Non-Radiative Electron Hole Recombination in Silicon ...and luminescence. Nonradiative relaxation of excitons induced by electron− phonon interactions is detrimental to these applications.

Non-Radiative Electron−Hole Recombination in Silicon Clusters: AbInitio Non-Adiabatic Molecular DynamicsJin Liu,† Amanda J. Neukirch,‡ and Oleg V. Prezhdo*,§

†Departments of Chemical Engineering, ‡Physics and Astronomy, and §Chemistry, University of Rochester, Rochester, New York14627, United States

ABSTRACT: Silicon clusters hold exciting potential for optoelectronic andsolar energy applications by yielding strong and tunable visible absorptionand luminescence. Nonradiative relaxation of excitons induced by electron−phonon interactions is detrimental to these applications. We combinenonadiabatic molecular dynamics (NAMD) with time-domain densityfunctional theory to study electron−hole recombination in Sin clusters (n =5−10, 15). The recombination is much faster in clusters than in bulk Si dueto enhanced electron−phonon coupling and transition from indirect todirect bandgap. By applying quantum-classical NAMD, we investigated theimportance of the decoherence correction in determining the nonradiative lifetime. The recombination rates decrease by anorder of magnitude due to decoherence, bringing the calculations in close agreement with experiment. We interpreted the resultsof the atomistic simulations analytically using Fermi’s golden rule, rationalizing the dependence of the relaxation rate on theelectron−phonon coupling and temperature. The electron−hole recombination time is roughly independent of the size of thesesmall clusters, with Si5 and Si7 exhibiting longer lifetimes due to enhanced stability to thermal fluctuations.

1. INTRODUCTION

Quantum confinement in nanocrystals imparts novel electronicproperties and leads to a variety of applications, including solarcells,1−3 light-emitting diodes,4,5 field-effect transistors,6 andbiosensors.7 Tunable bandgaps, enhanced exciton lifetimes andmultiple exciton generation in quantum dots (QDs) arefavorable for photovoltaics and photocatalysis, as the researchon CdTe,8 CdSe,9 PbSe,10,11 PbS,12 and InAs13 QDs hasshown. Strong and tunable radiative emission motivatesdevelopment of light-emitting diodes and related devices.4,5 Athorough experimental and theoretical understanding of thecharge pathways and optical properties of QDs is essential forrealizing their full potential. QDs differ from the parent bulkmaterials due to both quantum confinement and surface effects.Quantum confinement allows one to vary continuously QDproperties by changes in size. Surface regions deviate from bulkin structure and composition, containing ligands and defects. Asthe electronic energy levels become sparse with decreasing QDsize, the electron−phonon relaxation slows down, leading tothe expectation of a phonon bottleneck.14,15 The phononbottleneck is not as common as expected initially,14,16 becauseof a large density of electronic states, especially at highenergies,15,17,18 stronger electron−phonon coupling in QDscompared to bulk, and a broad spectrum of phonon modesparticipating in the nonradiative relaxation. QDs deviate fromthe ideal behavior due to symmetry breaking, defects,19,20

ligands,21,22 and anharmonicities.16,23 Auger-type exchange ofenergy between electrons and holes24,25 accelerate therelaxation further. The nonradiative relaxation representsinelastic electron−phonon scattering. Elastic scattering deter-mines homogeneous line widths of optical signals and lifetimes

of electronic state superpositions encountered during multipleexciton generation, fission and other excited state processes.26

Silicon is the most common material in electronics and solarenergy applications. Bulk Si is not suitable for optoelectronicsowing to its indirect bandgap (∼1.1 eV). As the size of Sicrystals decreases below the exciton Bohr radius, around 5nm,27 novel effects emerge. The bandgap becomes direct,28

stimulating development of silicon-based optoelectroniccomponents. In contrast to crystalline silicon, porous siliconproduces highly efficient, visible luminescence.29−31 TheFranck−Condon shift increases abruptly around 1 nm,manifesting transition from crystals to molecules.32 Siluminescence lifetime depends strongly on material size andgenerally decreases in smaller systems. The excited statelifetime of bulk Si is extremely long, around a millisecond,33

since radiative relaxation is forbidden by the indirect bandgap,and the electron−phonon coupling causing nonradiative energylosses is weak. The exciton lifetime reduces to 0.1 ms in 50 nmSi crystals.34 When the size decreases further to severalnanometers, the photoluminescence lifetime reduces to micro-seconds.31,35,36 Mason et al.30,37 spatially isolated and detectedluminescence from individual Si nanoparticles in the 5−20 nmrange, suggesting a 1 μs lifetime. Bechstedt38 investigatedelectron−hole relaxation using first-principles simulations andfound the radiative lifetime to be proportional to the Si clusterdiameter. When the Si cluster size decreased to 5 Å, which iscomparable to the size investigated in our work, the calculatedlifetime was around 4 ns. Recently, Wei Yu et al.39 created a Si-

Received: July 6, 2014Revised: August 15, 2014Published: August 15, 2014

Article

pubs.acs.org/JPCC

© 2014 American Chemical Society 20702 dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−20709

Dow

nloa

ded

via

UN

IV O

F SO

UT

HE

RN

CA

LIF

OR

NIA

on

Nov

embe

r 7,

201

9 at

19:

59:3

1 (U

TC

).Se

e ht

tps:

//pub

s.ac

s.or

g/sh

arin

ggui

delin

es f

or o

ptio

ns o

n ho

w to

legi

timat

ely

shar

e pu

blis

hed

artic

les.

