Exact and Closed Form Solutions for the Quantum Yield, Exciton

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Exact and Closed Form Solutions for the Quantum Yield, Exciton Diffusion Length, and Lifetime To Reveal the Universal Behaviors of the Photoluminescence of Defective Single-Walled Carbon Nanotubes Tao Liu* and Zhiwei Xiao High-Performance Materials Institute, Florida State University, 2525 Pottsdamer Street, Tallahassee, Florida 32310-6046, United States ABSTRACT: The exact and closed form solutions for the photoluminescence quantum yield (PL QY), effective exciton lifetime, and effective exciton diffusion length were derived for defective single-walled carbon nanotubes (SWCNTs) with randomly and uniformly distributed defects or quenching sites. Asymptotic analysis of the exact and closed form solution not only recovered but also corrected the analytical QY for defective SWCNTs recently derived by Hertel et al. (ACS Nano 2010, 4, 7161
7168). Using the exaction solutions, we reanalyzed the length-dependent QYs and exciton lifetimes of fractionated (6, 5) tubes and determined the intrinsic QY η∞0 = 0.106, exciton diffusion length L∞0 = 272 nm, internal quenching site density c = 0.0039 nm
1, exciton diffusion coefficient D = 2.2 cm2/s, radiative lifetime τr = 1.6 ns, and intrinsic exciton lifetime τ = 168 ps. The effects of F4TCNQ doping on the SWCNT PL intensity were quantitatively studied as well. Through numerical studies of the PL QYs of defective SWCNTs, we elucidated the critical role of the quenching site distance distribution in determining the absolute QYs of defective SWCNTs.

’ INTRODUCTION Since O’Connell et al.1 discovered the near-infrared photoluminescence (PL) of semiconducting tubes, significant progress has been made in the fundamental research of single-walled carbon nanotube (SWCNT) photophysics.2
7 In parallel with the fundamental research on the photophysical behaviors of SWCNTs, various SWCNT based optical and optoelectronic applications have also been proposed and developed,5,7 such as saturable optical absorbers,8 single-photon emitters,9 photodetector and sensors,10
12 molecular optical wires,13 and fluorescent-dye labeling in biological systems.14,15 In the fundamental investigation and technological application of the photophysical properties of SWCNTs, one of the major concerns is the photoluminescence quantum yield (PL QY). Subject to the sample preparation protocols and methods used for QY evaluation, the PL QY of SWCNTs reported in the literature varies significantly from ∼0.01% in refs 16
18 to ∼0.1
1% in refs 19
21 to ∼1
3% in refs 22 and 23 to 7
20% in refs 24
26 to greater than 30% in ref 27. Exciton, an excited state formed by the Coulomb binding of an excited electron to a hole, has been identified to play a critical role to dominate the optical behaviors of SWCNTs.2,6,28,29 On the basis of the excitonic picture, theoretical and experimental research efforts have been made in studying the intrinsic24,30
33 and extrinsic20,34
36 factors that are critical to understand the PL QY of SWCNTs. The radiative lifetime of the exciton in semiconducting SWCNTs has been theoretically estimated in the range of 1
10 ns.30,31 Multiphonon decay (MPD) and phonon-assisted indirect exciton ionization (PAIEI) were proposed as the efficient pathways for the nonradiative decay of the excited tubes to explain the low r 2011 American Chemical Society

QYs of SWCNTs.32 At high excitation fluence, the exciton
exciton Auger recombination has been identified as the key mechanism in limiting the QYs of SWCNTs.24,33 Rajan et al.34 proposed a simple and attractive 1D exciton diffusion model to understand the SWCNT length dependent QYs. They considered the tube ends as quenching sites to solve the 1D exciton diffusion problems and derived an infinite series solution for the QYs of defect-free SWCNT with finite length. The diffusional motion of exciton in SWCNTs was confirmed by the PL stepwise quenching experiments recently reported by Cognet et al.35 Taking into account both the tube ends and sidewall defects as the quenching sites, Hertel et al.20 recently reported an analytical solution for the QY of defective SWCNTs η¼


