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Interband optical absorptions in a parabolic quantum dot
ARTICLE IN PRESS
Physica E 28 (2005) 14–21 www.elsevier.com/locate/physe
Interband optical absorptions in a parabolic quantum dot Guanghui Wanga,, Kangxian Guob a
Laboratory of Light Transmission Optics, South China Normal University, Guangzhou 510631, PR China b Department of Physics, Guihuagang Campus, Guangzhou University, Guangzhou 510405, PR China Received 5 January 2005; accepted 10 January 2005 Available online 25 April 2005
Abstract The linear and nonlinear optical absorptions considering excitonic effect in a parabolic quantum dot are studied. Analytic forms of the linear and third-order nonlinear optical absorption coefficients are obtained for a symmetric parabolic quantum dot using the compact density matrix formalism. Based on this model, numerical results are presented for a typical CdS parabolic quantum dot. The calculated results show that the factors of the parabolic potential confinement and the incident optical intensity have great influence on the optical absorption coefficients. Furthermore, the optical absorption saturation intensity can be controlled by adopting a proper parabolic confinement potential. r 2005 Elsevier B.V. All rights reserved. PACS: 78.66.w; 42.65.k; 78.20.Ci; 71.35.Aa; 42.65.An Keywords: Optical absorption; Parabolic quantum dot; Exciton; Density matrix approach
1. Introduction In the past few years, the nonlinear optical properties of confined excitons in nanostructures (including quantum wells, quantum wires and quantum dots) have attracted much attention [1–5]. One of the reasons is that excitonic spectrum has been observed at the room temperatures [6,7], which is due to the enhancement of the excitonic binding energy caused by the quantum confineCorresponding author.
E-mail address: [email protected] (G. Wang).
ment effect in these structures, and another reason is that quantum confinement of carriers in these low-dimensional semiconductor nanostructures lead to the formation of discrete energy levels, and the drastic change of optical absorption spectra [8] and many novel optical properties, which is not in the bulk materials. Furthermore, with recently quick advances of modern technology, it has now become possible to produce quasi-zero-dimension systems that confine electrons in all three spatial dimensions by using techniques such as etching or molecular beam epitaxy [9] etc. They typically have a disk-like
1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.01.018
ARTICLE IN PRESS G. Wang, K. Guo / Physica E 28 (2005) 14–21
shape, a few hundred nanometers in diameter, and a few nanometers thick. In such nanometer structures, electrons not only are confined in all three spatial dimensions, but also are quantized into discrete energy levels, with energy spacings of a few meV or more [10]. Recently, the quantum dots with (quasi-) zero-dimension structures are more and more important because of their novel physical properties and promise for potential applications [11,12]. Recently, the linear intersubband optical absorption within the conduction band of a GaAs quantum well has been studied experimentally without an electric field [13], and with an electric field [14]. A very large oscillator strength and a narrow bandwidth were observed. These suggest that the intersubband optical transitions in a quantum well may have very large optical nonlinearities. Nonlinear intersubband optical absorption in a semiconductor quantum well also was calculated by Ahn and Chuang in 1987 [15]. In 1991, Rappen et al. [16] studied the nonlinear absorption for two-dimensional magnetoexcitons in Inx Ga1x As=Iny Al1y As quantum wells. In 1992, Bockelman and Bastard [17] discussed interband absorption in quantum wires with magnetic-field case and without magnetic field case [18]. Intersubband optical absorption in coupled quantum wells under an applied electric field was studied by Perng-fei Yuh and Wang [19]. In 1993, Da-fu Cui et al. [20] studied absorption saturation of intersubband optical transitions in GaAs=Alx Ga1x As multiple quantum wells in experiment, and they tested the saturation optical intensity for I s ¼ 0:67 MW=cm2 , electronic relaxation time for 0.1 ps. From fundamental and practical points of view, these linear and nonlinear size-quantized transitions have the potential for device applications in far-infrared laser amplifiers, photodetectors, and high-speed electrooptical modulators. However, up to the present, no one else have studied the linear and nonlinear optical absorption in parabolic quantum wires and dots, and in the previous work, the excitonic effects were rarely considered. In this paper, the linear, third-order nonlinear and total optical absorption coefficients in a disk-like parabolic quantum dot are studied
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theoretically. In Section 2, the electronic states in the quantum dot are described. The basic electronic excitations are taken to be excitonic states, without considering collective excitations and electronic dipole transition matrix element is obtained. In Section 3 an analytical formula for the linear, third-order nonlinear and total optical absorption coefficients are derived by using the compact-density-matrix approach and an iterative method. In Section 4 we calculate and discuss the linear, third-order nonlinear and total optical absorption coefficients in the CdS parabolic quantum dot, the results show that the incident optical intensity and the parabolic confinement frequency have great influence on the optical absorption coefficients. Furthermore, the optical absorption saturation intensity will reduce as increase in parabolic confinement frequency under the condition of resonance. A brief summary is given in Section 5.
2. Electronic excitations in a parabolic quantum dot In the following, we will only discuss weakconfinement regime, namely R0 baex , where R0 is the radius of the quantum dot, aex is exciton effective Bohr radius. According to this condition, the electronic excited state in a quantum dot can then be described by the Frenkel exciton as [2,3] X Y cn ¼ Rn ðjÞW cj ðrj Þ W vi ðri Þ, (1) j
ðiajÞ
where W vi and W cj are Wannier functions of the valence and the conduction bands, respectively, and Rn ðjÞ is the envelope function in a disk-like parabolic quantum dot, given as 2m o0 n! 1=2 jmj jmj 2 r2 =2 1 imy pffiffiffiffiffiffi e , r Ln ðr Þe Rn ðrÞ ¼ _ðjmj þ nÞ! 2p (2) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ðm o0 =_Þr ¼ uj; u is the unit-cell size, the site index j is chosen from positive integer between 1 and N, m ¼ 0; 1; 2; . . . ; n ¼ 0; 1; 2; . . . ; Ljmj are the Laguerre polynomials, o0 n is the frequency of the parabolic confining potential.
ARTICLE IN PRESS G. Wang, K. Guo / Physica E 28 (2005) 14–21
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The excitation energy spectrum of the conduction electrons can then be expressed as E nm ¼ ð2n þ jmj þ 1Þ_o0 .
