- Email: [email protected]
CdSe quantum dot quantum well
ARTICLE IN PRESS Physica E 42 (2010) 2178–2183
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Quantum size dependent optical nutation in CdSe/ZnS/CdSe quantum dot quantum well Yinan Fang a, Meng Xiao a, Duanzheng Yao a,b,n a b
Department of Physics, Wuhan University, Wuhan 430072, China Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education, Wuhan University, Wuhan 430072, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 October 2009 Received in revised form 30 March 2010 Accepted 31 March 2010 Available online 10 April 2010
The energy spectra and the optical nutation effect in CdSe/ZnS/CdSe quantum dot quantum well with core/shell/shell structure have been numerically studied and analyzed based on quantum theory and the effective mass approximation. For solving the uncertain coefficients A0, A1, B0, B1, A2, B2 and E, a new method has been used in the paper. It has been shown that the optical nutation signals of the CdSe/ZnS/CdSe quantum dot quantum well are dependent on the quantum size of the quantum dot quantum well and there exists an optimized condition for induced nutation signal, which is quite sensitive to the shells’ sizes of the QDQW with the core/shell/shell structure. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quantum dot quantum well Effective mass approximation Optical nutation Quantum size effect
1. Introduction
2. Model and theory
Quantum dot quantum well (QDQW) with the core/shell/shell structure is becoming one of the nano-scientists’ greatest interests because of its enormous wonderful properties that traditional semiconductor materials do not have. And those properties make QDQW a promising applicant on many areas [1]. Due to its small geometry size [2], it may replace today’s semiconductor units as the demand of minimizing the size of electronic unit is increasing. Some articles reported on how QDQW’s size influences its properties, for example, Haus [3] and Kim and Kish [4] reported the quantum confinement effect in semiconductor. Gong et al. [5] reported the optical nutation effect in a CdSe/ZnS QDQW with the core/shell structure, and Zhang et al. [6] reported the size dependent stimulated photon echo effect in the QDQW. However, there is still lack of reports on the photon echo in the QDQW with the core/shell/shell structure. Within the frame of the effective mass approximation and the density matrix method, the size dependent energy spectra and the optical nutation signals in CdSe/Zns/CdSe QDQW have been studied numerically and analyzed, and some useful results have been obtained in this paper.
Consider a CdSe/ZnS/CdSe QDQW with a core/shell/shell structure and potentials of spherical symmetrization, which depend on the radius size from the center of the QDQW. And r1, r2 and r3 stand for the radii of the core, the middle shell and the outer shell, respectively, shown in Fig. 1. We assume those potentials are constants [7]. ¨ The stationary Schrodinger equation of electrons used in the analyses reads " # _2 2 r þ VðrÞ cnlm ðr, y, jÞ ¼ Ecnlm ðr, y, jÞ, i ¼ 0,1 ð1Þ 2mi
n Corresponding author at: Department of Physics, Wuhan University, Wuhan 430072, China. Tel.: + 86 27 68752989; fax: + 86 27 68752569. E-mail address: [email protected] (D. Yao).
1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.03.036
where m0 and m1 are the effective masses of electrons in CdSe and ZnS, respectively, V(r) equals Vc (the conduction-band offset of ZnS [3]) when i¼1 and V(r) equals zero when i ¼0; cnlm(r, y, j) is the wave function of electron under spherical coordinates, E is the energy eigenvalue of electron. Using separating variables in Eq. (1), we have
cnlm ðr, y, jÞ ¼ Rnl ðrÞYlm ðy, jÞ
ð2Þ
where Rnl(r) is the solution of the radial part of Eq. (1), Ylm(y, j) the spherical harmonics function and it is the solution of the angular part of Eq. (1), n the principal quantum number, and l and m the angular momentum quantum numbers. Rnl(r) should satisfy the following boundary, convergence and normalization conditions [8,9]: Rnl,0 ð0Þ o 1
ð3Þ
ARTICLE IN PRESS Y. Fang et al. / Physica E 42 (2010) 2178–2183
8 pffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2m0 E=_ k > > < nl,0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi knl,1 ¼ 2m1 ðEVc Þ=_ > pffiffiffiffiffiffiffiffiffiffiffiffi > : knl,2 ¼ 2m E=_
2179
for
E4 Vc
ð13Þ
0
ðxÞ are the spherical Bessel where jl ðxÞ, nl ðxÞ, hðl þ Þ ðxÞ and hðÞ l function, Neumann function, and Hankel functions of the first and the second kind. _ is the Planck constant. If the coefficients satisfy Eqs. (3)–(9), then the left-hand side of those equations will be equal to the right-hand side of it. Thus the difference between them of those equations can be minimized. Inserting any five of those equations into the remaining one, an equation with only one variable E can be derived, which is called the idea equation. So, if E belongs to energy spectra, then the difference between left-hand and right-hand sides of the idea equation must minimized. If we define this difference as a function of E, this is called the target function T(E). Then, every intersected point between E and T(E) ought to be an energy eigenvalue. Therefore the whole spectra are got. There are many approaches to build target function. Now we illustrate a possible way to construct such a function here. We found that Eqs. (3)–(8) are linear combinations of A0, A1, B0, B1, A2 and B2, therefore, they can be expressed as
Fig. 1. CdSe/ZnS/CdSe QDQW.
