Effect of Aspect Ratio on Multiparticle Auger Recombination in Single

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Effect of Aspect Ratio on Multiparticle Auger Recombination in Single-Walled Carbon Nanotubes: Time Domain Atomistic Simulation Sougata Pal, David Casanova, and Oleg V. Prezhdo Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03150 • Publication Date (Web): 30 Nov 2017 Downloaded from http://pubs.acs.org on November 30, 2017

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Effect of Aspect Ratio on Multiparticle Auger Recombination in Single-Walled Carbon Nanotubes: Time Domain Atomistic Simulation Sougata Pal1,4, David Casanova2,3, Oleg V. Prezhdo4, ⃰ 1 2

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 3 4

IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Euskadi, Spain

Department of Chemistry, University of Southern California, Los Angeles, CA 90089, USA

Many-particle Auger-type processes are common in nanoscale materials due to a combination of high densities of states that can support multiple excitations and substantial Coulomb coupling between charges enhanced by quantum confinement. Auger decay dynamics in (10,5) semiconductor carbon nanotubes (CNT) with different aspect ratios and particle densities are simulated in time domain using global flux surface hopping, recently developed and implemented within Kohn-Sham tight-binding density functional theory. Despite an increasing density of states, the multiparticle Auger recombination rate decreases in longer CNTs. The atomistic simulation shows that the effect is directly 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-exciton recombination, and the calculated timescale ratio approaches the experimental value measured for long CNTs. Phononassisted transitions play a particularly important role during Auger recombination. Electron-phonon relaxation is faster than the recombination, and Auger transitions are assisted by phonons over a range of frequencies up to the G-mode. The involvement of phonons strongly enhances the probability of transitions involving asymmetric electron-hole pairs. The time-domain atomistic simulation mimics directly time-resolved optical experiments and provides a detailed, systematic analysis of the phononassisted Auger dynamics. Keywords: carbon nanotubes, multiparticle processes, Auger recombination, tight-binding density functional theory, nonadiabatic molecular dynamics TOC

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Carrier multiplication (CM) leading to generation of multiple electron-hole (e-h) pairs by a single photon in nano-systems motivates fundamental studies and potential applications for highly efficient 3rd-generation photovoltaics

1-4

. Auger recombination (AR), which can be regarded as the

inverse of CM, and other Auger-type processes play a major role in determining exciton dynamics in these nano-systems2,

5-8

. In semiconductor materials, Auger processes open up a new nonradiative

recombination channel in which the e-h recombination energy is transferred to a third particle (an e and or a h) that is excited to a higher energy state9. Such AR processes involve multi-carrier interactions and depend strongly on dimensionality and size of the nanostructure10,

11

. Auger-type

phenomena are responsible for energy exchange between electrons and holes, breaking the phonon bottleneck to the electron relaxation12, 13. Energy exchange between electron and hole produces a new mechanism of charge transfer, i.e., Auger-assisted electron transfer, that circumvents the Marcus inverted regime in the transfer rate dependence on the electron driving force14, 15. Due to kinematic restrictions imposed by energy and momentum conservations, Auger processes are strongly inhibited in bulk semiconductors16, 17. However, Auger type phenomena are much more prominent in quantum confined materials due to relaxation of the momentum conservation rule and increased overlap of carrier wave functions3, 18-20. Nanoscale materials span the gap between bulk and molecular systems and exhibit properties of both. Similarly to molecules, electrons and holes are confined to small volumes in nanoscale systems, and therefore, interact much more strongly than in the bulk. Just as in the bulk and in contrast to molecules, nanoscale materials have high densities of electronic states. Both strong interaction and high electronic density of states (DOS) are essential to obtain a large transition rate3. Quantum confinement enhances e-e Coulomb coupling much more than electron-phonon coupling, and as a result, the decay dynamics of multiparticle states are dominated by Auger processes in such materials16. As research and device fabrication move forward, a clear understanding of Auger processes involving multi-carrier interactions is pivotal to the development of efficient photovoltaic, photocatalytic, electronic, spintronic and related devices. Multiparticle AR occurs in a sequence of quantized steps starting from a multiparticle state N to N-1, N-2,…..3, 2 and finally to the 1-e-h pair state known as single exciton. The quantized recombination dynamics of multiparticle states can be described by a set of discrete recombination constants, τ3, τ2 …, characteristic of the decay lifetime of the 3-, 2-, . . . e-h pair states

16, 21

(see

schematic in Figure 1a). Multiparticle AR is well studied in semiconductor quantum dots (QD). Experiments establish a diameter dependence of multiparticle Auger lifetimes, in correlation with the


