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Size-dependent electronic energy relaxation in a semiconductor nanocrystal in the strong confinement limit
ARTICLE IN PRESS
Physica E 31 (2006) 48–52 www.elsevier.com/locate/physe
Size-dependent electronic energy relaxation in a semiconductor nanocrystal in the strong confinement limit S.-K. Honga,b,, K.H. Yeonb, S.W. Nama a
Department of Display Semiconductor, Korea University, Chungnam 339-700, Korea Department of Physics, Institute for Basic Science Research, College of Natural Science, Chungbuk National University, Chungbuk 361-763, Korea
b
Received 12 September 2005; accepted 15 September 2005 Available online 21 November 2005
Abstract The size dependence of the electronic energy relaxation in a quantum dot is considered. The Fo¨rster energy transfer from an exciton in a quantum dot to a surface state of the dot is considered as a dominant channel of the electronic energy relaxation in a quantum dot. For the consideration of the relaxation mechanism, a microscopic quantum mechanical description of the exciton in the quantum dot and a macroscopic description of the surface state of the quantum dot are employed. From this analysis, it is shown that the relaxation rate increases as the radius R of the quantum dot decreases and the rate has 1=Rc dependence in the strong confinement regime, where c is considered to be a fraction between 3 and 4. r 2005 Elsevier B.V. All rights reserved. PACS: 73.23.b; 78.67.n; 78.67.Hc; 72.10.d Keywords: Quantum dot; Fo¨rster energy transfer; Surface state; Strong confinement; Electronic energy relaxation
1. Introduction The energy relaxation in a semiconductor nanocrystal has been extensively studied due to the difference of its mechanism from that in bulk materials. Depopulation of the excited electronic state of a quantum dot occurs via a variety of radiative and nonradiative mechanisms [1]. The most important nonradiative processes competing with radiative recombination are supposed to be carrier trapping at the surface or the interface of the quantum dot and the Auger recombination. As a nonradiative multiparticle process, the Auger recombination leads to the recombination of electron–hole pairs through energy transfer to a third particle which is re-excited to a higher energy state. At low pump intensity which has less than one exciton per quantum dot on average, the role of Auger effects is negligible and nonradiative relaxation is primarily due to Corresponding author. Department of Display Semiconductor, Korea University, Chungnam 339-700, Korea. Tel./fax: +82 43 261 2276. E-mail address: [email protected] (S.-K. Hong).
1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.09.009
carrier trapping [1]. In real quantum dots, the channels of the relaxation process may involve Auger recombination, trapping by surface or defect states, and so on. As a channel to be added, Klimov et al. suggested a relaxation process which results from electron–hole interactions in intraband relaxation [1]. Since a semiconductor nanocrystal has a large surface-to-volume ratio, the wavefunctions of the electron and the hole in a semiconductor nanocrystal are strongly affected by the surface properties of the nanocrystal. The improvement in surface passivation leads to the suppression of the nonradiative decay indicating that the relaxation of the electronic energy is caused by trapping at surface defects which are likely associated with dangling bonds. Recently, Okuno et al. reported the experimental studies of the size-dependent picosecond energy relaxation in PbSe quantum dots [2,3]. They showed the result that smaller dots have faster relaxation, and which was explained by the fact that smaller dots occupy larger surface-to-volume ratio than larger dots. From their experimental data, they concluded that the relaxation rate has 1=R3 dependence
ARTICLE IN PRESS S.-K. Hong et al. / Physica E 31 (2006) 48–52
in the strong confinement regime of an exciton in a quantum dot. Another interband study of PbSe in glasses showed fast nonradiative relaxation indicating the role of trapping processes, as well [4]. In this paper, we deal with a simple model of the Fo¨rster type energy transfer from an exciton in a quantum dot to a surface state of the quantum dot in order to consider the size-dependent electronic energy relaxation in a semiconductor quantum dot. To calculate the relaxation rate we employ the semiclassical approach which was introduced by Basko et al. [5,6]. The next section describes the model of the energy relaxation rate. The numerical estimations of the results in Section 2 are given in Section 3. And in Section 4, we summarize our results and conclude briefly. 2. The model of the relaxation rate
where D is the thickness of the surface wall (boundary) and the dielectric constant of the surface of the nanocrystal ~ðoÞ is employed to introduce absorption in the frequency range of interest. From the power W the transfer rate w is given by w ¼ W =_o. The inverse of the transfer rate is the transfer time t ¼ 1=w. For comparison, Basko et al. obtained the electric field outside the quantum dot, i.e., in the host medium for the given polarization charge in the quantum dot and the effective surface charge induced by the difference of the dielectric constant between the quantum dot and the host material, where the surface charge density was not needed to be evaluated in explicit form [6]. Whereas we obtain the electric field on the surface of the quantum dot for the given polarization charge in the quantum dot and the induced surface charge which should be evaluated in explicit form. ~ rÞ corresponding to the exciton The electric field Eð~ polarization in the quantum dot is obtained by solving the Poisson equation ~ rÞ, ð~ rÞr2 fð~ rÞ ¼ 4pr Pð~
with appropriate boundary conditions. Here the exciton ~ rÞ in the quantum dot is written as polarization Pð~ ~ rÞ ¼ ~ r;~ rÞ, Pð~ mvc cð~
(2)
(3)
where ~ mvc is the matrix element of the electric dipole moment between the Bloch functions of the conduction and valence band extrema and cð~ re ;~ rh Þ is the envelope function which describes the electron–hole pair in the quantum dot, and ~ re , ~ rh are the electron and hole coordinates, respectively. In the spherical symmetry, the envelope wavefunction is given by the spherical harmonics Y lm ðOÞ, where l; m are angular quantum numbers and O ¼ ðy; jÞ. Letting the dipole vector ~ mvc be along z-axis, we can write the polarization as Pz ð~ rÞ ¼ PðrÞ ðrÞY lm ðOÞ.