(3)
The ground state of the system is Y W vi ðri Þ. cg ¼
(4)
quantum dot can be written as follows: X Rn ðjÞjðj r ÞhW cj ðrj Þjpj jW vj0 ðrj 0 Þi hcn jPjcg i ¼ jj 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pðMo 0 =_ÞR0 _ r dr ¼ pcv jð0Þ Mo0 0 2Mo0 n! 1=2 jmj jmj 2 ð1=2Þr2
r Ln ðr Þe _ðjmj þ nÞ! Z 2p 1 pffiffiffiffiffiffi eimy dy,
ð8Þ 2p 0
i
For the parabolic conduction and valence bands with effective mass me , and mh , respectively, the size quantization is dominated by the relationship among the sample size R0 ¼ Nu, and the effective Bohr radii of electrons ae ¼ 0 _2 =me e2 and holes ah ¼ 0 _2 =mh e2 , where 0 is the static dielectric constant. After taking into account the Coulomb interaction between the electrons and holes, adopting the relative coordinate and center-of-mass coordinate, the excitation states in Eq. (1) should be modified to [2] XX Y cn ¼ Rn ðjÞjðj r ÞW cj ðrj Þ W vi ðri Þ, (5) j
j0
iaj 0
where Rn ðjÞ is the envelope function given in Eq. (2), presenting the centre-of-mass motion of the exciton with j ¼ ðme j þ mh j 0 Þ=ðme þ mh Þ, and jðj r Þ describes the electron–hole relative motion with j r ¼ j j 0 . We also define the total mass M ¼ me þ mh , the reduced mass m ¼ me mh =M, and the effective Rydberg constant R ¼ me4 =20 _2 . For the lowest state of the exciton, we can describe the two-dimensional electron–hole relative motion of the exciton as 4 jðrÞ ¼ pffiffiffiffiffiffi e2r=aB , 2paex
(6)
where r ¼ uðj j 0 Þ, is the electron–hole separation. The energy spectrum of excitons in the parabolic quantum dot reads R E nm ¼ E g 2 þ ð2n þ jmj þ 1Þ_o0 , 2 n þ jmj þ 12 (7) where E g is the band-gap energy. The transition dipole moment from the ground state cg to the excited states cn in the parabolic
where P is a component of the dipole moment operator P: X P¼ pi , (9) i
and hW cj ðrj Þjpj jW vj0 ðrj0 Þi ¼ pcv djj 0 [2]. From Eq. (8), we can see that the transition dipole matrix element will be zero unless m ¼ 0. After setting m ¼ 0, we have ffi pffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffi ðMo0 =_ÞR0 2p_ hcn jPjcg i ¼ pcv jð0Þ Mo0 0 2Mo0 1=2 2
r dr Ln ðr2 Þeð1=2Þr . _ ð10Þ So the transition dipole moment from the ground state jcg i to the lowest state of the exciton jc1 i can be written as Mo0 p 1=2 hc1 jPjcg i ¼ pcv jð0ÞR20 , (11) _ pffiffiffiffiffiffiffiffiffiffiffi where pcv ¼ _e= 4mE g , obtained in kp perturbation theory [21].
3. Linear and nonlinear optical absorption in a parabolic quantum dot In this section we will study the linear and thirdorder nonlinear optical absorption coefficients in the parabolic quantum dot. Let us consider an monochromatic field with frequency o which is incident with a polarization vector normal to the quantum dot. The system is excited by the
ARTICLE IN PRESS G. Wang, K. Guo / Physica E 28 (2005) 14–21
q ðnþ1Þ 1 ðnþ1Þ ðnÞ ree ¼ Gee ree ðPeg rðnÞ ge Pge reg ÞEðtÞ, qt i_ (18)
incident field EðtÞ ¼ E 0 Cos ot e iot . e iot þ Ee ¼ Ee
ð12Þ
Let a sign r denote one-electron density matrix operator for this system. Then the evolution of the density matrix operator r obeys the following time-dependent equation qr=qt ¼ ði_Þ1 ½H 0 qrEðtÞ; r Gðr rð0Þ Þ,
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(13)
where ½ ; is the quantum mechanical commutator, H 0 is the Hamiltonian for this system without the electromagnetic field EðtÞ; q is the electronic charge; rð0Þ is the unperturbed density matrix operator; G is the phenomenological operator. Diagonal matrix element Gmm of operator G, which is called as intersubband relaxation rate, is the inverse of the relaxation time T m for the state jmi, namely Gmm ¼ 1=T m , whereas nondiagonal matrix element Gmn ðmanÞ is called as interband relaxation rate. Eq. (13) is solved using the usual iterative method [22,23]: X rðnÞ ðtÞ, (14) rðtÞ ¼
q ðnþ1Þ 1 ðnÞ r ¼ Ggg rðnþ1Þ ðPge rðnÞ gg eg Peg rge ÞEðtÞ, qt gg i_ (19) where Pij ¼ jhijqrjjij; ði; j ¼ g; eÞ. Ggg ¼ 1=T g ; Gee ¼ 1=T e , and Gge ¼ Geg ¼ 12ð1=T g þ 1=T e Þ [23]. Eqs. (16)–(19) are readily solved by expanding the density matrix elements as sums of terms proportional to expð iotÞ, thus the nth order perturbation term rðnÞ ðtÞ can be written as rðnÞ ðtÞ ¼ e rðnÞ ðoÞeiot þ e rðnÞ ðoÞeiot .
(20)
Due to inversion symmetry of the system, only odd n terms need to consider. In this paper, we only consider one- and three-order terms for simplicity. Using Eqs. (12), (16)–(20), and neglecting the off-resonance terms, we obtain e rð1Þ eg ðoÞ and ð3Þ e reg ðoÞ, after some mathematical calculations, as follows:
n
e rð1Þ eg ðoÞ ¼
with
e eg ðrð0Þ rð0Þ Þ EP gg ee , _oeg _o i_Geg
(21)
qrijðnþ1Þ =qt ¼ ði_Þ1 f½H 0 ; rðnþ1Þ ij i_Gij rðnþ1Þ g ij ði_Þ1 ½qr; rðnÞ ij EðtÞ.