1 2ðdqH

πdq2 π ¼ 2 2Dτr þ l
1 Þ Dτr

ð1Þ

In eq 1, l is the tube length, D is the exciton diffusion coefficient, τr is the radiative lifetime of the exciton, and dqH is the average quenching site distance in the absence of end quenching. For later discussion, we redefine the average quenching site distance dq equal to 1/(1/dqH + 1/l). Unlike dqH, dq is calculated without differentiating the quenching sites sitting in the interior of the tube from the one on the ends. Upon applying eq 1 to the lengthdependent QYs and exciton lifetimes of fractionated (6, 5) tubes,20 Hertel et al. were able to extract the fundamental Received: June 10, 2011 Revised: July 14, 2011 Published: July 19, 2011 16920

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photophysical parameters for the studied (6, 5) tube exciton diffusion coefficient and radiative lifetime. The derivation of eq 1 is based on a kinetic model of the bimolecular reaction of the diffusive exciton with the quenching sites. In this contribution, we presented a different approach from Hertel’s20 to study the PL behaviors of defective SWCNTs. Specifically, we solved the 1D diffusion problem of exciton in defective SWCNTs with finite length, where the quenching sites or defects are randomly and uniformly distributed along the tube length. Regardless of the nature of the defects, the tube ends, or single defect, the efficiency for each quenching site to quench the PL emission of the exciton is assumed to be 100%. Our major finding is an exact closed form solution (eq 5 presented in the next section) for the QY of defective SWCNTs. The exact solution takes a simple form—an intrinsic QY η∞0 multiplied by a universal √ function of a single dimensionless parameter νn = L∞0dq
1/ 2. η∞0 is the QY of an infinitely long SWCNT free of defects or quenching sites, L∞0 is the intrinsic exciton diffusion length to characterize the diffusion behaviors of the exciton in such a perfect tube, and dq is the average quenching site distance. The asymptotic expansion of the exact solution proves that eq 1 derived in ref 20 is an asymptotic solution of the exact one at large values of the dimensionless parameter νn. It further demonstrated that the numerical coefficient π/2 in eq 1 originally derived in ref 20 is incorrect. The correct value should be 1/6. The condition of using the asymptotic solution to appropriately approximate the exact one has been evaluated by comparing the QYs calculated with these two different approaches. Consequently, we noted that it is inappropriate to use the asymptotic solution, eq 1, in analyzing the length-dependent QYs of the fractionated (6, 5) tubes as presented in ref 20. We reanalyzed the length-dependent QYs and exciton lifetimes of fractionated (6, 5) tubes given in ref 20 using the exact solution for defective SWCNTs. The results we obtain are as follows: intrinsic QY η∞0 = 0.106, exciton diffusion length L∞0 = 272 nm, internal quenching site density c = 0.0039 nm
1, exciton diffusion coefficient D = 2.2 cm2/s, radiative lifetime τr = 1.6 ns, and intrinsic exciton lifetime τ = 168 ps. Using the QY solution established in this paper, we also quantitatively investigated and understood the F4TCNQ (tetrafluorotetracyano-p-quinodimethane) doping effects on the SWCNT PL intensity that were qualitatively studied in ref 37. Both the present and previous studies20,34 emphasized the critical role of the average quenching site distance dq in dictating the PL behaviors of defective SWCNTs. Besides the average value, the distribution of quenching site distance is very important too. In the last section of this paper, we will briefly discuss the effects of quenching site distance distribution on the PL behaviors of defective SWCNTs.

’ RESULTS AND DISCUSSION Derivation and Property Studies of the Closed-Form Solution for the PL QY of Defective SWCNTs. To obtain the

exact closed form solution for the QY of defective SWCNTs, we need to first consider the 1D diffusion problem of the exciton in a SWCNT of length l that is free of interior defects or quenching sites. Following the work of Rajan et al.,34 we have the corresponding governing equations ∂P ∂2 P ¼ D 2
kP ∂t ∂x B:C: : Pð0, tÞ ¼ Pðl, tÞ ¼ 0 I:C: :

