Emergence of New Materials for Exploiting Highly Efficient Carrier Multiplication in Photovoltaics

In conventional solar cell semiconductor materials (predominantly Si) photons with energy higher than the band gap initially generate hot electrons and holes, which subsequently cool down to the band edge by phonon emission. Due to the latter process, the energy of the charge carriers in excess of the band gap is lost as heat and does not contribute to the conversion of solar to electrical power. If the excess energy is more than the band gap it can in principle be utilized through a process known as carrier multiplication (CM) in which a single absorbed photon generates two (or more) pairs of electrons and holes. Thus, through CM the photon energy above twice the band gap enhances the photocurrent of a solar cell. In this review, we discuss recent progress in CM research in terms of fundamental understanding, emergence of new materials for efficient CM, and CM based solar cell applications. Based on our current understanding, the CM threshold can get close to the minimal value of twice the band gap in materials where a photon induces an asymmetric electronic transition from a deeper valence band or to a higher conduction band. In addition, the material must have a low exciton binding energy and high charge carrier mobility, so that photoexcitation leads directly to the formation of free charges that can readily be extracted at external electrodes of a photovoltaic device. Percolative networks of coupled PbSe quantum dots, Sn/Pb based halide perovskites, and transition metal dichalcogenides such as MoTe2 fulfill these requirements to a large extent. These findings point towards promising prospects for further development of new materials for highly efficient photovoltaics.


Abstract
In conventional solar cell semiconductor materials (predominantly Si) photons with energy higher than the band gap initially generate hot electrons and holes, which subsequently cool down to the band edge by phonon emission. Due to the latter process, the energy of the charge carriers in excess of the band gap is lost as heat and does not contribute to the conversion of solar to electrical power. If the excess energy is more than the band gap it can in principle be utilized through a process known as carrier multiplication (CM) in which a single absorbed photon generates two (or more) pairs of electrons and holes. Thus, through CM the photon energy above twice the band gap enhances the photocurrent of a solar cell. In this review, we discuss recent progress in CM research in terms of fundamental understanding, emergence of new materials for efficient CM, and CM based solar cell applications. Based on our current understanding, the CM threshold can get close to the minimal value of twice the band gap in materials where a photon induces an asymmetric electronic transition from a deeper valence band or to a higher conduction band. In addition, the material must have a low exciton binding energy and high charge carrier mobility, so that photoexcitation leads directly to the formation of free charges that can readily be extracted at external electrodes of a photovoltaic device. Percolative networks of coupled PbSe quantum dots, Sn/Pb based halide perovskites, and transition metal dichalcogenides such as MoTe 2 fulfill these requirements to a large extent. These findings point towards promising prospects for further development of new materials for highly efficient photovoltaics.

Introduction
A photon with energy hν exceeding the band gap (E g ) of a semiconductor can excite an electron from a valence band to a conduction band and create an electron-hole pair. In this way, a hot electron and hole are produced that usually thermalize quickly to the band-edge with the excess energy (hν − E g ) being lost as heat (Figure 1(a)). This poses a fundamental limitation to the efficiency of solar cells and one of the predominant reasons for the Shockley-Queisser limit of ~33% for single-junction solar cells. 1 Given sufficient excess energy, it can in principle be utilized to generate additional charge carriers through carrier multiplication (CM), as shown in Figure 1(b). [2][3][4][5][6][7] In this way, CM can enhance the photocurrent of a solar cell and help to surpass the Shockley-Queisser limit. 2 CM is also known as impact ionization (II) in bulk semiconductors and multi-exciton generation (MEG) in quantum confined nanomaterials when neutral excitons (Coulombically 5 bound electron-hole pairs) are formed rather than free charge carriers. The key factors characterizing CM are the threshold photon energy from which CM starts and the quantum yield (QY), i.e. the number of electron-hole pairs produced per absorbed photon. The ideal CM scenario is a staircase dependence of the QY on the photon energy where the QY reaches 2 (n) at twice (n-times) the band gap multiple (Figure 2(a)). The band gap multiple is the photon energy normalized to the band gap of the material; i.e. ℎ ⁄ . To effectively exploit CM in solar cells the band gap of the semiconductor should be 0.6-1.0 eV resulting in a maximum efficiency of ~ 44% for an ideal staircase scenario, see Figure   2(b). Due to their suitable band gap (0.7-1.0 eV), Pb-chalcogenide based nanomaterials have been widely investigated for CM. 3,5, In addition, CM has also been studied in nanoparticles consisting of Cd-chalcogenides, Si, Ag 2 S, CuInSe 2 , as well as in 2D graphene and 1D carbon 6 nanotubes. 12,[32][33][34][35][36][37][38][39][40][41][42][43][44] Extensive reviews of advances in CM research have appeared, with recent ones by Pietryga et al. 45 in 2016, and by Kershaw et al. 46 in 2017. In this review, we describe the general understanding of CM and focus on recent research in the past three years. The latter includes studies of CM in Pb-chalcogenide heterostructures and networks, Si nanorods, perovskites, and transition metal dichalcogenides (TMDCs). 11,28,36,[47][48][49][50][51][52][53][54] The current understanding of how the CM threshold is related to the band structure in terms of asymmetric optical excitations will be discussed in detail. 55 Recent results on the relatively high CM efficiencies found in weakly quantum confined and bulk perovskites and in TMDCs are of particular interest. Compared to quantum dots (QDs) the more facile charge transport in bulk perovskite and TMDCs are of particular interest for applications in photovoltaic devices. We include a brief discussion of CM-based solar cells and conclude with a future outlook.

