Optoelectronic Properties of Tin–Lead Halide Perovskites

Mixed tin–lead halide perovskites have recently emerged as highly promising materials for efficient single- and multi-junction photovoltaic devices. This Focus Review discusses the optoelectronic properties that underpin this performance, clearly differentiating between intrinsic and defect-mediated mechanisms. We show that from a fundamental perspective, increasing tin fraction may cause increases in attainable charge-carrier mobilities, decreases in exciton binding energies, and potentially a slowing of charge-carrier cooling, all beneficial for photovoltaic applications. We discuss the mechanisms leading to significant bandgap bowing along the tin–lead series, which enables attractive near-infrared bandgaps at intermediate tin content. However, tin-rich stoichiometries still suffer from tin oxidation and vacancy formation which often obscures the fundamentally achievable performance, causing high background hole densities, accelerating charge-carrier recombination, lowering charge-carrier mobilities, and blue-shifting absorption onsets through the Burstein–Moss effect. We evaluate impacts on photovoltaic device performance, and conclude with an outlook on remaining challenges and promising future directions in this area.

M etal halide perovskites have recently emerged as an exciting new class of semiconductors for solar energy generation, with device efficiencies now competing with those of commercial silicon cells. 1 To date, highest reported power conversion efficiencies (PCEs) for single-junction devices have relied on the exceptional performance of lead-based perovskites, which offer strong absorption, 2 long charge-carrier lifetimes and diffusion lengths, 3−5 and high defect tolerance. 6−8 However, the lowest bandgaps attainable for lead halide perovskites are around 1.5 eV, 9 higher than the value of ∼1.3 eV required for maximum theoretical efficiencies of single-junction devices. 10,11 Together with concerns about toxicity of lead in its soluble form, 12 these issues have led to increased research on alternative metal halide semiconductors. 13−16 Currently, the most promising materials to address these issues are mixed-metal tin−lead halide perovskites of stoichiometry ASn x Pb 1−x X 3 , where the A-site is typically occupied by formamidinium (FA + ), methylammonium (MA + ), cesium (Cs + ), or a mixture thereof, and the X-site mostly by iodide (to achieve lowest bandgaps), but bromide inclusion has also been reported. 17 Iodide-rich versions of these materials offer bandgap tunability between 1.2 and 1.6 eV (see below) and have become the leading choice for the narrow-bandgap absorber layer in all-perovskite tandem cells, as summarized in recent reviews. 17−19 Such devices combine the enhanced efficiency of a multi-junction architecture with lower processing temperatures and greater compositional tunability compared with perovskite-silicon tandem cells. The highest reported power conversion efficiencies of all-perovskite tandem cells incorporating mixed tin−lead halide perovskites have now reached 25% 20 for 4-terminal cells, and 25.6% for 2terminal cells, 21 while single-junction cells with PCEs near 21% have also just been reported. 21,22 While tin−lead halide perovskites have clearly excelled in photovoltaic devices, knowledge of their underlying optoelectronic properties is still emerging. Here, one obstacle has been the interplay of intrinsic (fundamental) effects and extrinsic effects deriving from their defect chemistry. Such issues are particularly prominent in the tin-rich stoichiometries, in which tin in its 2+ state is unstable to oxidation and vacancy formation, 23−25 resulting in a high background doping density of holes. 26−29 As a result, the field is still struggling to probe the truly fundamental limits of optoelectronic performance, which may be masked by such unintentional doping. This review aims to unravel such effects, providing a clear view of how fundamental and extrinsic mechanisms shape the optoelectronic properties of tin−lead halide perovskites. We explain the underlying scientific concepts governing the peculiar effect of bandgap bowing in these materials, and discuss how exciton binding energies, charge-carrier cooling, and maximum attainable charge-carrier mobilities vary along the tin−lead series. We further summarize how the defect chemistry of tin−lead halide perovskites often dominates their optoelectronic properties, accelerating charge-carrier recombination, lowering charge-carrier mobilities, and blue-shifting absorption onsets through the Burstein−Moss effect. For each of these phenomena, we discuss the impact on photovoltaic device performance, and conclude with an outlook on the remaining challenges and promising future areas of research for tin−lead halide perovskites. We hope the analysis provided will allow for a complete understanding of these materials that facilitates their implementation in photovoltaic and lightemitting devices.

OXIDATION OF Sn 2+ , TIN VACANCY FORMATION,
AND SELF-DOPING One profound difference between tin halide perovskites and their lead halide counterparts derives from the instability of Sn 2+ against oxidation to Sn 4+ and the propensity for tin vacancy formation. This unfavorable defect chemistry has a significant extrinsic influence on the optoelectronic properties of mixed tin−lead perovskites, in particular their tin-rich compositions, which will be examined in detail in this review. To aid discussion, we therefore first briefly review the mechanisms governing tin oxidation, as well as tin vacancy formation and its unintentional consequence of introducing a large density of background holes.
