Ab Initio Analysis of Auger-Assisted Electron TransferKim Hyeon-Deuk,*,†,‡ Joonghan Kim,§ and Oleg V. Prezhdo*,∥

†Department of Chemistry, Kyoto University, Kyoto 606-8502, Japan‡Japan Science and Technology Agency, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan§Department of Chemistry, The Catholic University of Korea, Bucheon, Gyunggi-do 420-743, Korea∥Department of Chemistry, University of Southern California, 920 Bloom Walk, Los Angeles, California 90089, United States

*S Supporting Information

ABSTRACT: Quantum confinement in nanoscale materialsallows Auger-type electron−hole energy exchange. We show bydirect time-domain atomistic simulation and analytic theory thatAuger processes give rise to a new mechanism of charge transfer(CT) on the nanoscale. Auger-assisted CT eliminates the renownMarcus inverted regime, rationalizing recent experiments on CTfrom quantum dots to molecular adsorbates. The ab initiosimulation reveals a complex interplay of the electron−hole andcharge−phonon channels of energy exchange, demonstrating avariety of CT scenarios. The developed Marcus rate theory forAuger-assisted CT describes, without adjustable parameters, theexperimental plateau of the CT rate in the region of large donor−acceptor energy gap. The analytic theory and atomistic insightsapply broadly to charge and energy transfer in nanoscale systems.

Q uantum confinement of charge carriers in semi-conductor nanoscale materials, such as quantum dots

(QDs) and nanotubes, leads to novel phenomena, giving rise tonew classes of systems utilized for light harvesting and chargeseparation during solar energy conversion.1−4 The rates andmechanisms of photogeneration and dissociation of coupledelectron−hole pairs, known as excitons, at interfaces ofnanoscale materials with various charge acceptors determineefficiencies of photovoltaic and photocatalytic devices.5−8 Onecustomarily invokes Marcus theory9−12 to describe electrontransfer (ET) to get the ET rate as a function of the electrondonor−acceptor (D−A) energy gap, known as the drivingforce, and the D−A couplings. Developed initially for intra- andintermolecular ET, Marcus theory assumes that ET isaccompanied by a large rearrangement of nuclear configuration,which brings the electronic donor and acceptor energies inresonance. The energy of the nuclear rearrangement is calledthe reorganization energy. The ET rate in traditional Marcustheory reaches a maximum when the reorganization energymatches the D−A energy gap. If the gap becomes too large, thenuclear rearrangement requires significant thermal activation,and the rate decreases, leading to the celebrated Marcusinverted region.10 The recent experimental studies on thecomplexes of QD donors with molecular acceptors reported ETrates over a broad range of driving energies.13 The rates of thephotoinduced ET from the QDs to the molecular acceptorsgrew with increasing driving force, finally reaching a plateau.The expected Marcus inverted region was not detected,

suggesting the possibility for ultraefficient charge transfers innanoscale-confined materials.14

We report time-domain ab initio simulation of ET from aCdSe QD to an adsorbed methylene blue (MB) molecule,faithfully representing the experimental system.14 The simu-lation has demonstrated that the ET process is accompanied byAuger-type electron−hole energy exchange, rationalizing thelack of the inverted regime in the experimental data. Thecurrent simulation reveals a complex interplay of the electron−hole and charge−phonon channels of energy exchange,demonstrating a variety of ET scenarios. It shows that thetraditional ET mechanism mediated by phonons competes withthe Auger-assisted ET process. A notable fraction of theelectron energy flows directly to phonons, bypassing the hole.Phonons dissipate the excited hole energy, allowing the hole toaccommodate additional electron energy. Having lost someenergy to phonons, the hole can be excited again, facilitatingthe ET further. In addition to the atomistic simulation, whichexhibits a complex interplay between the electron, hole, andphonon dynamics following QD photoexcitation, we derive ananalytic theory of Auger-assisted electron transfer (AAET). Thedeveloped theory successfully reproduces the experimentallyobserved plateaus in the ET rate at large driving forces basedsolely on ab initio input and without any adjustable parameters.

