Volume 1 photovoltaic solar energy 1 24 – upconversion

Volume 1 photovoltaic solar energy 1 24 – upconversion Volume 1 photovoltaic solar energy 1 24 – upconversion Volume 1 photovoltaic solar energy 1 24 – upconversion Volume 1 photovoltaic solar energy 1 24 – upconversion Volume 1 photovoltaic solar energy 1 24 – upconversion Volume 1 photovoltaic solar energy 1 24 – upconversion

1.24 Upconversion TW Schmidt and MJY Tayebjee, The University of Sydney, Sydney, NSW, Australia © 2012 Elsevier Ltd All rights reserved 1.24.1 1.24.2 1.24.3 1.24.4 1.24.5 1.24.5.1 1.24.5.2 1.24.6 1.24.6.1 1.24.6.1.1 1.24.6.1.2 1.24.6.2 1.24.6.2.1 1.24.6.2.2 1.24.6.2.3 1.24.6.2.4 1.24.6.3 1.24.7 References Further Reading Introduction Single Threshold Solar Cells Upconversion-Assisted Solar Cells: Equivalent Circuits Solar Concentration Definition of Upconversion Efficiency Monochromatic Optical Efficiency Photoelectrical Efficiency Practical Implementation Rare Earths Spectroscopy of rare-earth ions Upconversion with rare earths Upconversion with Organic Molecules Molecular photophysics Photochemical upconversion Annihilation statistics Experimentally achieved high efficiency Comparison between Organics and Rare Earths Prospects Glossary Centrosymmetric Characterized by being symmetric with respect to inversion through the center Rare earths The f-block lanthanoid elements, and in some cases scandium and yttrium Shockley–Queisser limit The limiting energy conversion efficiency of a single threshold solar cell under sun 533 534 535 538 538 539 540 540 540 540 540 541 542 543 543 543 546 546 547 548 Triplet–triplet annihilation A bimolecular reaction between two triplet states to yield one excited state and one ground state Upconversion The conversion of a stream of lower energy photons into one of a higher energy Upconvertor A device capable of upconversion 1.24.1 Introduction Upconversion (UC) is a general term used to describe the conversion of a stream of low-energy photons to a stream of higher-energy photons, the ‘up’ here referring to energy There are various ways in which this may be achieved For instance, a high-peak-power laser pulse propagating through a noncentrosymmetric medium may induce a second-order response in the polarization of the material Since the square of a sinusoidal wave may be written in terms of one of twice the frequency, second harmonic light can be emitted by the polarized medium This second harmonic generation (SHG) is widely used in laser applications [1] Where two light pulses of different frequencies interact in a similar medium to that which exhibits SHG, light at the sum and difference of the two frequencies may be produced Sum-frequency generation (SFG) is exploited in solid-state tunable lasers and also in ultrafast luminescence experiments The UC phenomena on which we will concentrate in this chapter are incoherent phenomena, described in terms of the absorption of multiple photons rather than the nonlinear response of a medium to an intense, coherent pulse of light In such a way, it is hoped that UC will become efficient under mW cm−2 illumination, comparable to unconcentrated solar radiation The recovery of subband gap light in a single threshold photovoltaic device by UC promises to increase solar energy conversion efficiencies, especially in high-band gap devices such as amorphous silicon and organic solar cells In this chapter, we revise the theory of single-junction solar cells and extend the treatment to cells enhanced with an upconvertor The limiting efficiencies are established for various cases, with several different equivalent circuits considered Following, the two leading technologies are reviewed, and their future application is considered Comprehensive Renewable Energy, Volume doi:10.1016/B978-0-08-087872-0.00130-X 533 534 Technology 1.24.2 Single Threshold Solar Cells Single threshold solar cells are photovoltaic energy conversion devices comprised of a material transparent to photons of energy below the threshold Eg, corresponding to the band gap in the case of a semiconductor Here we utilize equations based on a master equation model of the absorber, as described by Tayebjee et al [2] This differs from the usual treatment in the form of the emission rate, where stimulated emission is held to be negligible As discussed by Hirst and Ekins-Daukes [3], this approximation is found to be applicable for realistic band gaps, where the voltage of the solar cell differs from the Eg by many times, kBT The ideal current–voltage characteristic is given