Article pubs.acs.org/JPCC
Polarization-Controlled Two-Dimensional White-Light Spectroscopy of Semiconducting Carbon Nanotube Thin Films Randy D. Mehlenbacher,†,§ Thomas J. McDonough,†,∥ Nicholas M. Kearns,†,∥ Matthew J. Shea,‡ Yongho Joo,‡ Padma Gopalan,‡ Michael S. Arnold,*,‡ and Martin T. Zanni*,† †
Department of Chemistry and ‡Department of Materials Science and Engineering, University of WisconsinMadison, Madison, Wisconsin 53706, United States S Supporting Information *
ABSTRACT: Polarized two-dimensional white-light (2D-WL) spectra are reported for thin films of semiconducting carbon nanotubes. The orientational responses for 4-point correlation functions are derived for samples that are isotropic in two dimensions. Spectra measured using ⟨−45°,+45°,0°,90°⟩ polarizations eliminate the diagonal peaks in the spectra arising from S1 transitions to uncover cross peaks to a weaker transition that is assigned to radial breathing modes. In nanotubes purified by unwrapping PFO-BPY polymer using metal chelation, an absorption at 1160 nm is observed that is assigned to hole doping that forms trions. The trion peak may have a transition dipole nonparallel to the S1 transitions, and so its cross peak is prominent in polarized 2D-WL spectra. Energy transfer of photoexcitons to the trion peak occurs within 1 ps. Identifying and understanding the effects of purification on the electronic structure of thin films of semiconducting carbon nanotubes is important for learning how the inherent photophysics of individual carbon nanotubes translates to coupled nanotube thin-film materials.
I. INTRODUCTION Two-dimensional electronic spectroscopy (2D ES) is a powerful tool for studying electronic states and has already provided much insight into the dynamics and electronic couplings within many systems of interest in solar energy science,1 such as photosynthetic and rhodopsin proteins,2−5 quantum dots,6−8 quantum wells,9−11 carbon nanotubes,12−15 and molecular aggregates.16 In a typical 2D ES experiment, a sample interacts with three laser electric fields (with relative time delays t1 and t2) followed by signal emission in a phasematched direction.1,17 A 2D frequency correlation spectrum is formed by Fourier transforming the signal along the t1 delay and collecting the emitted electric field using a monochromator. Each electronic state produces a diagonal peak, and offdiagonal cross peaks appear between diagonal peaks whose electronic states are coupled. Together, the diagonal and cross peaks provide a wealth of information about a system, including dephasing time,18 frequency−frequency autocorrelations,19 frequency−frequency correlations between states,20,21 interstate couplings, dynamics of state evolutions, etc.17,22,23 However, cross-peaks are typically weaker than diagonal peaks. As a result, if two diagonal peaks have similar or degenerate frequencies, then the corresponding cross peaks will lie close to the diagonal and be obscured.21 When this occurs, much of the potentially interesting information available from the cross peaks is difficult to extract from the spectra. One way to circumvent this issue is to manipulate the polarizations of laser pulses to eliminate the diagonal peaks.24,25 In the mid-IR, polarized pulse sequences have been used to © 2016 American Chemical Society
collect 2D IR spectra that emphasize or eliminate the diagonal peaks, thereby exposing the weaker cross peaks.24,25 The diagonal peaks are eliminated by choosing pulse polarizations that generate two copropagating signals with 180° out-of-phase diagonal peaks so that they interfere on the detector and cancel. Interfering cross peaks do not cancel if the transition dipoles of the corresponding electronic states are nonparallel. There are several polarized pulse sequences that eliminate diagonal peaks, such as ⟨−60°,60°,0°,0°⟩ and ⟨−45°,+45°,0°,90°⟩, where the brackets contain the polarizations of the three electric fields (E1, E2, and E3) used to generate the emitted signal field and the polarization of the local oscillator (ELO) used to measure it, respectively, i.e., ⟨E1, E2, E3, ELO⟩ (Figure 1). One difficulty in utilizing these pulse polarizations is that their 2D spectra are difficult to phase. In this study, we utilize birefringent wedges to generate orthogonally polarized E1 and E2 pulses and a local oscillator trick used previously for collecting 2D spectra in the pump−probe phase matching geometry. Because we do not change any of the optics involved in the generation or timing of the pulse pairs, the cross-peak specific pathways use the same phasing as standard parallel polarization spectra. We discuss phasing and other aspects of this polarization pulse sequence below. We have also derived the orientational response for samples that are isotropic in 2dimensions, rather than 3, because the thin-film samples Received: May 16, 2016 Revised: June 28, 2016 Published: June 30, 2016 17069
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preparation alters the nature of the electronic states and the flow of charge and energy in these films is critical for their rational design into devices.35 In this article, we study two types of film preparation. The first builds on our previous work, in which we investigated energy transfer in thin films purified by polyfluorene wrapping. This purification method produces thin films of semiconducting carbon nanotubes, but the nanotubes still retain less than a monolayer of polymer wrapping. The second film preparation method was recently reported by Joo et al., which uses metal chelation to unwrap the remaining monolayer of polymer to create films of bare nanotubes.36 In both films, the cross-peak-specific 2D-WL spectra uncover cross peaks that are otherwise obscured by the strong diagonal peaks, thereby helping to better resolve the electronic structure and couplings in these films. We conclude that the samples of bare nanotubes prepared via polymer unwrapping are doped by residual metals that affect their energy transfer.
