DOI: 10.1002/cphc.201402720
Articles
Diffusion Length and Langevin Recombination of Singlet and Triplet Excitons in Organic Heterojunction Solar Cells David Ompong and Jai Singh*[a] We derived new expressions for the diffusion length of singlet and triplet excitons by using the Fçster and Dexter transfer mechanisms, respectively, and have found that the diffusion lengths of singlet and triplet excitons are comparable. By using the Langevin recombination theory, we derived the rate of recombination of dissociated free charges into their excitonic states. We found that in some organic polymers the probabilities of recombination of free charge carriers back into the sin-
glet and triplet states are approximately 65.6 and 34.4 %, respectively, indicating that Langevin-type recombination into triplet excitons in organic semiconductors is less likely. This implies that the creation of triplet excitons may be advantageous in organic solar cells, because this may lead to dissociated free charge carriers that can be collected at their respective electrodes, which should result in better conversion efficiency.
1. Introduction terface has been proposed[1] in which exciton relaxes first to a charge transfer (CT) exciton state by transferring its electron from the LUMO of the donor to the LUMO of the acceptor, which is at a lower energy and thus the excess energy is released in the form of molecular vibrational energy. The excess vibrational energy, if larger than the exciton binding energy, may impact back to the CT exciton to dissociate it into a pair of free electron and hole. It is therefore critically important that all generated excitons reach the interface to form CT exciton and dissociate into free charge carriers. As such, any recombination of CT excitons prior to their dissociation into free charge carriers will have a negative impact on the performance of an OSC. In low-mobility materials such as p-conjugated polymers (PCPs), the diffusion-controlled bimolecular recombination, or the so-called Langevin recombination, dominates. This type of recombination occurs when opposite charge carriers approach each other within the Coulomb radius, given by rc ¼ e2 =4per eo kT, at a temperature T , where e is the charge of an electron, er is the relative dielectric permittivity, eo is absolute permittivity, and k is Boltzmann’s constant.[9] Previous approaches involving the application of Langevin theory to study the recombination of CT excitons, prior to charge separation in OSCs,[10, 11] have not considered the spin; as such, no distinction is made between the recombination of singlet and triplet excitons. The introduction of spins to distinguish between the process of recombination of singlet and triplet excitons has been considered only recently.[7] In this paper, we have calculated the diffusion lengths of singlet and triplet excitons to study the influence of exciton diffusion lengths on the fraction of excitons that reaches the (D–A) interface in a heterojunction organic solar cell. Furthermore, by using the Langevin recombination theory, we have derived the rates of recombination of dissociated free charges back into their excitonic states. We have calculated the probability of re-
Research interest in organic solar cells (OSC) has increased recently because of the cost effectiveness, easy fabrication techniques, large-scale production,[1] and easy integration of such units in a wide variety of devices[2] such as watches, mobile phones, and laptop cases. When an organic molecule absorbs a photon, a pair of electron and hole is excited on the same molecule to form an exciton.[3] Excitons are usually excited in singlet (S) and triplet (T) spin configurations. An exciton consists of a pair of excited electron and hole bound in a hydrogenic state due to their Coulomb interaction. When the electron and hole in an exciton have opposite spins [›fl or fl›], it is a singlet exciton with multiplicity 1 and when the electron and hole have parallel spins [›› or flfl] then it is a triplet exciton with multiplicity 3.[3, 4] In organic materials, the selection rules for the electronic dipole transitions allow generation of only singlet excitons by exciting an electron from the singlet ground state. Therefore, the probability of generating a triplet exciton by spin inversion is extremely low in most molecules and mostly singlet excitons are created first in organic photovoltaic (OPV) cells when light is absorbed.[4–7] To achieve high conversion efficiency in OSCs, the following key processes must be optimized: 1) photon absorption and exciton generation; 2) efficient diffusion of excitons to the interface before their quenching; 3) dissociation and charge separation at the interface; and 4) carrier collection by the electrodes. Although each of these events has an influence on the overall conversion efficiency of an OSC, the efficiency of exciton dissociation and subsequent charge separation to the respective electrodes plays the most important role.[8] A mechanism for exciton dissociation at the donor–acceptor (D–A) in[a] D. Ompong, Prof. Dr. J. Singh School of Engineering and IT, Faculty of EHSE Charles Darwin University, Darwin, NT 0909 (Australia) An invited contribution to a Special Issue on Organic Electronics
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Articles combination into singlet and triplet excitonic states in some organic polymers. It is expected that the present work will advance the understanding of the roles of singlet and triplet excitons in OSCs. The results also show that the generation of not only singlet excitons but also triplet excitons can be useful in the operation of OSCs.
transfer rate decays exponentially with distance.[6] The Fçrster and Dexter transfer rates are given, respectively, as Equation (3):[2]
2. Exciton Generation in OSCs
and Equation (4):
kf ¼
When a photon is absorbed, an electron in the organic semiconductor is excited from the HOMO to the LUMO of a molecule. However, because of the low dielectric constant (er 3–4) in organic solids, a strong Coulomb attraction exists between a photo-excited electron–hole (e–h) pair,[12, 13] which binds them in a Frenkel exciton.[3] The excitonic Bohr radius is the radius of the orbital formed between the excited e–h pair in an exciton and it is different for singlet and triplet excitons as given by Equation (1):[14] aex ðS ¼ 0Þ ¼ aS ¼
a 2 er m ao ða 1Þ2 mex
ð1Þ
er m a mex o
ð2Þ
kd ¼
D¼
in which a is a material-dependent parameter representing the ratio of the magnitude of the Coulomb and exchange interactions between the electron and hole in an exciton, S is the spin of the exciton (S = 0 for singlet and S = 1 for triplet), m is the reduced mass of the electron in a hydrogen atom, ao is the Bohr radius, and mex is the reduced mass of an exciton, given by mex = 0.5me, with me being the free electron mass.[15] Singlet excitons have relatively small binding energy and larger excitonic Bohr radius compared with the triplet excitons.[2] Due to their favorable spin configuration, singlet excitons can easily recombine radiatively to the singlet ground state without reaching the interface.
