Highly efficient exciton-exciton annihilation process unique to one-dimensional systems is utilized for super-resolution imaging of air-suspended carbon nanotubes. Through the comparison of fluorescence signals in linear and sublinear regimes at different excitation powers, we extract the efficiency of the annihilation processes using conventional confocal microscopy. Spatial images of the annihilation rate of the excitons have resolution beyond the diffraction limit. We investigate excitation power dependence of the annihilation processes by experiment and Monte Carlo simulation, and the resolution improvement of the annihilation images can be quantitatively explained by the superlinearity of the annihilation process. We have also developed another method in which the cubic dependence of the annihilation rate on exciton density is utilized to achieve further sharpening of single nanotube images.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
As a result of strong Coulomb interaction arising from the one-dimensional (1D) nature of single-walled carbon nanotubes (CNTs), electron-hole pairs form excitons that are stable even at room temperature [1–3]. When multiple excitons exist in a CNT, an exciton can scatter with another exciton and decay nonradiatively. In this Auger process known as exciton-exciton annihilation (EEA) [4–6], the exciton recombines by giving its energy to theother exciton. Confinement and diffusion [7–10] of the excitons in a nanotube lead to efficient EEA process upon collision with one another, resulting in a peculiar cubic dependence of the EEA rate on the density of excitons [11, 12]. The efficient EEA can be, for example, utilized in room-temperature single photon generation at telecommunication wavelengths [13, 14]. The diameter-dependent wavelength of nanotube fluorescence also includes the near-infrared window, where scattering is small and absorption is weak, allowing for deep-tissue imaging using CNTs as fluorescent agents [15–18]. As advanced techniques of super-resolution imaging [19–22], such as two-photon excitation microscopy and stimulated emission depletion microscopy, rely on the nonlinear optical response in fluorescence agents [23–28], the EEA process could play a key role in the development of nanotube-based biological imaging as well.
Here we demonstrate subdiffraction imaging of air-suspended CNTs by extracting the nonlinear EEA component using a typical confocal microscopy system. By combining two fluorescence images obtained at different excitation powers, an EEA rate image with enhanced resolution can be constructed. Excitation power-dependence of the extracted EEA efficiency and the spatial resolution of the EEA imaging are experimentally investigated, and we perform Monte Carlo simulation of the EEA process to identify the resolution limit of this technique. In addition to the use of nonlinearity between the EEA rate and the exciton generation rate, the cubic dependence of the EEA rate on exciton density is utilized in another protocol for super-resolution imaging of CNTs. By measuring the excitation power required to establish a predefined exciton density, we are able to achieve even narrower widths for isolated nanotube images.
2. Air-suspended nanotubes and optical setup
The air-suspended nanotubes are synthesized by alcohol chemical vapor deposition [11, 29]. Trenches are formed on Si substrates through electron beam lithography and dry etching. We use Fe(III) acetylacetonate and fumed silica in ethanol as catalyst, where spin-coating and lift-off processes are used to deposit the catalyst in a region defined by another lithography process. After heating in air at 400C for 5 min, CNTs are synthesized at 800C for 1 min using Ar and H flowing through an ethanol bubbler. A schematic and a scanning electron micrograph of an air-suspended nanotube are shown in Figs. 1(a) and 1(b), respectively.
A homebuilt confocal microscopy system is used to perform photoluminescence (PL) measurements at room temperature [11, 30–32]. A wavelength-tunable Ti:sapphire laser is used for excitation after controlling its power and polarization by neutral density filters and a half-wave plate, respectively. The laser beam is focused on the samples using an objective lens with a numerical aperture of 0.8 and a working distance of 3.4 mm. PL is collected through the same objective lens and detected using a liquid-nitrogen-cooled InGaAs diode array attached to a spectrometer. With the laser beam position fixed, we scan the samples mounted on a motorized three-dimensional stage to achieve focusing, sample search over the entire chip, and imaging of the single tubes. All measurements are performed in dry nitrogen to avoid formation of oxygen-induced defects [9, 33].
3. Extraction of the influence of exciton-exciton annihilation
Figure 1(c) shows PL excitation spectroscopy data performed on a single air-suspended nanotube with a length m, and we determine the tube chirality to be (11,3). We then investigate laser polarization dependence of PL intensity with excitation energy at the E22 resonance [inset of Fig. 1(d)]. For the following PL measurements, the polarization angle and the excitation wavelength are fixed parallel to the tube axis and at the E22 resonance, respectively. Figure 1(d) shows an image of the integrated PL intensity IPL obtained with a ∼50-nm-wide spectral integration window centered at the E11 resonance. The width of the nanotube PL image is predominantly determined by the excitation laser beam profile, and thus much larger than the actual nanotube diameter of 1.0 nm.