ð15Þ
For simplicity, we will only pay our attention to two-level systems for electronic transitions. Hereafter, the ground state will be denoted by g, the excitonic state by e, respectively. Therefore, we can obtain q ðnþ1Þ 1 r ðE e E g Þ Gge rðnþ1Þ ¼ eg qt eg i_ 1 ðrðnÞ rðnÞ ee ÞPeg EðtÞ, i_ gg q ðnþ1Þ r qt ge
1 ðE g E e Þ Geg rðnþ1Þ ¼ ge i_ 1 ðrðnÞ rðnÞ gg ÞPge EðtÞ, i_ ee
ð16Þ
ð17Þ
e rð3Þ eg ðoÞ ¼
e Ej e 2 Peg jPeg j2 ðð1=Ggg Þ þ ð1=Gee ÞÞGeg ðrð0Þ rð0Þ Þ 2Ej gg ee ð_oeg _o i_Geg Þ½ð_oeg _oÞ2 þ ð_Geg Þ2
,
ð22Þ where oeg ¼ ðE e E g Þ=_. The polarization pðtÞ and susceptibility wðoÞ caused by the incident field EðtÞ can be expressed through the dipole operator P and the density matrix r as e iot þ e0 wðoÞE e eiot pðtÞ ¼ e0 wðoÞEe 1 ¼ TrðrPÞ, V
ð23Þ
where V is the volume of the system, e0 is the vacuum permittivity and Tr denotes the trace or summation over the diagonal elements of the matrix rP. The susceptibility wðoÞ is related to the
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absorption coefficient aðoÞ by [23] rffiffiffiffiffi m aðoÞ ¼ o Im½e0 wðoÞ, eR
the saturation intensity I s as follows: (24)
where eR is the real part of the permittivity, defined as eR ¼ n2r e0 , and wðoÞ is the Fourier component of wðtÞ; m is the permeability of the system. From Eqs. (21)–(24), we can obtain the linear and the third-order nonlinear optical absorption coefficients að1Þ ðoÞ and að3Þ ðo; IÞ, respectively, as follows: rffiffiffiffiffi m að1Þ ðoÞ ¼ o Im½e0 wð1Þ ðoÞ eR rffiffiffiffiffi jPeg j2 N_Geg m ¼o , ð25Þ eR ½ð_oeg _oÞ2 þ ð_Geg Þ2 rffiffiffiffiffi m Im½e0 wð3Þ ðoÞjEðoÞj2 eR rffiffiffiffiffi 2IjPeg j4 N_Geg m ¼ o , eR nr e0 c½ð_o _oeg Þ2 þ ð_Geg Þ2
að3Þ ðo; IÞ ¼ o
ð26Þ where nr is the medium refractive index, c is the speed of light in free space, N is the carrier density in this system. I is the incident optical intensity, defined as rffiffiffiffiffi eR 2nr I ¼2 jEðoÞj2 ¼ jEðoÞj2 . (27) m mc
Is ¼
e0 nr c½ðE eg _oÞ2 þ ð_Geg Þ2 . 4jPeg j2
(29)
4. Results and discussions In the following, we will discuss the linear, thirdorder and total optical absorption coefficients in the CdS parabolic quantum dot. The parameters used in our numerical work are adopted as [24]: _Geg ¼ 3:3 meV; me ¼ 0:2m0 ; mh ¼ 0:7m0 (m0 is the mass of a free electron), 0 ¼ 8:9; E g ¼ 2:5 eV; R0 ¼ 10 nm; nr ¼ 2:8; N ¼ 1014 cm3 . Fig. 1 illustrates the linear, and third-order nonlinear optical absorption (a1 and a3 ) as well as the total optical absorption coefficient ða1 þ a3 Þ at an incident optical intensity of 0.014 MW/cm2, o0 ¼ 5:0 1013 s1 . From Fig. 1, we can see that the large linear absorption coefficient a1 , which comes from the linear susceptibility wð1Þ ðoÞ term, is positive, whereas the third-order nonlinear optical absorption coefficient a3 generated by the wð3Þ ðoÞ term is negative. So the total optical absorption coefficient a is significantly reduced by the a3 contribution. Therefore, the third-order nonlinear optical absorption coefficient a3 should be considered when the incident optical intensity I is
So the total optical absorption coefficient aðo; IÞ can be written as aðo; IÞ ¼ að1Þ ðoÞ þ að3Þ ðo; IÞ rffiffiffiffiffi( jPeg j2 N_Geg m ¼o eR ½ð_oeg _oÞ2 þ ð_Geg Þ2 ) 2IjPeg j4 N_Geg . nr e0 c½ð_o _oeg Þ2 þ ð_Geg Þ2 2
ð28Þ
Since the third-order nonlinear optical absorption coefficient að3Þ ðo; IÞ is negative and is proportional to the incident optical intensity I, the total absorption coefficient aðo; IÞ decreases as I increases. aðo; IÞ is reduced by one-half when I reaches a value I s called the saturation intensity. So we have the relation aðo; I s Þ ¼ 12að1Þ ðoÞ, or equivalently að3Þ ðo; I s Þ ¼ 12að1Þ ðoÞ, so we obtain
Fig. 1. The linear, third-order and total optical absorption coefficients versus the incident photon energy hn with the parabolic confinement frequency o0 ¼ 5:0 1013 s1 ; I ¼ 0:014 MW=cm2 .
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Fig. 2. The total optical absorption coefficient a versus the photon energy hn for three different values of the parabolic confinement frequency o0 : (a) o0 ¼ 1:0 1013 s1 , (b) o0 ¼ 2:5 1013 s1 , (c) o0 ¼ 5:0 1013 s1 , with 2 I ¼ 0:01 MW=cm .
Fig. 3. The total optical absorption coefficient a versus the photon energy hn for five different values of the incident optical intensity I: (a) I ¼ 0, (b) I ¼ 0:01 MW=cm2 , (c) (d) I ¼ 0:015 MW=cm2 , (e) I ¼ 0:0125 MW=cm2 , 2 I ¼ 0:02 MW=cm , with o0 ¼ 5:0 1013 s1 .
comparatively strong, which can induce nonlinear absorption. Fig. 2 shows the total optical absorption coefficient aðo; IÞ as a function of the photon energy hn for three different values of parabolic confinement frequency o0 : (a) o0 ¼ 1:0 1013 s1 , (b) o0 ¼ 2:5 1013 s1 , (c) o0 ¼ 5:0 1013 s1 , with the incident optical intensity I ¼ 0:01 MW= cm2 . We observe three absorption peaks which appear at hn ¼ 2:40 eV; hn ¼ 2:41 eV; hn ¼ 2:43 eV, respectively. A very important feature is that the stronger the parabolic confinement is, the sharper the absorption peak will be and the bigger the absorption peak intensity will be. The reason is that the electronic dipolar transition matrix element will increase with the parabolic confinement potential, thus leading to the total optical absorption coefficient increase. As the parabolic confinement frequency o0 increases, the absorption peak will move to the right side of the curve, which predicts a strong confinement-induced blue shift of the exciton absorption in the semiconductor quantum dot in accordance with the recent report [25]. The physical origin of this shift is the quantum-confined stark effects. These features make the parabolic quantum dot become very promising candidates for nonlinear optical materials and devices.
