Photochemical Upconversion Theory: Importance of Triplet Energy Levels and Triplet Quenching
PPhotochemical Upconversion Theory: Importance of Triplet Energy Levels andTriplet Quenching
David Jefferies, Timothy W. Schmidt, and Laszlo Frazer ARC Centre of Excellence in Exciton Science, School of Chemistry, Monash University, Clayton, Victoria, Australia ARC Centre of Excellence in Exciton Science, School of Chemistry, UNSW Sydney, Sydney, NSW, Australia (Dated: August 15, 2019)Photochemical upconversion is a promising way to boost the efficiency of solar cells using tripletexciton annihilation. Currently, predicting the performance of photochemical upconversion devicesis challenging. We present an open source software package which takes experimental parametersas inputs and gives the figure of merit of an upconversion system, enabling theory-driven design ofbetter solar energy devices. We incorporate the statistical distribution of triplet excitons betweenthe sensitizer and the emitter. Using the dynamic quenching effect of the sensitizer on emittertriplet excitons, we show that the optimal sensitizer concentration can be below the sensitizersolubility limit in liquid devices. These theoretical contributions can explain, without use of heavyatom-induced triplet exciton formation or phenyl group rotation, the experimental failure of zincoctaethylporphyrin to effectively sensitize diphenylanthracene, where platinum octaethylporphyrinsucceeds. Our predictions indicate a change in direction for device design that will reduce tripletexciton losses.
I. INTRODUCTION
Solar cells have a transparent region below theirbandgap. The transparent region plays an important rolein limiting the efficiency of conventional solar cells illu-minated by sunlight [1, 2]. Photochemical upconversionis a phenomenon which converts light a solar cell can-not use into light that the cell can use [3–9]. The utilitycomes from the spontaneous increase in the energy perphoton. Owing to its exothermic nature, photochemicalupconversion can be relatively efficient [7, 10, 12, 13].Photochemical upconversion transfers energy througha series of energy levels, which are illustrated by an en-ergy level diagram in Fig. 1. The energy levels are intwo different molecules, the sensitizer [10, 14–18] and theemitter [19–23]. First, sunlight is absorbed by the sen-sitizer molecules. Second, the sensitizer undergoes inter-system crossing, which produces a triplet exciton. Third,the sensitizer molecule transfers energy to an emittermolecule [24, 25]. The triplet exciton state of the emit-ter is relatively long-lived [22, 26], enabling energy to bestored for conversion. Fourth, emitter excitons undergotriplet annihilation. Triplet annihilation converts a pairof triplet excitons to one singlet exciton. Fifth, the emit-ter molecules in the singlet excited state produce fluores-cence. This fluorescence has a higher energy per photonthan the light which was absorbed in the first step, so itcan be used by a solar cell.Photochemical upconversion cannot exceed 50% quan-tum yield because triplet annihilation converts twotriplet excitons into one singlet exciton [27]. 50% quan-tum yield is highly advantageous because upconversionenables use of a region of the solar spectrum where thesolar cell external quantum efficiency is zero [28]. In addi-tion, the output quanta have more energy than the inputquanta; the energy efficiency exceeds the quantum yield.In upconversion devices, the balance between desirabletriplet exciton annihilation and other forms of triplet loss determines the quantum yield of upconversion [24]. Here,we use simulations [4] to show how advanced triplet ex-citon physics can be applied to shift the fate of excitonstowards annihilation and away from other decay mech-anisms. In particular, we focus on the Boltzmann dis-tribution of triplets between molecules and the recentlydiscovered quenching action of the sensitizer [26, 29] onthe triplet exciton storage in the emitter.
