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[Daichi Kozawa](https://orcid.org/0000-0002-0629-5589), [Shun Fujii](https://orcid.org/0000-0002-0998-366X), [Yuichiro K. Kato](https://orcid.org/0000-0002-9942-1459)

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[Intrinsic process for upconversion photoluminescence via <math>  <mi>K</mi></math>-momentum–phonon coupling in carbon nanotubes](https://mdr.nims.go.jp/datasets/d844b87b-32c7-4fe9-a345-fcb9a8ce934d)

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Intrinsic process for upconversion photoluminescence via $K$-momentum&ndash;phonon coupling in carbon nanotubesPHYSICAL REVIEW B 110, 155418 (2024)Editors’ SuggestionIntrinsic process for upconversion photoluminescence via K-momentum–phononcoupling in carbon nanotubesDaichi Kozawa ,1,2,3,* Shun Fujii ,1,4 and Yuichiro K. Kato 1,2,†1Quantum Optoelectronics Research Team, RIKEN Center for Advanced Photonics, Wako, Saitama 351-0198, Japan2Nanoscale Quantum Photonics Laboratory, RIKEN Cluster for Pioneering Research, Wako, Saitama 351-0198, Japan3Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan4Department of Physics, Faculty of Science and Technology, Keio University, Yokohama, Kanagawa 223-8522, Japan(Received 4 April 2024; revised 29 August 2024; accepted 19 September 2024; published 10 October 2024)We investigate the intrinsic microscopic mechanism of photon upconversion in air-suspended single-walledcarbon nanotubes through photoluminescence and upconversion photoluminescence spectroscopy. Nearly linearexcitation power dependence of upconversion photoluminescence intensity is observed, indicating a one-photonprocess as the underlying mechanism. In addition, we find a strongly anisotropic response to the excitationpolarization, which reflects the intrinsic nature of the upconversion process. In upconversion photolumines-cence excitation spectra, three peaks are observed, which are similar to photoluminescence sidebands of theK-momentum dark singlet exciton. The features in the upconversion photoluminescence excitation spectraare well reproduced by our second-order exciton-phonon interaction model, enabling the determination ofphonon energies and relative amplitudes. The analysis reveals that the upconversion photoluminescence canbe described as a reverse process of the sideband emission linked to the K-momentum phonon modes. Thevalidity of our model is further reinforced by temperature-dependent upconversion photoluminescence excitationmeasurements reflecting variations in the phonon population. Our findings underscore the pivotal role of theresonant exciton-phonon coupling in pristine carbon nanotubes and present potential for advanced optothermaltechnologies by engineering the excitation pathways.DOI: 10.1103/PhysRevB.110.155418I. INTRODUCTIONUpconversion photoluminescence (UCPL) is a nonlinearoptical process in which photons with lower energy are ab-sorbed and reemitted as photons with higher energy [1–5].UCPL has profound implications in various fields, promi-nently in telecommunications [6], advanced photonics [7],renewable energy technologies [8], and optical cooling [9].Investigation of UCPL not only contributes to the theoreti-cal understanding of optical processes in materials but alsopaves the way for practical applications. Specifically, UCPLcan convert infrared photons, which are abundant but can-not be absorbed by Si-based optoelectronic devices, intohigher-energy photons that can be used for photovoltaic powergeneration [8] and bio-imaging [10]. This demonstrates thecapability of UCPL in bridging the gap between available andusable light spectra, especially in the near-infrared range.Near-infrared upconversion in single-walled carbon nan-otubes (SWNTs) stands out for its high efficiency and*Contact author: kozawa.daichi@nims.go.jp†Contact author: yuichiro.kato@riken.jpPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.remarkable energy gain exceeding several times the thermalenergy at room temperature [5,10]. Previous research has fo-cused on liquid-dispersed SWNTs with the limited