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Coordination Chemistry Reviews 263– 264 (2014) 161– 181

Contents lists available at ScienceDirect

Coordination Chemistry Reviews

jo ur nal ho me page: www.elsev ier .com/ locate /ccr

eview

ime-domain ab initio modeling of excitation dynamics inuantum dots

manda J. Neukircha, Kim Hyeon-Deukb, Oleg V. Prezhdoa,c,∗

University of Rochester, Department of Physics and Astronomy, Bausch & Lomb Hall, Rochester, NY 14627-0171, United StatesDepartment of Chemistry, Kyoto University, Kyoto 606-8502, JapanUniversity of Rochester, Department of Chemistry, Hutchison Hall, Rochester, NY 14627-0171, United States

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622. Theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

2.1. Hartree–Fock method provides a mean-field picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622.2. Configuration interaction incorporates electron correlations beyond the Hartree–Fock approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1632.3. Density functional theory includes electron correlations indirectly via an effective independent particle description . . . . . . . . . . . . . . . . . . . 1632.4. Time-domain density functional theory describes electron response to a perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1632.5. Nonadiabatic molecular dynamics and fewest switches surface hopping provide nuclear feedback to electronic evolution . . . . . . . . . . . . . 1642.6. Optical response functions and pure-dephasing times characterize elastic electron–phonon evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1642.7. Time-domain density functional theory for Auger processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

3. Multiple exciton generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.1. The photoexcitation mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.1.1. Photoexcitation of multiple excitons in small clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663.1.2. Dopants, defects and charges create new types of excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

3.2. The impact ionization mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.2.1. Single and multiple exciton state densities determine impact ionization threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.2.2. Real-time dynamics of impact ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.2.3. Interplay between impact ionization and exciton recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4. Phonon-induced dephasing of electronic excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.1. Luminescence, multiple exciton generation, and multiple exciton fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.2. Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.3. Size dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.4. Dephasing in metallic particles, plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5. Electron–phonon relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1735.1. Relaxation in semiconductor nanocrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1735.2. Relaxation in metallic particles, plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.3. Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.4. Phonon bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.5. Surface defects introduce additional states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.6. Ligands contribute high-frequency phonon modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

r t i c l e i n f o

rticle history:eceived 12 May 2013ccepted 26 August 2013vailable online 8 September 2013

a b s t r a c t

The review discusses the results of ab initio time-dependent density functional theory and non-adiabaticmolecular dynamics simulations of photoinduced dynamics of charges, excitons, plasmons, and phononsin semiconductor and metallic quantum dots (QDs). The simulations create an explicit time-domainrepresentation of the excited-state processes, including elastic and inelastic electron–phonon scattering,

∗ Corresponding author at: University of Rochester, Department of Chemistry, RC Box 270216, Rochester, NY 14627-0216, United States. Tel.: +1 585 275 4231;ax: +1 585 276 0205.

E-mail address: oleg.prezhdo@rochester.edu (O.V. Prezhdo).

010-8545/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.ccr.2013.08.035

162 A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181

Keywords:Semiconductor and metal nanoparticlesTime-domain density functional theoryNonradiative relaxationNonadiabatic molecular dynamicsMultiple-exciton generationElectron–phonon interactions

multiple exciton generation, fission, and recombination. These nonequilibrium phenomena control theoptical and electronic properties of QDs. Our approach can account for QD size and shape, as well aschemical details of QD structure, such as dopants, defects, core/shell regions, surface ligands, and unsatu-rated bonds. Each of these variations significantly alters the properties of photoexcited QDs. The insightsreported in this review provide a comprehensive understanding of the excited-state dynamics in QDs andsuggest new ways of controlling the photo-induced processes. The design principles that follow, guidedevelopment of photovoltaic cells, electronic and spintronic devices, biological labels, and other systemsrooted in the unique physical and chemical properties of nanoscale materials.

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. Introduction

Quantum dots (QDs) are nanoscale clusters of bulk materialhose electronic excitations are confined in all three spatial dimen-

ions. Every material has its own exciton Bohr radius, which is thehysical size of an excitation in the bulk of that particular mate-ial. When the dimensions of a nanocrystal reach this characteristicength, it starts to exhibit the effects of quantum confinement muchike a particle in a box. This allows for the properties of the QD to beuned continuously by changing the cluster’s size and shape. Oncehe system reaches the subnanometer scale, it begins to behave like

molecule; its structure varies from bulk and its properties changeiscontinuously with cluster size.

The unique physical and chemical properties [1–4] of semi-onducting and metallic nanocrystals form the basis for a varietyf applications, ranging from optical sensors [5,6] and probes7,8], to photovoltaic [9,10], optical [11], electronic [12], opto-lectronic [13], and spintronic [14] devices, and to light-emitting15] and imaging [16] technologies. The excited state dynamicsn semiconducting [17] and metallic [18] QDs are very intri-ate [19]. For instance, high absorption cross sections, decreasedlectron–phonon relaxation rates [20,21], and generation ofultiple electron–hole pairs [22–24] make QDs outstanding pho-

ovoltaic materials since all these features provide opportunitieso utilize photon energy in excess of the band-gap [20,25].lectron–hole and charge–phonon interactions have both funda-ental and practical importance, since both interactions contribute

o the overall efficiency of a photovoltaic device. Phonon-inducedephasing of spin and electron states influences charge and energyransfer processes. Inelastic scattering is responsible for energy lossuring charge tunneling though QDs.

Relaxation and pure-dephasing are two related but distincthenomena that result from electron–phonon interactions. Relax-tion is an inelastic process that leads toward energy losses.lectron–phonon pure-dephasing is an elastic process, and there-ore conserves electronic energy. Pure-dephasing determines theomogenous linewidths in optical spectra. It transforms a coherentuperposition of states into uncorrelated state ensembles. Whenonsidering MEG, electron–phonon relaxation competes with andnhibits it. At the same time, destruction of superpositions ofEs and MEs occur due to elastic electron–phonon scattering.honon-induced pure-dephasing of coherent superpositions of sin-le and ME states destroys the quantum mechanical entanglementetween the states and is an essential component of excited stateynamics in QDs.

Nanocrystals are sufficiently large objects that can support mul-iple excitations. Therefore, Auger-type processes, which creatend annihilate charges and excitons, play a particularly importantole. For instance, multiple exciton generation (MEG), also knowns carrier multiplication (CM), is the process in which a single high-nergy photon is absorbed and creates two or more electron–holeairs [23,26–28], This phenomenon provides potential for increas-

ng photovoltaic device efficiencies [24]. MEG happens in bulk,ut is typically more efficient in QDs [24,29]. This is because

n bulk materials crystal momentum, a pseudo-momentum

© 2013 Elsevier B.V. All rights reserved.

associated with electrons in a lattice, needs to be conserved. Thisrequirement enforces strict selection rules and causes the energyneeded to form a biexciton to be more than simply two times theband-gap. A lack of translational symmetry in QDs voids the needto conserve crystal momentum [30]. In addition, Coulomb inter-action between electrons and holes is enhanced due to the closerproximity of charge carriers inside a nanocrystal [31].

The current review presents a cutting edge ab initio descrip-tion of the time-dependent dynamics of photoexcited states insemiconductor and metallic QDs. The ab initio methods are usedto study excited state composition, evolution and relaxation. Thetime-dependent atomistic nature of the methods provides power-ful tools for studying the role of surface ligands, dopants, defects,unsaturated bonds, size, shape and other realistic aspects of QDs.The described simulations provide a comprehensive perspective onthe elastic and inelastic scattering dynamics of photoexcited chargecarriers in nanoscale materials, directly mimicking numerous time-resolved experiments, and offering insights into the fundamentalmechanisms underlying QD applications in optics, photovoltaics,electronics and related fields.

2. Theoretical approaches

We use a wide array of ab initio methods to study the photoex-cited dynamics of QDs. The simulations described in this reviewtreat many-body interactions in the electronic degrees of freedomfor either fixed nuclear coordinates, or nuclei evolving classi-cally or semiclassically. We refer readers interested in a detaileddescription of the methods and their numerical implementationto Prezhdo et al. [32] and Akimov et al. [33]. Symmetry adaptedcluster (SAC) theory with configuration interactions (CI) is used tostudy the nature of photoexcited states and to describe photoin-duced MEG for fixed nuclei. Exciton dynamics, including impactionization (II) and electron–phonon relaxation, are modeled usingtime-dependent density functional theory (TDDFT). Here, exci-tons are coupled to phonon motions with nonadiabatic moleculardynamics (NAMD) methods. The optical response formalism is usedto obtain information on electron–phonon dephasing, which has aninfluence in the exciton formation as well as relaxation.

2.1. Hartree–Fock method provides a mean-field picture

Even the smallest QDs consist of hundreds of atoms leadingto systems with tens of thousands of electrons. For example, aPb68Se68 QD with diameter of only 2 nm already has 8000 electrons.Solving for the exact many-body wavefunction for this systemwould be impossible. Fortunately, physically motivated approxi-mations have been shown to bring tractability to the many-bodyproblem. One of the simplest and earliest methods developed isthe Hartree–Fock (HF) approach. The HF theory was introducedto solve the electronic Schrödinger equation for a given set ofnuclear coordinates. The basic assumption behind the HF method

is that the many-electron wavefunction can be written as a prop-erly symmetrized product of one-electron orbitals. An individualelectron in the system feels an electrostatic field from the central

A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181 163

Fig. 1. Schematic of Hartree–Fock (HF), Hartree–Fock with configuration interaction (CI), and density functional theory (DFT). In HF each individual electron interacts witht weeni

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he mean-field formed by the remaining electrons. CI adds explicit correlations betn order incorporate electron correlations in a single particle description.

dapted from Accounts of Research 2009, 42, 2005. Copyright 2009 American Chem

otential of the nuclei and other electrons, Fig. 1. HF is a self-onsistent theory in that the electron orbitals are determined byhe mean-field, which depends on the orbitals. HF includes theauli exclusion principle by not allowing two electrons to occupyhe same location through the exchange interaction; however, nolectron correlation is incorporated.

.2. Configuration interaction incorporates electron correlationseyond the Hartree–Fock approximation

The non-interacting single-particle picture is provided by the HFpproximation, which excludes electron-correlation effects. Char-cterizing the excited states in the semiconductor QDs as singlexcitons (SEs) or multiple excitons (MEs) requires a proper ref-rence single-particle description and a rigorous account of thelectron–hole Coulomb interaction. This correlation can be system-tically added to HF using symmetry adapted cluster (SAC) theoryith configuration interactions (CI) [34,35]. Including explicit elec-

ron correlations allows for the exact many body wave functions toe obtained, although at such a large computational expense aso be feasible only for small clusters. Configurations of multi-bodyorrelations represent MEs. SAC-CI defines excited states as a super-osition of all potential excitations from the HF ground state. Thextent to which high-order configurations enter the wavefunctionllows for excited states to be quantified as SEs or MEs, because ofts ability to explicitly treat multiply excited electrons and holes35–37]. For example, the SAC-CI used below explicitly includeslectron correlation through cluster expansion of the ground stateavefunction.

