Bilayer Plots for Accurately Determining the Chirality of Single-Walled

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Bilayer Plots for Accurately Determining the Chirality of Single-Walled Carbon Nanotubes Under Complex Environments Juan Yang,* Daqi Zhang, Yuecong Hu, Chenmaya Xia, Sida Sun, and Yan Li Beijing National Laboratory for Molecular Sciences, Key Laboratory for the Physics and Chemistry of Nanodevices, State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China S Supporting Information *

ABSTRACT: The chirality (n,m) determines all structures and properties of a single-walled carbon nanotube (SWNT), therefore, accurate and convenient (n,m) assignments are vital in nanotube-related science and technology. Previously, a so-called Kataura plot that protracts the excitonic transition energies (Eii’s) of SWNTs with various (n,m) with respect to the tube diameter (dt) has been widely utilized by researchers in the nanotube community for all (n,m)-related studies. However, the facts that both Eii and the calculated dt are subject to interactions with the environments make it inconvenient to accurately determine the (n,m) under complex environments. Here, we propose a series of bilayer plots that take into account the interactions between the SWNTs and the environments so that the (n,m) of SWNTs can be accurately determined. These plots have more advantages than the Kataura plot in concision, less data overlapping, and the suitability to be used in complex environments. We strongly encourage the researchers in the carbon nanotube community to utilize the bilayer plots for all (n,m)-related studies, especially for accurate and convenient (n,m) determination. KEYWORDS: single-walled carbon nanotubes, chirality assignments, bilayer plots, Kataura plot, Raman spectroscopy tedious and inconvenient. In 1999, H. Kataura et al.8 first proposed a calculated Eii−dt plot to explain the absorption and Raman spectra of SWNTs. Later, this plot was named after him as the Kataura plot and was widely utilized by the researchers in the carbon nanotube community.4,6,7,9−11 Because the Kataura plot provides direct information on the two (n,m)-dependent parameters: Eii and dt, it is useful and helpful in various (n,m)related studies, especially in making (n,m) assignments.4,6,7 In some cases, 1/dt or the Raman shift of the radial breathing mode (ωRBM), which is directly related to dt by a formula of

T

he chiral index or chirality (n,m) of a single-walled carbon nanotube (SWNT) determines its geometric structure (diameter dt and chiral angle θ) and electronic structure (metallic and semiconducting) as well as all the structure-related properties.1 For example, a SWNT is metallic (M) if 2n + m is divisible by 3 and semiconducting (S) otherwise.1 The optical transitions of the SWNTs are allowed by symmetry selection rules to happen between the excitonic states, resulting in the (n,m)-characteristic excitonic transition energies (Eii).2 Both the bandgap of S-SWNTs and the splitting between the transition energies of M-SWNTs due to the trigonal warping effect3 are also (n,m)-dependent. Therefore, accurate and convenient (n,m) determination is vital in nanotube-related science and technology.4−7 As the SWNTs contain a large variety of (n,m) and each (n,m) possesses a different set of Eii’s, the analysis of the Eii’s of SWNTs becomes © 2017 American Chemical Society

Received: August 17, 2017 Accepted: October 3, 2017 Published: October 3, 2017 10509

DOI: 10.1021/acsnano.7b05860 ACS Nano 2017, 11, 10509−10518

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Cite This: ACS Nano 2017, 11, 10509-10518

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Figure 1. ERS process, ERS spectra, and the data points of air-suspended M-SWNTs shown in the Kataura plot and in the bilayer plot. (a) Schematic of the resonantly enhanced ERS+ and ERS− processes in a M-SWNT compared to typical resonant Raman processes. EL, Es, Ee‑h, and Eph represent the energies of laser, scattered photon, electron−hole pair, and phonon mode, respectively. (b) Raman spectra of (12,9) tubes in the ERS and RBM regions with two lowest and two highest ωRBM. A large ωRBM range of 8.5 cm−1 is observed. (c) 14 statistical data points of (12,9) shown in the Kataura plot. Blue and red rectangles indicate the corresponding data ranges for M11+ and M11− respectively. Red solid line indicates the 1.96 eV laser. (d) 14 statistical data points of (12,9) shown in the bilayer plot. Solid red squares are in M11 splitting vs M11 average scale, and hollow blue diamonds are in M11 splitting vs ωRBM scale. Blue rectangle indicates the corresponding data ranges for the blue diamonds. (e) 14 statistical data points of (12,9) shown in the enlarged bilayer plot. The red squares can be unambiguously classified into two subsets: 6 data points in the red rectangle corresponding to individual tubes and 8 data points in the orange rectangle corresponding to bundled tubes. (f) 12 different metallic chiralities reported in ref 23 shown in the bilayer plot. The data points for (14,8), (16,4), (12,9), (13,7), (14,5), (15,3), and (16,1) should correspond to individual tubes, and the data points for (17,5) and (11,8) should be bundled tubes. The M11+ and M11− values of a chirality can be read directly from the bilayer plot by introducing oblique guiding lines that are parallel to the corresponding laser lines (blue dashed lines).

