Super-Resolution Fluorescence Imaging of Carbon Nanotubes Using a Nonlinear Excitonic Process
SSuper-Resolution Fluorescence Imaging of Carbon Nanotubes Using a NonlinearExcitonic Process
K. Otsuka,
1, 2
A. Ishii,
1, 2 and Y. K. Kato
1, 2, ∗ Quantum Optoelectronics Research Team, RIKEN Center for Advanced Photonics, Saitama 351-0198, Japan Nanoscale Quantum Photonics Laboratory, RIKEN Cluster for Pioneering Research, Saitama 351-0198, Japan
Highly efficient exciton-exciton annihilation process unique to one-dimensional systems is uti-lized for super-resolution imaging of air-suspended carbon nanotubes. Through the comparison offluorescence signals in linear and sublinear regimes at different excitation powers, we extract theefficiency of the annihilation processes using conventional confocal microscopy. Spatial images ofthe annihilation rate of the excitons have resolution beyond the diffraction limit. We investigateexcitation power dependence of the annihilation processes by experiment and Monte Carlo simula-tion, and the resolution improvement of the annihilation images can be quantitatively explained bythe superlinearity of the annihilation process. We have also developed another method in which thecubic dependence of the annihilation rate on exciton density is utilized to achieve further sharpeningof single nanotube images.
As a result of strong Coulomb interaction arising fromthe one-dimensional (1D) nature of single-walled carbonnanotubes (CNTs), electron-hole pairs form excitons thatare stable even at room temperature [1–3]. Confinementand diffusion [4–7] of the excitons in a nanotube leadto their efficient annihilation process upon collision withone another [8–10], resulting in a peculiar cubic depen-dence of the exciton-exciton annihilation (EEA) rate onthe density of excitons [11, 12]. The efficient EEA can be,for example, utilized for room-temperature single photongeneration at telecommunication wavelengths [13, 14].The diameter-dependent wavelength of nanotube fluores-cence also includes the near-infrared window, where scat-tering is small and absorption is weak, allowing for deep-tissue imaging using CNTs as fluorescent agents [15–18].As advanced techniques for super-resolution imaging [19–22], such as two-photon excitation microscopy and stim-ulated emission depletion microscopy, rely on the nonlin-ear optical response in fluorescence agents [23–27], theEEA process could play a key role in the development ofnanotube-based biological imaging as well.Here we demonstrate subdiffraction imaging of air-suspended CNTs by extracting the nonlinear EEA com-ponent using a typical confocal microscopy system. Bycombining two fluorescence images obtained at differ-ent excitation powers, an EEA rate image with en-hanced resolution can be constructed. Excitation power-dependence of the extracted EEA efficiency and the spa-tial resolution of the EEA imaging are experimentallyinvestigated, and we perform Monte Carlo simulation ofthe EEA process to identify the resolution limit of thistechnique. In addition to the use of nonlinearity betweenthe EEA rate and the exciton generation rate, the cu-bic dependence of the EEA rate on exciton density isutilized in another protocol for super-resolution imagingof CNTs. By measuring the excitation power requiredto establish a predefined exciton density, we are able to ∗ Corresponding author. [email protected] achieve even narrower widths for isolated nanotube im-ages.
RESULTS AND DISCUSSION
Excitation Power Dependence.
Our samples are as-grown nanotubes suspended over trenches on silicon sub-strates. A schematic and a scanning electron micrographof an air-suspended nanotube are shown in Figs. 1(a) and1(b), respectively. We perform photoluminescence (PL)measurements on such samples using a homebuilt con-focal microscopy system [11], in which excitation laserpower is controlled in a wide range by a continuouslyvariable neutral density filter.Figure 1(c) shows PL excitation spectroscopy data per-formed on a single air-suspended nanotube with a length L ≈ . µ m, and we determine the tube chirality to be(11,3). We then investigate laser polarization dependenceof PL intensity with excitation energy at the E reso-nance [inset of Fig. 1(d)]. For the following PL mea-surements, the polarization angle and the excitation en-ergy are fixed parallel to the tube axis and at the E resonance, respectively. Figure 1(d) shows an image ofthe integrated PL intensity I PL obtained with a ∼ E resonance. The width of the nanotube PL image ispredominantly determined by the excitation laser beamprofile, and thus much larger than the actual nanotubediameter of 1.0 nm.The resolution enhancement will be achieved by usingthe efficient EEA process, whose effects can be observedin the power dependence of PL intensity [Fig. 2(a), blacksquares]. I PL is proportional to the excitation power atlow powers, but shows a cubic root dependence at highpowers due to EEA [11]. The blue line in Fig. 2(a) isa linear fit to the low-power results, corresponding tothe PL intensity expected in the absence of EEA. Thedeviation of actual PL intensity from the blue line wouldcorrespond to the EEA rate Γ EEA . a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Laser beam
Emission energy (eV)0.8 0.9 1.0 1.11.41.51.61.7 (c)
CNTCatalyst (a) (d)(b)
CNT
FIG. 1. (a) A schematic of an air-suspended nanotubesample. For optical imaging, the samples are scanned alongtrenches relative to the fixed laser beam. (b) A scanningelectron micrograph of a typical nanotube. (c) A PL exci-tation map for a (11,3) nanotube. The excitation power is0.1 µ W. (d) A PL image of the nanotube measured in (c).Inset: Polarization dependence of I PL . Excitation power andenergy used for imaging are 0.1 µ W and E = 1 .
