Time-Domain Ab Initio Study of Phonon-Induced Relaxation ofPlasmon Excitations in a Silver Quantum DotAmanda J. Neukirch,† Zhenyu Guo,§ and Oleg V. Prezhdo*,‡

†Department of Physics and Astronomy and ‡Department of Chemistry, University of Rochester, Rochester, New York 14627, UnitedStates§Jiangsu Key Laboratory for Carbon-based Functional Materials & Devices, Institute of Functional Nano & Soft Materials(FUNSOM), Soochow University, Suzhou, Jiangsu 215123, PR China

*S Supporting Information

ABSTRACT: Relaxation of plasmon excitations through thephonon channel is investigated in a silver nanocrystal with asurface-hopping Kohn−Sham density functional theory in thetime domain. Good agreement with the experimental data isobtained. Plasmons delocalize away from the nanocrystal coreand couple to a narrow range of low-frequency acousticphonons. Higher energy plasmon excitations tend to be moredelocalized, exhibit weaker coupling to phonons, and relaxmore slowly than lower energy plasmons. The phonon-induced plasmon relaxation occurs on a picosecond time scale. This is two orders of magnitude longer than the time scale ofelastic plasmon−phonon scattering, which contributes to the line-width of plasmon resonances via the pure-dephasingmechanism. The phonon-induced energy relaxation of plasmons in metallic particles is somewhat slower than that of chargecarriers in semiconducting nanoscale materials. The difference can be attributed to the extended nature of plasmon excitations,resulting in a weaker interaction with phonons and coupling only to low-frequency vibrations.

1. INTRODUCTIONThe novel electronic and optical applications of noble metalnanoparticles have received significant attention in a variety offields, including lasing, optical recording and data storage,biological imaging, electro-optics, and light harvesting for solarcells.1−11 The exciting optical physics of the metal nanoparticlesarise from the resonant interaction of conduction bandelectrons and the electromagnetic field.12−14 The collectiveexcitations, usually known as surface plasmons, are responsiblefor the specific light extinction and high local fields, followed bynontrivial electron dynamics. Relaxation of the surface plasmonresonance (SPR) includes several stages that occur on differenttime scales. The electron dephasing process determines theSPR lifetime and has a time constant of 10−100 fs. Theelectron−electron scattering creates a Fermi distribution andresults in the equilibration of the electron temperatures.15−17

Electron−electron scattering takes place on the 100−1000 fstime scale. It is followed by the electron−phonon interactionsthat cool the electron gas further until thermal equilibrium isreached. The electron−phonon equilibration requires 1 to 10ps. The energy relaxation is completed by interaction ofnanoparticle phonons with vibrational motions of the environ-ment, with times from 10 to 100 ps.18

Electron−phonon dynamics play an important role in manyapplications, including those mentioned above. The dynamicsare investigated in time-domain optical experiments.19−25

Plasmons have also been studied theoretically using time-dependent density functional theory (TDDFT) in the time

domain,26,27 jellium-type models,28,29 linear responseTDDFT,4,13 and Green function techniques.30,31 The real-time theoretical approaches developed in our group directlymimic the optical experiments and have allowed us to modelelectron−phonon interactions in a variety of nanoscalematerials.32−36 Focusing on plasmons, we have established37

that elastic scattering of plasmon excitations by phononsgenerates an important contribution to the SPR line-width,which can be determined in temperature-dependent stud-ies.38,39 Phonons are also responsible for the slower non-radiative relaxation of electronic excitations in a wide range ofmaterials, ranging from diatomic molecules to liquid, solid, softcondensed matter, and biological systems. The rate ofelectron−phonon relaxation is an important parameter inmany devices. For instance, it determines the response time ofelectro-optic switches.40 In optical, electronic, and photovoltaicdevices, electron−phonon coupling causes nonradiative energyloss and system heating and should be avoided, yet inphotothermal therapy metal nanoparticles have been usedsuccessfully because they rapidly convert the absorbed photonenergy into heat.41−44 Therefore, it is important to understandthe mechanisms that are responsible for the electron−phonondynamics in nanoscale materials, and plasmon−phonon