MðEÞC ¼ 0 Rnl,0 ðr1 Þ ¼ Rnl,1 ðr1 Þ
ð4Þ
1 dRnl,0 1 dRnl,1 ¼ r ¼ r1 r ¼ r 1 m0 dr m1 dr
ð5Þ
Rnl,1 ðr2 Þ ¼ Rnl,2 ðr2 Þ
ð6Þ
1 dRnl,1 1 dRnl,2 r ¼ r21 r ¼ r21 ¼ m1 dr m2 dr
ð7Þ
Rnl,2 ðr3 Þ ¼ 0
ð8Þ
Z
R1 0
Rnl,2 ðR3 Þ ¼ 0
Rnl,0 ðrÞRnl,0 ðrÞr 2 dr þ
Z
R2
R1
Z
R3
þ R2
H ¼ H0 þ H 0
Rnl,2 ðrÞRnl,2 ðrÞr 2 dr ¼ 1
ð9Þ
3
ð10Þ 8 A j ðk rÞ þB0 nl ðknl,0 rÞ > > > 0 l nl,0 > < A1 jl ðknl,1 rÞ þB1 nl ðknl,1 rÞ Rnl ðrÞ ¼ > A2 jl ðknl,2 rÞ þB2 nl ðknl,2 rÞ > > > :0
r o r1 r 1 o r o r2 r 2 o r o r3 r3 o r
ð15Þ
where H0 is the unperturbed Hamiltonian and H0 is the interaction Hamiltonian. We assume that the dipole approximation is satisfied and the amplitude of incident laser pulse is indicated by E0, then ð16Þ
where m is the electric dipole between two states, which is defined as
The extra sub-index here illustrates that Rnl(r) will take different forms when r belongs to different layers of the QDQW. Inserting Eq. (2) into (1), the motion equations of the radial part and the angular part can be obtained immediately. The solutions for the radial part of Eq. (1) read [10] 8 A0 jl ðknl,0 rÞ þB0 nl ðknl,0 rÞ r o r1 > > > > < A1 hðl þ Þ ðknl,1 rÞ þB1 hðÞ ðk rÞ r 1 or o r2 nl,1 l for E oVc Rnl ðrÞ ¼ > A j ðk rÞ þB n ðk rÞ r 2 l nl,2 2 or o r3 > 2 l nl,2 > > :0 r or
for
E 4Vc
and the coefficients in Eqs. (10) and (11) are as follows: 8 pffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2m0 E=_ k > > < nl,0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi knl,1 ¼ 2m1 ðVc EÞ=_ for E oVc > pffiffiffiffiffiffiffiffiffiffiffiffi > : knl,2 ¼ 2m E=_ 0
where C ¼(A0, A1, B1, A2, B2)T (B0 must equal zero, because Neumann function is singular at origin), M(E) is a matrix and only depends on E. If we define T(E)¼detM (E) (Appendix I), then E can be determined by solving equation T(E)¼ 0. The Hamiltonian for the two energy systems (i.e. states 9a 4 and 9b4) is
H0 ¼ mE0 Rnl,1 ðrÞRnl,1 ðrÞr 2 dr
ð14Þ
ð11Þ
m ¼ o a9er9b 4
ð17Þ
In order to calculate the polarization of atoms caused by the incident beam, the motion equation for the density operator r reads [11] dr 1 ¼ H, r dt i_
ð18Þ
To solve Eq. (18), we referred to the Bloch vector and its motion equation, which is an equivalent expression of Eq. (18). Inserting the relation between the density operator r and Bloch vector B (u, v, w) into Eq. (18) and based on the rotating-wave approximation [11], Eq. (18) can be rewritten as 10 1 0 1 0 0 1=T2 D u u dB C B CB C D 1=T O v ð19Þ 2 A@ v A @ A¼@ dt 0 O 1=T1 w w where T1 and T2 are longitudinal and transversal relaxation time;
O ¼ mE0 =_ the Rabi frequency, D ¼9o o09 in which o is frequency of the incident laser pulse and o0 is the eigentransition frequency, w0 reveals the electronic density difference between the states 9b 4 and 9a 4 at the initial time. Suppose T1 ¼T2 ¼T, then Eq. (19) has following solutions [11]: ð12Þ u ¼ egt
w0 OD ð1cos gtÞ g2
ð20Þ
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v ¼ e g t
Y. Fang et al. / Physica E 42 (2010) 2178–2183
w0 O sin gt g "
w ¼ egt w0 1
O2
ð21Þ # ð1cos gtÞ
ð22Þ g2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g ¼ O2 þ D2 , the exponential terms reveal the relaxation effect and g ¼1/T, u and v contribute to two orthogonal vectors in radiation field [12], and w stands for the particle density difference between states 9b4 and 9a4 [11]. Here we assume that values of w in each layer of QDQW are the same. Notice that the contribution to the optical nutation signal given by u is less than that given by v [12], hence we ignore it. If we assume that vz (projection of atoms’ velocity, which is caused by thermal vibration, at the beam transmission direction) obeys the same distribution at any time as it was in the equilibrium state then the effective polarization Peff, taking Doppler effect into account, can be derived based on the slowly varying amplitude approximation [12,13] Z 2 sin gt mO Peff ¼ pffiffiffiffi egt w0 eðvz =vr Þ ð23Þ dvz g vr p R
If we assume that the second term in the integration of Eq. (23) is independent of vz and D ¼0. After integrating Eq. (23) we get pffiffiffiffiffiffiffiffiffi Peff ¼ m Opt egt w0 J1=2 ðOtÞ ð24Þ where Ju(x) is the u-order Bessel function. According to the equations of signal field [13], the intensity of the nutation signal can be expressed as follows: Inutation ¼
1 oLPeff ðtÞE0 2
ð25Þ
where L is the effective interaction length.
3. Results and analyses The parameters used in the calculation are as follows: l ¼0, m¼0, Vc ¼0.9 eV [3], w0 ¼5 1024/m3, m0 ¼ 0:13me ,m1 ¼ 0:28me (here me is the rest mass of electron). The energy spectra (R1 ¼4 nm, R2 ¼4.5 nm, R3 ¼5 nm), solid for E10, E20 when EoVc, dashed for E10, E20 when E4Vc, are shown in Fig. 2 by looking at intersecting point (from left to right) with T(E)¼0. The comparison on the energy spectra between the QDQWs with different sizes have been depicted in Fig. 3(a)–(d).