Brief history of carrier multiplication
During the process of CM, a hot charge carrier with energy exceeding the band gap (either an electron in a conduction band or a hole in a valence band) relaxes by excitation of a valence band electron to the conduction band. CM occurs in competition with phonon emission (carrier cooling). In bulk materials the CM threshold is often as high as about 4 times the band gap. 40 In that case, CM is not useful for solar cell applications.
In 2002 Nozik theoretically proposed that CM in quantum confined nanomaterials can be more efficient than in bulk. 2 32,[56][57][58][59][60] Careful experimental procedures to avoid artifacts have shown the CM efficiency to be lower than the initial results in QDs, but still of promise for solar cell applications. 9,16,61 Later on the research of CM was extended to 1-D nanorods, 2-D nanosheets, complex heterostructures, and assemblies showing both a decrease of the CM threshold and an increase of the QY, see Section 6 of this paper. In the past few years efficient CM has been reported for (heterostructures of) Pb-chalcogenide based NCs of different shapes, with a CM threshold close to twice the band gap. More recently, efficient CM has also been observed in bulk perovskites and 2-D TMDCs.
Interestingly, recent research suggests quantum confinement may not be a necessary requirement for efficient CM, as will be discussed in Section 6.

Experimental techniques to investigate CM
The experimental techniques mostly utilized to investigate CM involve time-resolved pumpprobe laser spectroscopy with detection of transient optical absorption, photoluminescence, or microwave/terahertz conductivity.

a. Transient optical absorption measurements
Pump-probe transient optical absorption (TA) spectroscopy is the most widely used technique to characterize CM. In TA experiments, the sample is excited by a pump laser pulse creating Here, represents the cross-section of bleach, photoinduced absorption, and/or stimulated emission at the probe energy due to an electron-hole pair. (red arrow at B), indicating that Auger recombination after 3.1 eV excitation is complete and |∆ | is due to QDs containing a single exciton only. Adapted with permission from ref. 16 .

Copyright (2008) American Chemical Society.
For sufficiently high pump photon energy the hot electrons and holes can undergo CM or cooling by phonon emission. Hot charge carriers can lead to another magnitude and shape of the TA spectrum than relaxed charges at the band gap. 62 To exclude such effects in the determination of the QY, the value of |∆ | should be taken at a time when the hot carriers have relaxed and the spectral shape of the TA no longer varies with time. Then, for the same absorbed pump fluence, an increase of |∆ | at higher pump photon energies indicates the occurrence of CM ( Figure 3). After photogeneration of two or more excitons in a QD, the TA signal exhibits a rapid decay due to Auger recombination ( Figure 3). Consequently, the TA signal on longer times is due to QDs containing one exciton only. In this case, the initial QY of excitons can be determined by taking the ratio of |∆ | at an early time (A) when multi-excitons are still present and at a longer time (B) when the Auger process is complete leaving only one exciton in a QD ( Figure 3).

b. Transient photoluminescence measurements
Transient photoluminescence (PL) measurements have also been utilized to determine the CM threshold and QY. 32,59,63 For pump photon energies below twice the band gap (and at sufficiently low fluence so that each QD absorbs at most one photon) the PL reflects the radiative decay of single excitons. A faster decay of the PL at higher pump photon energy (due to Auger recombination of multi-excitons in a QD) is indicative of CM.

