Direct Observation of Cross-Polarized Excitons in Aligned Single-Chirality Single-Wall Carbon Nanotubes
Fumiya Katsutani, Weilu Gao, Xinwei Li, Yota Ichinose, Yohei Yomogida, Kazuhiro Yanagi, Junichiro Kono
DDirect Observation of Cross-Polarized Excitons inAligned Single-Chirality Single-Wall Carbon Nanotubes
Fumiya Katsutani, Weilu Gao, Xinwei Li, Yota Ichinose, Yohei Yomogida, Kazuhiro Yanagi, and Junichiro Kono
1, 3, 4, ∗ Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA Department of Physics, Faculty of Science and Engineering,Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, USA (Dated: August 28, 2018)Optical properties of single-wall carbon nanotubes (SWCNTs) for light polarized parallel to thenanotube axis have been extensively studied, whereas their response to light polarized perpendic-ular to the nanotube axis has not been well explored. Here, by using a macroscopic film of highlyaligned single-chirality (6,5) SWCNTs, we performed a systematic polarization-dependent optical ab-sorption spectroscopy study. In addition to the commonly observed angular-momentum-conservinginterband absorption of parallel-polarized light, which generates E and E excitons, we observeda small but unambiguous absorption peak whose intensity is maximum for perpendicular-polarizedlight. We attribute this feature to the lowest-energy cross-polarized interband absorption processesthat change the angular momentum along the nanotube axis by ± , generating E and E ex-citons. The energy difference between the E and E exciton peaks, expected from asymmetrybetween the conduction and valence bands, was smaller than the observed linewidth. Unlike previousobservations of cross-polarized excitons in polarization-dependent photoluminescence and circulardichroism spectroscopy experiments, our direct observation using absorption spectroscopy allowedus to quantitatively analyze this resonance. Specifically, we determined the energy and oscillatorstrength of this resonance to be 1.54 and 0.05, respectively, compared with the values for the E exciton peak. These values, in combination with comparison with theoretical calculations, in turnled to an assessment of the environmental effect on the strength of Coulomb interactions in thisaligned single-chirality SWCNT film. PACS numbers:
I. INTRODUCTION
Semiconducting single-wall carbon nanotubes (SWC-NTs) possess rich optical properties arising from one-dimensional excitons with extremely large binding ener-gies [1–9]. Although much has been understood aboutthe properties of excitons that are active for parallel-polarized light, excitons excited by perpendicular-polarized light have not been explored experimentally.Such cross-polarized excitons are predicted to exhibitstrong many-body effects due to a subtle interplay ofquantum confinement and Coulomb interactions [10–13].Figure 1 schematically shows the lowest-energy al-lowed interband optical transitions in a semiconductingSWCNT [14]. For absorption of light polarized parallelto the nanotube axis, the band index is preserved in anallowed optical transition (the E and E transitions).For light polarized perpendicular to the nanotube axis, atransition can occur when the subband index changes by1 (the E and E transitions). As first pointed out byAjiki and Ando [10], the E and E absorption peaksare expected to be suppressed because of the depolariza-tion effect. However, subsequent theoretical studies [11–13] taking into account the electron-hole Coulomb inter-actions indicated that a small absorption peak due to cross-polarized excitons should still appear.The E / E transitions were first observed in polar-ized photoluminescence excitation spectroscopy studieson aqueous suspensions of SWCNTs [15, 16]. By cross-ing the polarization of the excitation beam with respectto that of the collection beam, E photoluminescencedue to resonant absorption at the E / E transition wasobserved. More recently, in circular dichroism (CD) stud-ies [17–19], chirality-sorted nanotubes were further sepa-rated into enantiomers based on their “handedness,” i.e.,(6,5) and (5,6) SWCNTs. CD spectra for enantiomer-sorted nanotubes showed peaks due to E and E exci-tons. However, such cross-polarized exciton transitionshave never been directly identified in optical absorp-tion spectra. Therefore, quantitative characterization of E / E excitons has remained elusive.Here, we report the direct observation of cross-polarized excitons by absorption spectroscopy. Specifi-cally, we investigated the polarization dependence of op-tical absorption in a macroscopic film of aligned, single-chirality (6,5) SWCNTs. As the angle between the polar-ization of the incident beam and the nanotube alignmentdirection was increased from 0 ◦ to 90 ◦ , a peak due tothe E / E excitons appeared and grew in intensity atthe expense of the usual parallel-polarized excitons ( E a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug kE ⊥ ǁ E E E E ⊥ ǁ
12 12Conduction subband index:Valence subband index:
FIG. 1: Illustration of the lowest-energy allowed optical inter-band transitions in a semiconducting SWCNT. The numbersshown for the four subbands, two in the condiuction bandand two in the valence band, are their subband indices. E ij ( i = j ) denotes an allowed optical transition for parallel ( (cid:107) )polarization, whereas E ij ( i (cid:54) = j ) indicates an allowed opticaltransition for perpendicular ( ⊥ ) polarization. and E ). The energy of the E / E exciton peak was660 meV higher than the E exciton peak and 250 meVlower than the E exciton peak. Together with the ne-matic order parameter of the aligned SWCNT film deter-mined in the same analysis, these polarization-dependentabsorption measurements allowed us to determine the os-cillator strength of the E / E peak quantitatively. II. SAMPLES AND EXPERIMENTALMETHODSA. Preparation of an aligned single-chiralitySWCNT film
