Speed Limit for Triplet-Exciton Transfer in Solid-State PbS Nanocrystal-Sensitized Photon Upconversion Lea Nienhaus,† Mengfei Wu,‡ Nadav Geva,§ James J. Shepherd,† Mark W. B. Wilson,†,∥ Vladimir Bulović,‡ Troy Van Voorhis,† Marc A. Baldo,‡ and Moungi G. Bawendi*,† †
Department of Chemistry, ‡Department of Electrical Engineering and Computer Science, and §Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *
ABSTRACT: Hybrid interfaces combining inorganic and organic materials underpin the operation of many optoelectronic and photocatalytic systems and allow for innovative approaches to photon up- and down-conversion. However, the mechanism of exchange-mediated energy transfer of spin-triplet excitons across these interfaces remains obscure, particularly when both the macroscopic donor and acceptor are composed of many separately interacting nanoscopic moieties. Here, we study the transfer of excitons from colloidal lead sulfide (PbS) nanocrystals to the spin-triplet state of rubrene molecules. By reducing the length of the carboxylic acid ligands on the nanocrystal surface from 18 to 4 carbon atoms, thinning the effective ligand shell from 13 to 6 Å, we are able to increase the characteristic transfer rate by an order of magnitude. However, we observe that the energy transfer rate asymptotes for shorter separation distances (≤10 Å) which we attribute to the reduced Dexter coupling brought on by the increased effective dielectric constant of these solid-state devices when the aliphatic ligands are short. This implies that the shortest ligands, which hinder long-term colloidal stability, offer little advantage for energy transfer. Indeed, we find that hexanoic acid ligands are already sufficient for near-unity transfer efficiency. Using nanocrystals with these optimal-length ligands in an improved solid-state device structure, we obtain an upconversion efficiency of (7 ± 1)% with excitation at λ = 808 nm. KEYWORDS: upconversion, triplet exciton transfer, triplet−triplet annihilation, Dexter transfer, dielectric constant
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The basic process in excitonic upconversion is triplet−triplet annihilation (TTA) in organic semiconductors, termed as annihilators, where two spin-triplet excitons on neighboring molecules interact to generate one higher-energy singlet excited state.11−13 A fundamental advantage over direct nonlinear frequency conversion is that the incident energy is stored in long-lived triplet exciton states, which reduces the required light intensity for efficient upconversion.11 However, these triplets are generally dark (not directly optically accessible) due to spin-selection rules. Therefore, a sensitizer is required to absorb the incident light and transfer this energy to the triplet state of the annihilator prior to TTA. While this has conventionally been accomplished with phosphorescent metal−organic complexes,7,11,14,15 recent studies have shown that lead chalcogenide (PbS, PbSe) NCs are efficient sensitizers for the triplet state of the annihilator rubrene.1,4,6,16 The simple
ecent observations of efficient exchange-mediated energy transfer between organic molecules and inorganic colloidal nanocrystals (NCs) demand a detailed understanding of this archetypal organic/inorganic interface.1−4 In particular, it is important to refine our understanding of the dipoleless process by which “dark” spintriplet excitons transfer, and identify which hurdles, if any, must be overcome for the development of efficient devices. To address this question, we study the kinetics and distance dependence of triplet exciton transfer (TET) across the heterointerface in solid-state films for excitonic photon upconversion. These devices employ excitons as intermediates to enable two absorbed low energy photons to be converted into one emitted higher energy photon at low incident intensities.4−7 This technology has the potential to overcome the Shockley−Queisser limit8 in single-junction photovoltaics by sensitizing silicon to sub-bandgap light9 and also has other possible applications in biological imaging, cost-effective nightvision cameras, and photocatalysis.10 © XXXX American Chemical Society
Received: March 23, 2017 Accepted: July 20, 2017
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NC32 and rubrene,33,34 we expect a type-I heterojunction to form, and as a result, single electron or hole transfer should be disfavored thermodynamically. However, concerted transfer of the electron and hole (TET) to the first excited triplet state in rubrene (1.14 eV)35 is expected to be exothermic, as shown in Figure 1a. The device structures for time-resolved photoluminescence (TRPL) studies are depicted in Figure 1b. Devices are fabricated by spin coating a monolayer film (compare SI, Figure S2, for typical monolayer morphology determined by AFM) of PbS NCs with aliphatic ligands ranging from stearic acid (18 C atoms) to butyric acid (4 C atoms) on a
