Volume 1 photovoltaic solar energy 1 25 – downconversion

Volume 1 photovoltaic solar energy 1 25 – downconversion Volume 1 photovoltaic solar energy 1 25 – downconversion Volume 1 photovoltaic solar energy 1 25 – downconversion Volume 1 photovoltaic solar energy 1 25 – downconversion Volume 1 photovoltaic solar energy 1 25 – downconversion Volume 1 photovoltaic solar energy 1 25 – downconversion

1.25 Downconversion MJY Tayebjee and TW Schmidt, The University of Sydney, Sydney, NSW, Australia G Conibeer, University of New South Wales, Sydney, NSW, Australia © 2012 Elsevier Ltd All rights reserved 1.25.1 1.25.2 1.25.3 1.25.3.1 1.25.3.2 1.25.3.3 1.25.4 References Introduction Equivalent Circuits Practical Applications QC in Rare-Earth Materials MEG in Semiconductor Nanostructures SF in Organic Materials Prospects Glossary Carrier multiplication The generation of two low-energy charge carriers from one high-energy charge carrier Downconversion The conversion of a high-energy photon into two or more low-energy photons Downconverter A device capable of downconversion Exciton fission A general term for the generation of two or more low-energy excitons from one high-energy exciton 549 550 554 555 557 559 560 560 Multiple exciton generation Exciton fission within a semiconductor nanostructured system Quantum cutting Exciton fission within a rare-earth ion Shockley–Queisser limit The limiting energy conversion efficiency of a single threshold solar cell under one sun Singlet fission Exciton fission within an organic molecular system The initial exciton is in a singlet state, which then undergoes fission into two correlated triplet states 1.25.1 Introduction As explained in Chapter 1.24.2, and in detail in Chapter 1.14, one of the principal efficiency losses of all single threshold solar cells is that the energy absorbed in excess of the threshold is converted to heat This mechanism accounts for the greater part of energy losses in solar cells with lower thresholds such as crystalline silicon A strategy to counter this loss mechanism is to absorb photons well in excess of the threshold, and reradiate this energy with two photons at or above threshold In this way, the current of the cell is increased with no penalty to operating voltage This process is known as downconversion In this chapter we will consider the case where a photon with energy greater than an upper threshold, E > Eb, is converted into two or more photons with energy greater than a lower threshold, E > Er These photons are optically coupled to a single threshold solar cell (STSC), allowing for a reduction in the thermalization losses In practice, this may be achieved via exciton fission (EF) processes: multiple exciton generation (MEG), singlet fission (SF), or quantum cutting (QC) (We will adhere to popular definitions EF will be used a general term for the generation of two or more low energy excitons from one high energy exciton The term MEG will be reserved for inorganic semiconductor nanostructures; SF will refer to an organic molecule in an excited singlet state and a molecule in the ground state evolving into two molecules in lower energy triplet states; QC will refer to rare-earth ion systems that undergo EF.) In this chapter, we also consider carrier multiplication (CM), where, instead of multiple photon generation, an absorber directly creates multiple charge carriers from a single absorbed photon Apart from increasing the efficiency of a solar cell, one might wish to protect a solar cell from damaging effects of higher energy photons Organic cells are the only current solar technology which are constructed from materials that are abundant enough to scale up power generation to terawatt scales [1] They, however, suffer from photostability issues A downconverter placed at the front of these devices would play the dual role of increasing the photon flux and reduce degradation due to the absorption of high-energy photons Furthermore, compared to tandem devices, which require current matching [2], downconversion and CM solar cells not suffer from this restriction Before we proceed, be careful to note the difference between downconversion and downshifting Downconversion does not conserve exciton number Downshifting, on the other hand, is a linear process that alters the illumination spectrum such that its overlap with a device’s incident photon conversion efficiency (IPCE) curve is maximized In this chapter, we establish the efficiency limits of downconversion-assisted STSCs and solar cells with absorbers that undergo CM Different practical approaches, imple­ mentations, and developments within the field will then be discussed Comprehensive Renewable Energy, Volume doi:10.1016/B978-0-08-087872-0.00129-3 549 550 Technology 1.25.2 Equivalent Circuits The details of the efficiency limits of STSCs have been covered briefly in Chapter 1.24.2, and Reference Further details of the following derivation are given in Reference In keeping with the preceding chapter, we will use the STSC current–voltage   characteristic [5] I eVc Er ị ẵ1 ¼ kr − kr exp e kB T for a cell operating at a voltage Vc, where e, kB, and T = 300 K are the elementary charge, Boltzmann’s constant, and operating temperature, respectively The rate of absorption, kr = kS(Er, ∞, 1), from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO), with energy difference Er, is kS E1 ; E2 ; ị ẳ hc hc ð = E1 λFλ⊙ dλ ½2Š hc = E2 where σ is the step-function cross section of the absorber and Fλ⊙ is the solar spectrum irradiance in J m−2 nm−1 s−1 at a wavelength λ Planck’s constant and the speed of light in a vacuum are denoted h and c, respectively This rate is countered by the rate of emission kr′ into a medium