ωRBM = A/dt + B,12,13 can also be used as the abscissa in the Kataura plot instead of dt.14,15 However, further studies show that both Eii and the ωRBM−dt relation are strongly affected by the interactions between the SWNT and its local environments.13−20 Therefore, both coordinates in the Kataura plot are subject to appreciable shifts upon moderate interactions with the environments. In the ωRBM−dt relation, the parameter A is determined by the vibrational force constant of the RBM, and the parameter B is associated with various interactions between the SWNTs and the environments,13,21 including substrate−tube interaction,14,21,22 π−π interaction between tubes inside bundles,23,24 amorphous carbon coating,9,25 adsorption of gas molecules on the tube sidewalls,26 interaction between tubes and the dispersants in solution,10,27 static pressure difference,28 etc. The value of B is expected to be zero for individual vacuumsuspended SWNTs, which are free of interactions.16,29 The interactions between the SWNTs and the environments also cause significant shifts in Eii.30−32 In general, individual vacuumsuspended SWNTs give rise to the highest Eii values,33 and various environmental effects lead to red-shifted Eii values due to a dielectric screening effect.21,34−36 It has been reported that the entire sets of Eii observed in some specific conditions can be applied to samples in other environments by different offset values.15,21,37 These shifts in Eii become the major uncertainty in accurately determining the (n,m) under complex environ-

ments based on the Kataura plot, especially when resonant Raman spectroscopy is utilized because the shifts in Eii will cause significant changes in the resonance conditions of different tube species.21 Here, we report a series of bilayer plots that take into account the interactions between the SWNTs and the environments so that the (n,m) of SWNTs can be accurately determined even under complex environments. In the bilayer plots, we use the Mii splitting of M-SWNTs and the Sii difference of S-SWNTs as the vertical coordinates, and use the corresponding Eii average as the horizontal coordinate in one layer and the corresponding ωRBM as the horizontal coordinate in the other layer. We show that the Mii splitting and Sii difference are most insensitive to the interactions and thus can be nearly fixed in complex environments. Therefore, we only need to consider the shifts in the horizontal axes, which greatly simplifies the situation. Moreover, the difference in the data points shown in both layers can provide direct information about the degree of interactions between the SWNTs and the environments. By reanalyzing the data reported not only by our group but also by other research groups, we demonstrate how to accurately and conveniently determine the (n,m) of SWNTs and how the information about the degree of interactions can be obtained based on the bilayer plots. We have also launched a program on our group Web site (www.chem.pku.edu.cn/cnt_assign) for 10510


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ACS Nano Table 1. Statistical Raman Data of 14 Air-Suspended (12,9) Samples no.

ωRBM (cm−1)

calcd dt (nm)a

M11+ (eV)

M11− (eV)

M11+ − M11− (eV)

(M11+ + M11−)/2 (eV)

convt. ωRBM (cm−1)b

ΔωRBM (cm−1)

notec

1 2 3 4 5 6 7 8 9 10 11 12 13 14

162.5 164.7 166.0 166.1 167.1 167.2 167.3 167.8 168.2 169.0 169.5 170.3 170.7 171.0

1.465 1.442 1.429 1.428 1.417 1.417 1.415 1.411 1.406 1.399 1.393 1.386 1.382 1.379

1.881 1.883 1.877 1.882 1.879 1.882 1.857 1.857 1.849 1.849 1.864 1.862 1.865 1.856

1.810 1.814 1.809 1.807 1.808 1.813 1.792 1.789 1.789 1.791 1.796 1.796 1.797 1.793