II. OVERVIEW OF CALCULATIONS
We calculate the accepted figure of merit for photo-chemical upconversion, which is the photocurrent perunit area caused by upconversion, under the assumptionthat the solar cell has perfect quantum efficiency [30]. Asillustrated in Fig. 2, we simulate a device consisting of asolar cell, an anabathmophore layer which performs pho-tochemical upconversion [12], and a Lambertian diffusereflector [4]. Our simulations use random samples fromthe AM1.5G solar spectrum. For each sunlight sample,we use random sampling to determine if the light is ab-sorbed by the solar cell, absorbed by the sensitizer, ordiffusely reflected.It is established that, in well constructed systems, thesensitizer intersystem crossing, triplet energy transfer[22, 26, 31–35], and fluorescence have negligible losses.Therefore, we assume they are perfectly efficient. Whileour methods can be adapted to poorly constructed sys-tems, including low intersystem crossing rates, triplettransfer rates, and fluorescence yields, these possibilitiesare beyond the scope of this report. Triplet energy trans-fer is further discussed in Section III.The quantum yield of photochemical upconversionΦ UC was computed according to the accepted theory [24],which incorporates the triplet exciton annihilation rateconstant k , the triplet exciton concentration [ T ], and a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug FIG. 1. Photochemical upconversion energy level diagram. The system consists of sensitizer and emitter molecules. Thesensitizer molecules capture light and transfer the resulting exciton to the emitter. S n and T n indicate the n th singlet andtriplet spin energy levels, respectively. The triplet energy transfer double-ended arrows indicate that rapid triplet energytransfer achieves an equilibrium, rather than complete transfer. Figure adapted from [26]. the regular triplet loss rate constant k :Φ UC = k [ T ]2 ( k + k [ T ]) (1)This yield determines the quantity of fluorescence.Using random samples from the fluorescence spectrum,we calculate the rate at which fluorescence is absorbedby the solar cell. The figure of merit is calculated fromthis rate. We also include the self-absorption and photonrecycling [36], owing to both the sensitizer and the emit-ter, including diffuse reflections from the bottom surfaceof the anabathmophore. Self-absorption is typically smallfor well-designed systems.Our simulations use experimental solar spectral irra-diance, sensitizer absorption, emitter absorption, andemitter emission spectra. Therefore, they are readilyadapted to a wide range of illumination conditions andchemistries. For this paper, we use the sensitizer zincoctaethylporphyrin and the emitter diphenylanthracene. Previous work has investigated the relationship betweenthe chemical structure and upconversion properties ofclosely related sensitizers [17, 19, 22, 26, 37–40] and emit-ters [22, 23, 41–44]. The chemical structures are illus-trated in Fig. 3 and the spectra are presented in Fig.4. For the solar cell, we use the Tauc model of directbandgap absorption so that the bandgap of the solar cellis a free variable [45, 46]. Details of the algorithm are inSection V. III. TRIPLET KINETICS OWING TO TRIPLETENERGY LEVELS
The rate of energy transfer from the sensitizer to theemitter is much faster than the triplet decay rate of thesensitizer [22, 26, 31–35]. As a result, it is common todiscuss the decay rate of triplet excitons (excluding anni-hilation) k as if it were the same as the triplet decay rate FIG. 2. An illustration of the device, including the solarcell, light upconverting anabathmophore, and light distribut-ing diffuse reflector. The anabathmophore upconverts thelight which is not absorbed by the solar cell owing to thebandgap. The sensitizer and emitter are located in the an-abathmophore.FIG. 3. Molecular structures of sensitizer zinc octaethylpor-phyrin (top) and emitter 9,10-diphenylanthracene (bottom). of the emitter. However, we will show that the sensitizertriplet decay rate can play an important role in deter-mining the figure of merit, even though energy transferis faster than triplet decay.In equilibrium, the distribution of triplets between thesensitizer and the emitter is according to the Boltzmanndistribution [48]. Experiments show the triplet energytransfer is the fastest rate when the emitter concentra-tion is high [22]. We assume the sensitizer and emittertriplet exciton populations are in equilibrium. The equi-librium is illustrated in Fig. 1 by the triplet energy trans-fer arrows from the sensitizer to the emitter and from theemitter to the sensitizer.If k S is the triplet decay rate constant in the sensitizer E m i ss i on (r e l a t i v e ) S pe c t r a l i rr ad i an c e ( W m - n m - ) Wavelength (nm)EmitterSun 0 50000 100000 150000 200000 250000 300000 D e c ad i c M o l a r E x t i n c t i on ( M - c m - ) S o l a r C e ll A b s o r p t an c e Wavelength (nm)EmitterSensitizerSolar Cell