Received: April 8, 2012Revised: June 13, 2012Published: June 14, 2012

Article

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© 2012 American Chemical Society 15034 dx.doi.org/10.1021/jp303361y | J. Phys. Chem. C 2012, 116, 15034−15040

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relaxation in metal nanoclusters, in particular, to see if there isroom to tune for particular applications.The current article presents the first time-dependent ab initio

simulation of phonon-induced relaxation of plasmon excitationsin a metal particle. The results indicate that the relaxationoccurs on a picosecond time scale. The lower energy plasmonexcitations couple more strongly to phonon modes, causing therelaxation of these states to proceed faster than the relaxation ofhigher energy plasmons. These results are consistent with theexperimental data.19−23 We show that the phonon-induceddecay of plasmon excitation energy is facilitated by very low-frequency acoustic phonon modes45 with frequencies around100 cm−1. The simulation of inelastic plasmon-phononinteractions reported here is compared with our previouswork on elastic plasmon-phonon scattering37 as well as with thecorresponding data for phonon-induced electron and holedynamics in semiconductor quantum dots,35,36,46,47 carbonnanotubes,32,48 and graphene nanoribbons.49

The following section describes the simulation approach,which combines time-domain density functional theory withnonadiabatic molecular dynamics, and provides computationaldetails. The Results and Discussion section is organized intothree subsections, focusing on the geometric and electronicstructure of the silver nanoparticle under investigation, thephonon modes interacting with the plasmon excitations, andthe phonon-induced relaxation dynamics of plasmons. Thearticle concludes with a summary of the key observations.

2. THEORYThe quantum-classical fewest-switches surface-hopping (FSSH)approach50,51 within the time-dependent Kohn−Sham (TDKS)framework is utilized in the current study. The electrons aretreated quantum mechanically with DFT, and the slower,heavier nuclei are treated classically. FSSH is an atomisticnonadiabatic molecular dynamics approach that satisfies thedetailed balance between transitions up and down in energy.The detailed balance ensures that the time-domain simulationattains equilibrium Boltzmann populations of electronic statesin the long-time limit, making FSSH applicable to studyingelectron relaxation processes.51 FSSH can be viewed as aquantum master equation, where the state-to-state electronictransition rates depend on time through coupling to phonondynamics. Using FSSH-TDKS,48,52,53 electron−phonon relaxa-tion has been studied in many systems, including molecule−bulk interfaces,33,34,54 semiconductor quantum dots,35,36,46,47

carbon nanotubes,32,48 and graphene nanoribbons.49 DFT is acorrelated electron theory that maps an interacting many-bodysystem into a single-particle representation by includingelectron correlation effects into an effective Hamiltonian, thatis, a DFT functional. Real-time26,27 and linear response4,13,55

TDDFT have been used successfully to describe plasmonresonances in metallic nanoparticles. The current work is thefirst simulation of plasmon−phonon interactions within real-time TDDFT.2.1. Time-Dependent Kohn−Sham Theory. In the KS

representation of TDDFT, the electron density is writtenas56−58

∑ρ ϕ= | |=

t tr r( , ) ( , )p

N

p1

2e

(1)

where Ne is the number of electrons and ϕp(r,t) are single-electron KS orbitals. The evolution of ϕp(r,t) is determined by

applying the TD variational principle to the expectation valueof the Kohn−Sham density functional and leads to the systemof coupled equations of motion for the KS orbitals56

δϕ

δϕℏ = =i

t

tH t t p N

rr R r

( , )[ , ( )] ( , ), 1, ...,p

p e (2)

The Hamiltonian H is TD, both through the external potentialcreated by the nuclear trajectory R and through the electrondensity. The equations are coupled because H depends on thedensity, eq 1, and therefore, all KS orbitals are occupied by theNe electrons. Expanding the TD KS orbitals ϕp(r,t) in theadiabatic KS orbital basis ϕk(r;R(t)) gives

∑ϕ ϕ= | ⟩t c t tr r R( , ) ( ) ( ; ( ))pk

N

pk k

e

(3)

