Semi-Empirical Limiting Efficiency of Singlet-Fission-Capable

Article pubs.acs.org/JPCC

Semi-Empirical Limiting Efficiency of Singlet-Fission-Capable Polyacene/Inorganic Hybrid Solar Cells M. J. Y. Tayebjee,* A. Mahboubi Soufiani, and G. J. Conibeer School of Photovoltaic and Renewable Energy Engineering, UNSW, Sydney 2052, Australia S Supporting Information *

ABSTRACT: A detailed balance limiting efficiency calculation of a singlet fission capable organic/inorganic hybrid solar cell is presented. Incorporating experimental values for pertinent rates in the organic layer we find that the limiting efficiency of a tetracene/ideal inorganic semiconductor cell is 35.8%, exceeding the Shockley−Queisser limit. We further propose an “equivalent circuit” model of excitonic current, with three “cells” corresponding to the organic, inorganic, and interface species. By considering their relative contributions to the current density of the device, we discuss the major loss mechanisms and identify that the triplet lifetime and nonradiative decay of bound polaron pairs are major limitations to the development of a high-efficiency polyacene/ inorganic solar cell. The results suggest that the efficiency of a tetracene/silicon hybrid solar cell could exceed the Shockley−Queisser limit.

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INTRODUCTION Single threshold photovoltaic devices operate below the Shockley−Queisser (SQ) energy conversion efficiency limit of 311 or 33.7%2 under 6000 K blackbody or AM1.5G irradiation, respectively. These limits are derived using the principle of detailed balance, wherein rates of photon absorption and emission are related by the Einstein A and B coefficients. The SQ limit is derived assuming that the absorber in such a device has (i) surface absorption of photons with energy in excess of the band gap; (ii) infinite carrier mobility; and (iii) all recombination is radiative. This idealized system suffers from five unavoidable loss mechanisms: radiative recombination, Carnot losses, Boltzmann losses, transmission of sub-band gap energy photons, and the thermalization of photon energy absorbed in excess of the band gap.3 Previous reports have calculated the efficiency limits of idealized devices that circumvent the SQ limit by utilizing the solar spectrum more efficiently, such as up-4 or down-converters,5 hot carrier solar cells,6 intermediate band solar cells,7,8 tandem devices,9 and multiple exciton generation (MEG) solar cells.10 In the present report, we will focus on the latter process and consider MEG in an organic molecular crystal. MEG has been studied in inorganic quantum dots (QDs) wherein confinement discretizes the electronic eigenstates.11−13 Such systems have a limiting efficiency of 41.9% for an isothermic MEG process.10 MEG in inorganic systems has only been observed as an exothermic process.14−16 Singlet fission (SF) is an analogous process in molecular systems that has been shown to occur endothermically in bulk and thin-film tetracene (Tc),17−26 although SF is far more rapid in pentacene (Pn), where it is an exothermic process.17,20,27−29 SF occurs © 2014 American Chemical Society

when a chromophore is promoted into an excited singlet state via photon absorption and transitions to an excited triplet-pair on neighboring chromophores, which subsequently dissociate.30,31 By including endothermic fission, the thermodynamic limiting efficiency of an idealized system is increased to 45.9% under standard illumination.32 However, the Frenkel excitons generated in organic absorbers are highly localized (compared with their inorganic counterparts) and therefore have large binding energies (0.1 to 1 eV).33 These excitonic cells therefore operate in a fundamentally different way to conventional devices.34 Once generated, excitons diffuse to a donor/acceptor (DA) interface via a hopping mechanism. A free-energy driving force, ΔG, must exist for electron or hole transfer to occur at this interface.35 The resulting charges are still Coulombically bound at the DA interface, giving rise to a charge-transfer, or bound polaron pair (BP) state. The BP is entropically driven to dissociate into free charges that are collected at electrodes. BP recombination has recently been shown to limit the opencircuit voltage of excitonic solar cells.36,37 As such, a realistic approach to the limiting efficiencies of excitonic devices must account for the charge-separation process wherein exciton binding energy is overcome.38 Some reports have used Marcus theory to model the charge-transfer state,39−41 wherein a reduction of the reorganization energy can lead to substantial efficiency gains.42 Others have included a Received: October 13, 2013 Revised: January 8, 2014 Published: January 13, 2014 2298

