Effect of Aspect Ratio on Multiparticle Auger Recombination inSingle-Walled Carbon Nanotubes: Time Domain AtomisticSimulationSougata Pal,†,∥ David Casanova,‡,§ and Oleg V. Prezhdo*,∥

†Department of Chemistry, University of Gour Banga, Malda 732103, India‡Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC) 20018 Donostia,Euskadi, Spain§IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Euskadi, Spain∥Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States

ABSTRACT: Many-particle Auger-type processes are com-mon in nanoscale materials due to a combination of highdensities of states that can support multiple excitations andsubstantial Coulomb coupling between charges enhanced byquantum confinement. Auger decay dynamics in (10,5)semiconductor carbon nanotubes (CNT) with different aspectratios and particle densities are simulated in time domain usingglobal flux surface hopping, recently developed andimplemented within Kohn−Sham tight-binding density func-tional theory. Despite an increasing density of states, themultiparticle Auger recombination rate decreases in longerCNTs. The atomistic simulation shows that the effect isdirectly related to the coupling between electronic states,which decreases as the aspect ratio becomes larger. The dependence on tube length is stronger for three-exciton than two-excitonrecombination and the calculated time scale ratio approaches the experimental value measured for long CNTs. Phonon-assistedtransitions play a particularly important role during Auger recombination. Electron−phonon relaxation is faster than therecombination, and Auger transitions are assisted by phonons over a range of frequencies up to the G-mode. The involvement ofphonons strongly enhances the probability of transitions involving asymmetric electron−hole pairs. The time-domain atomisticsimulation mimics directly time-resolved optical experiments and provides a detailed, systematic analysis of the phonon-assistedAuger dynamics.

KEYWORDS: Carbon nanotubes, multiparticle processes, Auger recombination, tight-binding density functional theory,nonadiabatic molecular dynamics

Carrier multiplication (CM) leading to generation ofmultiple electron−hole (e−h) pairs by a single photon innanosystems motivates fundamental studies and potentialapplications for highly efficient third-generation photovol-taics.1−4 Auger recombination (AR), which can be regarded asthe inverse of CM, and other Auger-type processes play a majorrole in determining exciton dynamics in these nanosystems.2,5−8

In semiconductor materials, Auger processes open up a newnonradiative recombination channel in which the e−hrecombination energy is transferred to a third particle (an eand or an h) that is excited to a higher energy state.9 Such ARprocesses involve multicarrier interactions and depend stronglyon dimensionality and size of the nanostructure.10,11 Auger-typephenomena are responsible for energy exchange betweenelectrons and holes, breaking the phonon bottleneck to theelectron relaxation.12,13 Energy exchange between electron andhole produces a newmechanism of charge transfer, that is, Auger-assisted electron transfer, that circumvents the Marcus inverted

regime in the transfer rate dependence on the electron drivingforce.14,15

Because of kinematic restrictions imposed by energy andmomentum conservations, Auger processes are stronglyinhibited in bulk semiconductors.16,17 However, Auger typephenomena are much more prominent in quantum confinedmaterials due to relaxation of the momentum conservation ruleand increased overlap of carrier wave functions.3,18−20 Nanoscalematerials span the gap between bulk and molecular systems andexhibit properties of both. Similar to molecules, electrons andholes are confined to small volumes in nanoscale systems and,therefore, interact much more strongly than in the bulk. Just as inthe bulk and in contrast to molecules, nanoscale materials havehigh densities of electronic states. Both strong interaction and

Received: July 24, 2017Revised: November 27, 2017Published: November 30, 2017

Letter

pubs.acs.org/NanoLettCite This: Nano Lett. 2018, 18, 58−63

© 2017 American Chemical Society 58 DOI: 10.1021/acs.nanolett.7b03150Nano Lett. 2018, 18, 58−63

Dow

nloa

ded

via

UN

IV O

F SO

UT

HE

RN

CA

LIF

OR

NIA

on

Nov

embe

r 7,

201

9 at

16:

21:2

9 (U

TC

).Se

e ht

tps:

//pub

s.ac

s.or

g/sh

arin

ggui

delin

es f

or o

ptio

ns o

n ho

w to

legi

timat

ely

shar

e pu

blis

hed

artic

les.

