J. Chem. Phys. 139, 164303 (2013); https://doi.org/10.1063/1.4825401 139, 164303

© 2013 AIP Publishing LLC.

Phonon-induced pure-dephasing ofluminescence, multiple exciton generation,and fission in silicon clustersCite as: J. Chem. Phys. 139, 164303 (2013); https://doi.org/10.1063/1.4825401Submitted: 01 August 2013 . Accepted: 03 October 2013 . Published Online: 23 October 2013

Jin Liu, Amanda J. Neukirch, and Oleg V. Prezhdo

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THE JOURNAL OF CHEMICAL PHYSICS 139, 164303 (2013)

Phonon-induced pure-dephasing of luminescence, multiple excitongeneration, and fission in silicon clusters

Jin Liu,1 Amanda J. Neukirch,2 and Oleg V. Prezhdo3,a)

1Department of Chemical Engineering, University of Rochester, Rochester, New York 14627, USA2Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA3Department of Chemistry, University of Rochester, Rochester, New York 14627, USA

(Received 1 August 2013; accepted 3 October 2013; published online 23 October 2013)

The size and temperature dependence of the pure-dephasing processes involved in luminescence,multiple exciton generation (MEG), and multiple exciton fission (MEF) are investigated for Sin clus-ters (n = 5–10, 15) using ab initio molecular dynamics and optical response function theory. Thecluster bandgaps correlate with two types of binding energy, indicating that bandgaps can be used tocharacterize cluster stability. Ranging from 5 to 100 fs, the dephasing times are found to be longestfor MEF and shortest for MEG, with luminescence falling in the middle range. Generally, the de-phasing is fast, if the orbitals supporting the pair of states involved in the superpositions differ inenergy, atomic localization, and number of nodes. The dephasing accelerates with temperature, be-cause more phonon modes are activated, and lower frequency acoustic modes are able to explore theanhamonic part of the potential energy surface. The temperature dependence is stronger for largerclusters, since they possess a wider range of low-frequency anharmonic modes. Our research indi-cates that rapid dephasing in Si clusters favors generation of independent charge carriers from singleand multiple excitons, making the clusters a promising material for photon energy conversion. Thesimulations of the dephasing processes reported in this work assist in understanding of the excitonevolution pathways in inorganic semiconductor clusters and other nanoscale materials. © 2013 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4825401]

I. INTRODUCTION

Simulation of electron-phonon dynamics in nanoscaleclusters provides a valuable tool for studying the dynamicprocesses underlying cluster applications in photovoltaiccell,1–7 light-emitting diodes,8, 9 field-effect transistors,10

biosensors,11 and so on. For example, in solar cells it is de-sirable for the photoexcited electrons to relax slowly fromhigher to lower energy levels, in order to preserve the open-circuit voltage and the number of mobile charge carriers. Onthe contrary, rapid electron-hole recombination is needed togenerate a well-defined color in light-emitting diodes. Thus,study of electronic energy relaxation pathways is crucial torealizing the full potential of nanoscale materials.

Electron-phonon interaction leads to two qualitativelydifferent processes within the electronic subsystem – relax-ation and pure-dephasing. It is essential to characterize bothprocesses in order to understand the fate of the excess pho-ton energy and to reduce energy losses to heat. Relaxation re-sults from transitions between electronic energy levels, withthe lost electronic energy accommodated by phonons. Pure-dephasing destroys superpositions of electronic states that areformed during photo-excitation, electron-phonon relaxation,and other types of electronic transitions.12 Pure-dephasing oc-curs as a result of phonon-induced fluctuations in the energygaps between a pair of electronic states involved in the su-perposition. The Fourier transform of the energy gap charac-terizes the frequencies of the phonon modes coupled to the

a)Author to whom correspondence should be addressed. Electronic mail:oleg.prezhdo@rochester.edu

electronic subsystem. The rate of pure-dephasing affects therelaxation process. In general, the longer the two states existin a coherent superposition, the shorter is the quantum transi-tion time.13, 14 Pure-dephasing is elastic electron-phonon scat-tering, while relaxation is inelastic scattering.