Fig. 2. T(E) ¼ detM as a function of E when R1 ¼4 nm, R2 ¼ 4.5 nm and R3 ¼5 nm.
It has been seen from the results that the energy spectra are hardly changeable without modifying the QDQW’s sizes of the core and the outer shells. This indicates that variation of R2 will not induce big influences on the nutation signal. In Fig. 3(d), a new energy state (E4 Vc) has emerged obviously, which indicates that the electric dipole involved this state will be quite different between the situations in Figs. 2 and 3(d), and can lead to big difference in the radiation fields. Therefore, the nutation signals in those two cases will be quite different too. In order to get the optical nutation in CdSe/ZnS/CdSe QDQW, suppose the length of sample is 1 mm, the relaxation time is 300 fs and the amplitude of incident beam is 5 106 V m [5]. The electric dipole induced by the transition between 1s state in the inner layer of potential (inner) and 1s state in the outer layer of potential was then calculated, its value is 4.90342818792078 1028 C m for R1 ¼4 nm, R2 ¼4.5 nm, R3 ¼5 nm. The optical nutations signals as a function of time under different sizes of the QDQW at the 1s (inner)–1s (outer) transition have been depicted in Fig. 4(a)–(d). It has been shown from Fig. 4 that the QDQWs with different sizes will be different both in the first peak value of the nutation signals and the time length between two neighbored zero points. And the bigger peak value is corresponding to the shorter time. It is due to the fact that the Rabi frequency is proportional to the electric dipole and the peak values of the nutation signals are proportional to the effective polarization, which is proportional to the Rabi frequency. Therefore, the first peak value of the nutation signal is also proportional to the electric dipole, which is usually dependent on the QDQW’s size. So maximum amplitude of effective polarization (MAEP), which is related to the first peak value of the nutation signal by (25), could be used to characterize the total nutation signals. MAEP induced by the transition between 1s (inner) and 1s (outer) under different conditions has been calculated and shown in Fig. 5(a)–(c). As a function of radius, MAEP has a very sharp peak value. This result indicates that the nutation signal in the QDQW can be optimized and this is caused by separated and controllable energy spectra of the QDQW ultimately because MAEP is proportional to the electric dipole, which is decided by Eq. (17). Therefore, the bigger the integration in Eq. (17) is, the bigger the nutation signal will be. The squared radial part of the wave functions both at E 4Vc and at EoVc with different QDQW’s sizes has been shown in Fig. 6. The solid and long dashed-lines represent squared radial wave function of 1s (inner) and 1s (outer) while R1 ¼2.5 nm, R2 ¼4.5 nm and R3 ¼5 nm, respectively. Short dashed and dash– dot lines represent squared radial wave function of 1s (inner) and 1s (outer) while R1 ¼2.7 nm, R2 ¼4.5 nm and R3 ¼5 nm. In the inset of the figure, the solid line indicates nutation signal during 1s (inner)–1s (outer) transition while R1 ¼2.5 nm, R2 ¼4.5 nm and R3 ¼5 nm, dashed line indicates the nutation signal during 1s (inner)–1s (outer) transition while R1 ¼2.7 nm, R2 ¼4.5 nm and R3 ¼5 nm. If the two wave functions (especially those valleys) cover each other, i.e. short dashed and dash–dot lines in Fig. 6, then such two wave functions can contribute bigger value to Eq. (17). Therefore electric dipole will be larger in such situation, and the nutation signal will increase, i.e. the short dashed line in the inset of Fig. 6. Although the short dashed line in Fig. 6 has a valley around 1.5 nm while the dash–dot line has not, it will not contribute too much to the electric dipole because the function in the integration of Eq. (17) is proportional to the cubic square of r under spherical coordinates. There is a paper [14], entitled optical nutation induced by transition between levels inside and outside the well in a core-shell CdSe/ZnS quantum dot, published in Journal of Physics: Condensed Matter, in which the energy eigenvalues
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Fig. 3. T(E) ¼detM as a function of E under different conditions (intersecting point with T(E)¼ 0 indicates corresponding energy levels, solid for E10, E20 when E oVc and long dashed for E10, E20, etc. when E 4Vc): (a) R1 ¼ 3.5 nm, R2 ¼4.5 nm and R3 ¼ 5 nm; (b) R1 ¼4 nm, R2 ¼ 4.25 nm and R3 ¼ 5 nm; (c) R1 ¼ 4 nm, R2 ¼4.75 nm and R3 ¼5 nm; (d) R1 ¼ 4 nm, R2 ¼4.5 nm and R3 ¼5.5 nm.
Fig. 4. Rapid decay of nutation signal during 1s (inner) and 1s (outer) transmission as a function of time under different situations have been depicted in (a), (b), (c), and (d). (a) R1 ¼ 4 nm, R2 ¼4.5 nm and R3 ¼ 5 nm (solid); (b) R1 ¼4 nm, R2 ¼ 4.5 nm and R3 ¼5.5 nm (long dashed); (c) R1 ¼ 4 nm, R2 ¼ 4.25 nm and R3 ¼5 nm (long dashed) and R1 ¼4 nm, R2 ¼4.75 nm and R3 ¼ 5 nm (short dashed); (d) R1 ¼3.5 nm, R2 ¼ 4.5 nm and R3 ¼ 5 nm (long dashed).
and eigenfunctions of a core-shell CdSe/ZnS quantum dot structure have been obtained under an effective mass approximation. The electric transition dipole moment induced by the 1s
(inside the well)–1s (outside the well) transition has been calculated. The optical nutation signal induced by the transition has been calculated numerically based on the optical Bloch
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Fig. 5. Maximum amplitude of effective polarization (MAEP) during 1s (inner) and 1s (outer) transmission as a function of QDQW’s radius under different situations has been depicted in (a), (b) and (c). (a) R1 from 2.5 to 4.4 nm with step length of 0.1 nm (solid) and 0.05 nm (long dashed) while R2 ¼ 4.5 nm and R3 ¼ 5 nm; (b) R2 from 4.5 to 4.9 nm with step length of 0.1 (solid) and 0.05 nm (long dashed) while R1 ¼ 4 nm and R3 ¼ 5 nm; (c) R3 from 5 to 5.9 nm with step length of 0.1 (solid) and 0.05 nm (long dashed) while R1 ¼4 nm and R2 ¼ 4.5 nm.