FIG. 4. Example spectra, top: Solar spectral irradianceand diphenylanthracene emitter fluorescence (normalized topeak). Bottom: solar cell absorptance (with bandgap at480 nm), zinc octaethylporphyrin sensitizer molar extinction,and emitter molar extinction. The arrow indicates the ab-sorption peaks that sensitize upconversion. Since the solarcell is simulated as a single interface, it has a dimensionlessabsorptance [47]. k ( s - ) Temperature (K) ∆ E (eV)-0.04-0.0200.020.04
FIG. 5. k as a function of temperature for several valuesof ∆ E according to the statistical distribution of triplet exci-tons. Smaller k leads to a better Φ UC (Equation 1). Betterupconversion systems are exothermic (∆ E (cid:29) k B T ), so their k increases when heated by sunlight. Despite the detrimen-tal effect of heating, exothermic systems remain superior. Forthis figure, we use [ S ] = [ E ] = 1 m m . [22], k E is the triplet decay rate constant in the emitter[26], [ S ] is the sensitizer concentration, [ E ] is the emitterconcentration, k B is the Boltzmann constant, T is thetemperature, and ∆ E is the difference between the sen-sitizer triplet energy level and the emitter triplet energylevel, then the overall triplet decay rate is k = [ S ] k S e − ∆ EkBT + [ E ] k E [ S ] e − ∆ EkBT + [ E ] . (2)The rate constants are listed in Table I [22, 26].As shown in Fig. 5, if ∆ E >
0, then as temperatureincreases, k increases, which leads to reduced quantumyield. This result emphasizes that the interplay betweenenvironmental conditions, such as heating by the sun andcooling by the wind [49], with the exothermic nature ofupconversion must be considered when designing an en-ergy conversion system.In order to generate triplet excitons, the sensitizermust have a high intersystem crossing rate. As a re-sult, the sensitizer’s triplet decay rate constant k S is rel-atively large [22, 50, 51]. However, device designers havethe freedom to select an emitter molecule with a smalltriplet decay rate constant k E . Equations 1 and 2 showthat a large k S decreases the quantum yield.In addition, a pair of triplet states located in the sen-sitizer typically cannot produce upconversion because known sensitizer molecules lack a suitable first excitedsinglet spin state [52]. It is possible to harvest higherexcited states [53], a phenomenon which we do not sim-ulate. While annihilation of triplet excitons located indifferent molecules can be efficient [54], we assume thatsensitizers will not have this property. Therefore, theconcentration of usable triplet excitons is[ E ∗ ] = [ T ] [ E ][ S ] e − ∆ EkBT + [ E ] . (3)∆ E is the energy lost during transfer of a triplet excitonfrom the sensitizer to the emitter. It is indicated in Fig.1. The quantum yield is more precisely written asΦ UC = k [ E ∗ ]2 ( k + k [ E ∗ ]) . (4)In many cases, ∆ E much is larger than the thermalenergy. Then it is not necessary to consider k S . Anexcessively large ∆ E is detrimental to energy efficiencybecause it reduces the upward shift in the energy of theupconverted photons. If ∆ E is enhanced by shifting thesensitizer triplet energy upwards, then the portion of thesolar spectrum which could be captured by the sensitizerwill decrease. If ∆ E is enhanced by shifting the emittertriplet energy downwards, it may be necessary to alsoshift the fluorescence energy downwards to keep the sys-tem exothermic. This leads to the necessity of selectinga solar cell with a smaller bandgap. The solar cell willthen block more light from reaching the sensitizer.If ∆ E is negative, it is possible to increase the pho-ton energy by more than a factor of two [48]. However,more triplet excitons will be distributed in the sensitizermolecules, which makes the quantum yield small.Fig. 6 shows the simulated figure of merit, currentdensity, as a function of emitter concentration. For