From a basic chemical perspective, lead is considered to be most stable in the 2+ oxidation state; however, such stability decreases for lighter group 14 metals, with the result that tin favors the 4+ oxidation state and can be readily oxidized in tin halide perovskites 30,31 as well as in tin precursor solutions. 32,33 Such chemical instability of tin-rich metal halide perovskites introduces significant decomposition pathways, which also render them less stable to oxygen and moisture than lead halide perovskites. 24,25 The oxidation of Sn 2+ may form part of a chemical conversion that introduces secondary phases within an ASnI 3 perovskite, such as SnI 4 or the vacancy-ordered compound A 2 SnI 6 , in which tin exists in its Sn 4+ form. 24,34 Introduction of oxygen favors such processes, resulting in facile decomposition pathways with end products of AI, SnO 2 , and SnI 4 , 24 making these materials profoundly unstable in air. 31 Substitution of lead for tin will gradually raise hurdles to metal oxidation, such that majority-lead compositions instead revert to their precursor components as part of their favored decomposition pathway. 24,35 From an electronic perspective, the underperformance of tin halide perovskites is more readily understood within the concept of tin vacancy formation. Tin iodide perovskites exhibit much lower ionization energies than their lead-based counterparts, 35 because of their reduced spin−orbit coupling (tin is lighter than lead). 36,37 Density functional theory calculations have shown that tin vacancies are highly stable under these conditions, 23,35,38 and may also create a locally iodine-rich environment that promotes the oxidation of Sn 2+ and the chemical conversions described above. 35 Easily formed, both tin vacancies and iodide interstitials generate defect levels just below the valence band edge, where they capture valence band electrons, effectively releasing free holes. Such unintentional p-type doping is therefore ubiquitous in tin iodide perovskites, for which our literature survey 26−29,31,39−42 finds experimentally reported hole densities to range predominantly around 10 17 −10 20 cm −3 (see Figure 1 for some examples). When metal content is reduced to 70−90% tin by substitution with lead, reported values 42 fall to around 10 17 −10 18 cm −3 , and to 10 15 −10 17 cm −3 for 50−60% tin content. 5,20,21,32,42−46 Once lead is the predominant metal, background doping densities are rarely mentioned for tin−lead perovskites, meaning they most likely fall well below the density of photoexcited charge-carriers under solar illumination conditions (10 15 −10 16 cm −3 ) 47 and therefore have relatively little impact on photovoltaic device operation. Clearly, lead-rich tin−lead halide perovskites exhibit an inherently lower propensity for metal vacancy formation and hole doping, which has been ascribed to increased hurdles to oxidation 24,30 following a drop in the valence band maximum (a rise in ionization energy). 35 As discussed in detail below, the presence of tin vacancies and the resulting unintentional hole doping detrimentally affects optoelectronic performance of tin−lead halide perovskites. A wide range of strategies has therefore been explored to prevent such effects, including the use of additives to act as reducing agents or tin sources, control of crystallization, partial ion substitution, and reduced dimensionality. A full discussion of these is beyond the scope of this review, and we refer the reader to several existing reviews for full details. 25,48−50 Currently, the most frequently utilized approach involves the addition of SnF 2 to the precursor solution, which reduces the formation prospects of tin-poor stoichiometries. 28,29,42 As Figure 1 illustrates, SnF 2 addition to tin−lead halide perovskites considerably lowers the density of background holes (p-type doping), 28,29,42 in particular for tin-rich stoichiometries. We note that as such mitigation strategies against tin oxidation and vacancy formation are further refined, the intrinsic optoelectronic properties discussed in this review are expected to become more prominent and relevant to device performance.
The defect chemistry of tin−lead halide perovskites often dominates their optoelectronic properties, accelerating charge-carrier recombination, lowering charge-carrier mobilities, and blueshifting absorption onsets through the Burstein−Moss effect.
One profound difference between tin halide perovskites and their lead halide counterparts derives from the instability of Sn 2+ against oxidation to Sn 4+ and the propensity for tin vacancy formation.

BANDGAP TUNABILITY AND BOWING
One prominent reason for tin−lead iodide perovskites being particularly attractive for photovoltaic applications is their bandgap tunability in the range of 1.2−1.6 eV, which encompasses values required for optimum single-cell efficiencies (∼1.3 eV) 10,11 as well as suitable low-bandgap candidates for bottom cells in all-perovskite tandem devices. 17,18 As Figure  2 illustrates, the wide range of bandgap energies offered by the ASn x Pb 1−x I 3 series partly results from significant bandgap bowing, meaning that the alloyed perovskites exhibit lower bandgaps than either of the "parent" compositions APbI 3 or ASnI 3 . 51−57 In addition, smaller variations in bandgap can be obtained from substitution of A-cations (see Figure 2) or halide anions. Mixed tin−lead iodide perovskites thus offer a highly attractive bandgap tuning range for multi-junction photovoltaic devices, light-emitting diodes, and photodetectors.