The TDKS eq 2 transforms to the equation of motion for theexpansion coefficients

∑δδ

ε δℏ = − ℏ · it

c t c t i d R( ) ( )( )pkm

N

pm m km km

e

(4)

The adiabatic KS orbitals are obtained for the current valuesof the nuclear coordinates by solving the time-independent KSequations, which are implemented in many computer codes.The nonadiabatic coupling

ϕ ϕ

ϕ δδ

ϕ

ϕ ϕ ϕ ϕ

· = ⟨ |∇ | ⟩·

= ⟨ | | ⟩

≈Δ

⟨ | + Δ ⟩ − ⟨ + Δ | ⟩

d

t

tt t t t t t

R r R r R R( ; ) ( ; )

12

( ( ) ( ) ( )) ( ) )

km k R m

k m

k m k m

(5)

arises from the dependence of the adiabatic KS orbitals on thephonon dynamics.57,58 It is computed numerically according tothe right-hand-side of eq 5.59

Because of the inherent difference in the masses of electronsand nuclei, the electronic and nuclear motions are treated onindependent time scales. The TDKS equations, eq 4, describethe fast electronic dynamics that are propagated usingattosecond time steps. The significantly slower nuclear motionsare evolved with femtosecond time steps. This time-scaleseparation produces substantial computational savings, whilemaintaining the accuracy of the calculation.

2.2. Fewest Switches Surface Hopping in the Kohn−Sham Representation. The prescription for phonondynamics constitutes the quantum back-reaction problem.60

To define the effect of the quantum-mechanical electronevolution on the classical nuclear motion, FSSH uses astochastic algorithm that generates trajectory branching50 anddetailed balance.51 Trajectory branching mimics the splitting ofquantum-mechanical nuclear wave-packets in correlation withdifferent electronic states. Detailed balance ensures that theratio of the probabilities of transitions up and down in energy isequal to the Boltzmann factor. It is essential for studyingelectron−phonon relaxation and achieving thermodynamicequilibrium.FSSH assigns a probability for hopping between electronic

states. The probability is explicitly TD and is correlated to thenuclear evolution. The probability to hop between states k andm within the time interval Δt is50

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= Δ⎛⎝⎜

⎞⎠⎟dP

ba

tmax 0,kmkm

kk (6)

where

= − * · = *b a d a c cR2Re( );km km km km k m (7)

Here ck and cm are the coefficients evolving according to eq 4.The hopping probabilities depend explicitly on the non-adiabatic coupling defined in eq 5. The surface hoppingprobability is set to zero when dPkm is calculated to be negative.This ensures that a hop from state k to state m can only takeplace if the occupation of state k decreases, and the occupationof state m increases. The calculated probabilities are thencompared with a random number to determine if a hop ispredicted.In the original implementation of FSSH, the nuclear

velocities are rescaled50,59 along the direction of the electroniccomponent of the nonadiabatic coupling after each hop toconserve the total electron−nuclear energy. If eq 6 predicts atransition into a higher energy electronic state and there isinsufficient available nuclear kinetic energy, then the hop isrejected. The velocity rescaling and hop rejection providedetailed balance between upward and downward transitions.51

This work utilizes a simplified version of FSSH,61 where it isassumed that the energy exchanged between the electronic andnuclear degrees of freedom is rapidly distributed across allnuclear modes. The assumption allows velocity rescaling andhop rejection to be replaced by multiplication of the probabilityfor upward transitions by the Boltzmann factor. Elimination ofthe velocity rescaling removes the need for the explicit quantumback-reaction, while maintaining the detailed balance condition.As a result, one can use a predetermined nuclear trajectory toevolve the electronic subsystem and achieve great computa-tional savings. The excited states were represented in the KSorbital picture by promoting one electron from an occupiedorbital to an unoccupied orbital.2.3. Simulation Details. The TDKS-FSSH approach was

