The “energy gap law” for mid-infrared nanocrystals

In the early 1980s, Ekimov and Efros in Russia as well as Brus in the USA started to explore the quantum confinement in semiconductor nanocrystals produced by homogeneous nucleation of semiconductors in molten glasses and in liquids, respectively.¹ By the early 1990s, the nanocrystals synthesized in glasses showed better resolved absorption spectra² than those synthesized in liquids and both showed weak fluorescence. The brightest colloidal nanocrystals were made by Henglein and co-workers,³ and they were ∼5 nm CdS synthesized in water, capped with cadmium hydroxide, with up to 50% photoluminescence quantum yield, but with weak absorption features. This raised concerns that nanocrystals were inherently riddled with defects, impurities, affected by strong electron–phonon coupling, surface states, ultrafast trapping, and broad size dispersion, all these preventing sharp excitonic features. The 1993 paper by Murray et al. changed the perspective by showing an extended series of excitonic transitions tunable by size for colloidal Cd(S,Se,Te) nanocrystals.⁴ Prior attempts at capping and passivating CdSe nanocrystals with ZnS were not successful,⁵ but Hines showed that a couple monolayers of ZnS on the newer CdSe quantum dots led to a robust 50% fluorescence efficiency.⁶ The chemical synthesis of quantum confined colloidal semiconductor nanocrystals, also called colloidal quantum dots (CQD), which was started by Brus, was followed by a small community through the 1990s,⁷ and it has since been very successfully extended by many scientists. This has led to a wide range of different II-VI, III-V, and ternary semiconductor CQDs, with size distributions well below 10%, excitonic features from the UV to the infrared, and near 100% fluorescence in the visible.¹ In particular, InP CQDs, which were first reported by Micic et al.⁸ in 1994, are now the most used CQDs in the form of core/shell InP/ZnSSe CQDs, with narrow and bright green and red emissions that improve color rendering in commercial TV displays.

While being as bright as organic dyes in the visible spectrum, CQDs easily achieve much higher photoluminescence (PL) quantum yield (QY) at near-infrared wavelengths. This advantage increases further in the infrared where the fluorescence of dyes essentially disappears. The infrared ranges are often delimited by the transparency regions of the atmosphere, and they are the short-wave infrared (SWIR, 1–2.5 μm), the mid-infrared (MWIR, 3–5 μm) and the long-wave infrared (LWIR, 8–12 μm). These separate infrared bands are all important for chemical and thermal sensing and imaging. The state-of-the-art technology in these infrared bands is based on expensive fabrication of single crystal materials (InGaAs, InSb, and HgCdTe) or multiple quantum wells, and a delicate low throughput chip-bonding to silicon read-outs, which hinders high resolution imaging.⁹ 

Over the past two decades, there has been much progress on CQD optoelectronics.¹⁰ Due to their solution-processability, near-infrared CQDs can be coated on silicon chips with an arbitrarily small pixel pitch, leading to higher throughput and lower cost fabrication for spectroscopy and imaging sensors.¹¹ CQD detectors are now made from the near-IR to the LWIR, with wafer scale production of imaging chips being developed in the short-wave IR and the mid-wave IR. The performances are inching toward those of single crystal devices,^12–17 with commercial versions already available.¹⁸ 

Technological advances build on basic breakthroughs. After the synthesis of CQDs with a well-defined absorption, the most important properties are surface related, including photoluminescence, carrier transport, and doping. A key step for detector development was to go from non-Ohmic electrical properties^19,20 to conductive CQD films with a simple ligand exchange,^21,22 and a second key step has been to control carrier doping²³ from intrinsic to n- or p-doped CQD films.^24–26 

This work focusses on the PL QY of mid-infrared CQDs. In the visible, core/shell synthesis has been very successful and central to the applications of CQDs. Core/shells have not yet been so beneficial for infrared detectors because the shell decreases carrier mobility. It is clear that the PL QY determines the maximum possible quantum efficiency of a spontaneous emission light source. It is typically overlooked that it also fundamentally determines the maximum possible photon sensitivity of detectors.²⁷ The experiments to date show that the QY of mid-IR CQDs drops precipitously toward the long-wave infrared, and this is reminiscent of the behavior of organic dyes in the near-infrared. This work discusses the relevance of the PL QY for applications of infrared CQDs and the possible non-radiative mechanisms.