We first prepared an aqueous suspension of extremelypure (6,5) SWCNTs based on pH-controlled gel chro-matography [20, 21]. SWCNTs purchased from Sigma-Aldrich (Signis SG65i) were suspended in an aqueous so-lution of sodium cholate (SC). After ultracentrifugation,the supernatant was collected as an initial suspension.Sodium dodecyl sulfate (SDS) was added to the suspen-sion, which was used for a two-stage gel chromatogra-phy process. In the first-stage of gel chromatography toseparate the semiconducting SWCNTs by a difference inchiral angle, the suspension was loaded onto gel beads(GE Healthcare, Sephacryl S-200 HR) under surfactantenvironment of 2.0% SDS and 0.5% SC, and the nonad-sorbed fraction containing (6,5) nanotubes was collectedas a filtrate. This filtrate was used for the second-stageprocess to separate the semiconducting SWCNTs by a A b s o r ban c e Energy (eV) E E E E phonon E phonon FIG. 2: Absorbance spectrum in the near-infrared and visiblerange for the (6,5)-purified aqueous suspension of SWCNTswith an estimated chirality purity of 99.3%. See Appendix Afor more details on chirality purity determination. difference in diameter and to remove residual metallicSWCNTs. Before separation, the surfactant concentra-tions of the filtrate were adjusted to 0.5% SDS and 0.5%SC. The pH of the solution strongly influences the ad-sorption of residual metallic SWCNTs [21], and thus, weused pH-adjusted surfactant solutions. The pH-adjustedsolutions were loaded on gel beads, and the adsorbed(6,5) SWCNTs were eluted with a stepwise increase ofthe concentration of sodium deoxycholate (DOC).Figure 2 shows an absorbance spectrum for a puri-fied (6,5) suspension in a cuvette with a 10-mm pathlength. The assigned peaks are E (1.26 eV), E phononsideband (1.46 eV), E (2.17 eV), E phonon sideband(2.38 eV), and E (3.58 eV). Small unresolved peaks dueto residual metallic nanotubes exist in the range of 2.6–3.1 eV. We estimate the (6,5) chirality purity of the sam-ple to be 99.3% from this spectrum. See Appendix Afor more details about the method we used for chiralitypurity determination.The obtained suspension after surfactant exchange waspoured into a 1-inch vacuum filtration system with a 80-nm-pore filter membrane to obtain a wafer-scale film ofaligned SWCNTs [22]. The prepared suspension con-tained several surfactants, including SC, sodium dode-cylbenzenesulfonate (SDBS), and DOC. In order to havea thick film of highly-aligned (6,5) SWCNTs, we neededto have a mono-surfactant suspension. Therefore, weused ultrafiltration to exchange the mixed surfactants to0.04% (wt./vol.) DOC. The surfactant concentration wasalso adjusted to below the critical micelle concentrationof DOC through ultrafiltration, which is a necessary con-dition for the controlled vacuum filtration technique weused to prepare an aligned film [22]. The average lengthof SWCNTs in the prepared suspension before vacuumfiltration was ∼
200 nm.The suspension was poured into a funnel with a poly-carbonate filter membrane (Nuclepore track-etched poly-carbonate hydrophilic membrane). The pressure under-neath the membrane was lowered by a mechanical vac-uum pump connected to the side arm of a side-armflask. The filtration speed was adjusted to a rate of 1–2.5 mL/hour by controlling the valves in the vacuum line.Near the end of the filtration process, the filtration speedwas accelerated to ∼
10 mL/hour. In this procedure, thefiltration speed was also important to achieve sponta-neous alignment [22]. The obtained circular film had adiameter of ∼
20 mm. The thickness of the film grad-ually varied from the center ( ∼
10 nm) to the circumfer-ence ( ∼ B. Polarization-dependent visible–near-infraredabsorption spectroscopy
We performed optical transmission measurements onthe prepared SWCNT film using linearly polarized light.Our experimental setup consisted of a tungsten-halogenlamp (Thorlabs, SLS201L), a Glan-Thompson polarizer,and two spectrometers. One of the spectrometers covereda spectral range of 520–1050 nm, utilizing a monochro-mator (Horiba/JY, Triax320) equipped with a liquid-nitrogen-cooled CCD camera (Princeton Instruments,Spec-10). The other spectrometer, which covered a spec-tral range of 1050–1550 nm, consisted of a monochro-mator (Princeton Instruments, SP-2150) and a liquid-nitrogen-cooled 1D InGaAs detector array (Princeton In-struments, OMA V InGaAs System). Polarization de-pendence was achieved through changing the polarizationangle of the incident light beam by rotating the polarizer.The light beam was focused down to 30 µ m in diameterby a 50 × objective lens (Mitutoyo, M Plan NIR 50).A schematic diagram of the experimental geometryis shown in Fig. 3. The incident beam was polarizedalong the horizontal direction. The angle between thenanotube alignment direction and the light polarizationdirection is denoted by β throughout this manuscript.Polarization-dependent transmittance ( T ) spectra weretaken with a step size of 5 degrees. The measured spotwas ∼ ∼
10 nm at that spot. We calculatedattenuation spectra through A = − ln ( T ) . III. EXPERIMENTAL RESULTS