lowvalues of ∆ E , a high ratio of emitter molecules to sen-sitizer molecules is needed to produce upconversion. Anabundance of emitter molecules ensures some triplet exci-tons are distributed to the emitter. If ∆ E is large, thenthe Boltzmann distribution ensures triplet excitons arelocated in the emitter molecules even if those moleculesare scarce. We do not include the additional decrease inthe figure of merit which occurs at very low emitter con-centrations because the triplet excitons do not reach theequilibrium distribution before decaying, or the emitterbecomes saturated with excitations.Fig. 7 shows that upconversion fails to producephotocurrent when all the triplet decay occurs inthe sensitizer because ∆ E (cid:28)
0. Smaller sensitizertriplet decay k S makes the figure of merit more sen-sitive to ∆ E near ∆ E = 0. The zinc octaethylpor-phyrin/diphenylanthracene system is in this region. Both∆ E and k S can contribute to controlling triplet excitonloss. TABLE I. Physical rate constants assumed in the simulation.Name Symbol Compound Value ReferenceTriplet decay of sensitizer k S Zinc Octaethylporphyrin 8550 s − [22]Triplet decay of emitter, [ S ] = 0 k Diphenylanthracene 2000 s − [26]Quenching k q Both 4 . × m − s − [26]8 . × − cm s − Triplet annihilation k Diphenylanthracene 2 . × m − s − [22, 41]4 . × − cm s − - - - - F i gu r e o f M e r i t ( m A c m - ) [ s o li d ] D e c a y R a t e ( s - ) [ da s hed ] Emitter Concentration (M) ∆ E (eV)-0.04-0.0200.020.04
FIG. 6. Triplet decay rate constant k (dashed curves) andfigure of merit (solid curves) as a function of emitter concen-tration [ E ] for several values of ∆ E . The sensitizer concen-tration is 1 m m , temperature is 300 K and anabathmophorethickness is 0 . E or [ E ] are not big enough, up-conversion becomes inefficient because triplets decay in thesensitizer. A large triplet energy transfer rate cannot over-come this decay. In addition, triplet excitons in the sensitizerare not available for upconversion. IV. DYNAMIC TRIPLET QUENCHINGCAUSED BY SENSITIZER
Sensitizer concentration determines the excitation den-sity [55]. A high excitation density produces efficient an-nihilation because the triplet concentration is in the nu-merator of Equation 4. Therefore, one would expect thatthe highest achievable sensitizer concentration will pro-duce the highest possible upconversion figure of merit.We have recently shown that k E , the triplet decay inthe emitter, is dependent on sensitizer concentration [26].Here we show the resulting impact on the figure of meritand device design. - . - . - . - . . . . . F i gu r e o f M e r i t ( m A c m - ) ∆ E (eV)k (s -1 )50001000015000 FIG. 7. Figure of merit as a function of difference in tripletenergy levels ∆ E , for a variety of sensitizer triplet decay rateconstants k S and anabathmophore thickness of 0 . k S are mitigated if ∆ E (cid:29) k B T . Platinumoctaethylporphyrin with diphenylanthracene exhibits this ad-vantage. Zinc octaethylporphyrin does not. However, thespectral shift achieved by upconversion decreases as ∆ E in-creases. For this figure, we use [ S ] = [ E ] = 1 m m . A. Model
The decay rate of triplets in the emitter in the ab-sence of sensitizer and excluding annihilation is k , whichtypically ranges from 10 to 10 s − [26]. The rate con-stant quantifying emitter triplet quenching by the sensi-tizer, k q , is typically less than 5 × m − s − [26]. Totaltriplet losses in the emitter are k E = k + k q [ S ] . (5)The solubility limit on [ S ] is above 1 m m [26]. B. Quenching reduces figure of merit
Fig. 8 shows the calculated figure of merit withand without the quenching constant. Here, we assumestrongly exothermic triplet energy transfer. The devicethickness, which is important to achieving a high triplet (a) Excluding quenching constant - - - - Sensitizer Concentration (M)10 -4 -3 -2 -1 T h i ck ne ss ( c m ) F i gu r e o f M e r i t ( m A c m - ) (b) Including quenching constant - - - - Sensitizer Concentration (M)10 -4 -3 -2 -1 T h i ck ne ss ( c m ) F i gu r e o f M e r i t ( m A c m - ) FIG. 8. Figure of merit as a function of sensitizer concentra-tion and thickness, (a) without and (b) with the quenchingconstant k q = 4 . × m − s − [26]. White indicates zerofigure of merit. We assume ∆ E = 0 . concentration [4], is also included as a variable. Withoutthe quenching constant, the figure of merit increases withconcentration. The