rich oxide (SRO)/SiO2 multilayer structure and obtained a 1 nsphotoluminescence lifetime for the 1 nm SRO layer. Occurringtogether with radiative energy losses, nonradiative electron−phonon relaxation is investigated less frequently, because ofexperimental challenges in detecting nonradiative signals andadvanced theoretical tools required, such as nonadiabaticmolecular dynamics (NAMD).18,40,41

This work presents a time-domain ab initio study of phonon-induced nonradiative electron−hole recombination in Siclusters containing 5 through 10 and 15 atoms. Si6, Si7, andSi10 are considered magic clusters due to enhanced stability.They form basic subunits of larger clusters.42−44 The followingsection describes the theoretical methodology, combining time-dependent density functional theory (TDDFT) with NAMD,and including a semiclassical decoherence correction. TheResults and Discussion describes the geometric and electronicstructure of the Si clusters, and the nonradiative electron−holerecombination obtained with and without the decoherencecorrection. The dependence of the simulation data onelectron−phonon coupling and temperature is investigatedanalytically using Fermi’s golden rule.

2. METHODOLOGYThe electron−hole recombination is simulated in a mixedquantum-classical framework, in which the faster electronicdegrees of freedom are treated quantum mechanically, whilethe slower nuclear motions are treated semiclassically. Electronsmove in the classical external field created by nuclei. In turn,nuclei evolve under the influence of the quantum force due toelectrons. The evolution of the electronic structure of the Siclusters is described using TDDFT. The electron-vibrationaldynamics is modeled with the fewest switching surface hopping(FSSH)45,46 approach to NAMD. The classical-path approx-imation47,48 is employed. The quantum nature of nuclearmotions is captured using the semiclassical decoherencecorrection.49

2.1. Time-Domain Density Functional Theory. Bymapping the interacting many-body system onto a tractablesystem of noninteracting particles moving in an effectivepotential, the ground state energy can be expressed as afunctional of electron density. In practical implementations, thedensity is constructed from single-particle Kohn−Sham (KS)orbitals, φp(r,t),

ρ φ= Σ | |=t tr r( , ) ( , )pN

p( 1)( ) 2e

(1)

where Ne is the number of electrons. The evolution of φp(r, t)is determined by the time-dependent variational principle,leading to the equations of motion

φρ φℏ

∂

∂=i

t

tH t t

rr r

( , )[ ( , )] ( , )p

p (2)

These equations are nonlinear, since the Hamiltonian H[ρ(r,t)] is a functional of the electron density, which is obtained bysumming over occupied KS orbitals, eq 1. By expanding thetime-dependent KS orbitals φp(r, t) in the adiabatic KS orbitalbasis, obtained for a given nuclear configuration

φ φ= Σ | ⟩=t c tr r R( , ) ( ) ( ; )p kN

pk k( 1)( )e

(3)

one takes advantage of the computational efficiency of time-independent DFT codes.50 Substituting the expansion into eq 2gives equations for the expansion coefficients:

ε δℏ∂

∂= Σ − ℏ ·

=ic t

tc t i d R

( )( )( )pk

mN

pm m km km( 1)( )e

(4)

Here, εm is the energy associated with the adiabatic KS orbital,φm(r, t). The last term in the above equation is the NAcoupling:

φ φ

φ φ

= ℏ · = ℏ⟨ |∇ | ⟩·

= ℏ ∂∂

NA i i

it

d R r R r R R

r R r R

( ; ) ( ; )

( ; ) ( ; )

km k m

k m

R

(5)

If the off-diagonal matrix elements of the gradient with respectto the nuclear coordinates, encountered in the middle of eq 5,are not available analytically, one gains computational savingsby computing numerically the time derivative term on the right-hand-side of eq 5. The NA coupling arises because electronicwave functions depend on nuclear coordinates. Thus, it iselectron−phonon coupling.

2.2. Nonadiabatic Dynamics by Surface Hopping.Surface hopping (SH) is a stochastic algorithm for switchingelectronic states in a mixed quantum-classical simulation. It usessolutions to the time-dependent Schrodinger equation, eq 2, toobtain a master equation with time-dependent transition rates.Motived by failures of the mean-field approximation, known inquantum-classical dynamics as the Ehrenfest approach, SHintroduces correlations between the electronic and nuclearevolutions.51 Most importantly for the current electron−phonon relaxation study, SH gives detailed balance betweentransitions up and down in energy, properly reproducingBoltzmann statistics in the quantum-classical equilibrium.46

FSSH45,46 is the most popular SH technique, since it isapplicable to a broad range of systems.Common to most practical quantum-classical schemes,

including the Ehrenfest and FSSH approaches, the evolutionof the electronic subsystem is overcoherent. By comparing theoff-diagonal elements of the electronic density matrix obtainedby the mixed quantum-classical and fully quantum-mechanicaldescriptions of electron−nuclear dynamics, one observes long-lived coherences in the former description and a rapid decay tozero in the latter case. The off-diagonal elements of the reducedelectronic density matrix in the fully quantum description decaybecause the nuclear wave-packets associated with the electronicstates diverge. The decoherence effects are introduced toquantum-classical descriptions as semiclassical correc-tions.49,52,53

In this work, both fewest switches surface hopping (FSSH)and decoherence corrected FSSH (DC-FSSH) are used tocompute the probability of hopping between quantum states.The hopping probability is derived from the time-dependentwave function expansion coefficients, eq 3. The probability of ahop between k and m within the time interval dt is given inFSSH by

=Pba

td dkmkm

kk (6)

where

ε δ= ℏ * − * ·

= * ·

−b a a

a

d R

d R

2 Im( ) 2Re( )

2Re( )km km m km km km

km km

1

(7)