�SAC 〉 = exp

(M∑l=1

CISI

)|˚0〉

=(

1 +∑I

CI SI +12

∑I,J

CICJ SI SJ + · · ·)

|˚0〉. (1)

Here, |˚0〉 represents the closed-shell HF wavefunction, SI areymmetry adapted excitation operators, and CI are configura-ion coefficients. The excited state wavefunctions |� SAC−CI〉 are

electrons. DFT builds an effective potential that differs from the Coulomb law, 1/r,

ociety.

calculated from the electron-correlated ground state wavefunction|� SAC〉

|�SAC−CI〉 =N∑K=1

aKRK |�SAC 〉, (2)

where RK represents an excitation operator, and aK is the SAC-CI coefficient. A coefficient is associated with each configurationrepresenting that configuration’s contribution to the electronicstructure of the excited state. This allows the excited states to beclassified as either SE or ME.

2.3. Density functional theory includes electron correlationsindirectly via an effective independent particle description

In contrast to SAC-CI, DFT accounts for electron correlationsindirectly. The Hohenberg–Kohn theorem proves that the electrondensity, which only depends on three spatial coordinates, uniquelydetermines the ground state properties of a many body system.Electron correlation arising due to the Coulomb interaction, as wellas electron exchange reflecting the Pauli exclusion principle, arethen described through use of functionals of the electron density.Kohn–Sham (KS) DFT ensures that the system of non-interactingparticles in an effective potential can generate the same densityas the original system of interacting particles. KS DFT is thereforea single-particle representation that accounts for electron cor-relations, incorporating electron–hole interactions and excitoniceffects. HF can be thought of as a form of DFT that includes exactPauli exchange but excludes electron correlation. KS DFT is compu-tationally inexpensive and works particularly well with extendedstructures that do not experience complex chemical changes.

2.4. Time-domain density functional theory describes electronresponse to a perturbation

The Runge–Gross theorem shows that the three-dimensionaldensity of a many-body quantum system is sufficient to describe

the time-dependent response of the system to an external pertur-bation, including electro-magnetic field and vibrational motions,R(t) [38]. The resulting theory is known as time-dependent densityfunctional theory (TDDFT). Linear response TDDFT is regularly used

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64 A.J. Neukirch et al. / Coordination Ch

o evaluate the energies of electronic excitations. The full, real-timeDDFT employed in the studies discussed in this review explicitlyropagates the electron density in time. Solutions obtained fromime-independent DFT are used as the starting point for our TDDFTalculations [39].

The electron density in TDDFT is written in the KS representations [40]

(r, t) =Ne∑l=1

|ϕl(r, t)|2, (3)

here Ne is the number of electrons, and ϕl (r,t) are single-electronS orbitals. The evolution of ϕl (r,t) is determined by applying the

ime-dependent variational principle to the KS energy [40]

{ϕl} =Ne∑l=1

〈ϕl|K(r)|ϕl〉 +Ne∑l=1

〈ϕl|V(r; R)|ϕl〉

+ e2

2

∫ ∫�(r′)�(r)|r − r′| d

3rd3r′ + Exc{�}, (4)

hich contains the kinetic energy of noninteracting electrons K(r),he electron–nuclear attraction V(r;R) that depends on the phononoordinates R, the Coulomb repulsion of density �(r,t), and thexchange-correlation energy functional Exc that takes into accounthe many-body interactions. The result is a system of coupled equa-ions of motion for single particle KS orbitals [40–42]

�∂ϕl(r, t)∂t

= H(r; R){ϕ(r, t)}ϕl(r, t), p = 1, . . ., Ne. (5)

These time-dependent Kohn–Sham (TDKS) equations are cou-led, since the Hamiltonian H depends on the overall density, and,herefore, all occupied KS orbitals. The TDKS equation is solvedy expanding the TDKS orbitals ϕl (r,t) in the adiabatic KS orbital

˜ j(r; R) basis obtained from the ground state DFT calculation,

l(r, t) =Ne∑k

Clj(t)| ϕj(r, R)〉. (6)

Then, Eq. (5) transforms into a differential equation for thexpansion coefficients,

�∂clj(t)∂t

=Ne∑k=1

Clk(t)( ∈ kıjk − i�djk · R). (7)

The NA coupling describes the electron–phonon interaction,

jk · R = 〈 ϕj(r; R)|∇R | ϕk(r, R)〉 · R =⟨ϕj(r, R)

∣∣∣∣ ∂∂t∣∣∣∣ ϕk(r, R)

⟩(8)

hat arise from the dependence of the adiabatic KS orbitals on thehonon coordinates R(t) [43]. Nuclear trajectories for the groundlectronic state are used to sample initial conditions to creatensemble averages for the excited state dynamics.

.5. Nonadiabatic molecular dynamics and fewest switchesurface hopping provide nuclear feedback to electronic evolution

Non-adiabatic molecular dynamics (NAMD) provides a general-zation of ordinary MD to include transitions between electronictates. The nuclear motions drive the electronic evolution, asescribed, for instance, by TDDFT delineated in the previous sec-

ion. The influence of the electronic evolution on the classicalynamics of nuclei constitutes the quantum backreaction prob-

em. The latter can be solved in a mean-field manner, leading tohe Ehrenfest approximation. Correlations between the nuclear

y Reviews 263– 264 (2014) 161– 181

motions and the electronic states are built in using surface hoppingtechniques.

Fewest switches surface hopping (FSSH) minimizes the numberof surface hops by prescribing a probability for hopping betweenelectronic states that is based on changes in the electronic statepopulations, rather than populations themselves. The probabilityis explicitly time dependent and is correlated with the nuclear evo-lution. The probability of hopping between states j and k within thetime interval �t is equal to [44]

gjk(t, �t) = max

(0,bjk�t

amm(t)

), (9)

where

bjk = −2Re(a∗jkdkjk · R); ajk = cjc

∗k. (10)

The coefficients cj and ck evolve according to Eq. (7), djk · R isthe NA coupling in Eq. (8). If the calculated gjk is negative, the hop-ping probability is set to zero. This feature ensures that a hop fromstate j to state k only occurs if the electronic occupation of state jdecreases and the occupation of state k increases, minimizing thenumber of hops. In order to conserve the total electron nuclearenergy proceeding a hop, the nuclear velocities are rescaled alongthe direction of the electronic component, djk, of the NA coupling[43,44]. If the kinetic energy available to the nuclei along the direc-tion of the NA coupling is inadequate to allow for an increase in theelectronic energy, the hop is rejected. The hop-rejection creates thedetailed balance between upward and downward transitions [45].

In the simplified implementation of FSSH discussed in thisreview, it is assumed that the energy exchanged between theelectronic and vibrational degrees of freedom during a hop is redis-tributed swiftly among all of the vibrational modes. Under thisassumption, the velocity rescaling and hop rejection are substitutedwith multiplying the probability (Eq. (9)) for transitions upward inenergy by the Boltzmann factor. This allows for the use of a groundstate nuclear trajectory to determine the time-dependent poten-tial that drives the electron dynamics. The simplification leads tosubstantial improvements in the computational efficiency of FSSH.The majority of the FSSH simulations discussed here are performedseparately for electrons and holes in the basis of single-particleadiabatic KS orbitals. The single-particle representation is appro-priate for studies on QDs, since their electronic structure is wellrepresented by the independent electron and hole picture.

It is important to note that by treating nuclei classically, theoriginal FSSH scheme excludes coherence loss that occurs in theelectronic subsystem by coupling to quantum vibrations. Decoher-ence can be neglected if it is slower than the electronic transition.However, in some instances when the decay is slow, decoherencemust be explicitly included in the quantum-classical simulations.Decoherence is implanted within TDDFT-FSSH using a simple semi-classical approach [46]. The expansion coefficients of the KS wavefunctions are allowed to evolve coherently up to the decoherencetime, at this point they are reset to 0 or 1 with the probabilitiesgiven by the squares of the coefficients. The decoherence times,or vibrationally-induced dephasing times, are computed using theoptical response function formalism.

2.6. Optical response functions and pure-dephasing timescharacterize elastic electron–phonon evolution

In addition to facilitating electron relaxation, phonons alsocause electronic states to dephase from each other. Compared

to electron–phonon energy relaxation, electron–phonon dephas-ing is a more subtle effect. Dephasing is an elastic process thatconserves electronic energy. It results in a loss of the phase rela-tionship in a quantum-mechanical superposition between a pair of

emistr

eabttt

C

wtuei

C

ams

t

I

pis

pse

D

wbe

l

pdn

D

tditdt

2

AgTboit

A.J. Neukirch et al. / Coordination Ch

lectronic states. The pure-dephasing time is related to fluctuationsnd uncertainties in the energy levels that arise from the couplingetween electrons and phonons in the system. The fluctuations inhe energy levels are best described in terms of correlation func-ions [47]. The unnormalized autocorrelation function (ACF) for aransition of energy E is defined as

u(t) = 〈�E(t)�E(0)〉, (11)

here �E = E− 〈 E 〉, and the angular brackets denote taking the sta-istical average over a canonical ensemble. The initial value of thennormalized ACF gives the average fluctuation in the transitionnergy, Cu(0) =〈 �E2(0) 〉. Dividing Cu(t) by Cu(0) gives the normal-zed ACF:

(t) = 〈�E(t)�E(0)〉〈�E2(0)〉 . (12)

ACFs characterize periodicity and memory of the energy fluctu-tions. A rapid decay of an ACF indicates short memory and occurs ifultiple anharmonic phonon modes couple to the electronic tran-

ition.The Fourier transform of the ACF is known as the influence spec-

rum, or spectral density:

(ω) =∣∣∣∣ 1√

2�

∫ ∞

−∞dte−iωtC(t)

∣∣∣∣2

(13)

It identifies which phonon mode frequencies efficiently cou-le to the electronic subsystem. The amplitude of the peaks in an

nfluence spectrum corresponds to the electron–phonon couplingtrength at that frequency.

The optical response functions describing the dephasingrocesses between a pair of entangled states in a coherent superpo-ition can be obtained directly or with the second order cumulantxpansion [47]. The direct expression is:

(t) = exp(iωt)

⟨exp

(− i�

∫ t

0

�E()d

)⟩, (14)

here ω is the thermally averaged transition energy 〈�E〉 dividedy �. Often, it is difficult to achieve convergence with the directxpression since it requires averaging of a complex-valued oscil-

atory function, exp(

− i�

∫ t0�E()d

), whose real and imaginary

arts change signs. The 2nd order cumulant approximation to theephasing function is found by taking double integral and expo-ential of the unnormalized ACF, Eq. (11).

(t) = exp(−g(t)), g(t) = 1�2

∫ t

0

d1

∫ 1

0

d2Cu(2). (15)

The cumulant expression involves averaging over a real-valuedransition energy and its ACF, and converges more easily that theirect expression shown in Eq. (14). Analysis of the above formula

ndicates that rapid dephasing is facilitated by large fluctuations ofhe transition energy (i.e. large Cu(0) =〈 �E2(0) 〉). The timescale ofecay of the dephasing functions determines the pure-dephasingime.