implemented48,52,53 with the Vienna ab initio simulationpackage (VASP).62 The silver cluster Ag68 chosen in thepresent simulation has the diameter of 1.3 nm; see Figure 1.The cluster was cut from the silver bulk structure with aspherical cutoff. The cluster geometry was fully relaxed at 0 Kusing VASP. The optimized structure was slightly tetrahedral inshape. The PBE functional63 with projector-augmented-wave(PAW) pseudopotentials64 is employed in a converged plane-wave basis. The simulations are performed using periodic

boundary conditions, and neighboring silver clusters areisolated by 16 Å of vacuum in each direction. By repeatedvelocity rescaling, the cluster was heated to 300 K. Then, a 2 psmicrocanonical trajectory was generated using the Verletalgorithm. Five hundred initial conditions were sampled fromthis trajectory for the plasmon relaxation dynamics.

3. RESULTS AND DISCUSSION3.1. Geometric and Electronic Structure. Figure 1

presents the geometric structure of the Ag68 cluster and thecharge density distribution of three low-energy plasmon statesof different angular symmetries. The electron distributions ofthe plasmon excitation density extend spatially far beyond thenanoparticle. Note that the cluster geometry and state densitiesare shown on different scales. The electron density becomesmore delocalized for higher energy states. States withpredominant localization outside of the cluster start appearingat 3.9 eV above the Fermi energy. These states are identified asplasmon states in our calculation. A more detailed comparisonof the spatial localization of plasmon, surface and bulk-likestates can be found in ref 37.Figure 2 shows the evolution of the electronic density of

states (DOS) of the Ag68 cluster during a 1 ps part of the

molecular dynamics trajectory. We found the bandgap of Ag68to be ∼0.4 eV. However, the DOS is significantly sparser from−2.7 to 3.3 eV. The number of states is small in this region, andthe peaks are spaced apart from each other. The DOS increasessubstantially outside the low density region. The DOS increaseat −2.7 eV in the valence band corresponds to the d-electronband and agrees well with the experimental65,66 andtheoretical67 results for bulk silver. The increase in the DOSin the conduction band happens to coincide with the onset ofthe plasmon state band.The observed structure of the electronic DOS, Figure 2,

indicates that the phonon-induced relaxation dynamics ofelectrons and holes should undergo a substantial change oncethe electron or hole reaches the energy of −2.7 to 3.3 eV,respectively. In particular, the charge-phonon relaxation shouldslow down dramatically in the sparse region of the DOS. Theslowing down is known in the semiconductor quantum dotliterature as the phonon bottleneck.35,68−70 The bottleneckresults from the mismatch between the phonon frequencies andthe electronic energy gaps.TDDFT calculations were performed on Ag84 tetrahedral

clusters.83 The plasmon absorption peak was found at 3.62 eV,in agreement with our calculations. A study of the (Ag20-Ag)

+

complex found a band of highly delocalized states above the

Figure 1. Geometry of Ag68 and electron density of three low-energyplasmon states in the cluster. The plasmon states extend far outsidethe cluster.

Figure 2. Evolution of the electronic density of states (DOS), arbitraryunits, in the Ag68 cluster. Because silver is a metal, there is no bandgap, but the DOS is significantly higher below (above) −2.7(3.3) eV.

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plasmon band, leading to charge transfer from the cluster to theextra Ag atom.27 The increased delocalization of high energyplasmon states away from the cluster seen in our calculationscorrelates with this observation.3.2. Phonon Modes. To identify the phonon modes that

most strongly couple to the electronic degrees of freedom andinduce energy relaxation, we computed the Fourier transformsof the plasmon state energies. The resulting phonon spectra areshown in Figure 3. The nonadiabatic electron−phonon

coupling is directly related to the second derivative of theenergy along the nuclear trajectory.71 Therefore, the vibrationalmodes that most strongly modulate the energy levels will createthe largest coupling.35,47 The data presented in Figure 3indicate that the plasmon excitations in the Ag68 nanoparticlecouple exclusively to low-frequency phonon modes. The low-frequency modes are responsible for the energy decay of theplasmons in the present study. The slight differences in theparticipating frequencies for different plasmons can beexplained by variations in the localizations and symmetries ofthe plasmon state densities (Figure 1). In all cases, phononmodes with frequencies less than 100 cm−1 dominate in theplasmon−phonon relaxation. It should be noted that theresolution of the FT is 24 cm−1, as limited by the length of theMD trajectory. In general, the lower energy plasmon states tendto couple more strongly to phonons than the higher energystates. This observation can be explained by the fact that thehigher energy states are more strongly delocalized away fromthe nanoparticle; therefore, they are less affected by atomicmotions taking place inside the particle.Participation of slow acoustic modes in the phonon-induced