= *a c ckm k m

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−2070920703

If dPkm is negative, the hopping possibility is set to zero. A hopfrom k to m can occur only when the electronic occupation ofstate k decreases and the occupation of state m increases.48 Thisfeature of the algorithm minimizes the number of hops andgives rise to the “fewest switches” name of the technique. Everytime-step, a uniform random number between 0 and 1 isgenerated and compared to dPkm to determine whether to hopor not. To conserve the total energy after a hop, it is assumedthat electrons and nuclei exchange the energy instantaneously.The energy balance is achieved by nuclear velocity rescalingafter each hop. Only the component of the nuclear velocityvector in the direction of the NA coupling vector is modified. Ifthe magnitude of this component is too small to accommodatean increase in the electronic energy after a hop to a higherenergy state, the hop is rejected. This step gives rise to detailedbalance between transitions upward and downward in energy,leading to Boltzmann statistics and quantum-classical thermo-dynamic equilibrium.46

Making the calculation significantly more efficient, we applythe classical path approximation to FSSH.47 The approximationis valid in particular, if thermal nuclear displacements aregreater than differences in the electronic potential energysurfaces minima. Then, a ground state trajectory can be used todrive the nonequilibrium electron dynamics. The velocityrescaling procedure is replaced by multiplication of theprobability of upward hops with a Boltzmann factor. Thisstep is equivalent to assuming that energy is exchanged betweendifferent nuclear modes faster than the time between hops.Then, the probability that the nuclear velocity componentalong the NA coupling vector has kinetic energy E is given bythe Boltzmann expression, E = exp(−E/kT).The classical path approximation neglects the heating that

occurs when the electronic energy is deposited into vibrationalmodes. In experimental studies, the electronic energy isdispersed across both the cluster and the clusters environment,which serves as a heat bath to absorb the released energy.Therefore, it would be wrong to assume in the simulation thatthe released energy is fully constrained inside the cluster. Theassumption that the energy released into the vibrational degreesof freedom is dispersed rapidly is more justified. A proper studyof the heating effect requires a model for the clusterenvironment.FSSH is overcoherent, since a single classical nuclear

trajectory cannot represent reduction of the electronic wavefunction induced by the quantum nuclear bath. The problem isparticularly important if the wave function reduction, known asdecoherence, occurs faster than the quantum transition underinvestigation. This is true for the electron−hole recombinationin the silicon clusters, in which case decoherence, or pure-dephasing, occurs on a femtosecond time scale.54 Originally,Bittner, Rossky52 and co-workers55 introduced the decoherencecorrection to FSSH as decay of the off-diagonal elements of theelectronic density matrix. Prezhdo pointed out56 that at thewave function level, decoherence leads to electron−nuclearcorrelations, naturally leading to SH type algorithms.57 Here,we follow the simple, wave function level decoherencecorrection to FSSH (DC-FSSH) described and implementedwith TDDFT in ref.,49 and tested with several systems.49,58,59

The main idea is to reset the expansion coefficients to 0 or 1after evolving coherently up to the decoherence time. Theresetting is stochastic with the probabilities given by the squaresof wave function expansion coefficients.

2.3. Simulation Details. The electron−hole recombinationwas simulated in the TDDFT-FSSH framework50 implementedwith the Vienna ab initio simulation package.60 The projectoraugmented wave (PAW) method,61 the generalized gradientapproximation (GGA) functional of Perdew−Burke−Ernzer-hof62 (PBE), and a converged plane wave basis set were used.The simulations were carried out in a cubic unit cell usingperiodic boundary condition, with sufficient vacuum betweencluster images. Cluster geometry optimization was followed byheating to 100 and 300 K. Two ps microcanonical MDtrajectories were generated with a 1 fs nuclear time-step foreach cluster at each temperature. The trajectories were used tosample 500 initial conditions for NAMD.

3. RESULTS AND DISCUSSION3.1. Geometric and Electronic Structure of the Si

Clusters. We construct and optimize the clusters based onboth theoretical and experimental data,63,64 from which theoptimal geometry for each cluster size can be predicted. Figure1 shows that the structures of all clusters deviate from the

diamond-like face centered cubic (FCC) lattice. In a bulk SiFCC cell, every Si atom coordinates four other Si atoms. Inclusters, most Si atoms are located on the surface and form 3−5covalent bonds to attain a stable configuration. Deviations fromthe bulk structure indicate that surface Si atoms experiencesignificant reorganization during the optimization process. Thesmaller clusters (Si5, Si6, and Si7) enjoy additional D2 symmetrycompared to the larger clusters. The additional symmetryassists to stabilize the small clusters.The nonradiative electron−hole recombination occurs by an

electronic transition from the lowest unoccupied molecularorbital (LUMO) to the highest occupied molecular orbital(HOMO). Detailed information about the symmetry andlocalization of the HOMO and LUMO charge densities can befound in our previous work.54 Generally, the orbitals in thesmaller clusters are more localized, contain few nodes, andexhibit additional symmetry elements. HOMO and LUMO ofthe larger clusters contain more nodes, are less symmetric, andinvolve hybridization of many atomic orbitals. The differencebetween the LUMO to HOMO densities, Figure 2, character-izes redistribution of the electron density during the transition.The yellow and blue areas in Figure 2 represent positive andnegative changes in the charge density, respectively. Theisosurfaces are set to the same value in all plots. The HOMO−

Figure 1. Snapshots of Sin clusters, n = 5−10, 15, taken frommolecular dynamics simulation at 100 K. Surface Si atoms break thediamond-like crystal symmetry of bulk Si. Smaller clusters containmore symmetry elements, achieving structural stability and smallvolume.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−2070920704

LUMO density differences exhibit simpler nodal structure inSi5, Si7, and Si8 due to a smaller overlap between the HOMOand the LUMO in these cases.3.2. Nonradiative Electron−Hole Recombination.