dx.doi.org/10.1021/jp4101637 | J. Phys. Chem. C 2014, 118, 2298−2305

The Journal of Physical Chemistry C

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MODEL The rate scheme proposed for the SF capable polyacene/ inorganic hybrid cell considered is shown in Figure 2 (Figure

free-energy sacrifice to dissociate excitons at the DA interface by including the generation of an absorptive BP state.35,43 Inclusions of a free-energy sacrifice of 0.3 eV (0.5 eV) in an ideal SF organic solar cell lowers the limiting efficiency to 30.0% (23.9%).32 When considering SF in a device, it should be noted that the binding energy of the highly localized triplet exciton exceeds that of a singlet exciton by twice the exchange integral. However, the binding energy of a BP is typically much lower.35 Moreover, a recent experimental report has shown that the required driving energy for charge transfer via a BP is reduced when a more crystalline acceptor is incorporated (such as an inorganic lattice).44 Although external quantum efficiencies greater than 100% have been verified in solar cells that display SF,45 the advantages of using such materials remain controversial.31,46 There has been much recent interest in SF in molecular crystals of linear polyacenes,18−24,47−49 and it has been suggested that a tetracene/silicon hybrid cell may be able to achieve powerconversion efficiencies in excess of 30%.45 However, a model of the limiting efficiencies of a device with a polyacene organic donor layer and an inorganic acceptor layer is yet to be reported. We present a semiempirical detailed balance formalism to calculate the effect of applying a SF-capable organic layer to an idealized SQ absorber, as shown in Figure 1a. To account for

Figure 2. Energy levels of each state. (a) Radiative transitions for each of the three species considered. Absorption and emission are, respectively, shown in solid and dashed lines. (b) Nonradiative transitions. Transitions that involve more than one absorber have multiple arrows with the same color. Reverse processes are not shown for clarity.

2a,b shows the radiative and nonradiative processes, respectively), and the energy-transfer reactions are summarized in Table S1 of the Supporting Information. Forward and reverse rate constants are denoted k and k′, respectively. The population of ground-state organic absorbers, pdg, can absorb photons with rate constant kb, increasing the number of excited singlet donors, pds. The excited singlet decays to the ground state with rate constant k′b. Similarly, the ground-state acceptor population, p ag , absorbs photons with rate constant k r increasing the number of electronically excited acceptors, pae, and decays to the ground state with rate constant k′r (where b and r denote ‘blue’ and ‘red’ photons). Molecules that are part of the S1 population, pds, can undergo bimolecular SF with neighboring ground-state organic molecules with rate constant ksf, producing two T1 molecules in the population pdt. The reverse process, triplet−triplet annihilation (TTA), occurs with rate constant ksf′ with a fraction, f, of these events generating a molecules in the S1 state. All other TTA

Figure 1. Layers considered in the device. (a) Incident and emitted radiation. (b) Schematic representation of the DA species and a BP.

imperfect band alignment, we consider a DA species that exists at the interface between the organic and inorganic absorber (Figure 1b). Upon photon absorption, this species forms a BP. While the BP state has been studied in great detail at organic/ organic interfaces,50−52 it has received very little attention at organic/inorganic interfaces, with the notable exception of a very recent publication.44 As such, we model the interface species using very general properties: it may absorb and emit radiation with a given absorptivity. The resulting limiting efficiency of the organic/inorganic hybrid solar cell is calculated, and the issues that must be addressed for the development of such a device are discussed. 2299



