Photoinduced Dynamics in Carbon Nanotube Aggregates Steered by

Letter pubs.acs.org/JPCL

Photoinduced Dynamics in Carbon Nanotube Aggregates Steered by Dark Excitons Olena Postupna,† Heather M. Jaeger,‡ and Oleg V. Prezhdo*,§ †

Department of Chemistry, University of Rochester, Rochester, New York 14627, United States Department of Chemistry, Lehigh University, Bethlehem, Pennsylvania 18015-3102, United States § Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States ‡

ABSTRACT: Only optically active excitons can be identified by transient absorption spectroscopy, and the actual mechanisms of exciton relaxation in nanoscale systems remain unknown as dipole-forbidden transitions and charge-transfer states are not accounted for. Focusing on interacting (6,4) and (8,4) carbon nanotubes (CNTs), we show that dark excitons largely determine the relaxation pathways for photogenerated excitons in CNT bundles. New channels appear involving asymmetric electron−hole excitations within the same CNT and charge-transfer states, in which the electron and hole are confined to separate CNTs. The energy and charge transfers are facilitated by coupling to both low- and high-frequency phonons. Radial breathing modes are particularly important because they distort the CNT geometry, induce crossings of electronic states, and modulate coupling between CNTs. The time domain simulations reported herein uncover the quantum states and phonon modes that contribute to exciton relaxation in a CNT cluster, elucidating the complete relaxation mechanism. The established role of optically dark states pertains to nonequilibrium dynamics in nanoscale materials in general. SECTION: Physical Processes in Nanomaterials and Nanostructures

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and chirality.10,14,16,18−22 Exciton relaxation can be described as migration of electrons and holes from the photoexcited van Hove singularity to the lowest-energy exciton, followed by electron−hole recombination. The electronic energy lost during this process is accommodated by vibrational motions, which can be classified in CNTs as either high-frequency C−C bond stretching, known as G modes, or low-frequency radial breathing modes (RBMs) involving larger-scale motions distorting the CNT cylindrical symmetry. As the electrons and holes travel down the manifold of (quasi-)continuous conduction and valence bands and excitonic states, they visit states that are not optically active. These states cannot be studied by optical means, and their contribution to the excitedstate dynamics of CNTs and other nanoscale materials is hard to assess experimentally. Time domain theoretical studies can fill this important void by providing a detailed analysis of the variety of competing pathways available to the excess energy attained by a nanoscale material as a result of an external perturbation. The need for a thorough understanding of energy and charge dynamics in CNT systems has stimulated multiple experimental efforts. A variety of time-resolved optical experiments are used to study ultrafast energy transfer10,12,14,15,18 and free-carrier exciton generation21 in CNT aggregates and photoexcitation dynamics in thin films.16 For instance, Luer et al. investigated ultrafast excitation energy transfer in small semiconducting

arbon nanotubes (CNTs) possess unique optical and electronic properties, motivating a variety of applications.1−8 Success of the CNT applications relies on detailed understanding of and control over the CNT response to external perturbations, such as sunlight or electric current. The response is highly nonequilibrium in nature and involves flow of energy and charge between electronic and vibrational degrees of freedom of different CNTs and other device components. The nonequilibrium dynamics of large, nanoscale systems is best studied explicitly in the time domain, motivating numerous time-resolved spectroscopic investigations.9−17 Optical transitions in CNTs are brightest at the frequencies corresponding to van Hove singularities in the density of states (DOS). Strong excitonic transitions, S1 and S2 (Figure 1), involving symmetric excitations of electrons and holes, E11 and E22, have been readily observed in CNTs of varying dimension

Received: September 26, 2014 Accepted: October 21, 2014 Published: October 21, 2014

Figure 1. (Left) Electronic transitions seen in optical experiments. (Right) A more complete manifold of states involved in the dynamics. © 2014 American Chemical Society

