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On the role of spectral diffusion of excitons in sensitized photoconduction in conjugated polymers
Chemical Physics Letters 383 (2004) 166–170 www.elsevier.com/locate/cplett
On the role of spectral diffusion of excitons in sensitized photoconduction in conjugated polymers Vladimir I. Arkhipov
a,*
, Evgenia V. Emelianova
b,*
, Heinz B€ assler
c
a
b
IMEC, Kapeldreef 75, B-3001 Heverlee-Leuven, Belgium Institute of Physical, Nuclear and Macromolecular Chemistry and Material Science Center, Philipps University of Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany c Semiconductor Physics Laboratory, University of Leuven, Celestijnenlaan 200D, B-3001 Heverlee-Leuven, Belgium Received 3 February 2003; in final form 4 November 2003 Published online: 2 December 2003
Abstract An analytical model describing the kinetics of relaxed exciton dissociation into charge carriers is developed. It is based on the assumption that the dissociation of relaxed excitons occurs only on charge transfer centers and the dissociation rate is controlled by the probability for an exciton to encounter a charge transfer center in the course of spectral diffusion. Since the exciton jump rate decreases faster than the exciton density the characteristic time of the carrier photogeneration turns out to be shorter than the exciton lifetime. The developed model is used in order to fit experimental data on the kinetics of charge photogeneration. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Two possible mechanisms of charge carrier photogeneration in conjugated polymers have been suggested and extensively discussed over last decade. The first one suggests either direct photoexcitation of free charge carriers or carrier photoproduction via ultrafast dissociation of excitons whose binding energy is of the order of or less than kT [1,2]. This approach is supported by the fact that the onset of intrinsic photoconductivity, although weak, coincides with the optical absorption edge [3]. Measurements of the transient photoinduced optical absorption in poly(p-phenylenevinylene) [4] also indicated the occurrence of charged segments on the time scale of 100 fs after the onset of photoexcitation. On the basis of this observation the authors of [4] arrived at the conclusion that both charged polarons and neutral excitons are independently generated on the femtosecond time scale. It is worth noting that high excitation levels used in this study did not preclude a possibility of charge carrier production via on-chain *
Corresponding authors. Fax: +32-16-327987. E-mail address: [email protected] (V.I. Arkhipov).
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.11.008
singlet–singlet annihilation and/or two quantum absorption [5]. In addition, transient optical absorption does not allow distinguishing between free and geminately bound polarons. An alternative mechanism is based on the concept of strongly bound Frenkel-type molecular excitons, delocalized within conjugated segments, as primary optical excitations in conjugated polymers [6]. There is abundant evidence in favor of this hypothesis [7–9]. However, the question remains why and how a strongly bound optical excitation in a conjugated polymer can dissociate into free charge carriers. The model of hot exciton dissociation [7,10,11] suggests that the excess photon energy above the S1 S0 0–0 transition could have strongly facilitated on-chain dissociation of hot excitons before this energy is dissipated into the ambient phonon bath. Since the vibronic energy dissipation time is likely to be shorter than 1 ps the model of hot exciton dissociation predicts the time scale of free carrier photogeneration similar to that observed in [4]. However, intrinsic photocurrents, although small, can be observed in conjugated polymers even upon photoexcitation with wavelength close to the absorption edge, i.e., at zero excess photon energy [3,12]. This indicates that
V.I. Arkhipov et al. / Chemical Physics Letters 383 (2004) 166–170