Received: November 28, 2014Accepted: December 30, 2014Published: December 30, 2014

Letter

pubs.acs.org/JPCL

© 2014 American Chemical Society 244 DOI: 10.1021/jz502505mJ. Phys. Chem. Lett. 2015, 6, 244−249

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The electronic structure and adiabatic MD are computedwith VASP with the PW91 density functional and projector-augmented-wave pseudopotentials, as explained in the Support-ing Information. The QD/MB complex is fully optimized atzero temperature and heated up to an ambient temperature byrepeated velocity rescaling. A 4 ps microcanonical trajectory iscalculated on the ground electronic state by the ab initio MDusing the Verlet algorithm. It is hard to know how MBmolecules are attached to QDs in the experiments. It is likelythat the MB molecules are directly bound to QDs, while otherareas of the QD surface are covered by ligands. Our ab initioMD simulations show that the MB molecule is adsorbed andremains stable on the QD surface despite large thermal andstructural fluctuations at the ambient temperature.To account for the Auger hole excitation accompanying the

ET, we adopt the exciton representation. The driving force ofthe ET, ΔG, is defined as the energy difference between theQD LUMO and the MB LUMO. The nonadiabatic (NA)dynamics simulations for the AAET are performed bycombining the time-domain DFT (TDDFT) with theNAMD.15 The adiabatic KS orbitals are obtained with theDFT for atomic positions at each moment along the ab initioMD trajectory. We represent the original TDDFT equations inthe second quantization notation to include the ground andsingle exciton (SE) states, |Φg(r;R)⟩ and |ΦSE

i,j (r; R)⟩,respectively. The total wave function is then expanded as

∑|Ψ ⟩ = |Φ ⟩ + |Φ ⟩t C t C t( ) ( ) ( )g gi j

i j i j

,SE

,SE,

(1)

Substitution of eq 1 into the time-dependent Schrodingerequation leads to the EOMs for the expansion coefficients

∑

ℏ∂

∂= − ℏ ·

− ℏ · ′ ′

′ ′′ ′

iC t

tC t E i C t

i C t d

R

R

( )( ) ( )d

( )

XX X g X g

i j

i jX i j

;

,SE

,;SE, ,

(2)

where X corresponds to either the ground or SE state and EX isthe state energy. The NA couplings introduced by

· ≡ ⟨Φ |∇ |Φ ⟩· = ⟨Φ | ∂∂

|Φ ⟩dt

R RX Y X R Y X Y; (3)

induce NA transitions between the states in the exciton basis.The atomistic simulation of the ET dynamics is performed bydirectly solving eq 2 with the time-dependent NA couplings (eq3) and energies obtained by the ab initio MD simulation. Wefurther introduced energy dissipation to phonon modes by theexponential decay function, exp(−t/τdiss), where t is asimulation time and τdiss is a typical dissipation time scale.16

A schematic of the ET involving the Auger hole excitation isgiven in Figure 1. Photoexcitation promotes an electron fromthe QD HOMO to the QD LUMO, leaving a hole in the QDHOMO. Because the QD+MB complex is asymmetric, it isnatural that the orbitals are distributed asymmetrically. In thecurrent case, the CdSe QD attracts the electron more than theMB molecule, making the QD HOMO localized close to theMB and the QD LUMO localized far from the MB.We focus on the ET from the Cd33Se33 QD to the adsorbed

MB molecule. The ET involves an electronic transition fromthe QD LUMO to the MB LUMO; the electron traverses theentire QD. In the traditional ET, nuclear vibrational motionsaccommodate the excess energy lost by the electron. In thecurrent system, the energy lost by the electron is taken up by