optimal device thickness decreaseswith concentration, as the absorption length decreases.With the inclusion of a quenching constant, a maximumfigure of merit exists below 10 − m sensitizer concentra-tion.The relationship between the figure of merit and theemitter concentration is dramatically changed by theinclusion of the quenching constant. In our model,the triplet excitons in the sensitizer are protected fromconcentration-dependent quenching. As a result, in Fig.9, the triplet decay rate k increases with emitter con-centration. Unlike the results in Fig. 6, which ex-clude quenching, the figure of merit has a maximumwith respect to emitter concentration when quenchingis included. We omit the concentration quenching oftriplet excitons within triplet sensitizers, which can be - - - - F i gu r e o f M e r i t ( m A c m - ) [ s o li d ] D e c a y R a t e ( s - ) [ da s hed ] Emitter Concentration (M) ∆ E (eV)-0.04-0.0200.020.04
FIG. 9. Triplet decay rate constant k (dashed curves) andfigure of merit (solid curves) as a function of emitter concen-tration for several values of ∆ E with the quenching constantof 4 . × m − s − . The sensitizer concentration is 1 m m ,temperature of 300 K and thickness is 0 . ∼ m − s − , and may be important [56, 57]. C. Interplay of sensitizer quenching andBoltzmann statistics
The quenching effect of Equation 5 was experimentallydemonstrated in situations where ∆ E (cid:29) k B T [26]. Asthe sensitizer concentration increased, k increased in alinear fashion. If this analysis is performed on a systemthat is not strongly exothermic, then the contributionof Boltzmann statistics from Equation 2 will cause thequenching constant to be overestimated. For zinc oc-taethylporphyrin and diphenylanthracene, the reported∆ E is 0.02 eV [22, 58–61]. In our view, the experimentaland theoretical uncertainty on this value is enough thatthe sign is uncertain.In Fig. 10, we reanalyze zinc octaethylporphyrin anddiphenylanthracene data from Ref. [26]. We comparethe prediction of Equation 5 with the combined predic-tion of Equations 2 and 5. Equation 2 increases thenumber of free parameters, so its inclusion must improvethe accuracy of the model. While the experimental un-certainty is large enough that neither model can be re-jected, it does seem that ∆ E is not large enough to keepall the triplet excitons in the emitter. We suggest that∆ E = − . ∆ E (eV) k q ( m − s − ) J UC ( µ A / cm ) − .
02 4 . × . − .
02 0 370 . . . × . . . J UC under different assumptionsabout ∆ E and k q . The sensitizer concentration and devicethickness are optimized separately for each calculation. Withno k q the sensitizer concentration is constrained to 1 m m bysolubility. × × × × × - × - × - × - × - × - × - × - k ( s - ) Concentration (M)MeasurementBoltzmann modelQuenching model
FIG. 10. Experimental triplet decay rate in the emitter as afunction of sensitizer concentration. The Linear QuenchingModel 5 is compared against the Boltzmann Model includingEquations 2 and 5. The downward curvature of the datasuggests some triplet excitons remain in the sensitizer aftertriplet energy transfer reaches equilibrium. Data from [26]. endothermic.If the mechanism giving rise to k q were an externalheavy atom effect, then it should increase with atomicnumber Z . However, the opposite was observed [26].Zinc-containing sensitizer ( Z = 30) had the highest k q ,but palladium ( Z = 46) and platinum ( Z = 78) were sim-ilar to each other. Inclusion of ∆ E in the theory opensup the possibility that zinc-containing sensitizer does notreally have a higher k q . Both parameters can explain theexperimental increase in the emitter triplet exciton de-cay. Future measurements over a range of emitter con-centrations will eliminate this ambiguity. ∆ E and k q pro-vide theoretical explanations of the relationship betweenatomic number and upconversion performance that donot require models based on phenyl group rotation [22].Table II shows that both ∆ E and k q can change the fig-ure of merit, but that k q is more important. x x x x x F i gu r e o f M e r i t ( m A c m - ) S en s i t i z e r C on c en t r a t i on ( mM ) k q (M -1 s -1 )(a)Figure of MeritConcentration x x x x x T h i ck ne ss ( c m ) k q (M -1 s -1 )(b) FIG. 11. (a) Maximum figure of merit and optimal sensi-tizer concentration as a function of quenching constant. (b)Optimal thickness as a function of quenching constant. Weassume ∆ E = 0 and [ E ] = 10 − m . The width of the purplecurve indicates estimated Monte Carlo error. D. Selecting the optimal design
