18451
2009, 113, 18451–18454 Published on Web 09/30/2009
Stochastic Approach to Charge Separation in Multiexcited Quantum Dots Maria Hilczer*,† and M. Tachiya*,‡ Institute of Applied Radiation Chemistry, Technical UniVersity of Lodz, Wroblewskiego 15, 93-590 Lodz, Poland, and National Institute of AdVanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, Tsukuba, Ibaraki 305-8565, Japan ReceiVed: August 18, 2009; ReVised Manuscript ReceiVed: September 22, 2009
Recent time-resolved measurements of absorbance of CdSe quantum dots with methylviologen (MV2+) adsorbed on their surface have demonstrated ultrafast electron transfer from multiexcited quantum dot to MV2+ [Matylitsky et al. J. Am. Chem. Soc. 2009, 131, 2424.]. A stochastic approach to the carrier dynamics in quantum dots has been proposed that includes a Poissonian initial distribution of excitons in a quantum dot and the relaxation of multiple excitons by competing electron transfer and Auger recombination processes. The model has been successfully applied to analyze experimental data. Colloidal quantum dots can be used as light emitters in electroluminescent devices1 or light absorbers in photovoltaics.2 Efficient generation of multiexcitons in some quantum dots by charge carrier multiplication3 makes it possible to apply them in solar cells. Their successful application, however, requires the charge separation of electron-hole pairs to compete effectively with charge losses due to Auger recombination. Recently, Matylitsky et al.4 reported on the study of ultrafast electron transfer between multiexcited CdSe quantum dots and methylviologen (MV2+) adsorbed on the surface of the quantum dots. For the CdSe/MV2+ couple they identified the transient absorption band of MV•+ radicals that appears on the 0.1 ps time scale and results exclusively from electron transfer from photoexcited quantum dots to MV2+. Matylitsky et al. used a simplified model to reproduce the observed dependence of the relative transient absorption signal of the MV•+ radical on the average number nj0 of excitons generated initially per nanoparticle. They assumed in this model that (1) only Auger and electron-transfer processes play roles in the relaxation of multiple excitons and (2) excitons decay exclusively by Auger recombination as long as the number n of excitons in a quantum dot is larger than a certain number nmax, while for n e nmax, excitons decay exclusively by electron transfer. In this Letter, we present a stochastic model of carrier relaxation dynamics in quantum dots and apply it to analyze experimental data reported in ref 4. We do not need to implement assumption (2) into our model, but we assume that the excited electron-hole (exciton) population decays along two channels, by electron transfer to acceptor and by Auger recombination according to the following scheme
n(n - 1)γA /2
En 98 En-1
(2)
where En stands for a quantum dot with n excitons. The γ is the first-order electron transfer rate constant for a quantum dot with one exciton; thus, if a quantum dot contains n excitons, the rate constant is given by nγ. The γA is the firstorder Auger recombination for a quantum dot with two excitons. If a quantum dot contains n excitons, the rate constant is given by5,6 (1/2)n(n - 1)γA. Auger recombination involves an annihilation of an electron-hole pair with the released energy transferred to the electron (or hole) of another pair. Equation 2 assumes that the excited carrier relaxes back to the fundamental band edge by dissipating the transferred energy rapidly. The time-dependent probability density Fn(t) of finding n excitons in a quantum dot at time t obeys the following master equation
dFn(t) 1 ) - γ + (n - 1)γA nFn(t) + dt 2 1 γ + nγA (n + 1)Fn+1(t) (3) 2
[
]
(
)
With Fn(t), we can express the time evolution of the average number of excitons in a single quantum dot via ∞
n(t) )
nγ
En f En-1
(1)
* To whom correspondence should be addressed. E-mail: m.hilczer@ mitr.p.lodz.pl (M.H.) or
[email protected] (M.T.). † Technical University of Lodz. ‡ National Institute of Advanced Industrial Science and Technology.
10.1021/jp907969d CCC: $40.75
∑ nFn(t)
(4)
i)1
The solution of eq 3, using the generating function technique, is presented in ref 5. In the case in which a quantum dot contains n0 excitons at t ) 0, it has the form 2009 American Chemical Society
18452
J. Phys. Chem. C, Vol. 113, No. 43, 2009 n0
n(t) )
∑ (2γ/γ
A
+ 2i - 1)
i)1
Letters
Γ(2γ/γA + n0) n0 ! × (n0 - i)! Γ(2γ/γA + n0 + i)
(
[
)]
2γ 1 exp - iγA +i-1 t 2 γA
(5)
The yield of charge separation per quantum dot up to time t is given in terms of n(t) by
η(t;n0) )
∫0t γn(t)dt
(6)
By substituting eq 5 into eq 6, we obtain
{
η(t;n0) ) n0 2γ/γA + 2i - 1 2γ × γA i)1 i(2γ/γA + i - 1)
∑
Γ(2γ/γA + n0) n0 ! × (n0 - i)! Γ(2γ/γA + n0 + i)
[
{
(
) ]}
2γ 1 1 - exp - iγA +i-1t , 2 γA
0
for n0 g 1
(7)
Figure 1. Time evolution of the total yield of MV•+ radical formation per quantum dot, η(t), for a Poissonian distribution of the initial number n0 of excitons generated in a quantum dot. Calculations were performed for various values of the ratio of the electron-transfer rate constant γ to the Auger recombination rate constant γA and for the average initial number of excitons per quantum dot nj0 ) 0.5.