by the following [2]: I ¼ eẵk k expeV Eg ị=kB Tị ẵ1 where k is the rate of absorption of photons by the absorber and k′ is the rate of emission of photons The absorption rate is given by k¼ σ hc ð hc=Eg λFλ dλ ½2Š where σ is the (step-function) cross section of the absorber and Fλ⊙ is the solar spectral irradiance in Jm−2 nm−1 s−1 at wavelength λ In all that follows, we utilize the solar spectrum as defined by ASTM G173-03 [4] The rate of photon emission by the absorber is given by ð∞ σ E2 expð−ðE −Eg Þ=kB TÞdE 4π c2 ℏ3 Eg #   "     Eg Eg σ kB T ¼ 2 ỵ2 ỵ kB T kB T c k ẳ ẵ3 where emission is assumed to take place over only one hemisphere by a Lambertian surface Equation [1] is used to obtain the maximum power of a solar cell of given band gap under solar radiation This is plotted in Figure for an operating temperature of 300 K As can be seen, the maximum possible energy conversion efficiency for unconcentrated sunlight is 33.7% at a band gap of 1.34 eV This number is higher than the 32.9% derived by Brown and Green [5] who used a slightly different solar spectrum, but is identical to that found by Hanna and Nozik [6] This maximum single threshold efficiency is widely termed the Shockley–Queisser limit, for the authors of the original derivation [7] To understand this number, we should revise some of the following assumptions made in deriving eqn [1]: • All absorption takes place at a planar surface with step-function absorptivity All photons with energy above Eg are absorbed and those below Eg pass through • Carriers are thermalized to the lattice temperature, 300 K, with respect to the operating voltage, V • There is infinite carrier mobility, no shunting, and no series resistance • All recombination is radiative The contributions of the various efficiency losses have been recently reinvestigated by Hirst and Ekins-Daukes (see Chapter 1.15) [3] They elegantly plotted the contributions as a function of band gap, which is reproduced in Figure Immediately clear is that the Energy conversion efficiency 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.5 1.0 1.5 2.0 Band gap (eV) 2.5 3.0 Figure The energy conversion efficiency limit for a single threshold photovoltaic device operating at 300 K under sun (ASTM G173-03), as a function of the absorption threshold (band gap) Upconversion 535 1.0 étendue loss 0.9 EKE Energy conversion efficiency 0.8 Thermalization loss 0.7 0.6 Below Eg loss 0.5 0.4 FF 0.3 0.2 Electrical power out 0.1 0.0 0.5 0.9 1.3 1.7 2.1 2.5 2.9 3.3 Band gap, Eg (eV) 3.7 4.1 4.5 4.9 Figure Energy apportioning as a function of band gap for a cell illuminated by a 6000 K blackbody sun All energy received from the sun is accounted for EKE and FF stand for electron kinetic energy and fill factor losses, respectively Maximum efficiency is shown to be 0.31 for a band gap of 1.32 eV Adapted from Hirst LC and Ekins-Daukes NJ (2010) Progress in Photovoltaics: Research and Applications 19: 286–293 [3] major loss for cells with a band gap above about 1.3 eV is ‘below Eg loss’; that is, a large proportion of incoming energy with hν < Eg passes through the cell without being absorbed It is this loss mechanism that is addressed with intermediate band solar cells and UC, the latter being the focus of this chapter 1.24.3 Upconversion-Assisted Solar Cells: Equivalent Circuits Trupke, Green, Würfel (TGW), and co-workers considered a general upconverting device with two lower energy thresholds that conspire to upconvert to a higher energy threshold [8] This offers the greatest flexibility, with the AM1.5G spectrum affording an energy conversion efficiency exceeding 50% [9] Their highest efficiency was obtained with an upper threshold of 2.00 eV, with lower thresholds of 0.94 and 1.40 eV An intermediate thermalization of 0.34 eV enhances the efficiency considerably However, UC systems under current consideration for implementation in solar devices so from a single lower energy threshold Two excitations created at this lower threshold then interact to populate a higher level, bringing about UC As such, in the following we only consider such systems which, due to decreased flexibility, are not as efficient as multiple threshold devices such as those considered by TGW [8, 9] Indeed, the devices that follow are similar in spirit to the molecular intermediate band device put forward by Ekins-Daukes and Schmidt [10] Calculations are all performed with T = 300 K, using the AM1.5G spectrum (as defined by ASTM G173-03), except in calculations under concentrated sunlight, where only the direct