0.072 0.069 0.068 0.075 0.072 0.069 0.064 0.068 0.060 0.058 0.068 0.067 0.069 0.063

1.845 1.849 1.843 1.844 1.843 1.848 1.825 1.823 1.819 1.820 1.830 1.829 1.831 1.825

166.0 166.4 165.8 165.9 165.8 166.3 164.1 163.9 163.6 163.7 164.6 164.5 164.7 164.1

−3.5 −1.7 0.2 0.2 1.3 0.9 3.2 3.9 4.6 5.3 4.9 5.8 6.0 6.9

i i 1i i i 2i b 1b 2b b b b b b

a The dt value is calculated from dt = 200/(ωRBM − 26). bThe ωRBM value is converted from the corresponding (M11+ + M11−)/2 value by (M11+ + M11−)/2 = 3.170/dt − 0.764/dt2 and ωRBM = 200/dt + 26. cNote: i, individual tube; b, bundled tube. 1i and 2i are verified to be individual tubes and 1b and 2b are verified to be bundled tubes from the HRTEM images of two air-suspended Y-shaped bundles reported in ref 23.

range of 8.5 cm−1 in ωRBM. Converting ωRBM into dt, a standard deviation of ±0.024 nm and a range of 0.086 nm can be calculated. In this wide dt range and the ±100 meV laser resonance window,21,40 4 different chiralities are available. Therefore, the uncertainty in ωRBM will lead to ambiguous assignments. On the other hand, a standard deviation in the order of ±10 meV and a range in the order of 30 meV are obtained in Mii+ and Mii− from these 14 statistical data. The uncertainty in M11 can unambiguously distinguish (12,9). More importantly, we observe similar trends in both M11+ and M11− with respect to environmental changes, which means that the M11 splitting, M11+ − M11−, should be less sensitive to the environmental changes. Statistical data show a standard deviation of only ±4.6 meV and a small range of 17 meV in the M11 splitting. Heating the tubes with the laser to moderate temperatures to desorb the gas molecules only causes a negligible variation of 2 meV in the M11 splitting. Therefore, we conclude that the M11 splitting is nearly insensitive to the environmental changes and that this splitting value may serve as the universal criterion for the (n,m) determination of M-SWNTs under complex environments. From the above analysis, we feel confident to propose a bilayer plot that uses the environment-insensitive M11 splitting value as a coordinate for convenient (n,m) determination. From the reported equation from refs 23 and 37:

accurately determining the (n,m) of SWNTs under complex environments based on the bilayer plots.

RESULTS AND DISCUSSION Mii Splitting and the Bilayer Plots for M-SWNTs. As both Eii and the ωRBM−dt relation are strongly affected by the environments, if provided with an (n,m)-dependent parameter that is insensitive to the environmental changes, it is then possible to use this parameter as a universal criterion for the (n,m) assignments under complex environments. In our previous paper regarding the electronic Raman scattering (ERS) of M-SWNTs, we find that the Mii splitting value, that is, Mii+ − Mii−, can serve as such a parameter in M-SWNTs.23 Regular Raman processes involve phonon scattering, in which the phonon energy remains nearly constant, but the scattered photon energy varies with the excitation laser energy (EL).2 In the ERS process (Figure 1a), excitons are inelastically scattered by the low-lying electron−hole pairs near the Fermi level of the M-SWNTs.38,39 Owing to the linear dispersion near the Fermi level, whatever energies required for the electron− hole pairs can always be satisfied, resulting in resonant enhancements at the corresponding Mii’s. Therefore, the scattered photon energy remains constant at Mii+ or Mii−, but the energy of the electron−hole pairs varies with EL. Consequently, the ERS+ and ERS− can give direct information about the corresponding transition energies Mii+ or Mii−, respectively.20,23 We choose air-suspended (12,9) tubes as an illustrative example. This near-armchair chirality has narrowly split M11+ at 1.88 eV and M11− at 1.81 eV and thus can be unambiguously identified by its characteristic double-hump ERS bands at the 1.96 eV excitation. Due to the easy and unambiguous identification of this chirality, we observe altogether 14 different air-suspended (12,9) samples (Table 1). Those tubes are free of substrate interaction but may be coated with amorphous carbon and adsorbed with gas molecules at room temperatures. Some tubes are individual, and some are bundled with π−π interaction between the tubes inside bundles. We put the Raman spectra with two lowest and two highest ωRBM in Figure 1b for comparison and plot all 14 data points in the corresponding Kataura plot in Figure 1c. We observe a standard deviation of ±2.4 cm−1 and a surprisingly large