The resolution enhancement will be achieved by using the efficient EEA process, whose effects can be observed in the power dependence of PL intensity [Fig. 2(a), black squares]. IPL is proportional to the excitation power at low powers, but shows a cubic root dependence at high powers due to EEA . The blue line in Fig. 2(a) is a linear fit to the low-power results, corresponding to the PL intensity expected in the absence of EEA. The deviation of actual PL intensity from the blue line would correspond to the EEA rate .
The efficient EEA process also affects the imaging resolution. We measure 1D IPL profiles of the nanotube along the trench direction with various excitation powers, and the full-width at half-maximum (FWHM) wh of the profiles are plotted as red circles in Fig. 2(a) as a function of the excitation power. At the low power limit, PL intensity profiles reflect the excitation laser beam profile because IPL is proportional to the exciton generation rate g in the absence of EEA. At the other extreme where IPL is proportional to due to the cubic-law EEA process, we expect the effects of nonlinearity on the width of the PL intensity profile. If we approximate the laser beam profile by a Gaussian function with r being the laser 1/e2 radius, the intensity profile of PL that has dependence becomes , where α is the power exponent of the generation rate dependence. The FWHM changes by a factor
Although it may seem as if the EEA process has a negative effect on resolution, we can utilize the nonlinear power dependence on the generation rate to achieve enhanced spatial resolution. We extract the EEA component from PL intensities measured at two different excitation powers, where the EEA extraction power P1 is higher than the reference power , where . Figure 2(b) illustrates how the EEA component is extracted. The black line is the PL intensity profile W), showing slight broadening caused by EEA. The blue line is the intensity profile W) scaled by κ, which is equivalent to the expected PL intensity profile in the absence of the EEA process. The difference between the black and blue lines (red region) is proportional to the EEA rate in the nanotube.
Figure 2(c) shows normalized profiles of raw PL intensity (black and blue for P1 and P2, respectively) and the EEA component extracted by the protocol described above (red). The width of the intensity profile at the excitation power of 0.2 μW corresponds to the minimum width obtained in the power-dependent width in Fig. 2(a), but the profile of the extracted EEA component has an even smaller width. As increases superlinearly with g, the spatial profile of the EEA component peaks sharply when the laser is centered on the nanotube.
To characterize the resolution and signal-to-noise ratio, we perform experiments using a range of excitation power combinations for extracting the EEA rate . Measurements are repeated 7 times for each combination of the EEA extraction power P1 and the reference power P2, and the average width of the profiles is plotted in Fig. 3(a) as a function of P1 and P2. Roughly speaking, large P1 and P2 result in large wh of the profiles, while low powers are preferable for high-resolution imaging.
We consider the widths obtained at the lowest W to quantitatively evaluate the resolution improvement. Figure 3(b) shows the P1 dependence of wh obtained from the raw PL intensity profiles (black squares) and the extracted EEA profiles with P2 fixed at 0.05 μW (red circles). Error bars represent the standard deviation for the repeated measurements. The FWHM from raw PL reproduces the behavior observed in Fig. 2(a), and the error values are similar over the entire range of the excitation power. In comparison, the width for EEA rate profiles decreases from ∼500 to ∼350 nm as P1 decreases, which is approximately an improvement by a factor of . We note that the width of the EEA profiles has large error bars at low powers because of the small signals that depends superlinearly on g, becoming comparable to photon shot noise and detector noise.
4. Comparison with Monte Carlo simulation results
The minimum width achievable from the EEA profile in an ideal situation with negligible noise is investigated by conducting Monte Carlo simulation [11, 34] using parameters directly comparable with the experimental results. We use the same method as , and modify the excitation profile to reproduce the spatial scanning of the sample. Excitonic processes, such as exciton generation, diffusion, and decay, are evaluated at time intervals , where τ is the lifetime of the exciton intrinsic lifetime. For exciton generation, photons are supplied into the system at a rate with a spatial probability distribution given by a Gaussian function , where x and y are positions perpendicular to and along the nanotube, respectively. Excitons are generated from the photons on the nanotube with a photon absorption width of 5 nm. Note that a width thicker than the physical width of the nanotube does not affect the following discussion based on the number of generated excitons , and it is used to reduce computational load. Probability for the exciton displacement s due to diffusion is given by the normal distribution with D being the diffusion constant. In addition to the intrinsic decay process that occurs with the probability of , the excitons go through extrinsic decay (end quenching and EEA) after the diffusion. The tube length L and the laser radius r are 1.2 μm and 0.4 μm, respectively, while the diffusion length m is assumed.