c. Transient terahertz/microwave conductivity measurements
Free mobile charge carriers in assemblies of QDs, nanowires, 2-D, or bulk materials can be probed by time-resolved alternating current (AC) conductivity techniques at microwave or terahertz frequencies. 6,11,17,64,65 In the case of optical pump terahertz (THz) probe (OPTP) or microwave probe experiments the transient photoconductivity (∆ ) is obtained with picosecond and nanosecond time resolution, respectively. The magnitude of ∆ is given by: Here is the elementary charge, is the number of absorbed pump photons per unit volume, is the QY of charge carriers, and and ℎ are the electron and hole mobility, respectively. Therefore, the slope of the linear increase of ∆ vs.
gives the CM QY for the corresponding pump photon energy. If we excite below twice the band gap CM is impossible and the observed slope represents = 1. For excitation above twice the band gap an increase of the slope of a plot of ∆ vs.
gives the CM QY similar to the TA measurements discussed above.
With THz measurements, the magnitude of ∆ can be obtained on a picosecond timescale, which in most cases is sufficiently short to ensure recombination or trapping of charges has not yet occurred. Microwave conductivity measurements have a time resolution of nanoseconds and recombination/trapping of electrons and holes may already have taken place. The latter may be slower at higher pump photon energies and therefore care must be taken that a higher photoconductivity on a nanosecond timescale does indeed reflect CM.

Factors affecting the CM threshold and efficiency
For an ideal case scenario, the CM threshold appears at twice the band gap with QY of 2.
However, due to restrictions imposed by energy and momentum conservations the CM threshold is often far off from the ideal scenario. For parabolic bands with equal electron and hole effective masses, the threshold becomes 4 times the band gap as shown in Figure   4(a). 40 As momentum conservation rules are relaxed in QDs the CM threshold can be lower than for bulk material. In QDs with equal effective masses of electrons and holes, the CM threshold theoretically becomes 3 times the band gap Figure 4.(b)). Indeed, it has been shown experimentally that in Pb-chalcogenide QDs with almost equal effective masses of electrons and holes the CM threshold is close to thrice the band gap ( Figure 5). 8,9,55 The QY increases almost linearly above the threshold and the steeper the slope the higher is the CM efficiency. 12 In the context of solar cell applications, the CM QY is usually plotted vs. the photon energy normalized to the band gap, which is denoted as the band gap multiple, defined by ℎ ⁄ . The CM efficiency ( ) is defined as the change of the QY with the change of the band gap multiple ℎ ⁄ according to 9 : Reproduced with permission from ref. 9 . Copyright (2010) American Chemical Society.

a. The CM threshold is related to asymmetric optical excitations
If the excess photon energy above the band gap is almost entirely transferred to either the electron or the hole the CM threshold can be near twice the band gap. Such asymmetric photoexcitation is possible if the effective mass of the electron and hole are largely different, which is the case for InAs QDs ( ℎ ⁄~0.05). 40 In this case the excess photon energy is almost completely transferred to the electron and the CM threshold is close to twice the band gap. 40 Further permissions related to the material excerpted should be directed to the ACS.
Asymmetric photoexcitation as mentioned above is also possible if there is a second conduction (or valence) band with extremum at twice the band gap, as shown in Figure 6(a).
In that case, the excess photon energy is fully transferred to the electron (or hole), which can These asymmetric excitations involve higher valence and conduction bands, as shown in

Theory of carrier multiplication
The CM QY is the net result of the decay of a hot charge carrier via consecutive steps of CM However, direct comparison with experimental results was hindered by the fact that taking into account all exciton states was computationally too demanding.
A second approach known as the direct photogeneration model assumes a weak Coulomb coupling between single and biexciton states and was introduced by Klimov et al. 70,71 Photoexcitation is assumed to occur at an energy that is resonant with a biexciton state, but off-resonant with single exciton states. One mechanism involves an off-resonant 'virtual' single exciton state making photoexcitation from the ground state to the biexciton possible via an optical dipole transition. 71 This mechanism was used to explain the ultrashort timescale of CM in CdSe and PbSe QDs. Another mechanism corresponds to a ground state that is a mixture of a state with no excitons (vacuum state) coupled to a biexciton state by Coulomb interaction (this is analogous to CI in quantum chemistry). 70 The admixture of the biexciton in the ground state allows direct photoexcitation to a higher biexciton state that is resonant with the photon energy.
A third class of CM pathways has been introduced by Zunger et al., 72 Delerue et al., 73  A very general theoretical treatment of CM has been provided by Piryatinski and Velizhanin and is known as the exciton scattering model. 78 This model is applicable to cases ranging from weak to strong Coulomb coupling and includes the above described CM However, the results give qualitative insights that are useful to analyze the effects of material composition, phonons, and structural defects on CM and Auger recombination of electrons and holes.