with refractive index n, where   ð∞ −ðE − E1 Þ σn2 kA E1 ; n; ị ẳ 2 E2 exp kB T 4π c E1 " #      σn2 kB T E1 E1 ¼ 2 ỵ2 ỵ2 kB T kB T c σn kB T ≈ 2 E21 ½3Š 4π c ħ The rate of stimulated emission can be ignored given the low irradiance and the fact that Em ≫ kBT [3] As such, the lower order terms in E1 can also be omitted We will consider three devices and establish their energy conversion efficiency limits Figure shows their equivalent circuits and architectures Device A consists of two photocells with respective band gaps of Er and Eb The CM process is provided by a direct current (DC) transformer (Buck converter) that doubles the current and halves the voltage In practice, of course, the process would be provided by the absorber within a single photocell Devices B and C are luminescent downconverters which have the same equivalent circuits These are optically coupled to an STSC The difference arises from the overall device architecture: B is placed at the rear of the STSC, whereas C is placed at the front As we shall see, this gives rise to different absorption spectra In A, we must match the outgoing current of the cell, r, with twice the Buck converter’s incoming current:   eVc Er ị IA ẳ kr − kr exp e   kB T  ½4Š 2eVc − Eb ¼ kb − kb exp kB T When considering B and C, the operating voltage of the downconverters, Vd, differs from that of the STSC, Vc The current passing through the downconverters is determined by equating the currents passing through each photocell:   IC 2eVd Eb ị IB ẳ ẳ kb − kb exp e e kB T  ½5Š ð2eVd Er ị kr ỵ V c ị=2ị ẳ kr exp kB T The photon flux passing from the STSC to the downconverter is denoted (Vc) and is equally partitioned between the two identical light-emitting diodes (LEDs) The forms of the downconverted flux, kd, are shown in Table for B and C The I–V characteristic of the STSC assisted by luminescent downconversion is given by   ðeVc Er ị Ic ẵ6 ẳ kc ỵ kd − kc exp e kB T We are now in a position to calculate the efficiency of each device Pmax η ¼ ð∞ Fλ⊙ dλ where Pmax is the maximum power point obtained from IAVc for A and IcVc for B and C ½7Š Downconversion 2Vc 551 2Vd r Vc Vc Vd r b r b r A B/C Eb |3 STSC Eb |3 abs abs EF EF |2 |2 Er em abs abs |1 |1 A B/C ‘red’ ‘blue’ ‘blue’ STSC ‘blue’ ‘red’ ‘red’ ‘red’ b r - + b ‘red’ - + r A r r ‘red’ b STSC/B + r C/STSC Figure (Top) The equivalent circuits of A, B, and C The square component in A represents a DC step-down transformer, such as a Buck converter with an operating efficiency of 100% that links a higher threshold photocell, b, to a lower threshold photocell, r The luminescent circuits, B and C, consist of a high-threshold photocell, b, and two lower threshold LEDs, r, which drive an STSC (Middle) Generalized energy-level diagram of the three devices The labels |1>, |2>, and |3> will be used to describe systems undergoing EF Absorptive and emissive transitions are, respectively, labeled abs and em (Bottom) Schematics of the device architectures Table Parameter values for the three devices considered Parameter A B C kr kS(Er, Eb, 1) kS(Er, Eb, 1/2) kA(Er, 1, 1) kA(Er, nr, 1/2)  pffiffiffiffiffiffiffiffiffiffiffiffiffi  kA Er ; ỵ nr2 ; 1=2 kb kS(Eb, ∞, 1) kS(Eb, ∞, 1) kS(Eb, ∞, 1) kb′ kA(Eb, 1, 1) kA(Eb, nr, 1)  pffiffiffiffiffiffiffiffiffiffiffiffiffi  kA Eb ; ỵ nr2 ; kd 2kr exp ((eVd – Er)/kBT) F(2kr′ exp ((eVd – Er)/kBT) + kb′ exp ((2eVd – Eb)/kBT)) kc kS(Er, Eb, 1) k′c kA kA(Er, nr, 1)  Fkc′ exp ((eVc – Er)/kBT) ′ kr  p  Er ; ỵ nr2 ; kc′ exp ((eVc – Er)/kBT) Note that an internal refractive index, nr, of 3.6 was chosen for B and C in order to compare with Reference The fraction, f = nr2/(1 + nr2), denotes the fraction of photon flux that is directed into the device by virtue of its internal refractive index 552 Technology (a) 2.8 (b) 2.8 32 32 2.4 Eb /Er Eb /Er 2.4 2.0 40 44 28 32 36 24 1.6 36 1.2 0.8 2.0 1.0 1.2 1.4 1.6 1.8 1.2 0.8 2.0 (d) 24 28 32 Energy conversion efficiency (%) 2.4 Eb /Er 28 24 1.0 1.2 36 40 1.6 24 1.0 1.4 1.6 1.8 2.0 Band gap (eV) (c) 2.8 1.2 0.8 32 1.6 Band gap (eV) 2.0 36 40 1.2 1.4 1.6 Band gap (eV) 1.8 2.0 50 45 A 40 B C 35 30 SQ 25 20 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Band gap (eV) 44 Figure Contour plots of the limiting energy conversion efficiencies as a function of band gap and the ratio Eb/Er for (a) A, (b) B, and (c) C under the AM1.5G spectrum [7] (d) The maximum energy conversion efficiencies for each device The SQ is shown for reference Figure summarizes the energy conversion efficiencies of devices A, B, and C, under AM1.5G illumination, and their global maxima (45.9%, 42.9%, and 38.9%, respectively) are shown in Table The three devices have energy conversion efficiency limits that are significantly greater than the the Shockley–Quesser limit of 33.7% under AM1.5G illumination In all three cases, the greatest Table The maximum efficiency for devices operating under 6000 K black body and AM1.5G illumination Device A B C STSC 6000 K black body AM1.5G [7] η (%) η (%) 42.6 (39.6) 40.3 (39.6) 37.9 (36.8) 31.1 Eb/Er 1.73 1.93 1.89 - 45.9 (41.9) 42.9 (41.9) 40.5 (38.9) 33.7 Eb/Er 1.72 1.91 1.87 - In all cases, the optimal values of Er were 1.05 and 0.95 eV, for a device operating under black body and AM1.5G illumination, respectively The values for an unassisted STSC are shown for comparison, with optimal band gaps of 1.30 and 1.34 eV, respectively The values in parentheses denote the cases where the restriction Eb/Er = 2.0 has been applied Again, nr = 3.6 was used for B and C Downconversion 553 conversion efficiency is obtained when the ratio Eb/Er < We can equivalently say that the sum of the enthalpies of the two low-energy excitons is greater than the enthalpy of the high-energy exciton As such, the process is endothermic and must be driven by entropy This is best understood from a statistical thermodynamic viewpoint Given a system of N absorbers and the vector p containing , |2〉 , and |3〉(see middle of Figure 1), the total number of normalized state occupancies p1, p2, and p3, respectively, for |1〉 microstates in the system is     N N N pị ẳ ẵ8 p1 N p2 N p3 N   n where the definition of the binomial coefficient, ¼ n!