1D rate profiles are calculated by moving the nanotube position in the x direction, simulating the experiment performed to obtain the IPL profiles shown in Fig. 2(b). We compute the intrinsic decay rate from the time-averaged number of excitons that go through the intrinsic decay, which corresponds to the PL intensity in the experiments. The simulations are repeated for various photon incident rates, and the EEA rate profiles are computed from profiles in a manner similar to the measurements for Fig. 3(a). Figure 3(c) shows the FWHM of the extracted EEA rate profiles as a function of the EEA extraction and reference generation rates. Data are plotted in terms of a unitless parameter , representing the average number of excitons that are generated during τ. Note that we use the exciton generation rate evaluated with the nanotube at the center of the laser beam in Fig. 3, in which case it becomes equivalent to the excitation power in the experiments. The simulation results exhibit a trend similar to the experiments; large wh for combinations of large and vice versa.
Using the simulation data, it is possible to directly evaluate the EEA rates . In Fig. 3(d), the FWHM of the EEA profiles as well as the intrinsic decay profiles are plotted, and both reproduce the overall behavior of the experimental data. As the generation rate is decreased, the width of the intrinsic decay approaches that of the simulated laser profile as indicated by the solid gray line. The EEA profile width, in comparison, is already below the solid line at large generation rates. The width decreases with the generation rate, and approaches the broken gray line which corresponds to 1/ width of the laser profile. According to Eq. (1), the change in wh implies that the EEA rate has a nontrivial dependence on the exciton generation rate.
We experimentally confirm the g dependence of by measuring the PL intensity [Fig. 3(e), open circles] and subtracting a linear fit to the low power PL data (open diamonds). Similarly, the dependence of the intrinsic decay rate (filled circles) and the EEA rate (filled diamonds) obtained from the simulation is overlaid in Fig. 3(e) in terms of unitless parameters and , respectively. The simulation is in good agreement with the experiment, showing that our simple model accurately describes the behavior of excitons in nanotubes. In this log-log plot, has slopes of 1 and 1/3 at low and high generation rates, respectively, as also observed in Fig. 2(a). The slope of is of our interest, which is 2 at low powers and approaches 1 as the power is increased. Through Eq. (1), the observed values of the slope can quantitatively explain the power dependence of wh obtained from the EEA profiles [Fig. 3(d)].
The slope of 1 at high power is reasonable, because with EEA being the dominant decay process. The quadratic dependence of , however, seems to contradict with the cubic dependence expected for EEA . In order to resolve this apparent inconsistency, (filled diamonds) is plotted as a function of time-averaged exciton number [Fig. 3(f)], and we find a transition from the quadratic to cubic dependence near N = 1. Similarly, the experimentally extracted EEA rate is replotted as a function of the PL intensity (open diamonds) in Fig. 3(f), which coincides with the simulation result. The transition in the N dependence of the EEA rate can be explained by the fact that EEA process occurs only when multiple excitons coexist in a nanotube. When and the inter-exciton distance is small, the EEA rate in a 1D system is proportional to N3 as discussed in . When the exciton number , however, the situation changes. The probability for the instantaneous exciton number is negligible, while EEA does not occur for λ = 0 and 1. The EEA rate is then dominated by the case where two excitons coexist, whose probability is given by the Poisson distribution to be . In this regime, multiple exciton generation, rather than exciton diffusion, is the limiting factor of the EEA process.
We can further obtain an explicit expression for the EEA rate. After time t from the first exciton generation, the survival probability of the first exciton and the arrival probability of the second exciton are and , respectively. The expectation value of the arrival time interval for when two excitons coexist is
5. Two-dimensional images of exciton-exciton annihilation rates
Having understood the mechanism for the resolution improvement, we perform two-dimensional (2D) imaging of the raw PL intensity and the EEA rate. According to the analysis performed in Figs. 3(a) and 3(c), the reference power P2 should be small enough to avoid EEA while the EEA extraction power P1 also should be in the linear regime of IPL vs the excitation power in order to obtain high resolution images. We note, however, that P1 should not be too close to P2 to keep the signal-to-noise ratio sufficiently high. We thus choose W and W, where the excitation power is modulated by switching the neutral density filters at every step of the sample scan.
Figure 4 displays the optical images for three configurations of nanotubes. Because W used for Figs. 4(a), 4(d), and 4(g) is in the linear regime of IPL, the width of the raw PL image is solely limited by the laser beam profile and cannot be further reduced by lowering the power. In the simplest case of a single nanotube, the EEA image of the (9,7) nanotube [Fig. 4(b)] has even smaller width than the PL image [Fig. 4(a)]. We note that the EEA image represents the degree of PL intensity reduction through the EEA process. When two (9,7) tubes are lying closely, the tubes are more clearly resolved by extracting the EEA component, which is also apparent from the 1D profiles shown in Fig. 4(f). For a Y-shaped junction of (9,8) nanotubes, we show 1D profiles at the position where the IPL profile of the two tubes cannot be resolved [black line in Fig. 4(i)]. These nanotubes are successfully separated through the super-resolution imaging of EEA rates.