vibrationally relaxed excitons can also dissociate into free charge carriers although the dissociation yield is considerably smaller than that of hot excitons. Another evidence in favor of relaxed exciton dissociation came from the study of the time-resolved photoinduced optical absorption in a ladder-type methyl substituted poly(para-phenylene) at the photoexcitation intensities below the singlet–singlet annihilation threshold [13]. In these experiments, gradually increasing number of charged segments was observed throughout the entire lifetime of singlet excitons. Although this experiment gives practically no information about the yield of free carriers photogeneration it clearly shows that relaxed excitons can dissociate into either free or geminately bound charges. The recent study of photocurrents (PCs) induced by two femtosecond laser pulses in a ladder-type conjugated polymer [14] shone light on this problem. In these experiments, the first pulse with a photon energy of 3.1 eV excited singlet excitons. A time-delayed second pulse of a photon energy somewhat below the S1 S0 0–0 absorption edge lead to a decrease of the photocurrent by exciton depletion due to stimulated emission. The effect of the second pulse on the photocurrent gradually decreased with increasing delay time indicating the occurrence of both exciton dissociation and free carrier photogeneration during the entire lifetime of primary excitons. In parallel, the effect of exciton depletion on the prompt fluorescence (PF) intensity was studied. From the dependence of the PF quenching upon the delay time, the exciton lifetime s was estimated. An interesting result was that the characteristic time of free carrier photogeneration appeared to be 2.5 times shorter that the exciton lifetime. At first glance, this difference indicates that different groups of excitons contribute to PF and photoconductivity as it was suggested in [14]. However, it can be readily explained if one bears in mind that vibronically relaxed excitons are subjected to spectral diffusion in disordered organic materials [15,16]. The existence of spectral diffusion is primarily revealed as time-dependent redshift of the PF spectra [16–18] observed in a broad variety of conjugated polymers and oligomers. In the present Letter, we show this effect is also important as far as charge carrier photogeneration is concerned. As it was mentioned above, dissociation of relaxed excitons into free carriers is only possible at chargetransfer centers. One may expect that, in an undoped material, the concentration of such centers is rather low and, therefore, the probability of exciton dissociation is controlled by the probability to find a chargetransfer center near one of segments visited in the course of energy relaxation. Under these conditions, the number of energy-transfer events, i.e., exciton jumps between different segments, rather than the
167
exciton lifetime should control its dissociation probability. In other words, the dissociation potential of an exciton is determined by the number of segments that this exciton will visit during its lifetime. Since energetically downward jumps prevail during the exciton energy relaxation, every next F€ orster-type jump must, on average, be done towards a more distant segment and, concomitantly, the exciton jump rate decreases with time. Therefore, the average number of site to be visited after a time t and, concomitantly, the dissociation potential decrease faster than the density of excitons survived until this time. This notion qualitatively explains the observed difference in dependences of the PF quenching and PC reduction upon the delay time between the inducing and depleting laser pulses [14]. Below, we formulate a quantitative model of this effect.
2. Spectral diffusion of vibrationally relaxed excitons The F€ orster rate of energy transfer, mðrÞ, between molecules or conjugated segment separated by the distance r can be written as 1 rF 6 mðrÞ ¼ ; ð1Þ s r orster radius. During energy relaxawhere rF is the F€ tion, an exciton of an energy E can make its next jump to a segment on which its energy will be smaller than E. The density of such segments, N ðEÞ, is determined by the exciton density-of-states (EDOS) distribution gðEÞ as Z E N ðEÞ ¼ dE0 gðE0 Þ: ð2Þ 1
Since the F€ orster jump rate strongly decreases with increasing distance between segments most excitons will jump to nearest available neighbors. For a singlet of an energy E, the probability density of finding such a neighbor over the distance r, w(E,r) is given by the Poisson distribution 4p 3 2 wðE; rÞ ¼ 4pr N ðEÞ exp r N ðEÞ : ð3Þ 3 If an exciton has the nearest accessible segment over the distance r the probability pðr; tÞ that it did not jump to this segment until the time t is also determined by the Poisson formula t rF 6 pðr; tÞ ¼ exp½mðrÞt ¼ exp : ð4Þ s r Combining Eqs. (3) and (4) one obtains the probability density W ðE; t; rÞ that an excitation occupies a segment of the energy E at the time t having a nearest neighbor at distance r