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incident solar power. We now derive expressions for {p}, which are functions of the chemical potential μ = eV where V is the voltage across the cell. It is useful to define donor, acceptor, and DA complex molecular partition functions

events would generate a molecule in the T2 state (since the quintet state would not be energetically accessible), which, according to Kasha’s law, would rapidly decay back to the T1 state. Intersystem crossing from S1 to T1 and unimolecular decay of the T1 state have rate constants kisc and kt, respectively. Electron injection from a triplet donor giving rise to a BP, pbp, occurs with rate constant kd. Conversely, hole injection from an excited acceptor generating a BP occurs with rate constant ka. Using a similar model to Giebink et al., we include direct absorption (pda → pbp) and emission (pbp → pda) by DA species with rate constants kir and kir′ , respectively (where ir denotes an ‘infrared’ photon).35 The BP state undergoes charge separation and charge injection into the external circuit with rate constant ki. These processes are summarized in the following ordinary differential equations, where rate constants, {k}, are in units of s−1 cm−2; and the occupancy of each state, {p}, is unitless. dpdg

exp(−β(Es − 2μ))

− ksf pdg pds dpdt

= k iscpds + 2ksf pdg pds + kd′pdg pbp − (1 +

dt

(1)

f )ksf′ pdt2

− kdpdt pda − k tpdt dpds

(2)

= k bpdg + fksf′ pdt2 − k b′pds − ksf pdg pds − k iscpds

dt

dpag dt

= k r′pae + kapae pda − k rpag − ka′pag pbp

dpae

=−

dt dpda

(3)

(4)

dpag (5)

dt

− k i′pda − kdpdt pda − kapae pda

dpbp

=−

(6)

dpda (7)

dt

We have not included the terms k′t pdg and k′iscpdt (the thermal promotion from the ground state to T1 and reverse intersystem crossing from T1 to S1). While these processes are strictly required for a detailed balance analysis, they are both spinforbidden and sufficiently endothermic that they can be ignored at room temperature. Under steady-state conditions, the population of every state is constant and the LHS of eqs 1−7 equals 0; we derive an expression for the current density J = e(k ipbp − k i′pda ) = e[k irpda + k rpag + 2k bpdg − k ir′pbp − k r′pae −(2k b′ + k isc)pds − k tpdt ]

qa = Na + Na exp( −β(Eg − μ))

(10)

qda = Nda + Nda exp( −β(E bp − μ))

(11)

pdg = Nd /qd

(12)

pdt = 3Nd exp( −β(Et − μ))/qd

(13)

pds = Nd exp(−β(Es − 2μ))/qd

(14)

pag = Na /qa

(15)

pae = Na exp( −β(Eg − μ))/qa

(16)

pda = Nda /qda

(17)

pbp = Nda exp( −β(E bp − μ))/qda

(18)

By assuming a Boltzmann distribution and not explicitly including radical species (free charges), we have included an infinite rate of carrier and exciton mobility. While this is a limitation of the model, it should be noted that Pn and Tc have relatively high hole mobilities (on the order of several and 0.1 cm2 V−1 s−1, respectively).57 Moreover, reports of these values have differed significantly, and improvements are continually being realized.58 Indeed, an enhanced open circuit voltage, Voc, has been achieved by annealing Tc donor layers.57,59 Postannealing of Pn thin-film transistors has also been shown to increase hole mobility,60 suggesting that these values are currently limited by fabrication and processing techniques rather than an intrinsic property of the material. Furthermore, the thicknesses of the organic layers are limited by the triplet diffusion length rather than the hole diffusion length in a planar device. As such, we have restricted the thicknesses in our calculations accordingly. The absorbance, A0(λ), is defined such that the transmission is I = I0·10−A0(λ). Using experimental data (see Supporting Information Figure S1) and solving for the absorption coefficient, α(λ) = A0(λ)/z0, where z0 is the thickness of the sample, we simulate the absorption spectrum for an arbitrary thickness, z, using A(λ) = α(λ)z. The absorption rate constant is

= k ir′ pbp + k ipbp + kd′pdg pbp + ka′pag pbp − k irpda

dt

(9)

where β = 1/kbT (kb is the Boltzmann constant and T is the temperature of the cell). The energies of the triplet, singlet, excited acceptor and BP are, respectively, denoted Et, Es, Eg, and Ebp. Areal molecular densities are denoted Nd, Na, and Nda. Here we have explicitly defined a unique DA interface species. This is reasonable because the excitation of chromophores at the organic/inorganic interface will give rise to a hybrid excitons with different absorption and emission properties compared to the bulk.56 Given a Boltzmann distribution:

= k b′pds + kdpdt pda + ksf′ pdt2 + ktpdt − kbpdg − kd′pdg pbp

dt

dt

qd = Nd + 3Nd exp( −β(Et − μ)) + Nd

(8) 53

where e is the electronic charge, and we have set f = 1. There has been considerable debate in the literature over this value in tetracene,53−55 so we chose unity for simplicity. The maximum efficiency of the cell is η = (Jμ/e)max/Ptot, where Ptot is the

kb = 2300

∫0

∞

F(1 − 10−A(λ)) dλ

(19)



Article

The AM1.5G spectrum in photons s−1 cm−2 nm−1 is denoted F.61 We assume that the shape of the absorption spectrum does not vary significantly with Pn thickness.62 Tc has been shown to have significant enhancement of Davydov splitting with increasing thickness.18,63 As such, we use the data in figure 2b in ref 18 and a spline interpolation for intermediate thicknesses. The integrated absorbance in the wavelength range 300−650 nm is linearly dependent on the thickness. Other rates pertaining to the organic layer are included in Table S2 in the Supporting Information. The portion of the spectrum that is transmitted through the organic layer is F′ = F·10−A(λ), as shown in Figure 1a. We utilize the absorption rate constant of an inorganic surface absorber with unit absorptivity kr =

∫0

hc / Eg

′ pds F ′dλ + 0.5k b,rad

(20)

where kb,rad ′ is the radiative decay rate constant for S1, and the second term accounts for emission from the organic layer directed toward the inorganic surface. The corresponding emission rate constant is64 k r′ =

2 ⎤ 3 3 ⎡ Eg n2 kBT ⎢⎛ Eg ⎞ ⎥ + + 2 2 ⎜ ⎟ ⎥⎦ 4 ℏ3π 2c 2 ⎢⎣⎝ kBT ⎠ kBT

(21)

in the absence of nonradiative decay and stimulated emission. The former omission is required when establishing ultimate efficiencies, and the latter is a reasonable approximation because Eg − μ ≫ kBT.3 The refractive index of the organic layer, n, is included to account for the increased emission étendue of the acceptor into the organic layer.5 While the refractive index is a function of photon energy, it is almost constant for photon energies less than Es. We use a value of 1.6, which is reasonable for polyacenes. (See Figure S2 in the Supporting Information for Tc and ref 65 for Pn.) Using a similar formalism to Giebink et al., we define the radiative transitions of the BP state35 k ir = σ

Figure 3. (a) Limiting efficiencies of the hybrid cells when σ = 10−3. Solid blue and dashed red lines correspond to Pn (10 nm)/inorganic and Tc (100 nm)/inorganic cells, respectively, and the value of ΔG is marked in electronvolts. The SQ curve is also shown (dotted black line) for reference. (b) Limiting efficiency of a Tc (100 nm)/inorganic cell with ΔG = 0.5 eV (solid red lines) and different values of σ, which are marked on the plot. The limiting efficiency of a device with ΔG = 0.1 eV and σ = 10−3 (dashed red line) is approached as σ is decreased. The SQ curve is also shown (dotted black line) for reference.

hc / E bp

∫hc/min(E ,E ) F′dλ g

t

(22)

and ′ k ir,rad

2 ⎤ 3 3 ⎡ E bp n2σ kBT ⎢⎛ E bp ⎞ ⎥ 2 2 = + + ⎜ ⎟ ⎥⎦ 4 ℏ3π 2c 2 ⎢⎣⎝ kBT ⎠ kBT

Importantly, we note that the limiting efficiency of the Tc/ inorganic cell exceeds the SQ limit, reaching a maximum of 35.8% when Eg = 1.12 eV and a free-energy sacrifice of ΔG = 0.1 eV is made. The maxima of each plot depend on the relative values of Eg, Et, and Ebp. As Eg increases, more photons are transmitted, reducing the efficiency of the device. If Eg is too low, a decrease in Voc is observed. As Ebp is decreased (that is, as ΔG is increased), the Voc and the efficiency of the device decrease; however, the effect is gradual for ΔG < 0.3 eV. This gradual effect at low values of ΔG can be understood by considering the equivalent circuit formalism4,5 shown in Figure 4a, which is similar to that previously presented.32 We separate eq 8 into the current generated by the three absorptive species in the cell: organic (O), inorganic (I), and DA (DA), where the total current is J = JO + JI + JDA, and