chiralitiesof (6,5) and (8,3) where UCPL is attributed to one-phonon-assisted upconversion processes under one-photon excitationconditions [5]. The efficiency of UCPL is found to be en-hanced by localized states of intentionally introduced defects,suggesting that the UCPL is predominantly extrinsic. PristineSWNTs without such defects thus are presumed to exhibitnegligible UCPL.Here, we investigate UCPL in as-grown, air-suspendedSWNTs, which can be considered defect-free except for thetube ends [11–16]. By performing UCPL and photolumi-nescence (PL) measurements, the spectra and the excitationimages confirm that the emission originates from the identi-cal nanotube. The dependence of the UCPL intensity on theexcitation power reflects the one-photon excitation process,and the dependence on the excitation polarization shows ev-idence for the intrinsic characteristics. UCPL spectra acrossvarious chiralities of SWNTs are also examined, revealingthat UCPL is a universal phenomenon across the chiralities.Upconversion photoluminescence excitation (UCPLE) spec-tra are compared with sidebands in PL spectra to elucidatethe excitation process in UCPL. We develop a theoreticalmodel for upconversion that involves photon-exciton andexciton-phonon interactions, which is able to quantitativelyexplain the features in the UCPLE spectra. We conducttemperature-dependent UCPLE spectroscopy, verifying the2469-9950/2024/110(15)/155418(8) 155418-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-0629-5589https://orcid.org/0000-0002-0998-366Xhttps://orcid.org/0000-0002-9942-1459https://ror.org/01sjwvz98https://ror.org/01sjwvz98https://ror.org/026v1ze26https://ror.org/02kn6nx58https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.110.155418&domain=pdf&date_stamp=2024-10-10https://doi.org/10.1103/PhysRevB.110.155418https://creativecommons.org/licenses/by/4.0/KOZAWA, FUJII, AND KATO PHYSICAL REVIEW B 110, 155418 (2024)(a) (b) (c)FIG. 1. (a) PL and UCPL spectra for the same (9,8) SWNT with intensities normalized by the excitation power density. The PL spectrum iscollected with an excitation energy of 1.54 eV and a power of 10.0 µW. The UCPL spectrum is obtained with an excitation energy of 0.800 eVmarked by a broken line and a power of 1000 µW. Energy diagrams for (b) the PL and (c) the UCPL processes. Solid-horizontal lines representthe ground (G), the first sub-band (E11), and the second sub-band exciton states (E22), while the dotted line is an intermediate state in theupconversion process.model based on phonon-assisted upconversion in pristineSWNTs.II. PHOTOLUMINESCENCE AND UPCONVERSIONPHOTOLUMINESCENCE MEASUREMENTSThe air-suspended SWNTs are grown over trenches on Sisubstrates by chemical vapor deposition [13,17,18]. We fabri-cate the trenches with a depth of ∼1 µm and a width from 0.5to 4.0 µm by performing electron-beam lithography and dryetching. Another electron-beam lithography step is conductedto define catalyst areas near the trenches, onto which Co- orFe-silica catalysts dispersed in ethanol are spin-coated andthen lifted off. SWNTs are synthesized over the trenches usingalcohol chemical vapor deposition under a flow of ethanolwith a carrier gas of Ar/H2 at 800◦C for one minute.A home-built confocal microscopy system is used to col-lect PL spectra in dry N2 gas [13]. Si substrates are mountedon a motorized three-dimensional translation stage, allowingus to perform automated measurements over hundreds ofSWNTs. We utilize a continuous-wave Ti:sapphire laser forexcitation and a liquid-N2-cooled InGaAs photodiode arrayattached to a 30-cm spectrometer for detection. Laser polar-ization is kept perpendicular to the trenches unless specifiedotherwise. The excitation beam is focused via an objectivelens with a numerical aperture of 0.65 and a focal length of1.8 mm. The focused spot exhibits 1/e2 diameters of 1.46,1.31, and 1.06 µm for energies of 1.36, 1.46, and 1.60 eV,respectively, where these diameters are determined by PL linescans perpendicular to a suspended tube. The excitation polar-ization is rotated by a half-wave plate mounted on a