the hole that is promoted from the QD HOMO into the deepervalence band (VB) of the QD. The quantum confinement ofthe charge carriers in the QD enhances their Coulombinteraction, allowing this electron−hole energy exchange andopening up an alternative pathway for the energy flow. Inaddition, because the QD HOMO is located close to the MB, itis strongly coupled to the high-frequency MB phonon modes,inducing the efficient hole excitation. The excess energytransferred from the electron to the hole is ultimately dissipatedby phonon modes of the CdSe QD/MB complex; excited statesof the hole, located energetically deep inside the VB, tend todelocalize over the whole QD, in contrast with the QDHOMO. The Auger mechanism is inefficient in bulk semi-conductors because of weak, screened Coulomb interactions.There has been no observation of the AAET in any bulksemiconductor. The Marcus inverted region appears in suchcases. Electron−hole interactions are affected by material-dielectric properties. Electrons and holes weakly interact insemiconductors, forming excitons. The exciton Bohr radiusdetermines the material size at which the quantum confinementeffects appear. The Coulomb interactions can be strong inmolecules; however, the densities of electron and hole states arelow. Low state densities reduce Auger processes, making it hardto match properly the electron and hole energy levels.The VB density of states (DOS) is particularly important for

the AAET. We averaged ab initio molecular dynamics (MD)data of 2 ps on the VB energies fluctuating at the ambienttemperature and obtained the VB DOS shown in Figure 2a. Itexhibits nontrivial dependence, with growing and thensaturating away from the Fermi energy, EF = 0. Thedependence is neither square root, following from the simpleparabolic dispersion model, nor exponential, expected for adistribution of surface defect states. The saturation is essentialto reproduce the experimental plateau of the AAET rates,14 aswill be shown later. Figure 2b demonstrates that the energy ofthe QD HOMO fluctuates more than the energy of the QDLUMO. This is because the QD HOMO is spatially close to theMB molecule and is affected by the molecular vibrations. TheMB LUMO energy fluctuates most because the moleculeundergoes larger scale motions than the QD. The Fourier

Figure 1. (Upper) Densities of the key orbitals participating in theAAET. (Lower) Mechanism of the AAET. The electron and hole isinitially excited in the QD CB and VB, respectively. The holeexcitation in the QD VB accompanies the ET from the QD to the MB.This mechanism becomes possible in a QD due to strong Coulombinteraction between the confined electron and hole and a high densityof hole states. The excess energy that the hole gained is finallydissipated by phonons.

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transforms of the evolving orbital energies characterizing thephonon modes that couple to the electronic subsystem aregiven in Figure 2c. The phonon spectrum for the acceptor state,the MB LUMO, exhibits strong intensity for the high-frequencymodes, indicating that these modes actively participate in theET by the traditional mechanism, which competes with theAuger process. The large phonon intensity of the QD HOMOalso suggests that the high-frequency molecular modeseffectively assist relaxation and excitation of the hole.Figure 3a−c shows time evolutions of the electron, hole, and

phonon energies along representative nonequilibrium AAETtrajectories initiated by electron excitation from the QDHOMO to the QD LUMO. The D−A energy gap, that is,the energy difference between the QD LUMO and MB LUMO,is shown as well. As the ET proceeds, the electron energydecreases on a picosecond time scale, in agreement with theexperimental data.14 In the initial stage of Figure 3a, the energylost by the electron is gained exclusively by the hole, confirmingthe AAET mechanism. At a later time, after the electron hasbeen transferred and the hole has been excited, the holedeposits its excess energy to phonons. This Auger-typeelectron−hole energy exchange assisting the ET is not always

perfect, however. In the earliest stage in Figure 3b, the phononsgain more energy than the hole. By the time when the ETsignal has reached a plateau, around 0.2 ps, the hole hascaptured all of electron’s energy, while, on the longer time scale,the hole loses energy to phonons. Figure 3c shows an examplein which the hole and phonons participate equally in the energyexchange during the ET event. This example demonstrates thatphonons take active part in the AAET process. In neither casedoes the hole gain the energy equal to the D−A energy gapbecause of dissipation by phonons. In the long-time limit, theelectron energy decreases by the D−A energy gap, the holeenergy converges to zero, and the phonon energy approachesthe D−A energy gap.The above insights obtained in the single trajectory cases are

also confirmed by the averaged data. Panels d−f of Figure 3display the dynamics averaged over 51 nonequilibrium MDtrajectories for different D−A energy gaps. The excess energygenerated by the ET is only partially accommodated by thehole, and the rest of the electron energy is compensated byphonons. In a later stage, the ET continues at a slower pace,and the hole starts to lose energy to phonons by dissipation.The data shown in Figure 3d−f include both statistical