pubs.acs.org/NanoLetthttp://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.nanolett.7b03150http://dx.doi.org/10.1021/acs.nanolett.7b03150

high electronic density of states (DOS) are essential to obtain alarge transition rate.3 Quantum confinement enhances e−eCoulomb coupling much more than electron−phonon coupling,and as a result, the decay dynamics of multiparticle states aredominated by Auger processes in such materials.16 As researchand device fabrication move forward, a clear understanding ofAuger processes involving multicarrier interactions is pivotal tothe development of efficient photovoltaic, photocatalytic,electronic, spintronic, and related devices.Multiparticle AR occurs in a sequence of quantized steps

starting from amultiparticle state N toN-1, N-2,.....3, 2 and finallyto the 1-e−h pair state known as single exciton. The quantizedrecombination dynamics of multiparticle states can be describedby a set of discrete recombination constants, τ3, τ2..., character-istic of the decay lifetime of the 3-, 2-, . . . e−h pair states16,21 (seeschematic in Figure 1a). Multiparticle AR is well studied in

semiconductor quantum dots (QDs). Experiments establish adiameter dependence of multiparticle Auger lifetimes, incorrelation with the Coulomb interaction that scales as 1/Rwith the QD radius R.2,11,16 A linear dependence of multiparticlelifetimes on QD volume is observed in spherical CdSe QDs.16,22

For a fixed QD diameter, higher order excitonic states decaymuch faster, indicating that AR is much more effective at highcarrier densities. In particular, it has been found that themultiparticle lifetimes for 4-, 3- and 2-e−h pairs follow the fixedset of ratios 0.25:0.44:1 (τ4/τ3/τ2).

16,21 Themultiparticle lifetimeratios remain fixed as the volume of the spherical QDs is varied.All multiparticle decay constants increase linearly with increasingnanoparticle volume, provided that the particle aspect ratio isunchanged. The shape and dimensionality of a nanocrystal alsohave a great influence on the multiparticle decay times.21

Considering a series of elongated nanorods, Htoon et al. havefound that the ratio between biexciton (τ2) and triexciton (τ3)lifetimes, τ2/τ3, gradually decreases as the aspect ratio of theparticle increases.21

In contrast to inorganic semiconductors in which multiparticleAR dynamics have received considerable interest, there arerelatively few works devoted to Auger processes in single-wall

carbon nanotubes (SWCNT).23 Because of very high aspect ratioand low density of defects, SWCNTs provide an excellentphysical realization of a 1D confined system.24,25 E−h interactionenergies are quite large in SWCNTs in the range of 200 to 400meV,26,27 and as a consequence not only single excitons but alsohigher excitonic states are detectable and quite stable at roomtemperature.28 Carrier−carrier interactions in SWCNTs lead tonumerous interesting physical phenomena, including highlyefficient intraband relaxation via e−h energy transfer and ultrafastmultiparticle decay via AR.7,29 Huang et al. studied multiparticleAR on SWCNTs having a very high aspect ratio (approximately380 nm SWCNT length) and found the ratio of τ3/τ2 to be in itslowest limit of 1.5.30

The current Letter presents the first time-domain atomisticstudy of multiparticle AR dynamics in SWCNTs. Self-consistentcharge density functional tight binding (SCC-DFTB)theory31−34 combined with nonadiabatic molecular dynamics(NA-MD) allows us to mimic most directly the time-resolvedspectroscopic experiments and include both e−h and electron−phonon scattering events. Generally, the AR rate depends on thecoupling strength and the density of final states. The two factorsplay different roles in variation of the AR time with the number ofcharged particles and the SWCNT length. The single andmultiparticle electronic DOS grow with increasing particlenumber and SWCNT length. In contrast, the coupling decreaseswith the SWCNT length. We demonstrate that the couplingdecrease is more important than the electronic DOS increase,such that the AR dynamics is faster in shorter SWCNTs. Thedependence on the tube length is stronger for AR involving moreparticles, and hence the τ2/τ3 ratio decreases gradually with tubelength, approaching the experimental value found for longSWCNTs.30 We show that phonons play an important roleduring AR. Electron−phonon relaxation is faster than AR, andAR transitions are accompanied by energy losses to phonons ofvarious frequencies. The probability of phonon-assisted ARdecreases rapidly for energies exceeding the G-mode frequency,suggesting that phonon-assisted AR is first order in phononcoupling. Because phonons couple asymmetric e−h pairs, theprobability of asymmetric transitions is enhanced.Despite multiple ab initio and tight-binding calculations of