Figure 4(a) displays representative attenuation spec-tra for polarization angles β = 0 ◦ , 30 ◦ , 45 ◦ , 60 ◦ , and90 ◦ . The spectra are not intentionally offset. The ob-served peaks at 1.22 eV and 2.13 eV are due to the E and E exciton transitions, respectively. These peaks V H β AlignmentSubstrateSWCNTs
FIG. 3: Illustration of the geometry of the polarization-dependent transmission experiments performed on an alignedSWCNT film. The incident beam is linearly polarized alongthe horizontal axis, and the nanotube alignment direction isrotated from the horizontal axis by angle β . are red-shifted compared with the suspension spectrumin Fig. 2 by ∼
40 meV . The peak at 1.44 eV is the phononsideband of the E exciton peak. No other peaks areobserved due to any residual semiconducting chiralitieswithin this energy range. As the polarization angle β increases from 0 ◦ (parallel) to 90 ◦ (perpendicular), theseabsorption peaks decrease in intensity.The spectrum for perpendicular polarization ( β = 90 ◦ )shows a new peak around 1.9 eV, which we assign to the E /E transition. As stated above, this transition is ex-pected for light polarized perpendicular to the nanotubeaxis (Fig. 1). A closer look at the polarization-dependentspectra allowed us to identify this peak in all spectra forpolarization angles equal to or larger than 60 ◦ . Further-more, it should be noted that this peak exists even inthe suspension spectrum shown in Fig. 2, although peakassignment was impossible since the nanotubes in thesuspension are randomly oriented.Figures 4(b)–(d) compare the 0 ◦ ( A (cid:107) ) and 90 ◦ ( A ⊥ )spectra in more detail. In these figures, a polynomialbaseline was subtracted from each spectrum; see Sec. IVfor more details about this procedure. In Fig. 4(b), thered and black curves represent A (cid:107) and A ⊥ , respectively,where the A ⊥ spectrum is multiplied by 3.2 so that the E peak coincides in intensity between the two spectra.As a result, the two spectra deviate from each other onlyin the spectral region of the E /E peak. In Fig. 4(c), A ⊥ multiplied by 3.2 is plotted in the upper ( y > )plane, whereas A (cid:107) is plotted in the lower ( y < ) plane.The vertical dashed lines indicate the positions of the E peak, the E phonon sideband peak, the E /E peak, and the E peak, respectively. The blue curve is . A ⊥ − A (cid:107) , which is essentially zero everywhere exceptfor the E /E feature since the E /E feature onlyappears in A ⊥ . Finally, Fig. 4(d) shows a spectral differ- – l n ( T ) E phonon E E E / E -0.2-0.10.00.10.2 2.22.01.81.61.41.21.00.8 Energy (eV) -3-2-10123 E phonon E E E / E – l n ( T ) E E
0° = Parallel90° = Perpendicular E phonon 45° E / E (a)(c) (cid:1827) ∥ (cid:1827) (cid:2884) × ( . (cid:1827) (cid:2884) − (cid:1827) ∥ ) / (cid:1827) ∥ (d) − (cid:1827) ∥ . (cid:1827) (cid:2884) . (cid:1827) (cid:2884) − (cid:1827) ∥ (b) -2 × Energy (eV) E / E FIG. 4: (a) Polarization-dependent attenuation spectra for the aligned (6,5) SWCNT film for polarization angles ( β ) of ◦ , ◦ , ◦ , ◦ , and ◦ with respect to the nanotube alignment direction. (b) Comparison of attenuation spectra for ◦ ( A (cid:107) , blackline) and ◦ ( A ⊥ , red line). A ⊥ is multiplied by 3.2. They match except in the spectral region of E /E . (c) Comparison ofthe ◦ ( A (cid:107) ) and ◦ ( A ⊥ ) spectra. The blue line indicates . A ⊥ − A (cid:107) . (d) A normalized spectral difference ( . A ⊥ − A (cid:107) )/ A (cid:107) ,which shows a prominent peak due to the E /E exciton. ence ( . A ⊥ − A (cid:107) ) normalized by A (cid:107) . In this spectrum,the effects of the E peak, the E phonon sidebandpeak, and the E peak are nearly eliminated, leaving apronounced single peak due to the E /E exciton. IV. SPECTRAL ANALYSIS
To extract quantitative information from the obtainedpolarization-dependent spectra, we performed spectralanalysis. We fit each spectrum with a function consistingof Lorentzians representing the absorption peaks and apolynomial function representing the baseline: A ≡ − ln( T ) = or (cid:88) n =1 a n ( b n / ( E ph − c n ) + ( b n / + (cid:88) m =0 d m E m ph , (1)where E ph is the photon energy, acting as the indepen-dent variable, and a n , b n , c n , and, d m are the fittingparameters. a n , b n , and c n are the peak amplitude, fullwidth at half maximum, and peak position, respectively, of the n -th peak, while d m is the m -th polynomial coeffi-cient. We considered polynomials of order up to m = 4 .We performed fitting on all spectra with polarization an-gles from − ◦ to 90 ◦ with a step size of ◦ . The spectrafrom − ◦ to 30 ◦ were fit with a polynomial function andthree Lorentzians, to take account of the E peak, the E phonon sideband peak, and the E peak. The spec-tra from 35 ◦ to 90 ◦ were fit with four Lorentzians to takeinto account the E /E peak as well.Figure 5 shows fitting results for the spectra for β = ◦ , 45 ◦ , 60 ◦ , and 90 ◦ . The solid black lines are experi-mental data. The dashed red lines indicate the overall fitfunctions. The blue curves indicate the individual com-ponents of the fit function. Note that the spectrum for0 ◦ shown in Fig. 5(a) does not contain the E /E peak.Figures 6(a)-6(d) plot the extracted polarization-dependent spectra for the E peak, the E phononsideband peak, the E /E peak, the E peak, and thepolynomial baseline, respectively. The shape of the base-line slightly changes with the polarization angle. As theangle increases, the overall intensities of the baseline, the (b)(a)(c) (d) -2 – l n ( T ) Energy (eV)90° E phonon E E E / E data fit (total) individual fits – l n ( T ) E E phonon data fit (total) individual fits E – l n ( T ) Energy (eV)60° E phonon E E E / E data fit (total) individual fits – l n ( T ) E phonon E E E / E data fit (total) individual fits FIG. 5: Spectral analysis for the polarization-dependent extinction spectra for the aligned (6,5) SWCNT film using Eq.