Pðx, 0Þ ¼ δðx
x0 Þ

ð2Þ

In eq 2, P  P(x, t|x0) is the probability to locate the exciton at the position x and time t when it is initially created at x0 and t = 0. D is the exciton diffusion coefficient. k = knr + kr is the intrinsic exciton decay rate caused by the radiative (kr) and nonradiative (knr) mechanism(s) independent of the existence of quenching sites. τ = 1/k and τr = 1/kr are the related intrinsic and radiative lifetimes of the exciton,30 respectively. According to the definition of kr and k, the ratio of kr/k or τ/τr is equal to η∞0, the intrinsic QY of a defect-free or quenching site-free SWCNT with infinite length. The absorbing boundary condition (B.C.) in eq 2 reflects that the tube ends act as the exciton quenching sites with an efficiency of 100% to quench the PL emission of the exciton. An infinite series solution for P(x, t|x0) is available (see ref 34 and the reference therein for details). We can use it to calculate the QY of a SWCNT with finite length of l, in which the exciton is as Z ∞ Z l Z l 1 dx0 Pðx; tjx0 Þ dx dt ηðl; n ¼ 0Þ ¼ kr 0 0 l 0   2 ∞ kr 2 1
ð
1Þm ¼ m2 π 2 D k m¼1 mπ 1 þ 2 k l ð3Þ   m 2 ∞ 2 1
ð
1Þ ¼ η0∞ m2 π 2 D mπ m¼1 1 þ 2 k l

∑

∑

η(l, n=0) in eq 3 refers to the PL QY of SWCNT with length of l that is free of interior defects or quenching sites. It is noted that the infinite series solution for η(l, n = 0) originally derived in ref 34 can be recovered by replacing k with kr in eq 3. Using the following two infinite series identities38 8 ∞ 1 π2 > > ¼ > 2 < m ¼ 1 ð2m
1Þ 8

∑

> 4x > > :π

∞

¼ tanh ∑ 2 2 m ¼ 1 ð2m
1Þ þ x 1

πx 2

we can further simplify eq 3 to arrive at a closed form solution    1 0 ηðl, n ¼ 0Þ ¼ η∞ 1
2ν0 tanh ð4Þ 2ν0 rffiffiffiffi pffiffiffiffiffiffi 1 D1 L0 ffiffiffi ¼ Dτ ¼ p∞ ν0 ¼ kl l 2l L0∞ = (2D/k)1/2 = (2Dτ)1/2 is the intrinsic exciton diffusion length —the length scale to characterize the exciton diffusion behaviors in an infinitely long SWCNT free of defects or quenching sites. Equation 4 is a new result. It reveals that the effect of tube ends in reducing the QY of SWCNTs from its intrinsic value η0∞ can be concisely described by √ a universal function of a dimensionless variable ν0 = L0∞l
1/ 2. When the tube length l goes to infinity, this universal function takes a value of 1 to recover η0∞ from η(l, n = 0). Equation 4 is essential for the derivation of the closed form solution of the QY of defective SWCNTs. The derivation details are presented next. Let us consider a SWCNT of length l as a line segment. In the interior of this line segment, we randomly and uniformly place n quenching sites. Each quenching site is assumed to have an efficiency of 100% to quench the PL emission of the exciton. 16921

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Figure 1. Numerical verification of the closed form solution for the PL QY of (a) defect-free SWCNT of finite length, closed form solution (eq 4) vs infinite series solution (eq 3); (b) defective SWCNT of varied length, closed form solution (eq 5) vs computer simulated results (n = 1
196; L0∞ = 100 nm). A total of 5000 simulations were performed for each defective SWCNT to attain the average and standard deviation of the simulated relative QY. The error bar represents 0.1  standard deviations.

Taking into account the two ends of the segment as one quenching site, we have a total of n + 1 quenching sites to separate the original segment into n + 1 subsegments, the length of which is a random variable obeying the exponential distribution with parameter of 1/dq. As defined earlier, dq = l/(n + 1) is the average quenching site distance. With the knowledge of the subsegment length distribution and the closed form solution of the QY for each subsegment (eq 4), we can analytically derive a closed form solution for the QY of the defective SWCNT with length l and n interior quenching sites, η(l, n > 0). This is done by averaging the QY of the subsegments over their length distribution: Z