SWCNT electronic structure and excitations, as well as adiabaticground state MD studies of SWCNT interactions with othernanoscale and biological systems,10,35,36 NA-MD simulationscombining and extending the two techniques to model relaxationdynamics of photoexcited SWCNTs are very scarce. Habenichtet al. investigated phonon-induced intraband charge relaxation,intersystem crossing, and e−h recombination in severalSWCNTs using ab initio NA-MD.37−41 AR dynamics requiresignificantly larger calculations due to the strong dependence ofstate basis on the number of particles.42,43 In addition, fewestswitches surface hopping (FSSH)44,45 that is the most popularNA-MD technique excludes superexchange processes, in whichthe initial and final states are coupled via virtual high energy statesand which contribute notably to many-particle Auger dynam-ics.46,47 In order to circumvent these limitations, we havedeveloped the global flux surface hopping (GFSH) technique46

for NA-MD simulations and have implemented it48 within SCC-DFTB31−34 using the PYXAID (PYthon eXtension for Ab InitioDynamics) code.49,50 GFSH is a very simple generalization ofFSSH to higher order processes, such as superexchange andmultiparticle transitions. The simulations are performed in theadiabatic representation, which is the natural outcome ofatomistic electronic structure calculations. Adiabatic states are

Figure 1. (a) Diagram illustrating quantized steps involved inmultiparticle AR. Red and blue circles indicate electron and hole,respectively. (b) Geometry of the (10, 5) nanotube with 4.35 aspectratio at 300 K.

Nano Letters Letter

DOI: 10.1021/acs.nanolett.7b03150Nano Lett. 2018, 18, 58−63

59

http://dx.doi.org/10.1021/acs.nanolett.7b03150

eigenstates of the electronic Hamiltonian for fixed nuclearpositions, such that all off-diagonal Coulomb coupling terms arediagonalized out. As a consequence, the Coulomb (diabatic)coupling is included implicitly in the NA coupling that iscomputed during the NA-MD simulation.47

The time-dependent electron density is represented in thebasis of Kohn−Sham (KS) orbitals as

∑ψ ϕ| ⟩ = | ⟩=

r R t c t r R t( ; ( )) ( ) ( ; ( ))i

N

i i0 (1)

where ci(t) are time-dependent expansion coefficients, andϕi(r;R(t)) are adiabatic wave functions representing electronicstate i. The adiabatic wave functions depend parametrically onthe classical nuclear trajectory R(t). The time evolution ofcoefficient ci(t) is obtained by solving the time-dependentSchrödinger equations for the KS orbital expansion coefficients

∑ ωδℏ = − ℏ=

ic t

ti d c t

d ( )d

( ) ( )i

j

N

i ij ij j0 (2)

ϕϕ

=∂

∂d r R t

r R t

t( , ( ))

( , ( ))ij i

j

(3)

where dij is the NA coupling between states i and j, and ωi isenergy of adiabatic state i. The coefficients and NA coupling areutilized to calculate the transition probabilities in the GFSHsimulation. The many-particle generalization of the aboveequations and other details are provided in refs 49 and 50.The electronic structure calculations, geometry optimization,

and adiabatic MD are carried out using the SCC-DFTB methodas implemented in the DFTB+ code.31,51 The parameter set(Slater−Koster files) used in the calculation have been testedextensively for a broad range of compounds and can be foundelsewhere.33 The simulations are performed using periodicboundary conditions with 30 Å of vacuum added in the directionperpendicular to the axis of the tubes. The structures are fullyoptimized at 0 K and then heated to 300 K with repeated velocityrescaling. Five ps microcanonical MD trajectories are generatedfor each tube using the Verlet algorithm52 with a 1 fs time-stepand Hellman-Feynman forces. At each snapshot, the energies ofthe KS orbitals and the NA coupling constants are calculated, andthese time-dependent quantities are used to perform NA-MD.49,50

Motivated by the experiment of Huang et al.30 and aiming tominimize the size of the electronic basis and the simulation cell,we select the (10,5) semiconductor SWCNT with 1.05 nmdiameter. Following ref 21, we denote the aspect ratio by ε. Inorder to investigate the effect of the SWCNT aspect ratio on

Auger lifetime, we chose three (10,5) nanotubes having the samediameter and different aspect ratios ε = 1.08, 3.25, and 4.35. Thetotal number of carbon atoms in the simulation cells for the threesystems is 140, 420 and 560, respectively. The structure of the(10,5) SWCNT with ε = 4.35 is shown in Figure 1b.Figure 2 details the biexciton and triexciton Auger decay

dynamics in the (10,5) nanotubes for three different ε. Thedecays of the biexciton state populations due to AR are shown inFigure 2a. The curves are fitted with the sum of the Gaussian andexponential components, A exp[−t/τexp] + (1− A) exp [−0.5(t/τGau)