www.chemphyschem.org
L2D R2da ¼ t 6tH
LD ðSÞ ¼
ð4Þ
ð5Þ
pffiffiffiffiffiffiffiffi Rf 3 tS kf pffiffiffi 6
ð6Þ
and Equation (7):
LD ðTÞ ¼
rffiffiffiffiffiffiffiffiffi tT k d L Rd lnðtT kd Þ 6 2
ð7Þ
and the corresponding diffusion coefficients are obtained as Equation (8):
An exciton can move throughout the material as a neutral quasi particle. There are two general mechanisms by which excitons can diffuse in organic semiconductors and PCPs: 1) Fçrster resonance energy transfer (FRET) and 2) Dexter carriers transfer. In Fçrster transfer, an excited electron in an exciton on a molecule recombines with the hole and the energy is transferred nonradiatively to excite an exciton on another molecule. Such a recombination is only possible for singlet excitons due to their favorable spin configuration and hence FRET is applicable only to the transfer of singlet excitons. In the Dexter transfer mechanism, the excited electron and hole are transferred to excite a nearby molecule due to the overlap of electronic wavefunctions. Therefore, the Dexter transfer mechanism can occur only at short range and it applies to both singlet and triplet excitons. In Dexter’s approach, the exciton ChemPhysChem 0000, 00, 0 – 0
1 2Rd R exp 1 da tT L Rd
where tH is the exciton hopping time and LD is the exciton diffusion length. By using Equations (3)–(5) the diffusion lengths of singlet (S) and triplet (T) excitons are, respectively, obtained as Equation (6):
3. Exciton Diffusion
&
ð3Þ
where tS is the lifetime of singlet excitons, tT is the lifetime of triplet excitons, Rda is the D–A distance of separation, Rf (Rd ) is the Fçrster (Dexter) radius, and L is the average length of a molecular orbital, which is found to be 0.11 nm[16] in Ir(ppy)3. We have assumed the same value here for all triplet exciton calculations. As stated above, the rate in Equation (4) applies also to singlet excitons with the singlet lifetime; however, in this work, we have applied it only to triplet excitons. The diffusion coefficient (D) of an exciton of lifetime t is given by Equation (5):[2]
and Equation (2): aex ðS ¼ 1Þ ¼ aT ¼
1 Rf 6 tS Rda
R2 DðSÞ ¼ f 6
sffiffiffiffiffi 3 kf2 tS
ð8Þ
and Equation (9): DðTÞ ¼
2 kd L Rd lnðtT kd Þ 2 6
ð9Þ
Notably, in both Equations (6) and (7), the diffusion length is dependent on the product of the lifetime and the respective transfer rate.
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Articles 4. CT Exciton Recombination Dynamics
where N and Pare electron and hole concentrations, respectively. The recombination coefficients are evaluated by substituting the relative dielectric permittivity from Equations (1) and (2) into Equation (10), thereby giving the singlet (S) and triplet (T) recombination coefficients, respectively, as Equation (12):
Three main processes need to be considered from the time an exciton at the D–A interface generates a CT exciton to the time when charges are collected at the electrodes: geminate recombination, bimolecular recombination, and charge transport.[17] The geminate recombination phase starts when the electron of an exciton is transferred from a donor to an acceptor molecule at the interface to form a CT exciton. A singlet CT exciton can decay to the ground state with a rate constant kq or separate into free charge carriers with rate constant kd ; the latter can occur for both singlet and triplet excitons. Once separated, some fraction of the charge carriers may return to form CT excitons with a rate constant kr as shown in Figure 1. Therefore,
bS ¼ e
ðmn þ mp Þmao a2 a0 S mex eo ða 1Þ2
ð12Þ
and Equation (13): bT ¼ e
ðmn þ mp Þmao a0 T mex eo
ð13Þ
By using the coefficient of Equations (12) and (13) in Equation (11), we can obtain the rates of recombination of singlet (kS ) and triplet (kT ) excitons, which give the corresponding probability of recombination as Equation (14): kS a0 T a2 1 ¼ 0 2 ¼ kS þ kT a T a þ a0 S ða 1Þ2 1 þ a00 S a1 2 aT a
ð14Þ
and Equation (15): kT a0 S ða 1Þ2 1 ¼ 0 ¼ kS þ kT a S ða 1Þ2 þ a0 T a2 1 þ a00 T a 2 a S a1
Figure 1. Charge carrier transfer processes in organic solar cells. CT exciton may either decay to the ground state with a rate constant kq or dissociate with a rate kd . kr is the recombination rate of the charge carriers to form the original CT exciton after dissociation.
In deriving Equations (14) and (15), we have assumed the same electron and hole mobilities, and N and P for both singlet and triplet cases. a0 S ða0 T Þrepresents singlet (triplet) Bohr radius of CT excitons.
long-lived CT excitons can act as a precursor for free charge carriers.[11, 18] Among the fraction of reformed CT excitons, some may recombine and others may separate again. This can be repeated many times. When some fraction have already recombined and some separated to sufficiently long distances, the geminate recombination phase ends and the bulk recombination phase starts.[18] Recombination between electrons and holes is an energy loss process for the efficiency of OSCs, whether it occurs through geminate recombination or bulk recombination. Therefore, it is very important to suppress these recombinations in OSCs.[18] According to the Langevin theory, in an organic system, charge recombination occurs when an electron and a hole come close to each other within rc . In the Langevin bimolecular process, the recombination coefficient is given as Equation (10):[9] b¼e
ðmn þ mp Þ er eo
5. Results and Discussions A study of the diffusion length, diffusion coefficient, fraction of excitons that reach the D–A interface, and recombination of charge carriers into CT excitons have been carried out for both singlet and triplet excitons. We have found that triplet excitons have a lifetime about three orders of magnitude greater than singlet excitons; however, the transfer rate of singlet excitons is about three orders of magnitude larger than that of triplet excitons, which results in a higher diffusion coefficient for singlet excitons. The diffusion length as clearly shown in Equations (6) and (7) are dependent on the product of the lifetime and the transfer rate for singlet and triplet excitons, this product is the dominant term in both expressions. For calculating the diffusion lengths and diffusion coefficients for singlet and triplet excitons using Equations (6)–(9), we require some material-dependent input parameters. For this, we have selected a few materials used as donors and listed their available input parameters in Table 1. By using Equations (6)–(9) and the input parameters given in Table 1, the diffusion lengths and diffusion coefficients have been calculated for some of the donor materials; these are listed in Table 2. According to Table 2, the calculated diffusion
ð10Þ
where mn ðmp Þis the mobility of electrons (holes). The recombination rate is given by Equation (11):[7] kr ¼ bNP ChemPhysChem 0000, 00, 0 – 0
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ð15Þ
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Articles reach the D–A interface from Equation (16) as a function of d in Figure 2 for some organic semiconductors listed in Table 1. According to Figure 2, the comparable diffusion lengths allow a fairly good fraction between 0.99 to 0.64 of both singlet and triplet excitons to reach the D–A interface for d values of between 10 and 15 nm.