II. THEORETICAL The theoretical background behind the orientational response of 2D and 3D multidimensional spectroscopies has been summarized by a variety of authors for samples that are isotropic in 3-dimensions like a liquid.17,22,23,37−39 In this section, we report the 4-point orientational response for samples that are isotropic in 2-dimensions. This information is relevant for the samples studied here because the nanotubes are much longer (400 nm on average) than the thickness of the sample (5 nm) and so lie in a plane. These equations might also be useful in many novel two-dimensional materials such as graphene,40,41 transition metal chalcogenides,42 black phosphorus,43 perovskites,44 etc. with interesting optical responses.45,46 We begin with the following expression for the third-order signal, S(3), for a single Feynman pathway and in the semiimpulsive limit17
Figure 1. (Top) Pulse sequence for a multidimensional technique. Three pulses interact with a sample with relative delay times t1 and t2 followed by signal emission after a delay t3. Fourier transforming along t1 and t3 produces a 2D spectrum. (Middle) Assignment of Feynmen pathways that contribute to diagonal, rephasing, and nonrephasing pathways. (Bottom) Pathways that contribute for select polarization schemes.
S ∝ ⟨(μα⃗ ·Ea⃗ )(μβ⃗ ·E ⃗b)(μγ⃗ ·Ec⃗ )(μδ⃗ ·Ed⃗ )⟩
studied here are nominally planar. We have applied this polarization approach to collect cross-peak-specific 2D whitelight (2D-WL) spectra of thin films of polymer-wrapped and bare semiconducting carbon nanotubes. The unique electrical properties of semiconducting carbon nanotubes (CNTs) make them appealing materials for designing next-generation optoelectronic components.26 Their utility has already been demonstrated as high conductivity channels for field effect transistors,27,28 photoabsorbers in solar cells,29−32 and NIR light emitters.33,34 For all of these applications, semiconducting carbon nanotubes are required, which must be purified from their metallic counterparts. There are several methods for doing so, each of which creates different types of films. Characterizing and understanding how the film
(1)
where μ⃗ is the transition dipole as a function of both magnitude and direction and E⃗ is the polarization and magnitude of each of the laser pulse interactions in the laboratory frame (X and Z, corresponding to 0° and 90° laboratory polarizations, respectively, with pulses traveling along the Y-axis). The brackets represent an ensemble average over all possible molecular orientations. Thus, the final signal is a four-point correlation function that depends on the magnitude of the electric fields and the transition dipole strength, as well as the polarizations of each of the pulses. For most electronic states, rotational motions are decoupled from the magnitude of the electronic state transitions, and so we can separate the equation into
Table 1. Orientational Contribution to Feynman Pathways for a 2D Systema pathway
ZZZZ
ZZXX
ZXZX
ZXXZ
aaaa abab aabb abba
3/8 (2 + cos(2θab))/8 (2 + cos(2θab))/8 (2 + cos(2θab))/8
1/8 cos(2θab)/8 (2 − cos(2θab))/8 cos(2θab)/8
1/8 (2 − cos(2θab))/8 cos(2θab)/8 cos(2θab)/8
1/8 cos(2θab)/8 cos(2θab)/8 (2 − cos(2θab))/8
a The orientational contribution to the various Feynman pathways for a two-oscillator system with transition dipoles a and b, specifically for a system that is two dimensional. θab is the angle between the transition dipoles a and b. The pulses are propagating along the y-axis so that they have polarizations of either X or Z in the laboratory frame, corresponding to 0° and 90°, respectively.