We consider a spherical quantum dot of radius R with dielectric constant nc which is embedded in a matrix with dielectric constant m . Following the approach used by Basko et al. which was for the calculation of the Fo¨rster energy transfer from a quantum dot to the surrounding polymer matrix [6], we evaluate the relaxation rate of the electronic energy in a quantum dot to a surface state of the quantum dot. As in the approach, the energy relaxation rate via the Fo¨rster energy transfer is calculated simply from the Joule losses of the electric field energy of the quantum dot to the surface wall (boundary) of the ~ rÞ is induced by quantum dot, where the electric field Eð~ ~ rÞ in the quantum do. Thus the the exciton polarization Pð~ transfer rate is given from the calculation of the power dissipation W on the surface of the quantum dot by the relation [6] Z o Im ~ ðoÞ RþD ~ 2 2 W¼ jEð~ rÞj d ~ r, (1) 2p r¼R
49
(4)
Then the charge density corresponding to the polarization is given by the divergence ~ rÞ ¼ qPz =qz, rð~ rÞ ¼ r Pð~
(5)
i.e. [6], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 m2 qPðrÞ PðrÞ ðl þ 1Þ rð~ rÞ ¼ Y l1;m ðOÞ qr r 4l 2 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl þ 1Þ2 m2 qPðrÞ PðrÞ þ þ l Y lþ1;m ðOÞ qr r 4ðl þ 1Þ2 1 ðrÞ ¼ rðrÞ l1 ðrÞY l1;m ðOÞ þ rlþ1 ðrÞY lþ1;m ðOÞ.
ð6Þ
For the charge distribution of Eq. (6), the electric field is obtained indirectly through the electrostatic potential fð~ rÞ satisfying the Poisson equation ð~ rÞr2 fð~ rÞ ¼ 4prð~ rÞ.
(7)
The solution of the Poisson equation with Neumann boundary conditions on the bounding surface S for given charge density rð~ rÞ can be obtained by means of Green functions [7] Z Z 1 qf 0 0 3 0 fð~ rÞ ¼ rð~ r ÞGð~ r;~ r Þd ~ Gð~ r;~ r0 Þ da0 , (8) r þ 0 4p qn V S where the Green function Gð~ r;~ r0 Þ is given by Gð~ r;~ r0 Þ ¼
X rl 4p o Y ðO0 ÞY lm ðOÞ. lþ1 2l þ 1 lm r 4 lm
(9)
By substituting Eqs. (6) and (9) into Eq. (8) and utilizing the orthogonality of the spherical harmonics, for the given charge density rk and r4R the volume integral fV ð~ rÞ in Eq. (8) is obtained as [6] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Y k;m ðOÞ k2 m2 R ðr0 Þ 0 kþ1 0 rÞ ¼ 4p kþ1 P r dr fV ;k ð~ r 4k2 1 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p QV k Y k;m ðOÞ kþ1 , ð10Þ 2k þ 1 r
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where QV k
where
ffiZ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffi 4p 0 kþ1 2 2 ¼ k m Pðr Þ r0 dr0 , 2k 1 0
(11)
k ¼ l 1; l þ 1 and PðrÞ is given by the envelope function of the exciton in the quantum dot. For the charge density induced on the surface ðrÞ 1 qfk sk ¼ (12) , 4p qr
4ðl þ 1Þðl þ 2Þ R ln r;lþ1 RþD " 2lþ3 # ðl þ 1Þ R , þ 2 1 RþD r;lþ1
alþ1 ¼ ðl þ 2Þ
ð18Þ
and blþ1 ¼ ðl þ 2Þ.