synthetic bandgap tunability and cost-efficient scalability of these nanomaterials combined with their physical properties make them prime candidates as infrared sensitizers.17−19 It has been believed that the sensitization of the triplet state in rubrene occurs via exchange-mediated energy transfer (Dexter transfer).4,6 Such transfer is possible because of the electronic fine-structure of the NCs: the exchange interaction between singlet and triplet states at the band-edge is on the order of 1−25 meV.20,21 However, due to the strong spin−orbit coupling in PbS NCs, it is not spin, but total angular momentum, which is a good quantum number.20 This interaction mixes the electronic orbital and spin states so that the lowest exciton will have both singlet and triplet character.21 As a result, the initial directly optically excited exciton on the NC is able to undergo energy transfer into a dark, spinforbidden triplet state in rubrene. This occurs while conserving the spin state, thus making the transfer a spin-allowed process.22 This exchange-mediated energy transfer mechanism is known to be based on a direct wave function overlap between the donor and acceptor states,23 which allows an electronic coupling between the two states. Due to the requirement of a wave function overlap, the magnitude of this coupling is expected to be exponentially dependent on the distance between the donor and the acceptor.22,23 However, there are additional parameters that have the potential to influence the magnitude of the exchange interaction. (i) The exact nature of the electronic state on the PbS NC participating in the exchange mediated energy transfer is unknown. Due to the electronic fine-structure at the band-edge20,21 there are many states that can potentially electronically couple to the rubrene wave function and undergo energy transfer. Additionally, defects,24 midgap states,25 and surface states26 have been shown to participate in chargetransport and energy-transfer processes. (ii) Large donor− acceptor separations, exceeding the 10 Å maximum length scale usual for efficient Dexter transfer,27 have been observed in efficient exchange-mediated energy transfer involving NCs.28,29 This has been attributed to a large wave function leakage in the lead chalcogenide NC,30 or an incomplete surface ligand coverage resulting in a reduced spacing between the donor and acceptor.29 (iii) Lead chalcogenides have a very large dielectric constant, and the dielectric screening of the exciton prior to triplet transfer is expected to have a large impact on the electronic coupling between the donor and acceptor wave functions.31 (iv) The exact mechanism of triplet exciton transfer is still under debate for similar systems. An alternative mechanism to triplet exciton transfer via concerted electron and hole transfer (Dexter transfer) has been reported recently.32 Here, triplet generation on surface-bound TIPSpentacene occurs through the initial transfer of a single charge, yielding a long-lived charge-transfer intermediate followed by the delayed transfer of the second charge. To elucidate the underlying physics of the TET mechanism for the solid-state devices reported here, we investigate the effect of the NC surface-passivating ligand length, which functions as the spacer or tunneling barrier between the NC and rubrene, on the resulting TET rate.
Figure 1. Solid-state device structures (not to scale). (a) Energylevel diagram of the PbS NC/rubrene interface investigated. We represent the rubrene triplet as a bound electron (pink circle) and hole (white circle), separated by a triplet energy of 1.14 eV.33,34 (b) Upconversion device used for transient PL studies consists of a monolayer 790 nm PbS NCs (sensitizer) and 80 nm rubrene (annihilator) doped with 0.5% DBP (emitter). The PbS NCs are directly excited at 785 nm, and the photoexcitations undergo TET to sensitize the triplet state of rubrene. A singlet is formed by triplet−triplet annihilation (TTA), which then undergoes FRET to the dopant dye DBP, where it is readily emitted at 610 nm. (c) For efficiency measurements, the PbS NCs are excited at 808 nm, and two additional layers are added to the previous device structure: an optical spacer made of 20 nm thick tris(8-hydroxyquinoline) aluminum (AlQ3) and a 100 nm thick silver back reflector.41 For clarity, the QD ligands have been omitted.
