Efficient near-infrared up-conversion photoluminescence in carbon nanotubes

Photoluminescence phenomena normally obey Stokes' law of luminescence according to which the emitted photon energy is typically lower than its excitation counterparts. Here we show that carbon nanotubes break this rule under one-photon excitation conditions. We found that the carbon nanotubes exhibit efficient near-infrared photoluminescence upon photoexcitation even at an energy lying >100–200 meV below that of the emission at room temperature. This apparently anomalous phenomenon is attributed to efficient one-phonon-assisted up-conversion processes resulting from unique excited-state dynamics emerging in an individual carbon nanotube with accidentally or intentionally embedded localized states. These findings may open new doors for energy harvesting, optoelectronics and deep-tissue photoluminescence imaging in the near-infrared optical range.

On the other hand, unintentionally generated states may have shorter lifetimes in as-dispersed nanotubes, as suggested by the absence of clear photoluminescence peaks from these states. Therefore, the unintentionally doped states in as-dispersed nanotubes may present a smaller average exciton population than the artificially introduced optically-active 0D-like local states, possibly resulting in a dissimilar degree of state filling and thus, different nonlinear behaviors.

Supplementary Note 3: Comparison of light penetration depths into highly scattering media for upconversion and visible excitation conditions
Here we demonstrate that the penetration depth of the incident light into highly scattering media can be extended for near-infrared upconversion excitation conditions compared to normal visible excitation conditions. Supplementary Figure 5 shows (

Evaluation of the upconversion efficiency
This supplementary section addresses the efficiency of the upconversion processes of excitons trapped in the 0D-like localized states in carbon nanotubes. The ratio between normal and upconversion photoluminescence intensities I UCPL /I PL is expressed as, where ΔN L is the number of carbon atoms in the additional 0D-like local states. Here, a slight decrease in the normal photoluminescence intensity of free excitons after the doping treatment is neglected. Supplementary Equation (2) provides the rough estimate of the net quantum yield of the exciton upconversion from localized state to free exciton state r L-11 . Note that r L-11 obtained using Supplementary Eq. (2) is the upconversion quantum yield for excitons in the additionally (artificially) doped states, and not necessarily equal to that in the initially (unintentionally) doped ones. As will be demonstrated in the following discussion, we expect that the upconversion quantum yield is approximately proportional to the exciton lifetime in the localized state. From the enhanced nonlinear photoluminescence behavior of 0D-doped nanotubes ( Supplementary Fig. 4), we infer that the exciton lifetime in the additional local state is slightly longer than that in the unintentionally doped states (see Supplementary Note 2).
The r L-11 for the additional local states thus may be slightly higher than that for the unintentional ones.
Assuming that σ L is in the same order as that of first subband excitons (σ 11 ), a σ L /σ 22 ratio (≈ σ 11 /σ 22 ) of ca. 2 is obtained using the absorption spectra ( Supplementary   Fig. 1). The estimated density of the additional localized states in 0D-doped nanotubes is lower than 3 µm −1 2,3 . The small change in optical absorption spectra ( Supplementary   Fig. 1) and slight reduction in free exciton photoluminescence intensity (Fig. 4 in the main text) after localized state doping are consistent with the low density of the additional localized state. Next, ΔN L /N I < 0.006, assuming that the spatial size of the local site is at most as large as that of the free exciton (ca. 2 nm) in (6,5) nanotubes 6 .
The reported exciton relaxation quantum yield from state E 22 to state E 11 , r 22-11 , is of the order of 1 7 . Using these values, the evaluated exciton upconversion quantum yield r L-11 is on the order of 10 −1 at room temperature for an energy gain of 120 meV. The net upconversion photoluminescence quantum yield obtained by multiplying the normal photoluminescence quantum yield by r L-11 is thus only one order of magnitude smaller than its normal photoluminescence counterpart. Therefore, a net photoluminescence quantum yield of the upconversion photoluminescence is deduced to be about 10 -3 for an energy gain of 120 meV using a typical normal photoluminescence quantum yield of 10 -2 for an aqueous (6,5) carbon nanotube suspension 8 .