Fig. 6. Squared radial part of wave functions under different situations and corresponding nutation signal as a function of time (subplot) are depicted.
equations. It has been shown from the results that the optical nutation signal is sensitive to the size and structure, and that there is an optimal structure and size for the optical nutation phenomenon. The quantum size dependent Rabi frequency and intensity of the optical nutation are also discussed. Comparing with the paper, in this paper, the energy spectra and the optical nutation effect in CdSe/ZnS/CdSe quantum dot quantum well with
the core/shell/shell structure have been numerically studied and analyzed based on the quantum theory and the effective mass approximation. For solving the uncertain coefficients A0, A1, B0, B1, A2, B2 and E, a new method has been used. It is shown that the model and the structure are different and the method used in the paper is significant, which can be used for core-shell quantum dots with multiple-layers.
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4. Conclusion
References
Optical nutation effect in CdSe/ZnS/CdSe quantum dot quantum well (QDQW) with core/shell/shell structure has been numerically studied based on the quantum theory and the effective mass approximation. Energy spectra of this QDQW have been numerically calculated and analyzed. For solving the uncertain coefficients A0, A1, B0, B1, A2, B2 and E, a new method has been used in the paper. It is also shown from the final results that there exists an optimized nutation signal, which is quite sensitive to the shells’ sizes of the QDQW with core/shell/shell structure.
[1] Daniel Loss, David P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [2] Qu Quan Wang, in: Quantum Information and Computation of Semiconductor Quantum Dots, University of Science and Technology of China Press, 2009. [3] Joseph W. Haus, Phys. Rev. B. 47 (1993) 1359. [4] Jong U. Kim, Laszlo B. Kish, Phys. Lett. A 323 (2004) 16. [5] S.H. Gong, D.Z. Yao, X.B. Feng, H.L. Jiang, Microelectron. J. 37 (2006) 904. [6] Rongjun Zhang, Duanzheng Yao, Hongliang Jiang, Hua Xiao, J. Phys. Soc. Jpn. 78 (2009) 054402. [7] D. Schooss, A. Mews, A. Eychmuller, H. Weller, Phys. Rev. B. 49 (1994) 17072. [8] D.J. Ben Daniel, C.B. Duke, Phys. Rev. 152 (1966) 683. [9] L.E. Brus, J. Chem. Phys. 79 (1983) 5566. [10] J.W. Haus, H.S. Xhou, I. Honma, H. Komiyama, Phys. Rev B47 (1993) 1359. [11] Marlan.O. Scully, M. Suhail Zubairy, in: Quantum Optics, Cambridge Press, 2001 Chapter 5. [12] Rong Hui Xia, in: A Guidance to Spectra of Molecule and Laser, East China Normal University Press, 1980 Chapter 11. [13] Guiguang Xiong, in: Nonlinear Optics, Wuhan University Press, 1995 (Chapter 7). [14] Shaohua Gong, Duanzheng YaoJ. Phys.: Condens. Matter 18 (2006) 10989.
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant no. J0830310).
Appendix I: Expression of the target function in Section 2.2 Here are the expressions of the target functions T(E) for solving Eq. (1) with three different radii. For the first situation (EoVc), the target function T1(E) is
j ðk 0 r Þ l n ,0 1 m1 d m dr jl ðknl,0 rÞ9r ¼ R1 0 T1 ðEÞ ¼ 0 0 0
hðl þ Þ ðkn0 ,1 r1 Þ
hðÞ ðkn0 ,1 r1 Þ l
0
0
d ðÞ h ðknl,1 rÞ9r ¼ R1 dr l
0
0
m2 d ð þ Þ h ðknl,1 rÞ9r ¼ R2 m1 dr l
m2 d ðÞ h ðknl,1 rÞ9r ¼ R2 m1 dr l
hðl þ Þ ðknl,1 r2 Þ 0
hðÞ ðknl,1 r2 Þ l 0
jl ðknl,2 r2 Þ jl ðknl,2 r3 Þ
d ðþÞ h ðknl,0 rÞ9r ¼ R1 dr l
d j ðk rÞ9 dr l nl,2 r ¼ R2
d n ðk rÞ9 dr l nl,2 r ¼ R2
nl ðknl,2 r2 Þ nl ðknl,2 r3 Þ
And for the second situation (E4Vc), the target function T2(E) is
jl ðknl,0 r1 Þ m d 1 jl ðknl,0 rÞ9r ¼ R1 m0 dr T2 ðEÞ ¼ 0 0 0
jl ðkn0 ,1 r1 Þ d jl ðknl,1 rÞ9r ¼ R1 dr m2 d j ðk rÞ9 m1 dr l nl,1 r ¼ R2
nl ðkn0 ,1 r1 Þ d nl ðknl,1 rÞ9r ¼ R1 dr m2 d n ðk rÞ9 m1 dr l nl,1 r ¼ R2
0
0
0
0
jl ðknl,1 r2 Þ 0
nl ðknl,1 r2 Þ 0
jl ðknl,2 r2 Þ jl ðknl,2 r3 Þ
d j ðk rÞ9 dr l nl,2 r ¼ R2
d n ðk rÞ9 dr l nl,2 r ¼ R2
nl ðknl,2 r2 Þ nl ðknl,2 r3 Þ
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Quantum size dependent optical nutation in CdSe/ZnS/CdSe quantum dot quantum well Yinan Fang a, Meng Xiao a, Duanzheng Yao a,b,n a b
Department of Physics, Wuhan University, Wuhan 430072, China Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education, Wuhan University, Wuhan 430072, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 October 2009 Received in revised form 30 March 2010 Accepted 31 March 2010 Available online 10 April 2010