motorizedrotation stage for polarization-dependent PL measurements.All PL spectra are taken at the center of the nanotubes exceptfor hyperspectral PL imaging.UCPL spectra are also taken with the same microscope us-ing a continuous-wave laser diode with an energy of 0.800 eVwhere the 1/e2 diameter of the focused beam is 2.61 µm.The laser is coupled through a single-mode fiber, ensuring ahigh-quality beam. A short-pass dichroic filter with a trans-mission band between 0.855 and 1.24 eV is utilized to blockthe excitation light during UCPL collection. We ensure thatexcitation by laser sidebands does not occur by inserting aband-pass filter with a center energy of 0.800 eV, a bandwidthof 6.2 meV, and an optical density exceeding 5. Additionally,we carefully examine the possibility of laser emission beyondthe laser peak by analyzing a UCPL spectrum on an SWNTand a reflection spectrum on a Si substrate (see Fig. S1 withinSupplemental Material [19]). These spectra clearly show theabsence of any laser emission at energies higher than theshort-pass dichroic filter cutoff of 0.855 eV. The excitationpower is tuned by a fiber optic variable optical attenuator, andthe excitation polarization is rotated by the half-wave plate.III. UPCONVERSION PHOTOLUMINESCENCE INPRISTINE NANOTUBESWe conduct PL and UCPL measurements on the sameas-grown (9,8) SWNT where the chirality is determined byphotoluminescence excitation (PLE) spectroscopy (Fig. S2within the Supplemental Material [19]). In Fig. 1(a), bothspectra are obtained with an excitation polarization alignedto the nanotube axis and are displayed on a logarithmic scale.We observe UCPL in the pristine nanotube in contradictionto the expectation that UCPL is negligible in the absenceof localized states [5]. Here, the PL and UCPL spectra arecompared to confirm that the UCPL arises from E11 excitonemission. The emission peak mirrors the E11 PL emission interms of energy and linewidth. The UCPL intensity is lowerthan the PL intensity by a factor of 155 but with a significantenergy gain of 96.2 meV relative to the excitation energy.The observed PL and UCPL processes are illustrated byenergy diagrams in Figs. 1(b) and 1(c). The PL process beginswith excitation of an exciton to an energy of E22 and theexciton then relaxes to the E11 state, recombining to emit aphoton with an energy of E11. The UCPL involves excita-tion to an intermediate state and subsequent upconversion tothe E11 state before luminescence. We note that UCPL viareal states is enhanced by chemically introducing defects intosolution-processed SWNTs [5], but UCPL in pristine SWNTsis considerably strong. The intermediate state in air-suspendedSWNTs is likely to be virtual, given that the effect of defectsis minimal in as-grown nanotubes [13].Excitation imaging measurements are performed to char-acterize the spatial distributions of E11 emission for PL and155418-2INTRINSIC PROCESS FOR UPCONVERSION … PHYSICAL REVIEW B 110, 155418 (2024)(a) (b)(c) (d)FIG. 2. (a) PL and (b) UCPL excitation images of the identical(9,8) SWNT suspended over 3-µm trench, in which the excitationimages are obtained with the spectrally integrated intensity withina 3-meV window at the E11 emission peak. For PL, an excitationenergy of 1.54 eV and a power of 10.0 µW are used, while for UCPL,an excitation energy of 0.800 eV is used, with a power of 1000 µW.Scale bars represent 1 µm. (c) The excitation power dependence ofUCPL intensity with an excitation energy of 0.800 eV. The brokenline is a fit up to the power of 1090 µW. (d) Polarization dependenceof PL and UCPL intensity. Solid lines are fits to the expressiondescribed main text. PL and UCPL intensities in panels (c) and(d) are derived from the peak areas calculated via a Lorentzian fitto the emission spectra.UCPL. We perform a raster scan over the area containing thesame (9,8) SWNT and record an emission spectrum at eachposition. By extracting the emission intensity at the E11 energyfrom these spectra and replotting it in real space, we constructPL and UCPL images [Figs. 2(a) and 2(b), respectively]. Thespatial overlap