=ẵk!n kị!, is used The EF process involves a change in the number of k microstates: f pị ẳ f ðpÞ − Ωi ðpÞ         N N N N N N ¼ − p2 N ỵ p3 N1 p1 N p2 N p3 N p1 N−1 ½9Š The entropy generated in the EF process is ΔfS = kb ln (Δf Ω):   p1 Np3 NðN − p2 NÞðN − p2 N−1Þ f Spị ẳ kB ln N p1 N ỵ 1ịp2 N ỵ 1ịp2 N ỵ 1ịN p3 N þ 1Þ As N→ ∞ ; collecting all the terms in N ;   p1 p2 ðp2 −1 Þ ¼ kB ln p2 ð1− p1 Þð1− p3 ị ẵ10 Examining the above equation, we can see that as p1 or p3 approaches unity, the process becomes infinitely entropically favorable This is also shown in Figure 3, where the value of ΔfS is plotted on a ternary diagram as a function of p Horizontal lines correspond to iso-p2 values, which are symmetric over the occupancies of p1 and p3 EF corresponds to movement vertically up the diagram As such, entropy generation is maximized at the point (1/2, 0, 1/2) At the point (1/3, 1/3, 1/3), the EF process is at an entropic equilibrium (ΔfS = 0) Note that in the devices of interest the very large majority of the carrier population is in |1>, that is, the bottom right corner of the diagram Of course, the component values of p and therefore ΔfS depend on the operating voltage of the device The voltage of the device is a measure of the free energy (chemical potential) within each state of the system, thus, under steady-state conditions in A: ZA   eVc − Er p2 ¼ exp kB T ZA   2eVc − Eb p3 ¼ exp kB T ZA p1 ¼ 0.2 −0.2 ½12Š ½13Š 0.8 −0.1 0.6 p2 p3 0.4 ½11Š 0.6 0.4 0.0 0.8 0.2 0.1 0.2 0.4 0.6 0.8 p1 Figure Ternary contour plot of TΔfS in eV as a function of p EF (fusion) corresponds to moving vertically up (down) the plot Since |TΔfS| → ∞ as any component of p approaches unity, the vertices of the plot have been truncated Here T=300 K 554 Technology (a) 2.8 2.6 2.4 0.3 0.6 0.0 −0.9 −0.6 0.0 0.3 2.2 0.3 2.0 0.6 1.8 0.9 Eb /Er Eb /Er 2.6 0.0 2.4 2.2 (b) 2.8 −0.6 1.8 −0.3 −0.6 −0.9 1.2 1.6 2.0 1.6 1.5 1.4 1.4 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Voltage (eV) Voltage (eV) Figure The value of (a) TΔfS and (b) ΔfG in eV as a function of voltage and Eb/Er for A, B, and C, where Er = eV The white lines from left to right correspond to the operating voltage (at the maximum power point) of A, C, and B The partition function of the entire system is  ZA ¼     2eVc Eb eVc Er ỵ exp þ exp kB T kB T ½14Š We can establish similar equations for B and C: p1 ¼ Z   eVd − Er exp Z kB T   2eVd − Eb p3 ¼ exp Z kB T      eVd − Er 2eVd Eb Z ẳ ỵ exp ỵ exp kB T kB T p2 ẳ ẵ15 ẵ16 ẵ17 ẵ18 Figure shows the changes in free energy, ΔfG, and entropy, ΔfS, associated with the EF process for A, B, and C as a function of operating voltage and Eb/Er, where Er = eV The white lines from right to left show the voltages of A, C, and B when operating at the maximum power point We note that the ratio Eb/Er can be lowered without lowering the operating voltage since EF becomes more entropically favorable As such, EF is an endothermic process This effectively means that a greater portion of the solar spectrum can be absorbed into states that undergo EF, increasing the photocurrent without penalizing the device voltage This result is clarified in Figure 4(b), where isoenergetic contours are almost parallel to operating voltages In practice, this is probably only realizable in organic chromophores, since endothermic SF has been observed as a dominant relaxation pathway (cf Section 1.25.3.3) 1.25.3 Practical Applications A downconverter must be placed in front of a standard cell and can boost current by converting a UV photon to more than one photon just above the band gap of the solar cell, thus boosting current However, the downconverter does require that more lower energy photons are emitted than high-energy photons absorbed, that is, its quantum yield (QY) must be greater than Hence, there must be at least as many photons emitted at the lower energy as are absorbed at the higher, or else the DC layer will decrease the number of photons absorbed by the cell In fact, due to the partial transmission of photons from a low refractive index in air to a higher one in the DC layer, the QY must be greater than in order not to be parasitic – a QY of about for a refractive index of 3.6 – similar to that for a Si solar cell [8] DC devices have been attempted experimentally but so far an overall QY > has not been achieved, resulting in a downshifting of energies in, for example, Si nanocrystals [9] or porous silicon [10], which nonetheless can give a useful enhancement in spectral response at short wavelengths for some materials, due to better absorption of the downshifted photons This effective narrowing of the bandwidth of the absorbed spectrum allows the solar cell design to be optimized with respect to absorption coefficient as a function of wavelength, junction depth, and surface recombination [11] Similarly, lumines­ cent downshifting layers based on luminescent dyes have been demonstrated to boost the short wavelength response of CdTe cells in which short wavelengths are usually attenuated in the CdS window layer [12] True downconversion or QC, as first suggested by Downconversion 555 Dexter [13], requires absorption of a short wavelength photon and reemission of at least two photons of about twice the wavelength In turn, this