Measurement of a reaction-diffusion crossover in exciton-exciton recombination inside carbon nanotubes using femtosecond optical absorption

Exciton-exciton recombination in isolated semiconducting single-walled carbon nanotubes was studied using femtosecond transient absorption. Under sufficient excitation to saturate the optical absorption, we observed an abrupt transition between reaction- and diffusion- limited kinetics, arising from reactions between incoherent localized excitons with a finite probability of ~ 0.2 per encounter. This represents the first experimental observation of a crossover between classical and critical kinetics in a 1D coalescing random walk, which is a paradigm for the study of non- equilibrium systems.


dominated kinetics.
In exact theoretical methods the reaction rate is assumed to be infinite, i.e. the reaction probability p per encounter is unity. However in real experimental systems, the presence of energy barriers, exclusion mechanisms or orientation dependence act to reduce the reaction probability. In 1985, Kang and Redner [16] used scaling arguments to show that a finite reaction rate gives rise to a regime of reaction-limited behavior at early times and high excitation densities, with a crossover to diffusion-limited kinetics at later times. This was supported by approximate theoretical models [17][18][19] and Monte Carlo simulations [16,20]. A rigorous justification is found in field theoretic approaches, where the crossover corresponds to a trajectory between different fixed points of the renormalization group scaling transformation [21]. In experiments, a slow crossover might prevent the asymptotic regime from being reached in systems of finite size. In spite of this theoretical interest and experimental relevance, there has been no report (as far as we are aware) of a reaction-diffusion crossover in any experimental realization of a 1D coalescing random walk.
Here, we demonstrate for the first time an unambiguous crossover between reaction-and diffusion-limited scaling regimes, obtained by careful selection of experimental conditions and by the use of carbon nanotubes exhibiting strictly 1D diffusive transport. We illuminate the sample at sufficiently high intensity to saturate fully the optical absorption, avoiding the influence of spatially non-uniform excitation on the initial decay and allowing a study of the intrinsic kinetics at early times. We also show how the saturation condition allows determination of diffusion and reaction parameters without knowledge of the absolute exciton density, which is subject to uncertainty in the absorption cross-section and concentration of the probed species.
Semiconducting single walled nanotubes are ideal candidates for the study of 1D exciton recombination [15,22,23]. Nanotube diameters of ~ 1 nm lead to quantum confinement of the transverse motion and transport of excitons is highly one-dimensional [24]. The preparation of isolated nanotubes [25] has allowed extensive studies of exciton photophysics [26] which show that photoexcited excitons are compact, stable against dissociation at room temperature [27] and under high excitation [28], but diffuse with a high mobility [29]. They have sufficiently long radiative lifetimes [30] so that bimolecular interactions can be studied in the absence of single particle processes by using femtosecond pump-probe methods.
We produced an ensemble of isolated HiPco nanotubes [31] wrapped in single-strand DNA and dispersed in water; details of our sample preparation and characteristics are reported in reference [32]. The optical absorption spectrum shows features characteristic of isolated nanotubes and an excitation wavelength of 1133 nm was selected corresponding to nanotube species with diameters 0.8 -0.9 nm [24]. An average length L = 184 ± 5 nm was measured by atomic force microscopy. A transient population of excitons was generated by illumination with a short light pulse from an amplified Ti:sapphire laser with 250 kHz repetition rate, and the decay was studied by standard degenerate pump-probe techniques. The diameters of focused pump and probe beams were 130 and 95 µm respectively and the sample was contained in a cuvette with 1 mm path length. The energy of the pump pulse was varied between 0.2 and 104 nJ, with the maximum energy corresponding to a fluence of approximately 0.8 mJ cm -2 . The mean and standard error of multiple scans were determined; weak interference fringes between copolarised pump and probe beams gave rise to a peak in the standard error (inset of Fig. 1(a)), used to identify temporal overlap of the pump and probe beam to within ± 10 fs. The pulse width of 105 fs is a convolution of ≈ 60 fs pump and probe pulses with factors associated with the noncolinear geometry. Figure 1(a) shows the evolution of the differential transmission ΔT / T 0 , which for a given excitation condition is proportional to the total exciton population. The signal amplitudes have been normalized at times > 10ps, showing at long times a decay whose form is independent of excitation density, while at shorter times a rapidly decaying component emerges with increasing excitation. The correct identification of these two regimes is the main purpose of this paper. Figure 1b shows the data on a log-log plot where straight lines correspond to power-law decays.
At long times (t > 10 ps), ΔT / T 0 appears to decay with a diffusion-limited t −1/2 dependence, but the amplitude varies with excitation in contrast to the expectation for an asymptotic algebraic decay. We attribute this to an increase in the photoexcited volume, illustrating the experimental difficulty in relating n to ΔT / T 0 when the absorption is both nonlinear and non-uniform. The signal amplitude saturates with excitation strength at both long and short times (inset of Fig.  1(b)), as also observed in the early dynamics (t < 0.1 ps), where at low excitation the population builds continuously during illumination while at high excitation it saturates at earlier times. The saturation of the optical absorption has been attributed to phase-space filling due to Pauli exclusion of excitons [33].