Table 1. Input parameters for calculating the diffusion length (LD calc.) from Equations (6) and (7) and diffusion coefficient (D) from Equations (8) and (9) for excitons in several donor materials. Organic material
Exciton multiplicity
ti (i = S or T) [ns]
Rj (j = S or T) [nm]
Ref.
NPD CBP SubPc PTCDA DIP (upright) DIP (flat) PtOEP PtOEP Ir (ppy)3 Meo-POV PDHFV PTEH PDHFHPPV
S S S S S S T-monomer T-dimer T S S S S
3.5 0.7 1 3.2 1.8 1.8 800 2800 1330 0.055 0.05 0.23 0.71
1.9 2.2 1.5 1.4 2.0 1.7 0.7 0.7 1.1 1.9 2.2 2.4 4.3
[19] [19] [19] [19] [19] [19] [19] [19] [16] [20] [20] [20] [20]
Table 2. Calculated diffusion length (LD calc.) from Equations (6) and (7), diffusion coefficient (D) from Equations (8) and (9), and measured diffusion length (LD meas.) of excitons in some organic semiconductors. Organic material
Exciton multiplicity
D calc.[a] 104 [cm2 s1]
LD calc.[a] [nm]
LD meas. [nm]
Ref.
NPD CBP SubPc PTCDA DIP (upright) Dip (flat) PtOEP PtOEP Meo-POV PDHFV PTEH Ir(ppy)3 PDHFHPPV
S S S S S S T-monomer T-dimer S S S2 T S
3.6 43.2 3.0 0.6 9.5 2.6 0.000021 0.000006 228.1 604.7 21.6 0.069 2374.2
11.2 17.4 5.5 4.5 13.1 8.0 1.3 1.3 11.2 17.4 22.6 30.3 129.8
5.1 16.8 8.0 10.4 16.5 21.8 18.0 13.1
[19] [19] [19] [19] [19] [19] [19] [19]
Figure 2. Plot of fraction of the singlet (S) and triplet (T) excitons reaching the D–A interface as a function of the distance (d) calculated from Equation (10) for some selected organic semiconductors given in Table 1.
A typical exciton diffusion length in bulk heterojunction (BHJ) solar cells is in the 10–15 nm range;[11] however, in BHJ for which the donor and acceptor materials are mixed to form an interpenetrating nanoscale network of D–A blend, we will expect the interface to be at a distance less than the diffusion length. Thus, at a distance of d = 5 nm, an appreciable fraction (> 0.93) of excitons can make it to the interface in most organic molecules, as shown in Figure 2. However, in PTCDA, just 0.71 of excitons are able to reach the interface at the same 5 nm distance. The formation of CT excitons may mediate the exciton dissociation and charge separation due to the excess energy released in the process. However, in the event that a CT exciton does not dissociate, it may recombine due to the Coulomb interaction between e-h in the exciton. It is therefore critical to know the energy of the CT exciton formed. For instance, on the one hand, if the CT exciton energy is above the energy of the donor triplet exciton state, the CT exciton may go back to the donor triplet exciton state through intersystem crossing without being dissociated. On the other hand, if the CT exciton state energy is lower than the donor triplet exciton state then the CT exciton cannot be transferred back to the donor and it may dissociate.[7] A CT exciton state can be generated in three ways: 1) A singlet exciton from the donor can become a singlet CT exciton, 2) a triplet exciton excited in the donor can become a triplet CT exciton, and 3) a singlet exciton in the donor may go through an intersystem crossing to the donor triplet exciton state before going to the CT state. Thus, if a triplet CT exciton is formed and its energy is lower than that of the triplet donor
[a] Calculated values at Rda = 0.5 nm.
lengths in different donors for singlet excitons range from 4.5 to 129.8 nm, which agree reasonably well with the available experimental values, which range from 5.1 to 21.8 nm. Likewise, the calculated diffusion lengths for triplet excitons range from 1.3 to 30.2 nm and the experimental counterparts from 13.1 to 18.0 nm, which are also in agreement. Therefore, in view of these results, it may be concluded that the diffusion lengths of singlet and triplet excitons in most organic donors are comparable. The relation between the diffusion length and the fraction of excitons that reach the D–A interface is given by Equation (16):[21] nex ¼
LD ½1 expð2d=LD Þ , d ½1 þ expð2d=LD Þ
ð16Þ
where d is the distance between the donor and acceptor molecules. By using the diffusion lengths calculated from Equations (6) and (7), we have plotted the fraction of excitons that
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Articles exciton state, it cannot go back to the donor and may dissociate eventually into e-h pair. This is evident from observing a high photovoltaic performance in platinumacetylide polymer, which exclusively generated triplet excitons due to the introduction of heavy metal atom Pt, which induces strong spinorbit interaction and thus rapid singlet to triplet intersystem crossing (kisc > 1011 s1) with possible absorption to triplet excitons, resulting into triplet CT excitons at an energy sufficiently lower than the triplet state of the donor polymer to favor charge separation after excitation.[8, 17] A free e-h pair does not have a distinct singlet or triplet spin character;[14] however, prior to dissociation, a CT exciton will still be spin correlated to the singlet or triplet state. In Equations (12) and (13) we have derived spin-dependent recombination rate constants from which we can calculate how, prior to charge separation of dissociated exciton, the charges can still be drawn back to their original excitonic state and possibly decay to the ground state. The exciton Bohr radii found in most conjugated polymers are similar in magnitude, especially, for rigid-rod polymers. For singlet excitons, the calculated Bohr radius is 0.40 nm for ladder-type poly-p-phenylene oligomers and the experimental value is 0.72 nm, as reported for methylsubstituted ladder-type poly(para-phenylene) polymers.[22, 23] The triplet exciton Bohr radii are not known in these materials. As the dielectric constant of most organic solids is about the same, it is expected that the Bohr radius of an exciton and that of its corresponding CT state will be similar. The CT exciton Bohr radii for singlet and triplet states have been simulated in solid pentacene to be a0 S = 0.8 nm and a0 T = 0.1 nm, respectively.[24] Therefore, for calculating the probabilities of recombination from Equations (14) and (15) for different polymers, we have assumed that the CT exciton Bohr radii for singlet exciton as 0.72 nm, and 0.1 nm for triplet, which is the same as for solid pentacene. We have thus found the probability of recombination of singlet [Eq. (14)] and triplet excitons [Eq. (15)] to be approximately 65.6 and 34.4 %, respectively, by using a = 1.37.[2] This suggests that the Langevin-type recombination rate kr of triplet excitons is much lower in organic semiconductors.