2], and the computed biexciton Auger lifetimes, Aτexp + (1−A)τGau, are shown in Table 1. The observed trend in the decay

dynamics is perhaps surprising, because the single andmultiparticle electronic DOS increases progressively with thenanotube length and availability of more final states shouldfacilitate faster AR. But our results predict a significant increase ofτ2, from 11.70 to 23.03 ps, as ε of the nanotube increases from1.08 to 4.35. We further simulated 3-e−h pair AR dynamics,Figure 2b. The simulated decay curves are also composed of twocomponents, Gaussian and exponential, and the computedtriexciton lifetimes are listed in Table 1. Despite the increasingelectronic DOS, a significant increase of τ3, from 5.1 to 11.28 ps,is also found in this aspect ratio regime. All the decay constants(τ2 and τ3) are in picoseconds, which is consistent with theexperimental observations.30 The calculations show that for afixed ε, AR of higher-order excitons proceeds faster. This resultagrees with the experiments,21,30 providing a validation for ourcomputational methodology. The more rapid AR dynamics oftriexcitons, compared to biexcitons, can be attributed to thehigher density of final states for the triexciton annihilationprocess.The dependence of the biexciton and triexciton annihilation

times on the aspect ratio have different slopes, Figure 2c. Thedependence is stronger for triexcitons. The variation of the τ2/τ3ratio as a function of ε agrees with the experimental findings ofHtoon et al. for CdSe nanorods.21 They observed the τ2/τ3 ratioclose to 2.25 for low ε, ε∼ 1. As ε increased to 8 and even higher,the τ2/τ3 ratio gradually decreased and approached the limiting

Figure 2.Auger decay dynamics of (a) 2-e−h pairs and (b) 3-e−h pairs in the (10, 5) nanotubes with different aspect ratios (ε), ε = 1.08 (black), ε = 3.25(red), and ε = 4.35 (blue).The multiparticle lifetime increases with increasing ε, despite increasing electronic DOS, because the coupling decreases,Table 1. (c) Lifetime of 2-e−h pairs (τ2, black) and 3-e−h pairs (τ3, red) as functions of the nanotube aspect ratio. Note difference in the slopes.

Table 1. Two-e−h Pair (τ2) and 3-e−h Pair (τ3) DecayConstants, τ2/τ3 Ratio, and Root-Mean-Square NA Couplingbetween Initial and Final States As Functions of NanotubeAspect Ratio (ε)

ε τ2 (ps) τ3 (ps) τ2/τ3

coupling forelectron states

[meV]coupling for holestates [meV]

1.08 11.70 5.10 2.29 6.38 5.113.25 18.80 8.90 2.11 3.96 3.764.35 23.03 11.28 2.04 2.07 2.23

Nano Letters Letter

DOI: 10.1021/acs.nanolett.7b03150Nano Lett. 2018, 18, 58−63

60

http://dx.doi.org/10.1021/acs.nanolett.7b03150

value of 1.5. τ2/τ3 = 2.29 obtained in our calculations for ε = 1.08is very close to the experimentally determined τ2/τ3 = 2.25 for ε∼ 1. The calculated ratio decreases with increasing ε to 2.11 (ε =3.25) and 2.04 (ε = 4.35), approaching the experimentallydetermined limiting value of 1.5 for high ε. Because the aspectratio dependence of the AR dynamics in SWCNTs is analogousto that in CdSe nanocrystals, one can argue that Auger-typeprocesses exhibit properties that are similar for SWCNTs,inorganic nanocrystals, and other low-dimensional systems. Thedimensionality, aspect ratio and volume dependence of AR ratescan be characterized by scaling laws for exciton collisionfrequency and Coulomb interaction strength. In highly confined0D systems, the probability of multiparticle collisions is high, andthree-particle processes dominate.21 In 1D systems, includingSWCNTs, multiparticle collisions are less likely, and AR isbimolecular. Recent analysis of AR rates in 2D nanoplateletsshowed53 that the biexciton lifetimes do not scale with volume, asis the case for 0D crystals. The linear increase of biexciton ARwith nanoplatelet lateral area reflected the 1/area dependence ofthe binary collision frequency for 2D excitons, while thethickness-dependent biexciton recombination was attributed tostrong dependence of Coulomb interaction on quantumconfinement.53

