ¼ 0:28m0 (m0 is the rest mass of electron), V c ¼ 0:9 eV [13,17], the relaxation times T 1 ¼ T 2 ¼ 300 fs, and N 0a N 0b ¼ 5 1024 m3 . First of all, while R1 ¼ 4.5 nm, R2 ¼ 5 nm we have calculated the electric transition dipole moment for the 1s–2s transition using Eq. (14). When the electron locates outside the well, i.e., E4V c , mab ¼ 1:58817 1028 C m; and mab ¼ 1:48911 1028 C m for the electrons inside the well. The optical nutation signals are got from Eq. (18), and displayed in Fig. 2. Fig. 2 shows that the rapid decay of the nutation signals both outside the well (a) and inside the well (b) (arising due to the transient polarization induced in the QD, and decaying for the reason that the absorption coefficient is directly proportional to the population difference which is decreasing rapidly). Similar decay behavior was reported in QWRs and micro-cavity in Refs. [10,11,18]. Secondly, when one of R1 and R2 keeps steady and the other changes, we may see some interesting phenomena about the nutation signal change. In Fig. 3, we may see the three optical nutation curves almost overlap with each other. For compare and clarity, we plot the curve b alone which is almost covered by curve
Fig. 2. The optical nutation signals of the electrons outside (a) and inside (b) the well while R1 ¼ 4.5 nm and R2 ¼ 5 nm.
Fig. 3. The optical nutation signal of the electrons inside the well while the outer radius R2 ¼ 5 nm (a), R2 ¼ 6 nm (b), R2 ¼ 7 nm (c) and the inner radius does not change R1 ¼ 4.5 nm.
c completely. This illustrates that the optical nutation signal of the electrons inside the well almost does not change when the outer radius R2 changed, i.e., the change of the shell’s thickness has little influence on the nutation signal of the electrons inside the well. However, it’s different for the electrons outside the well. Fig. 4 reveals the fact that the Rabi frequency of the oscillation increases with the growth of the thickness of the shell while the size of the core does not change. According to Eqs. (16), (18), the immediate cause for this is the increase or decrease of their electric transition dipole moment with the change of the size and structure. It is owing to the QSCE in which most of the electrons are confined in the well and for the electrons inside the well, the change of R2 has little influence on their wave function. The electric transition dipole moment almost does not change (Fig. 3). But with the increase of R2, the probability of electrons presenting outside the well enhances. The wave function of the electrons outside the well will change a lot and the band gap will reduce [12]. Then the electric transition dipole increases [19] and the Rabi frequency will increase too (Fig. 4). For the electrons inside the well, the Rabi frequency of the optical nutation signal decreases rapidly with the reducing of the inner radius R1 while R2 ¼ 5 nm does not change, shown in Fig. 5. The reason is the same as the above. Due to the fact that the QSCE is enhanced inside the well and decreased outside the well, the probability of electrons presenting inside the well decreases greatly. The wave function of electrons inside the well changes a lot, and the electric transition dipole moment will reduce greatly. Thus the Rabi frequency will decrease according to Eq. (16). Compared with the results in Fig. 4, it can be found from Fig. 5 that the change of the signal’s Rabi frequency is more sensitive to the size of the core than to that of the shell.
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S. Gong et al. / Microelectronics Journal 37 (2006) 904–909
Fig. 4. The optical nutation signal of the electrons outside the well while the outer radius changes and the inner radius R1 ¼ 4.5 nm.
Fig. 6. The optical nutation signal of the electrons outside the well while the inner radius R1 changes and the outer radius R2 ¼ 5 nm.
shell will decrease and more electrons will enter into the shell. The wave function of the electrons outside the well will change a lot, and the band gap for 1s–2s transition will reduce too [12]. This will make the electric transition dipole moment improved greatly. So this phenomenon happens. 4. Conclusion
Fig. 5. The optical nutation signal of the electrons inside the well while the inner radius R1 changes and the outer radius R2 ¼ 5 nm.