590 eV,respectively, and the image is extracted at an emission en-ergy of E = 1 .
055 eV with a spectral integration window of ∼
50 meV. Scale bars in (b) and (d) are 500 nm.
The efficient EEA process also affects the imaging res-olution. We measure 1D I PL profiles of the nanotubealong the trench direction with various excitation pow-ers, and the full-width at half-maximum (FWHM) w h of the profiles are plotted as red circles in Fig. 2(a) asa function of the excitation power. At the low powerlimit, PL intensity profiles reflect the excitation laserbeam profile because I PL is proportional to the excitongeneration rate g in the absence of EEA. At the otherextreme where I PL is proportional to g / due to thecubic-law EEA process, we expect the effects of nonlin-earity on the width of the PL intensity profile. If weapproximate the laser beam profile by a Gaussian func-tion exp( − x /r ) with r being the laser 1/ e radius, theintensity profile of PL that has g α dependence becomes I PL ( x ) ∝ exp( − αx /r ) = exp[ − x / ( r/ √ α ) ], where α is the power exponent of the generation rate depen-dence. The FWHM changes by a factor r/ √ α √ r √ √ α , (1)and it is reasonable that the width of I PL profile increasesby a factor of √ Extracting the Influence of EEA.
Although it mayseem as if the EEA process has a negative effect on res- x position (μm) I P L ( a . u . ) N o r m a li z e d i n t e n s i t y ( a . u . ) Excitation power (μW)10 −2 −1 −2 I P L ( a . u . ) F W H M ( μ m ) −1 (a) (b) x13 (c) FIG. 2. (a) Excitation power dependence of PL intensity I PL with the laser focused at the center of the nanotube (blacksquares) and FWHM of PL intensity profiles along the trenchdirection (red circles). The nanotube is the same one as inFig. 1(c) and 1(d). I PL is obtained by calculating the peakarea of a Lorentzian fit to the emission spectrum. (b) PL in-tensity profiles scaled by excitation powers. Difference of theblack and the blue lines is proportional to the EEA rate Γ EEA .(c) Normalized profiles of I PL for the two different excitationpowers and the EEA rate obtained from the subtraction in(b). olution, we can utilize the nonlinear power dependenceon the generation rate to achieve enhanced spatial reso-lution. We extract the EEA component from PL inten-sities measured at two different excitation powers, wherethe EEA extraction power P is higher than the refer-ence power P . Figure 2(b) illustrates how the EEAcomponent is extracted. The black line is the PL in-tensity profile obtained with the EEA extraction power P = 2 . µ W, showing slight broadening caused by EEA.The blue line is the intensity profile at P = 0 . µ Wscaled by the ratio of the excitation powers, which isequivalent to the expected PL intensity profile in the ab-sence of the EEA process. The difference between theblack and blue lines (red region) is proportional to theEEA rate Γ
EEA in the nanotube.Figure 2(c) shows normalized profiles of raw PL in-tensity (black and blue for P and P , respectively) andthe EEA component extracted by the protocol describedabove (red). The width of the intensity profile at theexcitation power of 0.2 µ W corresponds to the mini-mum width obtained in the power-dependent width inFig. 2(a), but the profile of the extracted EEA compo-nent has an even smaller width. As Γ
EEA increases super-linearly with g , the spatial profile of the EEA componentpeaks sharply when the laser is centered on the nanotube.To characterize the resolution and signal-to-noise ra-tio, we perform experiments using a range of excitationpower combinations for extracting the EEA rate Γ EEA .Measurements are repeated 7 times for each combinationof the EEA extraction power P and the reference power P , and the average width of the Γ EEA profiles is plottedin Fig. 3(a) as a function of P and P . Roughly speak-ing, large P and P result in large w h of the Γ EEA pro-files, while low powers are preferable for high-resolutionimaging.We consider the widths obtained at the lowest P =0 . µ W to quantitatively evaluate the resolution im-provement. Figure 3(b) shows the P dependence of w h obtained from the raw PL intensity profiles (blacksquares) and the extracted EEA profiles with P fixed at0.05 µ W (red circles). Error bars represent the stan-dard deviation for the repeated measurements. TheFWHM from raw PL reproduces the behavior observed inFig. 2(a), and the error values are similar over the entirerange of the excitation power. In comparison, the widthfor EEA rate profiles decreases from ∼
500 to ∼
350 nmas P decreases, which is approximately an improvementby a factor of √
2. We note that the width of the EEAprofiles has large error bars at low powers because of thesmall Γ
EEA signals that depends superlinearly on g . Monte Carlo Simulation.