for n0 ) 0
The ultimate yield of charge separation per quantum dot is given by
{
η(∞;n0) ) n0 2γ/γA + 2i - 1 n0 ! 2γ × γA i)1 i(2γ/γA + i - 1) (n0 - i)!
∑
Γ(2γ/γA + n0) Γ(2γ/γA + n0 + i) 0
for n0 g 1
(8)
for n0 ) 0
If the initial distribution of excitons in a quantum dot is described by a Poisson distribution, the time evolution η(t) and the ultimate yield η(∞) of charge separation per quantum dot are given by ∞
nj0n0 η(t) ) η(t;n0) exp(-nj0) n0 ! n0)1
∑
(9)
Figure 2. Ultimate yield of MV•+ radicals per quantum dot, η(∞), for a Poissonian distribution of the initial number n0 of excitons generated in a quantum dot as a function of the average initial number of excitons per quantum dot nj0 calculated for various values of the ratio γ/γA.
in Figure 2. The curves for the limiting values of 2γ/γA are obtained in the following way. If the Auger recombination rate constant is zero, all excitons initially present in a quantum dot lead to charge separation. Therefore, if a quantum dot initially contains n0 excitons, we have
and
η(∞;n0) ) n0
∞
njnn0 η(∞) ) η(∞;n0) exp(-nj0) n0 ! n0)1
∑
(11)
(10)
where nj0 is the average initial number of excitons per quantum dot. It is instructive to consider dependence of the time evolution and the ultimate yield, η(t) and η(∞), of charge separation on the ratio of the electron-transfer rate constant γ to the Auger recombination rate constant γA. Figure 1 plots η(t) against γt for several values of the ratio 2γ/γA and for nj0 ) 0.5. As can be expected, the increase of γ or the decrease of γA results in the increase of the yield of charge separation. The dependence of the ultimate yield η(∞) of charge separation on nj0 is shown
and if the initial distribution of excitons in a quantum dot is described by a Poisson distribution, we obtain from eq 10
η(∞) ) nj0
(12)
On the other hand, if the Auger recombination rate constant is infinite, all excitons initially present in a quantum dot decay by Auger recombination, except the last one that is left behind alone, which decays by electron transfer to acceptor. Therefore, if a quantum dot initially contains n0 excitons, we have
Letters
J. Phys. Chem. C, Vol. 113, No. 43, 2009 18453
η(∞;n0) )
{
1 for n0 g 1 0 for n0 ) 0
(13)
and if the initial distribution of excitons in a quantum dot is described by a Poisson distribution, the ultimate yield η(∞) of charge separation per quantum dot takes the form
η(∞) ) 1 - exp(-nj0)
(14)
Equations 12 and 14 are also included in Figure 2. In Figure 3, the ultimate yield of charge separation predicted from eq 10 with eq 8 is compared with the observed one taken from Figure 2a of ref 4. Agreement between theory and experiment is obtained if we take 2γ/γA ) 4. In Figure 4, the time evolution of charge separation predicted from eq 9 with eq 7 is compared with the observed one taken from Figure 1a of ref 4. The theoretical curve fits to the observed one with the values of the parameters 2γ/γA ) 4 and γ ) 5.056 ps-1. For this value of γ, the Auger recombination rate constant can be estimated as γA ) 2.528 ps-1. In summary, we analyzed the experimental data of Matylitsky et al. on the basis of a detailed kinetic model and obtained 2γ/ γA ) 4 for the ratio of the electron-transfer rate constant to the Auger recombination rate constant. Matylitsky et al. assumed a simplified model, in which excitons decay exclusively by Auger recombination as long as the number n of excitons is larger than a certain value nmax but decay exclusively by electron transfer for n e nmax. By analyzing experimental data on the basis of that model, they obtained nmax ) 4 or 5 as the demarcation value. In their model, excitons decay exclusively by Auger recombination when n is larger than 4 or 5 and exclusively by electron transfer when n is smaller than 4 or 5. Let us consider the relation between their result and ours. In our model, the electron-transfer rate constant knET and the Auger recombination rate constant knA for a quantum dot with n excitons are given by n kET ) nγ
(15)
1 kAn ) n(n - 1)γA 2
(16)