decay of plasmon excitations is somewhat surprising, if notentirely unexpected. The nonadiabatic electron−phononcoupling, eq 5, depends on the phonon velocity R. At a giventemperature, higher-frequency modes with lighter effectivemasses will generate larger velocities. Therefore, one canassume a priori that high-frequency vibrations should generatestronger electron−phonon coupling. However, the non-adiabatic electron−phonon coupling, eq 5, also involves thederivative of the electronic wave function with respect to thephonon coordinate, ⟨ϕk(r,R)|▽R|ϕm(r;R)⟩. Low-frequencyvibrations, which are closely related to acoustic phonons inbulk materials, modulate the size and shape of the nanocluster.These large-scale motions are able to affect plasmon stateswhose densities are delocalized across and beyond the entiredot (Figure 1). In contrast, high-frequency optical modes

involve local displacements of atoms with respect to oneanother. They have little influence over global nanoclusterproperties. In particular, the integral over the whole spatialextent of the plasmon excitation tends to average out thepositive and negative changes induced by the gradient ▽Rtaken with respect to a high-frequency phonon. Ramanscattering72,73 and transient absorption45 experiments havedetected acoustic vibrations below 100 cm−1, verifying ourresults.It is instructive to compare the phonon modes that couple to

plasmons to those that couple to semiconductor quantum dots.Electrons and holes in semiconductor quantum dots couple tohigher frequency phonons than plasmons in metallic particles.This is because charge carriers are localized inside crystals,whereas plasmon excitation extend well beyond the crystalsurface. Charge carriers in PbSe interact strongly with low-energy acoustic modes, similarly to the Ag quantum dots.However, the participating modes have higher frequencies andinclude spheroidal acoustic phonons with frequencies around100 cm−1 and even longitudinal optical modes with frequenciesaround 200 cm−1.35 A similar trend is seen in the Sinanocrystals.46 Compared with inelastic relaxation processes,elastic pure-dephasing of electrons and holes in semiconductordots involve higher frequency modes covering a broader energyrange,74,75 following the same tendency as Ag plasmons. Thepresence of ligands55,76,77 containing much lighter elementsthan the core atoms of Ag, Si, or PbSe nanoparticles introducesvery high-frequency vibrations, up to 2000 cm−1.46,75,78,79 Theligand modes participate in the dynamics of electronicexcitations of solid-state nanoparticles in the energy rangeswhere ligand states contribute to the overall DOS.80

Carbon nanotubes32,48,81 and graphene nanoribbons49,82

show a much more pronounced participation of opticalphonons in the dephasing and relaxation of electronicallyexcited states. This fact can be attributed to a much largerdifference between the acoustic and optical frequencies ofcarbon-based nanoscale materials compared with the inorganicnanocrystals. The frequency difference emphasizes theimportance of the nuclear velocity term, R, over the electronicoverlap term, ⟨ϕk(r,R)|▽R|ϕm(r;R)⟩, in the electron−phononcoupling, eq 5.

3.3. Phonon-Induced Relaxation of Plasmon Excita-tions. The top and bottom panels of Figure 4 show the decayof the population and energy of the plasmon excitations startingfrom the eight plasmon states, whose average energies are givenin the second column of Table 1. The Table also presents thedecay time-scales, obtained by fitting the data shown in theFigure to the Gaussian function

τ= −f t A t( ) exp[ ( / ) ]2(8)

The normalization constant A was equal to 1 for thepopulation decay and to the initial excitation energy for theenergy decay. Examples of the fits are shown in the SupportingInformation. Each line corresponds to an initial excitation intoone of the eight plasmon states. The bold red line is the averageof all lines. The plasmon population decay is obtained bysumming TD populations of all plasmon states. The decayresults from transfer of population into nonplasmon states,which include both bulk and surface states.37 When thepopulation leaves all plasmon states, the data in the top panel ofFigure 4 decays to 0. The energy decay curves track the overallexcitation energy, including contributions from all states. Whenthe population transfers from the initial state to a lower energy

Figure 3. Fourier transforms of the energies of the eight plasmonstates (Table 1). The energy fluctuations are due to phonon motions.