(23)

The absorptivity, σ, is related to the effective absorption crosssection of the DA complex. A value of σ = 10−3 is used as an estimation to account for the relatively few interface states unless otherwise stated. Values that are typical for the ratio between DA and bulk absorption lie in the range 10−4 to 10−2 for bulk heterojunction devices.36,37 Values of σ are likely to be even lower in a planar structure because there are fewer DA interfaces. While the inclusion of σ introduces another parameter into the model, we will show that the choice of this value does not affect the ultimate efficiency of the system.

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RESULTS AND DISCUSSION Figure 3a shows the limiting efficiency of the devices considered where the energy of the BP state is defined such that Ebp = min(Et − ΔG, Eg). In an inorganic device, no freeenergy sacrifice is strictly required for charge transfer,35 and this gives rise to the form of Ebp. 2301

JO /e = 2k bpdg − (2k b′ + k isc)pds − k tpdt

(24)

JI /e = k rpag − k r′pae

(25)

JDA /e = k irpda − k ir′ pbp

(26)



Article

Figure 4. (a) Equivalent circuit model composed of an organic donor, inorganic acceptor, and DA “cells”. The organic cell contains three components: a photocell, a dc step-down transformer (marked with a square), and a reverse-biased diode which accounts for the power loss μkt exp(−β(Et − μ))/qd, which arises from triplet decay in O. The inorganic cell is modeled as an ideal photocell, and the BP is modeled as a photocell with nonideal absorption. The current density contributions from eq 8 are indicated. The J−V characteristics using the model in panel a are shown where ΔG = 0.1, 0.3, and 0.5 eV for (b) Pn (10 nm)/inorganic and (c) Tc (100 nm)/inorganic hybrid cells. The currents generated by O and I are independent of the value of ΔG.

approaches 0 (σ → 0), the limiting efficiency of the device approaches that of the device with ΔG = 0.1 eV. This is because both systems are not limited by BP recombination. That is, if the DA → BP transition is dark, the energetic sacrifice required to separate excitons will not reduce the energy conversion efficiency in an idealized system. However, one may reasonably argue that BP recombination would be largely nonradiative. Indeed, values of the electroluminescent external quantum efficiency EQEEL of organic bulk heterojunction devices have been reported in the range 10−7 to 10−5.37 Therefore, the BP recombination rate is adjusted to be ′ + kir,nr ′ , the sum of radiative and nonradiative rates kir′ = kir,rad where the radiative rate was defined in eq 23. We define k′ir,nr using Φ = k′ir,rad/(k′ir,rad + k′ir,nr), where Φ is the fluorescence quantum yield of the interface species and equals EQEEL if BP recombination is the only recombination process. The resulting efficiencies for a Tc (100 nm)/inorganic cell are plotted in Figure 5. Examining the resulting J−V diagrams (not shown) at values of ΔG, where the plots are parallel, recombination arises only at the interface and Φ = EQEEL. Importantly, if nonradiative recombination can be completely overcome, an energetic sacrifice of ΔG ≤ 0.25 eV will not limit the efficiency of the device. This is on the same order as that required to separate Frenkel excitons,35 suggesting that high-efficiency hybrid devices may be achieved if nonradiative recombination can be minimized. Returning to Figure 4, the recombination in O is more pronounced in the Pn hybrid cell than the Tc hybrid cell; this is largely due to the short lifetime of triplet excitons (1/kt = 10−