Figures 3 and 4 show decay of population of the first excitedstate obtained using FSSH and DC-FSSH, respectively. Thedecay is approximately exponential. Since the simulation can beperformed only for a short time, compared to the overallelectron−hole recombination time, we estimate the relaxationtime using the first-order expansion of the exponential:

τ τ= − ≈ −f t t t( ) exp( / ) 1 / (8)

The DC-FSSH data follow the linear dependence better thanthe FSSH results. Many FSSH lines exhibit negative curvature,indicating that the dynamics have not completely reached theexponential regime. In general, the earliest time of quantumdynamics is Rabi oscillation like, showing cosine (Gaussian,quadratic) time-dependence and leading, for instance, to thequantum Zeno effect.58,65 The experiment39 shows photo-luminescence decay on a 1 ns time scale at 300 K. FSSHunderestimates the electron−hole recombination time, whileDC-FSSH gives good agreement with the experiment, Table 1.Therefore, we focus on the DC-FSSH results and do notattempt to fit the FSSH data by a more complicated expression,e.g. a sum of Gaussian and exponential functions.The underestimation of the relaxation time by FSSH

highlights the need for the decoherence correction. Indeed,the decoherence time for a HOMO−LUMO superposition isabout 20 fs at 100 K and 10 fs at 300 K.54 This is several ordersof magnitude shorter than the relaxation time. A decoherencecorrection is required in such cases. The slowing down of

quantum transitions by decoherence is a manifestation of thequantum Zeno effect.58,65 Note that in contrast, intrabandrelaxation of electrons and holes involves rapid hops betweenclosely spaced states. Decoherence (elastic scattering) betweenclosely lying states takes a long time,26 while the hopping(inelastic scattering) time is short. Therefore, decoherencecorrections are not required for intraband relaxation, and FSSHgives reliable results.15,18

Both FSSH and DC-FSSH generate faster relaxation at thehigher temperature. This is expected, because the NAelectron−phonon coupling is proportional to the nuclearvelocity R, eq 5, and the velocity grows with temperature.Since the clusters are composed of heavy atoms, the phononquanta are smaller or on the order of kT, justifying the classicaldescription of the nuclear motions.In comparison, nonradiative electron−hole recombination in

carbon nanotubes is driven by high frequency C−C stretchingmodes and exhibits much less temperature dependence.66 Thepurely classical treatment of nuclear motion used in FSSHoverestimates the temperature dependence, while the decoher-ence correction in DC-FSSH restores the expected behavior.49

Similarly, the nonradiative relaxation of the hydrated electronshows little isotope effect, because the quanta of both OH andOD stretching modes are significantly larger than kT atexperimental temperatures. The decoherence correctionimproves agreement between theory and experiment in thiscase as well.53

The electron−hole recombination is longer in Si5 and Si7compared to the other clusters, see the DC-FSSH data in Table1. This is true for both 100 and 300 K. The effect is rationalizedby the difference in the NA electron−phonon coupling, Figure5. The following subsection provides a more detailed analysis ofthe simulation results employing the Fermi’s golden rule rateexpression. Rate expressions assume exponential decay, f(t) =exp(−kt) ≈ 1 − kt, where k is the decay rate. The DC-FSSHdata, Figure 4, are exponential, as required, and give goodagreement with experiment.39

3.3. Fermi’s Golden Rule Analysis. Fermi’s goldenrule40,67

π ρ=ℏ

⟨ | ′| ⟩|k f H i E2

( )F band2

(9)

provides an accurate description of quantum transitions in theweak coupling limit, which applies in the present case. The NAcoupling, Figure 5, is 3 orders of magnitude smaller than theHOMO−LUMO energy gaps. In eq 9, ρ(Eband) is the density ofstates, and ⟨f |H′|i⟩ is the matrix element of the perturbation H′

Figure 2. HOMO−LUMO energy gaps and density differences.Yellow and blue represent positive and negative values, respectively.

Figure 3. Decay of the LUMO population at 100 and 300 K computed using FSSH without decoherence. Note the difference in the y-axis scales inthe two panels.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−2070920705

between the initial and final states. The electron−holerecombination occurs by an electronic transition across thefundamental bandgap of the nanoclusters, and therefore, thedensity of states can be regarded as ρ(Eband) ∼ 1/Eg. Fermi’sgolden rule applies to cases with a dense manifold of finalstates. In present, there is only one final electronic state. Thefinal state manifold arises because the electronic energy isdeposited into multiple vibrational states. An estimate of thenumber of final states can be obtained by comparing the 2 eVelectronic energy to the 200−400 cm−1 (0.025−0.05 eV)vibrational quanta. The combinatorial number of ways theelectronic energy can be distributed between 20 and 40vibrational quanta justifies application of Fermi’s golden rule.The coupling in the golden rule is the NA coupling, eq 5.

Thus, Fermi’s golden rule can be represented as

∼ | |k

NAEF

g

2

(10)

or

+ ∼ | |k E NAln ln 2 lnF g

The bandgaps averaged at 100 and 300 K can be found in ourprevious work,54 while the average NA coupling are shown inFigure 5. We report the root-mean-square averages, because theNA coupling is defined up to a sign. Figure 6 plots the log−logversion of eq 10. Both FSSH and DC-FSSH at 100 and 300 Kgive slopes of approximately 2, in agreement with Fermi’sgolden rule.