.7. Time-domain density functional theory for Auger processes

MEG and ME recombination (MER) constitute examples ofuger-type processes. We simulate Auger dynamics involving sin-le and double exciton states coupled to nuclear motions withDDFT formulated in the adiabatic KS basis [48–50]. The adia-

atic representation is different from the more traditional picturef Auger dynamics where the Coulomb interaction causes non-nteracting electrons and holes to scatter off of one another. Inhe adiabatic representation, all Coulomb interactions are included

y Reviews 263– 264 (2014) 161– 181 165

in the Hamiltonian. Specifically, the electron–hole interactionis included in the exchange-correlation functional of DFT. TheCoulomb terms appearing in the electronic Hamiltonian are ‘diag-onalized out’ during the calculations of the adiabatic states, andtransitions between different SE and DE states occur due to theNA coupling. The adiabatic basis is the only representation that isreadily available in ab initio electronic structure calculations. Theadiabatic picture complements the phenomenological description,in which SE and ME states are coupled through the Coulomb terms.Arguably, the adiabatic basis provides the most physically rele-vant representation. For instance, continuous wavelength radiationexcites a system between adiabatic states.

Our simulation method includes the ground, SE, and DE states,|g(r; R)〉, |l,mSE (r; R)〉, |l,m,n,pDE (r; R)〉, respectively, and was formu-lated using second quantization with the ground state as a reference[49]. SEs and DEs are obtained as

|l,mSE 〉 = a†lam|g〉, |l,m,n,pDE 〉 = a†

lama

†nap|g〉, (16)

where the electron creation and annihilation operators, and am,generate and destroy an electron in the lth and mth adiabaticKS orbitals, respectively. The time-evolving wave function is thenexpressed by

| (t)〉 = Cg(t)|g〉 +∑l,m

Cl,mSE (t)|l,mSE 〉 +∑l,m,n,p

Cl,m,n,pDE (t)|l,m,n,pDE 〉.

(17)

Analogous to Eq. (7), the expansion coefficients appearing in Eq.(17), evolve by the first-order differential equations

i�∂Cx(t)∂t

= Cg(t)dX;g · R − i�∑l′,m′

Cl′m′SE (t)dX;SE,l′,m′ · R

− ih∑

l′,m′,n′,p′Cl

′,m′,n′,p′DE dX;DE,l′,m′,n′,p′ · R (18)

where X and Y now correspond to either ground, SE, or double exci-ton (DE) state, EX is the state energy, and the NA couplings aredefined by

dx,y · R ≡ 〈X |∇R|Y 〉 · R =⟨x

∣∣∣∣ ∂∂t∣∣∣∣Y

⟩. (19)

The atomistic simulation involving Auger-like transitionsbetween SE and DE states was accomplished by directly solvingEq. (18) with time-dependent NA couplings and energies [49,50].In this method, the two-particle electronic basis consists of theground, SE, and DE states. The NA coupling allows states to transi-tion into each other. The energies appear in the diagonal parts of theHamiltonian, while the NA couplings are located in the correspond-ing off-diagonal components. The NA coupling terms only connectstates that differ by a single electron or hole, resulting in sparseHamiltonian. This enabled us to use sparse matrix techniques todevelop an efficient simulation code that removes all zero compo-nents from the Hamiltonian, and then solves Eq. (18) using only theremaining non-zero components.

3. Multiple exciton generation

MEG is an interesting phenomenon that occurs in a varietyof nanoscale materials and can result in increased solar cell effi-

ciencies. Predicted [20] several years before discovery [22], MEGfrom high energy photons avoids energy losses associated withelectron–phonon relaxation to lower energy levels. MEG has drawnclose attention due to its potential for substantial improvement

166 A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181

F pact

oaQttaa[priattonsT

3

ppstbtf

sUoti

ig. 2. Diagram of the three proposed mechanisms of MEG: ME photoexcitation, im

f photovoltaic device efficiencies [20–25,27,51–58]. The ab initionalysis of the electronically excited states in the semiconductorDs allows for the different proposed mechanisms for the MEG

o be critically evaluated [20,23,55,59]. Different proposals echohe rapidly fluctuating views on MEs and the phonon bottleneck,nd reflect the variety in the electronic structure of the materi-ls exhibiting MEG. The ab initio electronic structure calculations56] discussed below will show two distinct pathways toward MEG,hotoexcitation and Auger processes [60]. Both of these pathwaysequire creating coherent superpositions of SE and MEs. Phonon-nduced dephasing destroys coherences between electronic statesnd converts them into ensembles of uncorrelated states. No mat-er how MEs are created, they dissociate into uncorrelated excitonshat coexist in the same QD and luminesce independently. This lossf correlation with in the coherent superposition of MEs has beenamed ME fission (MEF), in analogy to singlet fission in molecularystems [61], and can be explained by phonon-induced dephasing.he different processes important to MEG are shown in Fig. 2.

.1. The photoexcitation mechanism

Schaller et al. [22,25,54–56] suggested that a single absorbedhoton creates a bi-exciton instantaneously by a second ordererturbative process involving bi-exciton coupling to virtual SEtates. Photoexcitation of multi-electron states is forbidden inhe independent-particle description, but Coulomb interactionetween independent electrons and holes couples singly and mul-iply excited states, generating non-vanishing oscillator strengthor multi-electron excitations.

As explained in Section 2.2, CI describes each excited state as auperposition of all possible excitations from the HF ground state.

sing SAC-CI in the static, many-body picture, photoexcited statesf small clusters can be characterized as a superposition of elec-ronic configurations in a molecular orbital basis. The band-gaps taken to be difference between the ground state and the first

ionization, and phonon-induced dephasing of superpositions of SE and ME states.

excited state. Each electronic configuration is assigned CI coeffi-cient that denotes that configuration’s significance or contributionto the electronic structure of the excited state. Large coefficientsfor singly excited configurations indicate that the excited statecorresponds to the generation of a SE. Large coefficients for dou-ble excited configurations indicate nontrivial contributions frombi-excitonic excitation. Throughout this section SAC-CI is imple-mented within the Gaussian 03 [62] computational package usingthe B3LYP functional. The LANL2 relativistic effective core poten-tials were used for core electrons of all atoms, and the basisset employed for the valence electrons was the correspondingLANL2dz basis set [63,64]. MEG thresholds for lead selenide, cad-mium selenide, and silicon are determined for ideal, charged, andnonstoichiometric clusters.

3.1.1. Photoexcitation of multiple excitons in small clustersUsing a combination of the HF approximation and SAC-CI the

nature of excited states in different systems can be determined.SAC-CI is a rigorous and accurate ab initio approach. At the sametime, it is computationally very expensive, and therefore, cannot beapplied to systems exceeding about 20 atoms. Larger systems canbe investigated by semi-empirical techniques, such as the pseudo-potential approach [59]. Realistically sized quantum dots have onthe order of 1000 atoms leading to thousands of electronic orbitalsin the valence band alone. This poses a substantial computationalchallenge when taking into account all single and multiple electronstates. In order to surmount this limitation, the SAC-CI calculationswere only performed on model QDs that consist of about 10 atoms.As a justification for this approximation, the single-particle density-of-states (DOS) for lead selenide and cadmium selenide clusterswas calculated for systems ranging in size from 8 atoms to 360

atoms. In all cases, the different sized clusters exhibited maximaand minima in the DOS at the same energies [56]. Further, the opti-cal spectra of small silicon clusters agree well with the propertiesof bulk silicon [65].

A.J. Neukirch et al. / Coordination Chemistr

Fig. 3. Probability of observing MEs in photoexcited Pb4Se4, Cd6Se6, and Si7 as afunction of excitation energy normalized by the band-gap.

AC

ctqfilgbrlWcti

pttgobcbtttricMaamimaoc

ilar for positively charged p-type clusters, except it is the lack of a

dapted from J. Phys. Chem. C 2008, 112, 18294 and J. Phys. Chem. Lett 2010, 1, 232.opyright 2008 American Chemical Society.

The initial investigation focused on the Pb4Se4 and Cd6Se6lusters [56]. The two properties that determine the strength ofhe electron–electron interactions in semiconducting QDs are theuantum confinement and the dielectric screening. Quantum con-nement becomes a dominant feature as the size of the QD becomes

ess than the exciton Bohr radius, aB. The dielectric screening isiven by the dielectric constant, the larger it is the more screeningetween the charges there is. PbSe has a remarkably high Bohradius, aB = 46 nm, and a dielectric constant of ∼23. CdSe has a muchower Bohr radius at aB = 5.6 nm with a dielectric constant of ∼6.

hile the much larger Bohr radius of PbSe is indicative of strongeronfinement effects, the much larger dielectric constant is sugges-ive of shielded interparticle interactions and uncorrelated excitonsn the photoexcited state.

The results from this investigation are shown in the top twoanels of Fig. 3. The figure presents the fraction of multiple excita-ions as a function of energy. The fraction is computed by summinghe squares of the SAC-CI expansion coefficients describing MEs. Ineneral all excited stares are superpositions of SEs and MEs. In eachf case the calculation was performed using the singles and dou-les (SD) CI method using the double-� (dz) basis set. In the Pb4Se4luster, electronic excitations at energies lower that 2.5 times theand-gap, Eg, resulted in the production of predominantly SEs. Thehreshold for MEG was about 2.6 Eg. Based on energy conserva-ion, the lowest possible MEG threshold is 2 Eg. The less than idealhreshold in PbSe advocates that the large dielectric screening isestricting the MEG efficiency. The sharp transition from SEs to MEss additional evidence of uncorrelated excitons, as almost all opti-ally excited states between 2.5 and 3 times the band-gap becomeEs. This result indicates that in some cases all excited states above

certain energy contain a very high contribution from MEs and minor contribution from SEs. The results for the Cd6Se6 cluster,iddle panel of Fig. 3, show very different behavior. The MEG onset

s higher and exhibits a mixture of single and multiple excitons. Thisixture suggests that a significant percentage of CdSe excitations

re superpositions of single and multiple excitations and is a resultf unscreened many-body interactions due to the low dielectriconstant.

y Reviews 263– 264 (2014) 161– 181 167

Silicon QDs deserve particular attention, since much of thepresent photovoltaic industry is already based in Si. Experimen-tal measurements on Si QDs put the threshold of MEG at 2.4 timesthe first band-gap excitation energy, and found a quantum yield of2.6 excitons per photon at 3.4 times Eg [51]. In order for Si QDs tobecome a viable option in solar devices, a theoretical understandingof MEG in Si QDs is necessary. Si has a Bohr radius of 4.9 nm and adielectric constant of ∼12. First principles calculations show that at2–3 times the lowest excitation energy the majority of the opticallyexcited states in neutral Si7 take on ME character, bottom panel ofFig. 3, with an MEG threshold of about 2.4 Eg [66]. The transitionfrom SEs to MEs is not as sharp as for the PbSe clusters, but it is muchmore pronounced than in CdSe. Note that the dielectric constant ofsilicon is between that of lead selenide and cadmium selenide. Eventhough it has been shown that the dielectric constant is smaller forclusters than bulk [67], for clusters of similar size one expects it tobe largest for PbSe and smallest for CdSe. Many-body interactionsinfluence the nature of the phototexcited states more in cadmiumselenide QDs and less in lead selenide QDs compared to silicon QDs.