requires an appropriately located intermediate energy halfway between the excited state and the ground state Most work has focused on lanthanide materials because of their varied and discrete energy levels 1.25.3.1 QC in Rare-Earth Materials The unique properties of rare-earth ions lend themselves to QC applications A recent review by Zhang and Huang [14] detailed rare-earth ion QC phosphors, and the development of these systems has largely been inspired by applications in the lighting industry or in electronic displays Nevertheless, in light of their rich energy-level structure, the promise of rare-earth ion systems for use in downconverters has been recognized and remains an active area of research [15] The spectroscopy of rare-earth ions was discussed briefly in Chapter 1.24.6.1 Radiative electronic transitions in rare-earth ions are ‘quasi’-atomic due to the tight binding of 4f electrons Unlike molecular systems, electronic states are denoted by term symbols of the form 2S+1LJ Here the spin multiplicity is S = eu/2, where eu is the number of unpaired electrons with the same spin The total orbital quantum number L ¼ ∑enu¼1 jmℓ j, where mℓ is the magnetic orbital quantum number of a unpaired electron; in the case of rare-earth ions mℓ = − 3, − 2, − 1, 0, 1, 2, 3, as the highest energy electrons exist in the 4f orbital By convention, the value of L is denoted with a letter (S = 0, P = 1, D = 2, F = 3, G = 4, H = 5, I = …) The total angular momentum quantum number takes values in the range |L − S| ≤ J ≤ L + S in integer steps, where J = L + S corresponds to the lowest energy state There are several possible mechanisms for QC, which have been discussed by Wegh et al [16], and are summarized in Figure Rare-earth ions exist as pairs of sensitizers and emitters The trivalent ytterbium cation (Yb3+) was suggested as an emitter in QC systems in 1957 by Dexter [13] This is particularly applicable to solar cells since the Yb3+ $ 1.24 eV 2F5/2 → 2F7/2 transition could be used in conjunction with a crystalline silicon (Er $ 1.1 eV) STSC Luminescent QYs greater than unity were actually achieved in a Tb3+–Yb3+ couple in 2005 [17] Several other trivalent cationic rare-earth couples have been suggested, and they are summarized in Figure Praseodymium (Pr3+) is a good choice because of its widely dispersed energy levels well matched for photon cutting [27] (see Figure 7) The 3P2, 1I0, and P0 levels at between 440 and 490 nm can absorb blue photons which can then radiatively recombine via the 1G4 level at 1010 nm – at just greater than twice this wavelength, thus emitting two photons at just above the silicon band gap, although nonradiative recombination via the other levels at longer wavelengths than 1G4 is also likely Experiments indicating such photon cutting have been carried out on Pr3+ embedded in various phosphors [28, 29], and also for other lanthanide-doped materials [25, 30] Transition metals with their partially screened ‘d’ shell electrons also have partially discrete levels and work on photon cutting in transition metal-doped materials has also been carried out [31, 32] An alternative approach is to absorb a short wavelength photon high up in the conduction band of various semiconductors There is then the possibility of an impact ionization event (i.e., reverse Auger recombination) in which the high-energy electron excites an additional electron to the conduction band, thus creating two or more electron–hole pairs at the band gap energy [32] Luminescent recombination of these electron–hole pairs is then usually enhanced by choice of an appropriate doping level within the band gap and an increased number of photons emitted The QY of such impact ionization depends on the energy of the initial photon and is reduced by nonradiative thermalization and recombination QYs greater than have been achieved in some E I II III IV Eb 1 Er 1 S&E E S E S E S E Figure The mechanisms of QC with rare-earth ion sensitizers (S) and emitters (E) Scheme I shows QC within a single ion II shows a two-step energy transfer process, where part energy from an excited sensitizer is transferred to an emitter by cross-relaxation The emitter returns to the ground state through radiative recombination The sensitizer then transfers the remainder to a second emitter, which also radiatively returns to the ground state III and IV both show QC with only one energy step, that is, the sensitizer also emits one of the lower energy photons Redrawn and adapted from Wegh RT, Donker H, Oskam KD, and Meijerink A (1999) Visible quantum cutting in LiGdF4:Eu3+ through downconversion Science 283(5402): 663–666 [16] 556 Technology S0 40 D7/2 S 9/2 38 G9/2 36 7/2 D4 34 32 F3 D3 P0 D5/2 7/2 L8 M10 P3/2 K13/2 5/2 7/2 P3/2 30 D1/2 28 F2 G3 3/2 D2 26 P3/2 D2 24 D5/2 P1/2 D3 G5 D2 G3 G4 G 11/2 3/2 9/2 P0 D2 5/2 16 H11/2 F9/2 S3/2 2 K6 G5/2 F5 F9/2 F2 F1/2 7/2 I4 3/2 I9/2 5/2 10 S3/2 S2 3/2 4 S2 7/2 F11/2 H5/2 G4 9/2 9/2 F5/2 11/2 7/2 7/2 I8 F4 I15/2 11/2 3/2 H15/2 13/2 7 13/2 H6 11/2 1.1 eV 11/2 13/2 4 F7/2 9/2 11/2 5 7/2 F0 F6 13/2 H5 9/2 5/2 1/2 G4 F7/2 H11/2 F5 5/2 12 5/2 3 H9/2 F3/2 F3/2 H4 14 K7 D4 G7/2 7/2 G7/2 H9/2 G11/2 F1 K F9/2 82 G9/2 2 18 G11/2 5 I6 20 G4 D2 L8 P2 1 9/2 K7 3 22 G7/2 K15/2 P0 G5 L10 D3 H6 5 Energy (×103 cm−1) P2 I6 P 1 I 2 F5/2 H4 Ce Pr Nd Pm I9/2 I4 H5/2 Sm Eu Gd Tb F0 S F6 H15/2 Dy Ho I8 I15/2 Er Tm H6 F7/2 Yb Figure Dieke diagram showing the excited-state energies of trivalent rare-earth cations Excited states that can be used for downconversion are shown in color: blue and red, respectively, represent the upper and lower excited states involved in QC We consider rare earth couples where the emitter is Yb3+ in the 2F7/2 state To date, this includes