Fig. 3 shows the total optical absorption coefficient as a function of the incident photon energy hn with o0 ¼ 5:0 1013 s1 , while the incident optical intensity I for five various values: (a) I ¼ 0, (b) I ¼ 0:01 MW=cm2 , (c) I ¼ 0:0125 MW=cm2 , (d) I ¼ 0:015 MW=cm2 , (e) I ¼ 0:02 MW=cm2 . It can been seen from Fig. 3 that the total optical absorption coefficient will reduce significantly with the incident optical intensity I increasing, and at sufficiently highincident optical intensities the absorption at linear center will be strongly bleached. We also can see that the strong absorption saturation begins to occur at around I ¼ 0:0125 MW=cm2 . When the incident optical intensity I exceeds the value of 0:0125 MW=cm2 , the exciton absorption peak will be significantly split up into two peaks, which is in consequence of the absorption at linear center will be strongly bleached. Fig. 4 shows the optical absorption saturation intensity I s as a function of the parabolic confinement frequency o0 . Fig. 4 illustrates the optical absorption saturation intensity will reduce with parabolic confinement frequency o0 increasing under the condition of resonance. The reason is that the dipole transitive matrix element will increase with the parabolic confinement frequency o0 , and this leads to a decrease in the optical
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Fig. 4. The optical absorption saturation intensity I s versus the parabolic confinement frequency o0 .
absorption saturation intensity I s . Absorption saturation is mainly induced by nonlinear optical absorption of ground state energy level. It can supply theoretical foundation for experimental studies. Furthermore, from Fig. 4 we can see that the optical absorption saturation intensity I s will be invariable when parabolic confinement frequency o0 exceeds certain value (about 1:5 1014 s1 ). So the optical absorption saturation intensity can be controlled by parabolic confinement frequency, and very weak input light intensity can induce optical absorption saturation by adopting proper parabolic confinement frequency in the quantum dot.
optical absorption is reduced by half when I ¼ 0:0125 MW=cm2 , and very weak input light intensity can induce optical absorption saturation by adopting proper parabolic confinement frequency in the quantum dot. The contributions to absorption coefficient considering excitonic effects include (1) the transition dipole matrix element variations, and (2) the energy shifts (quantumconfined stark effects). Therefore, theoretical study on the nonlinear optical absorption may make a great contribution to experimental studies, may have profound consequences as regards practical application of electrooptical devices, for example, nonlinear optical absorption effects can be applied in Q switch, self mode-locking of lasers with saturable absorbers, laser stable frequence and absorption spectroscopy, and optical absorption saturation also has extensive application in optical communication etc.
Acknowledgements This work is supported by Natural Science Foundation of Guangdong Province, China, under Grant no. 031516 and no. 011835, and team project of Natural Science Foundation of Guangdong Province, China, under Grant no. 20003061.
References 5. Summary In this paper, the linear and nonlinear optical absorption coefficients (a1 and a3 ) as well as the total optical absorption coefficient ða1 þ a3 Þ for a nanometer-size CdS parabolic quantum dot are studied. The results show that the linear optical absorption coefficient a1 is not related to the incident optical intensity, whereas the incident optical intensity has great influence on the thirdorder nonlinear optical absorption coefficient a3 . Furthermore, the total optical absorption coefficient a will reduce when the incident optical intensity increases, and we find that the total
[1] V.M. Agranovich, D.M. Basko, et al., J. Phys.: Condens. Matter 10 (1998) 9369. [2] E. Hanamura, Phys. Rev. B 37 (1988) 1273. [3] T. Takagahara, Phys. Rev. B 36 (1987) 9293. [4] R. Chen, D.L. Lin, B. Mendoza, Phys. Rev. B 48 (1993) 11879. [5] Guanghui Wang, Kangxian Guo, J. Phys.: Condens. Matter 13 (2001) 8197. [6] S. Weiner, D.S. Chemla, D.A.B. Miller, T.H. Wood, D. Sivco, A.Y. Cho, Appl. Phys. Lett. 46 (1985) 619. [7] D.A.B. Miller, D.S. Chemla, P.W. Smith, A.C. Gossaard, W. Wiegmann, Appl. Phys. B 28 (1982) 96. [8] R. Dingle, W. Wiegman, C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. [9] M.A. Reed, et al., Phys. Rev. Lett. 60 (1988) 535. [10] Weiming Que, Phys. Rev. B 45 (1992) 11036. [11] S. Sauvage, P. Boucaud, Phys. Rev. B 59 (1999) 9830.
ARTICLE IN PRESS G. Wang, K. Guo / Physica E 28 (2005) 14–21 [12] D. Cotter, M.G. Burt, R.J. Manning, Phys. Rev. Lett. 68 (1992) 1200. [13] B.F. Levine, R.J. Malik, et al., Appl. Phys. Lett. 50 (1987) 273. [14] A. Harwitt, J.S. Harris Jr., Appl. Phys. Lett. 50 (1987) 685. [15] D. Ahn, S.L. Chuang, J. Appl. Phys. 62 (1987) 3052. [16] T. Rappen, et al., Phys. Rev. B 44 (1991) 13093. [17] U. Bockelmann, G. Bastard, Phys. Rev. B 45 (1992) 1700. [18] U. Bockelmann, G. Bastard, Phys. Rev. B 45 (1992) 1688. [19] Perng-fei Yuh, K.L. Wang, Phys. Rev. B 38 (1988) 8377.
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[20] Da-fu Cui, Zheng-hao Chen, Shao-hua Pan, et al., Phys. Rev. B 47 (1993) 6755. [21] L. Banyai, Y.Z. Hu, M. Lindberg, S.W. Koch, Phys. Rev. B 38 (1988) 8142. [22] E. Rosencher, Ph. Bois, Phys. Rev. B 44 (1991) 11315. [23] Doyeol Ahn, Shun-lien Chuang, IEEE J. Quantum Electron. QE-23 (1987) 2196. [24] Guanghui Wang, Kangxian Guo, Physica B 315 (2002) 234. [25] L. Banyai, S.W. Koch, Phys. Rev. Lett. 57 (1986) 2722.
