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The Role of Charge Transfer State on the Reduced Langevin Recombination in Organic Solar Cells: A Theoretical Study Yiming Liu, Karin Zojer, Benny Lassen, Jakob Kjelstrup-Hansen, Horst-Guenter Rubahn, and Morten Madsen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b08936 • Publication Date (Web): 26 Oct 2015 Downloaded from http://pubs.acs.org on October 31, 2015
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The Role of Charge Transfer State on the Reduced Langevin Recombination in Organic Solar Cells: A Theoretical Study Yiming Liu1, Karin Zojer2*, Benny Lassen1, Jakob Kjelstrup-Hansen1, Horst-Günter Rubahn1, and Morten Madsen1* 1
NanoSYD, Mads Clausen Institute, University of Southern Denmark, Alsion 2, DK6400 Sønderborg, Denmark 2
Institute of Theoretical and Computational Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria Abstract The reduced Langevin recombination has been observed in organic solar cells (OSCs) for many years, but its origin is still unclear. A recent work of T. Burke et al. was inspired by the reduced Langevin recombination, and proposed an equilibrium model of charge transfer (CT) states which correlates the open circuit voltage of OSCs to experimentally available device parameters. In this work, we extend Burke’s CT model further, and for the first time directly correlate the reduced Langevin recombination with the energetic and dynamic CT state behavior. The recombination via CT states straight-forwardly leads to a decrease in the Langevin reduction factor with increasing temperature, without explicitly considering the temperature-dependence of the mobility. To verify the correlation between the CT states and the reduced Langevin recombination, we incorporate this CT model and the reduced Langevin model into drift-diffusion simulations of a bilayer OSC. The simulations not only successfully reproduce realistic JV characteristics of the bilayer OSC, but also demonstrate the equivalence of these two models in the bilayer structure. 1. Introduction High-performance organic solar cells (OSCs) have demonstrated efficient photogeneration of charge carriers with >90% quantum efficiencies1, but their power conversion efficiencies are still lower than those of their inorganic counterparts. This can mainly be attributed to their relatively low open circuit voltages (Voc), compared to the optical bandgap, and their relatively low fill factors (FF), especially for optically thick devices. The donor/acceptor interface in OSCs plays an important role in the photocurrent generation process as it provides the energetic offset needed to dissociate the photogenerated excitons. The recombination at this interface dominates the loss of free charges and, hence, significantly affects Voc.2 Several experimental techniques have been developed to quantitatively extract recombination rates in organic bulk heterojunction (BHJ) devices.3–5 For many high-efficiency OSC systems, it has been found that recombination predominantly
occurs
via
bimolecular
recombination.
The
experimentally
measured
recombination rates in BHJ devices were found to be several orders of magnitude smaller than the ones predicted by the Langevin theory, which is the most commonly used model to describe *
Corresponding authors: Dr. Karin Zojer,
[email protected]; Assoc. Prof. Morten Madsen,
[email protected] ACS Paragon Plus Environment
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bimolecular recombination in homogeneous organic materials. The fundamental origin of such a reduced Langevin recombination rate is still under debate, see, e.g., Ref. [5]. To comprehend the foundation of this debate, it is useful to point out the starting point that is common to all explanation attempts: Onsager’s description of bimolecular recombination6. It states that recombination involves two stages, each being associated to a rate: the first rate describes the likelihood that two oppositely charged particles meet and form a pair of Coulomb-bound charges. This process can be counteracted by a possible dissociation back into independent mobile charges. The second rate describes the subsequent recombination into chargeneutral species. The multitude of suggested explanations stems from a different assessment inasmuch the two rates limit the net recombination. In the case that the first rate is the limiting one, the original Langevin theory applies.6 This is certainly valid for most pristine organic materials due to their rather low charge carrier mobilities. The fact, however, that the straight-forward Langevin theory overestimates the recombination rates in BHJ devices indicates that bimolecular recombination at interfaces formed between two organic semiconductors calls for a substantially extended description. A major line of research aims at clarifying the nature of the first-encounter at the organic-organic interface7–10 and accepts that recombination occurs instantaneously after pair formation. I.e., one assumes that mobile charges directly recombine. As a consequence, Voc is determined via the difference of the associated transport level energies, i.e., the effective gap, EDA, measuring the offset between the HOMO (highest occupied molecular orbital) level of the donor and the LUMO (lowest unoccupied molecular orbital) level of the acceptor.11,12 Another research direction is dedicated to a thorough assessment of the relative impact of the second rate.13 This direction is currently gaining momentum due to an increasing experimental evidence that the recombination process involves a charge-transfer (CT) state formed at the interface.14,15 In strong contrast to the encounter-limited model, the open-circuit voltage was found to be determined by the energy ECT of this interface state.2,16,17 Burke et al. recently suggested that the reduced Langevin recombination implies the existence of an equilibrium between free carriers and populated CT states.18 The necessary condition is that the re-dissociation rate of the populated CT states back to free carriers is several times larger than the recombination rate for the transition of the CT state into a neutral ground state, i.e., the CT states dissociate several times before recombining ultimately. Assuming an equilibrium situation, the density of populated CT states can be straightforwardly related to ECT and the quasi-Fermi levels of free carriers via a Boltzmann distribution. By deriving the interface recombination rate based on the CT state density for opencircuit conditions, one arrives at a concise equation that quantitatively relates Voc to ECT. This equation successfully explains the observed dependence of Voc on the interfacial energetic disorder, temperature, and dielectric constants.18 In this work, we explore the applicability of Burke’s model to other steady-state operating conditions of OSCs, i.e. beyond open-circuit conditions, since the underlying quasi-equilibrium
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should hold under these conditions as well. By utilizing drift-diffusion simulations, we can directly determine the participation of the CT state from the splitting between the quasi-Fermi levels of electrons and holes, and analyze its effects at each bias voltage. As this approach allows us to simulate the full current-voltage characteristics, we predict the impact of the CT states on device parameters, such as the FF and the temperature-dependent open-circuit voltage. 2. Interface recombination Before we analyze the effects of interface recombination on the device performance by driftdiffusion modeling, we first review the device physics of the interface recombination.