However, the Fig. 6 shows us a different result. In the case of R1 reduced from 4.5 to 3.5 nm and R2 kept steady, the Rabi frequency of optical nutation signal of the electrons outside the well reduces a little first, and then increases a lot. This tells us the fact that, for the optical nutation of the electrons outside the well, there is almost no influence while the inner radius changes in some region, and the influence becomes strong when the inner radius is less than some threshold value. It can be explained as follows. Though the change of optical nutation signal of the electrons outside the well is mainly determined by the change of the shell’s thickness (Fig. 4), to reduce the size of the core, the QSCE in the core will enhance, and the function of preventing the electrons from relaxing into the shell will be promoted, and the band gap will increase [12]. Then the electric transition dipole moment will decrease a little first. On the other hand, the thickness of the shell layer turns wider while R2 keeps steady. The QSCE in the
In conclusion, theoretical analysis of the optical nutation signal of a core-shell CdSe/ZnS QD has been presented in this paper based on the effective-mass approximation and two energy level model. Especially the influence of the change of the QDs size parameter on the optical nutation signals of the 1s–2s transition of the electrons inside the well and outside the well have been calculated respectively. Numerical calculation results reveal that the nutation signals are greatly dependent on the size and the structure of the QD. We think this work is helpful to study the relaxation mechanisms and hot-carrier effects in materials, and to apply this kind of QDs into optical devices and biologic probe field. Acknowledgment This work was financially supported by the National Natural Foundation of China under the granted No. 10534030. References [1] M.A. Haase, J. Qiu, J.M. DePuydt, H. Cheng, Appl. Phys. Lett. 59 (1991) 1272–1274. [2] S. Taniguchi, T. Hino, S. Itoh, et al., Electron Lett. 32 (1996) 552–553. [3] Z.H. Zheng, Z.P. Guan, L.C. Chen, et al., Japan J. Appl. Phys 34 (1995) 56–58. [4] T. Uematsu, J. Kimura, Y. Yamaguchi, Nanotechnology 15 (2004) 822–827.
ARTICLE IN PRESS S. Gong et al. / Microelectronics Journal 37 (2006) 904–909 [5] M. Azad, Malik, Paul O’Brien, Neerish Revaprasadu, Chem. Mater. 14 (2002) 2004–2010. [6] R.A.M. Hikmet, D.V. Talapin, et al., J. Appl. Phys. 93 (2003) 3509–3514. [7] G.B. Hocker, C.L. Tang, Phys. Rev. Lett. 21 (1968) 591–594. [8] R.L. Shoemaker, E.W. Van Stryland, J. Chem. Phys. 64 (1976) 1733–1740. [9] K. Singh, P. Sen, P.K. Sen, IEEE J. Quant. Electron QE-25 (1989) 67–72. [10] S. Kapoor, J. Kumar, P.K. Sen, J. Appl. Phys. 95 (2004) 4833–4839. [11] P.K. Sen, M.K. Bafna, S.K. Gupta, J. Appl. Phys. 96 (2004) 5552–5556.
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[12] D. Schooss, A. Mews, A. Eychmu¨ller, H. Weller, Phys. Rev. B 49 (1994) 17072–17078. [13] J.W. Haus, H.S. Zhou, I. Honma, H. Komiyama, Phys. Rev. B 47 (1993) 1359–1365. [14] D.J. Ben Daniel, C.B. Duke, Phys. Rev. 152 (1966) 683–692. [15] L.E. Brus, J. Chem. Phys. 79 (1983) 5566–5571. [16] M. Lindberg, S.W. Koch, Phys. Rev. B 38 (1988) 3342–3350. [17] A.N. Nethercot, Phys. Rev. Lett. 33 (1974) 1088–1091. [18] G. Bongiovanni, et al., Phys. Rev. B 55 (1997) 7084–7090. [19] L. Liu, J. Li, G. Xiong, Physica E 25 (2005) 466–471.