Physica E 28 (2005) 14–21 www.elsevier.com/locate/physe
Interband optical absorptions in a parabolic quantum dot Guanghui Wanga,, Kangxian Guob a
Laboratory of Light Transmission Optics, South China Normal University, Guangzhou 510631, PR China b Department of Physics, Guihuagang Campus, Guangzhou University, Guangzhou 510405, PR China Received 5 January 2005; accepted 10 January 2005 Available online 25 April 2005
Abstract The linear and nonlinear optical absorptions considering excitonic effect in a parabolic quantum dot are studied. Analytic forms of the linear and third-order nonlinear optical absorption coefficients are obtained for a symmetric parabolic quantum dot using the compact density matrix formalism. Based on this model, numerical results are presented for a typical CdS parabolic quantum dot. The calculated results show that the factors of the parabolic potential confinement and the incident optical intensity have great influence on the optical absorption coefficients. Furthermore, the optical absorption saturation intensity can be controlled by adopting a proper parabolic confinement potential. r 2005 Elsevier B.V. All rights reserved. PACS: 78.66.w; 42.65.k; 78.20.Ci; 71.35.Aa; 42.65.An Keywords: Optical absorption; Parabolic quantum dot; Exciton; Density matrix approach
1. Introduction In the past few years, the nonlinear optical properties of confined excitons in nanostructures (including quantum wells, quantum wires and quantum dots) have attracted much attention [1–5]. One of the reasons is that excitonic spectrum has been observed at the room temperatures [6,7], which is due to the enhancement of the excitonic binding energy caused by the quantum confineCorresponding author.
E-mail address: [email protected] (G. Wang).
ment effect in these structures, and another reason is that quantum confinement of carriers in these low-dimensional semiconductor nanostructures lead to the formation of discrete energy levels, and the drastic change of optical absorption spectra [8] and many novel optical properties, which is not in the bulk materials. Furthermore, with recently quick advances of modern technology, it has now become possible to produce quasi-zero-dimension systems that confine electrons in all three spatial dimensions by using techniques such as etching or molecular beam epitaxy [9] etc. They typically have a disk-like
1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.01.018
ARTICLE IN PRESS G. Wang, K. Guo / Physica E 28 (2005) 14–21
shape, a few hundred nanometers in diameter, and a few nanometers thick. In such nanometer structures, electrons not only are confined in all three spatial dimensions, but also are quantized into discrete energy levels, with energy spacings of a few meV or more [10]. Recently, the quantum dots with (quasi-) zero-dimension structures are more and more important because of their novel physical properties and promise for potential applications [11,12]. Recently, the linear intersubband optical absorption within the conduction band of a GaAs quantum well has been studied experimentally without an electric field [13], and with an electric field [14]. A very large oscillator strength and a narrow bandwidth were observed. These suggest that the intersubband optical transitions in a quantum well may have very large optical nonlinearities. Nonlinear intersubband optical absorption in a semiconductor quantum well also was calculated by Ahn and Chuang in 1987 [15]. In 1991, Rappen et al. [16] studied the nonlinear absorption for two-dimensional magnetoexcitons in Inx Ga1x As=Iny Al1y As quantum wells. In 1992, Bockelman and Bastard [17] discussed interband absorption in quantum wires with magnetic-field case and without magnetic field case [18]. Intersubband optical absorption in coupled quantum wells under an applied electric field was studied by Perng-fei Yuh and Wang [19]. In 1993, Da-fu Cui et al. [20] studied absorption saturation of intersubband optical transitions in GaAs=Alx Ga1x As multiple quantum wells in experiment, and they tested the saturation optical intensity for I s ¼ 0:67 MW=cm2 , electronic relaxation time for 0.1 ps. From fundamental and practical points of view, these linear and nonlinear size-quantized transitions have the potential for device applications in far-infrared laser amplifiers, photodetectors, and high-speed electrooptical modulators. However, up to the present, no one else have studied the linear and nonlinear optical absorption in parabolic quantum wires and dots, and in the previous work, the excitonic effects were rarely considered. In this paper, the linear, third-order nonlinear and total optical absorption coefficients in a disk-like parabolic quantum dot are studied
15
theoretically. In Section 2, the electronic states in the quantum dot are described. The basic electronic excitations are taken to be excitonic states, without considering collective excitations and electronic dipole transition matrix element is obtained. In Section 3 an analytical formula for the linear, third-order nonlinear and total optical absorption coefficients are derived by using the compact-density-matrix approach and an iterative method. In Section 4 we calculate and discuss the linear, third-order nonlinear and total optical absorption coefficients in the CdS parabolic quantum dot, the results show that the incident optical intensity and the parabolic confinement frequency have great influence on the optical absorption coefficients. Furthermore, the optical absorption saturation intensity will reduce as increase in parabolic confinement frequency under the condition of resonance. A brief summary is given in Section 5.
2. Electronic excitations in a parabolic quantum dot In the following, we will only discuss weakconfinement regime, namely R0 baex , where R0 is the radius of the quantum dot, aex is exciton effective Bohr radius. According to this condition, the electronic excited state in a quantum dot can then be described by the Frenkel exciton as [2,3] X Y cn ¼ Rn ðjÞW cj ðrj Þ W vi ðri Þ, (1) j
ðiajÞ
where W vi and W cj are Wannier functions of the valence and the conduction bands, respectively, and Rn ðjÞ is the envelope function in a disk-like parabolic quantum dot, given as 2m o0 n! 1=2 jmj jmj 2 r2 =2 1 imy pffiffiffiffiffiffi e , r Ln ðr Þe Rn ðrÞ ¼ _ðjmj þ nÞ! 2p (2) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ðm o0 =_Þr ¼ uj; u is the unit-cell size, the site index j is chosen from positive integer between 1 and N, m ¼ 0; 1; 2; . . . ; n ¼ 0; 1; 2; . . . ; Ljmj are the Laguerre polynomials, o0 n is the frequency of the parabolic confining potential.
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The excitation energy spectrum of the conduction electrons can then be expressed as E nm ¼ ð2n þ jmj þ 1Þ_o0 .