The minimum widthachievable from the EEA profile in an ideal situationwith negligible noise is investigated by conducting MonteCarlo simulation [11, 28] using parameters directly com-parable with the experimental results. Excitonic pro-cesses, such as exciton generation, diffusion, and decay,are stochastically evaluated. Excitation profile has a spa-tial distribution exp[ − x + y ) /r ], where x and y arepositions perpendicular to and along the nanotube, re-spectively. 1D rate profiles are calculated by moving thenanotube position in the x direction, simulating the ex-periment performed to obtain the I PL profiles shown inFig. 2(b). We compute the intrinsic decay rate Γ I fromthe time-averaged number of excitons that go through theintrinsic decay, which corresponds to the PL intensity inthe experiments. The simulations are repeated for vari-ous photon incident rates, and the EEA rate profiles arecomputed from Γ I profiles in a manner similar to the mea-surements for Fig. 3(a). Figure 3(c) shows the FWHMof the extracted EEA rate profiles as a function of theEEA extraction and reference generation rates. Data areplotted in terms of a unitless parameter gτ , representingthe average number of excitons that are generated duringthe exciton intrinsic decay lifetime τ . Note that we usethe exciton generation rate evaluated with the nanotubeat the center of the laser beam in Fig. 3, in which case itbecomes equivalent to the excitation power in the exper-iments. The simulation results exhibit a trend similar tothe experiments; large w h for combinations of large gτ and vice versa .Using the simulation data, it is possible to directlyevaluate the EEA rates Γ EEA . In Fig. 3(d), the FWHMof the EEA profiles as well as the intrinsic decay pro-files are plotted, and both reproduce the overall behaviorof the experimental data. As the generation rate is de-creased, the width of the intrinsic decay approaches thatof the simulated laser profile as indicated by the solidgray line. The EEA profile width, in comparison, is al-ready below the solid line at large generation rates. Thewidth decreases with the generation rate, and approachesthe broken gray line which corresponds to 1/ √ w h implies that the EEA rate has a nontrivial dependence −1 P (μW)0.30.50.7 F W H M ( μ m ) Raw PL EEA ( P =0.05 μW) (b) −1 −1 P (μW) P ( μ W ) gτ g τ F W H M ( μ m ) (d) Intrinsic decay EEA gτ −1 −1 PL intensity (a.u.) Γ EE A ( a . u . ) (f) N simulationexperiment 10 −1 −2 −3 −2 −1 Γ EE A τ Excitation power (μW) gτ −2 −1 −1 −1 −2 −1 −2 Γ τ I n t e n s i t y ( a . u . ) experiment (e) simulationEEA PL F W H M ( μ m ) (c) F W H M ( μ m ) (a) FIG. 3. (a) FWHM of 1D EEA profiles as a function of theEEA extraction power P and the reference power P usedfor the subtraction of I PL . The data are averaged for sevenmeasurements repeated with the same condition. (b) FWHMof I PL and the extracted EEA rate profiles for a fixed powerof P = 0 . µ W. Error bars represent the standard devi-ation. Solid and broken gray lines correspond to the laserbeam FWHM and that multiplied by 1 / √
2, respectively. (c)FWHM as a function of the EEA extraction and the refer-ence generation rate from the Monte Carlo simulations. (d)FWHM of the intrinsic decay rate profiles and the EEA rateprofiles from the simulations. (e) Excitation power depen-dence of the PL intensity and the extracted EEA componentfor the experiments and the simulations. (f) EEA rates as afunction of I PL and N from the experiments and the simula-tions, respectively. on the exciton generation rate. EEA Rate vs Generation Rate.
