Enhancing silicon solar cells with singlet fission: the case for Förster resonant energy transfer using a quantum dot intermediate

Abstract. One way for solar cell efficiencies to overcome the Shockley–Queisser limit is downconversion of high-energy photons using singlet fission (SF) in polyacenes like tetracene (Tc). SF enables generation of multiple excitons from the high-energy photons, which can be harvested in combination with Si. In this work, we investigate the use of lead sulfide quantum dots (PbS QDs) with a band gap close to Si as an interlayer that allows Förster resonant energy transfer (FRET) from Tc to Si, a process that would be spin-forbidden without the intermediate QD step. We investigate how the conventional FRET model, most commonly applied to the description of molecular interactions, can be modified to describe the geometry of QDs between Tc and Si and how the distance between QD and Si, and the QD bandgap affects the FRET efficiency. By extending the acceptor dipole in the FRET model to a 2-D plane, and to the bulk, we see a relaxation of the distance dependence of transfer. Our results indicate that FRET efficiencies from PbS QDs to Si well above 50% are possible at very short but possibly realistic distances of around 1 nm, even for QDs with relatively low photoluminescence quantum yield.


Introduction
The domination of the solar cell market by silicon led to the search of add-ons that could increase efficiency while also maintaining low cost. One possible way to increase efficiency is by downconverting high-energy light using an organic material that exhibits singlet fission (SF).
In a single-junction solar cell, photons with energy above the bandgap can excite an electron into the conduction band. Excess energy is lost, as the charge carriers quickly thermalize to the band edge. Downconversion schemes minimize the energy lost by thermalization, by converting high-energy photons to lower-energy charge carriers. Downconversion via SF can improve on the single-junction Shockley-Queisser 1,2 efficiency limit, raising it from 33.7% to 44.4%. 3 SF in organic semiconductors describes the conversion of a singlet exciton into two triplet excitons, conserving spin. In tetracene (Tc), SF is faster (90 ps) 4 than other decay channels, which leads to a yield of almost two triplet excitons per absorbed photon. The resulting triplet excitons cannot relax radiatively to the singlet ground state, as this process is spin-forbidden, leading to a long triplet lifetime. In Tc, the energy of the triplet excitons (1.25 eV) 5 is close to the bandgap of silicon (1.12 eV), allowing in principle for the triplet excitons to be injected into silicon (Si). In one possible realization, the triplet exciton energy is first transferred into a lead sulphide (PbS) quantum dot (QD) 6 interlayer and subsequently transferred into Si 7,8 (see Fig. 1). Once the triplet exciton is in the QD, the presence of lead with strong spin-orbit coupling leads to intersystem crossing of singlet and triplet states. The spin triplet and singlet excitons are energy degenerate (≈3 meV), 9 which makes the spin mixing efficient. Hence, the exciton can decay radiatively, in principle allowing for energy transfer into Si via photon emission, or Förster resonant energy transfer (FRET). Transfer into lead sulfide 6 and lead selenide 10 QDs was recently demonstrated with high efficiency (>90%). 6 While energy transfer from core/shell CdSe/ZnS QDs into c-Si 7 as well as inter-QD FRET for cases, where energy was transferred among the same QD species [11][12][13] and between two different QD species, 12,14 has been demonstrated, energy transfer from PbS QDs into Si with a QD bandgap close to the one of c-Si remains to be shown.
One of the processes competing with FRET is the emission of photons by the QDs and the reabsorption in Si. For that process to be efficient, careful light management to funnel photons into silicon is required. In addition, the low external quantum efficiency of the Si cell near the indirect band edge might somewhat limit the achievable efficiency. Direct energy transfer in the form of FRET would be an elegant solution to allow for higher efficiency, as FRET can outcompete radiative energy transfer at distances smaller than the system-specific Förster distance R 0 , which is around 8 nm in the case of FRET between PbS QDs. 13,15 Once the exciton resides in Si it will contribute to charge generation, as the extraction efficiency of state-of-the-art Si solar cells is close to unity. Thus, the SF-FRET geometry could lead to additional current in Si solar cells, if short distances between the donor and acceptor can be achieved.