168
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4p 3 t rF 6 W ðE; t; rÞ ¼ 4pr N ðEÞ exp r N ðEÞ : 3 s r 2
ð5Þ Multiplying Eq. (5) by the EDOS function yields the energy distribution of such excitations 4p t rF 6 ; f ðE; t; rÞ ¼ AðtÞr2 gðEÞN ðEÞ exp r3 N ðEÞ 3 s r ð6Þ where AðtÞ is the normalization constant. It is convenient to normalize the density of excitons such that their initial density be equal to one. Since energy relaxation does not change the intrinsic exciton lifetime s the normalization constant can be written as Z 1 t Z 1 dr r2 dE gðEÞN ðEÞ AðtÞ ¼ exp s 0 1 1 4p 3 t rF 6 exp r N ðEÞ : ð7Þ 3 s r By the use of Eq. (6) one can calculate the total jump rate mt as a function of time. Multiplying the exciton distribution function by the F€ orster jump rate and integrating over energy and distance yields Z 1 t Z 1 rF6 2 mt ðtÞ ¼ exp dr r dE gðEÞN ðEÞ s s 0 1 1 Z 1 4p t rF 6 dr exp r3 N ðEÞ 3 s r r4 0 Z 1 4p t rR 6 dE gðEÞN ðEÞ exp r3 N ðEÞ : 3 s r 1 ð8Þ The integrals over energy in Eq. (8) can be calculated analytically. The result reads Z 1 4p dE gðEÞN ðEÞ exp r3 N ðEÞ 3 1 9 1 4pNt 3 4pNt 3 ¼ r r 1 exp 16p2 r6 3 3 4pNt 3 r exp ; ð9Þ 3 where Nt is the density of intrinsic sites. Substituting Eq. (9) into Eq. (8) reduces the latter to dr 4pNt 3 r 1 exp r4 3 0 1 4pNt 3 4pNt 3 t rF 6 r exp r exp s r 3 3 Z 1 dr 4pNt 3 4pNt 3 r r 1 exp r10 3 3 0 4pNt 3 t rF 6 r exp exp : ð10Þ s r 3
t r6 mt ðtÞ ¼ F exp s s
Z
1
Eq. (10) proves that the exciton jump rate is not sensitive to the EDOS distribution over energy. It rather depends upon the total density of states that is typical for both exciton and charge carrier energy relaxations tacitly assuming that the F€ orster radius is independent of the exciton energy within the EDOS distribution.
3. Dissociation of excitons into charge carriers Integrating Eq. (10) over time from t to infinity yields the average number of jumps between different conjugated segments, nðtÞ, to be made by an exciton after the time t: 0 Z r6 1 0 t dt exp nðtÞ ¼ F s t s Z 1 dr 4pNt 3 r 1 exp r4 3 0 0 1 4pNt 3 4pNt 3 t rF 6 r exp r exp 3 3 s r Z 1 dr 4pNt 3 r 1 exp r10 3 0 0 4pNt 3 4pNt 3 t rF 6 r exp r exp : 3 3 s r ð11Þ In fact, Eq. (11) determines the probability that an exciton, that has survived to the time t, will make a jump that can result in dissociation after t, i.e., the exciton dissociation potential. This equation can be written in terms of dimensionless variables as Z 1 nðtÞ¼b dzexpðzÞ t=s
Z
1 dx bz ½1 expðxÞxexpðxÞexp 2 x2 x 0 Z 1 dx bz ½1 expðxÞxexpðxÞexp 2 ; x4 x 0 1
ð12Þ with only one dimensionless parameter b which combines the total density of intrinsic states and the F€ orster radius as 2 4p Nt rF3 : b¼ 3 Time dependence of the function nðtÞ is illustrated in Fig. 1 for different values of the F€ orster radius in comparison with the exciton density decay. It is assumed that dissociation occurs once an exciton jumps onto a segment that has an electron scavenger as its neighbor. For excitations, which survived until the time t, the probability to meet charge transfer centers and, thereby,
V.I. Arkhipov et al. / Chemical Physics Letters 383 (2004) 166–170
rate decreases with time in the course of energy relaxation when excitations are getting stronger localized on segments that belong to a deep tail of the EDOS distribution. In two-pulse experiments [14], the number of singlets, quenched by the second depleting pulse, is proportional to the density of excitations generated by the first inducing pulse and survived until the delay time td . Therefore, the quenching of prompt fluorescence is proportional to the density of excitations at the time td while the photocurrent quenching gives a direct measure for the exciton dissociation potential, i.e., for the number of intersegmental jumps that excitons can make after this time. Exactly this time dependence is described by the function nðtd Þ as illustrated in Fig. 2. The experimental points shown in this figure were taken from [14] and the solid line illustrates the fit of these data by Eq. (11) with the exciton lifetime s ¼ 180 ps determined in [14] from the delay-time dependence of the photoluminescence quenching.