Confinement of charge carriers in quantum dots (QDs)increases Coulomb interactions and enhanced yields ofAuger-type processes. As a result, more than one electron canbe excited from the valence band upon absorption of a singlephonon.15, 16 Known as multiple exciton generation (MEG),this phenomenon has drawn wide attention,1–3, 5, 15–18 since itcan help overcome energy losses associated with electron-phonon relaxation. External quantum efficiency has beenshown to exceed 100% in QD solar cells due to MEG.17, 19

The mechanisms responsible for MEG in confined semicon-ductors constitute an active area of research. In the photo-excitation mechanism,15 a photon excites superpositions ofsingle and MEs. The superpositions dephase by coupling tophonons.2, 12 Alternatively, MEs can be generated by impactionization.20–22 Once created, MEs can dissociate into un-correlated excitons by the pure-dephasing mechanism12 andevolve independently afterwards, ultimately separating intocharge carriers. The pure-dephasing process can be namedME fission (MEF), in analogy with the singlet fission ob-served in molecular systems.23

Si-based materials have been attracting considerable in-terest over many decades due to their dominance in semi-conductor electronics and photovoltaics. Synthesis of Si clus-ters allows one to control Si properties by size selection.24

The continuum of electronic states present in the bulk is

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164303-2 Liu, Neukirch, and Prezhdo J. Chem. Phys. 139, 164303 (2013)

reduced to a series of discrete levels in nanoscale systems.As a result, cluster properties are distinct from bulk prop-erties. Numerous studies have been carried out in the pastyears to examine the structural and electronic characteris-tics of Si clusters, using both first principles calculations25–41

and electronic spectroscopy.42–50 As the size of the Si clusterdecreases, the cluster surface can undergo a reconstruction,resulting in a cluster structural symmetry that differs fromthat of bulk Si. Raghavachari et al.26, 29 used the high-orderperturbation theory to study the equilibrium geometries anddemonstrated that the most stable conformations of small Siclusters are considerably different from those derived fromcrystal fragments. Some clusters, e.g., Si6, Si7, and Si10, enjoyadditional structural stability compared with clusters of othersizes. They are considered “magic clusters” and form stablesubunits entering clusters of larger sizes.51, 52 Si6 and Si10 areparticularly stable.39, 40 Further, photoelectron spectroscopy47

shows that bandgaps in Sin clusters (n = 3–45) are compara-ble or even smaller than the bandgap of bulk Si. These resultscontradict the prediction of the quantum confinement theoryand arise due to surface effects, which compete with quantumconfinement. First principle calculations on Si clusters havealready shown that they are capable of creating MEs.18, 53–55

However, systematic studies are required in order to realizethe full potential of MEG and MEF, in particular, with re-spect to cluster size and temperature. The present work reportsthe size and temperature dependence of the pure-dephasingprocess.

II. METHODS

The phonon-induced pure-dephasing times associatedwith the luminescence, MEG and MEF processes are com-puted by the optical response function approach. Consideringluminescence, its homogeneous linewidth � is inversely pro-portional to the dephasing time T2, which includes the excitedstate lifetime T1 and the pure-dephasing time T ∗

2 .

� = 1

T2= 1

2T1+ 1

T ∗2

. (1)

For a sufficiently long lifetime T1, which is common forbandgap excitations in clusters, � is determined solely byT ∗

2 .56

Pure-dephasing is associated with fluctuations of elec-tronic energy levels due to coupling to phonons. The fluc-tuations are characterized by the energy gap autocorrelationfunction (ACF). The normalized ACF is defined as

C(t) = 〈�E(t)�E(0)〉〈�E2(0)〉 = Cu(t)

〈�E2(0)〉 , (2)

where Cu(t) is the unnormalized ACF. The ACF represents anensemble average of the correlation of the energy gap fluctua-tion. In our study, the averaging is performed over the canon-ical ensemble. ACFs characterize the memory of the energygap fluctuation. A rapid decay of ACF implies fast dephasing.Fourier transform of an ACF produces the spectral density,

I (ω) =∣∣∣∣ 1√

2π

∫ ∞

−∞dt e−iωtC(t)

∣∣∣∣2

, (3)

which identifies the phonon modes involved in the dephasingprocess. The peaks of the spectra indicate the strength of theelectron-phonon coupling for the phonon modes of given fre-quencies. The presence of multiple frequencies in the spectraldensity usually leads to rapid ACF decay.