of the bright regions in the intensity maps con-firm that the origin of the emission is the same nanotube. Bothexcitation images visualize the long SWNT suspended over atrench, consistent with the picture that PL brightness is influ-enced by exciton diffusion and end quenching [12,13,20,21].The UCPL excitation image has a lower resolution becausethe focused laser spot for UCPL measurements is larger by afactor of ∼2 than for PL.Well-known mechanisms of upconversion involve one-photon and two-photon absorption processes characterized bylinear and quadratic response, respectively, in the UCPL in-tensity to the excitation power [22]. We collect UCPL spectrawith various powers up to ∼2000 µW (Fig. S3 for the spectraand Fig. S4 for the excitation power-dependent UCPL inten-sity at the higher power within the Supplemental Material[19]) and extract the peak area using a Lorentzian fit to theE11 emission peak [Fig. 2(c)]. Our observation of the mostlylinear response can be explained by the one-photon processfacilitated by exciton-phonon scattering. This finding is incontrast with the expected quadratic response in two-photonexcitation and Auger recombination-mediated upconversion[23]. A subtle shift towards a sublinear trend is observed atpowers above 1000 µW, indicating the onset of the exciton-exciton annihilation regime [12,13].Excitation polarization dependence of the UCPL intensitycan be an evidence for intrinsic characteristics of the upcon-version process. Figure 2(d) shows the excitation polarizationdependence of PL and UCPL intensities for the same (9,8)SWNT. We fit the data to Imin + (Imax − Imin) sin2(φ + φ0),where Imin and Imax correspond to the minimum and maxi-mum intensity, respectively, φ is the excitation polarizationangle, and φ0 is the suspended angle of the SWNT withrespect to the length of the trench. From the fit parameters, wecompute the degree of polarization (Imax − Imin)/(Imax + Imin)to be 0.82 ± 0.11 for PL and 0.77 ± 0.12 for UCPL. Theanisotropic response reflects the one-dimensional character ofSWNTs [11,24,25], supporting the intrinsic nature of UCPLin air-suspended SWNTs. We note that the polarization degreeof UCPL in solution-processed SWNTs is lower than that ofPL [10], which is attributed to the extrinsic nature of UCPLmediated by defect states.We now shift our focus to examining UCPL across variouschiralities of SWNTs. To assign the chiralities efficiently, weperform three sets of PL measurements with excitation ener-gies of 1.36, 1.46, and 1.60 eV, which are near-resonant tomany chiralities. Assuming that excitation is close to the E22energy, the chiralities are assigned based on the E11 emissionenergy. UCPL spectra are collected from over 300 individualSWNTs with an excitation energy of 0.800 eV, and repre-sentative spectra are shown in Fig. 3(a). We find that UCPLis a universal phenomenon across the chiralities and can bedetected up to an emission energy of 0.997 eV, correspondingto an energy gain of 197 meV. Only UCPL spectra with a fullwidth at half maximum of 4–40 meV are analyzed to excludeemission background from substrates. This approach yields 11chiralities for subsequent analysis of UCPL spectra.The intensities of UCPL and PL spectra are compared toquantify the dependency of quantum yield on the energy gainrelative to the excitation energy [5]. We define the intensityratio as (IUCPL/ρUCPL)/(IPL/ρPL) where I is the spectrallyintegrated intensity and ρ is the excitation power densitywith subscripts denoting the emission process. The intensitiesIUCPL and IPL are obtained by fitting the UCPL and PL spectra,respectively, with a Lorentzian function. The ratios across thechiralities are plotted in Fig. 3(b) in which an exponentialincrease is primarily observed as the energy gain approacheszero. We find that the UCPL intensity is within an order ofmagnitude compared to the PL intensity around Eexc − E11 =−50 meV where Eexc is the excitation energy and E11 is theemission energy. It is noteworthy that the intensity ratios forsolution-processed nanotubes are on the same order of thosefor air-suspended