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The Journal of Physical Chemistry C Table 2. Orientational Response for 2D Systems Including Transition Dipole Reorientationa pathway x exp(−D*(t1 + t3))
ZZZZ
ZZXX
ZXZX
ZXXZ
aaaa
2 + exp(−4Dt2) 8
2 − exp(−4Dt2) 8
exp(−4Dt2) 8
exp(−4Dt2) 8
abab
exp(−4Dt2) + 1 + cos(2θab) 8
−exp(−4Dt2) + 1 + cos(2θab) 8
exp(−4Dt2) + 1 − cos(2θab) 8
exp(−4Dt2) − 1 + cos(2θab) 8
aabb
2 + exp(−4Dt2) * cos(2θab) 8
2 − exp(−4Dt2) * cos(2θab) 8
exp(−4Dt2)cos(2θab) 8
exp(−4Dt2)cos(2θab) 8
abba
exp(−4Dt2) + 1 + cos(2θab) 8
−exp(−4Dt2) + 1 + cos(2θab) 8
exp(−4Dt2) − 1 + cos(2θab) 8
exp(−4Dt2) + 1 − cos(2θab) 8
a
The orientational contribution to the various Feynman pathways for a two-oscillator system with transition dipoles a and b, specifically for a system that is two dimensional and including 2D rotational diffusion of the transition dipole. θab is the angle between the transition dipoles a and b. The pulses are propagating along the y-axis so that they have polarizations of either X or Z in the laboratory frame, corresponding to 0° and 90°, respectively.
S ∝ ⟨(μα̂ ·Eâ )(μβ̂ ·E ̂b)(μγ̂ ·Eĉ )(μδ̂ ·Ed̂ )⟩⟨μα μβ μγ μδ ⟩EaE bEcEd
gives a different cross-peak pattern but the same pattern of diagonal peaks (one set of nonrephasing cross peaks also lies on the diagonal). Each of the peak positions is labeled by the set of 4 transition dipoles that contribute to that particular pathway. For example, the diagonal peaks are created by the four pulses interacting with the same electric field and so have the pathway aaaa (or bbbb). Cross peaks are generated when two laser fields interact with one electronic state and the other two laser fields interact with the second electronic state. Thus, cross peaks have combinations of a and b transition dipoles, i.e., aabb, abab, and abba. These four combinations form the four rows in Tables 1 and 2. The chart is generated by all the combinations of these pathways with the 4 combinations of orthogonal laboratory pulse polarizations. Consider first the orientational contributions to signals that generate diagonal peaks, which have aaaa pathways (Figure 1). If we want to measure the lifetime of an electronic state independent of reorientation dynamics, we would want to measure a signal that is independent of the rotational constant D in the top row of Table 2. However, no single term in the top row lacks D. However, linear combinations of 2D spectra can produce a D-independent signal. For example, measuring and adding spectra with polarizations ⟨Z,Z,Z,Z⟩ and ⟨Z,Z,X,X⟩ will give diagonal peaks with an intensity of ⟨Z,Z,Z,Z⟩ + ⟨Z,Z,X,X⟩ = 1/2. Thus, the rotational contribution is a constant, and any decay in the signal will come from the other terms in eq 2, like the lifetime. Instead of making two independent measurements, one can instead collect a single 2D spectra with pulse polarizations at ⟨45°,45°,Z,Z⟩ = 1/2 ⟨Z,Z,Z,Z⟩ + 1/2 ⟨Z,Z,X,X⟩, which produces an equivalent linear combination. (Notice that “magic angle” in 2-dimensional samples is at 45° in contrast to 53.7° in 3D samples.) Alternatively, if one wants to measure the reorientation diffusion constant, D, rather than the lifetime, then one can calculate the anisotropy, which in 2D is ⟨Z , Z , Z , Z⟩ − ⟨Z , Z , X , X ⟩ 1 S = ⟨Z , Z , Z , Z⟩ + ⟨Z , Z , X , X ⟩ = 4 exp( − 4Dt 2). This formula is
(2)
where the first term corresponds to the orientational contribution to the signal and the second part is the system response. Equation 2 holds for any 4 transition dipoles, but in practice most peaks in a 2D spectrum are created by interacting with just two transition dipoles, which we call a and b, separated by an angle θab. By evaluating the rotational response for each possible Feynman pathway and polarization combination for two transition dipoles, we create Table 1, the orientational contribution to the signal for a 2D isotropic sample. For this table, we have assumed there is no energy transfer to nonparallel transitions and no rotational motion. Given two transition dipoles with a relative angle of θab, Table 1 can be used to determine how the pulse polarizations scale the intensity of each peak in a 2D spectrum, which we explain in more detail below. To include the effects of transition dipole reorientation such as from rotational motions, we include 2D diffusion, which produces equations that are extensions of those in Table 1. Assuming angular reorientation of the transition dipoles is diffusive and isotropic in 2-dimensions, we model the reorientation using an angular version of Fick’s Second Law equation in 2 dimensions17 ∂G(ϑ, t ) 2 = −DJẐ G(ϑ, t ) ∂t