Apart from radiative energy transfer or FRET, other transfer mechanisms are also possible in the Tc-QD-Si geometry. The triplets from Tc could be transferred directly into silicon, bypassing the QDs. This would happen via the Dexter energy transfer mechanism, 8,16 which proceeds via correlated two-electron transfer. In this case, the excited electron of the triplet exciton would be transferred into the excited state in Si, while a ground-state electron from Si transfers into the Tc HOMO. Dexter energy transfer could also act as a transfer channel from the PbS QDs into Si. However, the Dexter transfer efficiency falls exponentially with distance from donor to acceptor due to the required wave function overlap. Thus, Dexter energy transfer is only relevant for short distances <1 nm. The QD ligands already contribute to a ∼1-nm separation between donor and acceptor. Hence, the overall contribution of Dexter energy transfer will presumably be negligible compared to FRET, which has a weaker distance dependency.
Sequential charge transfer from Tc or the PbS QDs to Si is another possible pathway for exciton dissociation, meaning that the electron would be transferred into Si and a hole would transfer from Si into Tc (or vice versa). This mechanism would require sandwiching the active layer between electrodes and is hence undesirable compared to the FRET or Dexter mechanisms.
Here, we establish the theoretical requirements for FRET between PbS QDs and Si, considering the QD bandgap, the distance between Si and QDs, and the geometry of the interface. We find that FRET can be efficient when the QDs are within 3.5 nm to the surface of Si, even for QDs with a bandgap close to the Si bandgap. This is a much shorter distance compared to inter-QD FRET or organic molecules, mostly because the molar absorption coefficient of Si is very low near the band edge. We further find that the distance dependence is somewhat relaxed when considering the Si surface as a plane or bulk acceptor. Finally, we lay out the path to prepare the Si surface to allow for efficient FRET from Tc into Si. Once efficient transfer of energy between QDs and Si can be achieved experimentally, SF could provide a direct path toward more efficient Si solar cells with minimal need for changes of the Si cell geometry.

Förster Resonant Energy Transfer
The FRET efficiency of excitons from QDs into Si, η FRET , is defined in Eq. (1). The main goal of this work is to determine how η FRET depends on donor-acceptor distance on the bandgap of the QDs and on the geometry of the system. The FRET efficiency η FRET compares the FRET rate k FRET to all the competing rates, defined as the exciton decay rate of the QD donor in absence of the silicon acceptor k D;0 : 17 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 6 ; 5 9 7 where k D;0 ¼ 1∕τ D;0 and τ D;0 represents the donor exciton lifetime in absence of an acceptor. FRET is a distance-dependent energy transfer mechanism between two molecules, which are approximated to be point dipoles. Förster derived an expression for the FRET rate, 18 which depends on the emission spectrum of the donor, absorption spectrum of the acceptor, donor lifetime, and donor acceptor distance. The classical as well as quantum mechanical approach both lead to Eq. (2): 17,18 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 6 ; 4 8 0 where R DA represents the distance between donor and acceptor and R 0 is the Förster distance. R 0 determines how strongly the FRET rate depends on the distance and is given by Eq. (3): 17 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 6 ; 4 1 1 In Eq. (3), the prefactor summarizes several numerical constants and Avogadro's number N A . Q D is the donor photoluminescence quantum yield (PLQY), κ 2 is a parameter that depends on the relative orientation between donor and acceptor dipole, and n represents the refractive index of the medium separating donor and acceptor. The parameter J is commonly referred to as spectral overlap integral as it represents the spectral matching of the donor emission and acceptor absorption spectra and is calculated as follows in Eq. (4): 17 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 6 ; 2 9 3 The overlap integral contains the normalized emission spectrum of the donor f D ðλÞ and the molar absorption coefficient of the acceptor α M;A ðλÞ, integrated over the wavelength λ [gray area in Fig. 2(a)]. We can use the far-field absorption coefficient of silicon for the near-field (Förster) coupling, because FRET has been measured to also be phonon assisted. 