Bandgap bowing is not uncommon in alloyed semiconductors, including, e.g., in GaN x As 1−x , GaAs x Sb 1−x , CdSe x Te 1−x , and ZnS x Te 1−x . 58 To quantify the extent of such bowing, the bandgap bowing parameter b is usually defined by the following equation: where E g (x) is the bandgap energy of the mixed tin−lead halide perovskites, E Sn and E Pb are the bandgaps of the tin-only and lead-only halide perovskites, respectively, and x is the fraction of tin included on the metal site. The first two terms represent a linear change in bandgap with composition between the two end points, while the third term captures any quadratic (parabolic) deviation from such linearity. While bandgap bowing is prominent in tin−lead halide perovskites, yielding experimental values of b for a range of A cation compositions of between 0.5 and 0.9 at room temperature, 42,53,57,59 it is almost absent for metal halide perovskites upon halide substitution. 9,60 Understanding the origins of bandgap bowing in tin−lead perovskites is therefore helpful to the realization of lowest achievable bandgaps. The emerging literature consensus points toward bandgap bowing in mixed tin−lead perovskites arising mostly from a combination of structural relaxation effects and chemical effects mediated by spin−orbit coupling. 53,57,61−64 Structural relaxation accommodates the random placement of differently sized lead and tin ions throughout the lattice by bond bending, as a result of which the structure varies locally and is not well described by an average value. 53 Because changes in the metal halide bond angle in these perovskites alter the bandgap, this structural relaxation contributes significantly to bandgap bowing, as has been identified both from first-principles calculations 61 and through experiments. 53,57,62 The magnitude of bowing varies somewhat with the choice of A cation and halide anion, which mediate structural relaxation effects through microstrain in the crystal structure. 57 Chemical effects, wherein the atomic orbitals of tin and lead that respectively form the valence band and conduction band edges are mismatched in energy, also have a significant influence on bandgap bowing. 57,63 The  contribution of spin−orbit coupling to bandgap bowing, 61 previously disputed, 63 can also be understood in relation to chemical effects as spin−orbit coupling enhances the influence of lead on the conduction band minimum so that the mismatch in energy between lead and tin orbitals becomes significant. 64 In addition to the local variations in structure which contribute to bandgap bowing, the metal ratio in mixed tin− lead perovskites affects overall crystal structure. The smaller size of tin cations compared to lead cations results in decreasing lattice parameter values as tin content increases, as has been observed via shifts in X-ray diffraction peak position with tin content. 42,52,55,56,65,66 In cases where the leadonly and tin-only perovskite compositions have different crystal structures, a transition between structures must occur at intermediate tin content. In MASn 1−x Pb x I 3 perovskites, a change from the tetragonal structure encountered for lead-rich compositions to a (pseudo)cubic structure at the tin-rich end has been observed to take place at 50% tin content. 30,54−56 By contrast, increasing tin content in perovskites with an otherwise pseudocubic structure can lead to increasing tetragonal distortion as a result of tin vacancy formation and associated lattice strain, an extrinsic effect which is suppressed by additives to control oxidation of tin. 42 The trends in crystal structure across the tin−lead compositional range are therefore influenced by the degree to which unwanted tin vacancy formation can be prevented, as well as by the structures of the tin-only and lead-only perovskites which can vary with A cation composition. 52 Further investigation of structural trends in tin−lead perovskites with A cations other than methylammonium, and with careful control of tin vacancies, may reveal whether 50% tin content is a common threshold for structural changes to occur and may develop greater understanding of the intrinsic influences of metal composition on crystal structure.

BURSTEIN−MOSS EFFECT
One peculiarity of tin−lead halide perovskites is their tendency to develop significant blue shifts of the absorption onset in the presence of strong tin vacancy formation. 29,42,67 Such permanent Burstein−Moss effects 68,69 arise because significant hole doping causes a downshift of the Fermi-level and a depletion of the electronic states near the top of the valence band (see schematic in Figure 3a). Consequently, electronic transitions between the highest occupied states in the valence band and the conduction band now occur at energies exceeding the intrinsic bandgap energy. Such effects are well known for classical inorganic semiconductors such as InSb, whose absorption onset has been found to blue shift considerably for high doping levels. 68  near their onset for MASnI 3 films produced with and without SnF 2 additive. Without SnF 2 present to mitigate tin vacancy formation, the resulting background hole density causes blueshifted onsets whose oscillator strength is weakened even for photon energies higher up into the band, 67 possibly as a result of exciton screening. Similar effects are observed for FASnI 3 , for which they are shown to disappear following the addition of as little as 1−5% SnF 2 during film processing (see Figure  3c,d). 29,42 Interestingly, the energy of the emission (photoluminescence) peak is hardly affected by the depletion of electronic states near the top of the valence band, because photoexcited electrons first relax to the conduction band edge, then subsequently recombine with holes to fill states near the valence band edge, as indicated schematically in Figure 3a. As a result, the Burstein−Moss effect causes sizable Stokes shifts between absorption and emission in heavily doped tin-rich tin−lead iodide films that may exceed a few hundred meV. 29,42,67 As Figure 3d illustrates, the most substantive Stokes shifts occur for materials with tin content in excess of 70%, because these are most susceptible to heavy tin vacancy formation. 42 Interestingly, the Burstein−Moss effect may also interfere with a correct assessment of band bowing from absorption measurements, because the blue-shift in absorption for tin-rich compositions artificially enhances the bowing parameter b and lowers the value of tin fraction at which the minimum bandgap is perceived to occur. 42 Such issues may thus partly contribute to the observed variation in bowing parameter across literature studies, evident in Figure 2.