(1) asthe fit function. The experimental spectra (black), overall fit (red dashed line), and individual components (blue lines) areshown for polarization angles of (a) 0, (b) 45, (c) 60, and (d) 90 degrees. (b)(a)(c) (d) – l n ( T ) E E -2 – l n ( T ) Energy (eV)0°30°45°60°90°Polynomial baseline -2 – l n ( T ) E phononsideband -3 – l n ( T ) Energy (eV) 30°45° 60°90° E / E FIG. 6: Detailed polarization dependence of the individual spectral components deduced from the fits: (a) E and E , (b) E phonon sideband, (c) E /E , and, (d) polynomial baseline for polarization angles of 0, 30, 45, 60, 90 degrees. (a)(c) (b)(d) × -3 × -2 I n t eg r a t ed i n t en s i t y ( e V ) × -2 × -3 E I n t eg r a t ed i n t en s i t y ( e V ) E (°) E / E (°) E phononsideband FIG. 7: Integrated peak intensity as a function of polariza-tion angle β extracted for (a) E , (b) E , (c) E phononsideband, and (d) E /E . E peak, the E phonon sideband, and the E peakdecrease, while the E /E peak grows in intensity. Thepeak widths of the E and E peaks are ∼
120 meV and ∼
190 meV, respectively.Finally, Figs. 7(a)-7(d) plot the integrated peak inten-sities of the E peak, the E phonon sideband, the E peak, the E /E peak, respectively, as a function ofpolarization angle β . While the integrated intensities ofthe E peak, the E phonon sideband, and the E peak decrease as the polarization angle β increases, theintegrated intensity of the E /E peak increases. V. DISCUSSIONA. Nematic order parameter
Since the average length of SWCNTs ( ∼
200 nm) ismuch larger than the film thickness ( <
10 nm) in our sam-ple, we use the two-dimensional (2D) theory of the opti-cal absorption by an ensemble of anisotropic molecules,described in Appendix B Section 2, to discuss our ex-perimental data. We assume that the nanotubes’ angu-lar distribution f ( θ ) can be represented by the followingGaussian function with θ = 0 as the alignment direction: f ( θ ) = 1 erf (cid:0) π/ √ σ (cid:1) √ πσ (cid:18) e − θ σ + e − ( θ − π )22 σ (cid:19) , (2)where θ is the angle between the macroscopic alignmentdirection and an individual nanotube and σ is the stan-dard deviation. Note that the nanotubes are distributed f ( ) (°) σ = 25 ° σ = 32 ° σ → ∞ f ( θ ) FIG. 8: Simulated nanotubes’ angular distribution f ( θ ) ,based on Eq. (2). The three traces correspond to σ = 25 ◦ , σ = 32 ◦ , and σ → ∞ , respectively. in an angular range of ≤ θ ≤ π , and f ( θ ) is normalizedin this range, i.e., ´ π f ( θ ) dθ = 1 .Figure 8 shows three examples of f ( θ ) for the cases of σ = 25 ◦ , 32 ◦ , and ∞ . When σ = 25 ◦ (shown as a blackdashed line), clear alignment along θ = 0 is observed.As σ increases, the distribution function f ( θ ) becomesflatter. Finally, when σ → ∞ , f ( θ ) → /π as indicatedby the black solid line.With the distribution function f ( θ ) given by Eq. (2),the 2D order parameter S , defined by Eq. (B30), can becalculated as S = ˆ π f ( θ ) (cid:0) θ − (cid:1) dθ = e − σ erf ( π/ √ σ ) (cid:20) erf (cid:18) π √ σ − i √ σ (cid:19) + erf (cid:18) π √ σ + i √ σ (cid:19)(cid:21) . (3)Since S and σ have one-to-one correspondence, we canplot S as a function of σ , as shown in Fig. 9(a). When σ → , S → , as expected. As σ increases, S monotoni-cally decreases, and finally, when σ → ∞ , S → .When the polarization angle is β with respect to thenanotube alignment direction (see Fig. 3), the absorptioncoefficient for incident light with photon energy E ph isgiven by α abs ( β ) = N E ph (cid:126) cn (cid:32) α (cid:48)(cid:48) ´ π f ( θ ) cos ( θ − β ) dθ ´ π f ( θ ) dθ + α (cid:48)(cid:48) ´ π f ( θ ) sin ( θ − β ) dθ ´ π f ( θ ) dθ (cid:33) = N E ph (cid:126) cn (cid:18) α (cid:48)(cid:48) ˆ π f ( θ − β ) cos ( θ − β ) dθ + α (cid:48)(cid:48) ˆ π f ( θ − β ) sin ( θ − β ) dθ (cid:19) , (4)where N is the total number of SWCNTs, (cid:126) is the re-duced Planck constant, c is the speed of light, n is therefractive index, and α (cid:48)(cid:48) ( α (cid:48)(cid:48) ) is the imaginary part of themolecular polarizability, α , of an individual SWCNT par-allel (perpendicular) to the tube axis. See Appendix Bfor more details.We assume that the polarizability of an E ii transitionis parallel to the nanotube axis ( ξ = 0 ◦ ) whereas thatof an E ij ( i (cid:54) = j ) transition is perpendicular to the nan-otube axis ( ξ = 90 ◦ ), where ξ = tan − (cid:16)(cid:112) α (cid:48)(cid:48) /α (cid:48)(cid:48) (cid:17) (see Appendix B Section 2). Namely, to consider the E transition, we assume ξ = 0 ◦ , i.e., α (cid:48)(cid:48) (cid:54) = 0 and α (cid:48)(cid:48) = 0 . With the distribution f ( θ ) given by Eq. (2), theabsorption coefficient for the E transition becomes α abs ,E ( β ) = N E (cid:126) cn α (cid:48)(cid:48) ˆ π f ( θ − β ) cos ( θ − β ) dθ = N E (cid:126) cn α (cid:48)(cid:48) (cid:20)
12 + e − σ erf ( π/ √ σ ) (cid:26) erf (cid:18) π √ σ − i √ σ (cid:19) + erf (cid:18) π √ σ + i √ σ (cid:19)(cid:27) cos 2 β (cid:21) . (5)Similarly, by assuming that ξ = 90 ◦ , we obtain theabsorption coefficient for the E /E transition as α abs ,E ( β ) = N E (cid:126) cn α (cid:48)(cid:48) ˆ π f ( θ − β ) sin ( θ − β ) dθ. = N E (cid:126) cn α (cid:48)(cid:48) (cid:20)