cients. These results are consistent with experimental observations.[19] Furthermore, we showed that triplet excitons can be considered to contribute to free charge carriers in OSCs if the CT exciton state is lower in energy than the triplet state of the donor polymer. Keywords: donor–acceptor systems · electronic structure · electron transfer · electron transport · energy conversion
[1] M. R. Narayan, J. Singh, J. Appl. Phys. 2013, 114, 073510-1 – 073510-7. [2] M. R. Narayan, J. Singh, Phys. Status Solidi C 2012, 9, 2386 – 2389. [3] J. Singh, Excitation Energy Transfer Processes in Condensed Matter: Theory and Applications, Plenum Press, New York and London, 1994. [4] J. D. Coyle, Introduction to Organic Photochemistry, John Wiley & Sons, Chichester, 1986. [5] Z. Xu, B. Hu, J. Howe, J. Appl. Phys. 2008, 103, 043909-1 – 043909-8. [6] K. Feron, W. Belcher, C. Fell, P. Dastoor, Int. J. Mol. Sci. 2012, 13, 17019 – 17047. [7] G. Lakhwani, A. Rao, R. H. Friend, Annual review of physical chemistry 2014, 65, 557 – 581. [8] F. Guo, Y.-G. Kim, J. R. Reynolds, K. S. Schanze, Chem. Commun. 2006, 1887 – 1889. [9] A. Pivrikas, G. Jusˇka, R. sterbacka, M. Westerling, M. Viliu¯nas, K. Arlauskas, H. Stubb, Phys. Rev. B 2005, 71, 125205. [10] R. Husermann, E. Knapp, M. Moos, N. A. Reinke, T. Flatz, B. Ruhstaller, J. Appl. Phys. 2009, 106, 104507. [11] V. D. Mihailetchi, L. J. A. Koster, J. C. Hummelen, P. W. M. Blom, Phys. Rev. Lett. 2004, 93, 216601. [12] A. C. Mayer, S. R. Scully, B. E. Hardin, M. W. Rowell, M. D. McGehee, Mater. Today 2007, 10, 28 – 33. [13] J. Singh, Harvesting Emission in White Organic Light Emitting Devices, in Organic Light Emitting Devices, Intech, 2012. [14] J. Singh, Koichi Shimakawa, Advances in Amorphous Semiconductors, Taylor & Francis, London, 2003. [15] J. Singh, H. Baessler, S. Kugler, J. Chem. Phys. 2008, 129, 041103. [16] F. S. Steinbacher, R. Krause, A. Hunze, A. Winnacker, Phys. Status Solidi A 2012, 209, 340 – 346. [17] B. C. Thompson, J. M. J. Frchet, Angew. Chem. Int. Ed. 2008, 47, 58 – 77; Angew. Chem. 2008, 120, 62 – 82. [18] M. Hilczer, M. Tachiya, J. Phys. Chem. C 2010, 114, 6808 – 6813. [19] R. R. Lunt, N. C. Giebink, A. A. Belak, J. B. Benziger, S. R. Forrest, J. Appl. Phys. 2009, 105, 053711. [20] J.-W. Yu, J. K. Kim, D. Y. Kim, C. Kim, N. W. Song, D. Kim, Curr. Appl. Phys. 2006, 6, 59 – 65. [21] P. Peumans, A. Yakimov, S. R. Forrest, J. Appl. Phys. 2003, 93, 3693 – 3723. [22] A. Kçhler, D. Beljonne, Adv. Funct. Mater. 2004, 14, 11 – 18. [23] M. Harrison, S. Mçller, G. Weiser, G. Urbasch, R. Mahrt, H. Bssler, U. Scherf, Phys. Rev. B 1999, 60, 8650 – 8658. [24] S. Sharifzadeh, P. Darancet, L. Kronik, J. B. Neaton, J. Phys. Chem. Lett. 2013, 4, 2197 – 2201.
6. Conclusions By using Fçster and Dexter transfer rates, we derived a relation for the diffusion length and diffusion coefficient of singlet and triplet excitons, respectively. We showed that excitons in both singlet and triplet states have comparable diffusion lengths, because the longer lifetime of triplet states are compensated for by their lower transfer rates, due to lower diffusion coeffi-
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Received: October 15, 2014 Revised: December 22, 2014 Published online on && &&, 2015
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ARTICLES D. Ompong, J. Singh*
Out for a spin: Expressions for the diffusion length of singlet and triplet excitons have been derived by using the Fçster and Dexter transfer mechanisms, respectively. The results show that the diffusion lengths of singlet and triplet excitons are comparable.