component is used (AM1.5D) A commonly used approach to calculating limiting efficiencies of solar cell configurations beyond the single threshold device is to invoke the equivalent circuit A simple device that recovers the ‘below Eg loss’ as identified by Hirst and Ekins-Daukes [3] is to place two lower-band gap solar cells in series behind the higher-band gap top cell, as illustrated in Figure In this configuration, the working voltage of the top device is V and that of the lower-band gap devices is, by symmetry, V/2 The currents are drawn together and thus add The I – V characteristic is given by I ¼ eẵkb k b expeV Eb ị=kB Tị ỵ kr k r expeV=2 Er ị=kB Tị ẵ4 where the subscripts refer to the lower threshold cell (r, red) and the higher threshold cell (b, blue), respectively Eb and Er are the thresholds of the cells The kinetic parameters are given by kb ¼ kr ¼ σ hc σ 2hc ð hc=Eb ð hc=Er hc=Eb λFλ⊙ dλ ½5Š λFλ⊙ dλ ½6Š 536 Technology Ir V I Ib “red” “blue” Er V/ Eb Eb − + Er Er Figure (Left) The equivalent circuit for a simple device that recovers ‘below Eg loss’ The thresholds are Er (red) and Eb (blue) 3.0 2.8 2.6 Eb (eV) 2.4 2.2 2.0 1.8 45 1.6 40 1.4 35 1.2 1.0 0.5 0.6 0.7 0.8 0.9 1.0 Er /Eb Figure The limiting efficiency for the device illustrated in Figure 3, as a function of the high threshold, Eb, and the ratio Er/Eb The highest efficiency is over 45%, with thresholds of 1.63 and 0.96 eV and k b′ ¼ k r′ σ 4π c2 σ ¼ 2 8π c   kB T ℏ kB T ℏ 3 " 3 " Eb kB T Er kB T 2  þ2 2 Eb kB T  # þ2 #   Er ỵ2 ỵ2 kB T ẵ7 ẵ8 The emission and absorption rates of the lower-threshold cells contain an additional factor of in the denominator as a result of their being half the size of the high-threshold cell The energy conversion efficiency of this device is shown in Figure The peak is over 45% at a large band gap of about 1.63 eV For this configuration, the optimal smaller band gap is about 0.59 that of the large one, 0.96 eV The left-hand edge of the plot is equivalent to the symmetric molecular intermediate device described by Ekins-Daukes and Schmidt [10], while the right-hand edge is equivalent to Figure This is not quite equivalent to an UC-assisted solar device The difference can be shown by considering particle flux Let the absorbed photon flux in the top cell be b and that in each bottom cell be r The emission from these cells is 0 b ðV Þ and r ðV Þ It holds that 0 I ẳ eb ỵ r b V ị r V =2ịị ẵ9 Now consider the situation where the two low-band gap cells pump a light-emitting diode that illuminates the rear of a bifacial top cell of the same band gap (Figure 5), as analyzed by Trupke et al [8] The incoming upconverted flux is uc Now, I ẳ eb ỵ uc 2b V ịị ẵ10 The emission from the top cell is doubled to take account of the bifacial emission In the absence of any external input, the UC circuit likewise gives Upconversion Iuc Vuc I V “blue” Er 537 “red” – Eb + Vuc / Eb Eb Er Er Figure The equivalent circuit for an UC-assisteilation yields, ηTTA, as a function of pulse energy Adapted from Cheng YY, Khoury T, Clady RGCR, et al (2010) Physical Chemistry Chemical Physics 12: 66–71 [27] fluorescence following 525 nm excitation of the rubrene emitter Comparison of the prompt and delayed fluorescence intensity shows that annihilation efficiencies ηTTA > 0.055 can be obtained, where ηUC = ηTTAΦF, ΦF being the fluorescence yield In the presently described experiment, the fluorescence quantum yield of the emitter is an internal reference [27] The best results indicate an annihilation efficiency of ηTTA ~ 0.16 However, these integrated decays include the many emitter triplet states that decay by first-order, unimolecular means As such, while the value ηTTA > 0.15 represents an experimentally attained annihilation efficiency, it neither does represent the actual upper limit for this system nor the highest instantaneously attainable annihilation efficiency To determine these factors, kinetic data were analyzed in terms of mixed first- and second-order kinetics [28] The delayed fluorescence solely originates from S1 emitters produced by TTA As such, its intensity IF is proportional to the square of the concentration of emitter triplets, [3M*]2, in the solution Since TTA occurs on a many microsecond timescale, it can be treated independently of the emitter triplet concentration buildup [38] The buildup illustrated on the left of Figure 15 is one Upconversion 545 1.0 1.0 [3M∗]t/[3M∗]0 [3M∗]t/[3M∗]0 0.8 0.6 0.4 kinetic fit 0.8 0.6 expon fit 0.4 0.2 0.0 10 Time (μs) 100 0.2 0.0 50 100 150 200 Time (μs) 250 300 