3 n(0)
1
2 1
n(t)/n(0)
0
1
10 β 100 1000
0.1
β -3
10 1 10 100 3 10
0.01 0.0
0.5
1.0
1.5
2.0
t/τ Fig. 1. Time dependence of the average number of jumps of an exciton with intrinsic lifetime s to be made after the time t. The dotted line shows an exponential decay of the exciton density, i.e., the prompt PL intensity in the absence of spectral relaxation.
4. Conclusions
1.0
Intrinsic dissociation of vibrationally relaxed optical excitations into charge carriers can occur at charge transfer centers visited by these excitations in the course of energy relaxation within an inhomogeneously broadened EDOS distribution. Under these circumstances, the dissociation rate must be proportional to the total F€ orster exciton transfer rate. The latter decreases with time due to (i) the decaying total density of photogenerated excitons and (ii) the decreasing average F€ orster energy transfer rate for survived excitons that occupy deeper states with more distant neighbors available for further jumps. As a result, the characteristic time of the intrinsic charge photogeneration from vibrationally relaxed excitons is smaller than the decay time of the prompt photoluminescence. Experimental observation of this difference in characteristic times does not therefore necessarily indicate that somehow different groups of excitons contribute to those processes.
0.5
Acknowledgements
0.0
This work was supported by the Deutsche Forschungsgemeinschaft.
to dissociate into pairs of free carriers is proportional to the average number of segments that these excitations will visit during the rest of their lifetimes. This number decreases with time for two reasons: (i) the density of excitations decreases with time due to both intrinsic decay and quenching by defects and impurities other than charge transfer centers, and (ii) the exciton jump
2.0
Photocurrent reduction, a.u.
169
β = 50 τ = 180 ps
1.5
0
50
100
150
200
250
300
Delay time, ps Fig. 2. The decrease of the photocurrent due to the second depleting laser pulse in the two-pulse experiment of [14]. The solid line is calculated from Eq. (12) with the exciton lifetime s ¼ 180 ps, as determined in [14] from the PF decay, and b ¼ 50 that corresponds to e.g. Nt ¼ 1021 cm3 and rF ¼ 1:2 nm. The dotted line shows for comparison an exponential decay of the exciton density with the intrinsic lifetime of 180 ps.
References [1] A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su, Rev. Mod. Phys. 60 (1988) 781. [2] A.J. Heeger, in: N.S. Sariciftci (Ed.), Primary Photoexcitations in Conjugated Polymers: Molecular Excitons versus Semiconductor Band Model, World Scientific, Singapore, 1997.
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[3] S. Barth, H. B€ assler, H. Rost, H.H. H€ orhold, Phys. Rev. B 56 (1997) 3844. [4] D. Moses, A. Dogariu, A.J. Heeger, Phys. Rev. B 61 (2000) 9373. [5] C. Silva, M.A. Stevens, D.M. Russell, S. Setayesh, K. M€ ullen, R.H. Friend, Synth. Met. 116 (2001) 9. [6] H. B€ assler, in: N.S. Sariciftci (Ed.), Primary Photoexcitations in Conjugated Polymers: Molecular Excitons versus Semiconductor Band Model, World Scientific, Singapore, 1997. [7] V.I. Arkhipov, E.V. Emelianova, H. B€assler, Phys. Rev. Lett. 82 (1999) 1321. [8] J.-W. van der Horst, P.A. Bobbert, M.A.J. Michels, H. B€assler, J. Chem. Phys. 114 (2001) 6950. [9] S.F. Alvarado, S. Barth, H. B€assler, U. Scherf, J.-W. van der Horst, P.A. Bobbert, M.A.J. Michels, Adv. Funct. Mat. 12 (2002) 117. [10] V.I. Arkhipov, E.V. Emelianova, S. Barth, H. B€assler, Phys. Rev. B 61 (2000) 8207.