Figure 2. (a) VB DOS of the Cd33Se33 QD. The VB DOS increasesaway from the band gap and then saturates. (b) Energy fluctuations ofthe QD HOMO (red line), LUMO (blue line), and MB LUMO(green line). The fluctuation amplitude reflects the strength of theelectron−phonon coupling. (c) Fourier transforms of the evolvingorbital energies characterize the phonon modes that couple to theelectronic subsystem. The MB LUMO exhibits the strongest phononintensity as expected, while the phonon intensity of the QD LUMO isweakest.

Figure 3. (a−c) Representative dynamics of the energies stored by theelectron (blue line), hole (red line), phonons (purple line), and theelectron D−A energy gap (green line) during the AAET whoseschematic illustration is given in Figure 1. The energy lost by theelectron is accommodated by both the hole and phonons. At a longertime, the hole relaxes by donating its energy to phonons. (d−f)Dynamics averaged over 51 nonequilibrium MD trajectories fordifferent D−A energy gaps. The excess energy generated by the ET isonly partially accommodated by the hole, and the rest of the electronenergy is compensated by phonons. In a later stage, the ET continuesat a slower pace, and the hole starts to lose energy to phonons bydissipation. The shown data include both statistical mechanical andquantum mechanical averaging.

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mechanical and quantum mechanical averaging, while the dataof Figure 3a−c are averaged only quantum mechanically. Theresults averaged quantum mechanically can be interpretedstatistically as well. The time-dependent data shown in theexamples, Figure 3a−c, can be viewed as probabilities fordifferent opportunities of the system to have the given energies.For instance, the energy of the electron in Figure 3a decreasesrapidly and nearly linearly within the first 0.15 ps. The resultcan be interpreted that there is a 1/3 chance that the electronhops within the first 0.05 ps, a 1/3 chance that it hops between0.05 and 0.1 ps, and a 1/3 chance that it hops between 0.1 and0.15 ps. The hole energy increases rapidly and linearly on thesame time scale, and the hole evolution can be interpreted inthe same way.Promotion of the hole from the QD HOMO into the deeper

QD VB during the ET is directly demonstrated in Figure 4. The

distributions of hole populations in different VB orbitals areshown for various elapsed times. The x axis denotes the VBorbital number, with 1 corresponding to the QD HOMO. Attime t = 0, the hole is in the QD HOMO. At 0.1 ps, the holestill populates the QD HOMO. The initial slow stage ofquantum population transfer is generic to all quantumtransitions. Demonstrated more explicitly in Figure S2 of theSupporting Information, this Gaussian, parabolic, or cosineregime can be viewed as part of a Rabi oscillation, giving rise tothe quantum Zeno effect.14,17−25 At later times, the holedistribution shifts to orbitals deeper inside the VB. Thepopulation shift directly demonstrates the Auger hole excitationduring the ET; the excess energy generated by the ET isactually accommodated by the hole. The hole excitation is notuniform because the energy levels of the VB states are discrete,and the NA couplings connecting each state pair vary in bothenergy and time. At long times, the hole distribution flows backto the upper VB states, reflecting hole energy dissipation byphonons.The ET rate is obtained as the inverse of the ET time

estimated as in Figure S2 in the Supporting Information, kET =1/τET. Figure 5a shows the ET rates for a wide range of ΔGvalues. Four distinct situations are compared: (1) AAETaccompanied by charge-phonon energy dissipation of the

realistic time scale, τdiss = 0.9 ps; (2) AAET accompanied bycharge-phonon energy dissipation of a longer time scale, τdiss =2.7 ps; (3) AAET without charge-phonon energy dissipation;and (4) conventional ET without the hole dynamics. The lastcase includes only the QD LUMO and MB LUMO electronstates, and the electronic energy is deposited into phononsonly.The AAET rates, cases 1−3, vary little with ΔG and reach