The optical response function considered below charac-terizes the pure-dephasing process for a pair of states entan-gled in a coherent superposition. It can be obtained directlyor via the second-order cumulant expansion.57 The direct de-phasing function is computed as

D(t) = exp(iωt)

⟨exp

(− i

¯

∫ t

0dτ �E(τ )

)⟩. (4)

Alternatively, the dephasing function can be rewritten usingthe cumulant expansion as

D(t) = exp(−g(t)), g(t) = 1

¯2

∫ t

0dτ1

∫ τ1

0dτ2Cu(τ2). (5)

The cumulant expression indicates that dephasing israpid if the integrated Cu(t) is large. This is achieved whenCu(t) shows no recurrences, and when its initial value Cu(0),corresponding to the standard deviation in the energy gap, islarge. The cumulant function (5) converges faster than the di-rect expression (4), because the latter involves averaging ofa complex-valued oscillatory function, whose real and imagi-nary parts change signs.

The atomistic simulation reported here were performedin the framework of density functional theory imple-mented with the Vienna Ab initio Simulation Package.The projector-augmented-wave pseudopotential, the PBEgeneralized-gradient density functional,58, 59 and a convergedplane-wave basis set were employed. The initial geometrieswere generated from both theoretical and experimental re-sults, which report the optimal geometry of Si clusters of dif-ferent size.50, 60 First, the structures were fully optimized at0 K. Then, the clusters were heated to 100 K, 300 K, 500 K,700 K, and 900 K with repeated velocity rescaling. Micro-canonical molecular dynamics (MDs) of 4 ps duration weregenerated for each temperature and for each cluster, usingthe Verlet algorithm with the 1 fs time step and Hellman-Feynman forces. The optimization, heating, and microcanoni-cal MD showed no major changes in the atomic and electronicstructure, such as bond breaking or fragmentation. The de-phasing times for the luminescence, MEG and MEF processeswere computed based on the MD trajectories using Eqs. (4)and (5).

III. RESULTS AND DISCUSSION

A. Geometry and electronic structureof the Si clusters

The optimized structures of the Si clusters under investi-gation are shown in Figure 1 together with their symmetries.Figure 2 shows the charge densities of the highest occupiedmolecular orbital (HOMO) and the lowest unoccupied molec-ular orbital (LUMO). The densities have different shapesdepending on the cluster symmetry. Similarly to the opti-mized geometries, both HOMO and LUMO exhibit higher

164303-3 Liu, Neukirch, and Prezhdo J. Chem. Phys. 139, 164303 (2013)

FIG. 1. Optimized structures for Sin clusters, n = 5–10, 15.

symmetry in smaller clusters. Compared to the larger clus-ters (n = 8, 9, 10, and 15), Si5, Si6, and Si7 possess D2 typesymmetry (Figure 1), indicating that they are invariant withrespect to the horizontal reflection. This additional symmetryfavors cluster stability, even though smaller volume reducesstability due to a higher contribution of the surface energy tothe total energy.

Figure 3 presents the cluster binding energy per atom andthe incremental binding energy. These properties character-ize cluster stability. The binding energy per atom, (nE(Si)− E

optn )/n, provides a straightforward tool to examine the

size dependence of cluster stability. Here, E(Si) is the energyof the Si atom and E

optn is the energy of the Sin cluster in

the optimized geometry. Figure 3 also shows the incremen-tal binding energy, which is defined as E

optn − E

opt

n−1 − E(Si),where both n and n − 1 clusters are in their optimized geome-tries. All incremental binding energies are below zero, indi-cating that it is always favorable to gain an atom in this rangeof Si cluster size. The value of incremental binding energy in-dicates the barrier that has to be overcome in order to fragmenta cluster. Si8 has the smallest incremental binding energy, i.e.,only a small amount of energy is gained by attaching a Si atomto Si7. Thus, Si8 is a relatively unstable structure. It may turnitself into Si7 by eliminating an atom or into Si9 by gainingan atom. A similar conclusion was suggested in Table VI ofMeloni et al.61

FIG. 2. HOMO and LUMO electron densities in Si clusters at 0 K.

FIG. 3. Bandgap, binding energy per atom, and incremental binding energyvs. cluster size for silicon clusters. The smallest bandgap in the Si8 clus-ter corresponds to relatively small binding energy per atom and incrementalbinding energy, indicating Si8 gains just a little energy when it forms.