a. Pb-chalcogenide 1-D nanorods and 2-D nanosheets
The characteristics of CM in 1-D Pb-chalcogenide nanorods (NRs) and 2-D nanosheets (NSs) differ from that in their 0-D QD counterparts. The CM QY in Pb-chalcogenide NRs with aspect ratio near 6 is about two times higher than for PbSe QDs with a similar band gap ( Figure   8.(a)). 23 The better performance of NRs can be due to enhanced Coulomb interaction between charge carriers resulting from electric field lines penetrating through the low dielectric medium surrounding the NRs. Interestingly, the Auger decay lifetime was longer in PbSe NRs than in QDs, which is beneficial for charge extraction in a solar cell.

b. Nanocrystal heterostructures
The usual symmetric optical excitations in Pb-chalcogenides can be made asymmetric in a heterostructure with a Cd-chalcogenide. This is possible due to the almost equal energy of the conduction band of these two materials, while the valence band of Cd-chalcogenides is lower in energy than for Pb-chalcogenides. Asymmetric excitations have been realized in core/shell QDs and Janus-like NCs, as discussed below.

i. Core/shell quantum dots
Asymmetric optical excitation has been demonstrated for core/shell PbSe/CdSe QDs and a CM threshold close to twice the band gap (~2.2 E g ) has been realized. 10 The CM QY was found to be higher than for PbSe NRs of a similar band gap (Figure 9). Core/shell QDs have several properties that are beneficial to CM: (i) the PbSe core and the CdSe shell share a common conduction band, but the valence band offset is 1.48 eV. This causes the hole to be strongly confined in the core, increasing the hole energy level spacing which can slow down the cooling rate. (ii) For photon energies more than twice the band gap the optical excitations mainly involve electrons from the CdSe-shell, which is due to the higher absorption cross-section of CdSe. Hence, above twice the band gap the hole is created in the CdSe-shell dominated state, which makes the optical excitation asymmetric with the hole having most of the excess energy. This leads to a CM threshold just above twice the band gap. These QDs also exhibit a higher CM QY due to the slow rate of hole cooling, resulting from the low density of hole states in the PbSe core. Indeed, the hot hole emission lifetime is as long as 6-10 picoseconds, which corroborates that cooling is much slower than CM. Reprinted with permission from ref. 10 . Copyright (2014) Macmillan Publishers Limited.

ii. Janus heterostructures
Kroupa et al. have shown that CdS/PbS Janus hetero-structures have a CM threshold close to twice the band gap (which is determined by the PbS component) and QY higher than core/shell QDs. 28 The Janus structure allows asymmetric optical excitations, see Figure 10. It was theoretically estimated that ~25% of the optical excitations above the CM threshold create hot holes with more excess energy than the electron. The holes get trapped at interfacial states at the CdS/PbS heterojunction within 1 picosecond and undergo CM rather than cooling by phonon emission (Figure 10). Note, that in the PbSe/CdSe core/shell QDs discussed above the hole is confined in the PbSe core and is difficult to extract. Reverse core/shell CdSe/PbSe would be ideal for charge extraction but are difficult to synthesize. In this regard, Janus structures where both charge carriers are accessible from the NC surface are promising candidates for photovoltaics. However, the difficulty is to deposit the Janus NCs so that all the CdS (and PbS) are selectively connected together so that the electron (hole) can move from one particle to another with ease and finally gets extracted at the electron (hole) contacts. CdS/PbS NCs, due to asymmetric optical excitations. Reprinted with permission from ref. 28 .