(3)
where D is the 2D rotational diffusion constant; JẐ is the angular momentum operator in the z direction; and ϑ is the direction of the transition dipole in polar coordinates. Solving this equation leads to the time propagator G(ϑ, t |ϑ0) =
1 2π
∞
∑
exp( −m2Dt )*exp( −im ϑ0)
m =−∞
*exp(im ϑ)
similar to the anisotropy calculated for a three-dimensional sample but has a factor of 1 in the denominator instead of 2, due to the reduced dimensionality of the system. We can also consider polarization combinations that emphasize cross peaks and remove diagonal peaks, which is the focus of this article. To accomplish this goal, we compare the pathways of the diagonal to the cross peaks and choose linear combinations that cancel the former but not the latter. Consider, for example, a signal generated by ⟨Z,Z,Z,Z⟩ − 3⟨Z,Z,X,X⟩. Neglecting reorientational dynamics for a moment, we see from Table 1 that the orientational contributions from a single transition dipole, i.e., aaaa, will vanish. Thus, this subtraction eliminates the diagonal peaks. In contrast, all of the
Table 2 shows the resulting polarization factors when orientational dynamics are taken into account within this model of orientational dynamics in two-dimensional samples. Using the results in Tables 1 and 2, we can examine several pulse sequences that will be useful for studying twodimensional samples. To do so, we need to match the pathways in Tables 1 and 2 to the Feynman paths for each of the peaks in a typical 2D spectrum. Shown in Figure 1 are typical diagonal and cross peaks found in 2D spectra, separated into diagonal and cross peaks. Rephasing and nonrephasing refer to two different types of spectra that are generated by the choice of pulse sequence and phase matching geometry. Each 17071
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heated to 60 °C and stirred overnight. Precipitates were collected by centrifuging at 10 000g for 20 min. Stripped PFOBPy and Re2(CO)10 were removed by washing the product with chloroform, THF, and methanol three times in a bath sonicator followed by centrifugation (10 000g, 20 min). Approximately 70% of the PFO-BPY is removed from the carbon nanotubes, quantified by the change in the optical absorption of the polymer. The remaining bundled nanotubes were dispersed in N-cyclohexyl-2-pyrrolidone (CHP) using a horn sonicator (5 min) to disrupt the bundles. Thin films were prepared by drop-casting either the CNT@PFO-BPy or the unwrapped nanotube solution on a SiO2 substrate at 60 °C under vacuum. The resulting films were characterized by UV/ vis/NIR absorption spectroscopy and PL spectroscopy. Thin films of nanotubes were also created with added dopants. Specifically, a solution of 0.1 mg/mL of triethyloxonium hexachloroantimonate in dichloroethane was prepared following procedures reported by Dowgiallo et al. and Chandra et al.52,53 Bare nanotube films were soaked in this solution for 1 h at 70 °C. The film was dipped in acetone to remove excess dopant. This resulted in films with dopant levels of ∼0.28 nm−1. Polarization-Controlled 2D-WL Spectroscopy. Many different experimental setups exist for measuring third-order spectroscopic signal.1,54−66 Because of its experimental compactness and simplicity, we collect our spectra in the pump probe geometry.67 To collect a 2D-WL spectrum, we split the output of a Spectra Physics Spitfire (800 nm, 150 fs, 1 kHz, 250 mW) equally to produce the pump and probe. We independently generate a white-light supercontinuum for both the pump and probe by spatially filtering the fundamental 800 nm light with an iris, attenuating