plateaus around ΔG = 1.05 eV. Because the hole excitationaccompanies the ET, the QD LUMO to MB LUMO gap isoffset by the hole excitation, and the true D−A gap changeslittle with increasing ΔG. The ET rate keeps growing becausethe density of final states, determined by the VB DOS,increases. The plateau is then reached because the VB DOSsaturates as in Figure 2a. Taking into account the experimentalfact that the ET rate increases with decreasing QD size, formost QD materials and molecular acceptors, the current ETrates compare well with the experimental ET rates.14 If the EThad followed traditional Marcus theory, the rates would havechanged by two orders for the shown ΔG range. Case 4supports this fact. Our time-domain ab initio simulationsconfirm that the Auger excitation of the hole eliminates theinverted region.Phonon dissipation plays a minor role in determining the

AAET rate. The ET rates with and without dissipation, cases 1

Figure 4. Distribution of the hole among different VB orbitals at 0.1(red line), 0.2 (green line), 0.5 (blue line), and 3.8 ps (purple line)following the photoexcitation. The highest five VB orbitals startingwith the QD HOMO (orbital number 1) are shown in the main panel,while the inset shows occupations of deeper orbitals. The orbitalnumber in the x axis correlates with the orbital energy below the QDVB edge. The Auger hole excitation during the ET is directlydemonstrated by the population shift to the deeper VB states. At 3.8ps, the hole is already relaxing back to the QD HOMO.

Figure 5. (a) ET rates as a function of the driving force ΔG. The panelcompares four cases with and without the Auger interaction,influenced by phonon dissipation on different time scales. TheAAET rates increase with ΔG and reach a plateau, while theconventional ET rate monotonically decreases with increasing ΔG,corresponding to the Marcus inverted region. Phonon dissipationslightly increases the ET rate because it quenches the hole energy andallows repeated hole excitation. (b) ET rates calculated from theextended Marcus theory. The AAET rates have no inverted region andreproduce the experimental plateau. (c) ET rates calculated with thesingle acceptor state; that is, ρ(E) = δ(E) in eq 4. The resulting ratesdecrease at large ΔG, clearly showing the Marcus inverted region.

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and 3, differ slightly, and the ET rates with the dissipation timeτdiss = 2.7 ps already converge to the ET rates without anydissipation. Energy dissipation by phonons accelerates the ETbecause the dissipation of the excited hole energy allowsadditional hole excitation. This fact is supported by the growingdifference in the ET rates of cases 1 and 3 with increasing ΔG.Larger ΔG requires more effective Auger hole excitation, andthe dissipation becomes more important in allowing additionalhole excitation during the ET.In addition to the time-domain ab initio simulation, we

derive the extended Marcus ET rate theory involving the Augerhole excitation based on Fermi’s golden rule expression startingfrom the model Hamiltonian, as explained in the SupportingInformation. Assuming that the average NA coupling constant |Hrp| is approximately independent of energy, we get the finalform of the rate

∫ππλ

ρ λλ

=ℏ

| |− + − Δ⎛

⎝⎜⎞⎠⎟k

H

kTE E

E GkT

24

d ( ) exp( )

4ETrp

2 2

(4)

with the temperature T = 300 K. k and λ are the Boltzmannfactor and a reorganization energy of phonons, respectively.The density of states ρ(E) is the density of the final acceptorstates as a function of the hole excitation energy E. Thus, − E +ΔG corresponds to the true driving force. Note that the currentsimulation and analytic model are formulated in the adiabaticrepresentation, in which all Coulomb interactions arediagonalized out to obtain the adiabatic state, that is, eigenstatesof the electronic Hamiltonian. Therefore, the coupling Hrp isthe NA coupling rather than the Coulomb coupling. Weemphasize that our final expression 4 includes neitheradjustable parameters nor scaling factors and is specifiedcompletely by the clear physical quantities obtained from thetime-dependent ab initio calculations. We numerically performthe integration in eq 4 using the averaged values of the DOSρ(E) and the NA coupling constant |Hrp(E)|