In order to understand why the AR slows down in longerSWCNTs, even though the electronic DOS increases, we analyzethe corresponding coupling matrix elements. Note that if thesingle-particle DOS increases, the corresponding many-particleDOS increases as well. Figure 3a−c presents contour plots of theroot-mean-square coupling between the electron and hole statesas a function of state energy for the (10,5) SWCNTs with the

three different aspect ratios. The data show that the couplingmagnitude decreases as the tube aspect ratio grows. Thesimulations predict strong coupling for e−h pairs that areasymmetric in energy. Involvement of phonon modes enhancethe asymmetric coupling channel54 and allow phonon-assistedAR. Figure 3d,e presents the root-mean-square coupling withinthe manifolds of electron and hole states, respectively, for thethree (10,5) SWCNTs. Similarly to the e−h couplings, Figure 3a-c, the e−e and h−h couplings decrease with increasing SWCNTaspect ratio.The e−e and h−h coupling (Figure 3d,e) is an order of

magnitude stronger than the e−h interaction (Figure 3a-c),indicating that many transitions within the electron and holemanifolds of states take place before electrons and holesannihilate. This is in agreement with the experimental literatureindicating that charge thermalization is the fastest process takingplace immediately after photoexcitation.55,56 Our data show thatfor the same number of charges, the e−e and h−h scattering andthermalization should be slower in longer SWCNTs. The averagecoupling between the initial (photoexcited) and final (near bandgap) states of electrons and holes is presented in Table 1. Thecoupling between the states separated by a large energy, Table 1,is weaker than the coupling averaged over all pairs of states(Figure 3d,e), suggesting that the e−e and h−h thermalizationoccurs by transitions between states that are close in energy.Similarly to other coupling measures, the coupling magnitudesshown in Table 1 decrease significantly for longer SWCNTs,explaining the slower AR dynamics with increasing ε, Figure 2.NA-MD provides a rather unique capability to include

electron−phonon interactions, with phonons treated anhar-

Figure 3.Contour diagram for the average root-mean-square NA coupling between electron and hole for (a) ε = 1.08, (b) ε = 3.25, and (c) ε = 4.35. Theelectron and hole energies are with respect to the conduction and valence band edges, respectively. The coupling decreases with increasing aspect ratio ε.The coupling is higher for asymmetric excitations. Root-mean-square NA coupling within (d) hole and (e) electron state manifolds with differentnanotube aspect ratios.

Nano Letters Letter

DOI: 10.1021/acs.nanolett.7b03150Nano Lett. 2018, 18, 58−63

61

http://dx.doi.org/10.1021/acs.nanolett.7b03150

monically and nonperturbatively, although classically. Figure 4presents the probability distribution of electronic energy loss tophonons during the simulated 2-e-h and 3-e-h AR transitions.The data show that the majority of AR events are accompaniedby energy losses to phonons. Electron−phonon energydissipation is a critical part of the AR dynamics.57 Thesimulations demonstrate that most phonons available inSWCNTs participate in phonon-assisted AR, with the probabilitydensity ranging from low frequencies to the high energy G-phonons. The phonon-assisted Auger transition probability ishigh in Figure 4 in the region between −0.2 to 0 eV, becauseenergy is transferred from charges to phonons (hence negativevalues), and because the highest frequency phonon mode, theC−C stretching G-mode, is at 1600 cm−1 which corresponds to0.2 eV. It has been established previously that the G-phononcouples particularly strongly to the electronic subsystem.37,38,58

The tails extending below −0.2 eV correspond to multiphononAR processes that are less probable. The data tail above 0 eVdemonstrates a small probability of exciting a phonon during AR.The phonon contribution to the Auger processes is important

in several ways. Phonons broaden the range of coupled states inboth energy and momentum spaces, lifting the strict energy andmomentum conservation requirements present for purelyelectronic transitions. Electronic energy is lost to phononsalready during Auger dynamics, accelerating equilibration ofelectron and phonon subsystems, and contributing to heating ofnanoscale devices. The current simulation shows that thephonon-assisted Auger scattering channel should be includedinto interpretation of experimental data on nonequilibriumelectron−phonon dynamics, for instance, by modification of thecommonly used two-temperature model.59 The current workconfirms the earlier observation by Shabaev et al.54 thatinvolvement of phonon modes facilitates coupling betweenasymmetric e−h pairs. It is interesting to note that the coupling isenhanced between electron and hole states that differ in energyby 50 to 200 meV, as can be seen in Figure 3a−c and most clearlyin Figure 3a. This observation is directly related to the fact thatthe SWCNT phonon spectrum stops at 0.2 eV. The enhance-ment is seen for states that differ by at least one phononquantum, with high frequency phonons playing the mostimportant role.The current work contains several limitations. The SWCNT