the power with a variable neutral density filter, then focusing the light into 4 mm YAG crystals.68 The iris aperture and attenuation are adjusted to optimize the stability of the white light. To minimize dispersion, only reflective optics are used to collimate and direct the probe to the sample. To create our pump frequency axis, we create a pump pulse pair, scan the time delay between the pump pulse pair, and Fourier transform this time delay to create a 2D frequency correlation map. We generate our pump pulse pair and control the relative timing using a birefringent wedge pair. This method has been described in detail elsewhere15,69 but briefly relies on differences in the refractive index for ordinary and extraordinary rays to control the time delay between pump pulses. By translating a mechanical stage, we are able to vary the thickness of one wedge and thereby control the t1 time. For a typical scan, we scan t1 from 0 to 160 fs in 0.8 fs steps. We use a prism compressor to compensate for dispersion introduced by the wedges. A razor blade is placed in the focal plane of the prism compressor to filter residual 800 nm fundamental from the white-light continuum. Both the pump and probe pulse are focused onto the sample via 12.5 cm off-axis parabolic mirrors. The pump fluence is ∼9 × 1013 photons cm−2.70−72 Following the sample, the probe light is directed into a grating spectrometer (Acton SP2150i) and detected using an InGaAs photodiode array (Princeton Instruments OMA-V 512:1.7). As discussed in the theory section, some Feynman pathways of particular interest are ⟨Z,Z,Z,Z⟩, ⟨Z,Z,X,X⟩, and ⟨−45°,45°,0°,90°⟩. Thus, we need independent experimental control over both pump pulse polarizations, the probe polarization, and the detection polarization. One advantage of the birefringent wedge system over other 2D optical spectros-
cross peaks, whether rephasing or nonrephasing, will have an intensity of (2 cos(2θab) − 2)/8, which will be nonzero as long as θab is not 0, i.e., as long as the transition dipoles are nonparallel. This linear combination can be done in a single measurement by using the pulse polarization ⟨−60°,+60°,0°,0°⟩ = 1/4(⟨Z,Z,Z,Z⟩ − 3⟨Z,Z,X,X⟩). Thus, 2D spectra collected using this polarization sequence eliminate the diagonal peaks and leave all cross peaks created between electronic states with nonparallel transition dipoles.47 Another polarization combination that eliminates the diagonal peaks is ⟨−45°, 45°, 0°, 90°⟩ = =
Z−X Z+X , , Z, X 2 2
1 (⟨Z , X , Z , X ⟩ − ⟨X , Z , Z , X ⟩) 2
However, there are several differences between the ⟨−60°,60°,0°,0°⟩ and ⟨−45°,45°,X,Z⟩ signals. First, rotational diffusion will contribute to 2D spectra collected with ⟨−60°,60°,0°,0°⟩ but not ⟨−45°,45°,X,Z⟩ signals, which can be seen by calculating the signal strengths using the terms in Table 2. Thus, if rotational motion or energy transfer occurs during the coherence times t1 or t3, then diagonal peaks will be created for ⟨−60°,60°,0°,0°⟩ but not ⟨−45°,45°,X,Z⟩ signals. Second, these two polarization schemes will have different cross-peak patterns and intensities. Spectra collected with ⟨−60°,60°,0°,0°⟩ will contain cross peaks from all of the pathway combinations, aabb, abba, and abab. Spectra collected with ⟨−45°,45°,X,Z⟩ will only contain cross peaks created with abba and abab pathways. An interesting consequence is that the ⟨−45°,45°,X,Z⟩ polarization scheme should be more sensitive to interstate coherences during the t2 time delay, which have been studied extensively to understand the role of electronic coherences in light-harvesting and energy transport proteins.2,48−51 While our discussion above is in regards to 2D samples, the same conclusions are reached with regards to ⟨−60°,60°,0°,0°⟩ and ⟨−45°,45°,X,Z⟩ signals for 3D samples. The conclusions are the same because it is the same four polarization schemes that generate the third-order signal. The relationships between the different polarizations do not depend on the sample (as long as it is isotropic), and thus the dimensionality of the sample will not change the fundamental relationships between them.