The averaged NA couplings are under 1 meV in all clusters atboth temperatures, Figure 5, rationalizing the slow nonradiativeelectron−hole recombination. One can use the Hellmann−Feynman theorem and express the NA coupling as,

= − ℏ⟨ |∇ | ⟩ NA i

f H iE

Rg

R

(11)

highlighting the fact that it is inversely proportional to thebandgap. Therefore, the NA coupling is smaller for relaxationacross large gaps, in particular for the electron−holerecombination, and larger for intraband relaxation, whichinvolves multiple transitions between nearly isoenergetic states.The smaller NA coupling in Si5 and Si7 compared to the otherSi clusters can be attributed to more stable structuresundergoing less thermal fluctuations during the MD trajectory.In addition, the HOMO−LUMO overlap is small for theseclusters, as evidenced by the simpler nodal structure of theHOMO−LUMO density differences in Si5 and Si7 compared tothe other clusters. Smaller wave function overlap translates into

Figure 4. Decay of the LUMO population at 100 and 300 K computed with DC-FSSH.

Table 1. Electron−Hole Recombination Times (ns) in the SiClusters

Si5 Si6 Si7 Si8 Si9 Si10 Si15

FSSH,100 K

2.7 0.30 5.1 1.0 1.2 0.25 0.40

FSSH,300 K

0.93 0.15 0.89 0.07 0.10 0.14 0.09

DC-FSSH,100 K

113 3.5 54 8.9 13 1.3 1.9

DC-FSSH,300 K

14 1.5 18 1.5 1.4 1.5 1.5

Figure 5. Average root-mean-square values of the NA couplingbetween HOMO and LUMO for the Si clusters as 100 and 300 K. TheNA electron−phonon coupling is larger at the higher temperature dueto enhanced nuclear motions. Si5 and Si7 show smaller NA couplingcompared to the other Si clusters.

Figure 6. Exponential fitting of the FSSH and DC-FSSH simulationresults to Fermi’s golden rule at 100 and 300 K, eq 10. The sloperepresents the power at the NA coupling.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−2070920706

weaker NA coupling. Because the NA coupling is proportionalto the nuclear velocity, it increases with temperature, Figure 5.Considering the temperature dependence of the electron−

hole recombination rate, we follow ref.68 and express Fermi’sgolden rule as

∼ | | ∼ ℏ⟨ |∇ | ⟩

| | ∼ ℏ⟨ |∇ | ⟩

k NAf H i

Ef H i

ETRF

g g

R R2

2

2

2

(12)

This representation explicitly isolates the temperature depend-ence stemming from the nuclear kinetic energy term, |R|. If thetemperature dependence of kF had arisen only from this term,then one would have kF ∼ T. In general, other terms alsoexhibit temperature dependence, including the (⟨f |∇RH|i⟩/Eg)part of the NA coupling and the decoherence correction, whichenters the nuclear wave function overlap part of the fullyquantum version of ⟨f |H′|i⟩, eq 9. As a result, the temperaturedependence of the electron−hole relaxation rate is quitecomplex, Table 2. The negative temperature dependence seen

in the DC-FSSH simulation of Si10 appears because thecoherence time decreases with increasing temperature, andshorter coherence gives longer relaxation, in line with thequantum Zeno effect.58,65 Temperature dependence of thedecoherence time is stronger in larger clusters, as more phononmodes are activated in large systems.54 Thus, it can be expectedat higher temperatures, that decoherence will slow down thetransition in larger clusters more efficiently.

4. CONCLUSIONSWe simulated the nonradiative electron−hole recombination inSi clusters using NAMD in the framework of time-domainDFT. The results demonstrate that the decoherence correctionto FSSH is essential to obtain good agreement with experiment.It increases the transition time by about an order of magnitude.The transition time obtained by DC-FSSH at room temper-ature corresponds well with the available experimental data.Compared to bulk Si and SiO2, the excited state lifetime in

small clusters is shorter by 6 orders of magnitude. The lifetimedecreases from milliseconds to nanoseconds, as a result ofstronger electron−phonon coupling and transformation of Sifrom indirect to direct bandgap semiconductor. Direct bandgaparises in nanoscale Si due to quantum confinement and surfaceeffects, and is favorable for optoelectronics applications, asdemonstrated with porous Si.The analysis of the simulation data using Fermi’s golden rule

rationalizes the dependence of the electron−hole recombina-tion rate on the NA coupling magnitude and temperature. BothFSSH and DC-FSSH give the square dependence on the NAcoupling, agreeing with Fermi’s golden rule. FSSH and DC-FSSH predict somewhat different temperature dependence ofthe transition rate. In large clusters, the temperature depend-ence obtained by DC-FSSH is weaker than that obtained byFSSH, because the decoherence correction counteracts thethermal increase in the nuclear velocity and NA coupling. The

differences are more significant in larger clusters, in which morephonon modes contribute to both elastic and inelasticelectron−phonon scattering. Structural stability to thermalfluctuations favors longer electron−hole recombination time.

■ AUTHOR INFORMATION

Corresponding Author*(O.V.P) E-mail: [email protected].

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSFinancial support of the U.S. Department of Energy, Grant DE-SC0006527, is gratefully acknowledged.