SWNTs when comparing at similar energygains (see Sec. 1 within the Supplemental Material [19]).We observe a large dispersion in the emission intensityratio even within the same chiralities [Fig. 3(b)]. In our PLexperiments, we fix the excitation energies at 1.36, 1.46, or1.60 eV while variations in the E22 energies of the order of155418-3KOZAWA, FUJII, AND KATO PHYSICAL REVIEW B 110, 155418 (2024)(b)(a)FIG. 3. (a) Normalized UCPL spectra for various chiralities ofSWNTs collected with an excitation energy of 0.800 eV as indicatedby the broken line and an excitation power of 1000 µW. (b) Theintensity ratio as a function of the energy separation between theexcitation and emission energies. The color code of the plot corre-sponds to the chiralities shown in panel (a). The mean values of theratio for each chirality along with the ratio for as-dispersed and sp3-doped (6,5) SWNTs in aqueous dispersion [5] are plotted together forcomparison. The solid line represents the exponential fit to the datausing the function IUCPL/ρUCPLIPL/ρPL∝ exp( Eexc−E11Ea). The activation energyEa is assumed to be 25 meV, corresponding to the thermal energy atroom temperature.several meV occur against a linewidth of 40 meV [13,19].Such variations in E22 likely contribute to the dispersion of IPLinfluenced by a range of factors. In addition, local strain canbe introduced in SWNTs during the growth, which reducesthe band gap by 100 meV/% [26]. The inhomogeneity ofthe dielectric environment because of adsorption of watermolecules also leads to redshifts [27–30]. Another factor is thevariation in the suspended lengths, altering IPL where the con-version from a parity-even dark to parity-odd bright excitonwith an efficiency higher than 50% is possible in micrometer-long air-suspended SWNTs [18]. Furthermore, the differenceof the focused beam diameters can amplify the dispersionin the intensity ratio, where the diameter of 2.61 µm for theUCPL measurements is larger by a factor of ∼2 than the di-ameter for the PL measurements. The larger beam illuminatesthe entire region of a longer nanotube, resulting in a strongerdependence of the emission intensity on the suspended length.IV. EXCITATION RESONANCES OF UPCONVERSIONPHOTOLUMINESCENCEThe large dispersion can be circumvented by examin-ing UCPLE spectra within single SWNTs. In the UCPLEmeasurements, we employ a frequency-tunable single-modeexternal cavity laser, sweeping the excitation energy within anavailable tuning range from 0.756 to 0.838 eV. Although thisrange of the excitation energy is limited, different ranges ofEexc − E11 can be obtained by consolidating UCPLE spectrawith various chiralities. When we sweep the excitation energy,we hold the excitation power constant with a fiber-optic vari-able optical attenuator.Figure 4(a) shows UCPL and UCPLE spectra for a (14,3)SWNT, in which UCPLE spectrum shows a complex be-havior including a slight dip at 0.79 eV and an exponentialincrease towards higher energy. The entire trend can be cap-tured by compiling UCPLE spectra for the various chiralities{Fig. 4(b); see Fig. S5 within the Supplemental Material forreproducibility [19]}. In this plot, the UCPL intensities areplotted on a common energy axis relative to the E11 emis-sion energy for each chirality. We find that UCPLE intensityexponentially increases up to −0.14 eV, exhibits a plateauup to −0.08 eV, and increases exponentially with a hump at−0.06 eV. This departure from a single-exponential functioncan be explained by multiple resonances of phonons involvedin the upconversion process.One prominent phonon sideband of interest has been ob-served in PL, PLE, and absorption spectra, corresponding tothe sideband of the K-momentum dark singlet exciton coupledwith an in-plane transverse optical (iTO) phonon [31–34].The energy of the K-momentum exciton EK is located 22.2–35.5 meV above the energy of the bright exciton E11 forSWNTs with diameters ranging from 0.829 to 0.747 nm [32].In further analysis, we approximate the energy differenceEK − E11 as constant at 25 meV across all the chiralities.A PL spectrum on a