Multiple exciton generation in quantum dots versus singlet fission in molecular chromophores for solar photon conversion

Abstract

Both multiple exciton generation (MEG) in semiconductor nanocrystals and singlet fission (SF) in molecular chromophores have the potential to greatly increase the power conversion efficiency of solar cells for the production of solar electricity (photovoltaics) and solar fuels (artificial photosynthesis) when used in solar photoconverters. MEG creates two or more excitons per absorbed photon, and SF produces two triplet states from a single singlet state. In both cases, multiple charge carriers from a single absorbed photon can be extracted from the cell and used to create higher power conversion efficiencies for a photovoltaic cell or a cell that produces solar fuels, like hydrogen from water splitting or reduced carbon fuels from carbon dioxide and water (analogous to biological photosynthesis). The similarities and differences in the mechanisms and photoconversion cell architectures between MEG and SF are discussed.

1. Introduction

There are two general approaches to exceed the Shockley–Queisser (S-Q) thermodynamic limit [1–14] for the efficiency of converting solar radiation into electrical- or chemical-free energy via the general mechanism of producing multiple photogenerated excitons (bound electron-hole pairs) from single absorbed photons. If the units for the input and output terms are watts/cm², then the conversion efficiency is called the power conversion efficiency (PCE) and the S-Q limit for the standard solar spectrum (AM1.5) at 1 sun intensity is 32% [14]. To complete the conversion process, the photogenerated excitons must subsequently be separated into free electrons and positive holes (free charge carriers), transported to external contacts, and the extracted separated charges used to generate electrical power (photovoltaic conversion) or chemical power (solar fuels production); thus the initial exciton multiplication process becomes a carrier multiplication (CM) process when the power conversion is complete. The two approaches are described below:

• (1) in semiconductor nanocrystals (quantum dots (QDs), quantum wires (QWires and quantum rods (QRs)), the exciton multiplication process occurs when the absorbed photons are at least twice the nanocrystal (NC) band gap (E_g) [1–5]; this requirement satisfies Conservation of Energy and the process is called multiple exciton generation (MEG) [2–5]. In the literature, MEG and CM are used interchangeably. In the ideal and most efficient PCE case, N excitons are formed when the absorbed photons have energies that are N times E_g; this behaviour generates a maximum conversion efficiency characterized as a staircase function in a plot of exciton quantum yield (QY) versus (photon energy/E_g) [7]. MEG has been reported to also occur in carbon nanotubes [8] and graphene [9,10]; and

• (2) in molecular chromophores that have a triplet state energy (T₁) that is close to 1/2 the energy of the first allowed optical transition (S₁–S₀), exciton multiplication can occur upon photoexcitation to produce two triplet states from the single singlet state [6,11,12,13]. This exciton multiplication process requires that the molecular entity consists of two monomers that are appropriately electronically coupled in either a dimer or oligomer configuration, or as two monomers appropriately coupled as neighbours in a molecular crystal; the multiplication process is called singlet fission (SF) [6,11–13]. Thus, the molecular chromophores undergoing SF that we refer to here consists of these coupled monomers with the appropriate singlet and triplet energy levels described above.

The differences and similarities between MEG and SF are discussed here with respect to (i) solar photon conversion device architectures for generating solar electricity and solar fuels, (ii) photoconversion efficiencies (QYs and PCE), and (iii) models and mechanisms for the exciton multiplication process.

2. The Shockley–Queisser limit and hot carriers in semiconductors

The S-Q limit [14] of 32% is derived from a thermodynamic detailed balance analysis of the conversion efficiency of solar irradiance and is based on four assumptions that are presented in figure 1; the restrictions imposed by the first three of these assumptions can be removed through appropriate cell design, resulting in theoretical conversion efficiencies above the S-Q limit of 32%; the last assumption of zero non-radiative recombination loss is not usually valid, which results in a decrease in efficiency of actual devices.

Figure 1.

Shockley–Queisser thermodynamic analysis for maximum PCE of solar photons based on detailed balance (adapted from [14]). (Online version in colour.)