Figure 16 The normalized triplet decay kinetics as inferred from the square root of the delayed fluorescence signal As can be seen, eqn [22] produces a fit to the data superior to a single exponential, revealing the importance of the second-order TTA kinetics, particularly at early times where the concentration of triplets is highest Redrawn from Cheng YY, Fückel B, Khoury T, et al (2010) Kinetic analysis of photochemical upconversion by triplet-triplet annihilation: Beyond any statistical limit Journal of Physical Chemistry Letters 1(12): 1795–1799 of the slowest observed, and is shown to illustrate the kinetics The highest efficiency experiments are associated with a much faster submicrosecond buildup For the time dependence of the concentration of emitter triplet molecules [3M*]t, one can write [34, 36, 39] pffiffiffiffiffiffiffiffiffi d IF tị dẵ3 M t ẵ21 ẳ k1 ½3 Mà Š t − k2 ½3 Mà Št2 dt dt where the first-order decay component k1 is a combination of the intrinsic phosphorescent and NRD of the triplet emitter, and other quasi first-order quenching processes The second-order component, k2, originates from the triplet quenching of the TTA process including all bimolecular pathways according to Figure 14 The analytical solution to [21] is given by [36] ½3 Mà Š t ẳ expk1 tị ẵ3 M ½22Š with β = α/(k1 + α) and α being the product of the TTA rate constant and the initial triplet concentration α = k2 [3M*]0 As pointed out by Bachilo and Weisman [36], β represents the initial fraction of decay that occurs in second-order TTA kinetics Figure 16 shows the excellent fit of eqn [21] to the date, from which we obtain β and thus can determine the proportions of first- and second-order decay, respectively, f1 and f2 By integrating eqn [21] ln ị ẵ23 f ẳ ẵ3 M k1 ½3 Mà Š t dt ¼ β and likewise f ẳ ẵ3 M k2 ẵ3 M t dt ẳ ln ị ẵ24 Hence, one can calculate the efficiency of bimolecular quenching of triplets to yield S1 emitter states ηconv = ηTTA/f2 The results of Cheng et al [28] show that, over a range of observed annihilation efficiencies, ηTTA and the proportion of triplets undergoing second-order decay, f2, is proportional The kinetic experiments reveal that emitter triplet states that undergo second-order decay produce S1 states with roughly 30% of the quantum efficiency of direct excitation at 525 nm This value of ηconv = 0.3 not only exceeds 0.055 but also comfortably exceeds the 0.2 limit imposed where the triplet channel is in operation Moreover, the initial annihilation efficiency observed immediately following the laser pulse exceeds 0.2 within the experimental uncertainty The quantity ηconv is directly related to the underlying TTA mechanism as depicted in Figure 14 Thus, in the current system, the conversion efficiency of encounter complexes to excited singlets exceeds the limit calculated assuming that the triplet channel is open, ηconv = 20% The operation of the triplet channel crucially depends on the energies of the product molecules It is known that the energy of the rubrene T2 state exceeds the energy of its S1 state by 87 meV in toluene [40] Indeed, for rubrene the energies of the T1 and T2 states are about 1.15 and 2.38 eV above the ground state, respectively [40] That is, the formation of a T2 rubrene state from two T1 states is endothermic by 74 meV, while the S1 state is exothermic by 12 meV The high ηconv values found in the above analysis are attributed to the inaccessibility of the T2 state Hence, it is proposed that emitter molecules with still larger T2 – S1 energy gaps than rubrene can yield ηconv values of up to 0.5 Indeed, when the excitation energy of the emitter T2 state exceeds double that of its T1 state, the amount of radiationless decay via the triplet channel is reduced 546 1.24.6.3 Technology Comparison between Organics and Rare Earths In efficiency experiments by Cheng et al [27, 28], ηUC = 0.1 was achieved using μJ pulses at kHz As the authors explain, since the triplets decay within about 30 μs, a power 30 times this (30 mW) would be required to reach an equivalent quasi-steady-state triplet concentration Since the spot illuminated was about mm2, this corresponds to W cm−2, or 300 suns if the light were distributed across the absorption band of the sensitizer The normalized efficiency for this process is therefore η UC ¼ 0:033 cm2 W − Despite the high reported efficiencies, this system is not as efficient as Fischer and co-workers’ phosphor, which achieved 0.27 cm2 W−1 [11] Baluschev et al report an external quantum yield of 0.04 with excitation intensity of