Overall, the susceptibility of tin-rich perovskites to the Burstein−Moss effect is clearly detrimental for photovoltaic operation, because it will directly translate into open-circuit voltage losses (from the perspective of Stokes shifts) or photocurrent losses (if viewed as an absorption bleach). In addition, any gradual shifts occurring through tin vacancy formation in the energy of the perovskite's valence band maximum over time may also detrimentally affect its alignment with the energy levels of the hole extractor layer. Therefore, such shifts ultimately present a hurdle to the long-term stability of tin−lead perovskite solar cells, unless they can be reliably prevented from occurring (e.g., by additives such as SnF 2 , or impermeable encapsulation) over the projected lifetime of the device.

CHARGE-CARRIER RECOMBINATION
The prevalence of tin oxidation and background hole doping in tin halide perovskites usually dominates their charge-carrier recombination dynamics. Additional charge-carrier recombination pathways introduced by these extrinsic impurities include both non-radiative Shockley−Read−Hall recombination 21,70 and pseudo-monomolecular recombination of photoexcited electrons with background holes. 5,20,29,53,71 The latter process has been shown to be radiative, 71 being fundamentally identical to that of band-to-band recombination of photogenerated electrons and holes, but scales linearly with the photogenerated charge-carrier density because the density of background holes typically exceeds that of photogenerated holes in these materials. 29,71 The introduction of such radiative recombination pathways means that tin halide perovskites often display relatively high photoluminescence quantum efficiencies, despite their short charge-carrier lifetimes. 72,73 Such luminescence enhancement with respect to lead-only counterparts is particularly prominent at low photo-generated charge-carrier densities, given the pseudo-monomolecular nature of the doping-induced radiative recombination. 71 While the high luminescence efficiency of tin halide perovskites is somewhat of a red herring for photovoltaic devices, it may however be advantageous for light-emitting and lasing applications. 72 The additional charge-carrier recombination pathways introduced by tin oxidation and vacancy formation may to a certain extent be ameliorated through variations in processing, such as the addition of SnF 2 29,42,74 (see, e.g., Figure 4a). To examine charge-carrier lifetimes thus achieved for tin−lead halide perovskites, we display in Figure 4b 5,20,21,32,44,79 to suppress unwanted tin vacancy formation. Only two of the plotted lifetimes correspond to thin-film samples prepared without such additives, 74,81 the other exceptions being based on use of a metallic tin precursor, 78 or tuned crystal growth methods. 77,80 Interestingly, the figure demonstrates that short charge-carrier lifetimes are prevalent for all tin−lead halide perovskites with tin content in excess of 60%, for which they rarely exceed a few nanoseconds. These findings suggest that current mitigation approaches still fail to fully address tin vacancy formation at the tin-rich end of the compositional range that is particularly prone to such effects. We note that for the popular mitigation technique of adding SnF 2 to precursor solutions, a cause may be that, while a tin-rich environment reduces the likelihood of tin vacancy formation and hole doping, it may unfortunately also create tin interstitials and iodide vacancies that constitute deep-level traps. 35 Therefore, neither a tin-poor nor a tin-rich environment will suffice, 42 and it is unlikely that this approach can ever fully suppress fast recombination pathways in tin-rich tin−lead halide perovskites. However, some promising alternative approaches based on metal substitution have recently been implemented 44 and examined 35 which may ultimately succeed in stabilizing these materials. In addition, for intermediate tin−lead compositions, such as the case of FAPb 0.5 Sn 0.5 I 3 highlighted in Figure 4c, avoiding oxygen exposure at all times through device encapsulation may well suffice to suppress tin vacancy formation. 44 Interestingly, lead-rich perovskites at the other end of the tin−lead compositional spectrum also exhibit deficiencies in optoelectronic properties, albeit unrelated to tin vacancy formation. Tin−lead iodide perovskites with tin content between 0.5 and 20% display short PL lifetimes, broadened spectra, increased Stokes shifts, a drop in PL quantum yield, and large Urbach tails, compared with their lead-only Tin−lead iodide perovskites are particularly attractive for photovoltaic applications because their bandgap is tunable in the range of 1.2−1.6 eV, which encompasses values required for optimum single-cell efficiencies and bottom cells in all-perovskite tandem devices.
counterparts. 42,53,82,84 Such effects can also be seen in our literature survey of mixed tin−lead halide perovskites ( Figure  4b) which indicates that for tin content between 0.5 and 20%, charge-carrier lifetimes rarely exceed 100 ns. Addition of minute fractions of tin to lead halide perovskites therefore appears to introduce non-radiative traps that are unlikely to be linked with changing band structure properties, as these vary only gradually from lead to tin halide perovskites. 35 Instead, such underperformance could potentially be related to the large mismatch in metal-iodide bond lengths deriving from the large ionic size discrepancy of tin compared with lead. 85 Such mismatch may cause increased energetic disorder, 53,82 or enhanced polaronic effects 85 when only small fractions of the much smaller tin are introduced into lead-only iodide perovskites.