&& – && Diffusion Length and Langevin Recombination of Singlet and Triplet Excitons in Organic Heterojunction Solar Cells
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Articles
Diffusion Length and Langevin Recombination of Singlet and Triplet Excitons in Organic Heterojunction Solar Cells David Ompong and Jai Singh*[a] We derived new expressions for the diffusion length of singlet and triplet excitons by using the Fçster and Dexter transfer mechanisms, respectively, and have found that the diffusion lengths of singlet and triplet excitons are comparable. By using the Langevin recombination theory, we derived the rate of recombination of dissociated free charges into their excitonic states. We found that in some organic polymers the probabilities of recombination of free charge carriers back into the sin-
glet and triplet states are approximately 65.6 and 34.4 %, respectively, indicating that Langevin-type recombination into triplet excitons in organic semiconductors is less likely. This implies that the creation of triplet excitons may be advantageous in organic solar cells, because this may lead to dissociated free charge carriers that can be collected at their respective electrodes, which should result in better conversion efficiency.
1. Introduction terface has been proposed[1] in which exciton relaxes first to a charge transfer (CT) exciton state by transferring its electron from the LUMO of the donor to the LUMO of the acceptor, which is at a lower energy and thus the excess energy is released in the form of molecular vibrational energy. The excess vibrational energy, if larger than the exciton binding energy, may impact back to the CT exciton to dissociate it into a pair of free electron and hole. It is therefore critically important that all generated excitons reach the interface to form CT exciton and dissociate into free charge carriers. As such, any recombination of CT excitons prior to their dissociation into free charge carriers will have a negative impact on the performance of an OSC. In low-mobility materials such as p-conjugated polymers (PCPs), the diffusion-controlled bimolecular recombination, or the so-called Langevin recombination, dominates. This type of recombination occurs when opposite charge carriers approach each other within the Coulomb radius, given by rc ¼ e2 =4per eo kT, at a temperature T , where e is the charge of an electron, er is the relative dielectric permittivity, eo is absolute permittivity, and k is Boltzmann’s constant.[9] Previous approaches involving the application of Langevin theory to study the recombination of CT excitons, prior to charge separation in OSCs,[10, 11] have not considered the spin; as such, no distinction is made between the recombination of singlet and triplet excitons. The introduction of spins to distinguish between the process of recombination of singlet and triplet excitons has been considered only recently.[7] In this paper, we have calculated the diffusion lengths of singlet and triplet excitons to study the influence of exciton diffusion lengths on the fraction of excitons that reaches the (D–A) interface in a heterojunction organic solar cell. Furthermore, by using the Langevin recombination theory, we have derived the rates of recombination of dissociated free charges back into their excitonic states. We have calculated the probability of re-
Research interest in organic solar cells (OSC) has increased recently because of the cost effectiveness, easy fabrication techniques, large-scale production,[1] and easy integration of such units in a wide variety of devices[2] such as watches, mobile phones, and laptop cases. When an organic molecule absorbs a photon, a pair of electron and hole is excited on the same molecule to form an exciton.[3] Excitons are usually excited in singlet (S) and triplet (T) spin configurations. An exciton consists of a pair of excited electron and hole bound in a hydrogenic state due to their Coulomb interaction. When the electron and hole in an exciton have opposite spins [›fl or fl›], it is a singlet exciton with multiplicity 1 and when the electron and hole have parallel spins [›› or flfl] then it is a triplet exciton with multiplicity 3.[3, 4] In organic materials, the selection rules for the electronic dipole transitions allow generation of only singlet excitons by exciting an electron from the singlet ground state. Therefore, the probability of generating a triplet exciton by spin inversion is extremely low in most molecules and mostly singlet excitons are created first in organic photovoltaic (OPV) cells when light is absorbed.[4–7] To achieve high conversion efficiency in OSCs, the following key processes must be optimized: 1) photon absorption and exciton generation; 2) efficient diffusion of excitons to the interface before their quenching; 3) dissociation and charge separation at the interface; and 4) carrier collection by the electrodes. Although each of these events has an influence on the overall conversion efficiency of an OSC, the efficiency of exciton dissociation and subsequent charge separation to the respective electrodes plays the most important role.[8] A mechanism for exciton dissociation at the donor–acceptor (D–A) in[a] D. Ompong, Prof. Dr. J. Singh School of Engineering and IT, Faculty of EHSE Charles Darwin University, Darwin, NT 0909 (Australia) An invited contribution to a Special Issue on Organic Electronics
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Articles combination into singlet and triplet excitonic states in some organic polymers. It is expected that the present work will advance the understanding of the roles of singlet and triplet excitons in OSCs. The results also show that the generation of not only singlet excitons but also triplet excitons can be useful in the operation of OSCs.
transfer rate decays exponentially with distance.[6] The Fçrster and Dexter transfer rates are given, respectively, as Equation (3):[2]
2. Exciton Generation in OSCs
and Equation (4):
kf ¼
When a photon is absorbed, an electron in the organic semiconductor is excited from the HOMO to the LUMO of a molecule. However, because of the low dielectric constant (er 3–4) in organic solids, a strong Coulomb attraction exists between a photo-excited electron–hole (e–h) pair,[12, 13] which binds them in a Frenkel exciton.[3] The excitonic Bohr radius is the radius of the orbital formed between the excited e–h pair in an exciton and it is different for singlet and triplet excitons as given by Equation (1):[14] aex ðS ¼ 0Þ ¼ aS ¼
a 2 er m ao ða 1Þ2 mex
ð1Þ
er m a mex o
ð2Þ
kd ¼
D¼
in which a is a material-dependent parameter representing the ratio of the magnitude of the Coulomb and exchange interactions between the electron and hole in an exciton, S is the spin of the exciton (S = 0 for singlet and S = 1 for triplet), m is the reduced mass of the electron in a hydrogen atom, ao is the Bohr radius, and mex is the reduced mass of an exciton, given by mex = 0.5me, with me being the free electron mass.[15] Singlet excitons have relatively small binding energy and larger excitonic Bohr radius compared with the triplet excitons.[2] Due to their favorable spin configuration, singlet excitons can easily recombine radiatively to the singlet ground state without reaching the interface.