(3)
The ground state of the system is Y W vi ðri Þ. cg ¼
(4)
quantum dot can be written as follows: X Rn ðjÞjðj r ÞhW cj ðrj Þjpj jW vj0 ðrj 0 Þi hcn jPjcg i ¼ jj 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pðMo 0 =_ÞR0 _ r dr ¼ pcv jð0Þ Mo0 0 2Mo0 n! 1=2 jmj jmj 2 ð1=2Þr2
r Ln ðr Þe _ðjmj þ nÞ! Z 2p 1 pffiffiffiffiffiffi eimy dy,
ð8Þ 2p 0
i
For the parabolic conduction and valence bands with effective mass me , and mh , respectively, the size quantization is dominated by the relationship among the sample size R0 ¼ Nu, and the effective Bohr radii of electrons ae ¼ 0 _2 =me e2 and holes ah ¼ 0 _2 =mh e2 , where 0 is the static dielectric constant. After taking into account the Coulomb interaction between the electrons and holes, adopting the relative coordinate and center-of-mass coordinate, the excitation states in Eq. (1) should be modified to [2] XX Y cn ¼ Rn ðjÞjðj r ÞW cj ðrj Þ W vi ðri Þ, (5) j
j0
iaj 0
where Rn ðjÞ is the envelope function given in Eq. (2), presenting the centre-of-mass motion of the exciton with j ¼ ðme j þ mh j 0 Þ=ðme þ mh Þ, and jðj r Þ describes the electron–hole relative motion with j r ¼ j j 0 . We also define the total mass M ¼ me þ mh , the reduced mass m ¼ me mh =M, and the effective Rydberg constant R ¼ me4 =20 _2 . For the lowest state of the exciton, we can describe the two-dimensional electron–hole relative motion of the exciton as 4 jðrÞ ¼ pffiffiffiffiffiffi e2r=aB , 2paex
(6)
where r ¼ uðj j 0 Þ, is the electron–hole separation. The energy spectrum of excitons in the parabolic quantum dot reads R E nm ¼ E g 2 þ ð2n þ jmj þ 1Þ_o0 , 2 n þ jmj þ 12 (7) where E g is the band-gap energy. The transition dipole moment from the ground state cg to the excited states cn in the parabolic
where P is a component of the dipole moment operator P: X P¼ pi , (9) i
and hW cj ðrj Þjpj jW vj0 ðrj0 Þi ¼ pcv djj 0 [2]. From Eq. (8), we can see that the transition dipole matrix element will be zero unless m ¼ 0. After setting m ¼ 0, we have ffi pffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffi ðMo0 =_ÞR0 2p_ hcn jPjcg i ¼ pcv jð0Þ Mo0 0 2Mo0 1=2 2
r dr Ln ðr2 Þeð1=2Þr . _ ð10Þ So the transition dipole moment from the ground state jcg i to the lowest state of the exciton jc1 i can be written as Mo0 p 1=2 hc1 jPjcg i ¼ pcv jð0ÞR20 , (11) _ pffiffiffiffiffiffiffiffiffiffiffi where pcv ¼ _e= 4mE g , obtained in kp perturbation theory [21].
3. Linear and nonlinear optical absorption in a parabolic quantum dot In this section we will study the linear and thirdorder nonlinear optical absorption coefficients in the parabolic quantum dot. Let us consider an monochromatic field with frequency o which is incident with a polarization vector normal to the quantum dot. The system is excited by the
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q ðnþ1Þ 1 ðnþ1Þ ðnÞ ree ¼ Gee ree ðPeg rðnÞ ge Pge reg ÞEðtÞ, qt i_ (18)
incident field EðtÞ ¼ E 0 Cos ot e iot . e iot þ Ee ¼ Ee
ð12Þ
Let a sign r denote one-electron density matrix operator for this system. Then the evolution of the density matrix operator r obeys the following time-dependent equation qr=qt ¼ ði_Þ1 ½H 0 qrEðtÞ; r Gðr rð0Þ Þ,
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(13)
where ½ ; is the quantum mechanical commutator, H 0 is the Hamiltonian for this system without the electromagnetic field EðtÞ; q is the electronic charge; rð0Þ is the unperturbed density matrix operator; G is the phenomenological operator. Diagonal matrix element Gmm of operator G, which is called as intersubband relaxation rate, is the inverse of the relaxation time T m for the state jmi, namely Gmm ¼ 1=T m , whereas nondiagonal matrix element Gmn ðmanÞ is called as interband relaxation rate. Eq. (13) is solved using the usual iterative method [22,23]: X rðnÞ ðtÞ, (14) rðtÞ ¼
q ðnþ1Þ 1 ðnÞ r ¼ Ggg rðnþ1Þ ðPge rðnÞ gg eg Peg rge ÞEðtÞ, qt gg i_ (19) where Pij ¼ jhijqrjjij; ði; j ¼ g; eÞ. Ggg ¼ 1=T g ; Gee ¼ 1=T e , and Gge ¼ Geg ¼ 12ð1=T g þ 1=T e Þ [23]. Eqs. (16)–(19) are readily solved by expanding the density matrix elements as sums of terms proportional to expð iotÞ, thus the nth order perturbation term rðnÞ ðtÞ can be written as rðnÞ ðtÞ ¼ e rðnÞ ðoÞeiot þ e rðnÞ ðoÞeiot .
(20)
Due to inversion symmetry of the system, only odd n terms need to consider. In this paper, we only consider one- and three-order terms for simplicity. Using Eqs. (12), (16)–(20), and neglecting the off-resonance terms, we obtain e rð1Þ eg ðoÞ and ð3Þ e reg ðoÞ, after some mathematical calculations, as follows:
n
e rð1Þ eg ðoÞ ¼
with
e eg ðrð0Þ rð0Þ Þ EP gg ee , _oeg _o i_Geg
(21)
qrijðnþ1Þ =qt ¼ ði_Þ1 f½H 0 ; rðnþ1Þ ij i_Gij rðnþ1Þ g ij ði_Þ1 ½qr; rðnÞ ij EðtÞ.