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Coulomb interaction that scales as 1/R with the QD radius R2,

11, 16

. A linear dependence of

multiparticle lifetimes on QD volume is observed in spherical CdSe QDs16,

22

. For a fixed QD

diameter, higher order excitonic states decay much faster, indicating that AR is much more effective at high carrier densities. In particular, it has been found that the multiparticle lifetimes for 4-, 3- and 2-eh pairs follow the fixed set of ratios 0.25:0.44:1 (τ4:τ3:τ2) 16, 21. The multiparticle lifetime ratios remain fixed as the volume of the spherical QDs is varied. All multiparticle decay constants increase linearly with increasing nanoparticle volume, provided that the particle aspect ratio is unchanged. The shape and dimensionality of a nanocrystal also have a great influence on the multiparticle decay times21. Considering a series of elongated nanorods, Htoon et al. have found that the ratio of the biexciton (τ2) and triexciton (τ3) lifetimes,τ2:τ3, gradually decreases as the aspect ratio of the particle increases21. In contrast to inorganic semiconductors, in which multiparticle AR dynamics have received considerable interest, there are relatively few works devoted to Auger processes in single-wall carbon nanotubes(SWCNT)23. Due to very high aspect ratio and low density of defects, SWCNTs provide an excellent physical realization of a 1D confined system24, 25. Electron-hole interaction energies are quite large in SWCNTs, in the range of 200 to 400 meV26, 27, and as a consequence, not only single excitons but also higher excitonic states are detectable and quite stable at room temperature28. Carrier-carrier interactions in SWCNTs lead to numerous interesting physical phenomena, including highly efficient intra-band relaxation via e-h energy transfer and ultrafast multiparticle decay via AR7, 29. Huang et al. studied multiparticle AR on SWCNTs having a very high aspect ratio (approx. 380 nm SWCNT length) and found the ratio of τ3/τ2to be in its lowest limit of 1.5 30. The current letter presents the first time-domain atomistic study of multiparticle AR dynamics in SWCNTs. Self-consistent charge density functional tight binding (SCC-DFTB) theory31-34 combined with nonadiabatic molecular dynamics (NA-MD) allows us to mimic most directly the timeresolved spectroscopic experiments, and include both e-h and electron-phonon scattering events. Generally, the AR rate depends on the coupling strength and the density of final states. The two factors play different roles in variation of the AR time with the number of charged particles and the SWCNT length. The single and multiparticle electronic DOS grow with increasing particle number and SWCNT length. In contrast, the coupling decreases with the SWCNT length. We demonstrate that the coupling decrease is more important than the electronic DOS increase, such that the AR dynamics is faster in shorter SWCNTs. The dependence on the tube length is stronger for AR involving more particles, and hence the τ2/τ3 ratio decreases gradually with tube length, approaching the experimental


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value found for long SWCNTs30. We show that phonons play an important role during AR. Electronphonon relaxation is faster than AR, and AR transitions are accompanied by energy losses to phonons of various frequencies. The probability of phonon-assisted AR decreases rapidly for energies exceeding the G-mode frequency, suggesting that phonon-assisted AR is first order in phonon coupling. Because phonons couple asymmetric electron-hole pairs, the probability of asymmetric transitions is enhanced. In spite of multiple ab initio and tight-binding calculations of SWCNT electronic structure and excitations, as well as adiabatic ground state molecular dynamics (MD) studies of SWCNT interactions with other nanoscale and biological systems10,

35, 36

, nonadiabatic MD (NA-MD)

simulations combining and extending the two techniques to model relaxation dynamics of photoexcited SWCNTs are very scarce. Habenicht et al. investigated phonon-induced intra-band charge relaxation, intersystem crossing, and e-h recombination in several SWCNTs using ab initio NA-MD37-41. AR dynamics require significantly larger calculations due to the strong dependence of state basis on the number of particles42, 43. In addition, fewest switches surface hopping (FSSH)44, 45 that is the most popular NA-MD technique excludes super-exchange processes, in which the initial and final states are coupled via virtual high energy states and which contribute notably to many-particle Auger dynamics46,

47

. In order to circumvent these limitations, we have developed the global flux

surface hopping (GFSH)technique46 for NA-MD simulations and have implemented it 31-34