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L2D R2da ¼ t 6tH
LD ðSÞ ¼
ð4Þ
ð5Þ
pffiffiffiffiffiffiffiffi Rf 3 tS kf pffiffiffi 6
ð6Þ
and Equation (7):
LD ðTÞ ¼
rffiffiffiffiffiffiffiffiffi tT k d L Rd lnðtT kd Þ 6 2
ð7Þ
and the corresponding diffusion coefficients are obtained as Equation (8):
An exciton can move throughout the material as a neutral quasi particle. There are two general mechanisms by which excitons can diffuse in organic semiconductors and PCPs: 1) Fçrster resonance energy transfer (FRET) and 2) Dexter carriers transfer. In Fçrster transfer, an excited electron in an exciton on a molecule recombines with the hole and the energy is transferred nonradiatively to excite an exciton on another molecule. Such a recombination is only possible for singlet excitons due to their favorable spin configuration and hence FRET is applicable only to the transfer of singlet excitons. In the Dexter transfer mechanism, the excited electron and hole are transferred to excite a nearby molecule due to the overlap of electronic wavefunctions. Therefore, the Dexter transfer mechanism can occur only at short range and it applies to both singlet and triplet excitons. In Dexter’s approach, the exciton ChemPhysChem 0000, 00, 0 – 0
1 2Rd R exp 1 da tT L Rd
where tH is the exciton hopping time and LD is the exciton diffusion length. By using Equations (3)–(5) the diffusion lengths of singlet (S) and triplet (T) excitons are, respectively, obtained as Equation (6):
3. Exciton Diffusion
&
ð3Þ
where tS is the lifetime of singlet excitons, tT is the lifetime of triplet excitons, Rda is the D–A distance of separation, Rf (Rd ) is the Fçrster (Dexter) radius, and L is the average length of a molecular orbital, which is found to be 0.11 nm[16] in Ir(ppy)3. We have assumed the same value here for all triplet exciton calculations. As stated above, the rate in Equation (4) applies also to singlet excitons with the singlet lifetime; however, in this work, we have applied it only to triplet excitons. The diffusion coefficient (D) of an exciton of lifetime t is given by Equation (5):[2]
and Equation (2): aex ðS ¼ 1Þ ¼ aT ¼
1 Rf 6 tS Rda
R2 DðSÞ ¼ f 6
sffiffiffiffiffi 3 kf2 tS
ð8Þ
and Equation (9): DðTÞ ¼
2 kd L Rd lnðtT kd Þ 2 6
ð9Þ
Notably, in both Equations (6) and (7), the diffusion length is dependent on the product of the lifetime and the respective transfer rate.
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Articles 4. CT Exciton Recombination Dynamics
where N and Pare electron and hole concentrations, respectively. The recombination coefficients are evaluated by substituting the relative dielectric permittivity from Equations (1) and (2) into Equation (10), thereby giving the singlet (S) and triplet (T) recombination coefficients, respectively, as Equation (12):
Three main processes need to be considered from the time an exciton at the D–A interface generates a CT exciton to the time when charges are collected at the electrodes: geminate recombination, bimolecular recombination, and charge transport.[17] The geminate recombination phase starts when the electron of an exciton is transferred from a donor to an acceptor molecule at the interface to form a CT exciton. A singlet CT exciton can decay to the ground state with a rate constant kq or separate into free charge carriers with rate constant kd ; the latter can occur for both singlet and triplet excitons. Once separated, some fraction of the charge carriers may return to form CT excitons with a rate constant kr as shown in Figure 1. Therefore,
bS ¼ e
ðmn þ mp Þmao a2 a0 S mex eo ða 1Þ2
ð12Þ
and Equation (13): bT ¼ e
ðmn þ mp Þmao a0 T mex eo
ð13Þ
By using the coefficient of Equations (12) and (13) in Equation (11), we can obtain the rates of recombination of singlet (kS ) and triplet (kT ) excitons, which give the corresponding probability of recombination as Equation (14): kS a0 T a2 1 ¼ 0 2 ¼ kS þ kT a T a þ a0 S ða 1Þ2 1 þ a00 S a1 2 aT a
ð14Þ
and Equation (15): kT a0 S ða 1Þ2 1 ¼ 0 ¼ kS þ kT a S ða 1Þ2 þ a0 T a2 1 þ a00 T a 2 a S a1
Figure 1. Charge carrier transfer processes in organic solar cells. CT exciton may either decay to the ground state with a rate constant kq or dissociate with a rate kd . kr is the recombination rate of the charge carriers to form the original CT exciton after dissociation.
In deriving Equations (14) and (15), we have assumed the same electron and hole mobilities, and N and P for both singlet and triplet cases. a0 S ða0 T Þrepresents singlet (triplet) Bohr radius of CT excitons.
long-lived CT excitons can act as a precursor for free charge carriers.[11, 18] Among the fraction of reformed CT excitons, some may recombine and others may separate again. This can be repeated many times. When some fraction have already recombined and some separated to sufficiently long distances, the geminate recombination phase ends and the bulk recombination phase starts.[18] Recombination between electrons and holes is an energy loss process for the efficiency of OSCs, whether it occurs through geminate recombination or bulk recombination. Therefore, it is very important to suppress these recombinations in OSCs.[18] According to the Langevin theory, in an organic system, charge recombination occurs when an electron and a hole come close to each other within rc . In the Langevin bimolecular process, the recombination coefficient is given as Equation (10):[9] b¼e
ðmn þ mp Þ er eo
5. Results and Discussions A study of the diffusion length, diffusion coefficient, fraction of excitons that reach the D–A interface, and recombination of charge carriers into CT excitons have been carried out for both singlet and triplet excitons. We have found that triplet excitons have a lifetime about three orders of magnitude greater than singlet excitons; however, the transfer rate of singlet excitons is about three orders of magnitude larger than that of triplet excitons, which results in a higher diffusion coefficient for singlet excitons. The diffusion length as clearly shown in Equations (6) and (7) are dependent on the product of the lifetime and the transfer rate for singlet and triplet excitons, this product is the dominant term in both expressions. For calculating the diffusion lengths and diffusion coefficients for singlet and triplet excitons using Equations (6)–(9), we require some material-dependent input parameters. For this, we have selected a few materials used as donors and listed their available input parameters in Table 1. By using Equations (6)–(9) and the input parameters given in Table 1, the diffusion lengths and diffusion coefficients have been calculated for some of the donor materials; these are listed in Table 2. According to Table 2, the calculated diffusion
ð10Þ
where mn ðmp Þis the mobility of electrons (holes). The recombination rate is given by Equation (11):[7] kr ¼ bNP ChemPhysChem 0000, 00, 0 – 0
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ð15Þ
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Articles reach the D–A interface from Equation (16) as a function of d in Figure 2 for some organic semiconductors listed in Table 1. According to Figure 2, the comparable diffusion lengths allow a fairly good fraction between 0.99 to 0.64 of both singlet and triplet excitons to reach the D–A interface for d values of between 10 and 15 nm.