ð15Þ
For simplicity, we will only pay our attention to two-level systems for electronic transitions. Hereafter, the ground state will be denoted by g, the excitonic state by e, respectively. Therefore, we can obtain q ðnþ1Þ 1 r ðE e E g Þ Gge rðnþ1Þ ¼ eg qt eg i_ 1 ðrðnÞ rðnÞ ee ÞPeg EðtÞ, i_ gg q ðnþ1Þ r qt ge
1 ðE g E e Þ Geg rðnþ1Þ ¼ ge i_ 1 ðrðnÞ rðnÞ gg ÞPge EðtÞ, i_ ee
ð16Þ
ð17Þ
e rð3Þ eg ðoÞ ¼
e Ej e 2 Peg jPeg j2 ðð1=Ggg Þ þ ð1=Gee ÞÞGeg ðrð0Þ rð0Þ Þ 2Ej gg ee ð_oeg _o i_Geg Þ½ð_oeg _oÞ2 þ ð_Geg Þ2
,
ð22Þ where oeg ¼ ðE e E g Þ=_. The polarization pðtÞ and susceptibility wðoÞ caused by the incident field EðtÞ can be expressed through the dipole operator P and the density matrix r as e iot þ e0 wðoÞE e eiot pðtÞ ¼ e0 wðoÞEe 1 ¼ TrðrPÞ, V
ð23Þ
where V is the volume of the system, e0 is the vacuum permittivity and Tr denotes the trace or summation over the diagonal elements of the matrix rP. The susceptibility wðoÞ is related to the
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absorption coefficient aðoÞ by [23] rffiffiffiffiffi m aðoÞ ¼ o Im½e0 wðoÞ, eR
the saturation intensity I s as follows: (24)
where eR is the real part of the permittivity, defined as eR ¼ n2r e0 , and wðoÞ is the Fourier component of wðtÞ; m is the permeability of the system. From Eqs. (21)–(24), we can obtain the linear and the third-order nonlinear optical absorption coefficients að1Þ ðoÞ and að3Þ ðo; IÞ, respectively, as follows: rffiffiffiffiffi m að1Þ ðoÞ ¼ o Im½e0 wð1Þ ðoÞ eR rffiffiffiffiffi jPeg j2 N_Geg m ¼o , ð25Þ eR ½ð_oeg _oÞ2 þ ð_Geg Þ2 rffiffiffiffiffi m Im½e0 wð3Þ ðoÞjEðoÞj2 eR rffiffiffiffiffi 2IjPeg j4 N_Geg m ¼ o , eR nr e0 c½ð_o _oeg Þ2 þ ð_Geg Þ2
að3Þ ðo; IÞ ¼ o
ð26Þ where nr is the medium refractive index, c is the speed of light in free space, N is the carrier density in this system. I is the incident optical intensity, defined as rffiffiffiffiffi eR 2nr I ¼2 jEðoÞj2 ¼ jEðoÞj2 . (27) m mc
Is ¼
e0 nr c½ðE eg _oÞ2 þ ð_Geg Þ2 . 4jPeg j2
(29)
4. Results and discussions In the following, we will discuss the linear, thirdorder and total optical absorption coefficients in the CdS parabolic quantum dot. The parameters used in our numerical work are adopted as [24]: _Geg ¼ 3:3 meV; me ¼ 0:2m0 ; mh ¼ 0:7m0 (m0 is the mass of a free electron), 0 ¼ 8:9; E g ¼ 2:5 eV; R0 ¼ 10 nm; nr ¼ 2:8; N ¼ 1014 cm3 . Fig. 1 illustrates the linear, and third-order nonlinear optical absorption (a1 and a3 ) as well as the total optical absorption coefficient ða1 þ a3 Þ at an incident optical intensity of 0.014 MW/cm2, o0 ¼ 5:0 1013 s1 . From Fig. 1, we can see that the large linear absorption coefficient a1 , which comes from the linear susceptibility wð1Þ ðoÞ term, is positive, whereas the third-order nonlinear optical absorption coefficient a3 generated by the wð3Þ ðoÞ term is negative. So the total optical absorption coefficient a is significantly reduced by the a3 contribution. Therefore, the third-order nonlinear optical absorption coefficient a3 should be considered when the incident optical intensity I is
So the total optical absorption coefficient aðo; IÞ can be written as aðo; IÞ ¼ að1Þ ðoÞ þ að3Þ ðo; IÞ rffiffiffiffiffi( jPeg j2 N_Geg m ¼o eR ½ð_oeg _oÞ2 þ ð_Geg Þ2 ) 2IjPeg j4 N_Geg . nr e0 c½ð_o _oeg Þ2 þ ð_Geg Þ2 2
ð28Þ
Since the third-order nonlinear optical absorption coefficient að3Þ ðo; IÞ is negative and is proportional to the incident optical intensity I, the total absorption coefficient aðo; IÞ decreases as I increases. aðo; IÞ is reduced by one-half when I reaches a value I s called the saturation intensity. So we have the relation aðo; I s Þ ¼ 12að1Þ ðoÞ, or equivalently að3Þ ðo; I s Þ ¼ 12að1Þ ðoÞ, so we obtain
Fig. 1. The linear, third-order and total optical absorption coefficients versus the incident photon energy hn with the parabolic confinement frequency o0 ¼ 5:0 1013 s1 ; I ¼ 0:014 MW=cm2 .
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Fig. 2. The total optical absorption coefficient a versus the photon energy hn for three different values of the parabolic confinement frequency o0 : (a) o0 ¼ 1:0 1013 s1 , (b) o0 ¼ 2:5 1013 s1 , (c) o0 ¼ 5:0 1013 s1 , with 2 I ¼ 0:01 MW=cm .
Fig. 3. The total optical absorption coefficient a versus the photon energy hn for five different values of the incident optical intensity I: (a) I ¼ 0, (b) I ¼ 0:01 MW=cm2 , (c) (d) I ¼ 0:015 MW=cm2 , (e) I ¼ 0:0125 MW=cm2 , 2 I ¼ 0:02 MW=cm , with o0 ¼ 5:0 1013 s1 .
comparatively strong, which can induce nonlinear absorption. Fig. 2 shows the total optical absorption coefficient aðo; IÞ as a function of the photon energy hn for three different values of parabolic confinement frequency o0 : (a) o0 ¼ 1:0 1013 s1 , (b) o0 ¼ 2:5 1013 s1 , (c) o0 ¼ 5:0 1013 s1 , with the incident optical intensity I ¼ 0:01 MW= cm2 . We observe three absorption peaks which appear at hn ¼ 2:40 eV; hn ¼ 2:41 eV; hn ¼ 2:43 eV, respectively. A very important feature is that the stronger the parabolic confinement is, the sharper the absorption peak will be and the bigger the absorption peak intensity will be. The reason is that the electronic dipolar transition matrix element will increase with the parabolic confinement potential, thus leading to the total optical absorption coefficient increase. As the parabolic confinement frequency o0 increases, the absorption peak will move to the right side of the curve, which predicts a strong confinement-induced blue shift of the exciton absorption in the semiconductor quantum dot in accordance with the recent report [25]. The physical origin of this shift is the quantum-confined stark effects. These features make the parabolic quantum dot become very promising candidates for nonlinear optical materials and devices.