■ REFERENCES(1) Kamat, P. V. Quantum Dot Solar Cells. SemiconductorNanocrystals as Light Harvesters. J. Phys. Chem. C 2008, 112,18737−18753.(2) Pi, X.; Zhang, L.; Yang, D. Enhancing the Efficiency ofMulticrystalline Silicon Solar Cells by the Inkjet Printing of Silicon-Quantum-Dot Ink. J. Phys. Chem. C 2012, 116, 21240−21243.(3) Long, R.; English, N. J.; Prezhdo, O. V. Photo-Induced ChargeSeparation across the Graphene−Tio2 Interface Is Faster Than EnergyLosses: A Time-Domain Ab Initio Analysis. J. Am. Chem. Soc. 2012,134, 14238−14248.(4) Coe, S.; Woo, W.-K.; Bawendi, M.; Bulovi, V. Electro-luminescence from Single Monolayers of Nanocrystals in MolecularOrganic Devices. Nature 2002, 420, 800−803.(5) Borner, R.; Kowerko, D.; Krause, S.; von Borczyskowski, C.;Hubner, C. G. Efficient Simultaneous Fluorescence Orientation,Spectrum, and Lifetime Detection for Single Molecule Dynamics. J.Chem. Phys. 2012, 137, 164202.(6) Wu, C.-C.; Jariwala, D.; Sangwan, V. K.; Marks, T. J.; Hersam, M.C.; Lauhon, L. J. Elucidating the Photoresponse of Ultrathin Mos2Field-Effect Transistors by Scanning Photocurrent Microscopy. J. Phys.Chem. Lett. 2013, 4, 2508−2513.(7) Dahan, M.; Levi, S.; Luccardini, C.; Rostaing, P.; Riveau, B.;Triller, A. Diffusion Dynamics of Glycine Receptors Revealed bySingle-Quantum Dot Tracking. Science 2003, 302, 442−445.(8) Kobayashi, Y.; Pan, L.; Tamai, N. Effects of Size and CappingReagents on Biexciton Auger Recombination Dynamics of CdteQuantum Dots†. J. Phys. Chem. C 2009, 113, 11783−11789.(9) Schaller, R. D.; Petruska, M. A.; Klimov, V. I. Effect of ElectronicStructure on Carrier Multiplication Efficiency: Comparative Study ofPbse and Cdse Nanocrystals. Appl. Phys. Lett. 2005, 87, 253102−253104.(10) Midgett, A. G.; Hillhouse, H. W.; Hughes, B. K.; Nozik, A. J.;Beard, M. C. Flowing Versus Static Conditions for Measuring MultipleExciton Generation in Pbse Quantum Dots. J. Phys. Chem. C 2010,114, 17486−17500.(11) Tisdale, W. A.; Williams, K. J.; Timp, B. A.; Norris, D. J.; Aydil,E. S.; Zhu, X.-Y. Hot-Electron Transfer from SemiconductorNanocrystals. Science 2010, 328, 1543−1547.(12) Sambur, J. B.; Novet, T.; Parkinson, B. A. Multiple ExcitonCollection in a Sensitized Photovoltaic System. Science 2010, 330, 63−66.(13) Schaller, R. D.; Pietryga, J. M.; Klimov, V. I. CarrierMultiplication in Inas Nanocrystal Quantum Dots with an OnsetDefined by the Energy Conservation Limit. Nano Lett. 2007, 7, 3469−3476.(14) Cooney, R. R.; Sewall, S. L.; Anderson, K. E. H.; Dias, E. A.;Kambhampati, P. Breaking the Phonon Bottleneck for Holes inSemiconductor Quantum Dots. Phys. Rev. Lett. 2007, 98, 177403.(15) Kilina, S. V.; Kilin, D. S.; Prezhdo, O. V. Breaking the PhononBottleneck in Pbse and Cdse Quantum Dots: Time-Domain Density