12 + e − σ erf ( π/ √ σ ) (cid:26) erf (cid:18) π √ σ − i √ σ (cid:19) − erf (cid:18) π √ σ + i √ σ (cid:19)(cid:27) cos 2 β (cid:21) . (6)Therefore, when the light polarization is parallel to themacroscopic alignment direction of the film, the absorp-tion coefficient of the E transition is given by α abs ,E (0 ◦ ) = N E (cid:126) cn α (cid:48)(cid:48) ˆ π f ( θ ) cos ( θ ) dθ = N E (cid:126) cn S α (cid:48)(cid:48) . (7) On the other hand, when the light polarization is per-pendicular to the alignment direction, the absorption co-efficient of the E transition is given by α abs ,E (90 ◦ ) = N E (cid:126) cn α (cid:48)(cid:48) ˆ π f (cid:16) θ − π (cid:17) cos (cid:16) θ − π (cid:17) dθ = N E (cid:126) cn − S α (cid:48)(cid:48) . (8)Hence, the absorption coefficient ratio between paralleland perpendicular polarization is given by α abs ,E (0 ◦ ) α abs ,E (90 ◦ ) = 1 + S − S . (9)By reversing Eq. (9), we can express S in terms of theabsorption coefficient ratio as S = α abs ,E (0 ◦ ) /α abs ,E (90 ◦ ) − α abs ,E (0 ◦ ) /α abs ,E (90 ◦ ) + 1 . (10)In Fig. 9(b), S is plotted as a func-tion of α abs ,E (0 ◦ ) /α abs ,E (90 ◦ ) . When α abs ,E (0 ◦ ) /α abs ,E (90 ◦ ) = 1 , there is no anisotropy,meaning that S = 0 . As the absorption ratio increases, S increases. As α abs ,E (0 ◦ ) /α abs ,E (90 ◦ ) → ∞ , S asymptotically approaches 1. B. Angular dependence of E and E /E absorption intensities When the reflection loss can be neglected, the quan-tity we measured experimentally, i.e., the attenuation A = − ln ( T ) is directly proportional to the absorptioncoefficient. Namely, A = α abs l , where l is the film thick-ness. Therefore, the experimentally determined E in-tegrated peak intensity ratio ( A (cid:107) /A ⊥ ) can be assumedto be equal to α e,E (0 ◦ ) /α e,E (90 ◦ ) . From Fig. 7(a), A (cid:107) /A ⊥ is determined to be 3.2, which, according to theplot in Fig. 9(b), corresponds to S = σ is determined to be 32 ◦ .Figure 8 plots the angular distribution of nanotubes forthis case as a red solid curve.Furthermore, we calculated the integrated intensity ofthe E peak in absorption coefficient as a function ofpolarization angle β for S = 0 . , as shown in Fig. 10 asa black dashed line together with the experimental data(red open circles). The calculated values are normalizedby the experimental value for 0 ◦ . The observed angulardependence is accurately reproduced by the theoreticalcurve, confirming the overall correctness of our theoreti-cal analysis. Finally, the blue dash dotted line in Fig. 10represents the angular dependence of the E integratedabsorption intensity calculated assuming perfect align-ment, i.e., S = 1 . (b)(a) S (°) S abs, E (0°) / abs, E (90°) FIG. 9: (a) Nematic order parameter S as a function of stan-dard deviation angle σ based on Eq. (3). (b) Nematic orderparameter S as a function of absorption ratio between paralleland perpendicular polarization based on Eq. (9). -2 I n t eg r a t ed i n t en s i t y ( e V ) (°) Experiment Simulation ( S = 0.52) Simulation ( S = 1) FIG. 10: Polarization angle dependence of the integrated in-tensity of the E peak. Black dashed curve: theoretical cal-culation assuming S = 0 . . Red open circles: experimentaldata. The experimental observation is well reproduced by thetheoretical curve. Blue dash-dotted curve: theoretical calcu-lation assuming perfect alignment, i.e., S = 1 . – l n ( T ) Energy (eV) E E E / E
0° fits 90° fit × 10
FIG. 11: Fit peak comparison of E and E for ◦ , and E /E for ◦ . The E /E peak is multiplied by 10. C. Energy and oscillator strength of the E /E transition Figure 11 shows a parallel-polarization spectrum ( β =0 ◦ ) exhibiting the E and E peaks, together with aperpendicular-polarization spectrum ( β = 90 ◦ ) exhibit-ing the E /E peak, which were extracted from theraw experimental data through the spectral analysis de-scribed in Section IV. The perpendicular-polarizationspectrum was multiplied by 10. The energy position ofthe E /E peak is 1.88 eV, which is 1.54 times that ofthe E peak (1.22 eV) and 0.88 times that of the E peak (2.13 eV). Previously, the E /E transition wasobserved through cross-polarized photoluminescence ex-citation experiments [16] and circular dichroism measure-ments [18, 19]. The reported energies range from 1.88 to1.93 eV. These fluctuations can be attributed to the dif-ferent dielectric constants of the surrounding of the nan-otubes studied under different conditions [11, 23]. Uryuand Ando calculated the energies of the E , E /E ,and E peaks for SWCNTs as a function of dielectricconstant κ and diameter [11]. While we found no singlevalue of κ that simultaneously makes the three calcu-lated energies match the experimental values, we foundreasonable overall agreement when . < κ < . .We next discuss the oscillator strength ratio of the E /E and E transitions. Directly from the tracespresented in Fig. 11, we can determine this ratio to be I /I = 0.05. Here, I ( I ) is the integrated intensityof the E ( E /E ) peak in the parallel-polarization(perpendicular-polarization) spectrum. It is importantto note that this ratio is independent of S . This can beeasily seen by comparing Eq. (7) and α abs ,E (90 ◦ ) = N E (cid:126) cn α (cid:48)(cid:48) ˆ π f (cid:16) θ − π (cid:17) sin (cid:16) θ − π (cid:17) dθ = N E (cid:126) cn S α (cid:48)(cid:48) . (11)Namely, α abs ,E (0 ◦ ) α abs ,E (90 ◦ ) = E α (cid:48)(cid:48) E α (cid:48)(cid:48) . (12)By equating this ratio to I /I , we can also obtain theratio of the imaginary part of the molecular polarizabilityfor perpendicular polarization at E to that for parallelpolarization at E α (cid:48)(cid:48) α (cid:48)(cid:48) = E E × .