It should be noted that these “cells” actually denote excitonic current; this model is simply a convenient construct for understanding the behavior of the device. Negative current densities correspond to net recombination in a respective layer. As such, the J−V characteristics shown in Figure 4b,c represent the net exciton current generated by a given species at a given chemical potential. The values of Eg were chosen at the maxima of the plots in Figure 3 when ΔG = 0.1 eV. In each of the J−V plots the contribution to the photocurrent by JDA is negligible when compared with that of the bulk layers. The Voc, and hence the efficiency, are limited by BP recombination for both the Tc and Pn hybrid cells when ΔG = 0.5 eV. At an intermediate value of ΔG = 0.3 eV, there is some recombination in all three species, and thus the effect of BP recombination on η is minimal for ΔG < 0.3 eV. (See Figure 3a.) Finally, when ΔG = 0.1 eV, the cell is completely limited by recombination in I and O. This means that any reasonable choice of σ will not alter the results for a device with ΔG = 0.1 eV because this will affect only the rate of BP recombination. As such, the proposed methodology is robust for calculating the ultimate efficiency of the system. As one departs from ultimate efficiencies, as is the case for larger values of ΔG, we must assign reasonable values to the absorptivity of the DA species. Figure 3b shows the effect of altering the value of σ for a Tc (100 nm)/inorganic cell with ΔG = 0.5 eV. A reduction in the rate of DA complex absorption, and the corresponding decrease in emission necessitated by detailed balance, increases the overall efficiency. Indeed, as the DA complex absorptivity 2302



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recombination. As a result, there is no minimum observed in the cross-section of the 2D plot in Figure 6b.

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CONCLUSIONS We have used a three-species model to calculate the semiempirical limiting efficiency of hybrid SF capable polyacene/inorganic solar cells. These results suggest that Tc is more suitable than Pn for use in a hybrid cell due to fewer triplet recombination losses. For low values of ΔG and large Φ, the limiting efficiency of a 100 nm Tc/inorganic cell exceeds the SQ limit, with a maxima very close to the band gap of bulk silicon (∼1.1 eV). In fact, using a silicon acceptor, one may make a free-energy sacrifice of ΔG = 0.35 eV and still exceed the SQ limit if nonradiative BP decay is not present. There are clear obstacles to realizing a Tc/silicon hybrid cell. In particular, band alignment, minimizing nonradiative decay at the interface yielding efficient charge separation, and efficient and selective charge transport pose practical barriers. However, we have shown that from a thermodynamic standpoint the incorporation of Tc could circumvent the SQ limit by exploiting exciton multiplication. The triplet recombination in Pn that arises from its low-lying triplet state is a significant drawback. It is also important to note that the advantages of fabricating an organic cell (roll-to-roll production, liquid processing, etc.) are negated when considering a hybrid cell because the inorganic component must also be made. So, while fairly high limiting efficiencies are calculated when compared with organic devices, the Pn/ inorganic device shows no advantage over a conventional silicon cell. That being said, the triplet lifetime can be significantly affected by the morphology of the organic layer, and perhaps the triplet decay rate considered here is a representation of what is achievable using current Pn processing techniques rather than an intrinsic material property. The “equivalent circuit” model presented here provides a simple construct to understand the effects of exciton recombination in different species on the limiting efficiency of solar cells. In the future, we hope to expand the model to include a more realistic acceptor species and incorporate the effects of carrier mobility more rigorously than by limiting the thickness of the organic layer. A detailed understanding of the

Figure 5. Efficiency of a Tc (100 nm)/inorganic hybrid device with Eg = 1.12 eV as a function of ΔG at several values of Φ.