The energy spectra and the optical nutation effect in CdSe/ZnS/CdSe quantum dot quantum well with core/shell/shell structure have been numerically studied and analyzed based on quantum theory and the effective mass approximation. For solving the uncertain coefficients A0, A1, B0, B1, A2, B2 and E, a new method has been used in the paper. It has been shown that the optical nutation signals of the CdSe/ZnS/CdSe quantum dot quantum well are dependent on the quantum size of the quantum dot quantum well and there exists an optimized condition for induced nutation signal, which is quite sensitive to the shells’ sizes of the QDQW with the core/shell/shell structure. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quantum dot quantum well Effective mass approximation Optical nutation Quantum size effect
1. Introduction
2. Model and theory
Quantum dot quantum well (QDQW) with the core/shell/shell structure is becoming one of the nano-scientists’ greatest interests because of its enormous wonderful properties that traditional semiconductor materials do not have. And those properties make QDQW a promising applicant on many areas [1]. Due to its small geometry size [2], it may replace today’s semiconductor units as the demand of minimizing the size of electronic unit is increasing. Some articles reported on how QDQW’s size influences its properties, for example, Haus [3] and Kim and Kish [4] reported the quantum confinement effect in semiconductor. Gong et al. [5] reported the optical nutation effect in a CdSe/ZnS QDQW with the core/shell structure, and Zhang et al. [6] reported the size dependent stimulated photon echo effect in the QDQW. However, there is still lack of reports on the photon echo in the QDQW with the core/shell/shell structure. Within the frame of the effective mass approximation and the density matrix method, the size dependent energy spectra and the optical nutation signals in CdSe/Zns/CdSe QDQW have been studied numerically and analyzed, and some useful results have been obtained in this paper.
Consider a CdSe/ZnS/CdSe QDQW with a core/shell/shell structure and potentials of spherical symmetrization, which depend on the radius size from the center of the QDQW. And r1, r2 and r3 stand for the radii of the core, the middle shell and the outer shell, respectively, shown in Fig. 1. We assume those potentials are constants [7]. ¨ The stationary Schrodinger equation of electrons used in the analyses reads " # _2 2 r þ VðrÞ cnlm ðr, y, jÞ ¼ Ecnlm ðr, y, jÞ, i ¼ 0,1 ð1Þ 2mi
n Corresponding author at: Department of Physics, Wuhan University, Wuhan 430072, China. Tel.: + 86 27 68752989; fax: + 86 27 68752569. E-mail address: [email protected] (D. Yao).
1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.03.036
where m0 and m1 are the effective masses of electrons in CdSe and ZnS, respectively, V(r) equals Vc (the conduction-band offset of ZnS [3]) when i¼1 and V(r) equals zero when i ¼0; cnlm(r, y, j) is the wave function of electron under spherical coordinates, E is the energy eigenvalue of electron. Using separating variables in Eq. (1), we have
cnlm ðr, y, jÞ ¼ Rnl ðrÞYlm ðy, jÞ
ð2Þ
where Rnl(r) is the solution of the radial part of Eq. (1), Ylm(y, j) the spherical harmonics function and it is the solution of the angular part of Eq. (1), n the principal quantum number, and l and m the angular momentum quantum numbers. Rnl(r) should satisfy the following boundary, convergence and normalization conditions [8,9]: Rnl,0 ð0Þ o 1
ð3Þ
ARTICLE IN PRESS Y. Fang et al. / Physica E 42 (2010) 2178–2183
8 pffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2m0 E=_ k > > < nl,0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi knl,1 ¼ 2m1 ðEVc Þ=_ > pffiffiffiffiffiffiffiffiffiffiffiffi > : knl,2 ¼ 2m E=_
2179
for
E4 Vc
ð13Þ
0
ðxÞ are the spherical Bessel where jl ðxÞ, nl ðxÞ, hðl þ Þ ðxÞ and hðÞ l function, Neumann function, and Hankel functions of the first and the second kind. _ is the Planck constant. If the coefficients satisfy Eqs. (3)–(9), then the left-hand side of those equations will be equal to the right-hand side of it. Thus the difference between them of those equations can be minimized. Inserting any five of those equations into the remaining one, an equation with only one variable E can be derived, which is called the idea equation. So, if E belongs to energy spectra, then the difference between left-hand and right-hand sides of the idea equation must minimized. If we define this difference as a function of E, this is called the target function T(E). Then, every intersected point between E and T(E) ought to be an energy eigenvalue. Therefore the whole spectra are got. There are many approaches to build target function. Now we illustrate a possible way to construct such a function here. We found that Eqs. (3)–(8) are linear combinations of A0, A1, B0, B1, A2 and B2, therefore, they can be expressed as
Fig. 1. CdSe/ZnS/CdSe QDQW.
