-
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
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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.
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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
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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.
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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:
[email protected] 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.
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mailto:[email protected]://orcid.org/0000-0002-5140-7500http://dx.doi.org/10.1021/acs.nanolett.7b03150
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Nano Letters Letter
DOI: 10.1021/acs.nanolett.7b03150Nano Lett. 2018, 18, 58−63
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