Table 1. Input parameters for calculating the diffusion length (LD calc.) from Equations (6) and (7) and diffusion coefficient (D) from Equations (8) and (9) for excitons in several donor materials. Organic material
Exciton multiplicity
ti (i = S or T) [ns]
Rj (j = S or T) [nm]
Ref.
NPD CBP SubPc PTCDA DIP (upright) DIP (flat) PtOEP PtOEP Ir (ppy)3 Meo-POV PDHFV PTEH PDHFHPPV
S S S S S S T-monomer T-dimer T S S S S
3.5 0.7 1 3.2 1.8 1.8 800 2800 1330 0.055 0.05 0.23 0.71
1.9 2.2 1.5 1.4 2.0 1.7 0.7 0.7 1.1 1.9 2.2 2.4 4.3
[19] [19] [19] [19] [19] [19] [19] [19] [16] [20] [20] [20] [20]
Table 2. Calculated diffusion length (LD calc.) from Equations (6) and (7), diffusion coefficient (D) from Equations (8) and (9), and measured diffusion length (LD meas.) of excitons in some organic semiconductors. Organic material
Exciton multiplicity
D calc.[a] 104 [cm2 s1]
LD calc.[a] [nm]
LD meas. [nm]
Ref.
NPD CBP SubPc PTCDA DIP (upright) Dip (flat) PtOEP PtOEP Meo-POV PDHFV PTEH Ir(ppy)3 PDHFHPPV
S S S S S S T-monomer T-dimer S S S2 T S
3.6 43.2 3.0 0.6 9.5 2.6 0.000021 0.000006 228.1 604.7 21.6 0.069 2374.2
11.2 17.4 5.5 4.5 13.1 8.0 1.3 1.3 11.2 17.4 22.6 30.3 129.8
5.1 16.8 8.0 10.4 16.5 21.8 18.0 13.1
[19] [19] [19] [19] [19] [19] [19] [19]
Figure 2. Plot of fraction of the singlet (S) and triplet (T) excitons reaching the D–A interface as a function of the distance (d) calculated from Equation (10) for some selected organic semiconductors given in Table 1.
A typical exciton diffusion length in bulk heterojunction (BHJ) solar cells is in the 10–15 nm range;[11] however, in BHJ for which the donor and acceptor materials are mixed to form an interpenetrating nanoscale network of D–A blend, we will expect the interface to be at a distance less than the diffusion length. Thus, at a distance of d = 5 nm, an appreciable fraction (> 0.93) of excitons can make it to the interface in most organic molecules, as shown in Figure 2. However, in PTCDA, just 0.71 of excitons are able to reach the interface at the same 5 nm distance. The formation of CT excitons may mediate the exciton dissociation and charge separation due to the excess energy released in the process. However, in the event that a CT exciton does not dissociate, it may recombine due to the Coulomb interaction between e-h in the exciton. It is therefore critical to know the energy of the CT exciton formed. For instance, on the one hand, if the CT exciton energy is above the energy of the donor triplet exciton state, the CT exciton may go back to the donor triplet exciton state through intersystem crossing without being dissociated. On the other hand, if the CT exciton state energy is lower than the donor triplet exciton state then the CT exciton cannot be transferred back to the donor and it may dissociate.[7] A CT exciton state can be generated in three ways: 1) A singlet exciton from the donor can become a singlet CT exciton, 2) a triplet exciton excited in the donor can become a triplet CT exciton, and 3) a singlet exciton in the donor may go through an intersystem crossing to the donor triplet exciton state before going to the CT state. Thus, if a triplet CT exciton is formed and its energy is lower than that of the triplet donor
[a] Calculated values at Rda = 0.5 nm.
lengths in different donors for singlet excitons range from 4.5 to 129.8 nm, which agree reasonably well with the available experimental values, which range from 5.1 to 21.8 nm. Likewise, the calculated diffusion lengths for triplet excitons range from 1.3 to 30.2 nm and the experimental counterparts from 13.1 to 18.0 nm, which are also in agreement. Therefore, in view of these results, it may be concluded that the diffusion lengths of singlet and triplet excitons in most organic donors are comparable. The relation between the diffusion length and the fraction of excitons that reach the D–A interface is given by Equation (16):[21] nex ¼
LD ½1 expð2d=LD Þ , d ½1 þ expð2d=LD Þ
ð16Þ
where d is the distance between the donor and acceptor molecules. By using the diffusion lengths calculated from Equations (6) and (7), we have plotted the fraction of excitons that
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Articles exciton state, it cannot go back to the donor and may dissociate eventually into e-h pair. This is evident from observing a high photovoltaic performance in platinumacetylide polymer, which exclusively generated triplet excitons due to the introduction of heavy metal atom Pt, which induces strong spinorbit interaction and thus rapid singlet to triplet intersystem crossing (kisc > 1011 s1) with possible absorption to triplet excitons, resulting into triplet CT excitons at an energy sufficiently lower than the triplet state of the donor polymer to favor charge separation after excitation.[8, 17] A free e-h pair does not have a distinct singlet or triplet spin character;[14] however, prior to dissociation, a CT exciton will still be spin correlated to the singlet or triplet state. In Equations (12) and (13) we have derived spin-dependent recombination rate constants from which we can calculate how, prior to charge separation of dissociated exciton, the charges can still be drawn back to their original excitonic state and possibly decay to the ground state. The exciton Bohr radii found in most conjugated polymers are similar in magnitude, especially, for rigid-rod polymers. For singlet excitons, the calculated Bohr radius is 0.40 nm for ladder-type poly-p-phenylene oligomers and the experimental value is 0.72 nm, as reported for methylsubstituted ladder-type poly(para-phenylene) polymers.[22, 23] The triplet exciton Bohr radii are not known in these materials. As the dielectric constant of most organic solids is about the same, it is expected that the Bohr radius of an exciton and that of its corresponding CT state will be similar. The CT exciton Bohr radii for singlet and triplet states have been simulated in solid pentacene to be a0 S = 0.8 nm and a0 T = 0.1 nm, respectively.[24] Therefore, for calculating the probabilities of recombination from Equations (14) and (15) for different polymers, we have assumed that the CT exciton Bohr radii for singlet exciton as 0.72 nm, and 0.1 nm for triplet, which is the same as for solid pentacene. We have thus found the probability of recombination of singlet [Eq. (14)] and triplet excitons [Eq. (15)] to be approximately 65.6 and 34.4 %, respectively, by using a = 1.37.[2] This suggests that the Langevin-type recombination rate kr of triplet excitons is much lower in organic semiconductors.