∞

ηðl; n > 0Þ ¼ 0

Z ¼

η0∞

∞ 0

 ¼ η0∞

  1 s ηðs; n ¼ 0Þ expð
Þ ds dq dq 

rffiffiffiffi    rffiffiffiffi D1 s k 1 s 1
2 ds exp
tanh ks 2 D dq dq

νn  Γð Þ  νn 1
2νn ln þ 2 ln ν 2þ 1 n 2 Γð Þ 2

reduction can be concisely described by √ a universal function of a dimensionless parameter νn = L0∞dq
1/ 2. Similar to QYs, the intrinsic exciton decay rate k or the intrinsic exciton lifetime τ = 1/k is also subject to modification when interior defects or quenching sites are introduced. The modified quantity is denoted as the effective exciton decay rate keff or the effective exciton lifetime τeff = 1/keff. They can be readily determined with eq 5 by using the relations η0∞ = kr/k = τ/τr and η(l, n > 0) = kr/keff = τeff/τr. The result is 8 2   39 νn > > > > Γ < 6 7 6 νn 7= 2 7 τeff ðl, n > 0Þ ¼ τ 1
2νn 6ln þ 2 ln  4 2 > νn þ 1 5> > > Γ ; : 2 The effective exciton diffusion length defined by Leff = √ keff) = (2Dτeff) can be similarly derived

rffiffiffiffi rffiffiffiffi rffiffiffiffi  Dn þ 1 D 1 D1 L0 ffi νn ¼ c þ ¼ ¼ pffiffi∞ ¼ k l k l k dq 2 dq

R, β > 0 was used to analytically evaluate the integration that appeared in eq 5. The parameter c = n/l in eq 5 is the interior quenching site density (excluding the tube ends). It is equivalent to 1/dqH in eq 1. As indicated by eq 5, the QY of defective SWCNTs is subject to reduction from its intrinsic value η0∞ when interior defects or quenching sites are introduced. The extent of QY

ð6Þ (2D/

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v8   39 2 u νn > u> > > Γ u< = 6 7 ν 6 7 2 u n 0 7 þ 2 ln  Leff ðl, n > 0Þ ¼ L∞ u 1
2νn 6ln 4 2 t> νn þ 1 5 > > > Γ : ; 2

ð5Þ

The dummy variable s in the integrand of eq 5 represents the length of the subsegments. The integration formula38 2    3 R β þ 1 Z ∞
Rx 7 6Γ 2 Γ e
e
βx 7 6 2 7 6    dx ¼ ln
x 4 β R þ 1 5 0 xð1 þ e Þ Γ Γ 2 2

√

ð7Þ To examine its correctness, the closed form solution for the QY of a defect-free SWCNT of length l (eq 4) was numerically checked against the corresponding infinite series solution (eq 3). Figure 1a shows the relative QY (η(l, n = 0)/η0∞) calculated by both equations as a function of the dimensionless parameter ν0 = √ L0∞l
1 2. The agreement is excellent. Similarly, the closed form solution for the QY of defective SWCNTs (eq 5) was also checked numerically by computer simulation. In the simulation, n random numbers created from a uniform distribution in the interval (0, l) were used as quenching site coordinates to divide the SWCNT into n + 1 subsegments. The relative QY for each segment was calculated first by eq 3 and then averaged over the total number of segments to give the relative QY (η(l, n > 0)/η0∞) of the defective SWCNTs. The relative QYs for the defective SWCNTs have been numerically determined by the simulation with varied l from 300 to 2000 nm, n from 1 to 196, and L0∞ = 100 nm. To determine the average and standard deviation of the simulated 16922

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Table 1. Summary of the Relative Errors and the Corresponding Conditions (νn or L0∞/dq) of Using the Asymptotic Expansions to Approximate the Exact Solutions for QY (η (l, n > 0)), Effective Exciton Lifetime (τeff(l, n > 0)), and Effective Exciton Diffusion Coefficient (Leff(l, n > 0)) of Defective SWCNTs relative errors = (asymptotic expansion
exact solution)/ exact solution is less than when νn is greater than

Figure 2. Comparison of the asymptotic expansions with the exact solutions for the QY, effective exciton lifetime and effective diffusion length of defective SWCNTs. η(l, n > 0)/η0∞: QY determined by eq 8a vs eq 5; τeff(l, n > 0)/τ: effective exciton lifetime determined by eq 8b vs eq 6; Leff(l, n > 0)/L0∞: effective diffusion length determined by eq 8c vs eq 7.