We experimentallyconfirm the g dependence of Γ EEA by measuring the PLintensity [Fig. 3(e), open circles] and subtracting a linearfit to the low power PL data (open diamonds). Similarly,the gτ dependence of the intrinsic decay rate (filled cir-cles) and the EEA rate (filled diamonds) obtained fromthe simulation is overlaid in Fig. 3(e) in terms of unitlessparameters Γ I τ and Γ EEA τ , respectively. The simulationis in good agreement with the experiment, showing thatour simple model accurately describes the behavior of ex-citons in nanotubes. In this log-log plot, Γ I has slopes of1 and 1/3 at low and high generation rates, respectively,as also observed in Fig. 2(a). The slope of Γ EEA is ofour interest, which is 2 at low powers and approaches1 as the power is increased. Through Eq. (1), the ob-served values of the slope can quantitatively explain thepower dependence of w h obtained from the EEA profiles[Fig. 3(d)].The Γ EEA slope of 1 at high power is reasonable, be-cause g ≈ Γ EEA with EEA being the dominant decayprocess. The quadratic dependence of Γ
EEA , however,seems to contradict with the cubic dependence expectedfor EEA [11]. In order to resolve this apparent inconsis-tency, Γ
EEA τ (filled diamonds) is plotted as a functionof time-averaged exciton number N = Γ I τ [Fig. 3(f)],and we find a transition from the quadratic to cubic de-pendence near N = 1. Similarly, the experimentally ex-tracted EEA rate is replotted as a function of I PL (opendiamonds) in Fig. 3(f), which coincides with the simula-tion result. The transition in the N dependence of theEEA rate can be explained by the fact that EEA processoccurs only when multiple excitons coexist in a nanotube.When N >
1, Γ
EEA in a 1D system is proportional to N as discussed previously [11]. When the exciton number N (cid:28)
1, however, the situation changes. The probabilityfor the instantaneous exciton number λ ≥ λ = 1. The EEArate is then dominated by the case where two excitonscoexist, whose probability is given by the Poisson distri-bution p ( λ ) = N λ e − N /λ ! to be p (2) ≈ N /
2. In thisregime, multiple exciton generation, rather than excitondiffusion, is the limiting factor of the EEA process.We can further obtain an explicit expression for theEEA rate. After time t from the first exciton gen-eration, the survival probability of the first excitonand the arrival probability of the second exciton are p s ( t ) = exp( − t/τ ) and p a ( t ) = gt exp( − gt ), respec-tively. The expected value of the arrival time interval (cid:82) tp s ( t ) p a ( t ) dt/ (cid:82) p s ( t ) p a ( t ) dt ≈ τ of coexisting excitonsis independent of N when N (cid:28)
1. The initial distancebetween the two excitons after diffusion is therefore con-stant ( ∼ (cid:112) Dτ /π ) with D being the diffusion constant.Despite the diffusion of the first exciton, the collision rateof such exciton pairs is then ∼ π/ τ under the conditionof λ = 2, and Γ EEA = p (2) × π/ τ = πN / τ .
2D EEA Imaging.