3.1.2. Dopants, defects and charges create new types ofexcitations

In order to successfully realize QD photovoltaic cells, investi-gations of excitations extending beyond the ideal cases in thesematerials are required. Our calculations establish that photoioni-zation, doping, and surface defects all influence MEG efficiencies,and that these alterations explain the variances observed inexperimentally measured MEG yields. In ideal semiconductors,excitations are relatively straightforward; an absorbed photonexcites an electron across the band-gap from the valence band(VB) to the conduction band (CB). This produces an interactingelectron–hole pair, or exciton, that is stabilized by the Coulombinteraction of the two charges in the semiconducting material.Additionally, if the energy of the incident photon is at least twicethe energy of the band-gap, then MEG is possible.

Deviations from the ideal structure can occur either by design,such as doping Si with phosphorous or boron to create an n- orp-type material, or by inherent surface defects such as danglingbonds, or through inadvertent photoionization of the material.This section summarizes SAC-CI results found for a negativelycharged and n-type doped silicon cluster. The optimized neutralstructure was used in the calculation on the charged cluster, seeFig. 4 (inserts) [66,68]. The dopant case is represented by tak-ing the ideal cluster substituting a phosphorus atom in place of asilicon atom. Fig. 4 shows the optical spectra, obtained by calculat-ing the transition dipole moment and oscillator strength betweenthe many-electron SAC-CI wavefunctions describing the ground anexcited electronic states. The electronic states are superpositionsof SEs and MEs. The SAC-CI wavefunctions go beyond the indepen-dent particle representation where only SEs are optically allowed,they include electron correlation effects and coupling between MEsand SEs. The calculations show that the main absorption peaks inthe optical spectra are blue-shifted when modifications to the idealcluster are made. We have observed a blue-shift in the calculatedoptical spectra with all types of defects including doping, charging,and dangling bonds in variety of systems [66,68,69]. Other groupshave also modeled the same spectral shifts in ionized species [70],as well as reduced photoluminescence in systems with danglingbonds [71]. The effect has been observed experimentally and canbe rationalized by Pauli blocking. This is where an electron occu-pies the lowest energy CB state in an n-type cluster, preventing a VBelectron from making a transition into this state. The effect is sim-

VB electron that prevents the lowest energy exciton transitions. Thelower energy IB transitions introduced by the cluster modificationsdo not contribute significantly to the absorption spectra.

168 A.J. Neukirch et al. / Coordination Chemistr

A

Spt

in these clusters are smaller than they are in the ideal cluster

Fma

A

Fig. 4. Optical absorption spectra for Si7. Si−7 , and Si6P.dapted from Chem. Sci., 2011, 2, 400. Copyright Royal Chemical Society 2011.

Materials containing imperfections display transitions beyond

Es and MEs. Examples of the different types of transitions areresented in the diagram on the left-hand side of Fig. 5. If the sys-em becomes charged, an extra charge carrier appears in the VB

ig. 5. Contributions from different electronic transitions to the excited state character

ultiple exciton (ME) transitions, red shows the contributions from intra-band (IB) transnd SE.

dapted from Chem. Sci., 2011, 2, 400. Copyright Royal Chemical Society 2011.

y Reviews 263– 264 (2014) 161– 181

or CB, and pathways for intra-band transitions open up. Dopantsoften introduce band edge states, and surface defects create dan-gling bonds that can produce gap-states, which can be occupiedor unoccupied and are not necessarily near the band edge. Each ofthese examples opens up the possibility of intra-band transitions(IB). IB transitions create excitons, but these excitons are containedwithin either the VB or CB, and do not involve promotion of an elec-tron across the band-gap. Photovoltaic applications require chargesto be separated across the band-gap. This renders IB transitionsuseless for photovoltaic applications. At sufficiently high energies,excitations in QDs containing dopants, defects or charges are com-plicated, since IB transitions occur in conjunction with SEs (IB + SE).This results in a multi-electron excitation that only creates one newelectron–hole pair spanning the band-gap. While formally thesetransitions are MEs, they are SEs as far as photovoltaic applicationsare concerned.

The contributions of each excited state configuration to theoverall character of the excited state in the ideal and modifiedclusters are presented along the right-hand side of Fig. 5. Thedata points shown are the squares of the expansion coefficientsfor a given excitation represented in the basis of the single- ormulti-electron configurations defined along the left-had side ofthe same figure. Excitations up to quadruples were calculated inLANL2DZ basis set. For simplicity only every fifth data point isdisplayed in the figure, and the line is a running average overthe entire data set. For the ideal Si7 cluster, the transforma-tion from predominantly SE to ME character occurs at about 2.8times the band-gap energy. In this ideal case the ME contribu-tion is substantial across the entire energy range. The Si−7 andSi6P clusters exhibit almost identical behavior to each other whichis to be expected, since doping Si with P is essentially equiva-lent to adding an extra electron into the CB. Excitation energies

because the lowest excitations in the modified clusters arise fromthe IB transitions. As the energy increases, the IB contributionsgradually decrease in magnitude, and SEs become the dominant

of Si7. Si−7 , and Si6P. Green indicates single exciton (SE) transition, black indicatesitions, and blue show the contribution from transitions that are a combination of IB

emistry Reviews 263– 264 (2014) 161– 181 169

cb

scTwpwwolltbftac

3

htpmgetswtrqtsraaiscppmb

3i

ipdfsmlTsitmSs

Fig. 6. Single and double exciton DOS of Si29H24. The SE DOS starts growing at lower

which agrees well with experiments [74–77]. The estimated Gauss-ian and exponential timescales are listed in Table 1.

The higher the initial excitation energy, the more quicklyMEs are formed, as seen in the rapid decrease of population

Fig. 7. (Top panel) The total population of SEs in the Si29H24 cluster, evolving start-ing from an initally excited SE at the displayed energy. The SE population decreasesbecause of MEG. The decay starts out Gaussian and becomes exponential. The cor-

A.J. Neukirch et al. / Coordination Ch

onfiguration. Most importantly, the MEG threshold is pushedeyond 4 Eg.

MEG thresholds were also investigated for charged and non-toichiometric lead selenide clusters [69,72,73]. It was found thatationic and anionic forms of Pb4Se4 do not generate MEs at all.he computed MEG threshold for these systems was about 5 Eg

hich is higher than the ionization energy. This cluster’s MEGroperties were also studied when either a lead or selenium atomere removed. These atomic vacancies were found to interfereith MEG. The extent to which the MEG was hindered depended

n the bonding properties of the electrons in the vicinity of theattice defect. In this case, lead’s valence electrons carry more angu-ar momentum and are more shielded from the nuclei comparedo selenium’s valence electrons, assisting in the reformation ofonds. This leads to MEG thresholds with excess lead on the sur-ace being comparable to that of the ideal system. However, whenhere is excess selenium on the surface, MEG is negatively impacted,lthough the effect is much less severe than that observed in thease of ionization and charging.

.2. The impact ionization mechanism

As in bulk materials, MEs in QDs can be created by relaxing aigh-energy carrier to its ground state and exciting valence elec-rons across the band-gap, thus producing additional electron–holeairs. This effect is known as impact ionization (II), and is a mainechanism for MEG. The inverse process, known as MER, is analo-

ous to the Auger process in bulk. The primary difference betweenxciton multiplication in bulk semiconductors and semiconduc-or QDs is the conservation of linear (crystal) momentum. In bulkemiconductors, both energy and momentum must be conserved,hich increases the ideal energetic threshold of carrier multiplica-

ion beyond 2 Eg. In QDs, this linear momentum constraint is lifted,educing the carrier multiplication threshold. Additionally, due touantum confinement, the Coulomb interactions between elec-rons and holes are stronger in small nanocrystals than in bulk. Theimulations reported in this section are performed in the adiabaticepresentation, which is the only representation readily available inb initio electronic structure theory. In the adiabatic representationll electronic (Coulomb) terms in the Hamiltonian are “diagonl-zed out’ to obtain the adiabatic states. Transitions between thetates arise due to the NA coupling, and in the absence of the NAoupling transitions cannot occur. This picture complements thehenomenological description, in which SE and ME states are cou-led by Coulomb terms. Generally, the adiabatic representation isore physically relevant. For example, sunlight excites a system

etween adiabatic states rather than SE and ME states.

.2.1. Single and multiple exciton state densities determinempact ionization threshold

We investigated the MEG dynamics caused by the II mechanismn the Si29H24 QD. As shown in the insert of Fig. 6, hydrogen atomsassivated the QD surface. The hydrogen surface passivation healsangling bonds on the bare QD surface, and also provides high-requency phonon modes due to the light hydrogen atoms. Fig. 6hows the SE and DE DOS of the Si29H24 QD [50]. The band-gap,arking the onset of the SE DOS, was calculated to be 2.1 eV. The

ower excitations, up to energies of twice the band-gap, are all SEs.he DE DOS starts at double the energy of the SE DOS, but increasesignificantly faster with energy compared to the SE DOS, due to thencreasing combinatorial number of DEs with energy. If the pho-

oexcitation energy can be exchanged freely between SEs and MEs,

ost of the initial population will flow into DEs at high energy, andEs can appear only at low energies. However, as will be demon-trated below, actual transitions are also influenced by dynamical

energies than the DE DOS, but the DE DOS rapidly overtakes the SE DOS as the energyincreases. The structure of the system is shown in the inset.

Adapted from ACS Nano 2012, 6, 1239. Copyright American Chemical Society 2012.

NA couplings between SE and DE states, and cannot be determinedsolely from the static electronic properties.

3.2.2. Real-time dynamics of impact ionizationThe upper panel of Fig. 7 shows the decrease of the total pop-

ulation of all SEs, started from an initially excited SE state of thedisplayed energy. This panel directly exhibits how the MEG appearsin the Si29H24 QD, because the lost SE population all flows into DEstates by the II mechanism. Our simulation includes the groundstate as well as SEs and DEs, but no population flows into theground state throughout the current simulation of picoseconds,

responding timescales are shown in Table 1. (Bottom panel) Time-energy domainrepresentation of the SE population dynamics. Photoexcited at a high energy, SEsdisappear due to MEG, while losing energy by coupling to phonons. SEs reappear ata later time and at a lower energy as a result of ME recombination.

Adapted from ACS Nano 2012, 6, 1239. Copyright American Chemical Society 2012.