relative QY, we have performed 5000 simulations for each defective SWCNT (l, n, L0∞). Figure 1b shows the simulated QY√results plotted against the dimensionless parameter νn = L0∞dq
1/ 2. For each simulated defective SWCNTs (l, n, L0∞), the relative QYs were evaluated by eq 5 as well. The results shown in Figure 1b demonstrated an excellent agreement between the simulated results and the closed form solution. As presented in Hertel’s work20 (see eq 1), the QY for defective SWCNTs is related to the average quenching site distance dq and the exciton diffusion coefficient D by a simple scaling relationship: η(l, n>0) µ dq2D
1. This scaling relationship can be recovered from eq 5 upon an asymptotic analysis at large νn by using the following known asymptotic series39 pffiffiffiffiffi Γðνn þ 1=2Þ 1 1 5 ¼ νn 1
þ þ Γðνn Þ 8νn 128ν2n 1024ν3n  21
þ ::: 32768ν4n The asymptotic expansion results for eq 5, eq 6, and eq 7 are respectively given by ηðl, n > 0Þ ≈

dq2 η0∞ η0∞ ¼ ¼ 2 2 6νn 6Dτr 3L0∞ ðc þ 1=lÞ2

ð8aÞ

dq2 τ 1 ¼ ¼ 6ν2n 6D 6Dðc þ 1=lÞ2

ð8bÞ

τeff ðl, n > 0Þ ≈

dq L0 1 ffi ¼ pffiffiffi ¼ pffiffiffi Leff ðl, n > 0Þ≈ pffiffi∞ 3ðc þ 1=lÞ 3 6 νn

ð8cÞ

It is noted that eq 8a provides the same scaling relationship of η(l, n > 0)µ dq2D
1 as eq 1.20 However, the numerical coefficient of this scaling relationship involved in eq 1 and eq 8a is qualitatively different. In eq 1, the coefficient is π/2, which leads to a greater √ than dq effective exciton diffusion length: Leff(l, n > 0) = dq π.20 In eq 8a, the coefficient is 1/6, which leads to a smaller √ than dq effective exciton diffusion length: Leff(l, n > 0) = dq/ 3 (eq 8c).

when L0∞/dq is greater than

5%

10%

20%

5%

10%

20%

η (l, n > 0) τeff(l, n > 0)

4.9 4.9

3.4 3.4

2.3 2.3

7.0 7.0

4.8 4.8

3.3 3.3

Leff(l, n > 0)

3.4

2.3

1.6

4.8

3.3

2.3

Apparently, in a defective SWCNT, it is unlikely for an exciton to have a diffusion length greater than the average quenching site distance. If it does, the exciton should have the chance to diffuse through the quenching sites without being quenched. The asymptotic expansions listed in eq 8 have been checked against the corresponding exact solutions. The results are shown in Figure 2. The asymptotic results agree remarkably well with the exact solutions at large νn. Nevertheless, there is a positive deviation at small values of νn. The smaller the νn, the larger the errors made by the asymptotic solution when it is used to calculate the QYs of defective SWCNTs. According to the relative errors at the level of 5%, 10%, and 20%, the corresponding conditions (νn and L0∞/dq) have been determined for safely use of the asymptotic solution as a good approximation. The results are summarized in Table 1. Application of the Closed-Form Solution for Investigating the PL Behaviors of Defective SWCNTs. Case Study I, LengthDependent QYs and Effective Exciton Lifetimes of (6, 5) Tubes. Assuming the interior quenching site density c = 1/dqH is independent of the tube length, Hertel et al.20 studied the lengthdependent QYs and the effective exciton lifetimes of fractionated (6, 5) tubes by eq 1. They determined the exciton diffusion coefficient (D = 10.7 cm2/s), radiative lifetime (τr = 1.6 ns), and the interior quenching site density (c = 1/dqH = 0.00083 nm
1). However, these reported values may be subject to errors due to a few problems associated with eq 1. First, the numerical coefficient of π/2 in eq 1 is incorrect. Second, the applicability of eq 1 for the length-dependent QYs of (6, 5) tube being studied in ref 20 is questionable. If eq √ 1 or eq 8a is applicable to the length-dependent QY data, 1/ η should be linearly proportional to 1/l under the assumption of constant c. Nevertheless, for the length-dependent QYs of √the (6, 5) tube presented in ref 20, the plot of 1/l versus 1/ η (not shown) shows strong nonlinearity. To resolve the problems aforementioned, the length-dependent QY data in ref 20 were reanalyzed and fitted by the exact solution for the QY of defective SWCNTs, eq 5. The length-dependent QY data (represented by a box symbol) for (6, 5) tube taken from ref 20 can be fitted well by eq 5 as shown in Figure 3a. The fit resulted in the intrinsic QY (η0∞ = 0.106), intrinsic exciton diffusion length (L0∞ = 272 nm) and the interior quenching site density (c = 0.0039 nm
1). The value of η0∞ = 0.106 agrees reasonably with the recently reported QYs for high-quality SWCNTs.24
26 It is worth mentioning that, η0∞ and L0∞ can be independently determined by using eq 5 to fit the length-dependent QY data when they are available in a broad 16923