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CNT aggregates.10 After an initial resonant excitation to the S2 level of the donor tube, intratube relaxation to the S1 level was observed on a 40 fs time scale, followed by an intertube transfer to an acceptor tube happening within 10 fs (Figure 1). Different aspects of energy transfer in CNTs were studied from the theoretical standpoint, covering the CNT band structure,23 the nature of cross-polarized excitons,19 excited carrier dynamics,24 and electronic transport in CNT junctions.25 However, the comprehensive mechanisms of exciton energy transfer occurring within a bundle of single-walled CNTs have not been investigated theoretically yet. In this Letter, we apply the time domain ab initio methodology developed in our group,29 combining time domain density functional theory (TDDFT) with nonadiabatic molecular dynamics (NAMD), to study the exciton and chargetransfer dynamics in a CNT aggregate following photoexcitation of the S2 bright exciton in one of the CNTs. We simulate atomistically the quantum mechanical evolution of the electronic degrees of freedom coupled to atomic motions and identify the key intermediate states and transfer pathways unknown to experimental observations. The obtained results show that exciton relaxation in CNT clusters not only involves optical transitions, visible in transient absorption spectroscopy experiments. It also heavily depends on dipole-forbidden transitions, as well as charge-transfer states, in which the electron and hole are spatially confined to separate CNTs. At shorter times, the dynamics involves dark states because they are closer in energy to the initially excited bright state than other bright states. At later times, populations of lower-energy bright states become significant. The CNTs used in the present work form a type-II heterojunction, with the lowest-energy excited state having a charge-transfer character. This state is populated at the end of the simulation. The electronic energy lost during the exciton relaxation is accommodated primarily by the RBMs of the CNTs. By changing CNT shape, RBMs modulate CNT energy levels and inter-CNT coupling, which is responsible for the exciton and charge transfer. High-frequency C−C stretching modes also participate in the relaxation, creating nonadiabatic coupling. Our simulations agree with the experimental findings and uncover the many quantum states involved in exciton relaxation, unveiling the relaxation mechanism. The reported study shows that optical experiments constitute only the first key step in the investigation of excitedstate dynamics and that a subsequent theoretical analysis supplementing the experiments gives a significantly more comprehensive description. We expect that this conclusion applies to most nanoscale materials. Using a system comprised of a pair of semiconducting CNTs with different chiralities, we study relaxation dynamics after the initial excitation to the E22 level of the donor tube. In the experiment by Luer et al.,10 (6,5) tubes are reported to be predominant in the samples, acting as a donor after the initial resonant excitation to their S2 level. The E11 energy levels of several possible acceptor tubes lie within kT of the donor’s levels. In our Letter, we consider a pair of (6,4) and (8,4) tubes (Figure 2) as their E11 and E22 energy distribution is comparable to the experimental one. (6,4) has a smaller unit cell than (6,5). In this ensemble, the (6,4) CNT acts as a donor, and (8,4) is an acceptor. The real-time energy-transfer dynamics are studied by means of TDDFT,26 as implemented in the Vienna ab initio simulation package (VASP),27,28 combined with NAMD.29

Figure 2. (a) Simulation cell of the (6,4)/(8,4) aggregate; (b) periodically replicated simulation cell; (c) HOMO and LUMO charge densities.