MðEÞC ¼ 0 Rnl,0 ðr1 Þ ¼ Rnl,1 ðr1 Þ
ð4Þ
1 dRnl,0 1 dRnl,1 ¼ r ¼ r1 r ¼ r 1 m0 dr m1 dr
ð5Þ
Rnl,1 ðr2 Þ ¼ Rnl,2 ðr2 Þ
ð6Þ
1 dRnl,1 1 dRnl,2 r ¼ r21 r ¼ r21 ¼ m1 dr m2 dr
ð7Þ
Rnl,2 ðr3 Þ ¼ 0
ð8Þ
Z
R1 0
Rnl,2 ðR3 Þ ¼ 0
Rnl,0 ðrÞRnl,0 ðrÞr 2 dr þ
Z
R2
R1
Z
R3
þ R2
H ¼ H0 þ H 0
Rnl,2 ðrÞRnl,2 ðrÞr 2 dr ¼ 1
ð9Þ
3
ð10Þ 8 A j ðk rÞ þB0 nl ðknl,0 rÞ > > > 0 l nl,0 > < A1 jl ðknl,1 rÞ þB1 nl ðknl,1 rÞ Rnl ðrÞ ¼ > A2 jl ðknl,2 rÞ þB2 nl ðknl,2 rÞ > > > :0
r o r1 r 1 o r o r2 r 2 o r o r3 r3 o r
ð15Þ
where H0 is the unperturbed Hamiltonian and H0 is the interaction Hamiltonian. We assume that the dipole approximation is satisfied and the amplitude of incident laser pulse is indicated by E0, then ð16Þ
where m is the electric dipole between two states, which is defined as
The extra sub-index here illustrates that Rnl(r) will take different forms when r belongs to different layers of the QDQW. Inserting Eq. (2) into (1), the motion equations of the radial part and the angular part can be obtained immediately. The solutions for the radial part of Eq. (1) read [10] 8 A0 jl ðknl,0 rÞ þB0 nl ðknl,0 rÞ r o r1 > > > > < A1 hðl þ Þ ðknl,1 rÞ þB1 hðÞ ðk rÞ r 1 or o r2 nl,1 l for E oVc Rnl ðrÞ ¼ > A j ðk rÞ þB n ðk rÞ r 2 l nl,2 2 or o r3 > 2 l nl,2 > > :0 r or
for
E 4Vc
and the coefficients in Eqs. (10) and (11) are as follows: 8 pffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2m0 E=_ k > > < nl,0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi knl,1 ¼ 2m1 ðVc EÞ=_ for E oVc > pffiffiffiffiffiffiffiffiffiffiffiffi > : knl,2 ¼ 2m E=_ 0
where C ¼(A0, A1, B1, A2, B2)T (B0 must equal zero, because Neumann function is singular at origin), M(E) is a matrix and only depends on E. If we define T(E)¼detM (E) (Appendix I), then E can be determined by solving equation T(E)¼ 0. The Hamiltonian for the two energy systems (i.e. states 9a 4 and 9b4) is
H0 ¼ mE0 Rnl,1 ðrÞRnl,1 ðrÞr 2 dr
ð14Þ
ð11Þ
m ¼ o a9er9b 4
ð17Þ
In order to calculate the polarization of atoms caused by the incident beam, the motion equation for the density operator r reads [11] dr 1 ¼ H, r dt i_
ð18Þ
To solve Eq. (18), we referred to the Bloch vector and its motion equation, which is an equivalent expression of Eq. (18). Inserting the relation between the density operator r and Bloch vector B (u, v, w) into Eq. (18) and based on the rotating-wave approximation [11], Eq. (18) can be rewritten as 10 1 0 1 0 0 1=T2 D u u dB C B CB C D 1=T O v ð19Þ 2 A@ v A @ A¼@ dt 0 O 1=T1 w w where T1 and T2 are longitudinal and transversal relaxation time;
O ¼ mE0 =_ the Rabi frequency, D ¼9o o09 in which o is frequency of the incident laser pulse and o0 is the eigentransition frequency, w0 reveals the electronic density difference between the states 9b 4 and 9a 4 at the initial time. Suppose T1 ¼T2 ¼T, then Eq. (19) has following solutions [11]: ð12Þ u ¼ egt
w0 OD ð1cos gtÞ g2
ð20Þ
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Y. Fang et al. / Physica E 42 (2010) 2178–2183
w0 O sin gt g "
w ¼ egt w0 1
O2
ð21Þ # ð1cos gtÞ
ð22Þ g2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g ¼ O2 þ D2 , the exponential terms reveal the relaxation effect and g ¼1/T, u and v contribute to two orthogonal vectors in radiation field [12], and w stands for the particle density difference between states 9b4 and 9a4 [11]. Here we assume that values of w in each layer of QDQW are the same. Notice that the contribution to the optical nutation signal given by u is less than that given by v [12], hence we ignore it. If we assume that vz (projection of atoms’ velocity, which is caused by thermal vibration, at the beam transmission direction) obeys the same distribution at any time as it was in the equilibrium state then the effective polarization Peff, taking Doppler effect into account, can be derived based on the slowly varying amplitude approximation [12,13] Z 2 sin gt mO Peff ¼ pffiffiffiffi egt w0 eðvz =vr Þ ð23Þ dvz g vr p R
If we assume that the second term in the integration of Eq. (23) is independent of vz and D ¼0. After integrating Eq. (23) we get pffiffiffiffiffiffiffiffiffi Peff ¼ m Opt egt w0 J1=2 ðOtÞ ð24Þ where Ju(x) is the u-order Bessel function. According to the equations of signal field [13], the intensity of the nutation signal can be expressed as follows: Inutation ¼
1 oLPeff ðtÞE0 2
ð25Þ
where L is the effective interaction length.
3. Results and analyses The parameters used in the calculation are as follows: l ¼0, m¼0, Vc ¼0.9 eV [3], w0 ¼5 1024/m3, m0 ¼ 0:13me ,m1 ¼ 0:28me (here me is the rest mass of electron). The energy spectra (R1 ¼4 nm, R2 ¼4.5 nm, R3 ¼5 nm), solid for E10, E20 when EoVc, dashed for E10, E20 when E4Vc, are shown in Fig. 2 by looking at intersecting point (from left to right) with T(E)¼0. The comparison on the energy spectra between the QDQWs with different sizes have been depicted in Fig. 3(a)–(d).