Table 2. Scaling Constant β in kF ∼ Tβ; β = 1 in Fermi’sGolden Rule

Si5 Si6 Si7 Si8 Si9 Si10 Si15

FSSH 0.97 0.63 1.59 2.41 2.31 0.53 1.37DC-FSSH 1.89 0.77 1.01 1.62 2.03 −0.13 0.22

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−2070920707

Functional Theory of Charge Carrier Relaxation. ACS Nano 2008, 3,93−99.(16) Schaller, R. D.; Pietryga, J. M.; Goupalov, S. V.; Petruska, M. A.;Ivanov, S. A.; Klimov, V. I. Breaking the Phonon Bottleneck inSemiconductor Nanocrystals Via Multiphonon Emission Induced byIntrinsic Nonadiabatic Interactions. Phys. Rev. Lett. 2005, 95, 196401.(17) Prezhdo, O. V. Multiple Excitons and the Electron−PhononBottleneck in Semiconductor Quantum Dots: An Ab InitioPerspective. Chem. Phys. Lett. 2008, 460, 1−9.(18) Kilina, S. V.; Craig, C. F.; Kilin, D. S.; Prezhdo, O. V. Ab InitioTime-Domain Study of Phonon-Assisted Relaxation of ChargeCarriers in a Pbse Quantum Dot. J. Phys. Chem. C 2007, 111,4871−4878.(19) Habenicht, B. F.; Kamisaka, H.; Yamashita, K.; Prezhdo, O. V.Ab Initio Study of Vibrational Dephasing of Electronic Excitations inSemiconducting Carbon Nanotubes. Nano Lett. 2007, 7, 3260−3265.(20) Habenicht, B. F.; Kalugin, O. N.; Prezhdo, O. V. Ab Initio Studyof Phonon-Induced Denhasina of Electronic Excitations in NarrowGraphene Nanoribbons. Nano Lett. 2008, 8, 2510−2516.(21) Kilina, S.; Ivanov, S.; Tretiak, S. Effect of Surface Ligands onOptical and Electronic Spectra of Semiconductor Nanoclusters. J. Am.Chem. Soc. 2009, 131, 7717−7726.(22) Kilina, S.; Velizhanin, K. A.; Ivanov, S.; Prezhdo, O. V.; Tretiak,S. Surface Ligands Increase Photoexcitation Relaxation Rates in CdseQuantum Dots. ACS Nano 2012, 6, 6515−6524.(23) Guyot-Sionnest, P.; Wehrenberg, B.; Yu, D. IntrabandRelaxation in Cdse Nanocrystals and the Strong Influence of theSurface Ligands. J. Chem. Phys. 2005, 123, 074709.(24) Zhu, H. M.; Yang, Y.; Hyeon-Deuk, K.; Califano, M.; Song, N.H.; Wang, Y. W.; Zhang, W. Q.; Prezhdo, O. V.; Lian, T. Q. Auger-Assisted Electron Transfer from Photoexcited SemiconductorQuantum Dots. Nano Lett. 2014, 14, 1263−1269.(25) Efros, A. L.; Kharchenko, V. A.; Rosen, M. Breaking the PhononBottleneck in Nanometer Quantum Dots - Role of Auger-LikeProcesses. Solid State Commun. 1995, 93, 281−284.(26) Madrid, A. B.; Hyeon-Deuk, K.; Habenicht, B. F.; Prezhdo, O.V. Phonon-Induced Dephasing of Excitons in SemiconductorQuantum Dots: Multiple Exciton Generation, Fission, and Lumines-cence. ACS Nano 2009, 3, 2487−2494.(27) Canham, L. T. Silicon Quantum Wire Array Fabrication byElectrochemical and Chemical Dissolution of Wafers. Appl. Phys. Lett.1990, 57, 1046−1048.(28) Lin, L.; Li, Z.; Feng, J.; Zhang, Z. Indirect to Direct Band GapTransition in Ultra-Thin Silicon Films. Phys. Chem. Chem. Phys. 2013,15, 6063−6067.(29) Delerue, C.; Allan, G.; Lannoo, M. Optical Band Gap of SiNanoclusters. J. Lumin. 1998, 80, 65−73.(30) Mason, M. D.; Credo, G. M.; Weston, K. D.; Buratto, S. K.Luminescence of Individual Porous Si Chromophores. Phys. Rev. Lett.1998, 80, 5405−5408.(31) Pavesi, L.; Dal Negro, L.; Mazzoleni, C.; Franzo, G.; Priolo, F.Optical Gain in Silicon Nanocrystals. Nature 2000, 408, 440−444.(32) Franceschetti, A.; Pantelides, S. T. Excited-State Relaxations andFranck-Condon Shift in Si Quantum Dots. Phys. Rev. B 2003, 68,033313.(33) Cuevas, A.; Macdonald, D. Measuring and Interpreting theLifetime of Silicon Wafers. Sol. Energy 2004, 76, 255−262.(34) Valenta, J.; Juhasz, R.; Linnros, J. Photoluminescence Spectros-copy of Single Silicon Quantum Dots. Appl. Phys. Lett. 2002, 80,1070−1072.(35) Linnros, J.; Lalic, N.; Galeckas, A.; Grivickas, V. Analysis of theStretched Exponential Photoluminescence Decay from Nanometer-Sized Silicon Crystals in Sio2. J. Appl. Phys. 1999, 86, 6128−6134.(36) Beard, M. C.; Knutsen, K. P.; Yu, P.; Luther, J. M.; Song, Q.;Metzger, W. K.; Ellingson, R. J.; Nozik, A. J. Multiple ExcitonGeneration in Colloidal Silicon Nanocrystals. Nano Lett. 2007, 7,2506−2512.

(37) Credo, G. M.; Mason, M. D.; Buratto, S. K. External QuantumEfficiency of Single Porous Silicon Nanoparticles. Appl. Phys. Lett.1999, 74, 1978−1980.(38) Weissker, H. C.; Furthmuller, J.; Bechstedt, F. StructuralRelaxation in Si and Ge Nanocrystallites: Influence on the Electronicand Optical Properties. Phys. Rev. B 2003, 67, 245304.(39) Fu, G.; Wang, X.; Feng, H.; Yu, X.; Dai, W.; Lu, W.; Yu, W. TheEvolution of Photoluminescence from Si Cluster to Nanocrystal in Si-Rich Oxide/Sio2Multilayer Films. J. Alloys Compd. 2013, 579, 284−289.(40) Habenicht, B. F.; Prezhdo, O. V. Time-Domain Ab Initio Studyof Nonradiative Decay in a Narrow Graphene Ribbon. J. Phys. Chem. C2009, 113, 14067−14070.(41) Akimov, A. V.; Neukirch, A. J.; Prezhdo, O. V. TheoreticalInsights into Photoinduced Charge Transfer and Catalysis at OxideInterfaces. Chem. Rev. 2013, 113, 4496−4565.(42) Phillips, J. C. Stability, Kinetics, and Magic Numbers of Si+N(N=7−45) Clusters. J. Chem. Phys. 1985, 83, 3330−3333.(43) Zhang, Q.-L.; Liu, Y.; Curl, R. F.; Tittel, F. K.; Smalley, R. E.Photodissociation of Semiconductor Positive Cluster Ions. J. Chem.Phys. 1988, 88, 1670−1677.(44) Yoo, S. H.; Zeng, X. C. Motif Transition in Growth Patterns ofSmall to Medium-Sized Silicon Clusters. Angew. Chem. Int. Edit 2005,44, 1491−1494.(45) Tully, J. C. Molecular-Dynamics with Electronic-Transitions. J.Chem. Phys. 1990, 93, 1061−1071.(46) Parandekar, P. V.; Tully, J. C. Mixed Quantum-ClassicalEquilibrium. J. Chem. Phys. 2005, 122, 094102.(47) Akimov, A. V.; Prezhdo, O. V. The Pyxaid Program for Non-Adiabatic Molecular Dynamics in Condensed Matter Systems. J. Chem.Theory Comput. 2013, 9, 4959−4972.(48) Prezhdo, O. V.; Duncan, W. R.; Prezhdo, V. V. PhotoinducedElectron Dynamics at the Chromophore-Semiconductor Interface: ATime-Domain Ab Initio Perspective. Prog. Surf. Sci. 2009, 84, 30−68.(49) Habenicht, B. F.; Prezhdo, O. V. Nonradiative Quenching ofFluorescence in a Semiconducting Carbon Nanotube: A Time-DomainAb initio Study. Phys. Rev. Lett. 2008, 100, 197402.(50) Craig, C. F.; Duncan, W. R.; Prezhdo, O. V. Trajectory SurfaceHopping in the Time-Dependent Kohn-Sham Approach for Electron-Nuclear Dynamics. Phys. Rev. Lett. 2005, 95, 163001.(51) Tully, J. C.; Preston, R. K. Trajectory Surface HoppingApproach to Nonadiabatic Molecular Collisions: The Reaction of H+