The simulation cell (Figure 2) comprises two semiconducting CNTs, (6,4) and (8,4), placed orthogonally over each other. The orthogonal position of the tubes is the most efficient approximation of CNT aggregates due to the restrictions imposed by the periodic boundary conditions necessary for the simulations. The current model is supported by the recent experimental work by Zanni and co-workers, showing that intertube transfer occurs at intersections between nonparallel CNTs.30 The simulation cell contains one unit cell of the (6,4) and (8,4) CNTs, composed of 152 and 112 carbon atoms, respectively. Because the CNT interactions and energy- and charge-transfer rates should depend strongly on the distance, the CNTs are put in close contact, as expected in a CNT aggregate. The Perdew−Burke−Ernerhof (PBE) DFT functional31,32 is used to describe the exchange−correlation energy. The interaction between atomic cores and valence electrons is treated with the projector-augmented wave (PAW) approach.33 The tube geometries are optimized at 0 K. Then, the two CNTs are placed in the simulation cell (Figure 2), and the system is heated to 300 K via repeated velocity rescaling. A 2000 fs MD production run follows in the microcanonical ensemble, with a 1 fs time step. The microcanonical MD trajectory obtained in the ground electronic state is used for geometry sampling in the NAMD calculation. Sufficient sampling of initial conditions is crucial as atomic motion results in a distortion of the equilibrium geometry of the system prior to the photoexcitation, thus producing a broad distribution of initial conditions. The nonadabatic dynamics are studied for 500 fs after the initial excitation to the E22 level of the (6,4) CNT. Further details of the TDDFT−NAMD approach used in the present can be found in refs 29 and 34−36. The optical properties of CNTs depend strongly on their electronic band structure, which can be represented by a onedimensional DOS. The shape of the DOS depends on the 3873

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computed Fourier transforms (FTs) of energy gaps between various pairs of states involved in the photoinduced dynamics. All FTs are similar qualitatively. Figure 4 presents the data for

nanotube structure, chirality, and the environment comprised of other CNTs. Intertube interaction affects the DOS of individual CNTs as can be seen from Figure 3. The top panel of

Figure 4. FTs of the energy differences between the pairs of states involved in the key nonradiative processes in the CNT aggregate. The dashed line characterizes relaxation between the S2 and S1 excitons of the (6,4) CNT. The solid line corresponds to charge transfer starting from the S1 state of the (8,4) CNT and creating a hole in the (6,4) CNT HOMO.

the key transitions, including relaxation between the S2 and S1 states of the (6,4) CNT (dashed line) and charge transfer starting from the S1 state of the (8,4) CNT and creating a hole in the (6,4) CNT HOMO (solid line). The charge-transfer state is the lowest-energy excitation according to the DOS of the combined system (Figure 3 bottom). The FTs exhibit prominent peaks at around 1600 cm−1 and at 300 cm−1 and below. These peaks are characteristic of the experimentally measured CNT Raman spectra and are attributed to the G mode and RBMs, respectively. The G mode arises due to inplane C−C stretching vibrations of carbon atoms in graphene, while RBMs are large-scale motions that alter the cylindrical symmetry of CNTs (Figure 4 inset). Unlike the signals seen in Raman spectra, the electron−phonon coupling spectra relevant for the nonradiative relaxation dynamics have stronger signals from the RBMs. The electron−phonon coupling selection rules are much weaker for the relaxation than those in the optical experiments, allowing participation of a broader range of phonon modes. The signal at around 1450 cm−1 seen in the S1CT spectrum in Figure 4 corresponds to an asymmetric version of the C−C stretch that is not observer in the Raman spectra. It is active during the CT process because the interaction of the electron and hole localized on different CNTs breaks the CNT spherical symmetry. The low-frequency RBMs alter the shapes of the wave functions globally, increasing the electronic contribution to the NA coupling matrix element29,36 and inducing crossings of electronic energy levels. The exciton basis represents the joint electron and hole dynamics occurring in the system. The exciton wave functions are defined as the direct product of electron and hole wave functions, as illustrated in Figure 5. The excitons can be characterized as optical van Hove transitions, E11 and E22, crossover excitons, E12 and E21, and charge-transfer states involving electron and hole excitations in different CNTs. Figure 6 presents the population of the key states as a function of time, following photoexcitation of the S2 (E22) exciton of the smaller-diameter (6,4) CNT. The top panel presents the populations of the states that are easily detected by

Figure 3. DOS of the noninteracting (top) and interacting (bottom) CNTs. The energy level alignment in the interacting system shows that the (6,4) and (8,4) CNTs form a type-II heterojunction. Apart from the relative energy alignment, the DOSs in the noninteracting and interacting systems are similar.