fragments considered here are short compared to the e−hcorrelation length60−62 and exclude effects arising at SWCNTends.63 By confining electrons and holes within spaces that aresmaller than their natural coherence lengths we enhance theirinteraction relative to that in long SWCNTs. End effects can beparticularly important for e−h annihilation, because they providetrap states and strong charge-phonon coupling, and because

charges are very mobile in SWCNTs and can reach the endsquickly.64,65 At the same time, transport in very long tubes isdiffusion limited, requiring modified modeling methodology66

and additional analysis of the experimental data, for example,fitting to stretched-exponentials. Nonexponential Auger decaycan also occur at long times that are greater than the system’slifetime.67

In summary, we have presented a comprehensive study ofmultiparticle AR dynamics in carbon nanotubes with a range ofaspect ratios, by performing a time-domain atomistic non-adiabatic molecular dynamics simulations for the first time. Ourresults show that the bi- and triexciton lifetimes increase in longertubes, even though the density of product states grows. Thisresult is rationalized by a faster decrease in the nonadiabaticcoupling for the multiparticle transitions in longer tubes.Biexcitons live longer than triexcitons, and the bi- to triexcitonlifetime ratio approaches the experimentally determined long-tube limit. Electron−phonon coupling is essential for the AR. Itlowers the overall electronic energy, driving the systems intofewer-particle and, eventually, ground state. The majority of ARtransitions are phonon-assisted and are first order in coupling tophonons. A broad range of phonons from low frequencies up tothe high frequency G-mode participate in the dynamics. Phononparticipation enhances coupling of e−h pairs that are asymmetricin energy, opening up additional energy exchange channels andaccelerating equilibration. Our results agree well with theavailable experimental observations, demonstrate generality ofAR scaling laws for SWCNTs andQDs, and provide new insightson phonon-assisted Auger processes.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: prezhdo@usc.edu.ORCIDOleg V. Prezhdo: 0000-0002-5140-7500NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSS.P. acknowledges financial support from SERB-DST, Govern-ment of India through Project Ref. No. CS-085/2014. O.V.P.acknowledges financial support from the U.S. Department ofEnergy, Grant DE-SC0014429 and is grateful to the RussianScience Foundation, project No. 14-43-00052, base organizationPhotochemistry Center RAS for hospitality during manuscriptpreparation. D.C. acknowledges financial support from theSpanish Government MINECO/FEDER, Project CTQ2016-80955.

Figure 4. Probability densities of phonon-assisted Auger transitions (a) from 2-e−h to 1-e−h, and (b) from 3-e−h to 2-e−h, as functions of transitionenergy.

Nano Letters Letter

DOI: 10.1021/acs.nanolett.7b03150Nano Lett. 2018, 18, 58−63

62

mailto:prezhdo@usc.eduhttp://orcid.org/0000-0002-5140-7500http://dx.doi.org/10.1021/acs.nanolett.7b03150