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Figure 3. (a) Experimental (open square, data taken from ref 20) and simulated (open tilted right triangle and open circle) length-dependent PL QY η(l, n > 0) and the corresponding results (smooth lines) fitted by eq 5. QY data fitting yields: L0∞ = 272 nm, η0∞ = 0.106, and c = 0.0039 nm
1 which were used as inputs to fit the lifetime√τeff (l, n > 0) data (solid down triangle, data taken from ref 20) with eq 6 to obtain the exciton diffusion coefficient D = 2.2 cm2/s. (b) νn = L0∞ dq
1/ 2 dependent universal behavior of the experimentally determined QYs from different sources.

Figure 4. F4TCNQ doping effects for (7, 5) and (8, 6) SWCNTs on (a) the relative change of the PL intensity (scattered data are taken from ref 39, smooth lines are the fitted results given by eq 9) and (b) the relative change of the average quenching site distance dq.

range of νn. If eq 1 or eq 8a is used, only the ratio of η0∞/L0∞ can be extracted. To strengthen this point, we first numerically simulated the length-dependent QYs for defective SWCNTs in a broad range of νn and then fit the data with eq 5. The parameters used for generating the simulated data are L0∞ = 272 nm, η0∞ = 0.106, and two different levels of quenching site density c = 0.01 and 0.001 nm
1. All can be faithfully extracted upon fitting. Figure 3a shows both the simulated data (triangle tilted to the right for c = 0.001 nm
1 and open circle for c = 0.01 nm
1) and the corresponding fitted results (smooth lines). Evidently, the agreement between the simulated data and the fitted results is excellent. With η0∞ = 0.106, L0∞ = 272 nm, and c = 0.0039 nm
1 as inputs, the length-dependent exciton lifetimes τeff for the studied (6, 5) tube (data taken from ref 20 and represented by a solid down triangle in Figure 3a) were also fitted by eq 6 to obtain the exciton diffusion coefficient D = 2.2 cm2/s. Furthermore, the intrinsic exciton lifetime τ and the exciton radiative lifetime τr can be as τ = 168 ps and τr = 1.6 ns by L0∞ = √ respectively estimated 0 (2Dτ) and η∞ = τ/τr. The exciton radiative lifetime determined above is comparable to the theoretically predicted 2.2 ns for (6, 5) tube when singlet excitons are the sole source for interband thermalization.30

The quality of SWCNTs may vary significantly subject to sample synthesis and preparation details. It is desirable to have a unified method in comparing the quality of SWCNTs from different sources. This can be done by constructing the νn dependent universal behavior for the QYs of different SWCNT samples. Using √ the value of η0∞ = 0.106, we constructed the 0
1 νn = L∞ dq / 2 dependent universal behavior of QYs for various SWCNT samples reported in the literature. The results are shown in Figure 3b. There is about 2 orders of magnitude variation on the νn values for the SWCNTs studied by different √ research groups. According to the definition of νn = L0∞ dq
1/ 2, the variation of νn could arise from the changes of L0∞ and/or dq. The intrinsic exciton diffusion length L0∞ of SWCNTs has a relatively weak dependence on the tube diameter and chiral angle40 as well as on the environments.41 Because of that, we concluded that the huge variation on the νn observed in Figure 3b is primarily due to the changes of the average quenching site distance dq, which gives a direct measure of SWCNT quality. In Figure 3b, it is also noted that the dimensionless parameter νn for the length-dependent QYs of fractionated (6, 5) tube given in ref 20 falls in the range of ∼1.0
1.9. According to Table 1, this value suggests that the asymptotic solution, eq 1 or eq 8a, is not a good approximation to the exact 16924