[11] D.M. Basko, E.M. Conwell, Phys. Rev. B 66 (2002) 155210. [12] S. Barth, H. B€assler, U. Scherf, K. M€ ullen, Chem. Phys. Lett. 288 (1998) 147. [13] V. Gulbinas, Y. Zaushitsyn, V. Sundstrom, D. Hertel, H. B€assler, A. Yartsev, Phys. Rev. Lett. 89 (2002) 107401. [14] J.G. M€ uller, U. Lemmer, J. Feldmann, U. Scherf, Phys. Rev. Lett. 88 (2002) 147401. [15] B. Movaghar, M. Gr€ unewald, B. Ries, H. B€assler, D. W€ urtz, Phys. Rev. B 33 (1986) 5545. [16] R. Kersting, U. Lemmer, R.F. Mahrt, K. Leo, H. Kurz, H. B€assler, E.O. G€ obel, Phys. Rev. Lett. 70 (1993) 3820. [17] S.C.J. Meskers, J. Hubner, M. Oestreich, H. B€assler, J. Phys. Chem. B 105 (2001) 9139. [18] C. Im, J.M. Lupton, P. Schouwink, S. Heun, H. Becker, H. B€assler, J. Chem. Phys. 117 (2002) 1395.
On the role of spectral diffusion of excitons in sensitized photoconduction in conjugated polymers Vladimir I. Arkhipov
a,*
, Evgenia V. Emelianova
b,*
, Heinz B€ assler
c
a
b
IMEC, Kapeldreef 75, B-3001 Heverlee-Leuven, Belgium Institute of Physical, Nuclear and Macromolecular Chemistry and Material Science Center, Philipps University of Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany c Semiconductor Physics Laboratory, University of Leuven, Celestijnenlaan 200D, B-3001 Heverlee-Leuven, Belgium Received 3 February 2003; in final form 4 November 2003 Published online: 2 December 2003
Abstract An analytical model describing the kinetics of relaxed exciton dissociation into charge carriers is developed. It is based on the assumption that the dissociation of relaxed excitons occurs only on charge transfer centers and the dissociation rate is controlled by the probability for an exciton to encounter a charge transfer center in the course of spectral diffusion. Since the exciton jump rate decreases faster than the exciton density the characteristic time of the carrier photogeneration turns out to be shorter than the exciton lifetime. The developed model is used in order to fit experimental data on the kinetics of charge photogeneration. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Two possible mechanisms of charge carrier photogeneration in conjugated polymers have been suggested and extensively discussed over last decade. The first one suggests either direct photoexcitation of free charge carriers or carrier photoproduction via ultrafast dissociation of excitons whose binding energy is of the order of or less than kT [1,2]. This approach is supported by the fact that the onset of intrinsic photoconductivity, although weak, coincides with the optical absorption edge [3]. Measurements of the transient photoinduced optical absorption in poly(p-phenylenevinylene) [4] also indicated the occurrence of charged segments on the time scale of 100 fs after the onset of photoexcitation. On the basis of this observation the authors of [4] arrived at the conclusion that both charged polarons and neutral excitons are independently generated on the femtosecond time scale. It is worth noting that high excitation levels used in this study did not preclude a possibility of charge carrier production via on-chain *
Corresponding authors. Fax: +32-16-327987. E-mail address: [email protected] (V.I. Arkhipov).