RESULTS AND DISCUSSION For the upconversion devices reported here, the PbS NCs are chosen to have the first excitonic absorption feature at λ = 790 nm (1.57 eV) and the emission peak at λ = 970 nm (1.28 eV) in solution (Figure S1). Based on the energy levels of the PbS B
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ACS Nano glass substrate. (see the Methods for details of NC synthesis and ligand exchange). Then, an 80 nm thick film of rubrene doped with 0.5 vol% of dibenzotetraphenylperiflanthene (DBP)36 is thermally evaporated to form a host−guest/ annihilator−emitter layer, an approach which was previously adopted in organic light-emitting diodes (OLEDs).37 The DBP enhances the quantum yield of the rubrene film, which is otherwise low due to singlet fission.38,39 Instead, the singlets formed in rubrene via TTA are rapidly transferred to DBP by Fö r ster resonance energy transfer (FRET), a process competitive with singlet fission (τ ≈ 110 ps) in rubrene.40 An improved device structure including an additional optical spacer and a silver back reflector is used for determination of the upconversion efficiency (Methods), shown in Figure 1c.41 As discussed previously, typical exchange-mediated TET decays exponentially with distance from donor to acceptor. Modulating the NC ligand length in C atoms will allow us to test whether transfer from PbS NCs follows the same exponential trend as in the reverse (downconversion) process.28 However, a detailed library translating the ligand length in C atoms (nC) to a real space ligand shell thickness is required. In a recent study, we showed that the ligand shell thickness LC follows a nonlinear trend with aliphatic chain length nC (Figure 2a).42 Using both atomistic molecular dynamics (MD) and transmission electron microscopy (TEM), we determined the NC spacing of close-packed films to be only approximately one extended ligand length, even though ligands from two NCs should both contribute to the spacing. Previous studies have attributed this reduced spacing to interdigitating ligands.43 However, our detailed atomistic simulations suggest a slumping of the surface-passivating ligands as the underlying cause. Despite the differences in the simulation and experiment, we find that the theoretical results are in agreement with the experimental distances measured by TEM (Figure 2a, inset).42 To further highlight the ligand behavior, we perform MD simulations on model NCs in the rock salt crystal structure. We continue to attach amine ligands due to the ease of simulation based on the single, uncharged binding group in comparison to the charged carboxylate binding group in the experiment. As in our previous study,42 we also observe a slumping behavior in the ligands on the highly faceted NC (Figure 2b). The ligand length ranges from C3 (left), to C9 (middle) and C17 (right). It is clear that the short ligands are mostly upright and extended (C3), while longer ligands (C9, C17) are no longer fully extended and collapse onto the NC. Since we find that the atomistic simulation and experiment are in close agreement despite the differences in the NCs and ligands, we use the equation found by Geva et al.42 for the relationship between the ligand shell thickness LC and the extended ligand length LC−C to translate the ligand length in C atoms into a real-space distance: ⎞ ⎛⎛ 3L ⎞1/3 LC = dNC − L + 12Å × ⎜⎜⎜ C − C + 1⎟ − 1⎟⎟ ⎠ ⎠ ⎝⎝ 12Å
Figure 2. Nonlinear modulation of the spacing between the NCs and rubrene by changing the aliphatic ligand length. (a) Theoretical relationship between the ligand shell thickness LC and the number of C atoms nC in the ligand (black) for CdSe in the zinc blende crystal structure. The teal circles show the experimental distances obtained for close-packed PbS NCs by TEM, in close agreement with the theoretical model. The NC−NC spacing measured by TEM is adapted from ref 42. The dashed and dotted gray lines function as guides to the eye and show the extended and half of the extended length LC−C, respectively. Inset: TEM image illustrating the experimental spacing between closepacked PbS NCs for a dodecanoic acid (12C) ligand. (b) Atomistic molecular dynamics simulations of the length-dependent ligand morphology of aliphatic amines (C3, left; C9, middle; C17, left) on a model CdSe NC in the rock salt crystal structure. VMD is used for image rendering.67
enable a direct comparison between the intrinsic decay dynamics of the NCs and the quenched dynamics in the presence of DBP-doped rubrene, each sample consists of an area of neat PbS NCs and an area containing the bilayer of the PbS NCs and DBP-doped rubrene. The PL dynamics of the close-packed NCs (Figure 3a,b, black curves) are consistent with previous reports,4,45−47 and a faster multiexponential decay gives way to a slow monoexponential decay with τPbS ≈ 2.4−2.8 μs (compare SI, Table S1), a lifetime similar to the PL decay of isolated NCs in solution (see SI, Figure S3). The initial fast quenching has been attributed to interactions between adjacent NCs in the form of energy transfer to nonemissive NCs19,48 or rapid nonradiative recombination.46−48 The addition of rubrene in the bilayer region adds a new exciton decay channel, and the PL dynamics are clearly accelerated for NCs with all ligand lengths (Figure 3a,b, red curves). As in our previous study,4 we find that the resulting decay dynamics cannot be wholly accounted for by simple firstorder kinetics and assert that there is a subpopulation of NCs not undergoing TET with unchanged PL dynamics. Thus, we subtract the multiexponential emission dynamics from these inactive NCs from the total PL decay to isolate the TET