Raising the PL efficiency of mid-IR CQDs is important for their potential applications. For light emission, recent experiments showed the possibility of making mid-IR light emitting devices (LED) with HgTe and HgSe CQDs based on single electron and hole recombination in p-n junctions²⁸ and single electron multiple “cascade” emission under high bias.²⁹ These cascade devices used the 1P_e-1S_e intraband emission of n-doped HgSe/CdSe CQDs films with an external PL QY of only ∼0.1%, but they already showed an external quantum efficiency, photon per electron, reaching 4.5% at 2 A cm⁻², and a power efficiency of 0.05%, which is close to the state-of-art epitaxial mid-IR LEDs. A moderate increase in PL efficiency would therefore make mid-IR CQD LEDs competitive, at least in terms of efficiency, while a further increase could allow equivalent brightness at lower electrical power.

For mid-IR photodetection with CQDs, detectors are now limited by noise more than by the quantum efficiency of detection.^30,31 The PL QY directly determines the thermal noise, which is the most fundamental noise source. Indeed, thermal noise originates from thermally generated carriers, which are indistinguishable from optically generated carriers. Faster non-radiative relaxation leads to faster thermal excitation and therefore greater thermal noise, and the minimum signal detectable is inversely proportional to $QY$⁠.²⁷ If the mid-IR PL QY of the CQDs could be increased from 10⁻³ to unity, the room temperature detectivity would increase 30-fold and that would make CQD detectors superior to any existing detectors. Other physical approaches to improve the detectivity include increasing the radiative rate or operating out of equilibrium, but this work only discusses the non-radiative rates.

Since the 1960s, it is known that molecular dyes show a strong decrease in the quantum yield past the near-infrared, which is called the “energy gap law” [Fig. 1(a)]. Organic molecules emitting at 1.5 μm can be expected to have a quantum yield (QY) approaching 10⁻⁴–10⁻⁵, with ps lifetimes. This low QY does not prevent making efficient pulsed infrared dye lasers, as demonstrated in pioneering studies in the early 1980s,³² but it prevents many other applications. For a fixed oscillator strength, the radiative rate scales as k_r ∼ ω², where ω is the emission frequency. Therefore, a constant non-radiative rate leads to lower quantum yields in the infrared, but this cannot explain an exponential dependence on ω. Instead, it is found empirically that the non-radiative rate increases exponentially at lower energy, as $knr∼e−ω/ω0$⁠. In Fig. 1(a), the lines are shown by $ω2ω2+Ae−ω/ωo$ with ω₀ = 800 cm⁻¹.

The characteristic energy ℏω₀ is of the order of vibrational mode energies, suggesting that the non-radiative process is due to molecular vibrations. Thus, for organic dyes, the PL decrease in the near-IR is generally attributed to electronic to vibrational relaxation.

The energy gap law can be qualitatively justified by a multiple-order perturbation theory. One assumes that the electronic excitation of energy ℏω couples to a vibration of lower energy ℏω₀, with a perturbation of order n, where n ∼ ω/ω₀. The relaxation rate is then taken to be of the form k ∼ βⁿ, where β is a coupling parameter that must be a small number compared to 1. This naturally leads to the observed exponential form for the relaxation rate as $knr∼elnβω/ω0$⁠. This argument explains why the characteristic frequency or energy is not that of a specific vibration, but it does not prescribe a specific value for the coupling parameter β. Moreover, it is not clear why all organic molecules would collectively fit to a particular value of the characteristic energy.

In the 1970s, the Fermi golden rule was applied to the problem^33–35 and led to quantitative expressions that provide the exponential dependence, with a characteristic energy that is an aggregate of different vibrational modes and their coupling parameters to the electronic transition. These derivations also showed that the characteristic energy is very sensitive to the coupling parameters. Even with the much better computational methods available today, the calculated rates are still fitted to the experiments and remain semi-quantitative.³⁶ There is a renewed interest in the energy gap law for short wave infrared dyes and bio-imaging. Deuteration experiments showed small benefits for near-infrared organic dyes³⁷ and greater benefits for some near-IR metal-organic complexes,³⁸ attributed to differences in coupling strength. Changing the skeletal atoms, from carbon to sulfur, with metal complexes, resulted in further improved QY, still of the order of 10⁻³ below 8300 cm⁻¹ (∼1.2 μm).³⁹ 