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.11.008
singlet–singlet annihilation and/or two quantum absorption [5]. In addition, transient optical absorption does not allow distinguishing between free and geminately bound polarons. An alternative mechanism is based on the concept of strongly bound Frenkel-type molecular excitons, delocalized within conjugated segments, as primary optical excitations in conjugated polymers [6]. There is abundant evidence in favor of this hypothesis [7–9]. However, the question remains why and how a strongly bound optical excitation in a conjugated polymer can dissociate into free charge carriers. The model of hot exciton dissociation [7,10,11] suggests that the excess photon energy above the S1 S0 0–0 transition could have strongly facilitated on-chain dissociation of hot excitons before this energy is dissipated into the ambient phonon bath. Since the vibronic energy dissipation time is likely to be shorter than 1 ps the model of hot exciton dissociation predicts the time scale of free carrier photogeneration similar to that observed in [4]. However, intrinsic photocurrents, although small, can be observed in conjugated polymers even upon photoexcitation with wavelength close to the absorption edge, i.e., at zero excess photon energy [3,12]. This indicates that
V.I. Arkhipov et al. / Chemical Physics Letters 383 (2004) 166–170
vibrationally relaxed excitons can also dissociate into free charge carriers although the dissociation yield is considerably smaller than that of hot excitons. Another evidence in favor of relaxed exciton dissociation came from the study of the time-resolved photoinduced optical absorption in a ladder-type methyl substituted poly(para-phenylene) at the photoexcitation intensities below the singlet–singlet annihilation threshold [13]. In these experiments, gradually increasing number of charged segments was observed throughout the entire lifetime of singlet excitons. Although this experiment gives practically no information about the yield of free carriers photogeneration it clearly shows that relaxed excitons can dissociate into either free or geminately bound charges. The recent study of photocurrents (PCs) induced by two femtosecond laser pulses in a ladder-type conjugated polymer [14] shone light on this problem. In these experiments, the first pulse with a photon energy of 3.1 eV excited singlet excitons. A time-delayed second pulse of a photon energy somewhat below the S1 S0 0–0 absorption edge lead to a decrease of the photocurrent by exciton depletion due to stimulated emission. The effect of the second pulse on the photocurrent gradually decreased with increasing delay time indicating the occurrence of both exciton dissociation and free carrier photogeneration during the entire lifetime of primary excitons. In parallel, the effect of exciton depletion on the prompt fluorescence (PF) intensity was studied. From the dependence of the PF quenching upon the delay time, the exciton lifetime s was estimated. An interesting result was that the characteristic time of free carrier photogeneration appeared to be 2.5 times shorter that the exciton lifetime. At first glance, this difference indicates that different groups of excitons contribute to PF and photoconductivity as it was suggested in [14]. However, it can be readily explained if one bears in mind that vibronically relaxed excitons are subjected to spectral diffusion in disordered organic materials [15,16]. The existence of spectral diffusion is primarily revealed as time-dependent redshift of the PF spectra [16–18] observed in a broad variety of conjugated polymers and oligomers. In the present Letter, we show this effect is also important as far as charge carrier photogeneration is concerned. As it was mentioned above, dissociation of relaxed excitons into free carriers is only possible at chargetransfer centers. One may expect that, in an undoped material, the concentration of such centers is rather low and, therefore, the probability of exciton dissociation is controlled by the probability to find a chargetransfer center near one of segments visited in the course of energy relaxation. Under these conditions, the number of energy-transfer events, i.e., exciton jumps between different segments, rather than the
167
exciton lifetime should control its dissociation probability. In other words, the dissociation potential of an exciton is determined by the number of segments that this exciton will visit during its lifetime. Since energetically downward jumps prevail during the exciton energy relaxation, every next F€ orster-type jump must, on average, be done towards a more distant segment and, concomitantly, the exciton jump rate decreases with time. Therefore, the average number of site to be visited after a time t and, concomitantly, the dissociation potential decrease faster than the density of excitons survived until this time. This notion qualitatively explains the observed difference in dependences of the PF quenching and PC reduction upon the delay time between the inducing and depleting laser pulses [14]. Below, we formulate a quantitative model of this effect.