± M11 = α /d t − β /d t2 ± γ cos(3θ )/d t2

(1)

we calculate that + − M11 − M11 = 2γ cos(3θ )/d t2

(2)

and put this M11 splitting as the vertical coordinate in the bilayer plot. We set the horizontal coordinate (bottom) in the first layer as the M11 average given by + − (M11 + M11 )/2 = α /d t − β /d t2

(3)

which is dependent only on dt. As the ωRBM given by

ωRBM = = A /d t + B

(4)

is also dependent only on dt, we can thus put the corresponding ωRBM as the horizontal coordinate (top) in the second layer of the bilayer plot and link it with the M11 average, although not in a linear scale. For the M11 transitions of air-suspended M10511


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Figure 2. Comparison between the bilayer plot and the Kataura plot for the M11 transitions of air-suspended M-SWNTs and M-SWNTs on silicon substrates. (a, b) The bilayer plot (a) and the Kataura plot (b) for the M11 transitions of air-suspended M-SWNTs with the 2n + m values from 21 to 48. (c, d) The bilayer plot (c) and the Kataura plot (d) for the M11 transitions of M-SWNTs on silicon substrates with the 2n + m values from 21 to 48. Gray, red, and green solid lines indicate the 1.58, 1.96, and 2.33 eV lasers, respectively. Red squares, green square, and blue diamond are data points reported in refs 38 and 40. See text for detail.

observed in ωRBM between the individual and bundled tubes. It is also worthy to notice that the ωRBM range of both individual and bundled air-suspended (12,9) tubes can be as large as 8.5 cm−1 and that even the ωRBM range of individual air-suspended (12,9) tubes can be as large as 4.7 cm−1. Therefore, the uncertainty in dt calculated from ωRBM alone will lead to ambiguous assignments. (3) The relatively large difference compared to the relatively small standard deviation in M11 average can unambiguously distinguish individual and bundled tubes. Further analysis of the 14 statistical data of air-suspended (12,9) suggests a criterion to distinguish individual and bundled tubes. For a bundled tube, the blue diamond should locate considerably to the right of the corresponding red square, and the (n,m) which lies in between these two points should be the corresponding chirality. For an individual tube, the red square should match the corresponding chirality, and the blue diamond should either coincide with or even locate to the left of the red square. In our previous paper, we unambiguously assign altogether 18 different metallic chiralities using the 1.96 and 2.33 eV lasers.23 We observe both M11+ and M11− for 12 chiralities, which allows us to calculate the M11 splitting and M11 average values. Figure 1f shows the corresponding data points in M11 splitting vs M 11 average scale (solid red squares) as well as in M11 splitting vs ωRBM scale (hollow blue diamonds) in the bilayer plot for these 12 chiralities. If multiple tubes of the same chirality are observed, we plot the one with the lowest ωRBM, which is most likely an individual tube. Using the abovementioned criterion, we conclude that the data points in Figure 1f for (14,8), (16,4), (12,9), (13,7), (14,5), (15,3), and (16,1) should correspond to individual tubes and that the data points

SWNTs, we use a set of fitting parameters of α = 3.170 eV nm, β = 0.764 eV nm2, γ = 0.286 eV nm2, A = 200 nm cm−1, and B = 26 cm−1, as reported in our previous paper.23 Upon interactions with the environments, we expect that the M11 splitting remains nearly constant, whereas the ωRBM increases and the M11 average decreases. We plot all 14 data points of air-suspended (12,9) both in M11 splitting vs M11 average scale (solid red squares) and in M11 splitting vs ωRBM scale (hollow blue diamonds) in the bilayer plot (Figure 1d,e). Compared to the data points in ωRBM, the data points in M11 average distribute a much narrower range. More importantly, we find that the values of M11 average can be unambiguously classified into two subsets: 6 data points in the 1.843−1.849 eV range (red rectangle in Figure 1e), in which 1i and 2i label two individual tubes given in Table 1 that are verified by high-resolution transmission electron microscope (HRTEM), and 8 data points in the 1.819−1.931 eV range (orange rectangle in Figure 1e), in which 1b and 2b label two bundled tubes verified by HRTEM.23 As the M11 is expected to redshift by ∼20 meV upon bundling,41 it is reasonable to speculate that the 6 data with higher M11 average correspond to individual tubes and the other 8 with lower M11 average are bundled tubes. The statistical data for the M11 splitting, M11 average, and ωRBM are 71 ± 3 meV, 1.846 ± 0.002 eV, and 165.6 ± 1.8 cm−1, respectively, for the individual tubes and 65 ± 4 meV, 1.825 ± 0.005 eV, and 169.2 ± 1.4 cm−1, respectively, for the bundled tubes. From the comparison of the above three sets of data, we conclude that: (1) The difference between individual and bundled tubes in M11 splitting is the smallest of all compared to the corresponding standard deviation. (2) Although the average ωRBM of bundled tubes is systematically higher than that of individual tubes, no obvious boundary is 10512