with D2. J. Chem. Phys. 1971, 55, 562−572.(52) Bittner, E. R.; Rossky, P. J. Quantum Decoherence in MixedQuantum-Classical Systems: Nonadiabatic Processes. J. Chem. Phys.1995, 103, 8130−8143.(53) Prezhdo, O. V.; Rossky, P. J. Evaluation of Quantum TransitionRates from Quantum-Classical Molecular Dynamics Simulations. J.Chem. Phys. 1997, 107, 5863−5878.(54) Liu, J.; Neukirch, A. J.; Prezhdo, O. V. Phonon-Induced Pure-Dephasing of Luminescence, Multiple Exciton Generation, and Fissionin Silicon Clusters. J. Chem. Phys. 2013, 139, 164303.(55) Schwartz, B. J.; Bittner, E. R.; Prezhdo, O. V.; Rossky, P. J.Quantum Decoherence and the Isotope Effect in Condensed PhaseNonadiabatic Molecular Dynamics Simulations. J. Chem. Phys. 1996,104, 5942−5955.(56) Prezhdo, O. V. Mean Field Approximation for the StochasticSchrodinger Equation. J. Chem. Phys. 1999, 111, 8366−8377.(57) Jaeger, H. M.; Fischer, S.; Prezhdo, O. V. Decoherence-InducedSurface Hopping. J. Chem. Phys. 2012, 137, 22A545.(58) Kilina, S. V.; Neukirch, A. J.; Habenicht, B. F.; Kilin, D. S.;Prezhdo, O. V. Quantum Zeno Effect Rationalizes the PhononBottleneck in Semiconductor Quantum Dots. Phys. Rev. Lett. 2013,110, 180404.(59) Nelson, T. R.; Prezhdo, O. V. Extremely Long NonradiativeRelaxation of Photoexcited Graphane Is Greatly Accelerated byOxidation: Time-Domain Ab Initio Study. J. Am. Chem. Soc. 2013, 135,3702−3710.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−2070920708

(60) Kresse, G.; Furthmuller, J. Efficient Iterative Schemes for AbInitio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys.Rev. B 1996, 54, 11169−11186.(61) Blochl, P. E. Projector Augmented-Wave Method. Phys. Rev. B1994, 50, 17953−17979.(62) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized GradientApproximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868.(63) Fielicke, A.; Lyon, J. T.; Haertelt, M.; Meijer, G.; Claes, P.;Haeck, J. d.; Lievens, P. Vibrational Spectroscopy of Neutral SiliconClusters Via Far-Ir-Vuv Two Color Ionization. J. Chem. Phys. 2009,131, 171105.(64) Haertelt, M.; Lyon, J. T.; Claes, P.; de Haeck, J.; Lievens, P.;Fielicke, A. Gas-Phase Structures of Neutral Silicon Clusters. J. Chem.Phys. 2012, 136, 064301−064306.(65) Prezhdo, O. V. Quantum Anti-Zeno Acceleration of a ChemicalReaction. Phys. Rev. Lett. 2000, 85, 4413−4417.(66) Berger, S.; Voisin, C.; Cassabois, G.; Delalande, C.; Roussignol,P.; Marie, X. Temperature Dependence of Exciton Recombination inSemiconducting Single-Wall Carbon Nanotubes. Nano Lett. 2007, 7,398−402.(67) Hyeon-Deuk, K.; Madrid, A. B.; Prezhdo, O. V. Symmetric BandStructures and Asymmetric Ultrafast Electron and Hole Relaxations inSilicon and Germanium Quantum Dots: Time-Domain Ab InitioSimulation. Dalton Trans. 2009, 45, 10069−10077.(68) Bao, H.; Habenicht, B. F.; Prezhdo, O. V.; Ruan, X. L.Temperature Dependence of Hot-Carrier Relaxation in Pbse Nano-crystals: An Ab Initio Study. Phys. Rev. B 2009, 79, 235306.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp5067296 | J. Phys. Chem. C 2014, 118, 20702−2070920709