■ REFERENCES(1) Gao, J.; Fidler, A. F.; Klimov, V. I. Nat. Commun. 2015, 6, 8185.(2) Padilha, L. A.; Stewart, J. T.; Sandberg, R. L.; Bae, W. K.; Koh, W.-K.; Pietryga, J. M.; Klimov, V. I. Nano Lett. 2013, 13, 1092−1099.(3) Rabani, E.; Baer, R. Nano Lett. 2008, 8, 4488−4492.(4) Schaller, R. D.; Agranovich, V. M.; Klimov, V. I. Nat. Phys. 2005, 1,189−194.(5) Li, Q. Y.; Lian, T. Q. Nano Lett. 2017, 17, 3152−3158.(6) Makarov, N. S.; Guo, S. J.; Isaienko, O.; Liu, W. Y.; Robel, I.;Klimov, V. I. Nano Lett. 2016, 16, 2349−2362.(7) Wang, F.; Dukovic, G.; Knoesel, E.; Brus, L. E.; Heinz, T. F. Phys.Rev. B: Condens. Matter Mater. Phys. 2004, 70, 241403.(8) Jain, A.; Voznyy, O.; Hoogland, S.; Korkusinski, M.; Hawrylak, P.;Sargent, E. H. Nano Lett. 2016, 16, 6491−6496.(9) Landsberg, P. T. Recombination in semiconductors; CambridgeUniversity Press: 2003.(10) Wang, F.; Wu, Y.; Hybertsen, M. S.; Heinz, T. F. Phys. Rev. B:Condens. Matter Mater. Phys. 2006, 73, 245424.(11) Klimov, V. I. Annu. Rev. Phys. Chem. 2007, 58, 635−673.(12) Efros, A. L.; Kharchenko, V. A.; Rosen, M. Solid State Commun.1995, 93, 281−284.(13) Kilina, S. V.; Neukirch, A. J.; Habenicht, B. F.; Kilin, D. S.;Prezhdo, O. V. Phys. Rev. Lett. 2013, 110, 6.(14) Zhu, H.; Yang, Y.; Hyeon-Deuk, K.; Califano,M.; Song, N.;Wang,Y.; Zhang,W.; Prezhdo, O. V.; Lian, T.Nano Lett. 2014, 14, 1263−1269.(15) Hyeon-Deuk, K.; Kim, J.; Prezhdo, O. V. J. Phys. Chem. Lett. 2015,6, 244−249.(16) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D.; Leatherdale, C.A.; Bawendi, M. G. Science 2000, 287, 1011−1013.(17) Carlson, L. J.; Krauss, T. D. Acc. Chem. Res. 2008, 41, 235−243.(18) Baer, R.; Rabani, E. Nano Lett. 2010, 10, 3277−3282.(19) Baer, R.; Rabani, E. Nano Lett. 2012, 12, 2123−2128.(20) Achermann, M.; Bartko, A. P.; Hollingsworth, J. A.; Klimov, V. I.Nat. Phys. 2006, 2, 557−561.(21) Htoon, H.; Hollingsworth, J.; Dickerson, R.; Klimov, V. I. Phys.Rev. Lett. 2003, 91, 227401.(22) Klimov, V. I. Annu. Rev. Condens. Matter Phys. 2014, 5, 285−316.(23) Hagen, A.; Steiner, M.; Raschke, M. B.; Lienau, C.; Hertel, T.;Qian, H.; Meixner, A. J.; Hartschuh, A. Phys. Rev. Lett. 2005, 95, 197401.(24) Wang, F.; Cho, D. J.; Kessler, B.; Deslippe, J.; Schuck, P. J.; Louie,S. G.; Zettl, A.; Heinz, T. F.; Shen, Y. R. Phys. Rev. Lett. 2007, 99, 227401.(25) Kanemitsu, Y. Acc. Chem. Res. 2013, 46, 1358−1366.(26) Avouris, P.; Freitag, M.; Perebeinos, V. Nat. Photonics 2008, 2,341−350.(27) Wang, F.; Dukovic, G.; Brus, L. E.; Heinz, T. F. Science 2005, 308,838−841.(28) Kammerlander, D.; Prezzi, D.; Goldoni, G.; Molinari, E.;Hohenester, U. Phys. Rev. Lett. 2007, 99, 126806.(29) Hirtschulz, M.; Milde, F.; Malic, E.; Thomsen, C.; Reich, S.;Knorr, A. Phys. Status Solidi B 2008, 245, 2164−2168.(30) Huang, L.; Krauss, T. D. Phys. Rev. Lett. 2006, 96, 057407.(31) Aradi, B.; Hourahine, B.; Frauenheim, T. J. Phys. Chem. A 2007,111, 5678−5684.(32) Elstner, M. J. Phys. Chem. A 2007, 111, 5614−5621.(33) Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.;Frauenheim, T.; Suhai, S.; Seifert, G. Phys. Rev. B: Condens. Matter Mater.Phys. 1998, 58, 7260.(34) Niehaus, T. A.; Suhai, S.; Della Sala, F.; Lugli, P.; Elstner, M.;Seifert, G.; Frauenheim, T. Phys. Rev. B: Condens. Matter Mater. Phys.2001, 63, 085108.(35) Spataru, C. D.; Ismail-Beigi, S.; Capaz, R. B.; Louie, S. G. Phys. Rev.Lett. 2005, 95, 247402.(36) Miyamoto, Y.; Rubio, A.; Tomańek, D. Phys. Rev. Lett. 2006, 97,126104.(37) Habenicht, B. F.; Craig, C. F.; Prezhdo, O. V. Phys. Rev. Lett. 2006,96, 187401.(38) Habenicht, B. F.; Prezhdo, O. V. Phys. Rev. Lett. 2008, 100,197402.