Microelectronics Journal 37 (2006) 904–909 www.elsevier.com/locate/mejo
Quantum size dependent optical nutation in a core-shell CdSe/ZnS quantum dot Shaohua Gong, Duanzheng Yao, Xiaobo Feng, Hongliang Jiang Department of Physics, Wuhan University, Wuhan 430072, China Received 3 December 2005; accepted 29 January 2006 Available online 3 April 2006
Abstract The energy eigenvalues and eigenfunctions have been obtained for a core-shell CdSe/ZnS quantum dot structure under effective-mass approximation. The electric transition dipole moment is calculated for the 1s–2s electronic transition. The optical nutation signal of the transition of electrons has been calculated numerically based on optical Bloch equations. Particularly, we have investigated the quantum size, the core’s radius and the shell’s thickness, dependent optical nutation. It is shown from calculation results that the optical nutation signal is sensitive to the size and structure change. And the reasons for the variation of the Rabi frequency have been discussed based on the theory of the quantum size confined effect (QSCE). r 2006 Elsevier Ltd. All rights reserved. PACS: 78.55.Et; 78.67. Hc; 42.60. Rn Keywords: Core-shell QDs; Optical nutation; Quantum size effect; Electric transition dipole moment
1. Introduction The rapid progress of nano-science is a result of its rich technology applications. The low-dimensional semiconductor structures, such as quantum wells (QWs), quantum wires (QWRs) and quantum dots (QDs), have been applied in optoelectronic materials and devices for optical communication, optical computing and biological probe [1–4]. Their wonderful optical properties, especially the optical nonlinear properties, have attracted many scientists’ interest. Many experimental and theoretical investigations are carrying on hotly. The core-shell CdSe/ZnS QD is one of them and it is a kind of wide direct gap semiconductor QDs. Its absorption and photoluminescence spectra were studied by Azad Malik et al. [5]. Its conduction mechanism and electro-luminescence spectra were studied by Mhikmet and Talapin et al. [6]. Optical coherent transient effect (OCTE) arises in an active medium when it is irradiated by a coherent radiation source over a time period shorter than the medium’s Corresponding author. Tel.: +86 27 68764026.
E-mail address: [email protected] (D. Yao). 0026-2692/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2006.01.015
relaxation time. Among these different coherent transient phenomena, optical nutation is an important tool for the study of relaxation mechanisms and hot-carrier effects in semiconductor materials. For example, it’s an important approach to calculate the matrix elements of transition dipole and the lifetime of excited states [7–8]. Thanks to the remarkable growth of femtosecond laser technology, we can observe them in QWs, QWRs and QDs of semiconductor materials. The optical nutation has been investigated in bulk semiconductor using two-level model by Singh [9]. Pranay K. Sen has studied the effect in semiconductor QWs and QWRs, even those low-dimension structures were put into a strong magnetic field [10–11]. However, there are few researches about it in core-shell QDs. In this paper, we make an attempt to theoretically investigate the optical nutation phenomena of the 1s–2s electronic transition in the core-shell CdSe/ZnS QD adopting two-level model. Particularly, we have studied the quantum size dependent optical nutation. In addition, we keep the radius of the QD smaller than the exciton Bohr radius of the materials in the calculation, i.e., it is a strong confined system. The calculation results have been discussed.
ARTICLE IN PRESS S. Gong et al. / Microelectronics Journal 37 (2006) 904–909
2. Model and theory The model used in calculation and analyses is an isolated CdSe/ZnS core-shell quantum dot with inner radius R1 and outer radius R2, shown in Fig. 1. Suppose the core-shell quantum dot forms a shell well and a centric barrier for the two kinds of materials have different potentials. The potential of the core is chosen to be referent zero point of energy, and the band-gap of ZnS is wider than that of CdSe, thus V C 40 [12]. On the premise of the effective mass approximation, the steady Schro¨dinger equation for electron can be written as