The EEA imaging technique may potentially be applied to three-dimensional biological imaging. We expect similar resolution enhancement in the depth direction as the laser intensity profile along this axis can also be approximated by a Gaussian function. It is also worth mentioning that the nanotube lengths in the EEA images are slightly shortened because the EEA rate is small near the nanotube ends where the exciton density is low. In principle, the FWHM along the tube axis could become smaller than the diffraction limit by using short CNTs, though the resolution improvement would be quantitatively different from the other axes. It is noteworthy that the EEA-derived nonlinearity in nanotubes enables subdiffraction imaging at a power density as low as ≈300 W/cm with a continuous wave laser, whereas considerably higher intensities are employed for other subdiffraction techniques that utilize similar algorithms relying on PL saturation in dyes [26, 28]. We also note that this method is compatible with dispersed nanotubes used for biological imaging [15, 16, 18]. No modification of the protocol should be needed, but the useful power range for EEA imaging will depend on the nanotube length and exciton diffusivity.
6. Image sharpening using the cubic-law excitonic process at high powers
We now consider whether it is possible to utilize the cubic dependence of the EEA rate on N to achieve even higher resolution images. In the imaging protocol employed in Figs. 2–4, the g2 dependence of is used [Protocol I, left panel of Fig. 5(a)]. In comparison, we introduce a protocol that exploits the N3 term of the EEA rate that appears at large N [Protocol II, right panel of Fig. 5(a)]. Instead of laser position dependent exciton generation rate resulting from a constant excitation power in Protocol I, we create laser position dependent exciton number that reproduces the laser beam profile. The EEA rate extracted from such should allow for resolution enhancement through the cubic dependence. As briefly illustrated in Fig. 5(b), we first obtain a PL intensity profile at a sufficiently low excitation power P0 to purely extract the generation rate profile . Then is multiplied by a constant , which will be the target PL intensity for the next scan. We reproduce the target PL intensity profile by controlling the excitation power to compensate for the intensity loss due to EEA. is tuned at every point until the intensity difference between the measurement and becomes less than 3%. The position-dependent exciton generation rate for this scan is used to extract the EEA rate . As the EEA rate is equal to the additional exciton generation rate compared with that expected in the absence of EEA, .
Figures 5(c) and 5(d) show the raw PL image and the EEA rate image, respectively, obtained through Protocol II. Since larger κ gives smaller wh, we choose κ = 25 and W to fully utilize the N3 nonlinearity at high powers, resulting in the maximum W. Cross sectional profiles of these images are shown in Fig. 5(e). Clearly, the EEA rate image shows a narrower profile. If we take horizontal cross sections of the images and average the width along the length of the tube, the EEA image gives an average wh of 290 nm, while that for the PL image is 520 nm. The width reduction corresponds to a factor of as expected from the N3 dependence. As shown in Fig. 5(f), this method can resolve multiple CNTs lying closely as a result of the improved spatial resolution. It should be noted, however, that the spatial resolution of the image does not always improve as much as the width observed for single tubes. Unlike the case in Protocol I, where the linear component of the PL is canceled out during the subtraction, the linear component from adjacent nanotubes is included in the signal during the power tuning step of Protocol II. The inclusion of the linear component inevitably causes a reduction of the nonlinear EEA component, resulting in a suboptimal resolution for high excitation powers.
In summary, we have performed super-resolution imaging of CNTs by visualizing the efficient EEA process using two protocols compatible with typical confocal microscopy systems. By subtracting the linear PL component to extract the nonlinear EEA rate, the spatial resolution improves by a factor of compared to the diffraction limit. The resolution improvement is determined by the N2 dependent EEA rate at low exciton numbers, as confirmed by Monte Carlo simulations. To utilize the N3 dependence unique to the 1D system of nanotubes, we have developed another protocol for super-resolution imaging in which the excitation power is adaptively tuned during the measurement. Using the second protocol, the width of a single nanotube can be as narrow as compared to the conventional PL imaging. The absorption and emission at the near-infrared window and the nonlinear response that appears at low excitation density would open up a pathway for the resolution enhancement in applications such as deep tissue imaging with suppressed damages.
JSPS (KAKENHI JP16H05962 and JP17H07359); RIKEN (Incentive Research Project); MEXT (Nanotechnology Platform).
We thank the Advanced Manufacturing Support Team at RIKEN for technical assistance.
The authors declare that there are no conflicts of interest related to this article.
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