the figure shows the (6,4) and (8,4) DOS calculated separately and superimposed by matching the centers of the band gaps. The energy gaps of both tubes are slightly underestimated with respect to the experiment, but the order is reproduced correctly. The experimental values of E11 are 1.420 and 1.116 eV for (6,4) and (8,4), respectively. The corresponding E22 values are 2.146 and 2.105 eV.37 When intertube interactions are taken into account (Figure 3 bottom), the exciton energies of the tubes change little, by about 30 meV, whereas their relative positions are shifted, restructuring pathways of the transfer. The CNTs are coupled through overlap of their πelectron systems. The change in the exciton energies upon contact characterizes the coupling strength. The modest strength of the donor−acceptor electronic coupling is indicative of a nonadiabatic process for exciton and charge transfer. The relative alignment of the energy levels of the two CNTs is determined by their chemical potentials, which are directly related to other measurable quantities such as work functions, ionization potentials, and electron affinities. Because the c2 peak positions of the two tubes are now extremely close to each other, several competing processes should be expected. For holes, the ν1 peaks of (6,4) and (8,4) CNTs are very close to each other, and the order of the peaks is shifted relative to the top panel of Figure 3. This situation creates a pathway for charge transfer at lower energies. In order to identify the phonon modes that couple to the electronic subsystem and create nonadiabatic coupling, we 3874

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are localized on different CNTs (Figure 2c). As can be seen from the top panel of Figure 6, after initial excitation to the E22 level of the (6,4) CNT, its occupation decreases drastically, whereas occupations of all lower-lying, optically active states slightly increase. At shorter times, population transfers to the E22 level of (8,4) and the E11 level of (6,4) are equally possible; over time, however, the E11 population on the photoexcited (6,4) CNT becomes greater than the E22 population on the adjacent CNT. This defines the pathway of the optically detectable transition in the system of the two tubes; the (6,4) intratube relaxation from E22 to E11 is more favorable than the intertube transfer to E22 of (8,4). The intertube transfer is less likely because the electronic coupling between the CNTs is relatively weak. The top panel of Figure 6 indicates clearly that considering only the optically active states is not sufficient for a thorough analysis of the transfer mechanisms taking place in a system of interacting tubes. At all times, the total occupation of all states in the system must be conserved and equal one. However, this rule is satisfied only at early times in Figure 6 top. This indicates that alongside with the optically visible excitons, other states, such as crossover excitons, are also involved in the energy and charge transfer. The crossover transitions arise from the distribution of the exciton over two asymmetric energy levels of the same tube. For example, E12 refers to a situation when a hole is localized on the ν1 level and the electron is on c2 (Figure 3). Figure 6 bottom shows evolution of the populations of the crossover excitations (E12 and E21) and charge-transfer states, neither of which are easily detectable in the optical experiments. As can be seen from the figure, contributions from the dark states cannot be neglected in the description of the overall energy transfer between the tubes. At shorter times, both crossover transitions and charge-transfer from the (6,4) CNT exhibit significant probability, implying that the highenergy dynamics are taking place within that tube. Moreover, the total population of the dark states is larger than that of the bright states. This is because the dark states are closer in energy to the photoexcited (6,4) S2 state than the bright states (Figure 1). The bright (8,4) S2 state is close in energy to the (6,4) S2 state; however, these states are coupled weakly. Generally, the coupling magnitude decreases every time an electron or a hole (or both) needs to hop to another tube. The NA coupling for intratube transitions is on the order of 20 meV, while the NA coupling for hops between CNTs is about 10 meV. At longer times, the charge-transfer state with the electron localized on the (8,4) CNT becomes prevalent, which agrees with the DOS of the combined system (Figure 3 bottom). On a technical note, couplings on the order of 10 meV allow us to use the standard surface hopping techniques. Numerical problems with crossings of weakly coupled states can arise if the coupling magnitude drops below 1 meV, requiring special treatments.38−41 Combination of the optically visible and invisible processes taking place in the bundle of (6,4) and (8,4) CNTs allows for a rigorous elucidation of the energy- and charge-transfer processes in the system. The initial excitation to the donor (6,4) tube is followed by intratube relaxation. Then, energy is transferred to the (8,4) tube. Finally, a charge-transfer excited state is formed on a longer time scale, Figure 7. Much later, beyond the time frames of both the experiment10 and the current calculation, the system returns to the ground state. This energy- and charge-transfer mechanism is in a good agreement with the experimental observations, thus providing an in-depth

Figure 5. Relationship between the electron−hole and exciton bases in the (6,4)/(8,4) CNT aggregate, demonstrating that the majority of excitons are dark, involving either asymmetric electron−hole excitations inside of the same CNT, E12 and E21, or charge-transfer excitations.