To quantify the algebraic decay ∝ n t α , the exponent is determined from α = t n ( ) dn dt and shown in Fig. 2 and a density n that decays anomalously slowly as t −1/2 [2]. The crossover between these regimes can be approximated by considering each coalescence event as a sequential diffusion and reaction; a similar separation was discussed for diffusion-limited chemical reactions [34]. Integration gives the time t to evolve to density n from the initial density n 0 t n The decay is algebraic once the population has decayed to a value n << n 0 and there are regimes of reaction-and diffusion-limited behavior at high and low density respectively. Equation (1) can be inverted to give an expression for n(t) equivalent to that obtained from scaling [18] and 'empty-intervals' [19] calculations, which have been shown to agree well with phenomenological models [17] and Monte Carlo simulations [16,20].
Since we do not know the absolute exciton density, we write (1) in terms of a relative density t s = t + t 0 + t 1 = t 0 n 0 n ( ) The constants of integration t 0 and t 1 have been absorbed into a time shift, yielding a 'scaling time' t s whose relation to n is independent of the initial condition and is purely algebraic in the high and low density limits. Crossover occurs at n 2 = t 0 t 1 ( ) n 0 and t 2 = t 1 2 / t 0 . Figure 3(a) shows a plot of n n 0 against t s under the highest excitation, where we believe the absorption to be fully saturated giving an initial density  n 0 . In the experiment the delay time is measured from the peak of the excitation pulse, whereas the decay process starts (roughly speaking) from when the excitation is turned off; an offset of 0.143 ± 0.005 ps was required to reproduce the reactionlimited region as observed in Fig. 2(b). Distinct regions of 1 − t and 1/2 − t decay are seen in Fig.   3(a). The characteristic timescales at initial density  n 0 are  t 0 = 23.9 ± 0.5 fs and  t 1 = 303 ± 3 fs , and crossover occurs at t 2 = 3.8 ± 0.2 ps . The corresponding decay exponent ′ α = t s n ( ) dn dt s 7 is shown in Fig. 3(b). Between 0.5 ps ≤ t ≤ 100 ps it exhibits intrinsic behavior with a crossover from ′ α = −1 to ′ α = −0.5 . This is compared in Fig. 3(b) to the calculated exponent from (2) The experimental crossover function is significantly more abrupt that that indicated by (3).
Mesoscopic parameters describing diffusion and reaction can be obtained from The reaction probability is given by Observation of crossover requires that the crossover density lies within limits imposed by particle exclusion and finite sample size l ex −1 >> n 2 >> L −1 , restricting the reaction probability to  [36], leading to dephasing and depopulation rates of similar magnitude [37].
Determination of the macroscopic coefficients r k and D requires a reliable relation between ΔT / T 0 and n. In its absence, upper bounds can be obtained by estimating the maximum exciton density. The t -1/2 decay associated with exciton-exciton recombination persists for times up to at least 100 ps, so that there remain at least 2 excitons per nanotube. The ratio of ΔT / T 0 at 100 ps and 0.1 ps yields an initial population N max ≥ 120, and hence the exciton length is l ex = L / N max ≤ 1.5 nm , similar to theoretical values [27]. The diffusion coefficient is indicating a diffusive environment comparable to earlier reports [29,33], i.e. exciton transport within the present samples is not exceptional. However, our reaction coefficient k r = l ex / t 1 ≤ 5.0 nm ps −1 is smaller than an earlier estimate by a factor > 60 [22].
In earlier studies of exciton-exciton recombination on nanotube samples similar to those reported here, an initial t -1 decay under high excitation was reported. This was followed by a slower decay which was attributed to exponential single exciton decay processes [22,23] rather than to diffusion-limited exciton-exciton recombination, leading to an assumption of low exciton densities. In order to explain the fast initial decay, Wang et al. proposed a giant enhancement of the exciton-exciton Coulomb interaction by 1D confinement [22]. Here we reach the opposite conclusion: it is a reduction in reaction probability (p < 1) that leads to reaction-limited classical kinetics. Ma et al. attributed the t -1 decay to coherent interactions between extensively delocalized excitons [23]. However we have determined the exciton length (< 1.5 nm) and exciton coherence time (< 80 fs) to be short, justifying the use of a classical stochastic description of these quantum-confined excitations. On the other hand, long-range electrostatic interactions are not ruled out; indeed Förster resonant energy transfer through intra-tube dipolar interactions seems the most likely interaction mechanism [38].
Existing theories and numerical simulations of reaction-diffusion systems are based on highly simplified models of particle interactions and transport. The abrupt crossover observed in our experiment suggests that the models may be insufficient away from the asymptotic limit; this is a topic for further research. Our results identify exciton reactions on carbon nanotubes as an experimental platform permitting precise investigations of anomalous reaction kinetics in a lowdimensional non-equilibrium stochastic system.