7 Figure 2(a) depicts α M;Si and f QD as a function of energy. The FWHM assumed for the QD PL is 200 meV, in agreement with literature. [20][21][22] The refractive index of the separating medium depends on how one accounts for the contributions of the dielectric functions of the QD itself, the surrounding ligand, and the spacer material. Following Yeltik et al., 7 we consider the average of refractive indices in a straight line from QD to the silicon surface. We approximate the refractive index as constant for different spacer thicknesses. As such, n SiO 2 ¼ 1.45 is used, which is the index of the SiO 2 spacer layer in between the QDs and the Si bulk. In fact, the QDs and the ligands will also influence the overall refractive index, as the light will be influenced by an effective medium given that the wavelength of emission is much larger than the distances involved in our system. The refractive index of the QDs is well above 1.45, and the refractive index of the organic ligands is between 1.45 for oleic acid (OA) 23 and 1.5 (3-mercatopropyonic acid) 24 for most organic ligands. Inorganic ligands like ZnI 2 are very short so we can neglect their influence on the electromagnetic field. However, since the ligands do not fill the entire volume, 21 we deem the approximation of n ¼ 1.45 valid for distances larger than 1 nm. The orientation parameter κ 2 depends on the relative transition dipole orientation of donor and acceptor. 17 Since the QDs have rotational symmetry, the dipole orientation in the QDs will be isotropic, which yields κ 2 iso ¼ 2 3 . 17 The quantum yield of PbS-QDs depends on various factors, including size, 25 excitation wavelength, 26 QD concentration, 26 ligands, 27 and whether they are in solution or in solid state. The choice of QD size is important because the corresponding bandgap has to be lower than the Tc triplet exciton energy and higher than the Si bandgap, to ensure that both transfer processes are downhill in energy. We choose QDs with emission centered at 1.2 eV, which corresponds to an average size of 3.4 nm. 25 The PLQY for these QDs ranges from 20% to 55% 26 in solution and up to 15% in films. 28 We determined the radiative lifetime of our 1.2 eV PbS QDs (see experimental methods for details on QD synthesis and PL lifetime measurement) in octane as τ PbS ¼ 2.4 μs [inset Fig. 2(b)], which is in good agreement with literature. 13,20,25 For a more accurate description of the FRET rate, the measured lifetime of the QDs in solution should be replaced by the QD lifetime measured after deposition on quartz, to obtain the reference value for "infinite" donor-acceptor separation τ D;0 . We exclude the effects of parasitic absorption in the QD layer because we assume a submonolayer QD coverage. To be specific, the ideal QD coverage to maximize transfer and minimize QD-absorption would be a submonolayer coverage, where the inter-QD spacing is far bigger than the inter-QD Förster radius of 8 nm. 13,15 Making this assumption allows us to neglect any significant contributions of inter-QD FRET. Inter-QD FRET should be regarded as an undesirable decay channel because screening more QDs increases the chance to find a surface trap state, and there will be a tendency to transfer toward lower energy QDs. The upper limit for the QD spacing is determined by the Tc triplet diffusion length. In the final geometry, the QD coverage has to be dense enough to allow all Tc triplets to diffuse toward a QD, meaning that the ideal QD separation corresponds to the triplet diffusion length of around 400 nm. 29,30 Such a QD coverage absorbs less than 0.01% of the solar spectrum, and thus, we can neglect absorption of the incident light by QDs (see Appendix for details on this estimate).