As shown in figure 2, the major efficiency loss of absorbed photons results from the third assumption that the excess kinetic energy of hot carriers is converted into heat as they relax to the conduction and valence band edges via carrier-phonon scattering. There are two general ways to avoid or reduce this loss: (i) collection of the hot carriers before they cool to the bandedges and extraction of the hot carriers from the solar cell to do useful electrical or chemical work at higher photovoltages, and (ii) conversion of the excess kinetic energy of hot carriers with energies at least twice the bandgap (2E_g) into a second exciton; a third exciton can be produced if the photon energy is greater than or equal to 3E_g. Other ways to exceed the S-Q limit are described by Green [15] and include (i) the use of a stack of multiple bandgaps of different specific values arranged in order of descending bandgaps in a tandem multi-junction solar cell, (ii) intermediate band solar cells wherein photon energies below E_g are absorbed in a band within the bandgap and the photogenerated electrons are excited into the conduction band through absorption of a second set of low energy photons, and (iii) restructuring the solar spectrum before it enters the solar cell via photon up- or down-conversion.

Figure 2.

Hot carrier energy loss due to relaxation/cooling via electron–phonon scattering. (Online version in colour.)

The principle of approach (i) above for a hot carrier solar cell is shown in figure 3, and the results of the S-Q-type PCE calculation are shown in figure 4 as a function of the hot electron temperature (for simplicity, it is assumed in this calculation that all excess photon energy resides in the photogenerated electron, and this would be the case if the effective mass of the electron is much greater than that of the positive holes in the semiconductor material). In the limit of a hot carrier cell that suffers no energy loss of hot carriers through phonon emission (i.e. no hot carrier cooling), the resulting electron temperature is 3000 K for a flat solar cell lying on the surface of the Earth; this results in maximum PCE of ≈66% and is equivalent to the PCE of a multi-junction tandem cell with the bandgaps closely matched to the solar spectrum (four optimum bandgaps in series produce most of the PCE gain).

Figure 3.

Hot carrier solar cell to produce higher photovoltages. (Online version in colour.)

Figure 4.

Maximum power conversion efficiencies as a function of bandgap for various hot electron temperatures (adapted from Ross & Nozik 16). (Online version in colour.)

3. Exciton multiplication

In approach (ii) above, the excess kinetic energy of photogenerated hot excitons is used to create additional excitons at the band edges (relaxed excitons cooled to the lattice temperature). This can occur efficiently in semiconductor QDs, QRs and Qwires. The first type of semiconductor NCs proposed [1–3] and first experimentally found [4] to exhibit efficient MEG were QDs. The reasons for this are (i) the requirement for conservation of crystal momentum (k) is relaxed because k is not a good quantum number; (ii) Coulomb coupling is greatly enhanced, which increases the rate of Auger-type processes, one of which is the inverse Auger process of MEG; (iii) hot exciton cooling is slowed relative to MEG, and (iv) the critical parameter for maximizing the MEG QY, which is the ratio of the MEG rate to the cooling rate, is greatly increased. Figure 5 shows the predicted MEG process in QDs.

Figure 5.

MEG from single absorbed photons in a quantum well (adapted from Nozik [1]). (Online version in colour.)

Another system that can generate multiple excitons (in the form of two triplet states) from single photons occurs in molecular chromophores via SF [6,11–13] when the energy of the transition from S₀ to S₁ is about twice the energy of the transition from S₀ to T₁. The SF process is presented in figure 6, which explains the consequences of the various energetic alignments of S₀, S₁ and T₁.

Figure 6.

Required energetic alignments of electronic states in molecules exhibiting SF (adapted Smith & Michl [12]). (Online version in colour.)

For SF to occur, two monomers with the energetic alignment in figure 6 must have an appropriate electronic coupling—not too weak and not too strong [6,11–13]. This can be achieved in molecular crystals where the monomers are arranged in a herringbone structure (slipped stack seems to be a good configuration), or in dimers or oligomers where the monomers are connected by optimum bridging moieties and have optimum spatial orientations with respect to each other [6]. Figure 1 in [6] also shows 24 molecules that could or already exhibit singlet fission.