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state, the curve in the bottom panel drops to a slightly lowerenergy. This difference explains the difference in the populationand energy relaxation time-scales shown in Table 1.Considering the plasmon population relaxation (top panel in

Figure 4), we detect a significantly faster decay for the lowerenergy excitations. The states that relax the quickest tend tocouple more strongly to the phonons (Figure 3). Thepopulation from the lowest energy plasmon excitations decaysprimarily into bulk states, rationalizing the faster decay andstronger coupling to phonons. Higher energy plasmonexcitations take time to traverse down the manifold of plasmon

states before decaying into the low energy bulk states. Theenergy relaxation exhibits little dependence on the initialexcitation energy (Table 1) because population transfer from aplasmon to a bulk state causes a small change in energy,regardless of the initial excitation.For comparison with the experimental results, we divided the

eight initial conditions into two categories, four lowerexcitations, and four higher excitations (Table 1). We focuson the population relaxation because it clearly differentiatesplasmon excitations studied experimentally from bulk states.The phonon-induced relaxation times of the lower and higherenergy plasmon excitations are 680 and 1210 fs in oursimulation. These are consistent with the lifetimes reported inthe ultrafast spectroscopy experiments including both theoverall time scale19,21−23 and the difference between the high-and low-energy excitations.21 The higher energy plasmon statesdecay more slowly than the lower energy states because ofdifferences in the electron−phonon coupling. Higher energyplasmons decay into other plasmon states, whereas lowerenergy plasmons decay into bulk-like states. The electron−phonon NA matrix element, eq 5, is larger when one of the twok, m states is a bulk state than when both k and m are plasmons.This is because the gradient with respect to the phonondisplacement, ▽R, is larger for bulk states that are localized onatoms than for plasmons that are localized away from the QD.Therefore, the relaxation of higher energy plasmon states canbe viewed as a two-step process, involving a slower relaxationthrough the manifold of plasmon states and a faster relaxationinto lower energy bulk-like states.Further details of the phonon-induced relaxation of plasmon

excitations in the Ag68 nanocluster are provided in Figure 5.Part a of the Figure gives a close up of the evolution of the Ag68electronic DOS in the energy range of the plasmon excitation(compare with Figure 2). Figure 5b shows the energy relaxationstarting from the lowest plasmon state, whereas Figure 5cshows the energy relaxation starting from one of the higherenergy plasmon excitations considered in this work. Thefunctions depicted in Figures 5b,c are obtained by multiplyingthe DOS by TD state occupations. Compared with the bottompanel of Figure 4, which shows the evolution of the averageenergy, Figure 5b,c presents explicitly the distribution of energyamong all states during the decay process. In both examples(Figure 5b,c), the excitation spreads from the initiallypopulated state to many intermediate states, diffuses quicklyacross the band down to the effective bandgap, and begins tocollect at around 3.3 eV, where the DOS decreases significantly(Figure 2), and one expects the phonon bottleneck.35,68,69 Thisresult indicates that the majority of the states present within therelevant energy range participate in the relaxation of plasmonexcitations and that no particular states are more importantthan other states.Compared with the charge carrier relaxation in semi-

conductor quantum dots,35,46,47,70,84,85 the phonon-induceddecay of plasmon excitations in the silver nanoparticle isnotably slower. In bare semiconductor crystals, electrons andholes lose energy on the subpicosecond time-scale. There, themajority of the relaxation can be described by a Gaussianfunction,35 as in eq 8. In the presence of ligands, the relaxationcan be as fast as 100 fs and is well-described by exponentialdecay.46 The charge-phonon relaxation in nanoscale carbonmaterials is also subpicosecond.32,49,86,87 The slower, few-picosecond decay of plasmon excitations can be rationalized bythe weaker coupling of plasmons to phonons, relative to the

Figure 4. Decay of the plasmon population (top) and energy(bottom) starting from the eight initial conditions (Table 1). The boldred dashed lines indicate the averages.