The bandgaps at 0 K (Figure 3) are an agreement withthe previously obtained results.35 Slight differences can be at-tributed to a different functional used in our research. Theclusters are sufficiently small for the details of their atom-istic structure to matter. As a result, the bandgaps deviatefrom the size dependence predicted by the quantum confine-ment theory,62–64 which shows that the bandgap for a spheri-cal particle is inversely proportional to the cluster diameter.Good correlation exists between the bandgap and the twotypes of binding energy, especially the incremental bindingenergy. Similar bandgap-stability correlations can be found inother systems,65 supporting our conclusion that clusters withlarger bandgaps are more stable. For instance, Si8 character-ized by the smallest incremental binding energy has the small-est bandgap. Thus, the bandgap provides an estimate of clus-ter stability. This result is quite useful, since bandgaps can beeasily measured experimentally by a variety of techniques.

The bandgaps decrease with temperature for all clustersconsidered here (Figure 4). In some cases, e.g., Si8 and Si15,the decrease is not entirely monotonous. Si8 and Si9 have lowbandgaps consistently for all temperatures, indicating theyare least stable over the whole temperature range. Si6, Si7,and Si10 exhibit large bandgaps, demonstrating they are moregeometrically stable, and hence are considered as “magic”clusters.51, 66 Note that the reported values represent canonicalaverages, and that thermal fluctuations in the bandgaps rangefrom 0.05 eV for low temperatures to 0.15 eV for high temper-atures. The temperature dependence of the bandgaps is gov-erned by a number of factors, and in particular, by the phononmodes coupled to the electronic subsystems. The electron-phonon interaction also determines the dephasing rates, whichare discussed in Subsection III B.

B. Pure-dephasing in luminescence, MEG and MEF

Figure 5 provides a detailed description of the dephasingprocesses considered here.12 The two states shown in each

164303-4 Liu, Neukirch, and Prezhdo J. Chem. Phys. 139, 164303 (2013)

FIG. 4. Temperature dependence of bandgaps for various Si clusters. Allclusters show decreasing bandgaps with the increasing temperature.

plot form a coherent superposition that loses coherence dueto elastic scattering with phonons. As a result, the states be-come uncorrelated. Luminescence is associated with a quan-tum transition across the bandgap (Eg). It involves a superpo-sition of the ground and lowest excited states, top-left panel inFigure 5. The inverse of the pure-dephasing time determinesluminescence linewidth, Eq. (1), which can be obtained ex-perimentally. Previous calculations showed excellent agree-ment with the experimental data, providing a benchmark forour calculations.67–73 The rest of the top row in Figure 5 rep-resents three examples of superpositions of single and MEs: asingle exciton at the energy of 2Eg and a bandgap bi-exciton,a 3Eg single exciton and a bandgap bi-exciton, and a 3Eg sin-gle exciton and a bandgap tri-exciton, respectively. In the 1stand 3rd cases, the single and multi-excitons have the same en-

ergy. In the 2nd case, the energies of the single and bi-excitondiffer by Eg. Dephasing between high energy single excitonsand low energy bi-excitons is associated with MEG.

The plots in the second row of Figure 5 illustrate the bi-exciton fission process. A superposition of two excitons withenergies close to the bandgap loses coherence, creating uncor-related single excitons. The process is similar to singlet fissionobserved in molecular systems.23, 74 We consider four exam-ples of superpositions of single excitons involved in MEF. Thefirst and second examples involve symmetric and asymmet-ric electron-hole excitations. The last two examples involvechanges in either hole or electron occupation. Since the num-ber of atoms in all clusters is relatively small, the energy spac-ings between HOMO/HOMO − 1 and LUMO/LUMO + 1 arefairly large, several tenths of eV.

The dephasing functions for the luminescence, MEG andMEF processes are evaluated using the optical response func-tions, which can be obtained directly or via the second-ordercumulant expansion, Eqs. (4) and (5), respectively. The datashown in Figures 6 and 7 are obtained using the cumulantapproximation due to its more rapid convergence. We com-puted direct dephasing functions for several clusters, confirm-ing that the 2nd-order cumulant expansion provides a reliabledescription. The pure-dephasing times are obtained by fittingthe dephasing functions with a Gaussian.