The interest in infrared CQDs started in the late 1990s, motivated by the impressive developments of infrared quantum well devices.^40,41 Inorganic CQDs are natural choices for infrared fluorophores because of their low phonon frequencies. Moreover, CQDs readily provide infrared transitions by using interband transitions with the few small gap semiconductors, such as the zero gap HgTe, or by using intraband transitions with a much broader range of semiconductors.⁴² 

In the near and short-wave infrared (<1.5 μm), InAs shelled with CdSe,⁴³ unshelled HgTe,⁴⁴ PbSe,⁴⁵ and PbS⁴⁶ all showed promising PL, with a PL QY exceeding 50% around 1.2 μm, which is 100-1000-fold brighter than molecular dyes. Mid-IR emission was later observed with PbSe⁴⁷ and HgTe CQDs.^48,49 However, it was noticeable that the quantum yield dropped dramatically, becoming as low as 10⁻³–10⁻⁴ around 5 μm (Fig. 1).⁵⁰ Therefore, CQDs also follow an “energy gap law.” Perhaps more surprisingly, Fig. 1(a) shows that the QY can be roughly scaled with the same characteristic energy as for the organic dyes.

Although the data are less extensive for intraband transitions, the “energy gap law” does not appear to differentiate between interband and intraband transitions. Early visible pump–probe experiments on CdSe CQDs showed a sub-ps 1P_e to 1S_e exciton relaxation.⁵¹ This fast excitonic relaxation was explained by an electron–hole energy transfer, an Auger process where the electron excess energy is relaxed to the hole, due to the Coulomb coupling and the much greater density of states for the more massive holes.⁵² Such sub-ps excitonic relaxation is rather general.⁵³ However, experiments specifically measuring the infrared intraband 1P_e to 1S_e relaxation pointed out the strong effect of ligands and showed that the mid-IR intraband lifetime could increase to ∼30 ps by avoiding the presence of a hole and by using ligands with lower vibrational absorption.⁵⁴ Thick ZnSe shells on CdSe CQDs slowed the intraband relaxation up to 1.5 ns,⁵⁵ while still being much shorter than the radiative lifetime, estimated at ∼700 ns. After permanent doping was observed in n-HgS and n-HgSe CQDs,⁵⁶ the intraband fluorescence was directly observed, and the PL lifetime was also measured by up-conversion.⁵⁷ Mid-IR intraband fluorescence was also observed with CdSe and CdSe/ZnSe CQDs but with double excitation by intense visible light.⁵⁸ Overall, the mid-IR PL QY of the CQDs has not been markedly different for interband or intraband transitions. At the present time, the brightest mid-infrared chromophore in solution at 5 μm is the intraband transition of lightly n-doped HgSe CQD with a thick ∼5 nm CdS shell,⁵⁹ with lifetimes longer than 10 ns and with a QY of 2%.

Figure 1(a) shows QY measured in solutions, but most applications require thin films. Making bright mid-IR films with any material is challenging. In our experience, polycrystalline bulk PbSe optimized for mid-IR detectors is the brightest mid-IR film at room temperature. The spectrum is shown in Fig. 1(b). Other materials, such as epitaxial HgCdTe, suffer from defects or a faster Auger recombination, which is the dominant PL quenching mechanism at room temperature. Using an integrating sphere and an InSb detector, the PbSe film has an absorption of ∼50% at 808 nm and a mid-IR emission QY ∼ 3%. The small QY is, in part, due to low photon extraction. For a thin smooth film on a reflector substrate, neglecting reabsorption/reemission, the extraction efficiency is ∼1/2n². For PbSe with n ∼ 4.3, this gives an extraction efficiency of 2.7%, close to the measured QY, noting that surface roughness and absorption/re-emission can increase the PL signal. Light extraction from CQD films should benefit from a lower index of refraction, and yet, as shown in Fig. 1(b), they are not yet bright. PL from films of mid-IR HgTe CQDs is about 100 times smaller than those from the PbSe diode. The PL QY in such close packed films on a reflector substrate is less than 10⁻³ and similar to the values measured in the tetrachloroethylene solution shown in Fig. 1(a). Films of HgSe/CdS CQD with ∼3 monolayers of CdS and embedded into an As₂S₃ sol-gel can achieve brighter PL. They show a small benefit from cooling, with the peak intraband PL increasing threefold at 80 K, the integrated PL increasing twofold, and a small 50 cm⁻¹ blueshift. These are the brightest films that we have yet obtained with mid-IR CQDs, and their QY is still less than 1%.