c. Pb-chalcogenide networks
We have discussed the individual Pb-chalcogenide NCs (QDs, NRs, NSs) and heterostructures (core/shell, Janus) in terms of CM threshold and QY. In heterostructures, the CM threshold is reduced to just above twice the band gap and the QY is higher than in NCs consisting of a Pbchalcogenide only. However, for photovoltaic device applications, the NCs must be coupled to allow charge carrier transport and extraction at external electrodes. This can be realized by mutually connecting NCs to form an assembly in which charges can move from one NC to 24 another. Therefore, from a practical point of view characterization of CM in solid films of coupled NCs is essential.
In the first instance, Pb-chalcogenide QDs were coupled by introducing short organic ligands on their surface or infilling the space between QDs with metaloxides. 13,17,25,26,82 While this yielded encouraging results, a breakthrough in terms of a low CM threshold and relatively high QY was realized by Kulkarni et al. in a percolative PbSe NC network with a band gap of 0.7 eV, which is suitable to exploit CM in a solar cell. 11 In this network, the original QDs are directly connected via strong crystalline PbSe bridges. 83,84 The efficiency of CM was studied using OPTP spectroscopy, see Section 3.c. Figure 11(a) shows that the THz conductivity increases with photoexcitation energy at twice the band gap. Interestingly, a stepwise behavior was found for the QY vs. the band gap multiple (Figure 11(b)), which has never been observed for uncoupled QDs in dispersion. The low CM threshold must be due to an asymmetric excitation where the excess energy ends up solely either in the electron or the hole. If a 2 nd VB or CB exists close to twice the band gap then a CM threshold at this energy is possible, as discussed in Section 4. Electronic structure calculations on percolative networks are needed to corroborate the occurrence of such asymmetric transitions.

d. Si nanorods
In bulk Si, the predominant semiconductor utilized in solar cells, the CM threshold is about 3.5 times the band gap and the QY becomes 140% at 4.5 times the band gap. 85  26 Importantly, the CM QY was found to be 1.6 at 2.9 times the band gap which is twice that of Si QDs (Figure 12).   The CM QY is shown as a function of band gap multiple in Figure 14.  to twice the band gap. Reproduced with permission from ref. 48 .

iii. Sn/Pb halide perovskites
Mixed Sn/Pb halide perovskites have a band gap as low as 1.28 eV, which is much more suitable for solar cell applications than the band gap of the perovskites discussed above. 50 Recently, Maiti et al. have shown efficient CM in a bulk Sn/Pb halide perovskite of the composition (FASnI 3 ) 0.6 (MAPbI 3 ) 0.4 . The CM threshold was found to be just above twice the band gap and the QY reaches 2 at 2.8 times the band gap ( Figure 15). 50 Asymmetric excitation in which the excess photon energy is transferred to the electron, is a plausible explanation for the low CM threshold and high QY, as a recent theoretical study showed the presence of a second conduction band close to 2.2 times the band gap. 87 The mixed Sn/Pb halide perovskite has a low exciton binding energy (~16 meV), so that photoexcitation will predominantly lead to the generation of free charges at room temperature, which is important for photovoltaic applications. Also, the bulk structure is better for charge transport than assemblies of NCs.   despite the fact that CM was observed by TA measurements. 28 Therefore, research is required to improve the device architecture for fast charge carrier transport and efficient extraction at the electrodes in a solar cell. 93 Recently, Kim et al. have developed a conductive atomic force microscope (CAFM) system to measure the local photocurrent in PbS QDs (5.4 nm diameter) for different photon energies. 94 The photocurrent was measured between an Au tip decorated with PbS QDs and a graphene layer on a SiO 2 /Si substrate. Interestingly, a step-like CM behavior was found with a threshold close to twice the band gap and near-ideal CM efficiency ( Figure 18). The advantage of this method is that it probes the local current between the QD and Au tip so that charge transport between QDs does not play a role. 34 Barati et al. investigated CM in a TMDC heterostructure consisting of MoSe 2 and WSe 2 . 54 Both in photocurrent and I SD -V G measurements CM was found to occur with QY up to 3.5. In this case, CM is due to an impact ionization-like process induced by the applied source-drain voltage.

Conclusion and future outlook
We have discussed recent advances of research on CM and findings of new materials exhibiting near-ideal CM. It is of interest that CM with low threshold and appreciable QY has been found to occur in two-dimensional and bulk materials. Conditions to be met for the significant impact of CM in photovoltaics are: (i) asymmetric photoexcitation in which the excess photon energy is transferred predominantly to the electron or the hole so that the CM threshold can be close to twice the band gap, (ii) the exciton binding energy must be sufficiently small to generate free charge carriers, and (iii) charge carrier mobilities need to be high enough for efficient charge carrier transport and collection at electrodes in a device.
These conditions have been realized to a large extent in percolative PbSe networks, a bulk Sn/Pb halide perovskite, and MoTe 2 . It appears that quantum confinement is not strictly required for efficient CM. Therefore, future research should also focus on two-dimensional and bulk-like materials.