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Table 2. Comparison Between the M11 Data Reported in Ref 40 by Raman Excitation Profile for Six M-SWNTs and Our Fitting Valuesa (n,m) 2n + m M11+ Our M11+ ΔM11+ M11− Our M11− ΔM11− M11+ − M11− Our M11+ − M11− Δ(M11+ − M11−) (M11+ + M11−)/2 Our (M11+ + M11−)/2 Δ(M11+ + M11−)/2 a

(15,6)

(13,10)

(11,5)

(12,3)

(13,1)

(10,7)

(10,7) correctedb

36

36 1.71 1.700 0.01 1.65 1.648 0.00 0.06 0.052 0.01 1.68 1.674 0.01

27

27

27

27 2.12 2.191 −0.07c 2.07 2.063 0.01 0.05 0.128 −0.08c 2.095 2.127 −0.03

27 2.19 2.191 0.00 2.07 2.063 0.01 0.12 0.128 −0.01 2.13 2.127 0.00

1.854 1.68 1.677 0.00 0.177

1.766

2.334

2.453

2.522

2.05 2.057 −0.01

2.07 2.038 0.03

2.02 2.022 0.00

0.277

0.415

0.500

2.196

2.245

2.272

All of the data are in eV. bCorrected (10,7) data by assuming the M11+ equals our value. cMarks the abnormally large values that suggest an error.

eV and ωRBM = 195 cm−1 but no M11+, we can unambiguously assign it to (14,2) by plotting an oblique line parallel to the right half of the V-shaped laser lines at a bottom x intercept of 1.948 eV and a vertical line at ωRBM = 195 cm−1, and reading the chirality at the intersection of both lines. Compared with the Kataura plot, the bilayer plot is more advantageous in several aspects: (1) concise; only one point is necessary for each chirality in the bilayer plot, whereas two points corresponding to M11+ and M11− are required in the Kataura plot. (2) Less data overlapping; we use the size of circles to represent the range of the corresponding values in the vertical axes for all the plots. For example, ∼15 meV for the M11 splitting is used in the bilayer plot, and ∼30 meV for M11 is used in the Kataura plot. The data overlap less in the bilayer plot than in the Kataura plot. Near zigzag chirality pairs such as (19,1)/(18,3), (16,1)/(15,3), and (15,0)/(14,2) overlap with each other in the Kataura plot but not in the bilayer plot. (3) Suitability to be used under complex environments. For SWNTs in various environments, both M11 and ωRBM−dt relation are strongly affected, and the shifts in both vertical and horizontal coordinates in the Kataura plot need to be considered simultaneously. However, the M11 splitting remains nearly constant, and only the shifts in the horizontal coordinates need to be considered in the bilayer plot, which greatly simplifies the situation. Upon interactions with the environments, the M11 average decreases, and the ωRBM increases. Consequently, the bottom horizontal axis shifts to the left, and the top horizontal axis shifts to the right. The (n,m) locating between these two shifted values is the corresponding chirality. The difference in the two shifted values indicates the degree of interactions between the SWNTs and the environments. Therefore, the bilayer plot can not only accurately and conveniently determine the (n,m) but also provide information about the degree of interactions between the SWNT and its local environments. All of the above data are for air-suspended M-SWNTs. SWNTs on SiO2/Si substrates are more interesting samples to analyze. Previously, a ∼40 meV difference for the Eii’s between air-suspended tubes and tubes on silicon substrates has been reported.36 We hereby construct the bilayer plot (Figure 2c) for the M11 transitions of M-SWNTs on silicon substrates by redshifting all of the M11’s by 40 meV from Figure 2a and by utilizing the reported parameters of A = 235.9 nm cm−1 and B = 5.5 cm−1 for such samples.21 For example, we observe M11+ =