170 A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181

anicalA .

ia[peTattoie2ppamObcst

3r

dsdtdatIibMtshp

TEd

Fig. 8. Illustration of the pairs of electronic states that form quantum-mechdapted from ACS Nano, 2009, 3, 2487. Copyright American Chemical Society 2009

n SE states. This strong energy dependence of MEG dynamicsgrees with the previous experimental and theoretical reports31,53,56,59,60,69,78–80]. However, most previous theories adoptertubative methods, such as Fermi’s golden rule, which assumexponential population decay by the MEG [31,53,55,59,60,79–82].he current simulation treats the NA couplings non-perturbativelynd describes the transformation from an initial Gaussian decayo the later exponential decay. This Gaussian to exponential decayransformation is generic for all quantum dynamical process andnly appears when sufficiently many quantum states are involvedn the dynamics. It should be remarked that the MEG is observedven with the initial energy is lower than the electronic threshold,

Eg, as seen in the 1.8 Eg case in the upper panel. The real-timehonon dynamics is now explicitly taken into account, allowinghonon-assisted Auger processes. The lack of the electronic energy,bout 0.2 Eg, is compensated by the high-frequency Si-H surfaceodes, which are around 2000 cm−1 (0.25 eV) in the Si29H24 QD.ur result indicates that the MEG dynamics is determined not onlyy the static electronic properties of the QD but also by the dynami-al NA couplings. It is expected that the effect of the surface ligandshould become smaller in a larger QD due to the decreased surface-o-volume ratio.

.2.3. Interplay between impact ionization and excitonecombination

The lower panel of Fig. 7 shows the first 3 ps of SE populationynamics as a 2D function of time and energy. The SE populationtarts from an initial excitation of 2.8 Eg and exhibits diffusiveynamics into other SEs at the initial stage before 1 ps. Thenhe total SE population decreases, reflecting the accelerated MEGrawn in the upper panel. However, the SE population recovers at

lower energy after 2 ps. This population recursion corresponds tohe MER, and the recursion energy is identical to 2 Eg, supporting theI mechanism. Since our simulation method simultaneously takesnto account both MEG and MER, it allows us to study the com-ined MEG/MER dynamics and the interplay between MEG andER. The MER timescale of a few ps agrees well with extrapola-

ion from experimental results [83–87]. If the MER started from aingly excited DE, we would have obtained a much longer MER time,undreds of ps, as we reported [50]. The many DEs produced by thereceding MEG prepare multiple pathways to SEs by couplings to

able 1stimated timescales for Gaussian and exponential decay caused by the MEG atifferent initial excitation energies. See Fig.7 for the time-dependent data.

Excitation energy MEG timescales (ps)

e g

1.8 Eg 48 192.8 Eg 9.0 133.5 Eg 5.5 8.1

superpositions during light absorption/luminescence, MEG, and ME fission.

a broader range of SEs, thereby accelerating MER significantly. Themultiple pathways are essential for the fast and accurate simula-tion of MER because, the density of final SE states is much lowerthan the density of initial DE states.

4. Phonon-induced dephasing of electronic excitations

4.1. Luminescence, multiple exciton generation, and multipleexciton fission

Electronic transitions in QDs involve coherent superpositions ofexcitons. For instance, light absorption and luminescence results intransient superposition of ground and excited states. Auger-typeprocesses, facilitated by strong Coulomb interactions, pass throughcoherent combinations of SEs and MEs. Such quantum coherenceis destroyed by coupling to phonons. The process is known asdecoherence or pure-dephasing, and it is generally completed insub-picoseconds. Fig. 8 shows a diagram of phonon-induced pure-dephasing processes in electronically excited QDs. The left panelillustrates the dephasing of a superposition of the ground elec-tronic state and a SE, whose time determines the homogeneouslinewidth in luminescence. The middle panel shows a superpositionof a high-energy SE and an ME. Elastic electron–phonons scatter-ing cause this superposition to dephase quickly, making the ME andSE states uncorrelated and evolve independently. The right panelof Fig. 8 pertains to the ME fission process (MEF), which is anal-ogous to singlet fission seen in polyacene crystals [88]. MEs areformed by coherent superpositions of SEs. With time, MEs breakinto incoherent combinations of SEs, which, for instance, can emitindependently. The right panel shows two SEs near the band-gap.When in a coherent superpositions, the two SEs form an ME. Elas-tic electron–phonon scattering destroys the superposition, and theME fissions into independent SEs.

The corresponding normalized ACFs (Eq. (12)), spectral den-sity obtained from the influence spectrum (Eq. (13)) and cumulantdephasing functions (Eq. (15)) for Si29H24 are shown in Fig. 9 [26].All the data are calculated at 300 K. A superposition of the groundand excited electronic states created during photoexcitation orluminescence dephases in an ultrafast time, as reflected by therapidly decaying ACF and dephasing function in Fig. 9. The deduceddephasing time is less than 10 fs. This ultrafast dephasing is mainlyattributed to the large energy fluctuation, Cu(0) =〈 �E2(0) 〉. Thelinewidth of the optical signal deduced from this dephasing life-time is 150 meV at room temperature, which is in good agreementwith the remarkably broad experimental luminescence linewidth[74,89]. The dephasing completes even faster during MEG, in afew fs. Such ultrafast dephasing of superpositions of MEs and SEs

contributes to MEG; an ME state can be generated by losing coher-ence with the initial photo-excited high-energy SE state. Finally,an ME decays into uncorrelated SEs by MEF. The phonon-induceddephasing time for MEF is an order of magnitude longer than the

A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181 171

Fig. 9. (Top panel) Autocorrelation functions describing phonon-induced pure-dephasing processes involved in luminescence, MEG, and ME fission for Si29H24 atrp

A

acrtlsctdei

psnuafmmqaet

Table 2Estimated dephasing times (fs) in the Si29H24 QD at different temperatures.

Dephasing times

300 K 80 K

Luminescence (SE/Eg) 4.00 ± 0.10 7.00 ± 0.20

oom temperature. (Middle panel) Fourier transform of the ACF shown in the topanel. (Bottom panel) Dephasing functions associated with the shown ACF.

dapted from ACS Nano, 2009, 9 2487. Copyright American Chemical Society 2009.

bove dephasing times for the luminescence and MEG. The cal-ulated timescale is about 54 fs. This significantly slower MEF isationalized by the long-lived coherence between the two SE stateshat have similar energy and orbitals. The small energy differenceeads to a small energy fluctuation, Cu(0) =〈 �E2(0) 〉, and thus to thelowly decaying dephasing function. After MEF, excitons have lostoherence and evolve quite independently [66]. In addition to thehree pure-dephasing processes discussed here, one can considerephasing of entangled electron–hole pairs forming excitons. Thelectron and hole lose coherence during electron transfer, resultingn uncorrelated charges [90–92].

Fourier transforms of the ACFs provide information abouthonon modes involved in dephasing between the two coherenttates of interest. The middle panel of Fig. 9 indicates that the lumi-escence and MEG involve both acoustic and optical phonon modesp to high frequencies, while MEF only involves low-frequencycoustic modes. Here, the low-dimensionality of the QD allowsor only approximate classification of acoustic and optical phonon

odes. The dephasing related to the luminescence and MEG isainly driven by the phonon modes within the 200–600 cm−1 fre-

uency range. Phonon modes pertaining to the ligand hydrogentoms do not contribute to the dephasing processes under consid-ration. It can be thus concluded that the main difference amonghe dephasing rates in the luminescence, MEG and MEF were caused

MEG (ME/SE) 1.60 ± 0.05 2.96 ± 0.09MEF (SE/SE) 54.0 ± 1.00 205 ± 0.90

only by the core phonon modes. The dephasing between two coher-ent SE states of similar energy and orbitals during MEF involvesonly lower-frequency phonon modes of the semiconductor core,with frequencies up to 200 cm−1.

4.2. Temperature dependence

We show the temperature dependence in the dephasingtimes of the Si QD in Table 2. High temperature activates abroader high-frequency spectrum of modes, inducing a strongerelectron–phonon coupling and thus faster dephasing. The temper-ature dependence of the MEF dephasing, related to low-frequencyacoustic phonon modes, is stronger than in the luminescence andMEG, whose electron–phonon couplings involve more modes, upto the high-frequency optical phonons. The MEF exhibits four timesfaster decoherence at 300 K than at 80 K, whereas the dephasing isonly twice faster for the luminescence and MEG at the higher tem-perature. The difference in the temperature dependence appearsbecause the low-frequency phonons require small amounts of ther-mal energy and are more susceptible to changes in temperature.The estimated luminescence linewidth at 80 K is 80 meV and stillagrees with the corresponding experimental data [74,89].

4.3. Size dependence

The upper panel of Fig. 10 shows the Fourier transforms ofthe ACF to demonstrate the phonon frequencies involved in thedephasing between the ground state and lowest SE, and betweena higher energy SE and bi-exciton, in the PbSe QD of differentsizes. The PbSe QDs couple to lower-frequency core modes com-pared the Si QD, since Pb and Se atoms are heavier atoms. Sincethe low-frequency acoustic modes stem from the QD cores, Fig. 10shows the strong QD size dependency. While the larger PbSe QDincludes a wider range of frequencies than the smaller PbSe QD,the intensity of each mode is stronger in the smaller QD. Thus,larger electron–phonon coupling is expected in the smaller PbSeQD, supporting the phenomenological elastic model of dephasingin QDs [93].

The corresponding dephasing functions of the larger PbSe QDare plotted in the lower panel of Fig. 10. The results were calculatedusing both the cumulant approximation and the direct approach,Eqs. (15) and (14), respectively. The results from the cumulant anddirect dephasing functions coincide well in spite of the long mem-ory in the second-order cumulant, Eq. (11). The estimated directdephasing times for the larger dot are 9.3 fs and 3.5 fs, respec-tively. The smaller PbSe QD exhibits shorter dephasing, 7.1 fs and3.2 fs respectively, due to its stronger electron–phonon couplings.The dephasing involving bi-excitons occurs faster than dephasinginvolving SEs even when the energy difference between the pairsof states, ground/Eg and 3 Eg/bi-exciton, is similar, about an Eg.Bi-excitons form superpositions with high-energy SEs, and suchSEs are more strongly influenced by perturbations caused by lat-

tice phonon modes than the low-energy SE, leading to the fasterdephasing. The homogeneous linewidths of the luminescence esti-mated from the pure-dephasing times again agree well with theexperimental results [74].

172 A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181

Fig. 10. (Top panel) Fourier transform of the autocorrelation functions associatedwith the phonon-induced fluctuations of the band-gap (solid line) and energy gapbetween SE at triple band-gap and lowest energy biexciton (dashed line) in Pb16Se16

and Pb68Se68. (Bottom panel) Dephasing functions for the above-mentioned pairso

A

4

ioodcttcFattodmptstopsibf

Fig. 11. (Top panel) Fourier transforms of the phonon-induced fluctuations of theenergies of bulk, surface, and plasmon states in the Ag104 cluster. The electron densi-ties of these states are shown in the insert. (Bottom panel) Direct dephasing functionfor the bulk, surface, and plasmon states respectively.

of three at 300 K and 50 K.

Table 3Estimated pure-dephasing times (fs) for bulk, surface and plasmon states in Ag104

at different temperatures.

States Dephasing times (fs)

f states computed both directly and using the second-order cumulant expansion.

dapted from Nano Lett., 2006, 6, 2295. Copyright American Chemical Society 2009.