(19)
S
the surface integral in Eq. (8) is obtained as Z 0 4p 2 k fS;k ð~ Y k;m ðOÞr skðr Þ 0 k dr0 rÞ ¼ 2k þ 1 r rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p Y k;m ðOÞrk QSk . 2k þ 1
3. Numerical results in strong confinement regime
ð13Þ
Here QSk is to be determined by applying the continuity of the potential and the continuity of the normal component of the displacement field on the interface between the media with different dielectric constants. Thus the potential on the boundary surface RoroR þ D with the wall thickness D is given by fk ð~ rÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 4p Qk S k Y k;m ðOÞ þ Q r ; k 2k þ 1 nc rkþ1
ð14Þ
where the dielectric screening effect by the polarization charges inside the dot is considered. Then from the relation derived by boundary conditions of the continuity of the potential and its derivative at the surface of the quantum dot, the coefficient QSk is given in terms of QV k by QSk ¼
QV 1 k 1 , nc r;k ðR þ DÞ2kþ1
(15)
rffiffiffiffiffiffi 2 j l ðaln r=RÞ jone ð~ Y lm ðOÞ, rÞ ¼ R3 j lþ1 ðaln Þ
where r;k ¼
knc þ ðk þ 1Þm , ðk þ 1ÞD
(16)
and D ¼ nc m . When we evaluate the potential, since the contribution of QV k with k ¼ l 1 is zero (due to the fact that PðrÞ ðrÞ vanishes at r ¼ R when it is integrated over the dot volume), only the contribution of QV k with k ¼ l þ 1 is involved in the evaluation of the potential [6]. Thus from the potential of Eq. (14) with Eq. (15), we obtain the ~ ¼ rf ~ and the substitution of the electric electric field E field into Eq. (1) leads to the decay rate 2 ~ 1 2 Q V lþ1 Im w¼ ¼ t ð2l þ 3Þ_2nc alþ1 blþ1 , ðR þ DÞ2lþ3 R2lþ3
In order to obtain the relaxation rate of Eq. (17), the wavefunction of the exciton cð~ re ;~ rh Þ for the polarization in Eq. (11) is needed. The wavefunction of an exciton in a quantum dot is determined by two interactions. One is the confinement potential and the other is Coulomb interaction of the electron and the hole. The solution of the Schro¨dinger equation which includes all the interaction terms is quite a complicated problem. Then in the limit where one of the two interactions dominates, the wavefunction is simplified. So when the quantum dot radius R is much less than the exciton bulk Bohr radius aB , Coulomb interaction can be completely ignored (strong confinement regime). Whereas when the exciton bulk Bohr radius aB is much less than the quantum dot radius R, the confinement potential can be completely ignored (weak confinement regime) and the exciton can be considered as a rigid particle moving in a spherical well [8]. In this section, the exciton wavefunction is considered in the strong confinement limit. In the strong confinement limit, the exciton wavefunction of the quantum dot is able to be approximated simply as the product of two one-particle wavefunctions, where the single-particle wavefunction is given by [8]
where j l ðxÞ is the lth spherical Bessel function, aln is its nth zero. In this regime, for l e ¼ l h ¼ l ¼ 0 and ne ¼ nh ¼ 1, i.e., for the lowest exciton state, the wavefunction of the exciton in a quantum dot is written as 2 j ða01 r=RÞ cðrÞ ðrÞ ¼ pffiffiffiffiffiffi 3 0 2 , 4pR j 1 ða01 Þ
(21)
and the effective charge density on the surface QS1 is given by QS1 ¼
ð17Þ
(20)
QV 1 , nc r;1 ðR þ DÞ3
(22)
where QV 1 ¼ mvc . Then the relaxation rate w for the lowest exciton in the quantum dot in strong confinement regime is
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obtained as 30
ð23Þ
In two extreme cases, when the wall thickness D of the quantum dot is much less than the radius R of the quantum dot, the dependence of the relaxation rate of Eq. (23) on the dot radius is approximated to be w/
D , R4
20
10
(24) 0 12
whereas when the wall thickness DR, the proportionality of the relaxation rate to the dot radius is 1 w/ 3, R
Energy transfer time (ps)
" " 2m2vc Im ~ 1 8 R 2 ln w¼ 3_2nc r;1 RþD ðR þ DÞ3 # 3 # 1 1 R 2 þ 2 2 3 . r;1 r;1 R þ D R
(25)
thus the actual exponent c in the dependence of the relaxation rate on the dot radius w / 1=Rc will be a number between 3 and 4. In Eq. (23), the quantity Im ~ðoÞ is the absorptive part of the complex dielectric function of the surface part of the quantum dot, which practically includes both the effects of the dielectric property of the material PbSe and the surface defects of the quantum dot. The difference of the dielectric constants between the quantum dot and the host material is included in D ¼ nc m of Eq. (16). From Eq. (17) when D ¼ 0 for the nanocrystal with idealized surface, the relaxation rate is zero, which indicates that there is no relaxation route to idealized surface via Fo¨rster type energy transfer when the host medium has the same dielectric constant as the quantum dot. For the numerical estimation which is for the comparison with Okuno et al.’s experimental results [2,3], the parameters which are typical for IV–VI semiconductors are used. Due to the large Bohr radius and dielectric constant, nanocrystals of the IV–VI semiconductors provide unique properties for investigating the effects of strong confinement [9]. The parameters for the numerical estimation are considered to be m ’ 5, nc ’ 23, mcv ’ 16 Debye, and Im ~ ’ 14 [10–15]. Several of the parameters have been inferred from other values in the references. First, m is assumed to be around 5 from the fact that the typical value of the dielectric constant for a glass material is 3.7–10 [10]. Second, since the typical transition dipole moment of an exciton in a semiconductor nanocrystal is considered to be tens of Debye, so mcv for an exciton in PbSe nanocrystal is assumed to be around 16 Debyes which is smaller than that for III–V semiconductor material [11–13]. Finally, the imaginary part of the dielectric constant for the surface of the quantum dot has been inferred from the imaginary part of the dielectric constant for the PbSe material [11,14,15]. Practically, the value Im ~ will be larger than the imaginary part of the dielectric constant of the material PbSe due to
16
20 24 Quantum dot radius (Å)
28
Fig. 1. Relaxation time versus radius of quantum dot, where m ’ 5, nc ’ 23, mcv ’ 16 Debye, and Im ~ ¼ 14. Dotted line, thick dashed line, ˚ 2 A, ˚ and 3 A, ˚ respectively. The and dashed-dotted line are for D ¼ 1 A, solid line is the best-fit curve with exponent c ¼ 3:5 for the curve t ¼ gRc . Data with closed circle are the experimental data shown in Okuno et al.’s report of Ref. [2].