The upper-left panel in Figure 6 shows temperature de-pendence of the pure-dephasing time that determines thehomogeneous linewidth of luminescence. The clusters stayfrozen and are less mobile at low temperatures, and therefore,the pure-dephasing time is long. Smaller clusters experiencestronger electron-phonon coupling, and thus exhibit faster de-phasing: the dephasing times of Si5, Si6, and Si7 are shorterthan those of Si10 and Si15. As temperature increases, the de-phasing times for the luminescence process decrease in allSi clusters, regardless of cluster size. Higher temperature en-ables the atoms to move faster, activating more phonon modes

FIG. 5. Illustration of the dephasing processes considered in the text. (Top row) Dephasing of the superposition between the first excited state and the groundstate, which determines the luminescence linewidth; MEG involves dephasing between single and multiple excitons, such as a double bandgap exciton and abiexciton (2Eg/biexc), a triple bandgap exciton and triexciton (3Eg/triexc), or a triple bandgap exciton and biexciton (3Eg/biexc). (Bottom row) MEF processesinvolve dephasing of various superpositions of single excitons near the bandgap.

164303-5 Liu, Neukirch, and Prezhdo J. Chem. Phys. 139, 164303 (2013)

FIG. 6. Pure-dephasing times for the luminescence and MEG processes in different size Si clusters as functions of temperature. The MEG processes showshorter dephasing times, because of the more substantial difference in the orbitals origins of single excitons and multiple excitons, relative to the differencebetween the ground state and lowest energy exciton.

FIG. 7. Pure-dephasing times for the MEF processes in different size Si clusters as functions of temperature. MEFs involve energy levels that are close to eachother and therefore, the dephasing times for the MEF processes are a longer than those for the MEG processes.

164303-6 Liu, Neukirch, and Prezhdo J. Chem. Phys. 139, 164303 (2013)

that can couple to electrons. At room temperature, the dephas-ing times for luminescence are around 12 fs in all clusters.The corresponding linewidth is about 55 meV. This value canbe compared to the experimental data75, 76 obtained for largerSi clusters. The experimental linewidth of 130 meV is largerthan the calculated value, likely because the calculation ig-nores the contributions arising from the chemically passivatedsurface. The oxidized surface of experimental Si nanocrys-tals contains light atoms, which possess high frequencyvibrational motions capable of strong electron-phononinteractions.

The dephasing times associated with superpositions ofsingle and bi-excitons, encountered in MEG, are shorter thanthose for luminescence, Figure 6. This is because the or-bitals supporting these states differ more significantly, e.g., inthe number of nodes, than the HOMO and LUMO involvedin luminescence. The temperature dependence of the pure-dephasing time in this case is the same as for luminescence:the time decreases with increasing temperature.

The pure-dephasing times associated with MEF areshown in Figure 7. The MEF times are notably longer thanthose for MEG, indicating that coherent superpositions ofsingle excitons comprising bi-excitons live longer than su-perpositions of single and bi-excitons, which are shown inFigure 5. The MEF dephasing times are relatively large, be-cause the orbitals supporting the single excitons involved inthe superpositions are close in energy and have similar elec-tronic densities and number of nodes. The similarity in thestate densities results in similar dependence of state energieson cluster geometry. Hence, the energies oscillate in phasewhen atoms thermally fluctuate, and the dephasing is slow.The extraordinarily long dephasing time for Eg/E

hg in Si5

comes from the near degeneracy of HOMO and HOMO + 1

in this case. As temperature increases, more vibrations are ex-cited thermally, accelerating the MEF dephasing in all cases.

The pure-dephasing time for the luminescence process isdirectly related to the single-particle luminescence linewidthsthat are available experimentally, as discussed above. Pure-dephasing for the MEG and MEF processes has not been ob-served directly, in particular since it is hard to deconvoluteexperimental data into the pure-dephasing and other compo-nents, such as relaxation, Eq. (2). The pure-dephasing timesare related to the experimental via the models12, 67, 77–79 usedto describe the experimental data.