Figure 2 shows that organic molecules and inorganic solids differ by many orders of magnitude in their infrared absorption. At 2000 cm⁻¹, inorganic solids can be 100 000 times more transparent than organic ligands. It is well appreciated that organic molecules have strong absorptions peaks at well-defined vibrational modes or overtones, but Fig. 2 emphasizes the broad exponential background absorption of the form $e−ωωabs$⁠.

From Fig. 2, the characteristic frequency of the background for dodecanethiol and oleylamine is ω_abs ∼ 1100 cm⁻¹, and this is similar for other organic ligands. This contribution is dominated by the aliphatic tail, but other organic groups, such as carboxylic acids, can significantly increase the absorption. Inorganic solids have a similar exponential absorption background, but with a much lower characteristic frequency. For ZnSe, ω_abs ∼ 60 cm⁻¹, which is about ¼ of the 253 cm⁻¹ LO phonon frequency. For the As₂S₃ glass used for mid-infrared transparent fibers, ω_abs ∼ 110 cm⁻¹, which is also about ¼ of the highest fundamental phonon frequencies of 485 cm⁻¹ (As–S–As stretch) and 345 cm⁻¹ (As₂S₃ stretch).⁶⁰ 

The strong reduction in the exponential background absorption of inorganic solids compared to organics stems from the lower phonon frequency and the lower oscillator strength associated with heavy ions. The exponential background absorption in inorganic solids has been well explained by Boyer et al.⁶¹ They modeled the bonds with a Morse potential, determining the ground state vibration energy from spectroscopy and the anharmonicity from the experimental thermal expansion. They then calculated the absorption of all higher order vibrational transitions. With an appropriate density of state that broadens the features, the absorption coefficient was shown to scale as $α=Ae−γωωo$⁠, where ω₀ is the phonon fundamental frequency and $γ=−lnℏω04D$⁠, where D is the bond dissociation energy. Boyer et al. further showed that γ ∼ 4–5 for many infrared transparent ionic crystals, therefore, similar to the values for ZnSe and As₂S₃. Moreover, the coefficient A is also semi-quantitative. It is approximately modeled as $A∼πNe212ε12ε0cμγω0f$⁠, where f is the oscillator strength of the diatomic ionic vibration, μ is the reduced mass of the ions, ɛ is the material optical dielectric constant above the reststrahlen band, and N is the volume density of the bonds. For the ionic solid KBr, using f = 1 gives A = 5.8 × 10⁵ cm⁻¹, which is convincingly close to the experimental value of A = 6.1 × 10⁵ cm⁻¹.

The infrared absorption of inorganic materials decreases at lower temperature, and experimental data are roughly consistent with the expected scaling of $A∼nωo+1ω/ωo$⁠, where $nωo$ is the thermal population. This can be a significant effect. For example, with ω = 2000 cm⁻¹ and ω_o = 200 cm⁻¹, the absorption at 300 K should be about 100 times larger than at 80 K. This is also in contrast with organics, where the frequencies of the relevant modes are much higher, leading to essentially no expected temperature effects, which are indeed typically not reported.

Figure 2 shows a clue to find the main issue with the mid-infrared quantum yield of colloidal quantum dots, noting that a few organic ligands/impurities on the surface of the dots can already lead to much greater optical absorption than the multiphonon absorption from the inorganic quantum dot core or matrix.

The relaxation mechanisms considered here are carrier trapping, Auger relaxation, energy transfer to resonant vibrations, and multiphonon relaxation. In addition, we also briefly discuss inter-quantum dot energy transfer.

Carrier trapping: carrier trapping is the traditional CQD PL quenching mechanism for visible CQDs. However, carrier trapping is likely of lesser importance for small gap quantum dots for the simple reason that the traps need to be within the gap, and this reduces their possible density of state. Carrier trapping is also usually very sensitive to the surface since it is associated with a very local atomic configuration. Carrier trapping is typically solved with epitaxial type I core/shells, with the shell acting as a barrier that exponentially suppresses the tunneling of carriers to surface states. This works very well for thin shells and visible emitting quantum dots. In contrast, mid-IR CQDs need much thicker shells to show improved QY.⁶² Carrier trapping is also typically associated with an Arrhenius activation energy due to the strongly localized trap charges and the associated energy barrier. To date, there is no report of strongly improved PL QY for mid-IR CQDs at lower temperatures. Therefore, carrier trapping is not yet considered a significant issue for mid-IR PL.