Having understood the mecha-nism for the resolution improvement, we perform two-dimensional (2D) imaging of the raw PL intensity andthe EEA rate. According to the analysis performed inFigs. 3(a) and 3(c), the reference power P should besmall enough to avoid EEA while the EEA extractionpower P also should be in the linear regime of I PL vsthe excitation power in order to obtain high resolutionimages. We note, however, that P should not be tooclose to P to keep the signal-to-noise ratio sufficientlyhigh. We thus choose P = 0 . µ W and P = 0 . µ W,where the excitation power is modulated by switching the I n t e n s i t y ( a . u . ) (c) (f) (i)(a) (d) (g)(b) (e) (h) (9,7) (9,7) (9,8) I P L ( a . u . ) Γ EE A ( a . u . ) FIG. 4. 2D images of a (9,7) nanotube for (a) the PL in-tensity and (b) the extracted EEA rate. (c) 1D profiles fromthe PL (black) and EEA (red) images at the same positionindicated by broken white lines. (d–i) Similar sets of 2D im-ages and the 1D profiles for the two nanotubes with (d–f) aparallel and (g–i) a Y-shaped configuration. The excitationenergies are fixed at the E of each nanotube, and all the im-ages were extracted at the E energy with a spectral windowof ∼
50 meV. All the images are normalized by their maximumintensity so that the image resolution can be easily compared.Scale bars are 500 nm. neutral density filters at every step of the sample scan.Figure 4 displays the optical images for three config-urations of nanotubes. Because P = 0 . µ W used forFigs. 4(a,d,g) is in the linear regime of I PL , the width ofthe raw PL image is solely limited by the laser beam pro-file and cannot be further reduced by lowering the power.In the simplest case of a single nanotube, the EEA im-age of the (9,7) nanotube [Fig. 4(b)] has even smallerwidth than the PL image [Fig. 4(a)]. We note that theEEA image represents the degree of PL intensity reduc-tion through the EEA process. When two (9,7) tubesare lying closely, the tubes are more clearly resolved byextracting the EEA component, which is also apparentfrom the 1D profiles shown in Fig. 4(f). For a Y-shapedjunction of (9,8) nanotubes, we show 1D profiles at theposition where the I PL profile of the two tubes cannotbe resolved [black line in Fig. 4(i)]. These nanotubes aresuccessfully separated through the super-resolution imag-ing of EEA rates. It is noteworthy that the nonlinearityin nanotubes enables subdiffraction imaging at a powerdensity as low as ∼
300 W/cm with a continuous wavelaser. This value is, for example, two-orders of magnitudesmaller than that used in the high-resolution microscopywhich uses fluorescence saturation of Rhodamine 6G [26],benefiting from the highly mobile excitons confined in1D [10]. Cubic-Law EEA at High Powers.
We now considerwhether it is possible to utilize the cubic dependence ofthe EEA rate Γ
EEA on N to achieve even higher res-olution images. In the imaging protocol employed inFigs. 2–4, the g dependence of Γ EEA is used [Proto-col I, left panel of Fig. 5(a)]. In comparison, we intro-duce a protocol that exploits the N term of the EEArate that appears at large N [Protocol II, right panelof Fig. 5(a)]. Instead of laser position dependent exci-ton generation rate g ( x ) resulting from a constant exci-tation power in Protocol I, we create laser position de-pendent exciton number N ( x ) that reproduces the laserbeam profile. The EEA rate extracted from such N ( x )should allow for resolution enhancement through the cu-bic dependence. As briefly illustrated in Fig. 5(b), wefirst obtain a PL intensity profile I ( x ) at a sufficientlylow excitation power P to purely extract the generationrate profile g ( x ). Then I ( x ) is multiplied by a con-stant κ >
1, which will be the target PL intensity κI ( x )for the next scan. We reproduce the target PL intensityprofile κI ( x ) by controlling the excitation power P ( x )to compensate for the intensity loss due to EEA. P ( x ) istuned at every point until the intensity difference betweenthe measurement and κI ( x ) becomes less than 3%. Theposition-dependent exciton generation rate for this scan g ( x ) = g ( x ) P ( x ) /P is used to extract the EEA rateΓ EEA ( x ). As the EEA rate is equal to the additional ex-citon generation rate compared with that expected in theabsence of EEA, Γ EEA ( x ) = ∆ g ( x ) = g ( x ) − κg ( x ).Figures 5(c) and 5(d) show the raw PL image and theEEA rate image, respectively, obtained through ProtocolII. Since larger κ gives smaller w h , we choose κ = 25 and P = 0 . µ W to fully utilize the N nonlinearity at highpowers, resulting in the maximum P ( x ) = 53 µ W. Crosssectional profiles of these images are shown in Fig. 5(e).Clearly, the EEA rate image shows a narrower profile. Ifwe take horizontal cross sections of the images and av-erage the width along the length of the tube, the EEAimage gives an average w h of 290 nm, while that for thePL image is 520 nm. The width reduction correspondsto a factor of √ N dependence.As shown in Fig. 5(f), this method can resolve multipleCNTs lying closely as a result of the improved spatialresolution. It should be noted, however, that the spatialresolution of the image does not always improve as muchas the width observed for single tubes. Unlike the case inProtocol I, where the linear component of the PL is can-celed out during the subtraction, the linear componentfrom adjacent nanotubes is included in the signal dur-ing the power tuning step of Protocol II. The inclusionof the linear component inevitably causes a reduction ofthe nonlinear EEA component, resulting in a suboptimalresolution for high excitation powers. CONCLUSIONS