05 = 0 . . (13)Finally, we can also use the obtained value of I /I =0.05 to get a value for the dielectric constant, κ , throughcomparison with the theoretical calculations of this ratioby Uryu and Ando [11]. The radiation power absorbedby a nanotube can be expressed as P (cid:107) = 12 σ (cid:48) D (14) P ⊥ = 14 σ (cid:48) D (15)for parallel and perpendicular polarizations, respectively.Here, σ (cid:48) ( σ (cid:48) ) is the real part of the optical conduc-tivity parallel (perpendicular) to the nanotube axis at E ph = E ( E ph = E ) and D is the amplitude of theelectric field of light. Note that these expressions takeinto account the fact that only the wavenumber compo-nents ± π/L (where L is the nanotube circumference)of the incident light can excite the E /E transitionwhereas only the zero-wavenumber component of the in-cident light can excite the E transition; the inclusionof the ± π/L components corresponds to the simultane-ous excitation of the E and E transitions [10]. Spec-trally integrated and properly normalized values of σ (cid:48) and σ (cid:48) (and thus those of P ⊥ and P (cid:107) ) can be found inFig. 7 of Ref. [11]. Hence, we compared the calculated ra-tio P ⊥ /P (cid:107) with our experimental value I /I = 0.10and obtained κ = 1.52. This value is slightly outsidethe range we deduced from the peak energy considera-tion above ( . < κ < . ). A better treatment of thesurrounding dielectrics [23] as well as inclusion of higher-order terms in the band structure calculation are neededto fully explain the experimental results quantitatively. VI. SUMMARY
We prepared a macroscopic film of highly alignedsingle-chirality (6,5) SWCNTs and performed a polarization-dependent optical absorption spectroscopystudy. In addition to the usual E and E excitonpeaks for parallel-polarized light, we observed a clearabsorption peak due to the E / E exciton peakfor perpendicular-polarized light. Unlike previousobservations of cross-polarized excitons in polarization-dependent photoluminescence and circular dichroismspectroscopy experiments, our direct absorption observa-tion allowed us to quantitatively analyze this resonance.We determined the energy of this peak to be 1.54 timesthat of the E peak and the oscillator strength of thisresonance to be 0.05 times that of the E peak. Thesevalues, in light of theoretical calculations available inthe literature, led to an assessment of the environmentaleffect on the strength of Coulomb interactions in thisaligned single-chirality SWCNT film. Acknowledgements
We thank Seiji Uryu, Tsuneya Ando, and KatsumasaYoshioka for useful discussions. This work was sup-ported by the U.S. Department of Energy Basic EnergySciences through grant no. DEFG02-06ER46308 (opti-cal spectroscopy experiments), the U.S. National ScienceFoundation through award no. ECCS-1708315 (model-ing), and the Robert A. Welch Foundation through grantno. C-1509 (sample preparation). K.Y. acknowledgessupport by JSPS KAKENHI through Grant NumbersJP16H00919, JP17K14088, JP25107003, JP17H01069,JP17H06124, and JP15K21722, JST CREST throughGrant Number JPMJCR17I5, Japan, and the YamadaScience Foundation.0
Appendix A: Chirality Purity Determination
To assess the chirality purity of our sample quantita-tively, we analyzed the absorption spectrum shown inFig. 2 using the method described in Ref. [24]. The spec-trum is reproduced in Fig. 12 with two spectral regions ofinterest expanded. In Region (i), we observe a shoulder,which we attribute to the E peak of residual (9,1) SWC-NTs. In Region (ii), there are three small peaks, whichcan be attributed to the E peaks of metallic SWCNTs.Through line-fitting analysis shown in Fig. 13, we deter-mined the relative peak intensities of the observed peaks,as summarized in Table I. From these values, neglectingany ( n , m ) dependence of oscillator strength, we can cal-culate the relative population of (6,5) SWCNTs to be(41.658/41.941) ×
100 = 99.3%.
TABLE I: Relative integrated peak intensities of the E peaks of (6,5), (9,1), and metallic SWCNTs in the sample.Chirality (6,5) (9,1) Metal 1 Metal 2 Metal 3 TotalArea 41.658 0.058 0.093 0.015 0.117 41.941% 99.33 0.13 0.22 0.04 0.27 100 A b s o r ban c e Energy (eV) E E E A b s o r ban c e Energy (eV) E (9,1) E phonon (i) (ii)(i) (ii) Energy (eV)
Metallic nanotubes E phonon FIG. 12: Absorbance spectrum for the SWCNT suspensionused for making the film studied in this study. Two spectralregions of interest – (i) and (ii) – are expanded in the bottomtwo panels. -2 A b s o r ban c e data fit A b s o r ban c e data fit (a)(c) (b)(d) A b s o r ban c e Energy (eV)(6,5) E (6,5) E phonon(9,1) E × 30 individual fits -3 A b s o r ban c e Energy (eV)Metal 1Metal 2 Metal 3 individual fits
FIG. 13: Spectral fitting analysis performed to determine therelative peak intensities of the E peaks of (6,5), (9,1), andmetallic SWCNTs in the sample. Appendix B: Optical Absorption and Nematic OrderParameter of an Ensemble of Anisotropic Molecules1. Three-dimensional (3D) case