20 ns29,31), which not only increases the rate of recombination but also limits the allowable thickness of the device, reducing kb. Indeed, from Figure 3, we note that the inclusion of a Pn organic layer actually reduces the limiting efficiency of the device when compared with an entirely inorganic cell, despite the generation of multiple excitons per absorbed photon. Figure 6a shows the effect of triplet lifetime and Pn thickness on the limiting efficiency of the device. As the thickness of the device is increased, an increased portion of the solar spectrum is available for exciton multiplication, increasing Jsc. However, this simultaneously increases the net rate of recombination in the pentacene layer, reducing Voc, resulting in an initial decrease in the limiting efficiency of the device as thickness is increased and a minima at ∼50 nm. Relatively long triplet exciton lifetimes have been observed in Tc thin films (40−200 ns18). Because the triplet state is higher in energy in Tc than in Pn, it is less populated at operating voltages, and thus less triplet recombination occurs. This gives rise to different trends in the contour in Figure 6b. As the thickness of the device is increased, the consequent increase in Jsc is able to offset decreases in Voc, which arise from triplet

Figure 6. Contour plot of the limiting efficiency as a function of the triplet lifetime and the thickness of the organic layer for a (a) Pn/inorganic cell and a (b) Tc/inorganic cell. Cross sections of the plots where 1/kt = 15 and 120 ns are shown, respectively, corresponding to the value used in all other calculations presented in this paper. In these calculations, ΔG = 0.1 eV, σ = 10−3, and the acceptor band gaps are, respectively, 0.86 and 1.13 eV. 2303



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(8) Ekins-Daukes, N. J.; Schmidt, T. W. A Molecular Approach to the Intermediate Band Solar Cell: The Symmetric Case. Appl. Phys. Lett. 2008, 93, 063507. (9) Bremner, S. P.; Levy, M. Y.; Honsberg, C. B. Analysis of Tandem Solar Cell Efficiencies Under AM1.5G Spectrum Using a Rapid Flux Calculation Method. Prog. Photovoltaics 2008, 16, 225−233. (10) Hanna, M.; Nozik, A. J. Solar Conversion Efficiency of Photovoltaic and Photoelectrolysis Cells with Carrier Multiplication Absorbers. J. Appl. Phys. 2006, 100, 074510−8. (11) Nozik, A. J. Quantum Dot Solar Cells. Physica E 2002, 14, 115− 120. (12) Shabaev, A.; Efros, A. L.; Nozik, A. J. Multiexciton Generation by a Single Photon in Nanocrystals. Nano Lett. 2006, 6, 2856−2863. (13) Hanna, M. C.; Beard, M. C.; Nozik, A. J. Effect of Solar Concentration on the Thermodynamic Power Conversion Efficiency of Quantum-Dot Solar Cells Exhibiting Multiple Exciton Generation. J. Phys. Chem. Lett. 2012, 3, 2857−2862. (14) Gabor, N. M.; Zhong, Z.; Bosnick, K.; Park, J.; McEuen, P. L. Extremely Efficient Multiple Electron-Hole Pair Generation in Carbon Nanotube Photodiodes. Science 2009, 325, 1367−1371. (15) Pijpers, J. J. H.; Ulbricht, R.; Tielrooij, K. J.; Osherov, A.; Golan, Y.; Delerue, C.; Allan, G.; Bonn, M. Assessment of CarrierMultiplication Efficiency in Bulk PbSe and PbS. Nat. Phys. 2009, 5, 811−814. (16) Trinh, M. T.; Limpens, R.; de Boer, W. D. A. M.; Schins, J. M.; Siebbeles, L. D. A.; Gregorkiewicz, T. Direct Generation of Multiple Excitons in Adjacent Silicon Nanocrystals Revealed by Induced Absorption. Nat. Photonics 2012, 6, 316−321. (17) Thorsmølle, V. K.; Averitt, R. D.; Demsar, J.; Smith, D. L.; Tretiak, S.; Martin, R. L.; Chi, X.; Crone, B. K.; Ramirez, A. P.; Taylor, A. J. Morphology Effectively Controls Singlet-Triplet Exciton Relaxation and Charge Transport in Organic Semiconductors. Phys. Rev. Lett. 2009, 102, 017401. (18) Burdett, J. J.; Muller, A. M.; Gosztola, D.; Bardeen, C. J. Excited State Dynamics in Solid and Monomeric Tetracene: The Roles of Superradiance and Exciton Fission. J. Chem. Phys. 2010, 133, 144506. (19) Burdett, J. J.; Gosztola, D.; Bardeen, C. J. The Dependence of Singlet Exciton Relaxation on Excitation Density and Temperature in Polycrystalline Tetracene Thin Films: Kinetic Evidence for a Dark Intermediate State and Implications for Singlet Fission. J. Chem. Phys. 2011, 135, 4508. (20) Chan, W. L.; Ligges, M.; Jailaubekov, A.; Kaake, L.; Miaja-Avila, L.; Zhu, X. Y. Observing the Multiexciton State in Singlet Fission and Ensuing Ultrafast Multielectron Transfer. Science 2011, 334, 1541− 1545. (21) Burdett, J. J.; Bardeen, C. J. Quantum Beats in Crystalline Tetracene Delayed Fluorescence Due to Triplet Pair Coherences Produced by Direct Singlet Fission. J. Am. Chem. Soc. 2012, 134, 8597−8607. (22) Chan, W. L.; Ligges, M.; Zhu, X. Y. The Energy Barrier in Singlet Fission Can Be Overcome Through Coherent Coupling and Entropic Gain. Nat. Chem. 2012, 4, 840−845. (23) Burdett, J. J.; Bardeen, C. J. The Dynamics of Singlet Fission in Crystalline Tetracene and Covalent Analogs. Acc. Chem. Res. 2013, 46, 1312−1320. (24) Chan, W.-L.; Berkelbach, T. C.; Provorse, M. R.; Monahan, N. R.; Tritsch, J. R.; Hybertsen, M. S.; Reichman, D. R.; Gao, J.; Zhu, X. Y. The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory. Acc. Chem. Res. 2013, 46, 1321−1329. (25) Tayebjee, M. J. Y.; Clady, R. G. C. R.; Schmidt, T. W. The Exciton Dynamics in Tetracene Thin Films. Phys. Chem. Chem. Phys. 2013, 15, 14797−14805. (26) Wilson, M. W. B.; Rao, A.; Johnson, K.; Gélinas, S.; di Pietro, R.; Clark, J.; Friend, R. H. Temperature-Independent Singlet Exciton Fission in Tetracene. J. Am. Chem. Soc. 2013, 135, 16680−16688. (27) Marciniak, H.; Fiebig, M.; Huth, M.; Schiefer, S.; Nickel, B.; Selmaier, F.; Lochbrunner, S. Ultrafast Exciton Relaxation in Microcrystalline Pentacene Films. Phys. Rev. Lett. 2007, 99.