DFTB

49, 50

using the PYXAID (PYthon eXtension for Ab Initio Dynamics) code

48

withinSCC-

. GFSH is a very

simple generalization of FSSH to higher order processes, such as super-exchange and multiparticle transitions. The simulations are performed in the adiabatic representation, which is the natural outcome of atomistic electronic structure calculations. Adiabatic states are eigenstates of the electronic Hamiltonian for fixed nuclear positions, such that all off-diagonal Coulomb coupling terms are diagonalized out. As a consequence, the Coulomb (diabatic) coupling is included implicitly in the NA coupling that is computed during the NA-MD simulation47. The time-dependent electron density is represented in the basis of Kohn-Sham (KS) orbitals as N

| ψ ( r ; R (t )) = ∑ ci (t ) | φi ( r ; R (t ))

(1)

i =0

where ci(t)are time-dependent expansion coefficients, and |φi(r;R(t))〉are adiabatic wave functions representing electronic state i. The adiabatic wave functions depend parametrically on the classical


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nuclear trajectory R(t). The time evolution of coefficient ci(t) is obtained by solving the timedependent Schrödinger equations for the KS orbital expansion coefficients

ih

dci (t ) N = ∑ (ωiδ ij − ihdij )c j (t ) dt j =0

d ij = φi (r , R (t ))

∂φ j (r , R (t ))

(2)

(3)

∂t

where dij is the NA coupling between states i and j, and ωi is energy of adiabatic state i. The coefficients and NA coupling are utilized to calculate the transition probabilities in the GFSH simulation. The many-particle generalization of the above equations and other details are provided in references49, 50. The electronic structure calculations, geometry optimization and adiabatic MD are carried out using the SCC-DFTB method as implemented in the DFTB+ code31, 51. The parameter set (SlaterKoster files) used in the calculation have been tested extensively for a broad range of compounds and can be found elsewhere33.The simulations are performed using periodic boundary conditions with30 Å of vacuum added in the direction perpendicular to the axis of the tubes. The structures are fully optimized at 0 K and then heated to 300 K with repeated velocity rescaling. 5 ps microcanonical MD trajectories are generated for each tube using the Verlet algorithm52 with a 1 fs time-step and HellmanFeynman forces. At each snapshot, the energies of the KS orbitals and the NA coupling constants are calculated, and these time-dependent quantities are used to perform NA-MD49, 50. Motivated by the experiment of Huang et al.

30

, and aiming to minimize the size of the

electronic basis and the simulation cell, we select the (10,5) semiconductor SWCNT with 1.05 nm diameter. Following ref21, we denote the aspect ratio by ε. In order to investigate the effect of the SWCNT aspect ratio on Auger lifetime, we chose three (10,5) nanotubes having the same diameter and different aspect ratios ε=1.08, 3.25 and 4.35. The total number of carbon atoms in the simulation cells for the three systems 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. The decays of the biexciton state populations due to AR are shown in Figure 2a. The curves are fitted with the sum of the Gaussian and exponential components, A exp[-t/τexp] + (1-A) exp


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[-0.5 (t/τGau)2], and the computed biexciton Auger lifetimes, Aτexp+(1-A)τGau, are shown in Table1. The observed trend in the decay dynamics is perhaps surprising, since the single and multiparticle electronic DOS increases progressively with the nanotube length, and availability of more final states should facilitate faster AR. But our results predict a significant increase of τ2, from 11.70 ps to 23.03 ps, as ε of the nanotube increases from 1.08 to 4.35. We further simulated 3-e-h pair AR dynamics, Figure 2b.The simulated decay curves are also composed of two components, Gaussian and exponential, and the computed triexciton lifetimes are listed in Table 1. Despite the increasing electronic DOS, a significant increase of τ3, from 5.1 ps 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 the experimental observations30. The calculations show that for a fixed ε, AR of higher-order excitons proceeds faster. This result agrees with the experiments,21,

30

providing a validation for our

computational methodology. The more rapid AR dynamics of triexcitons, compared to biexcitons, can be attributed to the higher density of final states for the triexciton annihilation process. The dependence of the biexciton and triexciton annihilation times on the aspect ratio have different slopes, Figure 2c. The dependence is stronger for triexcitons. The variation of the τ2/τ3 ratio as a function of ε agrees with the experimental findings of Htoon et al. for CdSe nanorods