coupling with Pr3+ [15, 18, 19], Nd3+ [20], Tb3+ [18, 21, 22], Er3+ [23, 24], or Tm3+ [18] sensitizers An extension of this diagram for energies greater than 40  103 cm− has been provided by Wegh et al [25] Reproduced and adapted from Dieke GH and Crosswhite HM (1963) The spectra of the doubly and triply ionized rare earths Applied Optics 2(7): 675–686 [26] Downconversion 20 P2 I0 P0 557 440 nm 490 nm D2 Energy (�103 cm−1) 15 10 G4 1010 nm 3 F4 F3 H6 H5 H4 Pr3+ 3+ Figure Energy levels for Pr based on data from Reference 27 and extracted detail from Figure Also shown is the possible photon-cutting mechanism for a photon absorbed around 450 nm and emitted as two photons via the 1G4 level materials, usually based on wide band gap oxides (at least eV) doped with lanthanide or transition metal atoms, but with incident photon energies of at least twice or thrice the band gap [30, 31] QYs up to about 1.20–1.40 have also been achieved with the relatively narrow gap (3.4 eV) ZnS doped with Zn, Cu, or Ag, but only with photons in excess of 20 eV [32] However, at the incident photon energies required for this impact ionization, there are almost no photons in the solar spectrum Also, the efficiency of the impact ionization mechanism is very low Hence, these are not good materials for downconversion for solar cells But, in some materials based on quantum dot (QD) arrays, this efficiency can be increased dramatically such that several excitons can be generated from one incident high-energy photon 1.25.3.2 MEG in Semiconductor Nanostructures MEG is a process whereby a high-energy electron–hole pair in a nanostructured inorganic semiconductor undergoes impact ionization, yielding two lower energy excitons (see Figure 8) Quantum confinement is a key concept for EF, since bulk semiconductors usually require Eb/Er = – due to (1) the conservation of carrier momentum and (2) extremely short hot carrier lifetimes as a result of rapid electron–phonon interactions [33–36] Nanostructured materials, on the other hand, lend themselves to EF because • • • • electron–hole pairs are correlated, giving rise to excitons rather than free carriers [2]; exciton cooling is slowed due to the discretization of electronic states [37]; momentum is no longer a good quantum number, and thus does not need to be conserved [37]; and Auger processes (such as EF) are enhanced due to the increased Coulomb interaction between electrons and holes [2] In the previous section, we suggested that EF could proceed even if Eb/Er < However, theoretical calculations of PbSe, CdSe, and InAs structures with quantum confinement suggest that MEG will only be appreciably observed when Eb/Er ≥ 2.2 [38, 39] In practice, time-resolved spectroscopy of lead sulfide, selenide, and telluride QDs has shown MEG thresholds in the range Eb/Er = 2.7–3.0 [40–43] More precisely, the efficiency of MEG has been defined in terms of the QY at a given photon energy, hν [44]:  QY ẳ  hv EHPM Er ẵ19 558 Technology E Eb MEG Er Figure The creation of two excitons via MEG MEG will occur in QD systems, where a band structure describes electronic states Closed and open circles represent electrons in the conduction band and holes in the valence band, respectively Table The MEG efficiency for a number of bulk QD and single-walled carbon nanotube (SWCNT) samples, as analyzed by Beard [45] Sample ηEHPM References Bulk Ge Bulk Si Bulk PbSe Bulk PbS PbSe QDs InP QDs SWCNTs 0.3 0.4 0.2 0.3 0.4 0.9 0.7 [46] [47] [48] [48] [49, 50] [51] [52] The quantity ηEHPM is the electron–hole pair multiplication efficiency and is defined as the minimum energy required to produce an electron–hole pair, divided by the actual energy required Very recently, Beard [45] quantified ηEHPM for a number of experimental results, which are tabulated in Table Much higher values are obtained in confined structures, suggesting that quantum confinement is indeed a promising method for the promotion of EF It should be noted, however, that there is still significant disagreement among researchers about the QY that can be obtained from such systems For instance, some report a QY ≫ in PbS and PbSe, while other investigations suggest that QY < 1.25 (see Reference 44 and references therein) These differences have chiefly been attributed to surface effects or charge delocalization, and a better understanding of these will stimulate further research into the field [45] For a MEG material to be used directly as a DC with an STSC, a few conditions are required The MEG QDs would need to have an appropriate band gap to illuminate a solar cell – for a Si STSC, Si QDs would be appropriate, and MEG has been shown in well-passivated Si QDs [41] But, much more challenging would be the need for a high luminescent efficiency of the multiple excitons such that there would be a net QY greater than unity However, it is not yet clear whether such a luminescent efficiency from these materials is feasible The currently observable rate of MEG and subsequent Auger decay processes back to a single exciton at about 200 ps [41, 45], and the long radiative lifetimes of these materials, at about 10 ns, would make efficient multiple photon emission unlikely An alternative approach is to directly incorporate the MEG QDs in a solar cell in order to boost the current in the device through direct CM Several attempts have been made to this, which until recently have not successfully shown an increase in current However, very recent results have shown a PbSe QD solar cell device with QYs greater than for 3.5 eV illumination [53] This very significant result demonstrates the feasibility of the approach and further significant improvement is likely to follow Downconversion 1.25.3.3 559 