In summary, we have performed super-resolution imag-ing of air-suspended CNTs by visualizing the efficientEEA process using two protocols compatible with typicalconfocal microscopy systems. By subtracting the linear
Generation rate Exciton number Γ EE A ∝ g ∝ g ∝ N ∝ N Prot. I Prot. II (i) Low power measurement I ( x ) ∝ g ( x ) κI ( x )(ii) Power tuning to get κI ( x ) P : const. P ( x ) κI ( x )(iii) Conversion to EEA rate Δg = g–κg = Γ EEA g = g P/P ∝ N ( x ) Γ EE A (c)(d) (a) (b) (cid:3400) 𝜅 x position (μm)0−0.5−1 0.5 1x position (μm)0−0.5−1 0.5 1 (e) N o r m a li z e d i n t e n s i t y (f) PLEEA PLEEA I PL Γ EEA
FIG. 5. (a) Illustration for utilizing the nonlinearity of Γ
EEA as a function of the exciton generation rate g or the excitonnumber N . While quadratic nonlinearity of Γ EEA at small g is used for Protocol I (used in Figs. 1–4), cubic nonlinearityof Γ EEA against N is used in protocol II. (b) Schematics ofthe protocol to extract the cubic nonlinearity. (c) PL and (d)EEA images of the (11,3) nanotube with P = 0 . µ W and κ = 25. Scale bars are 500 nm. (e) 1D intensity profiles ofthe nanotube shown in (c) and (d). (f) Intensity profiles ofthe nanotubes shown in Figs. 4(d–f). PL component to extract the nonlinear EEA rate, thespatial resolution improves by a factor of √ N dependent EEA rate at low exci-ton numbers, as confirmed by Monte Carlo simulations.To utilize the N dependence unique to the 1D systemof nanotubes, we have developed another protocol forsuper-resolution imaging in which the excitation poweris adaptively tuned during the measurement. Using thesecond protocol, the width of a single nanotube can be asnarrow as 1 / √ METHODS
Air-Suspended Carbon Nanotubes.
The air-suspendednanotubes are synthesized by alcohol chemical vapor de-position [11, 29]. Trenches are formed on Si substratesthrough electron beam lithography and dry etching. Weuse Fe(III) acetylacetonate and fumed silica in ethanolas catalyst, where spin-coating and lift-off processes areused to deposit the catalyst in a region defined by an-other lithography process. After heating in air at 400 ◦ Cfor 5 min, CNTs are synthesized at 800 ◦ C for 1 min usingAr and H flowing through an ethanol bubbler. Photoluminescence Microscopy.
A homebuilt confo-cal microscopy system is used to perform PL measure-ments at room temperature [11, 30–32]. A wavelength-tunable Ti:sapphire laser is used for excitation after con-trolling its power and polarization by neutral density fil-ters and a half-wave plate, respectively. The laser beam isfocused on the samples using an objective lens with a nu-merical aperture of 0.8 and a working distance of 3.4 mm.PL is collected through the same objective lens and de-tected using a liquid-nitrogen-cooled InGaAs diode ar-ray attached to a spectrometer. With the laser beamposition fixed, we scan the samples mounted on a motor-ized three-dimensional stage to achieve focusing, samplesearch over the entire chip, and imaging of the singletubes. All measurements are performed in dry nitrogento avoid formation of oxygen-induced defects [6, 33].
Monte Carlo Simulation.
We use the same method asin Ref. 11, and modify the excitation profile to reproducethe spatial scanning of the sample. Excitonic processesare evaluated at time intervals ∆ t ≤ − τ . For excitongeneration, photons are supplied into the system at arate with a spatial probability distribution given by a 2DGaussian function. Excitons are generated from the pho- tons on the nanotube with a photon absorption width of5 nm. Note that a width thicker than the physical widthof the nanotube does not affect the following discussionbased on the number of generated excitons gτ , and it isused to reduce computational load. Probability for theexciton displacement s due to diffusion is given by thenormal distribution √ πD ∆ t exp( − s D ∆ t ). In addition tothe intrinsic decay process that occurs with the probabil-ity of ∆ t/τ , the excitons that diffused beyond the tubeends disappear from the system. When two excitons passby one another, either one of them is eliminated, whilethe other exciton remains unchanged. The tube length L and the laser radius r are 1.2 µ m and 0.4 µ m, respectively,while the diffusion length √ Dτ = 1 µ m is assumed. ACKNOWLEDGMENTS
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