logarithmic scale for a (11,3) SWNTis displayed on an energy axis relative to EK {Fig. 4(c);see Fig. S6 within the Supplemental Material for the other155418-4INTRINSIC PROCESS FOR UPCONVERSION … PHYSICAL REVIEW B 110, 155418 (2024)Γ KMomentumBright excitonħωEexcEnergyK-momentumdark excitonEKE11(e) (f)(d)(b)(a) (c)FIG. 4. (a) UCPL and UCPLE spectra for a (14,3) SWNT with an excitation power of 1000 µW. (b) UCPLE spectra across 11 differentchiralities of SWNTs, plotted as a function of the energy gain with an excitation power of 1000 µW. (c) Merged UCPLE spectrum normalizedby nB plotted on an energy axis relative to EK . A PL spectrum for a (11,3) nanotube is shown for comparison. (d) An excitonic band diagramwhere the arrows indicate transitions. (e) The normalized UCPLE spectrum, fitted by a second-order exciton-phonon scattering model. Thesolid curve is the sum of the fit and the broken lines are the individual components of the fit. γ = 12 meV is used for fits to all the three peaks.(f) Temperature-dependent UCPLE spectra for a (9,7) SWNT at an excitation power of 1000 µW. The solid curves are the fits. SWNTs withthe chiralities of (10,5), (9,7), (12,4), (11,6), (9,8), (15,1), (14,3), (10,8), (13,5), (12,5), and (11,7) from left to right are measured in panels (b),(c), and (e).chiralities [19]}. Three sidebands are observed across thechiralities and the energy differences from EK are determinedto be 70, 125, and 159 meV by peak deconvolution (Fig. S7within the Supplemental Material [19]). The third sidebandis consistent with the previous reports of the sideband arisingfrom the K-momentum iTO phonon mode [32,33]. We notethat the remaining sidebands have appeared in Refs. [29,34–37], but their assignments are to be clarified.To facilitate analysis of the UCPLE spectral features, weseamlessly merge the spectra across the various chiralities bythe following procedure. Since the UCPLE spectra do notnecessarily use the same set of excitation energies, we firstdefine a common energy axis and use interpolated valuesfrom the UCPLE spectra to allow for a direct comparisonof the UCPL intensities. In order to match the spectra in thesemi-log plot, the logarithm of the spectra is computed. Thesquare of the difference is calculated at each energy and thenintegrated over the overlapping region. The coefficient cα ,which scales the intensity of the spectrum with chirality α, isadjusted to minimize the expression εαβ = ∫ E fEi[log10(cβIβ ) −log10(cαIα )]2dE , where εαβ is the intensity deviation betweenspectra with chirality of α and β while Ei and Ef are the lowerand upper bounds, respectively, of the overlapping region. Weset the coefficient c0 = 1 for the first spectrum and adjust theothers to minimize the total deviation across all combinations∑N−1β=0∑N−1α=0 εαβ . After determining the optimal coefficients,we combine the spectra into a single graph, ensuring a smoothtransition between different chiralities (Fig. S8 within theSupplemental Material [19]). The merged UCPLE spectrumrepresents what would be obtained if we had a laser capableof sweeping across the entire energy range.In order to compare the sidebands of the PL spectra withthe merged UCPLE spectrum, it is needed to take into ac-count the phonon population determining the exciton-phononscattering rate in the anti-Stokes process. We normalize theintensity of the UCPLE spectrum by the Bose-Einstein distri-bution of phonons,nB = 1exp(EK −EexckBT)− 1, (1)where kB is the Boltzmann constant, and T is the temperature.In the normalized UCPLE spectrum [Fig. 4(c)], three peaksare observed with their positions coinciding with the phononsidebands in the PL spectra. Notably, the relative intensitiesof these peaks in the normalized UCPLE spectrum are largelysimilar to those of the phonon sidebands in the PL spectra. Wefirst interpret the UCPLE peak corresponding to the PL side-band at 159 meV as a peak arising from the well-establishedsideband of the K-momentum exciton coupled with the K-momentum iTO phonon mode. The remaining two peaks are155418-5KOZAWA, FUJII, AND KATO PHYSICAL REVIEW B 