C. Phonon modes

In order to characterize the phonon modes causing pure-dephasing in luminescence, MEG and MEF, we report Fouriertransforms, Eq. (3), of the corresponding ACFs, Eq. (2).Figure 8 presents the data for the luminescence process.The results for MEG and MEF are given in supplementarymaterial.80 At lower temperature, the influence spectra forthe more stable clusters, i.e., Sin (n = 5, 6, 7, and 10), ex-hibit fewer phonon modes than the spectra of the less sta-ble clusters (n = 8, 9). This is because Si8 and Si9 are lesssymmetric, and therefore, the electron-phonon coupling se-lection rules are less strict. The largest cluster, Si15, displaysthe broadest range of frequencies at 100 K, likely because itcontains more atoms than the other clusters, and therefore, thelargest number of vibrations overall. When temperature rises,thermal fluctuations distort the optimized structures of theclusters. Consequently, the influence spectra of most clustersexhibit more phonon modes, especially low-frequency acous-tic modes that are easiest to activate thermally. Additionally,acoustic modes are most anharmonic, and therefore, they have

FIG. 8. Fourier transforms of autocorrelation functions for the transition energy between HOMO and LUMO in the Si clusters, corresponding to the lumines-cence process. The peaks indicate the modes that are responsible for the luminescence linewidth. The temperatures are labeled in the first plot.

164303-7 Liu, Neukirch, and Prezhdo J. Chem. Phys. 139, 164303 (2013)

less strict selection rules for the electron-phonon coupling.The appearance of the low-frequency modes is particularlyevident with the larger clusters, Si10 and Si15. The larger num-ber of modes involved in the elastic electron-phonon scatter-ing processes rationalizes the decrease in the pure-dephasingtimes with increasing temperature, Figures 6 and 7. Similartrends can be seen in the figures presented in the supplemen-tary materials,80 explaining the temperature dependence ofthe dephasing times for the MEG and MEF processes.

IV. CONCLUSIONS

The current work examined elastic electron-phonon scat-tering processes in small silicon clusters, Sin (n = 5–10, 15).The optimized structures for all clusters were studied, and thebandgap, binding energy per atom, and incremental bindingenergy were reported. These three properties show good cor-relation and can be utilized to characterize the stability of theclusters. In particular, Si8 has the lowest bandgap, binding en-ergy per atom and incremental binding energy. Hence, Si8 canbe considered least stable. A large specific surface area en-ables surface atoms to adjust their position in order to lowerthe total energy of the clusters. The surface reconstruction ef-fect compensates instability of the small clusters, as there ismore structural symmetry in Si5, Si6, and Si7.

The pure-dephasing times that determine the lumines-cence linewidth are in the range of 10–30 fs at 100 K. Smallerclusters dephase much faster due to stronger electron-phononcoupling. At higher temperatures, the dephasing times spana more narrow range, e.g., 10–15 fs at 300 K and 5–10 fsat 800 K. The dephasing times decrease with temperature,because more phonon modes are thermally activated and be-come involved in the dephasing process. The dephasing timesdrop more quickly in larger clusters, because they possess abroader variety of phonons, especially acoustic modes that areeasily activated by temperature and are more anharmonic.

Three MEG and four MEF examples were analyzed.The pure-dephasing of MEG was found to be the fastest, inparticular, because the orbitals supporting multiple excitonsand high-energy single excitons differ substantially in atomiclocalization and number of nodes. On the contrary, the sin-gle exciton states involved in the superpositions forming bi-excitons undergoing MEF have similar properties, and there-fore, the MEF pure-dephasing times are quite long.

At low temperature, only few vibrations are involvedin the dephasing processes. As the temperature rises, morephonon modes are activated, accelerating the dephasing.Thermal activation is particularly efficient for the low-frequency acoustic modes. These modes are least harmonic,and as a result, they are particularly efficient in destroying thelong-time correlations in the phonon-induced fluctuations ofthe electronic energy levels. Further, anharmonic modes haveweaker selection rules for the electron-phonon coupling. Be-cause larger clusters possess a broader range of low-frequencymodes, the temperature dependence of the dephasing time be-comes more pronounced with increasing cluster size. Thesetrends are seen for all three considered processes. By charac-terizing the electron-phonon dynamics in semiconductor clus-ters, the current study provides insights that are useful for

cluster applications in photovoltaic, photocatalytic, and othertypes of devices.

ACKNOWLEDGMENTS

Financial support of the U.S. Department of Energy,grant DE-SC0006527 is gratefully acknowledged.

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Fourier transforms, Eq. (3), of the ACFs, Eq. (2), associated with the MEGand MEF processes, Figure 5.