The differences in the initial early events between MEG and SF are presented in figure 7. In MEG, the two excitons are created in a single QD and form biexcitons; in SF the two triplets are formed separately on each of the two electronically coupled monomers, creating an effective overall singlet state thus allowing the transition from S₁ to T₁T₁. The dynamics of these early events are also quite different. The exciton multiplication in both systems occurs rapidly (femtosecond to picosecond time scales) but in MEG the lifetime of the biexciton is usually between 50 and 100 ps as it decays via Auger recombination to a single exciton with a lifetime of a few nanoseconds, while in SF the double triplet states have lifetimes of microsecond and the singlet has lifetimes of nanosecond [6,11–13].

Figure 7.

Comparison of SF process to the process of MEG. (a) ‘Hot’ and (b) relaxed singlet fission. (c) Typical MEG scheme in a semiconductor QD (adapted from [11]). (Online version in colour.)

4. Power conversion efficiencies

The differences in the QY of excitons versus photon energy for MEG and SF are presented in figure 8; the photon energy is normalized to the bandgap and presented as (hν/E_g) since this way of representing photon energy is the meaningful way for characterizing and comparing QYs. For MEG (top), the optimum characteristic of QY versus photon energy is the staircase function with an onset of 2E_g (labelled M_max). Present and common MEG characteristics are linear functions labelled L(n), where n is the onset of the MEG process in units of the number of bandgaps of photon energy required for the onset of MEG. The slope of the linear plots is the MEG efficiency (η_MEG), which is the number of additional excitons created per bandgap of photoexcitation beyond the MEG threshold energy for absorbed photons. A simple relationship between η_MEG and the threshold photon energy for MEG (hν/E_g)_th derived by Beard et al. [7] is (hν/E_g)_th=1+1/η_MEG; this expression is most applicable for a hard (i.e. sudden) MEG onset.

Figure 8.

Characteristic dependence of QY of photogenerated excitons on absorbed photon energy normalized to the quantum dot bandgap. (Online version in colour.)

The bottom section of figure 8 shows the MEG characteristic for conventional present-day solar cells (M1—only 1 exciton/absorbed photon independent of photon energy after hν=E_g, and no excitons are created with hν<E_g), and also for SF (2 triplet excitons at hν≥2E_g and no extra excitons for hν<2 E_g. In figure 9, the maximum thermodynamic efficiency calculations are presented for the various MEG characteristics of figure 8. The largest PCE is obtained for the M_max staircase, with greatly reduced gains in PCE obtaining with (hν/E_g)>2.5E_g. The PCE values versus E_g for MEG processes become about equal to the bulk S-Q results without MEG when the onset for MEG is about 5E_g (this is confirmed experimentally for many bulk semiconductors). Comparing figures 4 and 9, it is apparent that the maximum PCE for a hot carrier solar cell (66%) is significantly great than for MEG (45%). This is because for the former case all of the excess hot electron energy above E_g is extracted to do useful work (free energy), while for the latter case excess hot electron energy between 1 and 2E_g is still lost as heat and is unavailable for useful work.

Figure 9.

PCE of QDs with different MEG characteristics as defined in figure 8 versus the bandgap. The usual Shockley–Queisser result for bulk semiconductors is also shown for comparison (adapted from Beard et al. [7]). (Online version in colour.)

In figure 10, the PCE versus E_g (defined as T₁–S₀) for SF-based solar cells is presented. For a single SF chromophore that produces two triplets from a single singlet state, the max PCE is 33%. This is essentially the same as the S-Q PCE value for a single junction PV cell made from bulk semiconductor material; however, the peak PCE for the former occurs at 0.55–0.7 eV, red-shifted from the optimum bandgaps of 1.2–1.4 eV for the maximum PCEs for bulk-based single junction PV cells.

Figure 10.

PCE versus band gap for molecules undergoing SF used as sensitizers in a dye cell, for QDs showing a staircase characteristic and the M2 characteristic, and for regular bulk semiconductors (S-Q result) (adapted from Hanna & Nozik 17). (Online version in colour.)