2. Spectral diffusion of vibrationally relaxed excitons The F€ orster rate of energy transfer, mðrÞ, between molecules or conjugated segment separated by the distance r can be written as 1 rF 6 mðrÞ ¼ ; ð1Þ s r orster radius. During energy relaxawhere rF is the F€ tion, an exciton of an energy E can make its next jump to a segment on which its energy will be smaller than E. The density of such segments, N ðEÞ, is determined by the exciton density-of-states (EDOS) distribution gðEÞ as Z E N ðEÞ ¼ dE0 gðE0 Þ: ð2Þ 1
Since the F€ orster jump rate strongly decreases with increasing distance between segments most excitons will jump to nearest available neighbors. For a singlet of an energy E, the probability density of finding such a neighbor over the distance r, w(E,r) is given by the Poisson distribution 4p 3 2 wðE; rÞ ¼ 4pr N ðEÞ exp r N ðEÞ : ð3Þ 3 If an exciton has the nearest accessible segment over the distance r the probability pðr; tÞ that it did not jump to this segment until the time t is also determined by the Poisson formula t rF 6 pðr; tÞ ¼ exp½mðrÞt ¼ exp : ð4Þ s r Combining Eqs. (3) and (4) one obtains the probability density W ðE; t; rÞ that an excitation occupies a segment of the energy E at the time t having a nearest neighbor at distance r
168
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4p 3 t rF 6 W ðE; t; rÞ ¼ 4pr N ðEÞ exp r N ðEÞ : 3 s r 2
ð5Þ Multiplying Eq. (5) by the EDOS function yields the energy distribution of such excitations 4p t rF 6 ; f ðE; t; rÞ ¼ AðtÞr2 gðEÞN ðEÞ exp r3 N ðEÞ 3 s r ð6Þ where AðtÞ is the normalization constant. It is convenient to normalize the density of excitons such that their initial density be equal to one. Since energy relaxation does not change the intrinsic exciton lifetime s the normalization constant can be written as Z 1 t Z 1 dr r2 dE gðEÞN ðEÞ AðtÞ ¼ exp s 0 1 1 4p 3 t rF 6 exp r N ðEÞ : ð7Þ 3 s r By the use of Eq. (6) one can calculate the total jump rate mt as a function of time. Multiplying the exciton distribution function by the F€ orster jump rate and integrating over energy and distance yields Z 1 t Z 1 rF6 2 mt ðtÞ ¼ exp dr r dE gðEÞN ðEÞ s s 0 1 1 Z 1 4p t rF 6 dr exp r3 N ðEÞ 3 s r r4 0 Z 1 4p t rR 6 dE gðEÞN ðEÞ exp r3 N ðEÞ : 3 s r 1 ð8Þ The integrals over energy in Eq. (8) can be calculated analytically. The result reads Z 1 4p dE gðEÞN ðEÞ exp r3 N ðEÞ 3 1 9 1 4pNt 3 4pNt 3 ¼ r r 1 exp 16p2 r6 3 3 4pNt 3 r exp ; ð9Þ 3 where Nt is the density of intrinsic sites. Substituting Eq. (9) into Eq. (8) reduces the latter to dr 4pNt 3 r 1 exp r4 3 0 1 4pNt 3 4pNt 3 t rF 6 r exp r exp s r 3 3 Z 1 dr 4pNt 3 4pNt 3 r r 1 exp r10 3 3 0 4pNt 3 t rF 6 r exp exp : ð10Þ s r 3
t r6 mt ðtÞ ¼ F exp s s
Z
1
Eq. (10) proves that the exciton jump rate is not sensitive to the EDOS distribution over energy. It rather depends upon the total density of states that is typical for both exciton and charge carrier energy relaxations tacitly assuming that the F€ orster radius is independent of the exciton energy within the EDOS distribution.