110, 155418 (2024)likely similar sidebands associated with other K-momentumphonon modes.We note that the possibility of processes involving exci-tons other than the K-momentum exciton are implausible.Within the exciton manifold in carbon nanotubes, there existthe parity-odd bright excitons, parity-forbidden dark excitons,and spin-forbidden dark excitons. For the transitions involvingparity- and spin-forbidden dark states, they are unlikely tooccur because of their negligible absorption cross sections.In addition, phonons cannot change parity nor spin [38], andtherefore phonon-mediated absorption and emission do notsatisfy the selection rules. While the parity-odd bright ex-citon coupled with phonons at the � point can satisfy themomentum conservation rule, the experimentally observed PLsideband position is inconsistent with the phonon energies[31–33]. The similarities between UCPLE and PL emissionspectra imply that the � phonons cannot explain the upcon-version process.UCPL involving the K-momentum phonons can be under-stood as a reverse process of the sideband emission observedin PL as depicted in Fig. 4(d). This process begins withthe optical excitation of an exciton to an intermediate, vir-tual state. The process is followed by upconversion intothe K-momentum dark exciton state, facilitated by couplingwith a K-momentum phonon with the energy h̄ω. Rapidscattering then occurs between the K-momentum dark andthe �-momentum bright states, leading to population equi-librium [18]. The process ends with the recombination ofthe exciton and emission of a photon while in the brightstate.To quantitatively interpret the spectral features of UCPLE,we develop a theoretical model based on the second-orderperturbation theory [39]. This model accounts for the inter-actions between photons and excitons as well as betweenexcitons and phonons. The expression for UCPL intensities isgiven byI (Eexc)nB∝∣∣∣∣∣∣N∑j=1MjEexc − h̄ω j − EK − iγ∣∣∣∣∣∣2, (2)where N is the number of the phonon modes, Mj is the matrixelement including absorption, exciton-phonon coupling, andemission processes, h̄ω j is the phonon energy for the jthmode, EK is the energy of the K-momentum dark singlet exci-ton, and γ is the damping constant related to the finite lifetimeof the intermediate state. We apply N = 3 as we observe threepeaks in the detection range of the UCPLE [Fig. 4(c)] andoptimize the fit parameters Mj , ω j , and γ .The model is fitted to the normalized UCPLE spectrum andshows good agreement except for the region where EK − E11is close to −0.05 eV [Fig. 4(e)]. The observed discrepancyfor smaller EK − E11 can be quantitatively explained by con-sidering the direct excitation of the E11 state. The fit toI (Eexc)/nB yields phonon energies of 74 ± 0.5, 135 ± 0.8,and 164 ± 0.3 meV, revealing a close resemblance to thephonon modes at the K point when compared to the phonondispersion in graphene [40–42]. These UCPLE peaks can bethus explained by the same model where UCPL is a reverseprocess of the sideband emission from the K-momentum darkexciton observed in PL spectra. We are now able to assignthe peaks to the out-of-plane transverse optical (oTO)/out-of-plane transverse acoustic (oTA), in-plane transverse acoustic(iTA), and iTO phonon modes at the K point, and the relativeamplitudes of the peaks ∝ Mj are obtained as 0.286 ± 0.008,0.281 ± 0.013, and 0.433 ± 0.013, respectively. Notably, Mjextracted from the model indicate that the oTO/oTA andiTA phonon modes exhibit lower scattering rate than the iTOmode. The assignment is further supported by near-linearbehavior in the excitation power dependence of UCPL spectrathat involves the respective phonon modes (Fig. S9 within theSupplemental Material [19]). It should be noted that a signa-ture of the oTO/oTA phonon modes has been detected in PLEspectra [32] and predicted by tight-binding calculations [43],which is consistent with our observation. The shoulder at theenergy difference of 45 meV relative to E11 in the PL spectra[29,34–37] can therefore also be assigned to the oTO/oTAphonon sideband of the K-momentum exciton. We