In the following, we calculate R 0 , which is the distance for that the transfer efficiency reaches 50% in the dipole-dipole model. While this is not exactly the case for the plane and bulk geometries we will introduce later, R 0 is still a useful quantity to estimate separation distances. As can be seen in the upper plot of Fig. 3, the values for R 0 vary from 0.9 nm to up to 1.5 nm, depending on the QY and bandgap of the QDs. The steep loss of transfer efficiency below the bandgap of silicon (around 1.12 eV) can be attributed to the exponential decrease in the absorption coefficient. The largest QD bandgap for which energy transfer from triplet excitons in Tc was observed is 1.23 eV, 6 and we indicate the QD bandgap range by the gray area in Fig. 3. The bottom panel of the same figure shows the FRET efficiency, which obeys a relatively steep slope around 1.2 eV, compared to higher bandgaps, suggesting the importance of a careful choice of the QD bandgap. The bottom plot of Fig. 3 shows FRET efficiencies for 1-and 2-nm separation distances, with varying QY. Changes in distance by only 1 nm around R 0 lead to an efficiency increase of up to 75%. The efficiencies at 1-nm separation saturate for bandgaps slightly higher than required in the given geometry at values close to 100%. It is worth noting that high FRET efficiencies (>65%) can be achieved at realistic distances (1 nm) even for a low QY (20%).

Influence of Geometry
Up until now, we have calculated the FRET efficiencies according to a dipole-dipole model that does not take into account the extended nature of the silicon acceptor geometry. We introduce two potentially more accurate descriptions of the FRET rate in our system, in the following referred to as "dipole-infinite plane model" and "dipole-bulk model," similar to earlier approaches. 11,31 Our final geometry will probably be best represented by the bulk model, and in the following, we show how it differs from the more conventionally used dipole-dipole description laid out above.
The silicon acceptor occupies one half-space instead of being a point-dipole, leading to a modification of Eq. (2). 32,33 For the dipole-infinite plane model, the zero-dimensional dipole acceptor is substituted with a 2-D acceptor extended over the x-y plane, assuming that the acceptor dipole of FRET mainly resides on the surface of Si [see Eq. (5) where R DA ðr DA ; rÞ is the distance from the donor dipole to an infinitesimal acceptor dipole, and σ Si is the density of silicon atoms on a <111> silicon surface (σ Si ¼ 7.8∕nm 2 ). After integration over the Si surface (r is the radial component in polar/cylindrical coordinates), the rate only depends on the distance component perpendicular to the surface, thus on r DA . The parameterizations used are illustrated in Fig. 5(b). While this model is closer to the physical reality, it only considers the Si surface. In order to include the Si bulk, we can simply integrate Eq. (5) over the half space occupied by Si, which leads to Eq. (6): E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 6 ; 3 5 0 For the integration, z 0 ðz; r DA Þ is split into the integration variable for the half space z and the distance from the donor to the surface of the bulk acceptor r DA , and ρ Si is the density of silicon atoms (ρ Si ¼ 50∕nm 3 ). The additional prefactor n n Si arises because we have to consider the refractive index of the part of bulk silicon between the infinitesimal acceptor and the QD donor as part of the separating medium. We use a refractive index of 3.55 for silicon n Si , corresponding to the relevant energy region (1.2 eV). 19 For a derivation, see Appendix. We note that the prefactor is independent of distance between donor and acceptor. Mathematically, this is due to the choice of integration limits and leads to an effective Förster distance R 0;eff ¼ ð n SiO 2 n Si Þ 1 6 R 0 . Figure 4 shows the FRET efficiencies for both models introduced above. From comparison with the bottom panel of Fig. 3, it becomes obvious that for 2-nm separation, the FRET efficiencies are improved considerably up to around 85% for the dipole-infinite plane model in the relevant region compared to 15% for the dipole-dipole model, whereas the values for 1 nm do not change significantly. This occurs due to the different distance dependencies in different models and acceptor dipole densities (ρ Si and σ Si ) in different models, as shown in Fig. 5. Figure 5 shows that the point model shows the steepest distance dependency, which is relaxed in the planar geometry, and the efficiency drop with distance in the bulk model is the most shallow. The efficiency is unity for small separations r da for all models and drops of to zero at 2 nm for the point model, 8 nm for the plane model, and is nonzero even for separations exceeding 10 nm for the bulk model.