dynamics and energetics of the BP state is also required. Encouragingly, the results presented here are a reflection of the efficiency limits of a device with currently available empirical parameters, and these values may be significantly improved as fabrication techniques are enhanced. For example, the triplet lifetime of tetracene single crystals is 58 ± 5 μs66 − two orders of magnitude greater than that reported in thermally deposited thin films.18 This suggests that the limits presented here (with the empirical values given in Table S2 of the Supporting Information) could be significantly exceeded in the future.

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ASSOCIATED CONTENT

S Supporting Information *

Summary of the reactions considered in our model is included with the empirical values that were incorporated in the calculations. Absorption spectra of Tc and Pn thin films on quartz and fitted ellipsometry data and values of the refractive index of Tc on SiO2 are also included. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +61 (0)2 9385 6053. Fax: +61 (0)2 9385 5456. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Australian National Fabrication Facility for performing the thermal deposition of the Tc and Pn thin films on quartz that are presented in the Supporting Information. A.M.S. acknowledges receipt of a Tuition Fee Scholarship from the School of Photovoltaics and Renewable Energy Engineering, UNSW. A.M.S. thanks Dr. Hamid Mehrvarz and Dr. Ajay Pandey for assisting in Tc/SiO2/Si sample preparation and Tian Zhang and Xuguang Jia for assistance in fitting the ellipsometry data. (See the Supporting Information.) G.J.C. acknowledges the support of the Australian Research Council for his Future Fellowship. This project has been supported by the Australian Government through the Australian Renewable Energy Agency (ARENA). Responsibility for the views, information, or advice expressed herein is not accepted by the Australian Government.

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REFERENCES

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