(1)
Here, LC−C is the length of the extended ligand and dNC−L is the bond length between the NC and the ligand binding group (estimated at 2.3 Å for the bond length between the carboxylate and the surface lead atom).44 To reveal the transfer kinetics based on the ligand length, we measure the TRPL dynamics of hybrid films consisting of PbS NCs with varying ligand lengths, as seen in Figure 3a,b. To C
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emission from the organic film or the glass substrate (Figure S4) and inherent heterogeneity in the NC film. By accounting for the intrinsic PbS lifetime, we calculate the characteristic time of transfer for each ligand length (see the Methods). We observe slow transfer for long ligands (e.g., stearic acid (18C), τTET,18C ≈ 850 ns) (Figure 3a, dark blue curve), and more rapid transfer with shorter ligands (e.g., hexanoic acid (6C), τTET,6C ≈ 120 ns) (Figure 3b, light blue curve). In Figure 3c, we plot the obtained characteristic TET times (τTET) as a function of the ligand shell thickness LC (compare Figure S5 for additional aliphatic ligand lengths) on a semilogarithmic scale. The unsaturated native oleic acid ligand (OA) is highlighted in purple, showing slightly faster transfer (see Figure S6) in comparison to the saturated 18C ligand (dark blue) due to the slightly decreased ligand length resulting from the cis-double bond.42,43 As established previously, TET occurs via the Dexter mechanism, a simultaneous exchange between the ground and excited states of the donor and acceptor.6,23,28 Therefore, we anticipate a direct exponential relationship between the 1 characteristic TET rate k TET = τ and the spacing LC TET
between the rubrene and the NC k TET ∼ |V |2 exp−2LC / L TET
(2)
where LTET is the characteristic length of TET and V represents the electronic coupling between donor and acceptor states.20 While the transfer time follows the expected exponential trend for longer ligands, it appears to saturate at τ0 ≈ 100 ns for ligands with an effective length shorter than ca. 10 Å. This extracted TET rate is orders of magnitude slower than most previously reported rates for electron transfer,45 TET,3,29 or FRET49 in systems involving similar NCs. However, there are reports of TET on a ca. 70 ns time scale by Piland et al. using CdSe NCs.2 It is indeed not surprising that a single electron transfer is faster than a cooperative bound triplet exciton transfer, as the transfer rates for both the electron and the hole are multiplied to obtain the total observed rate.50 Interestingly, for many of the conflicting reports where much faster TET rates are reported, the energetics of the system allow for a single charge to transfer, resulting in allowed charge-separated intermediate states.29,32 In contrast to our study, these studies are based on ultrafast transient absorption results where the resulting kinetics are only followed for a few nanoseconds. Slower dynamics on the order of hundreds of nanoseconds as seen here would not be observed. To ensure that the slow extracted quenching times are indeed due to triplet transfer from the PbS to the rubrene triplet, we also investigate the dynamics of the visible emission for the bilayer-device. The visible rise time can be correlated to the combined time scale of TET, diffusion-mediated TTA and FRET to DBP (Figure S7).4 Increasing the repetition rate allows us to reduce the time scale of diffusion-mediated TTA due to a higher exciton density in the rubrene, and as a result, the TET will become the rate-limiting step. We observe a monoexponential rise-time on the order of τ8C,rise ≈ 250 ns, which is in fairly good agreement with the value found by the quenching study for the same ligand (τ8C ≈ 170 ns). To further investigate the unusually slow characteristic transfer rates kTET and the deviation from the exponential trend, we use constrained density functional theory (DFT) to calculate the maximum triplet transfer rate. For this, we place the rubrene perpendicular to the NC surface (surface-
Figure 3. Time-resolved photoluminescence decay of the infrared emission of a neat PbS NC film (black curve) and the quenched TRPL in the presence of rubrene/0.5% DBP (red curve) to extract the TET dynamics for different ligand lengths (blue curves). (a) TET dynamics for stearic acid (18C) showing a monoexponential decay lifetime τ18C = 651 ± 6 ns (dark blue). This results in an estimated characteristic transfer time of τTET,18C = 850 ns when accounting for the competition with the intrinsic decay channels (black curve, τPbS ≈ 2.8 μs). (b) Transfer dynamics for hexanoic acid (6C), with τ6C = 116 ± 3 ns (light blue). This corresponds to a characteristic transfer time of τTET,6C = 120 ns. Insets: cartoons to highlight the enhanced transfer rate (reduced transfer time) when replacing long ligands (d1) with short ligands due to the reduced spacing d2. (c) Deviation of the TET time τTET from the expected strictly exponential relationship (green dashed line) is apparent at ligand shell thicknesses LC below 10 Å, and the asymptote at τ0 ≈ 100 ns (pink dotted line) is highlighted. The unsaturated native oleic acid ligand is highlighted in purple. The error bars depict the standard deviation of multiple samples.