Overall, the interplay between these two effects means that mixed tin−lead halide perovskites currently exhibit the highest charge-carrier lifetimes and performance within the relatively narrow compositional range with tin content between 30 and 60%. 82 As Figure 4b shows, within this range, charge-carrier lifetimes in excess of 1 μs have been achieved on several occasions. 5,20,77 Fortunately, as Figure 2 illustrates, these compositions also offer the lowest achievable bandgaps for tin−lead halide perovskites, making them highly suitable for bottom cells in multi-junction photovoltaic devices, or highefficiency single-junction cells. 17−19 However, the use of tinonly halide perovskites as entirely lead-free absorber layers may well require radical new approaches to boost charge-carrier lifetimes above the currently reported maximum values of at most a few tens of nanoseconds. Progress here will require carefully balanced, stable control of tin content, given that a tin-poor environment causes tin vacancies and ensuing hole doping, while a tin-rich environment generates tin interstitials and iodide vacancies that constitute deep-level traps. 23,35 Any new approaches on film processing or additives will therefore need to eliminate such trade-offs between formation of these three most prominent defects in tin halide perovskites.

CHARGE-CARRIER MOBILITIES
A sufficiently high charge-carrier mobility is a prerequisite for efficient charge-carrier extraction in photovoltaic devices. Intriguingly, tin-rich metal halide perovskites offer the prospect of fundamental charge-carrier mobilities that are significantly higher than those of their lead halide counterparts. Unfortunately, such intrinsic advantages are all too often counteracted by the extrinsic lowering of mobilities in the presence of tin vacancy formation and the associated hole doping. We discuss below the intrinsic and extrinsic mechanisms that combine to govern the mobility of charge carriers in tin−lead halide perovskites.
The fundamental limit to the charge-carrier mobilities of tin−lead halide perovskites derives from interactions between charge carriers and longitudinal optical (LO) phonons of the polar metal halide lattice, 86,87 captured in Froḧlich's theory. 88−90 Experimental evidence for the dominance of this mechanism comes from analysis of the temperature dependence of spectral emission broadening 67,86 and charge-carrier mobilities. 29,91−93 As the example in Figure 5b shows, the charge-carrier mobility in FASnI 3 rises with decreasing temperature, suggesting that for sufficiently high-quality films, the charge-carrier mobility is governed by coupling to phonon modes. Within the Froḧlich model, charge-carrier motion is impeded because the macroscopic electric field generated by a longitudinal optical phonon interacts with charge carriers, leading to a local lattice distortion around the charge, termed a "large polaron". 88  energy, Ry = 13.606 eV the Rydberg constant, m*/m e the effective mass m* of the charge carrier as a fraction of the free electron mass m e , and ϵ Fr −1 = ϵ ∞ −1 − ϵ static −1 is determined by the static and high-frequency limits of the dielectric function with respect to the LO phonon resonance. 89,90,94,95 Therefore, expected trends across a series of tin−lead iodide perovskites can be readily estimated from changes in values of the dielectric function, LO phonon energies, and effective masses. From such considerations, a significant enhancement in charge-carrier mobilities should be expected toward the tinrich end of the series. 95 Effective masses of charge carriers drop appreciably 36,95 when moving from lead-to tin-based perovskites, commensurate with the lowering of the bandgap. In addition, optical phonon modes upshift in frequency toward the tin-rich compounds, as recently observed experimentally 42 and expected theoretically 36,95 for the lighter atomic mass of tin compared with lead. Since the Froḧlich mobility increases with the dimensionless parameter β = ℏω LO /k B T, an upshift in LO phonon frequencies will effectively shift temperaturedependent mobility curves to higher temperatures, thus the room-temperature mobility is enhanced. Such trends of increasing mobilities with increasing tin content can indeed be observed for carefully passivated tin−lead perovskite films made as part of a single fabrication and measurement series, as illustrated in Figure 5a. They may also be discerned in careful literature surveys, 87 which have highlighted higher cross-study averages of charge-carrier mobilities in tin iodide perovskites compared with lead iodide perovskites.
Despite such discernible underlying trends, charge-carrier mobility values reported for tin iodide perovskites vary by 3 o r d e r s o f m a g n i t u d e b e t w e e n d i ff e r e n t s t u dies, [26][27][28]30,31,38,42,51,96,97 highlighting a significant influence of extrinsic effects linked to fabrication techniques (as well as variations in measurement protocols 87 ). For such tin-rich compositions, a particularly strong influence again derives from tin vacancy formation, which, as discussed above, causes background hole densities as high as 10 17 −10 20 cm −3 . The remnant dopant site (the tin vacancy) must be negatively charged to preserve charge neutrality, and consequently acts as a scattering site to charge carriers, lowering their mobilities. Figures 1a and 5b demonstrate how addition of SnF 2 during the fabrication process causes a substantial increase in chargecarrier mobilities by lowering the materials' propensity toward tin vacancy formation and the resulting scattering. 28,29 Such effects are also evident in the temperature dependence of the charge-carrier mobility for thin films of FASnI 3 , illustrated in Figure 5b. The presence of ionized tin defects leads to shallower rises in mobility toward low temperature 29 because Coulombic interactions with such impurities become more effective as the thermal velocity of charge carriers is slowed. 88,98 Such lowering of charge-carrier mobility with increasing doping concentration is also well known to occur for a range of inorganic semiconductors. 98 Figure 5c contrasts the changes in charge-carrier mobility observed 42 across the tinrich end of a FA 0.83 Cs 0.17 Sn x Pb 1−x I 3 series with those recorded previously for GaAs and InP. 98 As indicated by the solid lines, all of these semiconductors can be well-described by the Hilsum formula 98 which assumes the mobility of charge carriers to be limited by coupling to phonons and scattering off ionized impurities.