note thatiLA/iLO phonons at the K point may also be involved, butthey are unresolved possibly because of the spectral vicinityto the iTO mode or insufficient intensity.We verify our model by examining the temperature depen-dence of UCPLE spectra. To control the sample temperatureduring the spectroscopy, the Si substrate is mounted on a flexi-ble resistive foil heater with an insertion of a thermal insulatorbetween the heater and the translation stage. The temperatureis monitored with a thermistor integrated into the heater and isensured to be within an error of 1% via proportional–integral–derivative feedback. Figure 4(f) shows the UCPLE spectra fora (9,7) SWNT at various temperatures. The spectra are fittedusing the same model as in Fig. 4(e), where only γ is variedwhile Mj and ω j are held constant across the temperatures.The model is able to reproduce the temperature dependenceof the UCPLE spectra that scale with nB, further reinforcingthe validity of our model. We note that this scaling has beenobserved in the solution-processed SWNTs [5]. The valuesof γ are found to be 11, 15, and 21 meV with temperaturesof 297.9, 319.0, and 339.6 K, respectively. This increasein γ with the temperature corresponds to shortening of theintermediate-state lifetime.We now discuss the mechanism underpinning the highefficiency of UCPL in our study. The key to this high effi-ciency lies in the resonant coupling with low-energy phonons,particularly the oTO/oTA modes, which are prevalent at roomtemperature. Moreover, the phonon-mediated upconversionprocess benefits from strong exciton-phonon coupling espe-cially near the K points in the Brillouin zone [31,32,43–46]. The coupling with the K-momentum phonons can bestronger when the dimensionality and the excitation energyare lower [46]. The high efficiency can also be explained byselective population of parity-odd bright excitons in the E11state. For excitation of E22 excitons, spontaneous dissociationof the excitons into free electron-hole pairs occurs during therelaxation process [47]. The free carriers are then distributedinto other singlet and triplet dark exciton states in additionto the bright exciton states [48,49]. For excitation below E11,excitons at the intermediate state are scattered into the Kpoint, and then reaches the population equilibrium betweenthe K-momentum dark and the �-momentum bright statescaused by the rapid transition process [18]. Considering that155418-6INTRINSIC PROCESS FOR UPCONVERSION … PHYSICAL REVIEW B 110, 155418 (2024)the lifetime of the K-momentum dark excitons is long andthe radiative quantum efficiency of the bright excitons is nearunity [50], almost all excitons in equilibrium should end upemitting light.V. CONCLUSIONIn conclusion, we have investigated the mechanism ofUCPL in as-grown SWNTs and shown that the UCPL is a re-verse process of the sideband emission from the K-momentumdark exciton observed in PL. The comparative analysis ofPL and UCPL measurements has confirmed that the emis-sion in the spectra and excitation images originates from thesame nanotube. We have attributed the nearly linear excita-tion power dependence to the one-photon and one-phononabsorption process. The polarization degree of UCPL has beenfound to be as high as that of PL, which is evidence for theintrinsic nature of the upconversion process. Our investiga-tion has covered eleven distinct SWNT chiralities exhibitingE11 emission in UCPL spectra, which indicates that UCPLis a universal phenomenon across the chiralities. ThroughUCPLE spectroscopy, we have detected three phonon-relatedpeaks, which arise from the sidebands of the K-momentumdark singlet excitons. We have successfully applied a second-order exciton-phonon scattering model, precisely identifyingphonon energies and relative matrix element amplitudes forthe scattering events with iTO, iTA, and oTO/oTA phononsinvolved in the upconversion processes. The robustness ofour model is further demonstrated by the ability to accuratelypredict the temperature-dependent UCPLE spectra. 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