21

. They

observed the τ2/τ3 ratio close 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 value of 1.5. τ2/τ3=2.29 obtained in our calculations for ε=1.08 is 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 experimentally determined limiting value of 1.5 for high ε. Since the aspect ratio dependence of the AR dynamics in SWCNTs is analogous to that in CdSe nanocrystals, one can argue that Auger-type processes exhibit properties that are similar for SWCNTs, inorganic nanocrystals, and other low-dimensional systems. The dimensionality, aspect ratio and volume dependence of AR rates can be characterized by scaling laws for exciton collision frequency and Coulomb interaction strength. In highly confined 0D systems, the probability of multi-particle collisions is high, and three-particle processes dominate21. In 1D systems, including SWCNTs, multi-particle collisions are less likely, and AR is bimolecular. Recent analysis of AR rates in 2D nanoplatelets showed53 that the biexciton lifetimes do not scale with volume, as is the case for 0D crystals. The linear increase of biexciton AR with nanoplatelet lateral area reflected the 1/area dependence of the binary collision frequency for 2D excitons, while the


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thickness-dependent biexciton recombination was attributed to strong dependence of Coulomb interaction on quantum confinement53. In order to understand why the AR slows down in longer SWCNTs, even though the electronic DOS increases, we analyze the corresponding coupling matrix elements. Note that if the single-particle DOS increases, the corresponding many-particle DOS increases as well. Figure 3a-c present contour plots of the root-mean-square coupling between the electron and hole states as a function of state energy for the (10,5) SWCNTs with the three different aspect ratios. The data show that the coupling magnitude decreases as the tube aspect ratio grows. The simulations predict strong coupling for electron-hole pairs that are asymmetric in energy. Involvement of phonon-modes enhance the asymmetric coupling channel54 and allow phonon-assisted AR. Figure 3 d, e present the root-meansquare coupling within the manifolds of electron and hole states, respectively, for the three (10,5) SWCNTs. Similarly to the e-h couplings, Figure 3a-c, the e-e and h-h couplings decrease with increasing SWCNT aspect 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 hole manifolds of states take place before electrons and holes annihilate. This is in agreement with the experimental literature indicating that charge thermalization is the fastest process taking place immediately after photo-excitation55, 56. Our data show that for the same number of charges, the e-e and h-h scattering and thermalization should be slower in longer SWCNTs. The average coupling between the initial (photo-excited) and final (near band-gap) states of electrons and holes is presented in Table 1. The coupling 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 thee-e and h-h thermalization occurs by transitions between states that are close in energy. Similarly to other coupling measures, the coupling magnitudes shown 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 anharmonically and non-perturbatively, although classically. Figure 4 presents the probability distribution of electronic energy loss to phonons during the simulated 2-e-h and 3-e-h AR transitions. The data show that the majority of AR events are accompanied by energy losses to phonons. Electron-phonon energy dissipation is a critical part of the AR dynamics 57. The simulations


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demonstrate that most phonons available in SWCNTs participate in phonon-assisted AR, with the probability density ranging from low frequencies to the high energy G-phonons. The phonon-assisted Auger transition probability is high in Figure 4 in the region between -0.2 eV to 0 eV, because energy is transferred from charges to phonons (hence negative values), and because the highest frequency phonon-mode, the C-C stretching G-mode, is at 1600 cm-1 which corresponds to 0.2 eV. It has been established previously that the G-phonon couples particularly strongly to the electronic subsystem37, 38, 58

. The tails extending below -0.2 eV correspond to multi-phonon AR processes that are less probable.

The data tail above 0 eV demonstrates 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 in both energy and momentum spaces, lifting the strict energy and momentum conservation requirements present for purely electronic transitions. Electronic energy is lost to phonons already during Auger dynamics, accelerating equilibration of electron and phonon subsystems, and contributing to heating of nanoscale devices. The current simulation shows that the phonon-assisted Auger scattering channel should be included into interpretation of experimental data on non-equilibrium electron-phonon dynamics, for instance, by modification of the commonly use two-temperature model59. The current work confirms the earlier observation by Shabaev et al.54 that involvement of phonon modes facilitates coupling between asymmetric electron-hole pairs. It is interesting to note that the coupling is enhanced between electron and hole states that differ in energy by 50 to 200 meV, as can be seen in Figures 3a-c, and most clearly in Figure 3a. This observation is directly related to the fact that the SWCNT phonon spectrum stops at 0.2 eV. The enhancement is seen for states that differ by at least one phonon quantum, with high frequency phonons playing the most important role. The current work contains several limitations. The SWCNT fragments considered here are short compared to the electron-hole correlation length60-62, and exclude effects arising at SWCNT ends63. By confining electrons and holes within spaces that are smaller than their natural coherence lengths we enhance their interaction relative to that in long SWCNTs. End effects can be particularly important for electron-hole annihilation, because they provide trap states and strong charge-phonon coupling, and because charges are very mobile in SWCNTs and can reach the ends quickly.64, 65 At the same time, transport in very long tubes is diffusion limited, requiring modified modeling methodology66 and additional analysis of the experimental data, e.g. fitting to stretched-exponentials. Non-exponential Auger decay can also occur at long times that are greater than the system’s lifetime67.