Overall, while charge-carrier mobilities are fundamentally enhanced with increasing tin content in tin−lead iodide perovskites, the concomitant susceptibility to tin vacancy formation and the resulting scattering of carriers with these ionized impurities may instead lower the mobilities for highly defective materials. To enhance and preserve charge-carrier extraction efficiencies in photovoltaic cells, it is thus essential for these materials to be stabilized against tin vacancy formation.

EXCITON BINDING ENERGIES
The exciton binding energy E b plays a crucial role for a semiconductor's suitability as a light absorber in photovoltaic applications. On the one hand, values of E b below thermal energies (26 meV at room temperature) are desirable because bound electron−hole pairs (excitons) may then self-dissociate, allowing efficient photocurrent collection of electrons and holes to their respective extraction layers. On the other hand, a low binding energy weakens the absorption coefficient strength near the band edge, because Elliott theory 99 dictates that Coulomb correlations also enhance the oscillator strength of above-gap continuum states that are populated by free charge carriers. As a result, materials with low E b have more gradual absorption onsets, requiring thicker layers that lower chargecarrier extraction efficiencies. On balance, exciton binding energies falling somewhat but not too far below thermal energies are therefore ideal for photovoltaic applications.
In metal halide perovskites, excitons are generally welldescribed by the hydrogenic model of the "Wannier" exciton, whose binding energy is given by 94 where Ry = 13.606 eV is the Rydberg constant, ϵ is the value of the dielectric function, and m r */m e is the reduced effective mass of the electron−hole system, expressed as a fraction of the free electron mass m e . Unsurprisingly, the exciton binding energy for the prototypical lead iodide perovskite MAPbI 3 has already been intensely investigated, 100−107 with a recent survey 94 compiling a room-temperature literature average of 12 ± 7 meV. However, experimental investigations of exciton binding energies in tin halide and tin−lead halide perovskites are still relatively scarce, 29,108,109 mostly because of issues with sample stability and self-doping discussed above. Given these experimental difficulties, we begin by discussing theoretical expectations of changes in exciton binding energy when tin is substituted for lead within a series of tin−lead halide perovskites. As eq 2 shows, E b depends on the reduced effective mass m r * and the value of ϵ, which may vary along the tin−lead perovskite series. First-principles calculations have suggested that the reduced effective mass of charge carriers should drop appreciably (by 20−50%) when moving from lead-iodide perovskites to their tin-iodide counterparts, 36,95 as typical for a lower bandgap material. 88 Magneto-absorption measurements conducted at 2 K have confirmed such trends experimentally, finding a fall of m r * by about 25% as tin content increases from x = 0.2 to 0.8 in MAPb 1−x Sn x I 3 films. 108 From effective-mass considerations alone, the exciton binding energy for tin iodide perovskite would therefore be expected to be lower than for lead-iodide perovskites.
An evaluation of the second critical parameter, ϵ, is considerably more complicated because ϵ is a particularly strong function of frequency in metal halide perovskites, 94 opening a debate 101,110,111 on which value of ϵ should enter eq 2. Self-consistency is an important criterion here, 94,101,112 which requires that when a directly determined value of the exciton binding energy is used to derive a value of ϵ according to eq 2, that value of ϵ must then indeed be encountered at the frequency E b /h. In this context, it is also important to assess whether the energies of optical phonons fall above or below the value of E b , since in the latter case, phonons may no longer be able to follow the motion of the electron−hole pair effectively 113 leading to lower screening and therefore lower effective values of ϵ entering eq 2. Optical phonon modes for mixed tin−lead iodide perovskites have recently been shown to increase in frequency with increasing tin content, as would be expected for lighter tin atoms. 42 As a result, excitons in tin iodide perovskites would thus experience more effective screening by the ionic tin halide lattice, leading again to lower exciton binding energies than those encountered in leadbased counterparts. 112,114 First-principles calculations by Umari et al. based on such self-consistent approaches have indeed suggested that ϵ should increase appreciably with increasing tin content along the tin−lead iodide perovskite series, with the exciton binding energy expected to reduce by over a factor of 2 from lead to tin iodide perovskites. 112 Direct experimental probes of exciton binding energies in tin−lead halide perovskites are still relatively scarce because of complications arising from spontaneous self-doping and the ensuing disorder in these materials. Galkowski et al. conducted low-temperature magneto-absorption measurements at high magnetic fields to examine how E b varies with tin content x in MAPb 1−x Sn x I 3 films. 108 Unfortunately, the authors found that the intrinsic inhomogeneity and instability of these materials meant that magneto-absorption features were less well resolved than in their earlier 101 work on lead halide perovskites. Therefore, discernible higher-lying excitonic features were only visible at very high magnetic field (≥40 T) which made the required extrapolation to zero fields through fan charts unreliable (see Figure 6a). The authors determined a constant value of E b = 16 meV for tin fractions x = 0.2, 0.6, and 0.8, similar to the value they had determined earlier for lead-only counterparts. Such invariance of E b with tin content would be surprising, given the theoretical considerations outlined above. However, as a result of the measurement uncertainties, the authors suggest that these values only comprise upper limits of the actual exciton binding energies in tin−lead perovskites. 