the additional contribution from the surface defects. Thus Im ~ is assumed to be around 14 or larger than this one. Using the parameters, the relaxation rate of Eq. (23) is plotted as the curves in Fig. 1. In the figure, dotted line, ˚ thick dashed line, and dashed-dotted line are for D ¼ 1 A, ˚ and 3 A, ˚ respectively. In the figure closed circle 2 A, indicates the data reported by Okuno et al. [2]. Using the method of least squares, the exponent c of the curve for the function of t ¼ 1=w ¼ gRc which fits best to the experimental data in Fig. 1 is given by c ¼ 3:5. The best-fit curve is shown as the solid line in Fig. 1. This fitting of the experimental data to the curve t ¼ gR3:5 coincides well with our analysis that the relaxation rate increases as the radius R of the quantum dot decreases and has 1=Rc dependence in the strong confinement limit, where c is practically a fraction between 3 and 4 depending on the surface wall thickness of the quantum dot as shown in Eqs. (23)–(25). As a reference to see the dependence of the rate on the values of the parameters, the thin dashed line is plotted for ˚ and other parameters the case with Im ~ ’ 10, D ¼ 2 A, fixed. The relaxation of the electronic energy in a quantum dot to the surface of the quantum dot strongly depends on the quality of the quantum dot. Then since nanocrystals synthesized by high-temperature precipitation in molten glasses usually have a large number of surface defects [1], they show the strong dependence of the relaxation of the electronic energy on the radius of quantum dots as shown in the figure. 4. Conclusions We have studied the size dependence of the electronic energy relaxation of PbSe quantum dots embedded in glasses to the surface (boundary) states of the quantum
ARTICLE IN PRESS 52
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dots. The nonradiative decay through the Fo¨rster type energy transfer mechanism from the exciton energy in a quantum dot to the surface state of the dot is considered as one of dominant channels of the electronic energy relaxation. For the consideration a microscopic quantum mechanical description of the exciton in the quantum dot and a macroscopic description of the surface of the quantum dot have been employed. From the analysis, it has been shown that the relaxation rate increases as the radius R decreases and has 1=Rc dependence in the strong confinement limit, where c is 4 when the thickness of the surface wall is much less than the radius of the quantum dot, and c is 3 when the wall thickness is close to the radius of the quantum dot. This means that practically the exponent c is a fraction between 3 and 4. References [1] V.I. Klimov, J. Phys. Chem. B 104 (2000) 6112.
[2] T. Okuno, A.A. Lipovskii, T. Ogawa, I. Amagai, Y. Masumoto, J. Lumin. 87–89 (2000) 491. [3] T. Okuno, Y. Masumoto, M. Ikezawa, T. Ogawa, A.A. Lipovskii, Appl. Phys. Lett. 77 (2000) 504. [4] K. Wundke, S. Potting, J. Auxier, Appl. Phys. Lett. 76 (2000) 10. [5] V.M. Agranovich, M.D. Galanin, Electronic Excitation Energy Transfer in Condensed Matter, North Holland, Amsterdam, 1982. [6] D.M. Basko, V.M. Agranovich, F. Bassani, G.C. La Rocca, Eur. Phys. J. B 13 (2000) 653. [7] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [8] F. Bassani, R. Buczko, Czajkowski, Nuovo Cimento D 19 (1997) 1565. [9] F.W. Wise, Acc. Chem. Res. 33 (2000) 773. [10] ohttp://www.clippercontrols.com/info/dielectric-constants.html4. [11] I. Kang, F.W. Wise, J. Opt. Soc. Am. B 14 (1997) 1632. [12] K. Li, X. Li, M.I. Stockman, D.J. Bergman, Phys. Rev. B 71 (2005) 115409. [13] K.L. Silvermana, R.P. Mirin, S.T. Cundiff, A.G. Norman, Appl. Phys. Lett. 82 (2003) 23. [14] A. Delin, P. Ravindran, O. Eriksson, J.M. Wills, Int. J. Quantum Chem. 69 (1998) 349. [15] G. Allan, C. Delerue, Phys. Rev. B 70 (2004) 245321.