Let us cosnider an ensemble of spheroidal moleculesand their anisotropic optical absorption properties. Asshown in Fig. 14(a), we define the molecular polarizabilityalong the long axis as α and the molecular polarizabilityalong the short axis as α . θ is the angle between thealignment direction of the ensemble and the long axis ofthe particular individual molecule that we examine.When an electric field is applied parallel to the align-ment direction (which is the z -direction in Fig. 14(a)),the expectation value (i.e., the ensemble average) of themolecular polarizability (cid:104) α (cid:105) is given by (cid:104) α (cid:105) (cid:107) , = α (cid:104) cos θ (cid:105) + α (cid:104) sin θ (cid:105) = α + ( α − α ) (cid:104) cos θ (cid:105) , (B1)where (cid:104) cos θ (cid:105) and (cid:104) sin θ (cid:105) are the expectation values of cos θ and sin θ , respectively.On the other hand, when the applied electric field isparallel to the y -axis in Fig. 14(a), that is to say, theelectric field is perpendicular to the alignment direction,the average molecular polarizability (cid:104) α (cid:105) ⊥ is given by (cid:104) α (cid:105) ⊥ , = α (cid:104) cos γ (cid:105) + α (cid:104) sin γ (cid:105) = α + ( α − α ) (cid:104) cos γ (cid:105) , (B2)where γ is the angle between the electric field, which isparallel to the y -axis in Fig. 14(a), and the long axis ofthe spheroidal molecule. Here, cos γ can be written as cos γ = sin θ sin φ, (B3)1where φ is the angle between the x -axis and the directionof α projected onto the xy -plane.Now, (cid:104) cos θ (cid:105) , which is the expectation value of cos θ when the molecules are randomly oriented, is given by (cid:104) cos θ (cid:105) = ´ π cos θd Ω ´ π d Ω , (B4)where d Ω is an infinitesimal solid angle, which is ex-pressed as π sin θdθdφ . Hence, by substituting d Ω =2 π sin θdθdφ into Eq. (B4), we obtain (cid:104) cos θ (cid:105) , = ´ π − π ´ π π cos θ sin θdθdφ ´ π − π ´ π π sin θdθdφ = 13 . (B5)Similarly, (cid:104) cos γ (cid:105) , which is the expectation value of cos γ when the molecules are randomly oriented, is givenby (cid:104) cos γ (cid:105) = ´ π − π ´ π π cos γ sin θdθdφ ´ π − π ´ π π sin θdθdφ = 13 . (B6)The mean polarizability of randomly oriented spheroidalmolecules can thus be obtained, through substitution ofEq. (B5) into Eq. (B1) or substitution of Eq. (B6) intoEq. (B2), as (cid:104) α (cid:105) , = 13 α + 23 α . (B7)When the system is uniaxial, the distribution dependsonly on θ . Since (cid:104) cos γ (cid:105) does not depend on φ in thiscase, (cid:104) cos γ (cid:105) is expressed as (cid:104) cos γ (cid:105) = 12 (cid:0) − (cid:104) cos θ (cid:105) (cid:1) . (B8)As a result, Eq. (B2) becomes (cid:104) α (cid:105) ⊥ , = 12 (cid:0) α + α − ( α − α ) (cid:104) cos θ (cid:105) (cid:1) . (B9)Therefore, the average polarizability for an ensemble ofrandomly orientated molecules in Eq. (B7) can be ex-pressed in terms of (cid:104) α (cid:105) (cid:107) and (cid:104) α (cid:105) ⊥ as (cid:104) α (cid:105) , = 13 (cid:104) α (cid:105) (cid:107) , + 23 (cid:104) α (cid:105) ⊥ , . (B10)Here, we introduce the nematic order parameter, S ,as a normalized degree of alignment. Namely, we requirethat S = 1 for a perfectly aligned ensemble and S = 0 fora randomly oriented ensemble. S can be expressed as anaverage of the long axis distribution of the angle θ , whichis the angle between a nanotube and the macroscopicalignment direction. For a 3D system [25], S = 12 (3 (cid:104) cos θ (cid:105) − (B11) (a)(c) Align-mentAlign-ment x yz θ Align-ment xy θ Align-ment α α α α (b) θ α α ϕ x yz ⊥ ǁ Light electric field yzx γ FIG. 14: (a) An ensemble of spheroidal molecules in 3D space.(b) Detailed illustration of a molecule in a 3D global coordi-nate system. The alignment direction is along the z -axis.(c) Schematic of a 2D ensemble of carbon nanotubes. Thealignment direction is along the y -axis. satisfies the requirements above. By reversing Eq. (B11),we obtain. (cid:104) cos θ (cid:105) = 13 (2 S + 1) . (B12)The average polarizabilities for parallel and perpendicu-lar electric fields, i.e., Eq. (B1) and Eq. (B2), can then bewritten in terms of S : (cid:104) α (cid:105) (cid:107) , = 13 { α + 2 α + 2 S ( α − α ) } . (B13) (cid:104) α (cid:105) ⊥ , = 13 { α + 2 α − S ( α − α ) } . (B14)Given the average molecular polarizability, we can nowobtain the susceptibility χ of the molecular ensemble as χ = N (cid:104) α (cid:105) , (B15)where N is the number of molecules. The absorptioncoefficient α abs for incident light with angular frequency2 ω is then obtained by α abs = ωcn χ (cid:48)(cid:48) = E ph (cid:126) cn χ (cid:48)(cid:48) = N E ph (cid:126) cn (cid:104) α (cid:48)(cid:48) (cid:105) , (B16)where χ (cid:48)(cid:48) is the imaginary part of χ , E ph = (cid:126) ω is the pho-ton energy of the incident light, c is the speed of light, (cid:126) is the reduced Planck constant, n is the refractive index,and α (cid:48)(cid:48) is the imaginary part of the molecular polarizabil-ity, α . When the molecules are randomly oriented, α abs can be obtained by substituting Eq. (B7) into Eq. (B16),i.e., α abs , , = N E ph (cid:126) cn ( α (cid:48)(cid:48) + 2 α (cid:48)(cid:48) ) , (B17)where α (cid:48)(cid:48) ( α (cid:48)(cid:48) ) is the imaginary part of α ( α ). UsingEq. (B13) and Eq. (B14), α abs , (cid:107) , and α abs , ⊥ , , whichare the