SF in Organic Materials SF can occur in molecular crystals, thin films, aggregates, dimers, or polymers An excited single-state organic chromophore undergoes fission with a nearby chromophore in the ground state giving rise to two triplet chromophores, as shown in Figure These triplets may either undergo phosphorescence or spatial separation and be directly injected into the external circuit A key advantage of using molecular systems is that triplet states are generally long lived due to the spin-forbidden transition to the ground singlet state It is important to note, however, that the SF process does not violate spin conservation: the two triplets produced will be correlated such that their tensor product is a singlet state Since phosphorence is a slow spin-forbidden transition, these organic molecules are more suited to CM than downconversion since the yields would be too low Alternatively, the system could be altered by the addition of a highly phosphorescent species, as shown in Figure 10 Sensitizers that undergo SF transfer their excited state energy to emitter molecules via triplet energy transfer (TET) In practice, the TET process would be exothermic; however, entropy could also be exploited by having a large emitter­ to-sensitizer ratio Unfortunately, to date, phosphorescent yields have not exceeded 50% - any gain from SF would be squandered SF first appeared in peer-reviewed literature in 1965, when Singh and coworkers [54] suggested that it occurred in anthracene Further evidence of SF in other polyacene structures, including anthracene [55, 56], tetracene [56–61], and pentacene [59, 62–65], has also been shown Investigations into other conjugated oligomer and polymer systems have also shown evidence of SF (see, for instance, the review by Smith and Michl [66]) Examining the body of literature, SF was initially avidly studied in the 1970s, before interest lapsed We have experienced renewed interest over the last decade in light of the development of novel, efficient, photovoltaic devices Many authors have introduced their studies with the requirements for efficient SF, and invariably state the restriction Eb ≥ 2Er, in spite of some of the earliest observations of SF occurring in systems where this was not the case (for instance, SF in anthracene is highly endothermic [55]) The fact that SF is nevertheless a dominant relaxation process cannot be explained by thermal activation alone A far more viable explanation is that entropy is acting as a driving force for the process, and the simple derivation given in Chapter 1.25.2 displays this Very recently, Jadhav and coworkers [67] have demonstrated a working solar cell containing the singlet fissile organic compound, tetracene (Tc) The Tc/copper phtalocyanine/C60 device displays SF with an efficiency of 71 Ỉ 18% E Eb S1 S1 Er T1 T1 S0 S0 Figure The creation of two excitons via SF SF proceeds in molecular systems where electronic states are described by molecular orbitals The first process (1) represents the absorption transition S1 ← S0, and the second (2) represents SF E S1 Eb S1 T1 Er T1 T1 S0 3 S0 Emitter T1 S0 Sensitizers S0 Emitter Figure 10 SF within sensitizer molecules, followed by phosphorescence of emitter molecules The first process (1) represents the absorption transition S1 ← S0, and the second (2) represents SF Process (3) is TET which leaves emitters in an excited triplet state from which they return to the ground state by phosphorescing (4) 560 Technology While there is an extremely wide body of experimental studies with evidence of SF in organic materials, a fundamental and universal understanding of the mechanism is still an active area of research In their review, Smith and Michl [66] presented quantum mechanical formalisms for SF by direct coupling and through a charge transfer state Greyson et al [68, 69] proposed SF mechanisms using density matrix theory and density functional theory Zimmerman et al [70] used ab initio molecular orbital calculations to show that SF proceeds in pentacene via a dark state 1.25.4 Prospects Chapter 1.24 dealt with the prospects of upconversion One of the major obstacles for the realization of a working upconverter is the fact that it is an unfavourable process from an entropic standpoint An enthalpic sacrifice is required in order for upconversion to be exergonic By contrast, downconversion and carrier multiplication involves the creation of two (quasi-) particles from one (quasi-) particle: an entropically favourable process That is, an exergonic reaction proceeds even where there is a gain in enthalpy Therefore downconversion may prove to be more effective than upconversion at circumventing the energy conversion efficiency limits associated with conventional photovoltaic devices A further advantage of a luminescent downconverter is that it can be added to the front face of existing photovoltaic installations This is especially advantageous when they are used to enhance an organic device as these often suffer from degradation due to photo-activated reactions with high energy photons A luminescent downconverter can play the dual role of decreasing the number of such photons, while increasing the photon number, and therefore the current of the device While downconversion technology is still currently in its infancy, the sheer extent of research can allow us to be optimistic about the realization of these devices in the near future References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] Grätzel M (2009) Recent advances in sensitized mesoscopic solar cells Accounts of Chemical Research 42(11): 1788–1798 Nozik AJ (2010) Nanoscience and nanostructures for photovoltaics and solar fuels Nano Letters 10(8): 