Another important feature of SF chromophores is that the lowest energy transition S₀–T₁ is forbidden and this energy difference is defined as the bandgap of the SF chromophore; this is because at the end of the SF process followed by charge separation and energy conversion, the two separated electron-hole pairs come from the triplet state. In order to avoid loss of input photon energy due to non-absorption in the S₀–T₁ transition, it is necessary to add a second light absorber with a bandgap equal to S₀–T₁ in tandem behind the SF chromophore. This results in an increase in the maximum PCE from 33 to 45%, which is about the same value for the maximum PCE of MEG using a single photomaterial; however, in the MEG case the optimum bandgap is between 0.70 and 0.95 eV. The energetic arrangement of this tandem SF is shown in figure 11 for two types of architecture along with their calculated maximum PCEs for both PV and H₂O splitting. It is noted in figure 11 that while in both tandem architectures the second absorber is optically in series with the first, in the top arrangement the photocurrent from the two absorbers is in parallel, while in the bottom arrangement the photocurrent is in series, thus requiring equalization of the photogenerated current in each photomaterial and recombination of 1/2 of the photogeneration electrons and holes in each photomaterial at the interface between them. Finally, PCE calculations show that there is no significant advantage for tandem architectures to have the second absorber undergo SF or MEG.

Figure 11.

Various tandem architectures for SF molecules used as sensitizers in dye cells for (a) photovoltaic cells (top), and for (b) water splitting cells (bottom). The power conversion efficiencies are shown in (a) for both PV and H₂O splitting (adapted from Hanna & Nozik 17). (Online version in colour.)

A final comparison of MEG and SF involves their earliest application in PV solar cells. For MEG, a simple low-temperature solution processed PV cell was fabricated from an array of PbSe QDs as shown in the top inset of figure 12. The fabrication started with a transparent glass/indium tin oxide (ITO) superstrate upon which a 40–60 nm nanocrystalline ZnO layer was deposited, then a 50–250 nm PbSe QD layer, finally followed by thermal evaporation of a gold contact. For the PbSe QD layer, a layer-by-layer ethanedithiol treatment was used to deposit the majority of the QD film, followed by layer-by-layer deposition of approximately 30 nm of PbSe QDs, but using 1 M hydrazine in acetonitrile to treat the film instead of ethanedithiol. The external QY (EQE) of the photocurrent was then measured for the PV cell and as shown in figures 12 and 13 the external and internal QYs exceeded 100% for photon energies more than 2.6E_g.

Figure 12.

External quantum efficiency (corrected for reflection) for PV PbSe QD solar cell and for standard bulk Si, CdTe and CIGS solar cells as a function of incident photon energy. Only the QD solar cell shows QYs exceeding 100% due to MEG; MEG occurs within the solar spectrum from 3 to 3.5 eV. MEG onset is at about 2.6E_g (adapted from Semonin et al. 18). (Online version in colour.)

Figure 13.

(a) External quantum efficiency for absolute photon energy for PbSe QD solar cells for several PV cells, only one of which has an anti-reflection coating; (b) internal quantum efficiencies versus normalized photon energy for three different bandgaps; (c) data for (b) normalized to 100% QY for incident photons having energies below the MEG threshold and compared to QYs obtained from time-resolved spectroscopic measurements of isolated PbSe QD in colloidal solutions. (Online version in colour.)

In [19], a PV cell based on SF was constructed from a pentacene/C60 donor–acceptor junction, and an external QY for photocurrent of 109% was measured, which resulted in an internal QY 160% at 2E_g. A second paper based on pentacene by this MIT group [20] showed a 126% external QY and a 137% internal QY.

To date, the PCE of PV cells based on QDs or SF chromophores are low (approx. 10%) and the contribution of MEG and SF to these efficiencies is small. This is because these cells are far from an optimized structure and suffer losses due to non-radiative recombination, incomplete absorption and incomplete photogenerated charge collection. Notwithstanding, they both demonstrate and confirm the principle of exciton multiplication in working photovoltaic cells, and offer the promise of efficiencies that can exceed the S-Q PCE limit of 32% upon future improvements and advances in cell design and minimization of loses with future advances.

The science of the mechanisms and characteristics of the MEG and SF processes to generate multiple excitons from single absorbed photons have been summarized here and show interesting similarities and differences.

Funding statement

The work on MEG and QDs was supported as part of the Center for Advanced Solar Photophysics, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under award DE-AC36-086038308. The work reported on SF was conducted by Justin Johnson and was supported by the Solar Photochemistry Program within the Division of Chemical Sciences, Geosciences, and Biosciences in the Office of Basic Energy Sciences, Office of Science, US Department of Energy.