Photochemical Upconversion Theory: Importance of Triplet Energy Levels and Triplet Quenching David Jefferies,1 Timothy W. Schmidt,2 and Laszlo Frazer1 1ARC Centre of Excellence in Exciton Science, School of Chemistry, Monash University, Clayton, Victoria, Australia 2ARC 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 triplet exciton annihilation. Currently, predicting the performance of photochemical upconversion devices is challenging. We present an open source software package which takes experimental parameters as inputs and gives the figure of merit of an upconversion system, enabling theory-driven design of better solar energy devices. We incorporate the statistical distribution of triplet excitons between the sensitizer and the emitter. Using the dynamic quenching effect of the sensitizer on emitter triplet excitons, we show that the optimal sensitizer concentration can be below the sensitizer solubility limit in liquid devices. These theoretical contributions can explain, without use of heavy atom-induced triplet exciton formation or phenyl group rotation, the experimental failure of zinc octaethylporphyrin to effectively sensitize diphenylanthracene, where platinum octaethylporphyrin succeeds. Our predictions indicate a change in direction for device design that will reduce triplet exciton losses. I. INTRODUCTION determines the quantum yield of upconversion [24]. Here, we use simulations [4] to show how advanced triplet ex- Solar cells have a transparent region below their citon physics can be applied to shift the fate of excitons bandgap. The transparent region plays an important role towards annihilation and away from other decay mech- in limiting the efficiency of conventional solar cells illu- anisms. In particular, we focus on the Boltzmann dis- minated by sunlight [1, 2]. Photochemical upconversion tribution of triplets between molecules and the recently is a phenomenon which converts light a solar cell can- discovered quenching action of the sensitizer [26, 29] on not use into light that the cell can use [3{9]. The utility the triplet exciton storage in the emitter. comes from the spontaneous increase in the energy per photon. Owing to its exothermic nature, photochemical upconversion can be relatively efficient [7, 10, 12, 13]. II. OVERVIEW OF CALCULATIONS Photochemical upconversion transfers energy through a series of energy levels, which are illustrated by an en- ergy level diagram in Fig. 1. The energy levels are in We calculate the accepted figure of merit for photo- two different molecules, the sensitizer [10, 14{18] and the chemical upconversion, which is the photocurrent per emitter [19{23]. First, sunlight is absorbed by the sen- unit area caused by upconversion, under the assumption sitizer molecules. Second, the sensitizer undergoes inter- that the solar cell has perfect quantum efficiency [30]. As system crossing, which produces a triplet exciton. Third, illustrated in Fig. 2, we simulate a device consisting of a the sensitizer molecule transfers energy to an emitter solar cell, an anabathmophore layer which performs pho- molecule [24, 25]. The triplet exciton state of the emit- tochemical upconversion [12], and a Lambertian diffuse ter is relatively long-lived [22, 26], enabling energy to be reflector [4]. Our simulations use random samples from stored for conversion. Fourth, emitter excitons undergo the AM1.5G solar spectrum. For each sunlight sample, triplet annihilation. Triplet annihilation converts a pair we use random sampling to determine if the light is ab- of triplet excitons to one singlet exciton. Fifth, the emit- sorbed by the solar cell, absorbed by the sensitizer, or ter molecules in the singlet excited state produce fluores- diffusely reflected. cence. This fluorescence has a higher energy per photon It is established that, in well constructed systems, the than the light which was absorbed in the first step, so it sensitizer intersystem crossing, triplet energy transfer can be used by a solar cell. [22, 26, 31{35], and fluorescence have negligible losses. arXiv:1908.04927v1 [cond-mat.mtrl-sci] 14 Aug 2019 Photochemical upconversion cannot exceed 50% quan- Therefore, we assume they are perfectly efficient. While tum yield because triplet annihilation converts two our methods can be adapted to poorly constructed sys- triplet excitons into one singlet exciton [27]. 