III. METHODS Carbon Nanotube Purification. The carbon nanotube preparation follows the processing described by Joo et al.36 Briefly, CoMoCAT CNTs (SouthWest NanoTechnologies, SG65i) were added to a 2 mg/mL solution of poly[(9,9dioctylfluorenyl-2,7-diyl)-alt-co-(6,60-[2,20-bipyridine])] (PFO-BPy) in toluene. The solution was homogenized using a horntip sonicator then centrifuged at 300 000g for 10 min to remove soot and aggregates. The supernatant was decanted and filtered (5 μm filter), and the toluene was evaporated. The resulting sediment was dissolved in hot chloroform and THF and centrifuged at 150 000g for 24 h, and the pellets were collected. This process was repeated four times. After the final centrifuge/dispersion cycle, the pellet was dissolved in toluene at a concentration of 10 μg/mL. To strip the polymer, excess dirheniumdecacarbonyl (Re2(CO)10) was added to the dispersed CNT@PFO-BPy solution, and the mixture was 17072
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Figure 2. (A) Absorption spectrum and (B) photoluminescence excitation spectrum for PFO-BPy wrapped (6,5) CNTs dispersed in solution. (C) ⟨Z,Z,Z,Z⟩ and (D) ⟨−45°,45°,0°,90°⟩ polarizations 2D-WL spectra collected at waiting time T = 0. Roman numerals are used to assign peaks as described in the text.
IV. RESULTS AND DISCUSSION Figure 2a shows the absorption spectrum for the initially sorted (6,5) PFO-BPy wrapped CNTs in solution. The absorption spectrum shows three features, a strong absorption peaked at 998 nm and two weaker absorptions that peak at 928 and 1130 nm. By measuring the photoluminescence excitation map (Figure 2b),74 we assign the peak at 998 nm to the S1 exciton of (6,5) CNTs. We tentatively assign the peak at 928 nm in the absorption spectrum to the S1 exciton on (9,1) CNTs, although the S2 excitation, S1 emission in the photoluminescence map is weak. From the photoluminescence map, we see contributions from majority CNT species (6,5) at Peak A = (λexcitation, λemission) = (575 nm,1005 nm) as well as three features at Peak B = (575 nm, 1120 nm); Peak C = (575 nm, 1160 nm); and Peak D = (650 nm, 1006 nm). Because there is no clear correspondence to a different chirality S2 excitation, we assign these to states on (6,5) CNTs. Peak B at (575 nm, 1120 nm) has previously been assigned to the S1− state in (6,5) nanotubes and reflects defects on the CNTs.75 Free carrier doping of carbon nanotubes produces new spectral features associated with trions (excitons bound to a free carrier) that emit light at 1170 nm in intentionally doped (6,5) CNT films,52 thus we assign Peak C at (575, 1160) to trions on the (6,5) CNTs. Intentionally introducing defects may lend further insight into this feature. Peak D at (650 nm, 1006 nm) most likely corresponds to excitation of S12 excitons, interband excitons where the electron and hole are in different valence and conduction sub-band levels, on (6,5) nanotubes, which relax to the S1 state and fluoresce.76 Figures 2c and 2d show the 2D-WL spectrum for PFO-BPy wrapped nanotubes that are isolated from each other in solution with polarization conditions ⟨Z,Z,Z,Z⟩ and ⟨−45°,45°,0°,90°⟩ polarizations. In the ⟨Z,Z,Z,Z⟩ spectrum (Figure 2c), we see a strong absorption at Peak I = (992 nm, 1001 nm) corresponding to ground state bleaching and stimulated emission from the S1 exciton on the (6,5) nanotubes. An excited state absorption is observed at Peak II
copy setups in the pump probe geometry is that the collinear pulses are orthogonally polarized.15 Thus, we are able to easily generate perpendicular pump pulses, necessary for pulse sequences such as ⟨−45°,45°,0°,90°⟩ and ⟨Z,X,X,Z⟩. To collect Feynman pathways that require parallel pump pulses, e.g., ⟨Z,Z,Z,Z⟩ and ⟨Z,Z,X,X⟩, we use a broadband polarizer set to 45° relative to both pump pulses. Accessing the different Feynman pathways based on laser polarization requires carefully controlling the polarizations of the pump, probe, and the detected polarization. To control the probe polarization, we use an 800 nm waveplate to control the initial laser polarization. This polarization is transferred to the generated supercontinuum. Because YAG is not birefringent, unlike other common supercontinuum generation media,68 changing the 800 nm polarization does not change the whitelight spectrum or stability. We ensure a linear polarization at the sample by using a thin, broadband polarizer (Meadowlark OWL). Although this introduces dispersion in our probe