Table 1. Phonon-Induced Relaxation Times of Populationand Energy of Plasmon Excitations in the Silver QuantumDot, Figure 1, Computed from Gaussian Fit, Equation 8, ofthe Population and Energy Decay Curves Shown in Figure4a

state no. ⟨E⟩, eV τpop, ps τenergy, ps

1 3.93 0.40 3.222 3.96 0.59 2.763 4.03 0.81 2.664 4.05 0.91 2.845 4.10 0.98 2.786 4.11 1.02 2.897 4.13 1.11 3.088 4.24 1.74 4.681−4 3.99 0.68 2.875−8 4.15 1.21 3.351−8 4.07 0.95 3.11

a⟨E⟩ is the canonically averaged energy of the correspondingexcitation. The energies are given relative to the Fermi energy. (1-4), (5-8), and (1-8) represent the average relaxation times for firstfour, latter four, and all eight initial conditions, respectively.

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charge-phonon coupling, as well as by the involvement of lowerfrequency phonons in the case of plasmons,45 Figure 3,compared with other systems.32,35,46−49,85 In turn, the weakplasmon−phonon coupling and involvement of low-frequencyphonons are explained by delocalization of plasmon excitationsaway from the nanoparticle (Figure 1) and hence by smalloverlap of plasmon and phonon wave functions.

4. CONCLUSIONSUsing a time-domain ab initio approach, we have investigatedthe phonon-induced relaxation dynamics of plasmon excitationsin a silver nanoparticle. The relaxation occurs on a picosecondtime scale, in agreement with the experimental data. Weestablish that plasmon excitations are weakly coupled tophonons, in particular, because they are spatially delocalizedaway from the nanoparticle core. Only low-frequency acousticmodes are able to facilitate the energy decay because only thesemodes can modulate plasmon wave functions and energies. Theeffects of high-frequency vibrations on plasmon states tend toaverage out, when integrated over the delocalized plasmonwave functions. The higher energy plasmon excitations are bothmore delocalized and tend to exhibit weaker coupling tophonons. This explains the different decay time scales for low-and high-energy excitations observed in experiments.The elastic phonon-induced pure-dephasing of plasmon

excitations is two orders of magnitude faster than the inelasticplasmon-phonon scattering and involves higher frequencyphonons. The charge−phonon interactions in semiconductorquantum dots and nanoscale carbon materials are stronger than

the plasmon−phonon interactions. As a result, both elastic andinelastic charge−phonon scattering processes are faster thantheir corresponding plasmon−phonon scattering counterparts.Whereas elastic phonon-induced pure-dephasing creates anotable contribution to the line width of plasmon resonances,the contribution of the phonon-induced energy decay to theline width is negligible. At the same time, the energy ofplasmon excitations is ultimately deposited into the phonons.Our simulations show that the transfer of energy from plasmonexcitations into acoustic phonons of metallic nanoparticlesproceeds within a few picoseconds.

■ ASSOCIATED CONTENT

*S Supporting InformationExamples of fits of the data from Figure 4, with the timeconstants reported in Table 1. This material is available free ofcharge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: oleg.prezhdo@rochester.edu.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe wish to acknowledge the support of the NSF Grant CHE-1050405 dedicated to the theory development and the DOEGrant DE-FG02-05ER15755 supporting the nanoclusterstudies.

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Figure 5. 3D representation of the plasmon energy relaxationcompared with the bottom panel of Figure 4. (a) Closeup of theevolution of the plasmon DOS, see Figure 2, in the plasmon excitationenergy range. (b) Decay of a lower energy plasmon excitation. (c)Decay of a higher energy plasmon excitation. The plots in parts b andc were obtained by multiplying the DOS by time-dependent stateoccupations. The z axis in all plots has arbitrary units.

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