Fig. 2. T(E) ¼ detM as a function of E when R1 ¼4 nm, R2 ¼ 4.5 nm and R3 ¼5 nm.
It has been seen from the results that the energy spectra are hardly changeable without modifying the QDQW’s sizes of the core and the outer shells. This indicates that variation of R2 will not induce big influences on the nutation signal. In Fig. 3(d), a new energy state (E4 Vc) has emerged obviously, which indicates that the electric dipole involved this state will be quite different between the situations in Figs. 2 and 3(d), and can lead to big difference in the radiation fields. Therefore, the nutation signals in those two cases will be quite different too. In order to get the optical nutation in CdSe/ZnS/CdSe QDQW, suppose the length of sample is 1 mm, the relaxation time is 300 fs and the amplitude of incident beam is 5 106 V m [5]. The electric dipole induced by the transition between 1s state in the inner layer of potential (inner) and 1s state in the outer layer of potential was then calculated, its value is 4.90342818792078 1028 C m for R1 ¼4 nm, R2 ¼4.5 nm, R3 ¼5 nm. The optical nutations signals as a function of time under different sizes of the QDQW at the 1s (inner)–1s (outer) transition have been depicted in Fig. 4(a)–(d). It has been shown from Fig. 4 that the QDQWs with different sizes will be different both in the first peak value of the nutation signals and the time length between two neighbored zero points. And the bigger peak value is corresponding to the shorter time. It is due to the fact that the Rabi frequency is proportional to the electric dipole and the peak values of the nutation signals are proportional to the effective polarization, which is proportional to the Rabi frequency. Therefore, the first peak value of the nutation signal is also proportional to the electric dipole, which is usually dependent on the QDQW’s size. So maximum amplitude of effective polarization (MAEP), which is related to the first peak value of the nutation signal by (25), could be used to characterize the total nutation signals. MAEP induced by the transition between 1s (inner) and 1s (outer) under different conditions has been calculated and shown in Fig. 5(a)–(c). As a function of radius, MAEP has a very sharp peak value. This result indicates that the nutation signal in the QDQW can be optimized and this is caused by separated and controllable energy spectra of the QDQW ultimately because MAEP is proportional to the electric dipole, which is decided by Eq. (17). Therefore, the bigger the integration in Eq. (17) is, the bigger the nutation signal will be. The squared radial part of the wave functions both at E 4Vc and at EoVc with different QDQW’s sizes has been shown in Fig. 6. The solid and long dashed-lines represent squared radial wave function of 1s (inner) and 1s (outer) while R1 ¼2.5 nm, R2 ¼4.5 nm and R3 ¼5 nm, respectively. Short dashed and dash– dot lines represent squared radial wave function of 1s (inner) and 1s (outer) while R1 ¼2.7 nm, R2 ¼4.5 nm and R3 ¼5 nm. In the inset of the figure, the solid line indicates nutation signal during 1s (inner)–1s (outer) transition while R1 ¼2.5 nm, R2 ¼4.5 nm and R3 ¼5 nm, dashed line indicates the nutation signal during 1s (inner)–1s (outer) transition while R1 ¼2.7 nm, R2 ¼4.5 nm and R3 ¼5 nm. If the two wave functions (especially those valleys) cover each other, i.e. short dashed and dash–dot lines in Fig. 6, then such two wave functions can contribute bigger value to Eq. (17). Therefore electric dipole will be larger in such situation, and the nutation signal will increase, i.e. the short dashed line in the inset of Fig. 6. Although the short dashed line in Fig. 6 has a valley around 1.5 nm while the dash–dot line has not, it will not contribute too much to the electric dipole because the function in the integration of Eq. (17) is proportional to the cubic square of r under spherical coordinates. There is a paper [14], entitled optical nutation induced by transition between levels inside and outside the well in a core-shell CdSe/ZnS quantum dot, published in Journal of Physics: Condensed Matter, in which the energy eigenvalues
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Fig. 3. T(E) ¼detM as a function of E under different conditions (intersecting point with T(E)¼ 0 indicates corresponding energy levels, solid for E10, E20 when E oVc and long dashed for E10, E20, etc. when E 4Vc): (a) R1 ¼ 3.5 nm, R2 ¼4.5 nm and R3 ¼ 5 nm; (b) R1 ¼4 nm, R2 ¼ 4.25 nm and R3 ¼ 5 nm; (c) R1 ¼ 4 nm, R2 ¼4.75 nm and R3 ¼5 nm; (d) R1 ¼ 4 nm, R2 ¼4.5 nm and R3 ¼5.5 nm.
Fig. 4. Rapid decay of nutation signal during 1s (inner) and 1s (outer) transmission as a function of time under different situations have been depicted in (a), (b), (c), and (d). (a) R1 ¼ 4 nm, R2 ¼4.5 nm and R3 ¼ 5 nm (solid); (b) R1 ¼4 nm, R2 ¼ 4.5 nm and R3 ¼5.5 nm (long dashed); (c) R1 ¼ 4 nm, R2 ¼ 4.25 nm and R3 ¼5 nm (long dashed) and R1 ¼4 nm, R2 ¼4.75 nm and R3 ¼ 5 nm (short dashed); (d) R1 ¼3.5 nm, R2 ¼ 4.5 nm and R3 ¼ 5 nm (long dashed).
and eigenfunctions of a core-shell CdSe/ZnS quantum dot structure have been obtained under an effective mass approximation. The electric transition dipole moment induced by the 1s
(inside the well)–1s (outside the well) transition has been calculated. The optical nutation signal induced by the transition has been calculated numerically based on the optical Bloch
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Fig. 5. Maximum amplitude of effective polarization (MAEP) during 1s (inner) and 1s (outer) transmission as a function of QDQW’s radius under different situations has been depicted in (a), (b) and (c). (a) R1 from 2.5 to 4.4 nm with step length of 0.1 nm (solid) and 0.05 nm (long dashed) while R2 ¼ 4.5 nm and R3 ¼ 5 nm; (b) R2 from 4.5 to 4.9 nm with step length of 0.1 (solid) and 0.05 nm (long dashed) while R1 ¼ 4 nm and R3 ¼ 5 nm; (c) R3 from 5 to 5.9 nm with step length of 0.1 (solid) and 0.05 nm (long dashed) while R1 ¼4 nm and R2 ¼ 4.5 nm.