Physica E 31 (2006) 48–52 www.elsevier.com/locate/physe
Size-dependent electronic energy relaxation in a semiconductor nanocrystal in the strong confinement limit S.-K. Honga,b,, K.H. Yeonb, S.W. Nama a
Department of Display Semiconductor, Korea University, Chungnam 339-700, Korea Department of Physics, Institute for Basic Science Research, College of Natural Science, Chungbuk National University, Chungbuk 361-763, Korea
b
Received 12 September 2005; accepted 15 September 2005 Available online 21 November 2005
Abstract The size dependence of the electronic energy relaxation in a quantum dot is considered. The Fo¨rster energy transfer from an exciton in a quantum dot to a surface state of the dot is considered as a dominant channel of the electronic energy relaxation in a quantum dot. For the consideration of the relaxation mechanism, a microscopic quantum mechanical description of the exciton in the quantum dot and a macroscopic description of the surface state of the quantum dot are employed. From this analysis, it is shown that the relaxation rate increases as the radius R of the quantum dot decreases and the rate has 1=Rc dependence in the strong confinement regime, where c is considered to be a fraction between 3 and 4. r 2005 Elsevier B.V. All rights reserved. PACS: 73.23.b; 78.67.n; 78.67.Hc; 72.10.d Keywords: Quantum dot; Fo¨rster energy transfer; Surface state; Strong confinement; Electronic energy relaxation
1. Introduction The energy relaxation in a semiconductor nanocrystal has been extensively studied due to the difference of its mechanism from that in bulk materials. Depopulation of the excited electronic state of a quantum dot occurs via a variety of radiative and nonradiative mechanisms [1]. The most important nonradiative processes competing with radiative recombination are supposed to be carrier trapping at the surface or the interface of the quantum dot and the Auger recombination. As a nonradiative multiparticle process, the Auger recombination leads to the recombination of electron–hole pairs through energy transfer to a third particle which is re-excited to a higher energy state. At low pump intensity which has less than one exciton per quantum dot on average, the role of Auger effects is negligible and nonradiative relaxation is primarily due to Corresponding author. Department of Display Semiconductor, Korea University, Chungnam 339-700, Korea. Tel./fax: +82 43 261 2276. E-mail address: [email protected] (S.-K. Hong).
1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.09.009
carrier trapping [1]. In real quantum dots, the channels of the relaxation process may involve Auger recombination, trapping by surface or defect states, and so on. As a channel to be added, Klimov et al. suggested a relaxation process which results from electron–hole interactions in intraband relaxation [1]. Since a semiconductor nanocrystal has a large surface-to-volume ratio, the wavefunctions of the electron and the hole in a semiconductor nanocrystal are strongly affected by the surface properties of the nanocrystal. The improvement in surface passivation leads to the suppression of the nonradiative decay indicating that the relaxation of the electronic energy is caused by trapping at surface defects which are likely associated with dangling bonds. Recently, Okuno et al. reported the experimental studies of the size-dependent picosecond energy relaxation in PbSe quantum dots [2,3]. They showed the result that smaller dots have faster relaxation, and which was explained by the fact that smaller dots occupy larger surface-to-volume ratio than larger dots. From their experimental data, they concluded that the relaxation rate has 1=R3 dependence
ARTICLE IN PRESS S.-K. Hong et al. / Physica E 31 (2006) 48–52
in the strong confinement regime of an exciton in a quantum dot. Another interband study of PbSe in glasses showed fast nonradiative relaxation indicating the role of trapping processes, as well [4]. In this paper, we deal with a simple model of the Fo¨rster type energy transfer from an exciton in a quantum dot to a surface state of the quantum dot in order to consider the size-dependent electronic energy relaxation in a semiconductor quantum dot. To calculate the relaxation rate we employ the semiclassical approach which was introduced by Basko et al. [5,6]. The next section describes the model of the energy relaxation rate. The numerical estimations of the results in Section 2 are given in Section 3. And in Section 4, we summarize our results and conclude briefly. 2. The model of the relaxation rate
where D is the thickness of the surface wall (boundary) and the dielectric constant of the surface of the nanocrystal ~ðoÞ is employed to introduce absorption in the frequency range of interest. From the power W the transfer rate w is given by w ¼ W =_o. The inverse of the transfer rate is the transfer time t ¼ 1=w. For comparison, Basko et al. obtained the electric field outside the quantum dot, i.e., in the host medium for the given polarization charge in the quantum dot and the effective surface charge induced by the difference of the dielectric constant between the quantum dot and the host material, where the surface charge density was not needed to be evaluated in explicit form [6]. Whereas we obtain the electric field on the surface of the quantum dot for the given polarization charge in the quantum dot and the induced surface charge which should be evaluated in explicit form. ~ rÞ corresponding to the exciton The electric field Eð~ polarization in the quantum dot is obtained by solving the Poisson equation ~ rÞ, ð~ rÞr2 fð~ rÞ ¼ 4pr Pð~
with appropriate boundary conditions. Here the exciton ~ rÞ in the quantum dot is written as polarization Pð~ ~ rÞ ¼ ~ r;~ rÞ, Pð~ mvc cð~
(2)
(3)
where ~ mvc is the matrix element of the electric dipole moment between the Bloch functions of the conduction and valence band extrema and cð~ re ;~ rh Þ is the envelope function which describes the electron–hole pair in the quantum dot, and ~ re , ~ rh are the electron and hole coordinates, respectively. In the spherical symmetry, the envelope wavefunction is given by the spherical harmonics Y lm ðOÞ, where l; m are angular quantum numbers and O ¼ ðy; jÞ. Letting the dipole vector ~ mvc be along z-axis, we can write the polarization as Pz ð~ rÞ ¼ PðrÞ ðrÞY lm ðOÞ.