for (17,5) and (11,8) should be bundled tubes. We could not tell for sure the data points for (13,10), (12,6), and (13,4) because they lie very closely to the borderline. Figure 2a,b compares our bilayer plot with respect to the corresponding Kataura plot, both containing only the M11 transitions of air-suspended M-SWNTs with the 2n + m values from 21 to 48. In the bilayer plot, the chiralities with the same 2n + m values are also connected with broken lines for convenience. For each 2n + m family, armchairs always locate on the bottom-left corner with an M11 splitting of zero, and θ always decreases from the bottom-left to the up-right. In the Kataura plot, the laser lines can be indicated by solid lines, and the laser resonance windows can be represented by rectangles of 200 meV in width over the corresponding solid laser lines. In the bilayer plot, from + − ⎧ ⎪ x = (M11 + M11)/2 ⎨ ⎪ + − ⎩ y = M11 − M11

(5)

we have + ⎧ ⎪ M11 = x + y /2 ⎨ ⎪ − ⎩ M11 = x − y/2

(6)

+ ⎧ ⎪ y = − 2x + 2M11 = − 2x + 2E L ⎨ − ⎪ ⎩ y = 2x − 2M11 = 2x − 2E L

(7)

or

Therefore, the laser lines can be represented by the V-shaped lines consisting of two lines with a bottom x intercept of EL and slopes of −2 (for M11+ = EL) and +2 (for M11− = EL), as indicated in Figure 2a. The laser resonance window can be represented by the V-shaped regions over the corresponding Vshaped lines with an x intercept range from EL−100 meV to EL+100 meV. All the chiralities in the left wing of the V-shaped region are in resonance with M11+, and all the chiralities in the right wing are in resonance with M11−. Consequently, the M11+ and M11− values of a data point can be read directly on the M11 average axis in the bilayer plot by introducing oblique guiding lines that are parallel to the corresponding laser lines through the data point. For example, it can be read directly via the blue dashed lines in Figure 1f that M11+ = 1.89 eV, M11− = 1.72 eV, and ωRBM = 162 cm−1 for (15,6). For a tube with M11− = 1.948 10513


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Figure 3. Bilayer plots for the M22 (a) and M33 (b) transitions of air-suspended M-SWNTs. Gray, red, and green solid lines indicate the 1.58, 1.96, and 2.33 eV lasers, respectively. Red squares and blue diamonds are data points reported in refs 29 and 37. See text for detail.

2.140 eV and M11− = 1.804 eV from the ERS spectra for a tube on silicon substrates. By calculating an M11 splitting of 336 meV and an M11 average of 1.972 eV, we can assign it to (16,1) unambiguously based on the bilayer plot. Next, we use this bilayer plot to reanalyze some old M11 data reported by other groups and show how to accurately and conveniently determine the (n,m). Note that the ∼15 meV range in the M11 splitting and the ∼30 meV range in M11 are estimated from the statistical data of air-suspended (12,9) tubes, in which the environmental effects consist of π−π interaction inside bundles, amorphous carbon coating, and gas molecules adsorption. It is possible that the range in M11 splitting becomes wider for SWNTs in other environments, however, we do not expect it to be significantly wider than ∼15 meV. In 2010, the RBM resonance profiles of several M-SWNTs in the 2n + m = 36 and 27 families on silicon substrates have been reported, although no ωRBM value has been provided.40 The authors have observed M11− for (15,6), (11,5), (12,3), and (13,1) and both M11+ and M11− for the near armchair chiralities (13,10) and (10,7). Plotting the data points of (13,10) and (10,7) as the red squares in Figure 2c, we find that the data point of (13,10) matches the circle representing (13,10), however, there is no chirality located near the data point of (10,7), suggesting an imperfection either in the data analysis or in the assignment. To further analyze the data, we put the available data of all six chiralities in Table 2 and compare them with our values in the bilayer plot. It turns out that all their data agree with our values except for the M11+ of (10,7), the difference in which is 70 meV and is unacceptably large. Plotting the data point calculated with our M11+ = 2.191 eV as the green square in Figure 2c, we find that this data point matches nicely with the circle representing (10,7). Since their RBM resonance profile of (10,7) acquires only in the range of 2.04−2.18 eV, our analysis speculate that they do not cover the entire profile of M11+. In the first ERS work reported in 2011, an M-SWNT on substrates with M11+ = 1.76 eV, M11− = 1.63 eV, and ωRBM = 160 cm−1 is assigned to (15,6).38 With an M11 splitting of 130 meV and an M11 average of 1.69 eV, we plot the data points both in M11 splitting vs M11 average scale (solid red square) and in M11 splitting vs ωRBM scale (hollow blue diamond) in the bilayer plot in Figure 2c. We find that the data points are somehow away from the circle representing (15,6). The difference in M11 splitting is ∼50 meV, which is significantly larger than the ∼15 meV range for the M11 splitting. Therefore, we speculate that the assignment to (15,6) is inappropriate.