.4. Dephasing in metallic particles, plasmons

Metallic nanoparticles exhibit collective electronic excitations,.e. surface plasmon. We consider the phonon-induced dephasingf three types of electronic states in the Ag104 cluster, as the insertf Fig. 11 illustrates. The gray distribution indicates the spatialensities of the lowest energy states of each type. The states arelassified by their localization with respect to the silver core atoms;he bulk state where the charge density is delocalized over the clus-er (∼0.25 eV above the Fermi energy), the surface state where theharge density is localized on the cluster surface (∼1.5 eV above theermi energy), and the plasmon state where the charges are locatedway from the cluster (∼4 eV above the Fermi energy). The charac-eristic phonon modes coupled to each state can be deduced fromhe Fourier transforms of the time-dependent fluctuating energyf the corresponding electronic excitations. The excitations areescribed as the energy differences between the Fermi level of theetallic QD, and the corresponding excited state level. The upper

anel of Fig. 11 shows that the low-frequency acoustic modes con-ribute to the excitation energy fluctuations in all three types oftates. The slow acoustic modes modulate the size and shape ofhe cluster. Such cluster-scale motions easily affect all three typesf electronic excitations. However, different electron–phonon cou-ling strengths are found in each of the three cases. The plasmon

tate exhibits a much smaller amplitude in the spectral density,ndicating weaker electron–phonon couplings. This is rationalizedy the charge distribution of the plasmon state being localized awayrom the cluster core. The largely detached charges can only poorly

Adapted from Phys. Rev. B, 2010, 81, 125415. Copyright American Physical Society2010.

couple to phonon modes of the silver nanocluster. The phonon cou-plings are the strongest for the bulk excitation, since its chargeis delocalized over the whole silver cluster, and it can efficientlycouple to cluster’s phonon modes.

The direct dephasing functions for the lowest-energy bulk, sur-face, and plasmon excitations are shown in the lower panel ofFig. 11. The bulk state excitation, which couples to the phononsmost strongly, dephases faster than the other two excitations. Theplasmon excitation exhibits the slowest dephasing, reflecting theweak coupling between plasmon states and phonons. Table 3 com-pares the dephasing times for the plasmon excitation at 300 Kand 50 K. The pure-dephasing times for bulk, surface, and plasmonstates at 300 K directly reflect the differences in the magnitudes ofthe electron–phonon coupling, as indicated in the upper panel ofFig. 11. The estimated timescale of the plasmon state is in agree-ment with the experimental data [94]. The temperature affects thephonon dynamics of Ag104, and thus the dephasing timescales. Theplasmon pure-dephasing times for Ag104 differ by almost a factor

Bulk (300 K) 8.7Surface (300 K) 15.6Plasmon (300 K) 27.6Plasmon (50 K) 82.4

emistry Reviews 263– 264 (2014) 161– 181 173

5

towmttarHptapat

anprmparvhceaSca

thscmtTd[

5

atbairitcttech[ce

Fig. 12. (Top panel) DOS evolution in Pb68Se68 at room temperature. The DOS fluc-tuates because of thermal atomic motions. (Bottom panel) The electron and holeenergy relaxation in the photoexcited Pb68Se68. Both electrons and holes relaxwithin a picosecond.

A.J. Neukirch et al. / Coordination Ch

. Electron–phonon relaxation

High-energy excitons rapidly lose excess energy to the latticehrough a series of electron–phonon scattering events. The processccurs, for instance, under solar radiation, which has a broad band-idth capable of exciting electrons in photovoltaic semiconductoraterials into many different energy levels within the conduc-

ion band. It is desirable to find ways of controling and reducinghe relaxation rate in photovoltaic devices, in order minimize themount of energy lost to thermalization [20,25,95]. In bulk mate-ials, the intrinsic electron–phonon coupling cannot be modified.owever, QDs offer the opportunity of decoupling electrons andhonons. Due to quantum confinement effects, the electronic struc-ure of QDs display discrete energy levels, and the spacing betweendjacent levels may be tuned to larger than the energy of a singlehonon. In this case, only multi-phonon processes would be able tollow the electrons to energetically relax, potentially slowing downhe relaxation rate compared to bulk materials [20,96,97].

Experiments have demonstrated that both electron–phononnd Auger-type processes participate in the electron relaxation inanocrystals [98]. Depending on the materials involved, the surfaceassivation, and the types of surface ligands, the Auger processesesulting in energy exchange between electrons and holes, may oray not contribute to the overall relaxation. For instance, charge-

honon relaxation is faster for holes than electrons in CdSe QDs,nd therefore, electrons can lose energy by transferring it to holes,ather than directly to phonons. The Auger process can be circum-ented if electrons and holes are separated, or if the VB and CBave a similar DOS. Current experimental methods have been suc-essful at reducing the Auger pathway by separating the excitedlectrons and holes, making it possible to decouple the Augernd electron–phonon relaxation pathways in nanocrystals [98,99].lowing down electron–phonon coupling is favored for MEG effi-iency as well as hot electron extraction, leading to higher currentsnd voltages of photovoltaic devices.

Electron–phonon coupling is interesting from both fundamen-al and practical points of view, motivating the need to understandow factors, such as material, temperature, nanoparticle size andhape, surface termination, and ligands, influence electron–phononoupling in nanoscale systems [100]. Recently, a nonadiabaticolecular dynamics method was established [39,101] in order

o simulate the electron relaxation process in QDs [102–104].he method has been used to investigate different systems, asiscussed below. For further reading, please, consult references39,46,102–112].

.1. Relaxation in semiconductor nanocrystals

Among various semiconducting nanomaterials, lead salts, suchs PbS and PbSe, show some of the most unique electronic andransport properties [83,113–116]. Their conduction and valenceands are more symmetric than in other semiconductor materi-ls, such as CdSe, potentially rendering Auger relaxation processesnefficient [117], and making it easier to study the electron–phononelaxation pathway. The effective masses of electrons and holesn lead salts are similar, as a result of the symmetric band struc-ure, and small [118]. The latter feature leads to strong quantumonfinement effects that should create wide gaps between elec-ron and hole energy levels [114]. This discretization was expectedo produce a disparity between the electronic and vibrationalnergy quanta and to produce a phonon bottleneck in the charge-arrier relaxation [20]. These qualities favor MEG, and indeed

igh quantum yields of MEG in PbSe QDs have been observed22,23,27,54,55]. However, PbSe QDs exhibit ultrafast intrabandharge-phonon relaxation signifying that no phonon bottleneckxists [119,120]. Taking steps beyond static effective-mass theory

Adapted from ACS Nano, 2009, 3, 99. Copyright American Chemical Society 2009.

and pseudopotential calculations, theoretical time-domain stud-ies where able to directly mimic time-resolved experiments andprovide a detailed atomistic description of the electron–phononrelaxation process.

The top panel of Fig. 12 is an example of how phonon motionsinduce fluctuations in electronic DOS. The fluctuations are smoothand less than 0.1 eV in both the CB and VB. Phonon motionsmix states of different symmetries and reduce the energy gaps.The bottom panel of Fig. 12 presents the relaxation dynamicsof electron and hole, by coupling to phonons [25,102,104]. Thecharge carriers visit many states during relaxation, and none ofthe intermediate states plays a special role. The excited popula-tion created by a photoexcitation relaxes to the band edge within apicosecond, agreeing well with the experimental results [119,121].Multi-phonon relaxation was initially proposed to rationalize theultrafast experimental data [119], and indeed our simulations pro-duced instances where up to 0.3–0.6 eV of electronic energy waslost in a in a single even [25,102,104]. Nevertheless, most transi-tions involve small amounts of energy that are close to a phonon

−1

energy of 100–200 cm (12–25 meV), indicating that electronicenergy gaps are small at high energies, explaining the absence ofthe phonon bottleneck.

174 A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181

Fig. 13. Fourier transforms of the phonon-induced evolution of the energy levels ofeight lowest plasmon states in Ag68.

AS

5

ottomoteTdatgapietpic

nmcnIe1ahatnmodsec

Table 4Phonon-induced relaxation times of populations and energy for plasmon excitationsin Ag QDs [109]. The bottom three rows indicate averages over lower energy, higherenergy, and all plasmon excitations.

State number Initial energy (eV) Population relaxationtime (ps)

Energy relaxationtime (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 4.68

1–4 3.99 0.68 2.87

The temperature dependence of charge carrier relaxation differ-entiates phonon-induced processes from other channels, such asAuger scattering [133]. Experiments have demonstrated [99,119]

Fig. 14. Fourier transforms of the phonon-induced evolution of lowest energy unoc-

dapted from J. Phys. Chem. C, 2012, 116, 15034. Copyright American Chemicalociety 2012.

.2. Relaxation in metallic particles, plasmons

Metallic nanoparticles have been studied for a wide varietyf applications, with the focus on noble metals that are robusto photo-oxidation. Metallic systems exhibit collective excitationshat are known as plasmons and that are responsible for manyf the novel electronic and optical properties exhibited by nobleetal nanoparticles. Examples include data storage, lasing, electro-

ptics, biological imaging, and light harvesting [122–132]. Similaro their semiconductor counterparts, metallic QDs can exhibit rapidlectron–phonon dynamics, strongly influencing the applications.he electron–phonon relaxation rate is a central parameter in manyevices. Electron–phonon coupling causes nonradiative relaxationnd heating in electronic, optical, and photovoltaic devices. Inhese systems, the relaxation is an unwanted attribute and theoal is to minimize this effect. In contrast, in photothermal ther-py metal nanoparticles ability to rapidly convert an absorbedhoton into energy and heat is a desirable feature to capital-

ze on. Similarly, rapid electron–phonon relaxation is desirable inlectro-optic switches, since its rate controls the switch responseime. Therefore, it is necessary to learn the mechanisms behindlasmon–phonon relaxation in metal nanoclusters, in order to

nvestigate the possibility of tuning nanocluster properties for spe-ific applications.

The Fourier transforms of the plasmon state energies of an Ag68anoparticle are shown in Fig. 13 [109]. The vibrational modes thatodulate the energy levels, create the largest electron–phonon

ouplings. It was found that the plasmon excitations in the Ag68anoparticles couple exclusively to low-frequency phonon modes.

t is these low-frequency modes that cause plasmons to relax innergy. In all cases, the dominant modes have frequencies less than00 cm−1. This is in contrast to PbSe and Si nanocrystals, where inddition to acoustic phonons, optical modes around 200 cm−1 alsoave strong influence on the charge carrier relaxation. The slowcoustic modes dominate the plasmon relaxation process, becausehe low frequency vibrations modulate the size and shape of theanoclusters. High-frequency optical modes involve local displace-ents of atoms relative to each other, and have little influence

ver the global nanoclusters properties. Since plasmon states areelocalized around the entire QD they couple most strongly to the

low acoustic modes. The variations in the symmetry of the differ-nt plasmon states explains the differences seen the frequenciesontributing to electron–phonon relaxation. The overall trend

5–8 4.15 1.21 3.351–8 4.07 0.95 3.11

observed is that lower energy plasmon states tend to couple morestrongly to phonons compared to the higher energy states. Thehigher energy states are more delocalized from the nanoparticle,and are less affected by atomic motions taking place inside theparticle.