We consider a spherical quantum dot of radius R with dielectric constant nc which is embedded in a matrix with dielectric constant m . Following the approach used by Basko et al. which was for the calculation of the Fo¨rster energy transfer from a quantum dot to the surrounding polymer matrix [6], we evaluate the relaxation rate of the electronic energy in a quantum dot to a surface state of the quantum dot. As in the approach, the energy relaxation rate via the Fo¨rster energy transfer is calculated simply from the Joule losses of the electric field energy of the quantum dot to the surface wall (boundary) of the ~ rÞ is induced by quantum dot, where the electric field Eð~ ~ rÞ in the quantum do. Thus the the exciton polarization Pð~ transfer rate is given from the calculation of the power dissipation W on the surface of the quantum dot by the relation [6] Z o Im ~ ðoÞ RþD ~ 2 2 W¼ jEð~ rÞj d ~ r, (1) 2p r¼R
49
(4)
Then the charge density corresponding to the polarization is given by the divergence ~ rÞ ¼ qPz =qz, rð~ rÞ ¼ r Pð~
(5)
i.e. [6], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 m2 qPðrÞ PðrÞ ðl þ 1Þ rð~ rÞ ¼ Y l1;m ðOÞ qr r 4l 2 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl þ 1Þ2 m2 qPðrÞ PðrÞ þ þ l Y lþ1;m ðOÞ qr r 4ðl þ 1Þ2 1 ðrÞ ¼ rðrÞ l1 ðrÞY l1;m ðOÞ þ rlþ1 ðrÞY lþ1;m ðOÞ.
ð6Þ
For the charge distribution of Eq. (6), the electric field is obtained indirectly through the electrostatic potential fð~ rÞ satisfying the Poisson equation ð~ rÞr2 fð~ rÞ ¼ 4prð~ rÞ.
(7)
The solution of the Poisson equation with Neumann boundary conditions on the bounding surface S for given charge density rð~ rÞ can be obtained by means of Green functions [7] Z Z 1 qf 0 0 3 0 fð~ rÞ ¼ rð~ r ÞGð~ r;~ r Þd ~ Gð~ r;~ r0 Þ da0 , (8) r þ 0 4p qn V S where the Green function Gð~ r;~ r0 Þ is given by Gð~ r;~ r0 Þ ¼
X rl 4p o Y ðO0 ÞY lm ðOÞ. lþ1 2l þ 1 lm r 4 lm
(9)
By substituting Eqs. (6) and (9) into Eq. (8) and utilizing the orthogonality of the spherical harmonics, for the given charge density rk and r4R the volume integral fV ð~ rÞ in Eq. (8) is obtained as [6] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Y k;m ðOÞ k2 m2 R ðr0 Þ 0 kþ1 0 rÞ ¼ 4p kþ1 P r dr fV ;k ð~ r 4k2 1 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p QV k Y k;m ðOÞ kþ1 , ð10Þ 2k þ 1 r
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where QV k
where
ffiZ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffi 4p 0 kþ1 2 2 ¼ k m Pðr Þ r0 dr0 , 2k 1 0
(11)
k ¼ l 1; l þ 1 and PðrÞ is given by the envelope function of the exciton in the quantum dot. For the charge density induced on the surface ðrÞ 1 qfk sk ¼ (12) , 4p qr
4ðl þ 1Þðl þ 2Þ R ln r;lþ1 RþD " 2lþ3 # ðl þ 1Þ R , þ 2 1 RþD r;lþ1
alþ1 ¼ ðl þ 2Þ
ð18Þ
and blþ1 ¼ ðl þ 2Þ.