Figure 1. Energy band diagram of heterointerfaces for (a) inorganic solar cells and (b), (c) organic
solar cells. (a) The recombination in the bulk is Shockley-Read-Hall (SRH) recombination via localized trap states. Ec, Ev denote the conduction band and the valence band of each material, respectively. The recombination at the interface consists of four paths: two paths are the SRH recombination between local carriers, and the other two SRH-like paths are between the electrons and holes from different sides of the interface (marked by red arrows) (b) Bimolecular recombination in the bulk of OSC is described by Langevin theory (represented by green arrows).
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At the interface two additional recombination paths (represented by red arrows, similar to the inorganic case) are illustrated. ELA and EHD denote the position of the LUMO level of the acceptor and of the HOMO level of the donor at the interface, respectively. The energy difference between ELA and EHD is the effective band gap EDA of this OSC. (c) Schematic illustration of interface recombination via CT states.
2.1 Inorganic model In the field of inorganic solar cells, the interface recombination at the heterojunction has been studied extensively.19–22 The recombination in the bulk of inorganic solar cells is usually dominated by the Shockley-Read-Hall (SRH) process, which characterizes the carrier recombination via localized trap states.23 The recombination via the interface states can be treated by an extension of the SRH formalism, adding two more recombination paths (represented by the red arrows shown in Fig. 1a) between electrons and holes of two adjacent semiconductors.24 The addition of these paths is crucial in the case illustrated in Fig. 1a, where the dominant interface recombination pathway involves holes of semiconductor I and electrons of semiconductor II. R. Scheer has systematically analyzed the impact of interface recombination paths on Voc of heterojunction devices.21 Generally, Voc can be expressed as:21
=
−
(
)
( )
(Eq. 1)
where Ea is the activation energy of the heterojunction diode, q is the electron charge, A is the ideality factor, k is Boltzmann constant, T is temperature, J00 is the reference current density,25 Jsc is the short-circuit current, and η(Voc) is the voltage-dependent collection function. The particular temperature dependence of Eq. 1 suggests a straight-forward experimental way to determine the activation energy Ea. The value of Ea corresponds to the intercept of the linearly extrapolated portion of the Voc-T plot with 0 K. The comparison between Ea and the semiconductor band gaps reveals the dominant recombination path within inorganic cells.26 For instance, for the interface band alignment shown in Fig. 1a, Ea will be equal to the energy gap between Ec(II) and Ev(I) if interface recombination dominates over bulk recombination; otherwise, Ea will be equal to the smaller of the two semiconductor band gaps. 2.2 Organic model In organic solar cells, losses in Voc are known to be related to the properties of the donor/acceptor interface. However, also in OSCs, Voc can be cast into an expression similar to Eq. 1. Thus, Voc ought to possess an explicit linear dependence on the temperature. Consequently, Ea can be attained by the extrapolation approach similar to the inorganic case.16 The remaining debate to be settled is whether Ea corresponds to the effective gap at the donor-acceptor interface, EDA = ELAEHD, or to the energy of the charge transfer state ECT.12 The case associated to the effective gap is described through a reduced Langevin model, while the latter case demands the CT recombination model.