dynamics in active NCs (Figure 3a,b, blue curves).4 This results in largely monoexponential decay dynamics, with an initial fast multiexponential decay, which we attribute to parasitic IR D
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Hence, we postulate that the saturation of the characteristic TET rate kTET points to an additional parameter that influences the transfer rate other than the direct donor−acceptor spacing determined by the ligands. To investigate the plateauing of the observed characteristic transfer rate, we examine the ligand dependent properties of single NCs and the macroscopic NC arrays. Incomplete passivation during the ligand exchange process can result in dangling bonds which can act as rapid nonradiative recombination centers.46,47 The TET must indeed compete with the nonradiative and radiative decay pathways. However, the neat PbS NC kinetics are already accounted for via subtraction to first order and should not play a role in the TET kinetics. As a result, the overall dynamics of the exciton in a single NC undergoing TET are not expected to be greatly affected by a change solely in the ligand length. Indeed, we do not observe a significant change in the PbS NC PL lifetime in solution upon ligand exchange (Figure S8). Changing the ligand length, however, not only changes the spacing between adjacent NCs and between the NCs and rubrene but also changes the volume fraction of PbS in the NC monolayer,47 which has a large effect on the dielectric screening of the exciton prior to TET. Studies have previously shown a solvatochromic redshift in the optical properties of CdSe NCs54 and PbSe NC arrays55 due to a change in the dielectric environment. We observe a similar redshift in the emission of our PbS NCs when going from solution to solid state (Figure S9a). This redshift is often attributed to energy transfer to larger NCs56 and not to the dielectric effect. However, the nearly symmetrical line shape and lack of narrowing of the line width of our emission peak is inconsistent with energy transfer (Figure S9b), as the enhancement of the low energy emission and the quenching of the higher energy emission will yield narrower and asymmetric emission line shapes.46 On the other hand, the long (μs) lifetime of PbS NCs has been attributed to the large dielectric constant of PbS,57 and a change in the PL lifetime as a result of a change in the refractive index of the NC solvent has been previously reported.56 To investigate the potential dependence of the TET rate on the dielectric screening effect, we estimate the dielectric constant of the PbS NC monolayer using an effective medium approximation following the Bruggeman model58,59
passivated with propylamine (C3) ligands) at a donor− acceptor spacing of 3.5 Å, which is the closest possible spacing based on short-range repulsive interactions. We compute the Kohn−Sham wave functions of an unconstrained exciton in the PbS NC (Figure 4, top) and the triplet state in rubrene (Figure
Figure 4. DFT calculations of the electron density distribution in the excitonic energy levels involved in the TET process. Kohn− Sham wave functions of the initial state, the unconstrained exciton on the PbS NC (top), and the final state, the triplet exciton on rubrene (bottom). The coupling V between the wave functions is computed using constrained density functional theory−configuration interaction, and the rate of TET (ktheory = 0.1 ns−1) is estimated from Marcus theory. VMD is used for image rendering.67
a
4, bottom), evaluate the coupling between the exciton in the NC and the triplet state, and then obtain the TET rate ktheory from a Marcus-type expression.51−53 We are obliged to make several general assumptions for the terms in the Marcus equation (see the Methods), and so the resulting rates are useful as order-of-magnitude estimates. Our approach yields a computed coupling between the NC and rubrene of V = 5.4 × 10−5 eV, which corresponds to a rate of ktheory = 0.1 ns−1 (τ ≈ 10 ns) for TET transfer. While this calculated rate is an order of magnitude faster than the fastest experimental transfer rate, it is also the rate that is calculated from the Marcus-type expression (compare Methods, eq 7), assuming a donor−acceptor spacing of only 3.5 Å, shorter than the ligand spacing in the experiments. Nevertheless, the simulation does not explain the experimentally observed asymptotic behavior for short ligands. The calculations are consistent with an uninterrupted exponential rise in the TET rate for short ligands, whereas the experimental results show a clear deviation from this Dexter-like expectation.