50% quan- tems, including low intersystem crossing rates, triplet tum yield is highly advantageous because upconversion transfer rates, and fluorescence yields, these possibilities enables use of a region of the solar spectrum where the are beyond the scope of this report. Triplet energy trans- solar cell external quantum efficiency is zero [28]. In addi- fer is further discussed in Section III. tion, the output quanta have more energy than the input The quantum yield of photochemical upconversion quanta; the energy efficiency exceeds the quantum yield. ΦUC was computed according to the accepted theory [24], In upconversion devices, the balance between desirable which incorporates the triplet exciton annihilation rate triplet exciton annihilation and other forms of triplet loss constant k2, the triplet exciton concentration [T ], and 2 FIG. 1. Photochemical upconversion energy level diagram. The system consists of sensitizer and emitter molecules. The sensitizer molecules capture light and transfer the resulting exciton to the emitter. Sn and Tn indicate the nth singlet and triplet spin energy levels, respectively. The triplet energy transfer double-ended arrows indicate that rapid triplet energy transfer achieves an equilibrium, rather than complete transfer. Figure adapted from [26]. the regular triplet loss rate constant k1: Previous work has investigated the relationship between the chemical structure and upconversion properties of k2[T ] ΦUC = (1) closely related sensitizers [17, 19, 22, 26, 37{40] and emit- 2 (k1 + k2[T ]) ters [22, 23, 41{44]. The chemical structures are illus- trated in Fig. 3 and the spectra are presented in Fig. This yield determines the quantity of fluorescence. 4. For the solar cell, we use the Tauc model of direct Using random samples from the fluorescence spectrum, bandgap absorption so that the bandgap of the solar cell we calculate the rate at which fluorescence is absorbed is a free variable [45, 46]. Details of the algorithm are in by the solar cell. The figure of merit is calculated from Section V. this rate. We also include the self-absorption and photon recycling [36], owing to both the sensitizer and the emit- ter, including diffuse reflections from the bottom surface of the anabathmophore. Self-absorption is typically small III. TRIPLET KINETICS OWING TO TRIPLET for well-designed systems. ENERGY LEVELS Our simulations use experimental solar spectral irra- diance, sensitizer absorption, emitter absorption, and The rate of energy transfer from the sensitizer to the emitter emission spectra. Therefore, they are readily emitter is much faster than the triplet decay rate of the adapted to a wide range of illumination conditions and sensitizer [22, 26, 31{35]. As a result, it is common to chemistries. For this paper, we use the sensitizer zinc discuss the decay rate of triplet excitons (excluding anni- octaethylporphyrin and the emitter diphenylanthracene. hilation) k1 as if it were the same as the triplet decay rate 3 1.2 1.8 1.6 1 Spectral irradiance (W m 1.4 0.8 1.2 1 Emitter 0.6 Sun 0.8 0.4 0.6 -2 Emission (relative) nm 0.4 -1 0.2 ) FIG. 2. An illustration of the device, including the solar 0.2 cell, light upconverting anabathmophore, and light distribut- ing diffuse reflector. The anabathmophore upconverts the 0 0 light which is not absorbed by the solar cell owing to the 300 350 400 450 500 550 600 650 700 bandgap. The sensitizer and emitter are located in the an- abathmophore. Wavelength (nm) 300000 1 ) -1 cm 250000 Solar Cell Absorptance -1 0.8 200000 0.6 150000 0.4 100000 0.2 50000 Decadic Molar Extinction (M 0 0 300 350 400 450 500 550 600 650 700 Wavelength (nm) Emitter FIG. 3. Molecular structures of sensitizer zinc octaethylpor- Sensitizer phyrin (top) and emitter 9,10-diphenylanthracene (bottom). Solar Cell FIG. 4. Example spectra, top: Solar spectral irradiance of the emitter. However, we will show that the sensitizer and diphenylanthracene emitter fluorescence (normalized to triplet decay rate can play an important role in deter- peak). Bottom: solar cell absorptance (with bandgap at mining the figure of merit, even though energy transfer 480 nm), zinc octaethylporphyrin sensitizer molar extinction, is faster than triplet decay. and emitter molar extinction. The arrow indicates the ab- In equilibrium, the distribution of triplets between the sorption peaks that sensitize upconversion. Since the solar sensitizer and the emitter is according to the Boltzmann cell is simulated as a single interface, it has a dimensionless absorptance [47]. distribution [48]. Experiments show the triplet energy transfer is the fastest rate when the emitter concentra- tion is high [22]. We assume the sensitizer and emitter triplet 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 the emitter to the sensitizer. S If k1 is the triplet decay rate constant in the sensitizer 4 ∆E (eV) known sensitizer molecules lack a suitable first excited -0.04 singlet spin state [52]. It is possible to harvest higher -0.02 excited states [53], a phenomenon which we do not sim- 0 0.02 ulate. While annihilation of triplet excitons located in 0.04 different molecules can be efficient [54], we assume that 8000 sensitizers will not have this property. Therefore, the concentration of usable triplet excitons is 7000 3 ∗ [E] 6000 [ E ] = [T ] ∆E : (3) − k T ) [S]e B + [E] -1 5000 (s 1 k ∆E is the energy lost during transfer of a triplet exciton 4000 from the sensitizer to the emitter.