Auger relaxation: An exciton can relax by Auger if the CQD contains at least one free carrier. The thermal carrier occupation being much larger for a small gap implies that Auger is more relevant at longer wavelengths. In intrinsic CQDs, the number of thermal carriers can be estimated as $NeNhe−Eg2kT$⁠, where N_e and N_h are the number of thermally accessible states for electrons and holes in the CQDs. For the mid-IR HgTe CQDs with a 5 μm energy gap, this leads to about 1% of the CQDs having a free carrier at 300 K for HgTe CQDs, and it increases to about 30% for 10 μm energy gap.⁶³ Nevertheless, the majority of dots do not have free carriers. With isolated CQDs in solution, the effect of Auger on PL quenching is, therefore, expected to be limited. For HgTe and HgSe solutions in the mid-IR, infrared up-conversion and pump–probe measurements also show that Auger is slower than the exciton non-radiative relaxation.^57,64 Another indication that Auger is not limiting is the lack of significant brightness improvement at low temperatures. Nevertheless, Auger might become an issue when brighter and conductive films are eventually produced.

Equation (1) can be extended to a thicker shell or a matrix by integration. Figure 3(a) shows the calculated QY for HgTe CQDs using the exponential absorption background coefficient of the ligands as $α=600e−ν̄1100$ for oleylamine shown in Fig. 2. The imaginary dielectric constant of the ligand shell is taken as $Imε=αn2πν̄$⁠, where $ν̄$ is the frequency in wavenumbers (cm⁻¹) and α is the absorption coefficient (cm⁻¹). The core size as a function of frequency is taken as $R=13001+ν̄400+120.5$ in nm, using a fitting to three sizes (1000 cm⁻¹, 10 nm), (2000 cm⁻¹, 6 nm), and (6600 cm⁻¹, 2 nm). The calculations were performed for different thicknesses of ligand shells, for core and core/shell CQDs. Figure 3(a) shows a qualitative agreement with the data shown in Fig. 1, as previously reported. A striking effect of the near-field energy transfer is that the QY decreases by orders of magnitude even though the far-field transmission of the ligand shell is nearly unity. For example, at 2000 cm⁻¹, taking the ligand absorption coefficient of 80 cm⁻¹, a 2 nm ligand shell leads to near-unity far-field transmission (0.999 984) even though it gives QY ∼ 4 × 10⁻⁴.

The model can be applied to CQDs directly immersed in nearly transparent infrared materials without the intermediate ligand shell. This is relevant to quantum dots grown in glass and salt matrices.^69–73 Integrating the FRET model to infinity gives $TrTs,F=Imε32π3nλvacR3$⁠, where R is the core or core/shell radius and ɛ and n are the dielectric properties of the matrix. Since FRET decays quickly, distances beyond about 10 particle radii have a negligible effect. Figure 3(b) shows the expected QY for the HgTe core and core/shell CQDs emitting at 1.5, 5, and 10 μm, as a function of the matrix absorption coefficient. The figure shows that the absorption coefficient of the matrix needs to be smaller than 6.6 × 10⁻³ cm⁻¹ to get a 50% QY at 10 μm while 0.1 cm⁻¹ is sufficient at 1.5 μm. This difference is due to the $λvacR3$ term in Eq. (1). In addition, since matrix absorption typically decreases exponentially for shorter wavelengths, it is relatively much easier to get bright emission in the SWIR. In contrast, mid-IR and LWIR matrix materials will need to be very pure. For example, the absorption coefficient of pure chalcogenide glass As₂S₃ at 2000 cm⁻¹ is about 10⁻³ cm⁻¹,⁷⁴ but sol-gel films are closer to 1 cm⁻¹,⁷⁵ likely due to a small amount of trapped organic impurities.

The temperature dependence of the QY from FRET arises from the changes of the spectral overlap and acceptor absorption. For energy transfer to a background arising from high frequency modes, such as organic impurities, the temperature dependence is expected to be small, which is consistent with the observations.