By analyzing the experimental data on the basis of this model we obtained 2γ/γA ) 4. By combining this result with eqs 15 n for n > 5, equal and 16, we can show that kAn is larger than kET n n to kET for n ) 5, and smaller than kET for n < 5. Therefore, their demarcation value (nmax ) 4 or 5) approximately coincides with ours (nmax ) 5). However, this does not justify their assumption that if n is larger than nmax, excitons decay exclusively by Auger recombination. This can be easily seen if one considers the case of n ) 6. In this case, kAn is larger than n , but the former is not exceedingly larger than the latter; kET therefore, it is not exact that excitons decay exclusively by Auger recombination. The assumption that excitons decay exclusively by Auger recombination for n > nmax cannot be considered as appropriate for n which is larger than nmax but not exceedingly larger than nmax. Similarly, the assumption that excitons decay exclusively by electron transfer for n e nmax is inappropriate for n which is smaller than nmax but not exceedingly smaller than nmax. Matylitsky assumed that the unimolecular exciton population decay is exclusively due to electron transfer, and we followed
Figure 3. Dependence of the transient absorption signal for the MV•+ radical probed at 403 nm (i.e., nearly at the maximum of the absorption band of the MV•+ radical) on the average initial number of excitons per quantum dot nj0. The solid squares are the experimental data of ref 4 with the ordinate expressed in an arbitrary unit. The solid line corresponds to the theoretical predictions based on eqs 8 and 10. The ordinate value of the theoretical curve was scaled up with a factor of 2.344 to fit to the observed curve.
Figure 4. Comparison of the transient absorption kinetics probed at 411 nm (i.e., in the absorption band of the MV•+ radical) after photoexcitation of CdSe quantum dots with the MV2+ ions adsorbed on the surface of the quantum dots. Symbols are the experimental data of ref 4 with the ordinate expressed in an arbitrary unit. The solid line corresponds to the theoretical predictions based on eqs 7 and 9. The ordinate value of the theoretical curve was scaled up with a factor of 4.717 to fit to the observed curve.
this assumption. However, this may not be correct. If the unimolecular decay process consists of electron transfer with the rate constant γET and other processes with the rate constant γO, γ in our equations is replaced by γET + γO, and the timedependent and ultimate radical formation yields are replaced by η(t) and η(∞) multiplied by γET/γ. There are some implicit assumptions involved in the present treatment. First, the electron-transfer rate is assumed to be proportional to the number of excitons in a quantum dot. This assumption is correct if all excitons in a quantum dot have the same property. However, the quantized electronic structure of a quantum dot puts constraints on the available energy levels for excitons, and excitons occupying different energy levels are
18454
J. Phys. Chem. C, Vol. 113, No. 43, 2009
Letters
likely to have different radiative decay rates7 and different electron-transfer rates.
Acknowledgment. We thank Prof. Wachtveitl and Dr. Matylitsky for useful discussions.
When electron transfer occurs from an exciton to a MV2+ molecule, a hole is left in the quantum dot. This hole may affect electron transfer and Auger recombination8 of the remaining excitons. This effect is not taken into account in the present treatment. We have assumed the absence7 of carrier multiplication and the Poisson distribution4-6,9 of excitons generated in a quantum dot. In the presence of carrier multiplication, the distribution of excitons generated in a quantum dot is not described by a Poisson distribution.
References and Notes
The implicit assumptions involved in the present treatment should be experimentally and theoretically examined in more detail in the future.
(1) See, for example: Coe, S.; Woo, W.-K.; Bawendi, M.; Bulovic, V. Nature 2002, 420, 800–803. (2) See, for example: Robel, I.; Subramanian, V.; Kuno, M.; Kamat, P. V. J. Am. Chem. Soc. 2006, 128, 2385–2393. (3) Klimov, V. I. Annu. ReV. Phys. Chem. 2007, 58, 635–673. (4) Matylitsky, V. V.; Dworak, L.; Breus, V. V.; Basche´, T.; Wachtveitl, J. J. Am. Chem. Soc. 2009, 131, 2424–2425. (5) Barzykin, A. V.; Tachiya, M. Phys. ReV. B 2005, 72, 075425/1– 075425/5. (6) Barzykin, A. V.; Tachiya, M. J. Phys.: Condens. Matter 2007, 19, 065105/1–065105/9. (7) Nair, G.; Bawendi, M. G. Phys. ReV. B 2007, 76, 081304/1–081304/4. (8) Jha, P. P.; Guyot-Sionnest, P. ACS Nano 2009, 3, 1011–1015. (9) Schaller, R. D.; Petruska, M. A.; Klimov, V. I. Appl. Phys. Lett. 2005, 87, 253102/1–253102/3.
JP907969D