101 Another direct approach to determining an exciton binding energy was made by Milot et al. 29 for FASnI 3 thin films through examining photoinduced changes in the THz spectral range at low temperature (5 K). They observed a Lorentzian oscillator resonance feature in the complex transmission spectra that they ascribed to inter-excitonic transitions, given that these features only appeared at low temperature, were independent of excitation fluence, and disappeared in heavily doped films in which excitons would be screened (see Figure  6b). By evaluation of the resonance energy, an exciton binding energy of only 3.1 meV was extracted. Finally, attempts have been made 109  To summarize these considerations, exciton binding energies are theoretically expected to fall as tin content is increased along the tin−lead halide perovskite series, because of a lowering of charge-carrier masses, and an increase in optical phonon frequencies that facilitates effective screening of Coulomb interactions by the polar sublattice. The resulting positive correlation between exciton binding energy and bandgap energy is well known from the case of classic inorganic semiconductors. 88 Further direct measurements of the exciton binding energies in tin-rich tin−lead iodide perovskites would be helpful, given that present studies in the field still appear to be inconclusive, with some pointing toward lower exciton binding energies for tin-rich perovskites, 29,109 while others suggest that the binding energy may potentially be unchanged along the tin−lead iodide perovskite series. 108 Finally, we note that both lead and tin iodide perovskites appear to exhibit exciton binding energies sufficiently below thermal energies at room temperatures to induce efficient charge-carrier separation.

CHARGE-CARRIER COOLING
For tin halide perovskites, the cooling dynamics of charge carriers following above-gap excitation and thermalization have been an interesting subject of debate, following early reports 115 of unusually long (nanoseconds) cooling times in FASnI 3 . Prolonged cooling dynamics could facilitate the long-term goal of hot-carrier solar cells, in which charge carriers are extracted before the excess energy supplied during above-gap photoexcitation has been transferred to the lattice, permitting PCEs above the Shockley−Queisser limit. 116 The time scale on which hot carriers return to the ambient lattice temperature has been observed to vary widely across different metal halide perovskite compositions. 117 The unusually slow cooling 115 initially proposed for tin iodide perovskites prompted suggestions that these could be exceptional candidate materials for realizing hot carrier extraction in solar cells. 118 However, a subsequent investigation instead attributed the nanosecond dynamics observed at the high-energy end of FASnI 3 to slow relaxation between energetically disordered states connected to the oxidation of tin, an extrinsic effect. 119 It is therefore still debated whether these nanosecond dynamics truly reflect a population of hot carriers that could be extracted to increase device voltage.
Meanwhile, when such dynamics are examined in tin−lead halide perovskites over the picosecond time scales during which charge-carrier cooling has typically been found to occur in lead halide perovskites 105,117 and inorganic semiconductors, such as GaAs, 120 the small number of studies conducted so far have reached somewhat inconclusive results. While Savill et al. 119 and Verma et al. 121 reported slowed cooling for tin halide perovskites compared with their lead-based counterparts, Monti et al. 97 reported the opposite, and Ma et al. 74 similar time scales. Verma et al. found slowed picosecond cooling rates for increasing tin content along a mixed tin−lead perovskite series and attributed these effects to slower phonon emission deriving from two effects, stronger screening as tin addition increases the dielectric constant and deformation potential scattering arising from changes in lattice distortion and band structure. 121 However, Monti et al. observed the opposite trend in initial cooling dynamics despite the use of similar excitation conditions, finding shorter cooling times with increasing tin concentration up to 75%. They stipulated that cooling in this Froḧlich regime was dominated by longitudinal optical phonon emission as the dominant mechanism, and attributed the observed acceleration of cooling to an increasing frequency of phonon modes as the lighter tin cation was introduced. 97 Overall, it is clear that further investigation will be required to provide a complete understanding of how composition influences the time scales of charge-carrier cooling in tin−lead halide perovskites. Experimental studies have yielded highly disparate results to date, 74,97,115,119,121 most likely because of differences in both sample quality and the methods by which charge-carrier temperatures are extracted from measured data. Since the propensity of tin-rich perovskites toward tin vacancy formation creates large background charge-carrier densities, additional inter-carrier scattering pathways may become available that will accelerate the loss rate of excess energy.
Here it is worth noting that such large densities of background holes comprise already thermalized "cold" charge carriers which thus offer rapid thermalization pathways to newly photogenerated ("hot") charge carriers. In addition, the energetic disorder caused by the presence of ionized tin vacancies may cause energetic relaxation of charge-carriers slowly migrating through high-energy tails of the available density of states. Such relaxation dynamics may mimic chargecarrier cooling dynamics when examined at the high-energy end of emission peaks, despite being unrelated to the actual cooling processes. 119 Disentangling such disorder-related effects from true charge-carrier cooling dynamics is thus important for a correct experimental determination of cooling time scales. Finally, theoretical investigations may further help to elucidate charge-carrier cooling in these materials, drawing, e.g., on the knowledge available for trends of charge-carrier mobilities across the tin−lead perovskite series, which are also governed by electron−phonon interactions.