Figure 6. Exciton dynamics after initial excitation to the S2 level of the (6,4) CNT: populations of bright (top) and dark (bottom) excitons. CTel (8,4) and CTel (6,4) refer to charge-transfer states with the electron located on the (8,4) and (6,4) tubes, respectively.

electronic optical spectroscopies. The bottom panel presents the populations of the optically inactive states, including crossover transitions19 E12 and E21 between asymmetric van Hove singularities and charge-transfer states. The asymmetric excitations have small transition dipole moments pointing perpendicular to the tube axes. Charge-transfer excitations are dark, because the wave functions of the initial and final states 3875

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G modes induce strong nonadiabatic electron−phonon coupling. Compared to Raman measurements, a larger range of modes is observed in the electron−phonon coupling spectra due to more relaxed selection rules. The intertube interactions relax the selection rules further by lowering the symmetry of excitons in individual CNTs. The work highlights the importance of time domain atomistic simulation in interpreting experimentally measured electron vibrational dynamics in nanoscale materials.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS Funding from the U.S. Department of Energy, Grant DESC0006527, is gratefully acknowledged. Financial support for contributions made by H.M.J. is generously provided by startup funds from Lehigh University.

Figure 7. Energy transfer mechanism, obtained computationally.

explanation of the underlying processes. The theoretical time scale is larger than the experimental time due to the unit cell setup. Only two tubes are involved in the simulation, while the experiments are performed with tube bundles. The influence of the media is not taken into account. The presence of surrounding CNTs in the experiment is likely to create additional energy pathways and thus speed up the overall dynamics. Moreover, the tubes are placed in the orthogonal configuration (Figure 2) because matching tube periodicities in the same direction requires huge simulation cells or significant distortions to tube geometries. This arrangement minimizes the overlap of wave functions between different CNTs. Our results also agree with the experimental results obtained on CNTs by Graham et al.22 using two-dimensional electronic spectroscopy. The authors report energy decay from the E22 level of similar systems to take about 120 fs. Our work makes important contributions to the evergrowing compendium on energy transfer within nanoscale systems and offers valuable insights toward tailoring CNTs for solar energy conversion. Considering photovoltaic applications, the simulation suggests that hot electron transfer is less likely in the E22 band and that the photovoltaic efficiency is determined by the transfer of the excited electron within the E11 band. The reported simulation complements experiment by taking into consideration the states and interactions that are hard to detect and analyze in the experimental investigations, thus providing a rigorous description of the energy-transfer mechanisms present in nanoscale systems. To recapitulate, the reported state-of-the-art simulation provides a unique time domain ab initio perspective on the photoinduced energy- and charge-transfer dynamics in CNT aggregates. It mimics directly the recently reported timeresolved optical experiments while uncovering key details that are not observable in such experiments. The calculations show that after the initial excitation to the S2 level of the donor tube, the exciton relaxation not only depends on the optically active intermediates but also is largely governed by dipole-forbidden and charge-transfer states, neither of which are detected by transient absorption. At shorter times, the relaxation is significantly influenced by the dark states as they are closer in energy to the initially excited bright state. The intratube transfer from the S2 to the S1 level of the (6,4) CNT occurs first. Then, the population is transferred to the lower-lying bright state of the (8,4) CNT. The electronic energy lost during the relaxation is accommodated by both RBM and G modes. RBMs distort CNT geometries, affecting the electronic energy level distribution and modulating the intertube coupling. The faster

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