(19)
S
the surface integral in Eq. (8) is obtained as Z 0 4p 2 k fS;k ð~ Y k;m ðOÞr skðr Þ 0 k dr0 rÞ ¼ 2k þ 1 r rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p Y k;m ðOÞrk QSk . 2k þ 1
3. Numerical results in strong confinement regime
ð13Þ
Here QSk is to be determined by applying the continuity of the potential and the continuity of the normal component of the displacement field on the interface between the media with different dielectric constants. Thus the potential on the boundary surface RoroR þ D with the wall thickness D is given by fk ð~ rÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 4p Qk S k Y k;m ðOÞ þ Q r ; k 2k þ 1 nc rkþ1
ð14Þ
where the dielectric screening effect by the polarization charges inside the dot is considered. Then from the relation derived by boundary conditions of the continuity of the potential and its derivative at the surface of the quantum dot, the coefficient QSk is given in terms of QV k by QSk ¼
QV 1 k 1 , nc r;k ðR þ DÞ2kþ1
(15)
rffiffiffiffiffiffi 2 j l ðaln r=RÞ jone ð~ Y lm ðOÞ, rÞ ¼ R3 j lþ1 ðaln Þ
where r;k ¼
knc þ ðk þ 1Þm , ðk þ 1ÞD
(16)
and D ¼ nc m . When we evaluate the potential, since the contribution of QV k with k ¼ l 1 is zero (due to the fact that PðrÞ ðrÞ vanishes at r ¼ R when it is integrated over the dot volume), only the contribution of QV k with k ¼ l þ 1 is involved in the evaluation of the potential [6]. Thus from the potential of Eq. (14) with Eq. (15), we obtain the ~ ¼ rf ~ and the substitution of the electric electric field E field into Eq. (1) leads to the decay rate 2 ~ 1 2 Q V lþ1 Im w¼ ¼ t ð2l þ 3Þ_2nc alþ1 blþ1 , ðR þ DÞ2lþ3 R2lþ3
In order to obtain the relaxation rate of Eq. (17), the wavefunction of the exciton cð~ re ;~ rh Þ for the polarization in Eq. (11) is needed. The wavefunction of an exciton in a quantum dot is determined by two interactions. One is the confinement potential and the other is Coulomb interaction of the electron and the hole. The solution of the Schro¨dinger equation which includes all the interaction terms is quite a complicated problem. Then in the limit where one of the two interactions dominates, the wavefunction is simplified. So when the quantum dot radius R is much less than the exciton bulk Bohr radius aB , Coulomb interaction can be completely ignored (strong confinement regime). Whereas when the exciton bulk Bohr radius aB is much less than the quantum dot radius R, the confinement potential can be completely ignored (weak confinement regime) and the exciton can be considered as a rigid particle moving in a spherical well [8]. In this section, the exciton wavefunction is considered in the strong confinement limit. In the strong confinement limit, the exciton wavefunction of the quantum dot is able to be approximated simply as the product of two one-particle wavefunctions, where the single-particle wavefunction is given by [8]
where j l ðxÞ is the lth spherical Bessel function, aln is its nth zero. In this regime, for l e ¼ l h ¼ l ¼ 0 and ne ¼ nh ¼ 1, i.e., for the lowest exciton state, the wavefunction of the exciton in a quantum dot is written as 2 j ða01 r=RÞ cðrÞ ðrÞ ¼ pffiffiffiffiffiffi 3 0 2 , 4pR j 1 ða01 Þ
(21)
and the effective charge density on the surface QS1 is given by QS1 ¼
ð17Þ
(20)
QV 1 , nc r;1 ðR þ DÞ3
(22)
where QV 1 ¼ mvc . Then the relaxation rate w for the lowest exciton in the quantum dot in strong confinement regime is
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51
obtained as 30
ð23Þ
In two extreme cases, when the wall thickness D of the quantum dot is much less than the radius R of the quantum dot, the dependence of the relaxation rate of Eq. (23) on the dot radius is approximated to be w/
D , R4
20
10
(24) 0 12
whereas when the wall thickness DR, the proportionality of the relaxation rate to the dot radius is 1 w/ 3, R
Energy transfer time (ps)
" " 2m2vc Im ~ 1 8 R 2 ln w¼ 3_2nc r;1 RþD ðR þ DÞ3 # 3 # 1 1 R 2 þ 2 2 3 . r;1 r;1 R þ D R
(25)
thus the actual exponent c in the dependence of the relaxation rate on the dot radius w / 1=Rc will be a number between 3 and 4. In Eq. (23), the quantity Im ~ðoÞ is the absorptive part of the complex dielectric function of the surface part of the quantum dot, which practically includes both the effects of the dielectric property of the material PbSe and the surface defects of the quantum dot. The difference of the dielectric constants between the quantum dot and the host material is included in D ¼ nc m of Eq. (16). From Eq. (17) when D ¼ 0 for the nanocrystal with idealized surface, the relaxation rate is zero, which indicates that there is no relaxation route to idealized surface via Fo¨rster type energy transfer when the host medium has the same dielectric constant as the quantum dot. For the numerical estimation which is for the comparison with Okuno et al.’s experimental results [2,3], the parameters which are typical for IV–VI semiconductors are used. Due to the large Bohr radius and dielectric constant, nanocrystals of the IV–VI semiconductors provide unique properties for investigating the effects of strong confinement [9]. The parameters for the numerical estimation are considered to be m ’ 5, nc ’ 23, mcv ’ 16 Debye, and Im ~ ’ 14 [10–15]. Several of the parameters have been inferred from other values in the references. First, m is assumed to be around 5 from the fact that the typical value of the dielectric constant for a glass material is 3.7–10 [10]. Second, since the typical transition dipole moment of an exciton in a semiconductor nanocrystal is considered to be tens of Debye, so mcv for an exciton in PbSe nanocrystal is assumed to be around 16 Debyes which is smaller than that for III–V semiconductor material [11–13]. Finally, the imaginary part of the dielectric constant for the surface of the quantum dot has been inferred from the imaginary part of the dielectric constant for the PbSe material [11,14,15]. Practically, the value Im ~ will be larger than the imaginary part of the dielectric constant of the material PbSe due to
16
20 24 Quantum dot radius (Å)
28
Fig. 1. Relaxation time versus radius of quantum dot, where m ’ 5, nc ’ 23, mcv ’ 16 Debye, and Im ~ ¼ 14. Dotted line, thick dashed line, ˚ 2 A, ˚ and 3 A, ˚ respectively. The and dashed-dotted line are for D ¼ 1 A, solid line is the best-fit curve with exponent c ¼ 3:5 for the curve t ¼ gRc . Data with closed circle are the experimental data shown in Okuno et al.’s report of Ref. [2].