Fig. 11 gives the optimized figure of merit as a func-tion of quenching constant. This shows the harmful effectof the sensitizer quenching the emitter on device perfor-mance. In addition, k q makes it necessary to increase thedevice thickness and decrease the sensitizer concentrationto generate the most current from an upconversion de-vice. The quantity of sensitizer used may be importantto cost effectiveness. V. FIGURE OF MERIT ALGORITHMA. Sampling Sunlight
The device was modeled as an infinite plane with thesunlight incident perpendicular to the surface. The so-lar spectral irradiance was stochastically sampled 10 times using the cumulative distribution function of theAM1.5G spectrum. To determine if a sample was ab-sorbed into the solar cell, the Tauc model of directbandgap semiconductors [45, 46] was scaled so there wasa 99% probability of the solar cell absorbing the sample0 . B. Light Transmission and Scattering
The interior of the anabathmophore was divided into10 bins arranged vertically. Bins were used to modelthe inhomogeneous distribution of excitons within theanabathmophore.If a sample reached the anabathmophore, spline inter-polation was used to calculate the absolute value of thecorresponding sensitizer and emitter molar absorptivitiesfrom the experimental spectra shown in Fig. 4. Theemitter absorption spectrum was filtered so any molarextinction below 1000 m − cm − was set to 0 m − cm − to mitigate instrument noise. This reduces nonphysi-cal anti-Stokes shifts. Using the sum of the molar ab-sorptivities, the distance travelled by the sample wasstochastically determined from the Beer-Lambert Law[55]. The concentrations and molar absorptivities wereused to stochastically assign the sample to be absorbedby the sensitizer or emitter. From the distance travelled,the bin the sample was absorbed into was determined.For each bin, the number of samples absorbed and reab-sorbed by the sensitizer and the emitter were recorded.If the distance travelled exceeded the thickness of theanabathmophore, Lambertian reflection was simulatedand the distance travelled was recalculated. The up-ward component of the distance travelled was used todetermine the bin. If the distance travelled upward wentbeyond the region occupied by the anabathmophore, thesample was not absorbed by the anabathmophore anddid not contribute to the figure of merit. C. Sampling Fluorescence
Samples absorbed by the sensitizer and emitter weremodelled separately and had different fluorescence yields.For sensitizer excitation, we assumed the singlet yield oftriplet annihilation was perfect [27, 62]. The quantumyield Φ UC was calculated using Equation (4) with valuesfrom Table I. Temperature was assumed to be 300 K. Since triplet energy transfer is rapid [22], triplet energytransfer was assumed to be in equilibrium. The tripletconcentration was calculated using [24][ T ] = − k + (cid:112) k + 4 k φ k [ S ]2 k , (6)where k φ is the excitation rate computed under the usualassumption that the irradiance of the sun is 1 kW m − .The upconversion yield Φ UC was different for each binbecause k φ was different. Samples absorbed by theemitter had perfect fluorescence yield, corresponding toan assumption of perfect fluorescence quantum yield.The quantum yield assumption also applied to convertedtriplet excitons.To simulate emission, fluorescence samples generatedaccording to the quantum yield were each stochasticallyassigned a wavelength. The wavelength was determinedusing the cumulative distribution function of the exper-imental emitter fluorescence spectrum in Fig. 4. Thefluorescence was propagated in a random direction fromthe middle of the bin according to the Beer-Lambert Law.If the direction of travel was downward, the sample couldundergo Lambertian reflection. The vertical componentof distance travelled determined the bin where the sam-ple was reabsorbed. If the sample escaped from the top ofthe anabathmophore, the Tauc model was used again todetermine if the solar cell absorbed the sample [45, 46]. Ifit did, then the sample contributed to the figure of merit.Reabsorption and emission were recalculated five timesto account for photon recycling [36]. During each cycle,the number of samples reabsorbed in each bin and Φ UC were recalculated. Typically, reabsorption was small.Here, we chose a sensitizer which was mostly transparentto fluorescence; our previous results suggest that molarextinction is more important to the figure of merit thanreduced reabsorption, so it is important to account forphoton recycling [4]. D. Figure of Merit