The relaxation times of both population and energy are shownin Table 4. The plasmon population decay is obtained by sum-ming time-dependent populations of all plasmon states, and thedecay results from population transfer into non-plasmon states.We see that the lower energy plasmon states relax out of the plas-mon band more quickly than the high-energy states. The higherenergy plasmon excitations take time to traverse the manifold ofplasmon states before decaying into the low energy bulk states.The states that undergo the quickest population relaxation are alsothe states that couple most strongly to the phonons. The energyrelaxation exhibits little dependence on the initial conditions. In allcases, relaxation takes place on the picosecond timescale, similarlyto semiconductor QDs.

5.3. Temperature dependence

cupied orbital (LUMO) in the Cd33Se33 quantum dot at high and low temperatures,red solid and black dashed lines respectively.

Adapted from J. Phys. Chem. C, 2011, 115, 11400. Copyright American ChemicalSociety 2011.

A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181 175

0 500 1000 1500 2000 2500 3000 35000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6E

nerg

y (

eV

)

Time (fs)

63 K 100 K 151 K 211 K 293 K 417 K 501 K

Fig. 15. Electron energy decay in the Pb16Se16 quantum dot for various tempera-tures.

A2

tPemceCi

oeins[ttoFtsTatiicaf

adissitsotaa

�

0 100 200 300 400 500

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Electron

Hole

Experiment

Re

laxa

tio

n T

ime

(p

s)

Temperature (K)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

DO

S

Energy (eV)

Fig. 16. The calculated electron and hole relaxation times at different temperaturesfor Pb16Se16. The experimental results are the temperature-dependent hot-carrier

decouple the Auger and electron–phonon relaxation processes in

dapted from Phys. Rev. B 2009, 79, 235306. Copyright American Physical Society009.

hat relaxation is significantly more temperature-dependent inbSe QDs compared to CdSe QDs, where Auger processes are moreffective. The differences in the temperature dependence betweenaterials indicate that the importance of the role phonons play

an vary from material to material. Temperature dependence oflectron–phonon relaxation was studied for Pb16Se16 [105] andd33Se33 [106] QDs. Several facts were ascertained from these stud-

es, many ubiquitous for both systems.The phonon modes that induce charge carrier relaxation are

btained by taking the Fourier transform of the initially populatednergy level. The electron–phonon coupling, djk in Eqs. (7) and (8),s directly related to the second derivative of the energy along theuclear trajectory, and therefore the vibrational modes that mosttrongly modulate the energy levels generate the largest coupling134]. Fig. 14 shows the Fourier transform of the energy vibra-ions in the lowest unoccupied state for the CdSe QD [106]. All ofhe conclusions drawn from Fig. 14 hold for the spectral densityf PbSe as well. Temperature changes the coupling in two ways.irst, the Fourier transforms show a high-frequency tail at higheremperatures, suggesting that a larger fraction of high-frequencyurface modes are involved in the carrier relaxation dynamics.his can be rationalized by bearing in mind that elevated temper-tures can activate higher-frequency modes. Second, the Fourierransform curves are widened at higher temperature, demonstrat-ng that more modes are modulating the energy levels. Both thencrease in the number of participating modes and the increase inoupling to high-frequency modes would lead to quicker relaxations the system’s temperature is raised. This is indeed what has beenound for both CdSe and PbSe, and is shown in Fig. 15 for PbSe.

The studies allowed for a quantitative model on the temper-ture dependence of hot-carrier relaxation in nanocrystals to beeveloped [105,106]. It follows from Eq. (8), that if we assume djk

s purely implicitly dependent on temperature, the NA couplingtrength is proportional to the ion velocities R, and therefore thequare root of kinetic energy. Due to the classical treatment ofons in MD calculations the temperature, TMD, is proportional tohe system’s kinetic energy. Time-dependent perturbation theoryhows that the transition probability is proportional to the squaref the off-diagonal element of the perturbation matrix, which ishe non-adiabatic coupling, |NA|2, in our case. Based on the abovessumptions the temperature dependence of the hot carrier relax-

tion rate could be simply written as

∼|NA|2∼|djk|2|R|2∼|djk|2TMD, (20)

relaxation times for PbSe quantum dots with an average radius of 1.9 nm.

Adapted from Phys. Rev. B 2009, 79, 235306. Copyright American Physical Society2009.

where � , NA, djk, R, and TMD represent the relaxation rate, nona-diabatic coupling, electron–phonon coupling, ion velocity andtemperature, respectively. Note that the total NA coupling is theproduct of the electronic component djk and phonon velocity,R, Eq. (8). However, in both PbSe and CdSe QDs the calcu-lated results diverged significantly from the linear temperaturedependence, predicted by Eq. (20). For both systems, relaxationrates were better fitted to T0.4

MD rather than TMD, see Fig. 16for PbSe. This weaker than predicted temperature dependencemeans that the electron–phonon coupling strength, djk, has aninherent temperature dependence. To test this, we calculatedthe NA coupling strengths between pairs of states for eachsystem, and found that they decreased with temperature asT−0.3. Although the QD geometry is only weakly temperature-dependent, the coupling can strongly depend on QD expansion.Fig. 16 also shows that the holes relax more quickly thanthe electrons. This is explained by a higher density of holestates.

Experimental results show no temperature dependence inCdSe QDs, and a much weaker temperature dependence forPbSe at low temperatures than at high temperatures [119]. Atlow temperatures, the Auger channel plays an important role,accounting for the weaker temperature dependence. Note thatthe holes exhibit weaker temperature dependence than elec-trons, Fig. 16. Thus, if electrons are able to transfer their energyto holes by the Auger channel, the sensitivity of electron relax-ation to temperature is small. As the temperature rises, thedirect electron–phonon channel becomes more efficient and dom-inates the overall relaxation rate, creating stronger temperaturedependence. The lack of temperature dependence observed inCdSe QDs confirms that the Auger relaxation channel is extremelyimportant, as long as nothing is done do shut down thatpathway.

5.4. Phonon bottleneck

Current experimental methods have become successful ateffectively separating electrons and holes, making it possible to

NCs. For CdSe nanocrystals, with a deep hole-trapping surfactantmolecule, such as pyridine, the electron–phonon interaction isdominant, and a slow relaxation of several hundred picoseconds

176 A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181

Fig. 17. (Top panel) Optimized geometries for the Cd33Se33 and core–shellCd33Se33/Zn78S78 clusters. (Bottom panel) Electronic structure of the CdSe andCdSe/ZnS QDs. The bold blue line corresponds to the Cd33Se33 system while the redline shows DOS for the Cd33Se33/Zn78S78 system. The vertical lines represent indi-vidual electronic states in the CdSe/ZnS QD; the gray part identifies the localizationof the state on the CdSe core and the magenta part corresponds to the percentagelocalized on the ZnS shell.

A2

iiceTataCVns

Tsobtwiwsottdrhttqsa

0 0.40 0.80 1.20 1.60 2.000.9944

0.9952

0.9960

0.9968

0.9976

0.9984

0.9992

1.0000

trelax~ 670 ps

trelax ~300

ps

trelax ~ 420 ps

Po

pu

latio

n o

f 1

Pe

Time (ps)

τdeph

= 36 fs

54 fs

90 fs

Fig. 18. Electron relaxation from the photoexcited 1Pe state into the 1Se state in theCd33Se33 QD calculated for different dephasing times.

dynamics. Ab initio calculations demonstrated [103] that quantum

dapted from Phys. Rev. Lett. 2013, 110, 18404. Copyright American Physical Society013.

s observed [98,99]. This is orders of magnitude longer than thatn the bulk system. Panday and Guyot-Sionnest have found thatoating CdSe QDs with thick shells of multiple ZnS and ZnSe lay-rs protected the surface of the QD core from the environment [21].his serves to suppress the ligand mediated relaxation channel. Theuthors also suggested that the shells trapped the holes, decouplinghem from the core-bound electrons and suppressing Auger relax-tion. Upon performing a DOS analysis on both a CdSe QD and adSe/ZnS core/shell QD, we find that the ZnS shell creates additionalB states near the edge of the band-gap. The electronic structureear the CB edge remains unaffected, supporting the electron–holeeparation concept (Fig. 17).

When the semiclassical decoherence correction was included inDDFT-NAMD [46], a phonon bottleneck between the 1Pe and 1Se

tates in a CdSe QD was observed in the simulation [135]. Using theptical response formalism, Section 2.6, the pure-dephasing timeetween these states was found to be 36 fs. When this dephasingime was incorporated into the NAMD, the lifetime of the 1Pe stateas calculated to be 670 ps, in good agreement with the exper-

mentally measured lifetimes of hundreds of picoseconds [21]. Itas also found that if the dephasing time were increased, the

ystem relaxed more quickly, Fig. 18. Dephasing can be thoughtf as a measurement of the electronic subsystem performed byhe phonon environment. A fast decoherence then correspondso frequent measurements of the quantum-mechanical state. Asemonstrated, a fast decoherence also leads to reduced relaxationates. This principle, where a system in a known initial state canave its evolution “frozen” by frequent measurements, is known ashe quantum Zeno effect. It is the suppression of unitary time evolu-ion caused by quantum decoherence within the system [109]. The

uantum Zeno effect rationalized the phonon bottleneck seen inemiconductor QDs [135]. Note that decoherence can also acceler-te the dynamics [136].

Adapted from Phys. Rev. Lett. 2013, 110, 18404 in press. Copyright American PhysicalSociety 2013.

5.5. Surface defects introduce additional states

Another example of controlling the optical properties of a sys-tem by tuning its surface composition was demonstrated by Weiet al. [137]. It was found that when CdS QDs were terminated withsulfur, the band edge emission was quenched; the photolumine-scence was recovered when the QDs were cadmium terminated. Inboth cases, the absorption spectra remained largely unaffected. Thehypothesis was that the termination of the CdS QDs with a sulfurshell introduces surface trap states that provide for effective non-radiative recombination pathways. To test this conjecture, the Cdand S atom contribution to the electronic DOS were investigatedfor representative CdS QDs. The dots had either a stoichiometricsurface, or were terminated with predominately Cd or S. The Cdand S-rich clusters were simulated by using the coordinates for theideal QD and substituting one, three or five Cd(S) atoms with S(Cd)atoms. These modified structures were then each relaxed to theirlowest energy state.

When the larger Cd atoms were used to replace the smaller Satoms, the QD got slightly larger. The energy gap narrowed becauseof decreased quantum confinement and the appearance of defectstates near the edges of the VB and CB. Despite the band-gapnarrowing, a well-defined energy gap remained, with no evidenceof a midgap surface trap states. On the contrary, as the surfacewas made increasingly S-rich, numerous midgap surface statesappeared in the DOS, Fig. 19. Time-domain ab initio simulationsdemonstrate that these midgap states should provide an effi-cient nonradiative recombination pathway for electrons and holes[46,104]. Thus, it was concluded that in the S-rich QDs, efficientnonradiative recombination was mediated by the midgap surfacestates, quenching fluorescence. Linear response TDDFT calcula-tions were performed to calculate the absorption spectra, and itwas found that the midgap states were largely optically inactive,explaining the very little change in the absorption spectra observedexperimentally.