Usually the characteristic length for FRET, the distance at which the transfer efficiency is 50%, is in the order of 10 nm (QD-QD FRET of 8 nm, 13,15 ) which is considerably longer than in the case of QD-silicon energy transfer which we discuss in this paper. However, the FRET distance becomes larger going from point (1.8 nm) to plane (2.5 nm) to bulk (3.5 nm) model. The slope is mainly determined by the distance dependence of the FRET rate [Eq. (1)] which changes from r −6 (point model) to r −4 (plane model) to r −3 (bulk model). The absolute efficiencies going from point to plane to bulk model are larger because the FRET rate is dominant compared to base rate k D;0 [Eq. (1)]. The underlying reason for the larger efficiencies is that there are more acceptors available in bulk (ρ Si ) compared to plane (σ Si ) and point (one acceptor) models.
With increasing distance, first dipole-plane and then dipole-bulk interactions become relatively stronger as they take into account more area/volume. Which model most accurately describes the distance dependence in our QD-silicon geometry? While the bulk-model represents the geometry more accurately, one could argue that due to the strong distance dependence of FRET, the majority of the interaction occurs already at the surface, so the plane-model might be valid after all. However, the spatial extend of the Bloch waves in silicon will ultimately govern the transition geometry.
We note that the mathematical treatment shown here does not take into account that part of the electromagnetic field is reflected by silicon, which leads to a reduced donor lifetime for small distances according to CPS theory. 34 Furthermore, the exciton in the QD could be more accurately described as an extended dipole. The point-dipole approximation is no longer valid if the distance between donor and acceptor is on the order of the exciton (QD) size. If the separation between electron and hole (1.8 nm for PbS QDs 26 ) is taken into account, the near field will no longer be accurately described by the r −3 dependence used in the FRET derivation. The final step would be the addition of a quantitative description of Dexter transfer, 16 which is a possibly competing charge-mediated energy transfer. Dexter transfer has an exponential distance dependence, which leads to transfer distances of around 1 nm but it does not depend on the absolute molar absorption coefficient of silicon (only on the spectral shape), which could make Dexter rates comparable with FRET rates in this case.
A factor that greatly affects k FRET is the overlap between QD emission and Si absorption spectra. The QD absorption energy must be lower than the Tc triplet exciton energy and the emitted energy of the QD must be above the Si bandgap. The broadening of the QD emission spectrum leads to additional losses when the emission spectrum broadens beyond the given limits. Sharper QD emission spectra could be achieved with a QD ensemble with sharper size distribution. 35 Apart from that, the Stokes shift might influence the choice of QD size strongly. We now assumed emission at 1.2 eV, which means that the absorption of the QDs would occur at a higher energy. However, the absorption is limited by the fact that Tc triplet states impose an upper boundary for the QD bandgap of around 1.25 eV.

QD Synthesis and Passivation
The colloidal PbS QDs were synthesized via the hot injection method. 36 In order to obtain the 1.2-eV QDs we measured, the following recipe was followed. 37 Most chemicals were purchased from Sigma-Aldrich. For those that were not, the distributor will be indicated.
The octadecene is degassed heating to 80°C. A 20-mL syringe is filled with 0.213mL of bis(trimethylsilyl)sulphide (synthesis grade) together with 10 mL of octadecene (technical grade 90%) in a glove box (< 0.5 ppm H 2 O, <0.5 ppm O 2 ) environment. The 0.45 g of PbO (99.999%, Alpha Aesar), 1.34 g of OA (technical grade 90%), and 14.2 g of octadecene are mixed together in a three-necked Schlenk flask. At a temperature of 95°C and under vacuum, this forms a clear solution. Then, the temperature is increased to around 170°C in a nitrogen environment. Now, the Schlenk flask containing the lead precursor is transferred to a heating mantle, which is at room temperature. As soon as the temperature has reached the injection temperature of 150°C (for 1.2-eV QDs), the sulphur precursor is injected into the flask with the solution being vigorously stirred. When the solution has cooled down to 35°C, 20 mL of acetone are added.