The total radiant exposure entering the system wascalculated by summing the photon energies of samplesstochastically generated from the solar spectrum. Theradiant exposure was then divided by the standard solarirradiance, 1 kW m − , to find the simulation duration t .The figure of merit, current density, was ent (7)where e is the fundamental charge and n is the area den-sity of emitter fluorescence samples absorbed by the solarcell. VI. CONCLUSIONS
Previously, we argued that 0 . − is a mean-ingful figure of merit [4]. Reaching this goal requires im-provement. The efficiency of photochemical upconversionrelies on the exothermic nature of each process involvedin the steps of light conversion. Here, we relate a de-crease in the potential energy which drives triplet energytransfer to a reduction in the figure of merit. We quanti-tatively demonstrated that the photocurrent is improvedwhen the energy loss ∆ E (cid:29) k B T . The quenching oftriplets located in the emitter but caused by the sensitizeralso inhibits the figure of merit, even when exothermicoperation is successfully achieved. In the future, upcon-version can be advanced by creating sensitizer/emitterpairs which have a large ∆ E and a small quenching k q .The poor performance of zinc octaethylporphyrin withdiphenylanthracene, compared to alternate sensitizerspaired with diphenylanthracene, can be explained by thequenching action of the sensitizer on the emitter andthe alignment of triplet energy levels. These propertiescannot be determined by only measuring triplet energytransfer rate constants. The Boltzmann distribution oftriplet excitons inhibits upconversion even when energytransfer is efficient.Our results highlight the value of measuring triplet de-cay rates as a function of concentrations. This type ofexperiment provides information about exciton conver-sion processes which lead to major changes in the figureof merit. When ∆ E ≈
0, theoretical and phosphores-cence methods may not be precise enough. In addition,a pulsed experiment showing a high triplet energy trans-fer rate is not sufficient to show complete triplet transferin equilibrium. If the Boltzmann factor is in doubt, thenboth the sensitizer and emitter concentration should be explored to find ∆ E and k q .The simulation program, which is available from http://laszlofrazer.com , can readily be used to pre-dict the relative merits of different compounds with re-spect to their usefulness as sensitizers and emitters. Thisis particularly useful for the search for sensitizers whichhave absorption spectra that efficiently capture sunlight.In addition, different solar cell bandgaps, illuminationconditions, and rate constants can be conveniently mod-eled. The manual is included as supplemental material[63].We have shown that, using knowledge of the sensitizerand emitter physics, the energy conversion performanceof complete devices can be simulated. These simula-tions enable prediction of the best device design with-out requiring construction of many devices with differentmolecular concentrations and device geometries. ACKNOWLEDGMENTS
This work was supported by the Australian Re-search Council Centre of Excellence in Exciton Science(CE170100026). D. J. acknowledges a ResearchFirst fel-lowship from Monash University. This research was un-dertaken with the assistance of resources and servicesfrom the National Computational Infrastructure (NCI),which is supported by the Australian Government. Wethank Elham Gholizadeh and Rowan MacQueen for ex-perimental spectra used in this work. [1] W. Shockley and H. J. Queisser, Detailed balance limitof efficiency of p-n junction solar cells, Journal of AppliedPhysics , 510 (1961).[2] L. C. Hirst and N. J. Ekins-Daukes, Fundamental lossesin solar cells, Progress in Photovoltaics: Research andApplications , 286 (2011).[3] J. Pedrini and A. Monguzzi, Recent advances in theapplication triplet–triplet annihilation-based photon up-conversion systems to solar technologies, Journal of Pho-tonics for Energy , 022005 (2017).[4] L. Frazer, J. K. Gallaher, and T. 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