■ SUMMARY AND FUTURE DIRECTIONS
Our analysis shows that, from a fundamental perspective, mixed tin−lead halide perovskites have much to offer compared with their lead-only counterparts. Toward the tinrich end of the compositional range, fundamentally attainable mobilities rise, exciton binding energies are expected to fall, and charge-carrier cooling may potentially slow, all of which are potentially beneficial for photovoltaic applications. In addition, significant bandgap bowing allows for attractive nearinfrared bandgaps at intermediate tin content, ideally suited for applications in high-efficiency single-junction photovoltaic cells, or in bottom cells for multi-junction devices. Several challenges still remain with regard to our understanding of the underlying optoelectronic properties of these materials. A full literature consensus on experimentally determined exciton binding energies and hot-carrier cooling dynamics has not yet been reached. A theoretical evaluation of exciton binding energies fully from first-principles approaches is still a complex task because it requires evaluation of the difference between electronic transitions in the presence and absence of Coulomb correlations, which is difficult to achieve accurately when such differences are comparatively small. First-principles calculations specifically for mixed perovskite stoichiometries are computationally demanding, because of the need for large supercells to accurately reflect different compositions and the resulting configurational disorder. Nevertheless, the general picture emerging at this point is that the intrinsic optoelectronic properties of tin−lead halide perovskites are highly suited to efficient solar cell operation.
In reality, the most challenging aspect, therefore, remains attaining control over the extrinsic defect chemistry of tin−lead halide perovskites. As our review has highlighted, the propensity of tin-rich compositions toward tin oxidation, vacancy formation, and the ensuing unintentional background doping with holes has many adverse effects on their real-world optoelectronic performance. At high tin vacancy density, the Burstein−Moss effect leads to blue shifts of absorption onsets that lower light-harvesting efficiencies and potentially cause energetic misalignment with extraction layers, with adverse effects on the open-circuit voltages and photocurrents of solar cells. Tin vacancy formation further accelerates charge-carrier recombination significantly, by causing enhanced non-radiative Shockley−Read−Hall recombination and radiative pseudomonomolecular recombination of photogenerated electrons with a large pool of background holes. Moreover, when tin vacancies release holes into the valence band, they become negatively charged, acting as local scattering sites and lowering the mobility of charge carriers. Therefore, the combined lowering of charge-carrier lifetimes and mobilities results in significantly reduced charge-carrier diffusion lengths in tin− lead halide perovskites exhibiting high tin vacancy densities.
Nevertheless, impressive performance of solar cells has been achieved when mixed tin−lead halide perovskites of intermediate tin content (30−60%) have been incorporated into photovoltaic devices. 17−21 Our review shows that in this intermediate range, tin−lead perovskite materials offer the best of both worlds: lowest attainable bandgaps as a result of bandgap bowing, charge-carrier lifetimes similar to those of lead-only counterparts (often in excess of microseconds) and moderate charge-carrier mobility enhancements over those of lead halide perovskites. However, it remains to be seen whether such enhanced performance will indeed be stable over decades, in particular given the instability of the Sn 2+ oxidative state in the presence of oxygen. In addition, as the prevalence of tin vacancy mediated defects recedes with better material processing and passivation protocols emerging, other defectmediated recombination pathways may come to our attention. In this context, it is interesting to note that inclusion of only a low fraction of tin into lead halide perovskite leads to highly defective materials, 82 for reasons that are not yet fully understood. In addition, it is becoming apparent that the currently most popular technique to prevent tin vacancy formation, i.e., SnF 2 addition, may unfortunately introduce additional non-radiative recombination pathways 29 through local tin-rich environments that cause tin interstitials and iodide vacancies, which constitute deep-level traps. 35 Since this approach may therefore fail to fully suppress fast charge-carrier recombination pathways, new passivation strategies are urgently required, which may for example include metal substitution. 35,44 Finally, fully lead-free tin iodide perovskites (ASnI 3 ) still suffer from tin vacancy formation that is relatively illcontrolled. Our literature survey discussed above indicates that charge-carrier lifetimes rarely exceed a few nanoseconds, and perhaps unsurprisingly, best single-cell PCEs have so far been much lower than those for lead-only counterparts, with recently reported values reaching slightly above 13%. 83,122 While such PCEs are far below the theoretically attainable The general picture emerging at this point is that the intrinsic optoelectronic properties of tin−lead halide perovskites are highly suited to efficient solar cell operation.
single-junction limit, 10,11 it is worth acknowledging that betterperforming lead-free contenders have yet to emerge from the plethora of metal halide semiconductors explored to date. 13 Attaining long-term, sustainable control over tin vacancy formation in lead-free tin iodide perovskites, and its detrimental effects on bandgaps, charge-carrier lifetimes, and mobilities, is therefore perhaps the most challenging goal of all.