Fig. 3 shows the total optical absorption coefficient as a function of the incident photon energy hn with o0 ¼ 5:0 1013 s1 , while the incident optical intensity I for five various values: (a) I ¼ 0, (b) I ¼ 0:01 MW=cm2 , (c) I ¼ 0:0125 MW=cm2 , (d) I ¼ 0:015 MW=cm2 , (e) I ¼ 0:02 MW=cm2 . It can been seen from Fig. 3 that the total optical absorption coefficient will reduce significantly with the incident optical intensity I increasing, and at sufficiently highincident optical intensities the absorption at linear center will be strongly bleached. We also can see that the strong absorption saturation begins to occur at around I ¼ 0:0125 MW=cm2 . When the incident optical intensity I exceeds the value of 0:0125 MW=cm2 , the exciton absorption peak will be significantly split up into two peaks, which is in consequence of the absorption at linear center will be strongly bleached. Fig. 4 shows the optical absorption saturation intensity I s as a function of the parabolic confinement frequency o0 . Fig. 4 illustrates the optical absorption saturation intensity will reduce with parabolic confinement frequency o0 increasing under the condition of resonance. The reason is that the dipole transitive matrix element will increase with the parabolic confinement frequency o0 , and this leads to a decrease in the optical
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Fig. 4. The optical absorption saturation intensity I s versus the parabolic confinement frequency o0 .
absorption saturation intensity I s . Absorption saturation is mainly induced by nonlinear optical absorption of ground state energy level. It can supply theoretical foundation for experimental studies. Furthermore, from Fig. 4 we can see that the optical absorption saturation intensity I s will be invariable when parabolic confinement frequency o0 exceeds certain value (about 1:5 1014 s1 ). So the optical absorption saturation intensity can be controlled by parabolic confinement frequency, and very weak input light intensity can induce optical absorption saturation by adopting proper parabolic confinement frequency in the quantum dot.
optical absorption is reduced by half when I ¼ 0:0125 MW=cm2 , and very weak input light intensity can induce optical absorption saturation by adopting proper parabolic confinement frequency in the quantum dot. The contributions to absorption coefficient considering excitonic effects include (1) the transition dipole matrix element variations, and (2) the energy shifts (quantumconfined stark effects). Therefore, theoretical study on the nonlinear optical absorption may make a great contribution to experimental studies, may have profound consequences as regards practical application of electrooptical devices, for example, nonlinear optical absorption effects can be applied in Q switch, self mode-locking of lasers with saturable absorbers, laser stable frequence and absorption spectroscopy, and optical absorption saturation also has extensive application in optical communication etc.
Acknowledgements This work is supported by Natural Science Foundation of Guangdong Province, China, under Grant no. 031516 and no. 011835, and team project of Natural Science Foundation of Guangdong Province, China, under Grant no. 20003061.
References 5. Summary In this paper, the linear and nonlinear optical absorption coefficients (a1 and a3 ) as well as the total optical absorption coefficient ða1 þ a3 Þ for a nanometer-size CdS parabolic quantum dot are studied. The results show that the linear optical absorption coefficient a1 is not related to the incident optical intensity, whereas the incident optical intensity has great influence on the thirdorder nonlinear optical absorption coefficient a3 . Furthermore, the total optical absorption coefficient a will reduce when the incident optical intensity increases, and we find that the total
[1] V.M. Agranovich, D.M. Basko, et al., J. Phys.: Condens. Matter 10 (1998) 9369. [2] E. Hanamura, Phys. Rev. B 37 (1988) 1273. [3] T. Takagahara, Phys. Rev. B 36 (1987) 9293. [4] R. Chen, D.L. Lin, B. Mendoza, Phys. Rev. B 48 (1993) 11879. [5] Guanghui Wang, Kangxian Guo, J. Phys.: Condens. Matter 13 (2001) 8197. [6] S. Weiner, D.S. Chemla, D.A.B. Miller, T.H. Wood, D. Sivco, A.Y. Cho, Appl. Phys. Lett. 46 (1985) 619. [7] D.A.B. Miller, D.S. Chemla, P.W. Smith, A.C. Gossaard, W. Wiegmann, Appl. Phys. B 28 (1982) 96. [8] R. Dingle, W. Wiegman, C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. [9] M.A. Reed, et al., Phys. Rev. Lett. 60 (1988) 535. [10] Weiming Que, Phys. Rev. B 45 (1992) 11036. [11] S. Sauvage, P. Boucaud, Phys. Rev. B 59 (1999) 9830.
ARTICLE IN PRESS G. Wang, K. Guo / Physica E 28 (2005) 14–21 [12] D. Cotter, M.G. Burt, R.J. Manning, Phys. Rev. Lett. 68 (1992) 1200. [13] B.F. Levine, R.J. Malik, et al., Appl. Phys. Lett. 50 (1987) 273. [14] A. Harwitt, J.S. Harris Jr., Appl. Phys. Lett. 50 (1987) 685. [15] D. Ahn, S.L. Chuang, J. Appl. Phys. 62 (1987) 3052. [16] T. Rappen, et al., Phys. Rev. B 44 (1991) 13093. [17] U. Bockelmann, G. Bastard, Phys. Rev. B 45 (1992) 1700. [18] U. Bockelmann, G. Bastard, Phys. Rev. B 45 (1992) 1688. [19] Perng-fei Yuh, K.L. Wang, Phys. Rev. B 38 (1988) 8377.
21
[20] Da-fu Cui, Zheng-hao Chen, Shao-hua Pan, et al., Phys. Rev. B 47 (1993) 6755. [21] L. Banyai, Y.Z. Hu, M. Lindberg, S.W. Koch, Phys. Rev. B 38 (1988) 8142. [22] E. Rosencher, Ph. Bois, Phys. Rev. B 44 (1991) 11315. [23] Doyeol Ahn, Shun-lien Chuang, IEEE J. Quantum Electron. QE-23 (1987) 2196. [24] Guanghui Wang, Kangxian Guo, Physica B 315 (2002) 234. [25] L. Banyai, S.W. Koch, Phys. Rev. Lett. 57 (1986) 2722.