εPbS − εligand εPbS + 2εligand
+ (1 − a)
ε′ligand − εligand ε′ligand + 2εligand
=0 (3)
where a is the volume fraction of PbS in the film, εPbS represents the static dielectric constant of bulk PbS, ε′ligand is the dielectric constant of the carboxylic acid ligand, and εligand is the calculated effective medium dielectric constant. Decreasing the ligand length from 18C to 6C (4C) results in a 2-fold increase in the PbS volume fraction a from 8.5% to 17.3% (21.2%), based on a hexagonal close-packing model of the PbS NCs in a monolayer. The high static dielectric constant of PbS εPbS = 169 combined with the much lower dielectric constant of the ligand ε′ligand ranging from ε′18C = 2.3 to ε′6C = 2.6 (ε′4C = 3) results in a large change in the effective medium dielectric constant εligand from ε18C = 3 to ε6C = 5.2 (ε4C = 7.1) when exchanging the NC ligand from 18C to 6C (4C) (Figure 5a; for a full list of the dielectric constants used for the calculation, compare Table S2). The coupling term or exchange integral V in exchangemediated energy transfer is inversely proportional to the 1 dielectric constant of the medium: V ∼ ε , which in turn yields E
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Figure 5. Correction of the extracted transfer rate by the calculated dielectric constant. (a) Calculated dielectric constant for a PbS monolayer with different ligands based on the Bruggeman model. Inset: cartoon highlighting the change in the volume fraction of PbS vs organic ligand when shortening the ligand (not to scale). (b) Saturation of the extracted rates (gray line) and deviation from the exponential trend (green dashed line). (c) Multiplication of characteristic TET rates kTET (b, gray diamonds) by the square of the effective dielectric constant εligand: kTT = kTET(εligand/ε18C)2 (black diamonds) recovers the expected purely exponential trend (solid green line). Error bars are based on an error propagation with an estimated error for the dielectric constant Δε = 0.2 and the standard deviation of the extracted rate. Hexanoic acid, oleic acid, and stearic acid (6C, OA, 18C) are highlighted in light blue, purple, and dark blue, respectively.
a lower coupling as the dielectric constant increases.22,23 The energy-transfer rate in exchange-mediated TET is proportional to the square of the coupling term k ∼ |V|2 (compare eq 2 and Methods, eq 7) and, therefore, inversely proportional to the square of the dielectric constant.20 We multiply the extracted characteristic TET rates kTET (Figure 5b) by the square of their respective effective medium dielectric constants normalized by that of the stearic acid (C18) NC film as a reference value: k TT
⎛ εligand ⎞2 = k TET⎜ ⎟ ⎝ ε18C ⎠
This effect can also be observed in the calculated quenching efficiencies based on the counts detected during the IRPL quenching studies. We observe a slight quenching of the detected PL intensity for a bilayer sample consisting of NCs and bathocuproine (BCP), a wide-bandgap material too high in energy to act as an acceptor for rubrene triplets (Table S3). Nevertheless, the PL dynamics of NCs in this control sample are unchanged (Figure S10), as expected due to the lack of an energy-transfer pathway. We use the quenching value for the NC/BCP control as an estimate of the error caused by scattering in the organic film and obtain corrected quenching efficiencies of 94−96% for short ligands (4C, 6C) and 88% for long ligands (18C) (Table S4). These values are comparable to the expected transfer efficiencies based on the extracted TET lifetimes. The discrepancy for the long ligands is likely explained by a bias to thinner ligand shell thicknesses caused by inhomogeneity in the ligand coverage, where thinner regions will result in faster transfer and a higher transfer efficiency. This is likely also the cause of the initial fast multiexponential decay (