pulse, we find that it is necessary to ensure a clean polarization at the sample. A polarizer is placed after the sample to control the detected signal polarization. For measuring pathways with orthogonal probe/signal polarizations, we find that setting the polarizer off by ∼3° provides enough local oscillator to heterodyne the signal while also being selective for only the pathway of interest. Smaller angles could be achieved with a more intense probe pulse. Therefore, we are able to independently select the polarizations for all four interactions with the sample and control which orientational pathways we are measuring. Another nice feature of using the birefringent wedges is that we can phase the ⟨−45°,45°,0°,90°⟩ spectra by using the phasing of the ⟨Z,Z,Z,Z⟩ spectra because we do not change any of the optics involved in the generation or timing of the two pump pulses, in contrast to the phasing for other polarization pulse sequences.47,48,56,73 17073
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Figure 3. (A) Absorption and (B) photoluminescence pumped at 580 nm for thin films created from PFO-BPy wrapped, bare, and doped (6,5) nanotubes. The peaks near 1000 nm correspond to the (6,5) nanotube S1 absorption. In the photoluminescence spectrum, the peak at 1000 nm corresponds to emission from (6,5) nanotubes; the peak at 1050 nm corresponds to emission from minority (7,5) nanotubes; and the peaks near 1200 and 1300 nm correspond to emission from trions on (6,5) and (7,5) tubes, respectively.
= (985 nm, 948 nm).77 The ⟨−45°,45°,0°,90°⟩ signal appears qualitatively different from the ⟨Z,Z,Z,Z⟩ polarization 2D-WL spectrum. A negative feature is observed at Peak III = (990 nm, 1050 nm), and a positive feature is found at Peak IV = (980 nm, 975 nm). While the pump frequencies closely match the excitations in the ⟨Z,Z,Z,Z⟩ spectrum, indicating that S1 excitons are being photoexcited, the probe frequencies are significantly red-shifted. Furthermore, the peaks in the ⟨−45°,45°,0°,90°⟩ spectrum are significantly broader than in the ⟨Z,Z,Z,Z⟩ spectrum. Because these nanotubes are well dispersed from one another, the peaks we measure must come from transitions on the same tube. Furthermore, because all pathways with parallel transition dipoles are suppressed by this choice of polarizations, the peak must be a cross peak that couples two (or more) nonparallel transition dipole moments. Finally, we assume that we are in the transition dipole limit because our laser pulse electric fields are weak, and thus we do not consider effects from higher-order electric multipoles nor from magnetic multipoles. There are several possible electronic states in carbon nanotubes that these signals may arise from such as dark or quasi-dark excitons and vibronic states. Interband excitons, such as S12 and S21 states, are known to have transition dipole moments perpendicular to the nanotube axis.78,79 These states, however, are separated from the S1 excitons by >300 meV (∼200 nm) and therefore cannot be the source of the peaks that we observe in the ⟨−45°,45°,0°,90°⟩ spectra.76,80,81 Coupling of S1 excitons to the free carrier continuum in carbon nanotubes and direct excitation of S12p have been both been proposed as explanations for peaks observed in crosspolarized photoluminescence experiments, but these states are shifted by >100 meV (∼75 nm) from the S1 exciton and thus cannot explain our results.76,81,82 Longitudinal and transverse dark exciton states, arising from coupling between the valley degeneracies of the K and K′ points in the reciprocal space within the S1 and S12 energy manifolds, exist ∼10 meV (∼10 nm) lower in energy and ∼130 meV (∼100 nm) higher, respectively, than the optically bright S1 exciton.78,82,83 While the longitudinal dark exciton state has approximately the correct energy level to explain our observed signal its dipole moment is parallel to the tube, and thus the signal is suppressed by the polarization scheme we are using (Table 2).82 Furthermore, the longitudinal dark exciton is optically forbidden within the transition dipole approximation.79,83 The energy level for the transverse dark exciton is ∼100 nm blueshifted from the S1 exciton in isolated nanotubes, thus it cannot contribute to our observed signal.79 A signal may also be
observed in the polarized 2D-WL spectrum if the nanotube is not straight over the coherence length of the exciton. Several reports show that that exciton coherence length is