2735–2741 Hirst LC and Ekins-Daukes NJ (2010) Fundamental losses in solar cells Progress in Photovoltaics: Research and Applications 19(3): 286–293 Tayebjee MJY and Schmidt TW (submitted) Entropy recycling in exciton fission solar cells Physical Review Letters Tayebjee MJY, Hirst LC, Ekins-Daukes NJ, and Schmidt TW (2010) The efficiency limit of solar cells with molecular absorbers: A master equation approach Journal of Applied Physics 108(11): 124506 Trupke T, Green MA, and Würfel P (2002) Improving solar cell efficiencies by down-conversion of high-energy photons Journal of Applied Physics 92(3): 1668–1674 ASTM (2008) Astm Standard G173 Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37° Tilted Surface ASTM G173 West Conshohocken, PA: ASTM Würfel P (2005) The Physics of Solar Cells Weinheim, Germany: Wiley González-Diaz B, Guerrero-Lemus R, Haro-González P, et al (2006) Down-conversion properties of luminescent silicon nanostructures formed and passivated in HNO3-based solutions Thin Solid Films 511: 473–477 Švrcˇek V, Slaoui A, and Muller JC (2004) Silicon nanocrystals as light converter for solar cells Thin Solid Films 451: 384–388 Strümpel C, McCann M, Beaucarne G, et al (2007) Modifying the solar spectrum to enhance silicon solar cell efficiency – An overview of available materials Solar Energy Materials and Solar Cells 91(4): 238–249 Klampaftis E, Ross D, McIntosh KR, and Richards BS (2009) Enhancing the performance of solar cells via luminescent down-shifting of the incident spectrum: A review Solar Energy Materials and Solar Cells 93(8): 1182–1194 Dexter DL (1957) Possibility of luminescent quantum yields greater than unity Physical Review 108(3): 630–633 Zhang QY and Huang XY (2010) Recent progress in quantum cutting phosphors Progress in Materials Science 55(5): 353–427 van der Ende BM, Aarts L, and Meijerink A (2009) Near-infrared quantum cutting for photovoltaics Advanced Materials 21(30): 3073–3077 Wegh RT, Donker H, Oskam KD, and Meijerink A (1999) Visible quantum cutting in LiGdF4:Eu3+ through downconversion Science 283(5402): 663–666 Vergeer P, Vlugt TJH, Kox MHF, et al (2005) Quantum cutting by cooperative energy transfer in YbxY1 −xPO4:Tb3+ Physical Review B 71(1): 014119 Zhang QY, Yang GF, and Jiang ZH (2007) Cooperative downconversion in GdAl3(BO3)4:RE3+,Yb3+ (RE = Pr, Tb, and Tm) Applied Physics Letters 91(5): 051903 Aarts L, van der Ende B, Reid MF, and Meijerink A (2010) Downconversion for solar cells in YF3:Pr3+, Yb3+ Spectroscopy Letters 43(5): 373–381 Meijer JM, Aarts L, van der Ende BM, et al (2010) Downconversion for solar cells in YF3:Nd3+, Yb3+ Physical Review B 81(3): 035107 Zhang QY, Yang CH, Jiang ZH, and Ji XH (2007) Concentration-dependent near-infrared quantum cutting in GdBO3:Tb3+,Yb3+ nanophosphors Applied Physics Letters 90(6): 061914 Zhang QY, Yang CH, and Pan YX (2007) Cooperative quantum cutting in one-dimensional (YbxGd1 −x)Al3(BO3)4:Tb3+ nanorods Applied Physics Letters 90(2): 021107 Aarts L, van der Ende BM, and Meijerink A (2009) Downconversion for solar cells in NaYF4:Er, Yb Journal of Applied Physics 106(2): 023522 Aarts L, Jaeqx S, van der Ende BM, and Meijerink A (2011) Downconversion for the Er3+, Yb3+ couple in KPb2Cl5-a low-phonon frequency host Journal of Luminescence 131(4): 608–613 Wegh RT, Meijerink A, Lamminmaki RJ, and Holsa J (2000) Extending Dieke’s diagram Journal of Luminescence 87–9: 1002–1004 Dieke GH and Crosswhite HM (1963) The spectra of the doubly and triply ionized rare earths Applied Optics 2(7): 675–686 Dieke GH (1968) Spectra and Energy Levels of Rare Earth Ions in Crystals New York: Wiley Meijerink A, Wegh R, Vergeer P, and Vlugt T (2006) Photon management with lanthanides Optical Materials 28(6–7): 575–581 Piper WW, Ham FS, and Deluca JA (1974) YF-Pr3+, a phosphor with a quantum efficiency greater than one for emission of visible light Bulletin of the American Physical Society 19(3): 257–258 Michels JJ, Huskens J, and Reinhoudt DN (2002) Noncovalent binding of sensitizers for lanthanide(III) luminescence in an EDTA-bis(beta-cyclodextrin) ligand Journal of the American Chemical Society 124(9): 2056–2064 Ilmas ER and Savikhina TI (1970) Investigation of luminescence excitation processes in some oxygen-dominated compounds by to 21 eV photons Journal of Luminescence 1–2: 702–715 Berkowitz JK and Olsen JA (1991) Investigation of luminescent materials under ultraviolet excitation energies from to 25 eV Journal of Luminescence 50(2): 111–121 Downconversion [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] 561 Werner JH, Kolodinski S, and Queisser HJ (1994) Novel optimization principles and efficiency limits for semiconductor solar cells Physical Review Letters 72(24): 3851 Nozik AJ (2001) Spectroscopy and hot electron relaxation dynamics in semiconductor quantum wells and quantum dots Annual Review of Physical Chemistry 52: 193–231 Nozik AJ (2002) Quantum dot solar cells Physica E: Low-Dimensional Systems and Nanostructures 14(1–2): 115–120 Nozik AJ (2009) Nanophotonics: Making the most of photons Nature Nanotechnology 4(9): 548–549 Nozik AJ, Beard MC, Luther JM, et al (2010) Semiconductor quantum dots and quantum dots arrays and applications of multiple exciton generation to third-generation photovoltaic solar cells Chemical Reviews 110(11): 6873–6890 Franceschetti A, An JM, and Zunger A (2006) Impact ionization can explain carrier multiplication in PbSe quantum dots Nano Letters 6(10): 2191–2195 Rabani E and Baer R (2008) Distribution of multiexciton generation rates in CdSe and InAs nanocrystals Nano Letters 8(12): 4488–4492 Ellingson RJ, Beard MC, Johnson JC, et al (2005) Highly efficient multiple exciton generation in colloidal PbSe and PbS