(39) Habenicht, B. F.; Prezhdo, O. V. J. Am. Chem. Soc. 2012, 134,15648−15651.(40) Habenicht, B. F.; Kamisaka, H.; Yamashita, K.; Prezhdo, O. V.Nano Lett. 2007, 7, 3260−3265.(41) Habenicht, B. F.; Kalugin, O. N.; Prezhdo, O. V. Nano Lett. 2008,8, 2510−2516.(42)Hyeon-Deuk, K.; Prezhdo, O. V.Nano Lett. 2011, 11, 1845−1850.(43) Hyeon-Deuk, K.; Prezhdo, O. V. ACS Nano 2012, 6, 1239−1250.(44) Tully, J. C. J. Chem. Phys. 1990, 93, 1061−1071.(45) Parandekar, P. V.; Tully, J. C. J. Chem. Phys. 2005, 122, 094102.(46) Wang, L.; Trivedi, D.; Prezhdo, O. V. J. Chem. Theory Comput.2014, 10, 3598−3605.(47) Trivedi, D. J.; Wang, L.; Prezhdo, O. V. Nano Lett. 2015, 15,2086−2091.(48) Pal, S.; Trivedi, D. J.; Akimov, A. V.; Aradi, B.; Frauenheim, T.;Prezhdo, O. V. J. Chem. Theory Comput. 2016, 12, 1436−1448.(49) Akimov, A. V.; Prezhdo, O. V. J. Chem. Theory Comput. 2014, 10,789−804.(50) Akimov, A. V.; Prezhdo, O. V. J. Chem. Theory Comput. 2013, 9,4959−4972.(51) Frauenheim, T.; Seifert, G.; Elstner, M.; Niehaus, T.; Köhler, C.;Amkreutz, M.; Sternberg, M.; Hajnal, Z.; Di Carlo, A.; Suhai, S. J. Phys.:Condens. Matter 2002, 14, 3015.(52) Verlet, L. Phys. Rev. 1967, 159, 98.(53) Li, Q. Y.; Lian, T. Q. Nano Lett. 2017, 17 (5), 3152−3158.(54) Ellingson, R. J.; Beard, M. C.; Johnson, J. C.; Yu, P. R.; Micic, O. I.;Nozik, A. J.; Shabaev, A.; Efros, A. L. Nano Lett. 2005, 5 (5), 865−871.(55) Rajauria, S.; Luo, P. S.; Fournier, T.; Hekking, F.; Courtois, H.;Pannetier, B. Phys. Rev. Lett. 2007, 99, 047004.(56) Liao, B.; Zhou, J.; Chen, G. Phys. Rev. Lett. 2014, 113, 025902.(57) Velizhanin, K. A.; Piryatinski, A. Phys. Rev. Lett. 2011, 106, 4.(58) Song, D. H.; Wang, F.; Dukovic, G.; Zheng, M.; Semke, E. D.;Brus, L. E.; Heinz, T. F. Phys. Rev. Lett. 2008, 100 (22), 225503.(59) An, M.; Song, Q. C.; Yu, X. X.; Meng, H.; Ma, D. K.; Li, R. Y.; Jin,Z. L.; Huang, B. L.; Yang, N. Nano Lett. 2017, 17 (9), 5805−5810.(60)Mann, C.; Hertel, T. J. Phys. Chem. Lett. 2016, 7 (12), 2276−2280.(61) Luer, L.; Hoseinkhani, S.; Polli, D.; Crochet, J.; Hertel, T.;Lanzani, G. Nat. Phys. 2009, 5 (1), 54−58.(62) Kilina, S.; Badaeva, E.; Piryatinski, A.; Tretiak, S.; Saxena, A.;Bishop, A. R. Phys. Chem. Chem. Phys. 2009, 11 (21), 4113−4123.(63) Kilina, S.; Kilin, D.; Tretiak, S. Chem. Rev. 2015, 115 (12), 5929−5978.(64) Crochet, J. J.; Duque, J. G.; Werner, J. H.; Lounis, B.; Cognet, L.;Doorn, S. K. Nano Lett. 2012, 12 (10), 5091−5096.(65) Ishii, A.; Yoshida, M.; Kato, Y. K. Phys. Rev. B: Condens. MatterMater. Phys. 2015, 91 (12), 125427.(66) Wang, L. J.; Beljonne, D. J. Phys. Chem. Lett. 2013, 4 (11), 1888−1894.(67) Ishkhanyan, A. M.; Krainov, V. P. Phys. Lett. A 2015, 379 (36),2041−2043.

Nano Letters Letter

DOI: 10.1021/acs.nanolett.7b03150Nano Lett. 2018, 18, 58−63

63

http://dx.doi.org/10.1021/acs.nanolett.7b03150