2 obtained for thecurrent QD/MB complex at the ambient temperature by ourtime-dependent ab initio MD simulations, |Hrp(E)| = 1.00 ×10−2 eV. The calculated AAET rates are shown in Figure 5b.The rates exhibit no inverted region and saturate at large ΔG,reproducing the plateau observed in both the experiment anddirect time-domain ab initio calculations. The analytic modelindicates that dense VB states and strong Coulomb electron−hole interaction in a donor nanomaterial can induce an efficientAAET. As shown in Figure 5c, the ET rates calculated with thesingle acceptor state, that is, ρ(E) = δ(E), are qualitativelydifferent; they decrease at large ΔG, clearly showing the Marcusinverted region.We conclude that Auger-type hole excitation in the electron

donor material eliminates the Marcus inverted region. We haveexamined the AAET mechanism by the direct time-domain abinitio simulation performed on the CdSe QD/MB complex.14

We have demonstrated that the ET is accompanied bypromotion of the hole from the QD HOMO deeper into theQD VB. The simulated ET dynamics exhibits two time-scales.The initial Gaussian dynamics is associated with the directelectron−hole energy exchange, while the slower exponentialcomponent corresponds to the charge-phonon energyrelaxation.We have also derived an extended Marcus rate theory for the

ET to include the Auger hole excitation and have reproducedanalytically the experimental plateau of the AAET rate at large

driving forces. The analytic results are based solely on ab initioinput without any adjustable parameters.We have found that phonons play competitive and important

roles. A notable fraction of the electron energy flows directly tophonons, bypassing the hole, thereby enabling the traditionalphonon-assisted ET mechanism. In addition, phonons dissipatethe excited hole energy, allowing the hole to accommodateadditional electron energy. Having lost some energy tophonons, the hole can be excited again, facilitating AAET.In the particular system under investigation, the QD electron

donor state is localized opposite to the adsorbed molecularacceptor. By forcing the electron to travel across the QD, thissituation further strengthens the electron−hole interaction,which is already enhanced by the quantum confinement.It is possible that acceptor molecules are separated from QD

donors by ligands. Ligands would decrease the donor−acceptorelectronic coupling and create an energy barrier, slowing downthe ET. The electron transferred to the acceptor molecule willinteract with the hole much less; the electron−hole interactionwill be strong only in the initial stage of the ET, making theAuger process less efficient. In addition, removing the holefrom the QD core by ligand trapping should eliminate theAAET mechanism altogether and restore the Marcus invertedregime. The Marcus inverted region should be restored as theQD size approaches and exceeds the exciton Bohr radius of thecorresponding bulk semiconductor. The VB DOS would be toolow to support AAET in very small nanoclusters, whoseproperties are similar to properties of molecules. Thequalitative aspects of AAET should not change in the presenceof multiple acceptor molecules adsorbed on the QD surface.Assuming that these molecules do not form strongly interactingdimers and larger clusters, they should just provide parallelchannels for AAET.Auger-type phenomena are very common in nanoscale

systems due to strong Coulomb interactions induced byquantum confinement and due to high densities of states.Molecules also exhibit strong electron−hole interactions buthave low state densities. Bulk materials have high state densitiesbut exhibit weak Coulomb interactions due to dielectricscreening. The reported insights into the AAET processstimulate ideas on how this recently discovered phenomenoncan be utilized for controlling charge transfer and relaxationprocesses in nanoscale materials.

■ ASSOCIATED CONTENT*S Supporting InformationComplete theoretical derivations, simulation details, andadditional figures. This material is available free of charge viathe Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Authors*K.H.-D.: E-mail: kim@kuchem.kyoto-u.ac.jp.*O.V.D.: E-mail: prezhdo@usc.edu.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSK.H.-D. acknowledges financial support from JST (PRESTO)and Grant-in-Aids for Scientific Research from Japan Societyfor the Promotion of Science (KAKENHI), Grant No.24750016. O.V.P. acknowledges funding from the Office of

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Basic Energy Sciences of the U.S. Department of Energy, GrantDE-SC0006527.

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