quantum dots Nano Letters 5(5): 865–871 Nozik AJ (2008) Multiple exciton generation in semiconductor quantum dots Chemical Physics Letters 457(1–3): 3–11 Schaller RD and Klimov VI (2004) High efficiency carrier multiplication in PbSe nanocrystals: Implications for solar energy conversion Physical Review Letters 92(18): 186601-1–186601-4 Schaller RD, Agranovich VM, and Klimov VI (2005) High-efficiency carrier multiplication through direct photogeneration of multi-excitons via virtual single-exciton states Nature Physics 1(3): 189–194 Beard MC, Midgett AG, Hanna MC, et al (2010) Comparing multiple exciton generation in quantum dots to impact ionization in bulk semiconductors: Implications for enhancement of solar energy conversion Nanon Letters 10(8): 3019–3027 Beard MC (2011) Multiple exciton generation in semiconductor quantum dots Journal of Physical Chemistry Letters 2(11): 1282–1288 Koc S (1957) The quantum efficiency of the photo-electric effect in germanium for the 0.32 μ wavelength region Czechoslovak Journal of Physics 7(1): 91–95 Kolodinski S, Werner JH, Wittchen JH, and Queisser HJ (1993) Quantum efficiencies exceeding unity due to impact ionization in silicon solar-cells Applied Physics Letters 63(17): 2405–2407 Pijpers JJH, Ulbricht R, Tielrooij KJ, et al (2009) Assessment of carrier-multiplication efficiency in bulk PbSe and PbS Nature Physics 5(11): 811–814 Beard MC, Knutsen KP, Yu PR, et al (2007) Multiple exciton generation in colloidal silicon nanocrystals Nano Letters 7(8): 2506–2512 Midgett AG, Hillhouse HW, Huges BK, et al (2010) Flowing versus static conditions for measuring multiple exciton generation in PbSe quantum dots Journal of Physical Chemistry C 114(41): 17486–17500 Stubbs SK, Hardman SJO, Graham DM, et al (2010) Efficient carrier multiplication in InP nanoparticles Physical Review B 81(8): 081303 Wang SJ, Khafizov M, Tu XM, et al (2010) Multiple exciton generation in single-walled carbon nanotubes Nano Letters 10(7): 2381–2386 Semonin OE, Luther JM, Choi S, et al (2011) Peak External Photocurrent Quantum Efficiency Exceeding 100% via MEG in a Quantum Dot Solar Cell Science 334(16): 1530–1533 Singh S, Jones WJ, Siebrand W, et al (1965) Laser generation of excitons and fluorescence in anthracene crystals Journal of Chemical Physics 42(1): 330–342 Klein G, Voltz R, and Schott M (1972) Magnetic field effect on prompt fluorescence in anthracene: Evidence for singlet fission Chemical Physics Letters 16(2): 340–344 Klein G, Voltz R, and Schott M (1973) Singlet exciton fission in anthracene and tetracene at 77K Chemical Physics Letters 19(3): 391–394 Yarmus L, Rosenthal J, and Chopp M (1972) EPR of triplet excitons in tetracene crystalsspin polarization and the role of singlet fission Chemical Physics Letters 16(3): 477–481 Müller AM, Avlasevich YS, Müllen K, and Bardeen CJ (2006) Evidence for exciton fission and fusion in a covalently linked tetracene dimer Chemical Physics Letters 421(4–6): 518–522 Thorsmølle VK, Averitt RD, Demsar J, et al (2009) Morphology effectively controls singlet-triplet exciton relaxation and charge transport in organic semiconductors Physical Review Letters 102(1): 017401 Burdett JJ, Muller AM, Gosztola D, and Bardeen CJ (2010) Excited state dynamics in solid and monomeric tetracene: The roles of superradiance and exciton fission Journal of Chemical Physics 133(14): 144506 Grumstrup EM, Johnson JC, and Damrauer NH (2010) Enhanced triplet formation in polycrystalline tetracene films by femtosecond optical-pulse shaping Physical Review Letters 105(25): 257403 Jundt C, Klein G, Sipp B, et al (1995) Exciton dynamics in pentacene thin films studied by pump-probe spectroscopy Chemical Physics Letters 241(1–2): 84–88 Marciniak H, Fiebig M, Huth M, et al (2007) Ultrafast exciton relaxation in microcrystalline pentacene films Physical Review Letters 99(17): 176402 Johnson JC, Reilly TH, Kanarr AC, and van de Lagemaat J (2009) The ultrafast photophysics of pentacene coupled to surface plasmon active nanohole films The Journal of Physical Chemistry C 113(16): 6871–6877 Wilson MWB, Rao A, Clark J, et al (2011) Ultrafast dynamics of exciton fission in polycrystalline pentacene Journal of the American Chemical Society 133(31): 11830–11833 Smith MB and Michl J (2010) Singlet fission Chemical Reviews 110(11): 6891–6936 Jadhav PJ, Mohanty A, Sussman J, et al (2011) Singlet exciton fission in nanostructured organic solar cells Nano Letters 11(4): 1495–1498 Greyson EC, Vura-Weis J, Michl J, and Ratner MA (2010) Maximizing singlet fission in organic dimers: Theoretical investigation of triplet yield in the regime of localized excitation and fast coherent electron transfer The Journal of Physical Chemistry B 114(45): 14168–14177 Greyson EC, Stepp BR, Chen XD, et al (2010) Singlet exciton fission for solar cell applications energy aspects of interchromophore coupling Journal of Physical Chemistry B 114(45): 14223–14232 Zimmerman PM, Zhang ZY, and Musgrave CB (2010) Singlet fission in pentacene through multi-exciton quantum states Nature Chemistry 2(8): 648–652  ... 11 /2 3/2 H15/2 13 /2 7 13 /2 H6 11 /2 1. 1 eV 11 /2 13 /2 4 F7/2 9/2 11 /2 5 7/2 F0 F6 13 /2 H5 9/2 5/2 1/ 2 G4 F7/2 H 11/ 2 F5 5/2 12 5/2 3 H9/2 F3/2 F3/2 H4 14 K7 D4 G7/2 7/2 G7/2 H9/2 G 11/ 2 F1 K F9/2... 0.6 1. 8 0.9 Eb /Er Eb /Er 2.6 0.0 2.4 2.2 (b) 2.8 −0.6 1. 8 −0.3 −0.6 −0.9 1. 2 1. 6 2.0 1. 6 1. 5 1. 4 1. 4 1. 2 0.0 0.2 0.4 0.6 0.8 1. 0 1. 2 1. 4 1. 6 1. 8 2.0 1. 2 0.0 0.2 0.4 0.6 0.8 1. 0 1. 2 1. 4 1. 6 1. 8... 24 1. 6 36 1. 2 0.8 2.0 1. 0 1. 2 1. 4 1. 6 1. 8 1. 2 0.8 2.0 (d) 24 28 32 Energy conversion efficiency (%) 2.4 Eb /Er 28 24 1. 0 1. 2 36 40 1. 6 24 1. 0 1. 4 1. 6 1. 8 2.0 Band gap (eV) (c) 2.8 1. 2 0.8 32 1. 6