the additional contribution from the surface defects. Thus Im ~ is assumed to be around 14 or larger than this one. Using the parameters, the relaxation rate of Eq. (23) is plotted as the curves in Fig. 1. In the figure, dotted line, ˚ thick dashed line, and dashed-dotted line are for D ¼ 1 A, ˚ and 3 A, ˚ respectively. In the figure closed circle 2 A, indicates the data reported by Okuno et al. [2]. Using the method of least squares, the exponent c of the curve for the function of t ¼ 1=w ¼ gRc which fits best to the experimental data in Fig. 1 is given by c ¼ 3:5. The best-fit curve is shown as the solid line in Fig. 1. This fitting of the experimental data to the curve t ¼ gR3:5 coincides well with our analysis that the relaxation rate increases as the radius R of the quantum dot decreases and has 1=Rc dependence in the strong confinement limit, where c is practically a fraction between 3 and 4 depending on the surface wall thickness of the quantum dot as shown in Eqs. (23)–(25). As a reference to see the dependence of the rate on the values of the parameters, the thin dashed line is plotted for ˚ and other parameters the case with Im ~ ’ 10, D ¼ 2 A, fixed. The relaxation of the electronic energy in a quantum dot to the surface of the quantum dot strongly depends on the quality of the quantum dot. Then since nanocrystals synthesized by high-temperature precipitation in molten glasses usually have a large number of surface defects [1], they show the strong dependence of the relaxation of the electronic energy on the radius of quantum dots as shown in the figure. 4. Conclusions We have studied the size dependence of the electronic energy relaxation of PbSe quantum dots embedded in glasses to the surface (boundary) states of the quantum
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dots. The nonradiative decay through the Fo¨rster type energy transfer mechanism from the exciton energy in a quantum dot to the surface state of the dot is considered as one of dominant channels of the electronic energy relaxation. For the consideration a microscopic quantum mechanical description of the exciton in the quantum dot and a macroscopic description of the surface of the quantum dot have been employed. From the analysis, it has been shown that the relaxation rate increases as the radius R decreases and has 1=Rc dependence in the strong confinement limit, where c is 4 when the thickness of the surface wall is much less than the radius of the quantum dot, and c is 3 when the wall thickness is close to the radius of the quantum dot. This means that practically the exponent c is a fraction between 3 and 4. References [1] V.I. Klimov, J. Phys. Chem. B 104 (2000) 6112.
[2] T. Okuno, A.A. Lipovskii, T. Ogawa, I. Amagai, Y. Masumoto, J. Lumin. 87–89 (2000) 491. [3] T. Okuno, Y. Masumoto, M. Ikezawa, T. Ogawa, A.A. Lipovskii, Appl. Phys. Lett. 77 (2000) 504. [4] K. Wundke, S. Potting, J. Auxier, Appl. Phys. Lett. 76 (2000) 10. [5] V.M. Agranovich, M.D. Galanin, Electronic Excitation Energy Transfer in Condensed Matter, North Holland, Amsterdam, 1982. [6] D.M. Basko, V.M. Agranovich, F. Bassani, G.C. La Rocca, Eur. Phys. J. B 13 (2000) 653. [7] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [8] F. Bassani, R. Buczko, Czajkowski, Nuovo Cimento D 19 (1997) 1565. [9] F.W. Wise, Acc. Chem. Res. 33 (2000) 773. [10] ohttp://www.clippercontrols.com/info/dielectric-constants.html4. [11] I. Kang, F.W. Wise, J. Opt. Soc. Am. B 14 (1997) 1632. [12] K. Li, X. Li, M.I. Stockman, D.J. Bergman, Phys. Rev. B 71 (2005) 115409. [13] K.L. Silvermana, R.P. Mirin, S.T. Cundiff, A.G. Norman, Appl. Phys. Lett. 82 (2003) 23. [14] A. Delin, P. Ravindran, O. Eriksson, J.M. Wills, Int. J. Quantum Chem. 69 (1998) 349. [15] G. Allan, C. Delerue, Phys. Rev. B 70 (2004) 245321.