Equation (2) can be used to estimate the multiphonon relaxation rate for II-VI nanocrystals. We used an upper limit for the phonon frequency ω_o ∼ 250 cm⁻¹, a coupling frequency of ω₁ ∼ 10 cm⁻¹, the electronic transition of ω ∼ 2000 cm⁻¹, and a phonon width Δ ∼ 10 cm⁻¹. The most important parameter is λ. The value is not known for HgTe but likely lies in the range 0.05–0.3. Using three values of λ, the calculated nonradiative rates are shown in Fig. 4(b) as a function of the gap frequency. This demonstrates the sensitivity of the multiphonon rate on the coupling parameter and shows the difficulty in predicting the multiphonon limit based on Eq. (2). However, it is clear that multiphonon relaxation should play a much smaller role in the mid-IR compared to the long-wave IR. When multiphonon relaxation is the only mechanism that takes place, such as with the mid-IR laser crystals ZnSe:Fe, the QY can approach ∼100% below 150 K.⁷⁷ However, the surface of CQDs is complex. Higher frequency vibrations and broader density of states could arise from surface atoms and increase multiphonon relaxation.

Apart from the strong scaling of the non-radiative rate with the energy gap, multiphonon relaxation should show a temperature dependence identical to the temperature dependence of the absorption described earlier. The expected effect of cooling from 300 to 77 K is shown in Fig. 4(c) for several phonon frequencies, along with the data point of the experimental PL integrated intensity ratio for the HgSe/CdS film shown in Fig. 1(b). This suggests that the mid-IR PL QY of CQDs is currently far from the multiphonon relaxation limit.

Input from the theory will be very useful. Han and Bester used a simplified electron–phonon coupling model and predicted a sub-100 ps multiphonon relaxation lifetime for the mid-IR 1S_e-1P_e intraband transitions for CdSe CQDs.⁷⁸ This is 100× faster than the slowest intraband relaxation lifetime measured so far for HgSe/CdS. Prezdho and co-workers used time-dependent density functional theory on very small Cd₃₃Se₃₃ clusters and no ligand to obtain a phonon-mediated lifetime of 210 ps for the intraband transition in CdSe,⁷⁹ but there are no experimental data for such small particles.

The large ratio of $TrTFRET$ shown in Fig. 5(a) is an overestimate for several reasons. The oscillator strength may be shared between different states near the band edge. The FRET is assumed to be resonant, and it will slow down with a Stokes shift, while the disorder will lead to sub-diffusive transport.⁸¹ 

Among the implications, a high QY in films of isolated dots might benefit from decreasing the packing. In conductive films, one needs to compare the dipole–dipole FRET time to the electron–hole separation time, which can be faster for high mobility films. Therefore, FRET complicates the relation between the QY and the CQD properties as one explores longer wavelengths.

Colloidal quantum dots can be used in an increasing number of ways with potential technological relevance. Their application as solution processable materials seems promising and complementary to the presently exclusive solid state epitaxial materials for infrared emission and detection. The low phonon energies of inorganic quantum dots should favor long-lived electronic states even in the mid-infrared, but it has been a challenge to obtain quantum yields above 10⁻³ at 5 μm, with a present record of only 2% with a thick shell. At present, the low quantum yield is limiting the efficiency of CQD mid-IR light emitting devices and the sensitivity of CQD mid-IR detectors.

For mid-infrared CQDs, there appears to be an energy gap law similar to the one describing the photoluminescence quenching of near-IR dyes. The near-field energy transfer to organics on the surface of CQDs is a most likely culprit because the absorption of organics exceeds that of inorganics by orders of magnitude. Coupled with the exponential dependence of the absorption of organics with energy, this energy transfer accounts rather well for the energy gap law of mid-IR quantum dots. Other relaxation mechanisms, such as carrier trapping, Auger relaxation, and multiphonon relaxation, are likely less relevant at this time. With proper surface passivation and removal of all organic contamination, Auger and multiphonon relaxation should then become the ultimate relaxation mechanisms that cannot be avoided. At present, multiphonon relaxation for mid-infrared colloidal quantum dots has not been experimentally identified, and it would be useful to have accurate theoretical estimates.

The supplementary material contains FRET to organic internal molecular vibrational absorption and literature photoluminescence quantum yield values.

This work was supported by the U.S. DOE under Award No. DE-SC0023210.

The authors have no conflicts to disclose.

Ananth Kamath: Conceptualization (equal); Writing – original draft (equal). Philippe Guyot-Sionnest: Conceptualization (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.