Fig. 6. Squared radial part of wave functions under different situations and corresponding nutation signal as a function of time (subplot) are depicted.
equations. It has been shown from the results that the optical nutation signal is sensitive to the size and structure, and that there is an optimal structure and size for the optical nutation phenomenon. The quantum size dependent Rabi frequency and intensity of the optical nutation are also discussed. Comparing with the paper, in this paper, the energy spectra and the optical nutation effect in CdSe/ZnS/CdSe quantum dot quantum well with
the core/shell/shell structure have been numerically studied and analyzed based on the quantum theory and the effective mass approximation. For solving the uncertain coefficients A0, A1, B0, B1, A2, B2 and E, a new method has been used. It is shown that the model and the structure are different and the method used in the paper is significant, which can be used for core-shell quantum dots with multiple-layers.
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4. Conclusion
References
Optical nutation effect in CdSe/ZnS/CdSe quantum dot quantum well (QDQW) with core/shell/shell structure has been numerically studied based on the quantum theory and the effective mass approximation. Energy spectra of this QDQW have been numerically calculated and analyzed. For solving the uncertain coefficients A0, A1, B0, B1, A2, B2 and E, a new method has been used in the paper. It is also shown from the final results that there exists an optimized nutation signal, which is quite sensitive to the shells’ sizes of the QDQW with core/shell/shell structure.
[1] Daniel Loss, David P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [2] Qu Quan Wang, in: Quantum Information and Computation of Semiconductor Quantum Dots, University of Science and Technology of China Press, 2009. [3] Joseph W. Haus, Phys. Rev. B. 47 (1993) 1359. [4] Jong U. Kim, Laszlo B. Kish, Phys. Lett. A 323 (2004) 16. [5] S.H. Gong, D.Z. Yao, X.B. Feng, H.L. Jiang, Microelectron. J. 37 (2006) 904. [6] Rongjun Zhang, Duanzheng Yao, Hongliang Jiang, Hua Xiao, J. Phys. Soc. Jpn. 78 (2009) 054402. [7] D. Schooss, A. Mews, A. Eychmuller, H. Weller, Phys. Rev. B. 49 (1994) 17072. [8] D.J. Ben Daniel, C.B. Duke, Phys. Rev. 152 (1966) 683. [9] L.E. Brus, J. Chem. Phys. 79 (1983) 5566. [10] J.W. Haus, H.S. Xhou, I. Honma, H. Komiyama, Phys. Rev B47 (1993) 1359. [11] Marlan.O. Scully, M. Suhail Zubairy, in: Quantum Optics, Cambridge Press, 2001 Chapter 5. [12] Rong Hui Xia, in: A Guidance to Spectra of Molecule and Laser, East China Normal University Press, 1980 Chapter 11. [13] Guiguang Xiong, in: Nonlinear Optics, Wuhan University Press, 1995 (Chapter 7). [14] Shaohua Gong, Duanzheng YaoJ. Phys.: Condens. Matter 18 (2006) 10989.
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant no. J0830310).
Appendix I: Expression of the target function in Section 2.2 Here are the expressions of the target functions T(E) for solving Eq. (1) with three different radii. For the first situation (EoVc), the target function T1(E) is
j ðk 0 r Þ l n ,0 1 m1 d m dr jl ðknl,0 rÞ9r ¼ R1 0 T1 ðEÞ ¼ 0 0 0
hðl þ Þ ðkn0 ,1 r1 Þ
hðÞ ðkn0 ,1 r1 Þ l
0
0
d ðÞ h ðknl,1 rÞ9r ¼ R1 dr l
0
0
m2 d ð þ Þ h ðknl,1 rÞ9r ¼ R2 m1 dr l
m2 d ðÞ h ðknl,1 rÞ9r ¼ R2 m1 dr l
hðl þ Þ ðknl,1 r2 Þ 0
hðÞ ðknl,1 r2 Þ l 0
jl ðknl,2 r2 Þ jl ðknl,2 r3 Þ
d ðþÞ h ðknl,0 rÞ9r ¼ R1 dr l
d j ðk rÞ9 dr l nl,2 r ¼ R2
d n ðk rÞ9 dr l nl,2 r ¼ R2
nl ðknl,2 r2 Þ nl ðknl,2 r3 Þ
And for the second situation (E4Vc), the target function T2(E) is
jl ðknl,0 r1 Þ m d 1 jl ðknl,0 rÞ9r ¼ R1 m0 dr T2 ðEÞ ¼ 0 0 0
jl ðkn0 ,1 r1 Þ d jl ðknl,1 rÞ9r ¼ R1 dr m2 d j ðk rÞ9 m1 dr l nl,1 r ¼ R2
nl ðkn0 ,1 r1 Þ d nl ðknl,1 rÞ9r ¼ R1 dr m2 d n ðk rÞ9 m1 dr l nl,1 r ¼ R2
0
0
0
0
jl ðknl,1 r2 Þ 0
nl ðknl,1 r2 Þ 0
jl ðknl,2 r2 Þ jl ðknl,2 r3 Þ
d j ðk rÞ9 dr l nl,2 r ¼ R2
d n ðk rÞ9 dr l nl,2 r ¼ R2
nl ðknl,2 r2 Þ nl ðknl,2 r3 Þ