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In summary, we have presented a comprehensive study of multiparticle Auger recombination dynamics in carbon nanotubes with a range of aspect ratios, by performing a time-domain atomistic nonadiabatic molecular dynamics simulations for the first time. Our results show that the bi- and triexciton lifetimes increase in longer tubes, even though the density of product states grows. This result is rationalized by a faster decrease in the nonadiabatic coupling for the multi-particle transitions in longer tubes. Biexcitons live longer than triexcitons, and the bi- to triexciton lifetime ratio approaches the experimentally determined long-tube limit. Electron-phonon coupling is essential for the Auger recombination. It lowers the overall electronic energy, driving the systems into fewerparticle and, eventually, ground state. The majority of Auger recombination transitions are phononassisted and are first order in coupling to phonons. A broad range of phonons from low frequencies up to the high frequency G-mode participate in the dynamics. Phonon participation enhances coupling of electron-hole pairs that are asymmetric in energy, opening up additional energy exchange channels and accelerating equilibration. Our results agree well with the available experimental observations, demonstrate generality of Auger recombination scaling laws for SWCNTs and QDs, and provide new insights on phonon-assisted Auger processes.

Author Information *Corresponding Author, E-mail: [email protected]

Acknowledgements. S. P. acknowledges financial support from SERB-DST, Government of India through project Ref. No. CS-085/2014.O.V.P. acknowledges financial support from the U.S. Department of Energy, grant DE-SC0014429. D.C. acknowledges financial support from the Spanish Government MINECO/FEDER, project CTQ2016-80955.


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References 1.

Gao, J.; Fidler, A. F.; Klimov, V. I. Nat. Comm. 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 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. Cambridge University Press: 2003.

10.

Wang, F.; Wu, Y.; Hybertsen, M. S.; Heinz, T. F. Phys. Rev. B 2006, 73, 245424.

11.

Klimov, V. I. Ann. Rev. Phys. Chem. 2007, 58, 635-673.

12.

Efros, A. L.; Kharchenko, V. A.; Rosen, M. Solid State Comm. 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.


Page 11 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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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. Ann. Rev. Cond. Matt. 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. Phot. 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. Stat. Sol. 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.


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33.

Page 12 of 20

Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G.

Phys. Rev. B 1998, 58, 7260. 34.

Niehaus, T. A.; Suhai, S.; Della Sala, F.; Lugli, P.; Elstner, M.; Seifert, G.; Frauenheim, T. Phys. Rev. B

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.; Tománek, 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.; Yarnashita, 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. Theor. Comp. 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. Theor.

Comp.. 2016, 12, 1436-1448. 49.

Akimov, A. V.; Prezhdo, O. V. J. Chem. Theory Comp. 2014, 10, 789-804.

50.

Akimov, A. V.; Prezhdo, O. V. J. Chem. Theory Comp. 2013, 9, 4959-4972.


Page 13 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

51.

Frauenheim, T.; Seifert, G.; Elstner, M.; Niehaus, T.; Köhler, C.; Amkreutz, M.; Sternberg, M.; Hajnal,

Z.; Di Carlo, A.; Suhai, S. J. Phys. Cond. Matt. 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). 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. Nature 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 2015, 91, (12).


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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.


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Table 1:

2-e-h pair (τ2) and 3-e-h pair (τ3)decay constants,τ2/τ3 ratio, and root-mean-square NA coupling between initial and final states as functions of nanotube aspect ratio (ɛ). ɛ

τ2(ps)

τ3(ps)

τ2 / τ3

Coupling for electron

Coupling for hole

states [meV]

states [meV]

1.08

11.70

5.10

2.29

6.38

5.11

3.25

18.80

8.90

2.11

3.96

3.76

4.35

23.03

11.28

2.04

2.07

2.23


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Figure Captions

Figure 1: (a) Diagram illustrating quantized steps involved in multiparticle Auger recombination. Red and blue circles indicate electron and hole, respectively. (b) Geometry of the (10, 5) nanotube with 4.35 aspect ratio at 300K.

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, in spite of 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.

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. The electron 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 different nanotube aspect ratios.

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 transition energy.


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Figure 1:

(a)

(b)


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Figure 2:


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Nano Letters

Figure 3:


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Figure 4:


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Effect of Aspect Ratio on Multiparticle Auger Recombination in Single

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