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2.2.1 Langevin model For low mobility materials typically used in OSCs, the free charge recombination in the bulk is well described by the Langevin theory. The Langevin recombination rate RL is expressed as:27
= ( − )
(Eq. 2)
where n, p are the densities of electrons and holes, respectively, and ni is the intrinsic carrier density. The Langevin recombination prefactor β is:
=
! "
(Eq. 3)
#
where µn and µp are the carrier mobilities for electrons and holes, respectively, and ε=ε0εr is the permittivity of the material with εr being the dielectric constant. Staudigel et al. extended Eq. 2 to describe the recombination Rint at the interface by introducing, similar to the inorganic case (Fig. 1a), two further bimolecular recombination pathways:28
$ = % & ' + ' &
(Eq. 4)
where the superscripts D, A denote the carriers located at the donor side and the acceptor side of the interface, respectively. Each of the two pathways is assumed to possess an individual Langevin recombination prefactor
% =
&! ' "
#)
, =
' ! &"
#)
(Eq. 5)
with *̅ being the average dielectric constant of the two adjacent organic layers. However, this formulation, initially proposed for organic light emitting diodes, overestimates the recombination rates obtained for interfaces in OSCs.5 To be able to nevertheless analyze the recombination rates in terms of Langevin theory, a Langevin reduction factor γ has been introduced.5,7 A typical arrangement of transports levels at the OSC interface is presented in Fig. 1b. The first recombination term in Eq.4 (corresponding to an electron in the donor material recombining with a hole in the acceptor material) is negligible in comparison to the second term. Moreover, it is physically more reasonable to define recombination as a process aiming at re-establishing thermal equilibrium, i.e., to subtract the intrinsic thermal generation rate to guarantee the net recombination is zero at thermal equilibrium. Then, the recombination rate Rint for direct bimolecular recombination between electrons and holes at the interface is proportional to the carrier concentrations at the interface , - , ./ in excess of the equilibrium concentrations ,0- , .0/ . To preserve a Langevin-type prefactor in the rate constant, the former must be multiplied with the reduction factor γ. Then, the rate Rint takes a final form:
$ = 1 ' & − ' & ".
(Eq. 6)
The value of γ has been empirically determined to adopt values between 10-4~10-1 in bulk heterojunction OSCs and was found to decrease with increasing temperature.7 The evaluation of the temperature-dependence of the open circuit voltage Voc due to Eq. 1 reveals an activation
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energy Ea given by the effective donor acceptor band gap EDA. 29 2.2.2 CT model In the scenario of interface CT states participating in recombination (Fig. 1c), free carriers meeting at the donor/acceptor interface will neither be ultimately bound to form a CT state, nor will this CT state instantaneously recombine. Rather, the populated CT states can either dissociate back into free carriers, or recombine directly to the ground state15. An apparently reduced Langevin recombination is consistent with the situation that the CT states split back to free carriers several times before a single recombination occurs. It is possible to reach an equilibrium between the free carrier densities and the population of CT states whose density, NCT, is given at open-circuit voltage as:18 8; 9:
345 = 630 7 ;(?@9 ABC= 0 is present, the dependence of Voc on temperature is no longer linear as for the case of direct bimolecular recombination. As pointed out in Ref. [18], ECT cannot be simply found by extrapolating the Voc (T ) curve down to 0 K; rather, it is necessary to consider interfacial disorder when interpreting the Voc (T ) data. To explore the impact of CT-state mediated recombination on the entire current-voltage characteristics, we will modify Eq. 7 to account for any external bias. Inspired by the fact that the CT is in equilibrium with free carriers,18 we replace qVoc with the splitting Efn-Efp of the quasiFermi levels for electrons, Efn, and holes, Efp, at the interface at each bias voltage. As a result, the net interface recombination rates via CT states for bilayer structures can be expressed as:
VW =
XY ZVW
[
\ $ (]W)
VW
X AX
[^ ]W _[
]W
− %`
(Eq. 9)
3. Temperature dependence of the interface recombination As mentioned before, the determination of temperature dependence of the open circuit voltage is
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an important pathway to reveal the nature of interface recombination. To distinguish between the two mechanisms proposed above, it is necessary to investigate which of the models is capable of reproducing the specific temperature dependence of the recombination. To that aim, experiments directly extracting the recombination rate as a function of temperature are an excellent starting point. Juška et al. and Deibel et al. performed photo-CELIV experiments on P3HT:PCBM blends sandwiched between ITO and Al contacts7. These experiments reveal a recombination being substantially reduced with respect to Langevin theory. The temperature-dependent reduction factor γ, that relates the actual interface recombination rate R to the Langevin recombination rate via R = γRL, was found to decrease with increasing temperature.7 This marked decrease in temperature cannot be ascribed to an encounter-limited recombination Rint (Eq. 6) occurring between uniformly distributed charges.7 Rather, one has to assume temperature-dependent mobilities to recover at least the qualitative trend in temperature. 7 As also many other experiments are commonly evaluated in terms of a reduced Langevin recombination, it appears to be convenient to carry out the comparison between different recombination models in terms of the model-inherent reduction factor γ. Below, we will demonstrate under which conditions it is indeed possible to cast model-dependent expressions R into apparent reduction factors. Even though not explicitly mentioned in Ref. [18], the rate expression of the above-described CT model can be readily cast into a Langevin reduction factor γ. We will elaborate this point as follows: According to the Boltzmann distribution, the carrier densities satisfy: , = 30 7
30 7
Bf e ABag