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Figure 5. Effects of quenching site distance distribution on the QYs of defective SWCNTs. (a) dq/L0∞ dependent universal behavior for the QYs of defective SWCNTs. Expd-QY is for exponential distribution, Normd-QY is for normal distribution, and Constd-QY is for constant distribution. (b) The QYs of defective SWCNTs with normal and constant quenching site distance distribution normalized by Expd-QY. Solid lines are for guiding the eye.

one (eq 5) for analyzing the given QY data. This finding is consistent with the conclusion drawn previously based on √ an observation of the strong nonlinearity between 1/l and 1/ η for the QY data presented in ref 20. Case Study II, F4TCNQ Doping Effects on the QYs of SWCNTs. Matsuda et al.37 investigated the doping effects on the PL behavior of SWCNTs. A monotonic decrease of the PL intensity was observed when the concentration of F4TCNQ (tetrafluorotetracyano-p-quinodimethane) was increased. The decrease of PL intensity induced by the F4TCNQ doping can be quantitatively understood by eq 5 or eq 8a. Assuming that the doping process introduces extra interior quenching site density ca in addition to the originally existed ones c0, we have, to a first-order approximation, ca = kdCd. Cd is the dopant concentration and kd is the rate constant for creation of the quenching sites due to doping. The change of the relative PL intensity due to doping, IPLdoped/IPLundoped can be derived based on either the asymptotic QY solution, eq 8a, or the exact one, eq 5. Both formulas were applied to analyze the PL intensity data presented in ref 37 and similar results were obtained. For this reason, we only present the result derived based on eq 8a sffiffiffiffiffiffiffiffiffiffiffiffiffiffi doped IPL undoped IPL

¼¼

dq ðCd > 0Þ 1 ¼ 1 þ kd dq0 Cd dq ðCd ¼ 0Þ

ð9Þ

In eq 9, dq0 = dq(Cd = 0) and dq = dq(Cd > 0) are the average quenching site distance before and after doping, respectively. Shown by the scattered points in Figure 4a are the experimental results of the relative PL intensity for (7, 5) and (8, 6) tubes as a function of F4TCNQ concentration (data taken from ref 37). In the same figure, the smooth lines are the corresponding fitted results by eq 9. The decrease of PL intensity induced by F4TCNQ doping can be reasonably explained by eq 9. Moreover, using eq 9, we can also quantify the effects of doping on the changes of the relative average quenching site distance, dq(Cd> 0)/dq(Cd = 0). Figure 4b shows the results. With an increase of the dopant concentration, the average quenching site distance for the (8, 6) tube drops more rapidly than that for the (7, 5) tube. This indicates that F4TCNQ doping introduces defects more easily in the (8, 6) tube than in the (7, 5) tube.

Effects of Quenching Site Distribution on the QYs of Defective SWCNTs. Equation 5, eq 6, and eq 7 reveal that the

average quenching site distance dq plays a critical role in dictating the PL behaviors of defective SWCNTs. In addition to the average value, the quenching site distance distribution is also an important factor in influencing the PL behaviors of defective SWCNTs. Having constructed the defective SWCNTs with different quenching site distance distribution and numerically evaluated the corresponding QYs with the methods established previously, we will briefly discuss the effects of quenching site distance distribution on the QYs of defective SWCNTs in this section. The defective SWCNT constructed for this study has a length of l and is divided by n interior quenching sites into n + 1 subsegments. It gives the same average quenching site distance dq = l/(n + 1) irrespective of the length distribution of the subsegments. Constant, exponential, and normal distributions are considered for the subsegment length in the calculation of the QY for the defective SWCNTs. In the constant distribution, each of the subsegments has the same length dq. Denoted by ConstdQY, the QY for the corresponding defective SWCNTs was calculated by eq 4. The exponential distribution has been discussed previously. The mean value and variance of the quenching site distance are dq and dq2, respectively. We used eq 5 to calculate the QY for the defective SWCNTs with an exponentially distributed subsegment length. It is represented by Expd-QY. The QY of defective SWCNTs, in which the quenching site distance follows a normal distribution, is evaluated by computer simulation and denoted by Normd-QY. In brief, n + 1 random numbers to represent the subsegment length are generated from a normal distribution with mean value of dq and variance of dq2. The QY for each segment was first calculated by eq 4 and then averaged over the total number of segments n + 1 to give NormdQY for the corresponding defective SWCNTs. For each given dq, we performed 5000 simulations to obtain the average and standard deviation of Normd-QY. Figure 5a shows the dq dependent Constd-QY, Expd-QY, and Normd-QY for defective SWCNTs of l = 2000 nm at L0∞ = 100, 200, and 600 nm, respectively. For each type of quenching site distance distribution, the QYs of the corresponding defective SWCNTs can be described by a universal function of the √ dimensionless parameter dq/L0∞ (= 1/ 2νn). It reminds us of 16925