cients. These results are consistent with experimental observations.[19] Furthermore, we showed that triplet excitons can be considered to contribute to free charge carriers in OSCs if the CT exciton state is lower in energy than the triplet state of the donor polymer. Keywords: donor–acceptor systems · electronic structure · electron transfer · electron transport · energy conversion
[1] M. R. Narayan, J. Singh, J. Appl. Phys. 2013, 114, 073510-1 – 073510-7. [2] M. R. Narayan, J. Singh, Phys. Status Solidi C 2012, 9, 2386 – 2389. [3] J. Singh, Excitation Energy Transfer Processes in Condensed Matter: Theory and Applications, Plenum Press, New York and London, 1994. [4] J. D. Coyle, Introduction to Organic Photochemistry, John Wiley & Sons, Chichester, 1986. [5] Z. Xu, B. Hu, J. Howe, J. Appl. Phys. 2008, 103, 043909-1 – 043909-8. [6] K. Feron, W. Belcher, C. Fell, P. Dastoor, Int. J. Mol. Sci. 2012, 13, 17019 – 17047. [7] G. Lakhwani, A. Rao, R. H. Friend, Annual review of physical chemistry 2014, 65, 557 – 581. [8] F. Guo, Y.-G. Kim, J. R. Reynolds, K. S. Schanze, Chem. Commun. 2006, 1887 – 1889. [9] A. Pivrikas, G. Jusˇka, R. sterbacka, M. Westerling, M. Viliu¯nas, K. Arlauskas, H. Stubb, Phys. Rev. B 2005, 71, 125205. [10] R. Husermann, E. Knapp, M. Moos, N. A. Reinke, T. Flatz, B. Ruhstaller, J. Appl. Phys. 2009, 106, 104507. [11] V. D. Mihailetchi, L. J. A. Koster, J. C. Hummelen, P. W. M. Blom, Phys. Rev. Lett. 2004, 93, 216601. [12] A. C. Mayer, S. R. Scully, B. E. Hardin, M. W. Rowell, M. D. McGehee, Mater. Today 2007, 10, 28 – 33. [13] J. Singh, Harvesting Emission in White Organic Light Emitting Devices, in Organic Light Emitting Devices, Intech, 2012. [14] J. Singh, Koichi Shimakawa, Advances in Amorphous Semiconductors, Taylor & Francis, London, 2003. [15] J. Singh, H. Baessler, S. Kugler, J. Chem. Phys. 2008, 129, 041103. [16] F. S. Steinbacher, R. Krause, A. Hunze, A. Winnacker, Phys. Status Solidi A 2012, 209, 340 – 346. [17] B. C. Thompson, J. M. J. Frchet, Angew. Chem. Int. Ed. 2008, 47, 58 – 77; Angew. Chem. 2008, 120, 62 – 82. [18] M. Hilczer, M. Tachiya, J. Phys. Chem. C 2010, 114, 6808 – 6813. [19] R. R. Lunt, N. C. Giebink, A. A. Belak, J. B. Benziger, S. R. Forrest, J. Appl. Phys. 2009, 105, 053711. [20] J.-W. Yu, J. K. Kim, D. Y. Kim, C. Kim, N. W. Song, D. Kim, Curr. Appl. Phys. 2006, 6, 59 – 65. [21] P. Peumans, A. Yakimov, S. R. Forrest, J. Appl. Phys. 2003, 93, 3693 – 3723. [22] A. Kçhler, D. Beljonne, Adv. Funct. Mater. 2004, 14, 11 – 18. [23] M. Harrison, S. Mçller, G. Weiser, G. Urbasch, R. Mahrt, H. Bssler, U. Scherf, Phys. Rev. B 1999, 60, 8650 – 8658. [24] S. Sharifzadeh, P. Darancet, L. Kronik, J. B. Neaton, J. Phys. Chem. Lett. 2013, 4, 2197 – 2201.
6. Conclusions By using Fçster and Dexter transfer rates, we derived a relation for the diffusion length and diffusion coefficient of singlet and triplet excitons, respectively. We showed that excitons in both singlet and triplet states have comparable diffusion lengths, because the longer lifetime of triplet states are compensated for by their lower transfer rates, due to lower diffusion coeffi-
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Received: October 15, 2014 Revised: December 22, 2014 Published online on && &&, 2015
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ARTICLES D. Ompong, J. Singh*
Out for a spin: Expressions for the diffusion length of singlet and triplet excitons have been derived by using the Fçster and Dexter transfer mechanisms, respectively. The results show that the diffusion lengths of singlet and triplet excitons are comparable.
&& – && Diffusion Length and Langevin Recombination of Singlet and Triplet Excitons in Organic Heterojunction Solar Cells
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