_2 q 2 q 1 q q r sin y þ qr sin y qy qy 2mi r2 qr 2 1 q þ 2 þ V i ðrÞ Fnlm ðrÞ ¼ EFnlm ðrÞ sin y qj2
905
relative to the position in the model and expressed as follows: ( m1 ; rpR1 ; (2) mi ¼ m2 ; R1 orpR2 ; 8 > < 0; V i ðrÞ ¼ V c ; > : 1;
0orpR1 ; R1 orpR2 ; r4R2 :
(3)
Due to the fact that the mass and the potential are symmetric spherically, the separation of radial and angular coordinates leads to Fnlm ðr; y; fÞ ¼ Rnl ðrÞY lm ðy; fÞ,
ð1Þ
where _ ¼ h=2p, h is Planck’s constant; E the energy eigenvalue; and Fnlm ðrÞ the responding eigenfunction. n is the principal quantum number, and l and m are the angular momentum quantum numbers. mi is the effective mass of electron in the ith region, and V i ðrÞ the potential. They are
(4)
where Rnl ðrÞ is the radial wave function, and Y lm ðy; fÞ is the spherical harmonics. For the spherical potential consists of three parts, the radial eigenfunction Rnl ðrÞ consists of three parts too, according to the position of the electron in the model. Consequently, to solve the Eq. (1), two cases need to be distinguished. In the region where E4V c , the solution of radial wave function Rnl ðrÞ is a linear combination of spherical Bessel function j l ðxÞ and spherical Neumann function nl ðxÞ [12,13]. 8 > < A1 j l ðknl;1 rÞ þ B1 nl ðknl;1 rÞ; rpR1 ; Rnl ðrÞ ¼ A2 j l ðknl;2 rÞ þ B2 nl ðknl;2 rÞ; R1 orpR2 ; (5) > : 0; r4R2 ; with knl;1 ¼ knl;2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mn1 E=_2 ;
(6)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2mn2 ðE V c ÞE=_2 ;
(7)
A1, A2, B1 and B2 are normalized constants. The boundary conditions [14,15] for the function are Rnl;1 ðR1 Þ ¼ Rnl;2 ðR1 Þ,
(8)
1 dRnl;1 1 dRnl;2 ¼ . m1 dr r¼R1 m2 dr r¼R2
(9)
For the wave function is limited when r ! 0, there is B1 ¼ 0. At the same time, the wave function has to vanish rapidly when r ! 1, namely Rnl;2 ðR2 Þ ¼ 0.
(10)
And the wave function must satisfy the normalization condition Z r1 Z r2 r2 Rnnl;1 Rnl;1 dr þ r2 Rnnl;2 Rnl;2 dr ¼ 1 (11) 0
Fig. 1. The core-shell CdSe/ZnS QD model and its schematic diagram of potential.
r1
The four coefficients A1, A2, B2 and E can be got by solving Eqs. (8)–(11), then the wave function and eigenenergy are obtained.
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In the region where EoV c , the radial wave function will keep the same as the above where rpR1 and r4R2 , but in the range of R1 orpR2 , the radial wave function is a linear combination of two Hankel functions hðþÞ and hðÞ l l . Rnl ðrÞ 8 0 A1 j l ðknl;1 rÞ þ B01 nl ðknl;1 rÞ; > > < ðþÞ ðÞ ¼ A02 hl ðiknl;2 rÞ þ B02 hl ðiknl;2 rÞ; > > : 0;
rpR1 ; R1 orpR2 ;
ð12Þ
r4R2 ;
with knl;2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mn2 ðV c E ÞE=_2 ;
(13) A01 ,
B01
A02 ,
B02
¼ 0, and and E can be It is obvious that got by solving Eqs. (8)–(11). Of course, we can restrict the calculations to 1s and 2s states. For l ¼ m ¼ 0 the spherical harmonics pffiffiffiffiffiffi Y0,0 (y, j) yields only a constant factor of 1= 4p. And these calculations can be performed for not only electrons but also holes. We consider a single photon transition process in a twolevel system. Due to the fact that the radius of the semiconductor QD is kept smaller than its Bohr radius, the kinetic energy of those charge carriers is bigger a lot than the Coulomb interaction energy between electron and hole. The energy of electrons and holes is quantized respectively. (the exciton Bohr radius is 5.6 nm in bulk CdSe materials.) Therefore the Coulomb interaction term can be neglected, and we can use the particle-in-the-box-model (PBM) to analyze. The Hamiltonian for this system irradiated with a femtosecond laser pulse can be written as H ¼ H M þ H I þ H R, where H
M
¼
"
Ea 0
# 0 , Eb "
H I ¼ m E 0 ¼
R
½H ; r ¼ i_
0
mab E 0
mba E 0
0
raa ga
rab gab
rba gab
rbb gb
# ,
! ,
where HM is the unperturbed Hamiltonian, Ea and Eb are the eigen-energies of a and b levels, respectively; HI is the interaction Hamiltonian between the light field and the QD; E0 is the electric component of the polarized light field, of transition, and ! 0 mab m¼ mba 0 is the electric dipole moment and mab ¼ hFa j erjFb i.