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The Journal of Physical Chemistry C the critical role played by the average quenching site distance dq in the PL behaviors of defective SWCNTs. In addition, the detailed behaviors for the universal function to response to the change of νn obviously vary from one to another type of quenching site distance distribution. The results shown in Figure 5a clearly indicate that not only the average value of the quenching site distance but its underlying distribution also play important roles in determining the absolute QYs of defective SWCNTs. To have a better visualization on the effects of quenching site distance distribution for the QYs of defective SWCNTs, we present in Figure 5b the Normd-QY and Constd-QY values at different dq normalized by the corresponding Expd-QY. Apparently, when dq/L0∞ is smaller than 0.1, the QYs are almost the same for the defective SWCNTs with exponential (Expd-QY) and normal (Normd-QY) quenching site distance distribution. Expd-QY and Normd-QY are significantly higher (∼2 Constd-QY) comparing to the Constd-QY. When dq/L0∞ (>0.1) increases, the Normd-QY shows a positive deviation from Expd-QY with a maximum value (∼1.3 Expd-QY) at dq/L0∞ ≈ 2.0. Similar behavior is also observed for Constd-QY. When dq/L0∞ increases, the Constd-QY gradually approaches the valu of Expd-QY. Ultimately it exceeds Expd-QY at dq/L0∞ ≈ 1.0 and then reaches a maximum (∼1.4 Expd-QY) at dq/L0∞ ≈ 4.0.

’ CONCLUSIONS In consideration of 1D diffusion of exciton in a SWCNT of finite length, where the quenching sites are randomly and uniformly distributed, we derived and verified the exact and closed form solutions for the photoluminescence quantum yield (eq 5), effective exciton lifetime (eq 6), and effective exciton diffusion length (eq 7) of defective SWCNTs. The closed form solutions are featured by a simple universal function of a single dimensionless parameter νn ∼ L0∞/dq, which is the ratio of the intrinsic exciton diffusion length L0∞ to the average quenching site distance dq. Asymptotic analysis of the exact solution for the QY of defective SWCNTs not only recovered but also corrected the result previously derived by Hertel et al.20 Using the exaction solutions, we reanalyzed the length-dependent QYs and exciton lifetimes of fractionated (6, 5) tubes given in ref 20 and determined the intrinsic QY η0∞ = 0.106, exciton diffusion length L0∞ = 272 nm, internal quenching site density c = 0.0039 nm
1, exciton diffusion coefficient D = 2.2 cm2/s, radiative lifetime τr = 1.6 ns, and intrinsic exciton lifetime τ = 168 ps. The effects of F4TCNQ doping on the PL intensity of SWCNTs37 were also quantitatively studied. Through the case studies above, we demonstrated an analytic framework which provided a convenient and unified approach to study the PL behaviors of defective SWCNTs. Lastly, we elucidated the critical role of the quenching site distance distribution in determining the absolute QYs of defective SWCNTs by constructing the defective SWCNTs with different quenching site distance distribution and numerically evaluating the corresponding QYs. ’ AUTHOR INFORMATION Corresponding Author

*E-mail address: [email protected].

’ ACKNOWLEDGMENT We acknowledge the financial support by the Air Force Office of Scientific Research (AFOSR) and the High-Performance Materials Institute at Florida State University.

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