3. Dissociation of excitons into charge carriers Integrating Eq. (10) over time from t to infinity yields the average number of jumps between different conjugated segments, nðtÞ, to be made by an exciton after the time t: 0 Z r6 1 0 t dt exp nðtÞ ¼ F s t s Z 1 dr 4pNt 3 r 1 exp r4 3 0 0 1 4pNt 3 4pNt 3 t rF 6 r exp r exp 3 3 s r Z 1 dr 4pNt 3 r 1 exp r10 3 0 0 4pNt 3 4pNt 3 t rF 6 r exp r exp : 3 3 s r ð11Þ In fact, Eq. (11) determines the probability that an exciton, that has survived to the time t, will make a jump that can result in dissociation after t, i.e., the exciton dissociation potential. This equation can be written in terms of dimensionless variables as Z 1 nðtÞ¼b dzexpðzÞ t=s
Z
1 dx bz ½1 expðxÞxexpðxÞexp 2 x2 x 0 Z 1 dx bz ½1 expðxÞxexpðxÞexp 2 ; x4 x 0 1
ð12Þ with only one dimensionless parameter b which combines the total density of intrinsic states and the F€ orster radius as 2 4p Nt rF3 : b¼ 3 Time dependence of the function nðtÞ is illustrated in Fig. 1 for different values of the F€ orster radius in comparison with the exciton density decay. It is assumed that dissociation occurs once an exciton jumps onto a segment that has an electron scavenger as its neighbor. For excitations, which survived until the time t, the probability to meet charge transfer centers and, thereby,
V.I. Arkhipov et al. / Chemical Physics Letters 383 (2004) 166–170
rate decreases with time in the course of energy relaxation when excitations are getting stronger localized on segments that belong to a deep tail of the EDOS distribution. In two-pulse experiments [14], the number of singlets, quenched by the second depleting pulse, is proportional to the density of excitations generated by the first inducing pulse and survived until the delay time td . Therefore, the quenching of prompt fluorescence is proportional to the density of excitations at the time td while the photocurrent quenching gives a direct measure for the exciton dissociation potential, i.e., for the number of intersegmental jumps that excitons can make after this time. Exactly this time dependence is described by the function nðtd Þ as illustrated in Fig. 2. The experimental points shown in this figure were taken from [14] and the solid line illustrates the fit of these data by Eq. (11) with the exciton lifetime s ¼ 180 ps determined in [14] from the delay-time dependence of the photoluminescence quenching.
3 n(0)
1
2 1
n(t)/n(0)
0
1
10 β 100 1000
0.1
β -3
10 1 10 100 3 10
0.01 0.0
0.5
1.0
1.5
2.0
t/τ Fig. 1. Time dependence of the average number of jumps of an exciton with intrinsic lifetime s to be made after the time t. The dotted line shows an exponential decay of the exciton density, i.e., the prompt PL intensity in the absence of spectral relaxation.
4. Conclusions
1.0
Intrinsic dissociation of vibrationally relaxed optical excitations into charge carriers can occur at charge transfer centers visited by these excitations in the course of energy relaxation within an inhomogeneously broadened EDOS distribution. Under these circumstances, the dissociation rate must be proportional to the total F€ orster exciton transfer rate. The latter decreases with time due to (i) the decaying total density of photogenerated excitons and (ii) the decreasing average F€ orster energy transfer rate for survived excitons that occupy deeper states with more distant neighbors available for further jumps. As a result, the characteristic time of the intrinsic charge photogeneration from vibrationally relaxed excitons is smaller than the decay time of the prompt photoluminescence. Experimental observation of this difference in characteristic times does not therefore necessarily indicate that somehow different groups of excitons contribute to those processes.
0.5
Acknowledgements
0.0
This work was supported by the Deutsche Forschungsgemeinschaft.
to dissociate into pairs of free carriers is proportional to the average number of segments that these excitations will visit during the rest of their lifetimes. This number decreases with time for two reasons: (i) the density of excitations decreases with time due to both intrinsic decay and quenching by defects and impurities other than charge transfer centers, and (ii) the exciton jump
2.0
Photocurrent reduction, a.u.
169
β = 50 τ = 180 ps
1.5
0
50
100
150
200
250
300
Delay time, ps Fig. 2. The decrease of the photocurrent due to the second depleting laser pulse in the two-pulse experiment of [14]. The solid line is calculated from Eq. (12) with the exciton lifetime s ¼ 180 ps, as determined in [14] from the PF decay, and b ¼ 50 that corresponds to e.g. Nt ¼ 1021 cm3 and rF ¼ 1:2 nm. The dotted line shows for comparison an exponential decay of the exciton density with the intrinsic lifetime of 180 ps.
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