absorption coefficients for parallel polarizationand perpendicular polarization, respectively, can then bewritten as α abs , (cid:107) , = N E ph (cid:126) cn { α (cid:48)(cid:48) + 2 α (cid:48)(cid:48) + 2 S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) } . (B18) α abs , ⊥ , = N E ph (cid:126) cn { α (cid:48)(cid:48) + 2 α (cid:48)(cid:48) − S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) } . (B19)respectively. From Eqs. (B17), (B18), and (B19), the fol-lowing relation can also be derived: α abs , , = 13 α abs , (cid:107) , + 23 α abs , ⊥ , . (B20)When the reflection loss is negligible, the absorbanceis given as α abs l/ ln(10) , where l is the sample thickness.Therefore, the linear dichroism LD is written as LD = l ln(10) ( α abs , (cid:107) , − α abs , ⊥ , ) (B21) = N lE ph (cid:126) cn ln(10) S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) . (B22)The reduced linear dichroism LD r , which is the lin-ear dichroism normalized by α abs , , l/ ln(10) , where α abs , , is given by Eq. (B17) or Eq. (B20). Thus, LD r = 3 (cid:0) α abs , (cid:107) , − α abs , ⊥ , (cid:1) α abs , (cid:107) , + 2 α abs , ⊥ , . (B23)Substituting Eq. (B18) and Eq. (B19) here, we obtain LD r = 3 S D ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) α (cid:48)(cid:48) + 2 α (cid:48)(cid:48) . (B24)Defining an angle ξ ≡ tan − (cid:16)(cid:112) α (cid:48)(cid:48) / α (cid:48)(cid:48) (cid:17) , LD r = 12 S (cid:0) ξ − (cid:1) . (B25)
2. Two-dimensional (2D) case
We apply the above-developed 3D theory to an en-semble of planar or 2D aligned nanotubes. As shown inFig. 14(c), we define the polarizability along the tube axisas α and the polarizability perpendicular to the tubeaxis as α . As before, θ is the angle between the macro-scopic alignment direction and the individual nanotubeunder question.The expectation value of the polarizability of this 2Densemble (cid:104) α (cid:105) for an electric field parallel to the align-ment direction is given by (cid:104) α (cid:105) (cid:107) , = α (cid:104) cos θ (cid:105) + α (cid:104) sin θ (cid:105) = α + ( α − α ) (cid:104) cos θ (cid:105) , (B26)and that for an electric field perpendicular to the align-ment direction is given by (cid:104) α (cid:105) ⊥ , = α (cid:104) sin θ (cid:105) + α (cid:104) cos θ (cid:105) = α + ( α − α ) (cid:104) cos θ (cid:105) . (B27)When the nanotubes are randomly oriented, the ex-pectation value of cos θ is given by (cid:104) cos θ (cid:105) , = ´ π cos θdθ ´ π dθ = 12 . (B28)The mean polarizability of randomly oriented nanotubescan then be obtained by substituting Eq. (B28) intoEq. (B26) or Eq. (B27) as (cid:104) α (cid:105) , = 12 α + 12 α . (B29)The order parameter S in 2D is expressed as [26–28], S = (cid:104) θ − (cid:105) . (B30)By reversing this equation, we obtain (cid:104) cos θ (cid:105) = 12 ( S + 1) . (B31)The average polarizabilities for parallel and perpendicu-lar electric fields, obtained as Eq. (B26) and Eq. (B27),respectively, can then be expressed in terms of S as (cid:104) α (cid:105) (cid:107) , = 12 { α + α + S ( α − α ) } (B32)and (cid:104) α (cid:105) ⊥ , = 12 { α + α − S ( α − α ) } . (B33)respectively.When the nanotubes are randomly oriented, the ab-sorption coefficient α abs can be obtained, by substitutingEq. (B29) into Eq. (B16), as α abs , , = N E ph (cid:126) cn ( α (cid:48)(cid:48) + α (cid:48)(cid:48) ) , (B34)3where α (cid:48)(cid:48) ( α (cid:48)(cid:48) ) is the imaginary part of α ( α ). FromEq. (B32) and Eq. (B33), the absorption coefficients forparallel and perpendicular polarizations are given, re-spectively, by α abs , (cid:107) , = N E ph (cid:126) cn { α (cid:48)(cid:48) + α (cid:48)(cid:48) + S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) } , (B35)and α abs , ⊥ , = N E ph (cid:126) cn { α (cid:48)(cid:48) + α (cid:48)(cid:48) − S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) } . (B36)The absorption coefficient for randomly orientated nan-otubes is also expressed by α abs , , = 12 α abs , (cid:107) , + 12 α abs , ⊥ , . (B37)and α abs , (cid:107) , α abs , ⊥ , = α (cid:48)(cid:48) + α (cid:48)(cid:48) + S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) α (cid:48)(cid:48) + α (cid:48)(cid:48) − S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) . (B38)In a manner similar to the 3D case, the linear dichro-ism, LD , is expressed as LD = l ln(10) ( α abs , (cid:107) , − α abs , ⊥ , ) (B39) = N lE ph (cid:126) cn ln(10) S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) . (B40)The reduced linear dichroism LD r is given by LD r = 2( α abs , (cid:107) , − α abs , ⊥ , ) α abs , (cid:107) , + α abs , ⊥ , . (B41)Substituting Eq. (B35) and Eq. (B36) here, we obtain LD r = 2 S ( α (cid:48)(cid:48) − α (cid:48)(cid:48) ) α (cid:48)(cid:48) + α (cid:48)(cid:48) . (B42)Defining an angle ξ ≡ tan − (cid:16)(cid:112) α (cid:48)(cid:48) /α (cid:48)(cid:48) (cid:17) , LD r = 2 S (cid:0) cos ξ − (cid:1) . (B43)Finally, we consider absorption coefficients for twocases: (i) ξ = 0 ◦ ( α (cid:48)(cid:48) (cid:54) = 0 , α (cid:48)(cid:48) = 0 ), and (ii) ξ = 90 ◦ ( α (cid:48)(cid:48) = 0 , α (cid:48)(cid:48) (cid:54) = 0 ). 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