(14)
HR is the relaxation Hamiltonian; ga and gb are longitudinal relaxation velocities of a and b energy levels; gab is the transverse relaxation velocity. The Liouville equation for the density matrix is qr qr i_ ¼ H M þ H I þ r þ i_ qt qt damping Then, the optical Bloch equation can be derived using rotating-wave approximation and slowly changing approximation of amplitude. u u_ ¼ Dv gab u ¼ Dv , (15a) T2 v_ ¼ Du þ ww gab v ¼ Du þ ww
w_ ¼ wv g½w w0 ¼ wv
v , T2
w w0 , T1
(15b)
(15c)
where u, v and w are the three parameters of Bloch vector M ðu; v; wÞ. w is the Rabi frequency, in the case of nearresonant excitation, m E0 w ¼ ab , (16) _ where g ¼ 1=T 1 , gab ¼ 1=T 2 ; T1 and T2 are the longitudinal and transversal relaxation time of energy level respectively. D is the deviation of frequency between the irradiating light field frequency and the resonant frequency of the material, usually it is zero in the special case of a near-resonant excitation by a polarized quasi-monochromatic field. We can solve these three Eqs. (15a–c); the analytic solution as following: 0 wD 1 0 1 ð1 cos gtÞ g2 u B C w B C B C (17) g sin gt @ v A ¼ egt w0 B C, @ A w2 w 1 g2 ð1 cos gtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g ¼ w2 þ D2 is the rotational velocity of processional motion. We can calculate the induced effective polarization in the QD as Z 1 m w gt0 pffiffiffi e Peff ¼ mab v dvz ¼ ab v r p Z11 sin gt0 dvz , ðN 0a N 0b Þ expðv2z =v2r Þ ð18Þ g 1 where vz is the velocity component of atoms’ thermal motion at the direction of the laser pulse. vr is the rootmean-square value of the velocity of atoms’ thermal motion, and N 0a N 0b is the difference of particle population in unit volume between two energy levels in initial state. The numerical solution of the induced polarization can obtain by solving Eqs. (17), (18). The transient nutation
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signal can be calculate as 1 I S ¼ E 0 oLPeff ðt tÞ (19) 2 where o is the frequency of the incident laser pulse. L is the thickness of the sample and t is the time that the laser pulse takes to cross the sample. 3. Results and discussion Suppose the QD is interacted with a 150 femtosecond laser pulse under a near-resonant excitation, whose electric field amplitude E 0 ¼ 5 106 V=m, which is under weak excitation regime and low temperature [16], and N 0b ¼ 0. The parameters used in the calculation are as follows: mne;CdSe ¼ 0:13m0 , mne;ZnS ¼ 0:28m0 (m0 is the rest mass of electron), V c ¼ 0:9 eV [13,17], the relaxation times T 1 ¼ T 2 ¼ 300 fs, and N 0a N 0b ¼ 5 1024 m3 . First of all, while R1 ¼ 4.5 nm, R2 ¼ 5 nm we have calculated the electric transition dipole moment for the 1s–2s transition using Eq. (14). When the electron locates outside the well, i.e., E4V c , mab ¼ 1:58817 1028 C m; and mab ¼ 1:48911 1028 C m for the electrons inside the well. The optical nutation signals are got from Eq. (18), and displayed in Fig. 2. Fig. 2 shows that the rapid decay of the nutation signals both outside the well (a) and inside the well (b) (arising due to the transient polarization induced in the QD, and decaying for the reason that the absorption coefficient is directly proportional to the population difference which is decreasing rapidly). Similar decay behavior was reported in QWRs and micro-cavity in Refs. [10,11,18]. Secondly, when one of R1 and R2 keeps steady and the other changes, we may see some interesting phenomena about the nutation signal change. In Fig. 3, we may see the three optical nutation curves almost overlap with each other. For compare and clarity, we plot the curve b alone which is almost covered by curve
Fig. 2. The optical nutation signals of the electrons outside (a) and inside (b) the well while R1 ¼ 4.5 nm and R2 ¼ 5 nm.