For surface passivation with I 2 , we follow Lan et al. 38 After the completed synthesis, the QDs are precipitated with acetone in a glovebox. After centrifuging for 4 to 10 min at 4000 to 5000 rpm, the residual liquid is disposed of, which is followed by vacuum-drying of the precipitate overnight. The QDs are then redispersed in toluene (≥99.9%) to obtain a concentration of 150 mg ml . Now, a 25-mM iodine (99.999%) in toluene solution is added to the QD solution at a 1∶5 ratio and stirred for 24 h. Afterward, the QDs are precipitated with methanol and centrifuged at 1500 to 5000 rpm for 2 to 5 min. The residual fluid was disposed of, and after a night of vacuum-drying, the QDs were dispersed in octane to obtain a 37.5 mg ml solution. Eventually, this solution was diluted with octane to obtain a 4.4 mg ml solution, which was used in the lifetime measurements.

PL Lifetime Measurement
The photoluminescence decay of the 1.2 eV bandgap PbS QD was measured on a home-built time-correlated single-photon counting (TCSPC) system consisting of a 640-nm pulsed laser (PicoQuant LDH-D-C-640) with a repetition rate of 0.2 MHz as an excitation source controlled by a PicoQuant PDL 828 "Sepia II." The signal was collected by a single-photon avalanche diode (SPAD) detector (Micro Photon Devices, MPD-5CTD) connected to a PicoQuant HydraHarp 400 multichannel picosecond event timer. The laser has a power of 14.6 μW at the used repetition rate. The laser light was filtered out of the collection path by a Chroma ZET 642-nf notch filter and a Chroma ET 655lp long-pass filter. The TCSPC decays were collected for 5 min.

Conclusion
In conclusion, we showed that FRET from PbS QDs to silicon is possible with sufficiently high FRET efficiencies, even for QDs that have a bandgap close to silicon and low PLQY. While efficient FRET is only possible over small separation distances in the order of a few nanometers, those distances are physically feasible, given careful engineering of the interface.
It is of great importance that the emission and absorption peak of the QDs are between the Tc triplet exciton energy and the bandgap of Si, with a narrow emission spectrum.
The equations above show the derivation of the n n Si prefactor in Eq. (6) of the main text. For the integration, substitution of variables was used with u ¼ r DA þ n Si n z.

A.2 Estimate of the Fraction of Light That is Absorbed by the PbS QD Layer
As a conservative estimate, we assume that the QDs are separated by 50 nm on a square lattice, which is well below the triplet diffusion length. 29,30 This means that one QD occupies an area of 50 nm × 50 nm. The QDs are approximated as spheres with a radius of 1.75 nm, which corresponds to a bandgap of 1.2 eV. The volume of the QDs was calculated and divided by the area occupied by one QD, which gives an effective QD layer thickness across the whole geometry of d QD;eff ¼ 9 Ã 10 −3 nm. With this effective layer thickness, we estimated the relative absorption of incident light by the QDs by using the Beer-Lambert law, as shown in Eq. (10): E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 0 ; 1 1 6 ; 4 4 3 I QD ðλÞ ¼ I 0 ðλÞ Ã expð−α QD ðλÞd QD;eff Þ: Here, I QD ðλÞ stands for the intensity of light behind the QD layer. I 0 ðλÞ is the incident light intensity for which we used the AM1.5 solar spectrum. 41 α QD ðλÞ denotes the wavelengthdependent absorption coefficient of PbS QDs. 42 The relative intensity loss due to the QDs can then be calculated, as shown in Eq. (11), with the integrals going over the whole wavelength range: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 1 1 6 ; 3 5 0 This leads to a relative intensity loss of ΔI rel ¼ 0.006%, which confirms our assumption that QD absorption is negligible in our geometry.