5.6. Ligands contribute high-frequency phonon modes

The hydrogen atoms terminating Si and Ge QDs provide goodexamples of the contributions that ligands make to excitation

confinement makes the electron and hole DOS more symmetric in Siand Ge QDs compared with bulk. Despite the symmetric DOS, elec-trons decay more quickly than holes as shown in Fig. 20 for Si. This

A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181 177

F show

a a non

A 2.

oepsFpdi

ig. 19. (Top panel) DOS and geometry of the perfect Cd33S33 QD. Panels a-c and d-f

series of midgap states span the band-gap, facilitating luminescence quenching vi

dapted from Nano Lett. 2012, 12, 4465. Copyright American Chemical Society 201

bserved asymmetric relaxation can be explained by the strongerlectron–phonon coupling in the CB that stems from a stronger cou-ling to high-frequency phonons linked with the Ge H and Si Hurface bonds. As can be seen in the inserts in the right panel of

ig. 21, the CB states for Si are more delocalized on the surface com-ared to the VB states. The remainder of Fig. 21 displays the spectralensities of the phonon modes that couple to CB and VB states

n Si29H24. The low-frequency motions originate from Si-Si bond

the DOS for increasingly S- and Cd- rich systems, respectively. In the S-rich systems,radiative relaxation.

vibrations and vary little between VB and CB, left panels. However,an asymmetry is clearly observed in the high-frequency compo-nents that stem from Si H surface bonds (right panels of Fig. 21).The electrons in the CB couple much more strongly to these high-

frequency modes, as compared to the counterpart holes in the VB.The higher frequency modes have higher velocities, acceleratingcharge relaxation through the velocity dependence of NA coupling,Eq. (8), allowing electrons to relax faster than holes [39].

178 A.J. Neukirch et al. / Coordination Chemistr

Fig. 20. (Top panel) DOS of Si29H24. (Bottom panel) Relaxation times of electrons(squares) and holes (diamonds) in the Si QD. In spite of the nearly symmetric DOS,the electron relaxation is significantly faster than the hole relaxation. The calculatedrelaxation times are correlated with the state energy spacing averaged over therelaxation energy range. The slope of the correlation depends on the charge-phononcoupling.

Adapted from Dalton Trans, 2009, 45, 10069. Copyright Royal Chemical Society 2009.

Fig. 21. Frequencies of the phonon modes that couple to the CB and VB states in Si29H24. TSi QDs, electrons decay faster than holes because they couple more strongly to high-frequby the larger surface delocalization of CB orbitals compared to VB orbitals.

Adapted from Dalton Trans, 2009, 45, 10069. Copyright Royal Chemical Society 2009.

y Reviews 263– 264 (2014) 161– 181

A more recent study of CdSe confirms that ligands tendto increase relaxation rates [110]. Photoexcited dynamics werestudied in a Cd33Se33 cluster passivated by commonly used lig-ands, a trioctylphosphine oxide (TOPO) mimic (OPMe3) and aprimary amine (NH2Me), and compared to the ligand-free system.It was found that the ligands introduce a manifold of high-energyhybridized electronic states. Fig. 22 shows that relaxation occursnoticeably faster in the ligated QDs. This fast relaxation in the pas-sivated dots originates from the strong electron–phonon couplingsprovided by the hybridized states. The hybridized states coupleto both high-frequency vibrations associated with the ligands andlow-frequency phonon modes of the cluster core. Even if an elec-tron or a hole is not initially excited into a hybridized orbital, the NAcouplings between the electronic states forces the charge to jump toone of the neighboring hybridized orbitals, thus opening new relax-ation channels. These hybridized states start closer to the band-gap

in the NH2Me passivated dot compared to the OpMe3 passivatedone, allowing the amine litigated system to relax faster at the 2.5Eg excitation, Fig. 22. At 3Eg the hybridized states in both VB and CBcontributed equally to electron–phonon couplings in both ligated

Fig. 22. Evolution of exciton energy in the ligand-free and ligated Cd33Se33 QDsinitially excited to 2.5 times the band-gap energy. Ligands notably accelerate therelaxation process.

Adapted from ACSNano, 2012, 6, 6515. Copyright American Chemical Society 2012.

he Inserts on the right show the HOMO and LUMO orbital densities. In both Ge andency surface ligand modes. The stronger coupling to the ligand modes is explained

A.J. Neukirch et al. / Coordination Chemistry Reviews 263– 264 (2014) 161– 181 179

Ψ0

Dephasing El-Ph Relaxation

El-Ph Relaxation

Impact

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Exciton

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+

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Ψ(ME ,SE )Ψ(ME )

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Ψ(SE ) Ψ(ME ) Ψ(SE )

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10 fs 100 fs 1 ps 100 ps

hv

El-Ph Relaxation

Dephasing

hv

Fig. 23. A comprehensive scheme of excited state dynamics in QDs. Quantum-mechanical superpositions of SE and ME states are generated upon absorbing photons withenergies greater than two times the band-gap. Within 10 fs, the quantum coherence between the SEs and MEs is destroyed. Additional MEs are created by high energyexcitons through impact ionization. Continuously excitons lose energy on a picosecond timescale via electron–phonon interactions. MEs can undergo Auger recombinationr ate re

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dapted from Accounts of Chemical Research, http://dx.doi.org/10.1021/ar3002365

ystems, and the relaxation rates became nearly identical betweenhe two systems at this excitation energy. It may be interestingo apply these methods to the experimentally created magic sizeddSe clusters terminated with thiols and amines [58,138].

. Conclusions

Valuable insights into semiconductor QD properties can bestablished from ab initio modeling of the excited-state dynamics inhe energy and time domains. Elastic and inelastic electron–phononcattering, and Auger-type processes dominate the excitation andharge dynamics in nanoscale systems. The atomistic methodseviewed in this paper are ideal for modeling defects, ligands,harges, dopants, dangling bonds and other chemical aspects of QDtructure. These details greatly influence the excited-state dynam-cs of QDs and should be taken into account when developinganoscale devices. The studies lead to an enhanced fundamentalnderstanding of the QD properties, creating valuable insights fordvancing applications.

Many-body correlations between electrons, arising fromoulombic interactions, open up the possibility of MEG upon thebsorption of a single photon. Configuration interaction methods,uch as SAC-CI, directly account for these correlations. The naturef the excited states of different materials, such as CdSe, PbSe, andi, depend on both quantum confinement and dielectric shielding.t has been found that QD charging gives rise to intraband transi-ions that exhibit little optical activity but dramatically modify theature and energy of the excited electronic states. Charging alsolue-shifts absorption spectra and increases ME thresholds. Simi-

ar changes are seen with QDs containing dopants and unsaturatedhemical bonds. Surface reconstruction is particularly important inhe latter case, since it significantly changes QD band-gaps.

Dephasing and relaxation are the two distinct processes thatre induced by electron–phonon interactions. Superpositions oflectronic states, created by the Coulomb interaction during pho-oexcitation and the subsequent time evolution, dephase intoncoherent mixtures of states on a timescale of tens of femtosec-nds, Fig. 23. Dephasing is ultrafast, if it includes electronic statesith considerably different energies and spatial densities. Exam-les of this include superpositions of SEs and MEs, and ground andxcited states. ME fission into independent SEs occurs by dephas-ng that is far slower, provided that MEs are formed by SEs that are

imilar in energy, for instance, in the lowest energy bi-exciton.

Photo-driven exciton multiplication and phonon-induced pure-ephasing occur very quickly in semiconductor QDs. A combinationf impact ionization and exciton thermalization allows the system

quires hundreds of picoseconds to nanoseconds.

right American Chemical Society 2012.

to reach the band edge within 10 ps in most systems, Fig. 23. Untilvery recently, II had only been modeled in the time-independentapproach. The newly developed sparse matrix techniques dis-cussed in this review allow for the study of time-domainsimulations of Auger processes that inherently involve many elec-tronic states. These studies have helped to corroborate standardrate theory models that are based on perturbation theory andassume exponential decay. Since the coupling between SEs andMEs permits both forward and backward processes, the direction-ality is highly influenced by the relative DOS for SEs and MEs. Athigh energies, SEs generate MEs, while at low energies MEs anni-hilate to form SEs. The timescale of transition from the II to theexciton annihilation regime is determined by the electron–phononrelaxation. The developed technique includes electron–phononrelaxation simultaneously with the Auger processes, and incorpo-rates phonon-assisted Auger events.

TDDFT combined with NAMD simulates the complex evolutionsof coupled electronic and vibrational degrees of freedom as it occursin nature. Time-domain modeling of electron–phonon relaxationexplained two seemingly contradicting experimental observations:despite large spacing between optical lines, a phonon bottleneckof electron–phonon relaxation exists only under very special con-ditions. QD spectra consist of multiple individual excitations thatcombine into distinct bands. Even though relatively few excitationsare strongly optically active, most excited electronic states par-ticipate in phonon relaxation. Close to the band edges, quantumdecoherence effects have to be included into the quantum-classicalrelaxation dynamics. This semiclassical technique predicts a bottle-neck, which can be found experimentally when a shell is added tothe QD to separate electron and hole states. Through these studies,it has been established that ligand and shell layers saturate surfacedangling bonds and alter high-energy regions of CB and VB. Whileshells can isolate electrons and holes from high-frequency ligandmodes, ligands do create strong electron–phonon coupling whichspeeds up electron–phonon relaxation. Studies described in thisreview elucidate processes dominant in specific systems, allowingfor rational design of new applications

With improved computational power, the existing methodswill be implemented on larger QDs from a more diverse pool ofmaterials. Systems involving realistic ligands, surface defects andviable core/shell compositions are currently under investigation. Abetter understanding of luminescence quenching and the phonon

bottleneck can be obtained by doing longer simulations. Pho-tovoltaic assemblies of QDs interacting with molecular chro-mophores, polymers, and inorganic semiconductors present newtheoretical challenges. Ab initio approaches in the time and energy

1 emistr

des

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80 A.J. Neukirch et al. / Coordination Ch

omains will significantly progress our understanding of QD prop-rties that govern solar energy harvesting, photovoltaics, lasing,pintronics, and biological imaging.

cknowledgements

The authors express deepest gratitude to the colleagues andollaborators for their contributions to the studies reviewed inhis work, including, in alphabetic order: Bradley Habenicht, Huaao, Liangliang Chen, Colleen Craig, Sean Fischer, Christine Isborn,eather Jaeger, Hideyuki Kamisaka, Dmitri Kilin, Svetlana Kilina,iaosong Li, Run Long, Angeline Madrid, Xiulin Ruan, Taizhi Tan,nd Koichi Yamashita. The research was supported by grants fromhe US National Science Foundation, US Department of Energy, andAKENHI of Japan.

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