Phenomenological model for the efficient exciton upconversion in a system combining a 1D carbon nanotube and 0D-like local states
Let us now discuss the physical mechanism underlying efficient exciton upconversion phenomena in carbon nanotubes presenting local states. When considering the exciton upconversion from the 0D-like local state into the free exciton state mediated by large energy phonons in carbon nanotubes, it is reasonable to assume that the upconversion rate is proportional to the phonon number n ph . In this case, the upconverted excitons can be backscattered into the original local state at a rate proportional to 1+n ph , which is much larger than n ph <0.01 for high-energy optical phonons (E ph ≥ 120 meV) at room temperature. This suggests that efficient phonon-assisted upconversion hardly occurs solely by a simple phonon scattering mechanism. However, for quasi-1D carbon nanotubes with embedded 0D-like local states, upconverted excitons in localized states can be scattered toward spatially distant parts along the 1D nanotube axis through ultrafast exciton scattering processes driving rapid exciton diffusion 9,10 in carbon nanotubes. This 1D exciton diffusion process enables efficient exciton upconversion in carbon nanotubes, as will be discussed below.
We first consider a system shown in Supplementary Fig. 10. For the studied sp 3 -doped nanotubes, a deep exciton state below the intrinsic E 11 state has been predicted as the origin of the local state by density functional theory 11 . Similar local state has also been predicted for oxygen-doped nanotubes 12,13 . In this model, the local part with a deep exciton state |L> is seamlessly connected to the intrinsic part of the 1D carbon nanotube with a length L. Equations for the exciton density n(x,t) in the 1D states |X 11 > and the exciton population N * in the local state |L> are written as: The net upconversion quantum yield of the localized exciton r L-11 can be obtained by calculating Supplementary Figure 11b shows the calculated upconversion yield r L-11 as a function of temperature T for various γ 0 . The experimentally deduced value of the net upconversion yield r L-11 ~ 10 -1 at T ~ 300 K is reproduced for γ 0 on the order of (10 ps) -1 . Under this condition, the numerical results of r L-11 below T ~ 300 K can be well approximated by an Arrhenius-type behavior ( Supplementary Fig. 11c) as: consistent with the experimental observations of the temperature dependence well fitted by the Arrhenius equation. We found that the upconversion yield nearly scales with γ 0 Γ *-1 and the coefficient A is well approximated as A ~ γ 0 Γ *-1 for γ 0 < ~ (5 ps) -1 as shown in Supplementary Fig. 11c. The above analysis thus indicates that the upconversion yield increases when localized exciton lifetime and/or phonon scattering rate expand, and the ratio γ 0 /Γ * is a determining factor of the net upconversion yield of the localized excitons.
For a more intuitive understanding of the exciton upconversion physics, here we consider a simplified model as shown in Supplementary Fig. 12a. In this model, we focus on the exciton dynamics only within the local part (shaded region in Supplementary Fig. 12a) explicitly as: where N is defined as an effective number of X 11 excitons in the local part, corresponding to ∫n(x)g(x-x 0 )dx, and J rem is introduced as an effective exciton removal rate from the local part via diffusive exciton migration along the 1D axis. J rem is the difference of exciton numbers flowing out of and into the local part per unit time, thus the ratio J rem /G * corresponds to the net upconversion quantum yield r L-11. The intrinsic recombination rate of N excitons is neglected because it is considered much slower than other processes. J rem should vary depending on N in general. Under conditions in which J rem is a simple function of N, an analytical expression of the upconversion rate r L-11 can be obtained, which will be helpful for a more intuitive understanding of the mechanism.
Here we examine the dependence of J rem on N. According to Fick's first law, an exciton flux J ex (x) under steady state condition is expressed as: The net exciton flux J rem removed from the local part is evaluated as: where 2l is defined as an effective width of the local part and we set l = 3σ in the following analysis (variations in l ≥ σ do not affect the conclusion of the following discussion).
Using Supplementary Eqs. conditions G * ≤ Γ * at T = 300 K. As shown in Supplementary Fig. 12b, the results clearly indicate that J rem is proportional to N, and can be well approximated as J rem = Γ rem N with a proportionality constant Γ rem ~ (0.1 ps) -1 .