Fig. 3. The optical nutation signal of the electrons inside the well while the outer radius R2 ¼ 5 nm (a), R2 ¼ 6 nm (b), R2 ¼ 7 nm (c) and the inner radius does not change R1 ¼ 4.5 nm.
c completely. This illustrates that the optical nutation signal of the electrons inside the well almost does not change when the outer radius R2 changed, i.e., the change of the shell’s thickness has little influence on the nutation signal of the electrons inside the well. However, it’s different for the electrons outside the well. Fig. 4 reveals the fact that the Rabi frequency of the oscillation increases with the growth of the thickness of the shell while the size of the core does not change. According to Eqs. (16), (18), the immediate cause for this is the increase or decrease of their electric transition dipole moment with the change of the size and structure. It is owing to the QSCE in which most of the electrons are confined in the well and for the electrons inside the well, the change of R2 has little influence on their wave function. The electric transition dipole moment almost does not change (Fig. 3). But with the increase of R2, the probability of electrons presenting outside the well enhances. The wave function of the electrons outside the well will change a lot and the band gap will reduce [12]. Then the electric transition dipole increases [19] and the Rabi frequency will increase too (Fig. 4). For the electrons inside the well, the Rabi frequency of the optical nutation signal decreases rapidly with the reducing of the inner radius R1 while R2 ¼ 5 nm does not change, shown in Fig. 5. The reason is the same as the above. Due to the fact that the QSCE is enhanced inside the well and decreased outside the well, the probability of electrons presenting inside the well decreases greatly. The wave function of electrons inside the well changes a lot, and the electric transition dipole moment will reduce greatly. Thus the Rabi frequency will decrease according to Eq. (16). Compared with the results in Fig. 4, it can be found from Fig. 5 that the change of the signal’s Rabi frequency is more sensitive to the size of the core than to that of the shell.
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Fig. 4. The optical nutation signal of the electrons outside the well while the outer radius changes and the inner radius R1 ¼ 4.5 nm.
Fig. 6. The optical nutation signal of the electrons outside the well while the inner radius R1 changes and the outer radius R2 ¼ 5 nm.
shell will decrease and more electrons will enter into the shell. The wave function of the electrons outside the well will change a lot, and the band gap for 1s–2s transition will reduce too [12]. This will make the electric transition dipole moment improved greatly. So this phenomenon happens. 4. Conclusion
Fig. 5. The optical nutation signal of the electrons inside the well while the inner radius R1 changes and the outer radius R2 ¼ 5 nm.
However, the Fig. 6 shows us a different result. In the case of R1 reduced from 4.5 to 3.5 nm and R2 kept steady, the Rabi frequency of optical nutation signal of the electrons outside the well reduces a little first, and then increases a lot. This tells us the fact that, for the optical nutation of the electrons outside the well, there is almost no influence while the inner radius changes in some region, and the influence becomes strong when the inner radius is less than some threshold value. It can be explained as follows. Though the change of optical nutation signal of the electrons outside the well is mainly determined by the change of the shell’s thickness (Fig. 4), to reduce the size of the core, the QSCE in the core will enhance, and the function of preventing the electrons from relaxing into the shell will be promoted, and the band gap will increase [12]. Then the electric transition dipole moment will decrease a little first. On the other hand, the thickness of the shell layer turns wider while R2 keeps steady. The QSCE in the
In conclusion, theoretical analysis of the optical nutation signal of a core-shell CdSe/ZnS QD has been presented in this paper based on the effective-mass approximation and two energy level model. Especially the influence of the change of the QDs size parameter on the optical nutation signals of the 1s–2s transition of the electrons inside the well and outside the well have been calculated respectively. Numerical calculation results reveal that the nutation signals are greatly dependent on the size and the structure of the QD. We think this work is helpful to study the relaxation mechanisms and hot-carrier effects in materials, and to apply this kind of QDs into optical devices and biologic probe field. Acknowledgment This work was financially supported by the National Natural Foundation of China under the granted No. 10534030. References [1] M.A. Haase, J. Qiu, J.M. DePuydt, H. Cheng, Appl. Phys. Lett. 59 (1991) 1272–1274. [2] S. Taniguchi, T. Hino, S. Itoh, et al., Electron Lett. 32 (1996) 552–553. [3] Z.H. Zheng, Z.P. Guan, L.C. Chen, et al., Japan J. Appl. Phys 34 (1995) 56–58. [4] T. Uematsu, J. Kimura, Y. Yamaguchi, Nanotechnology 15 (2004) 822–827.
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