Although the M11 splitting matches nicely with (16,7), it could not be (16,7) because the coincidence of the red square and the blue diamond indicates that this is an individual tube with little interaction with the environments, and the ωRBM for such a tube should not be significantly higher than the corresponding circle. Both the M11 average and the ωRBM match (14,8). With an acceptable difference in M11 splitting of ∼20 meV from the fitting circle, we speculate that (14,8) is a more reasonable assignment for this tube. Here, we emphasize that the appropriate parameters as well as the appropriate link between the two layers, that is, between the M11 average and the ωRBM, for specific samples are important in telling the interactions between the SWNT and its local environments. Figure 2c is specifically suited for SWNTs on silicon substrates, in which the interaction between the tube and the silicon substrates is already taken into account by the redshifts in M11’s and by the appropriate parameters A and B. Therefore, the coincidence of red square and blue diamond indicates negligible additional interactions, implying that this is an individual tube. Plotting the same set of values in Figure 2a instead, we find that the blue diamond is located considerably to the right of the red square, suggesting the existence of additional interactions. This is because Figure 2a is meant for air-suspended SWNTs, in which the interaction between the tube and the substrates is not considered by default and thus is regarded as additional interactions. The corresponding Kataura plots for the M11 transitions of air-suspended SWNTs and SWNTs on silicon substrates are given in Figure 2b,d, respectively, for comparison. Note that although experienced researchers can make the assignments for (13,10), (10,7), and (14,8) in the above examples based on the correctly shifted Kataura plot in Figure 2d, it remains a challenge for them to make those assignments based on the incorrectly shifted Kataura plot in Figure 2b. However, even if we use the bilayer plot in Figure 2a that is meant for airsuspended SWNTs to make assignments for SWNTs on silicon substrates, we can still obtain straightforward and reasonable results, demonstrating the advantage in suitability of the bilayer plots to be used under complex environments. In addition to the plot containing only the M11 transitions of SWNTs, we can also construct the plots considering higher Mii (i = 2, 3, ...) transitions in a similar manner. Since we do not have our own data for the higher Mii transitions, we use the sets of fitting parameters reported in refs 29 and 37 from Wang’s group, in which A = 228 nm cm−1 and B = 0. From the reported formula:29 10514


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Figure 4. Bilayer plots for the Sii transitions of S-SWNTs. (a) The bilayer plot for the S33 and S44 transitions of air-suspended S-SWNTs. (b) The bilayer plot for the S11 and S22 transitions of ssDNA-dispersed S-SWNTs. Gray, red, and green solid lines indicate the 1.58, 1.96, and 2.33 eV lasers, respectively. Red squares and blue diamonds are data points reported in refs 27, 29, 32, and 37. See text for detail.

Mii± = α(i)/d t − β(i)/d t2 ± γ(i)cos(3θ )/d t2

where α, β, and γ are all i-dependent parameters. For two different transition energies Sii and Sjj, we calculate that

(8)

we calculate that α = 6.508 eV nm, β = 2.768 eV nm2, γ = 0.928 eV nm2 for the M22 transitions and α = 9.857 eV nm, β = 6.228 eV nm2, γ = 1.692 eV nm2 for the M33 transitions. The resulting bilayer plots for the M22 and M33 transitions of air-suspended SWNTs are shown in Figure 3a,b. From these two papers,29,37 we find altogether 7 chiralities in common with both experimental M22+, M22−, and ωRBM values and plot these 7 data points in Figure 3a. The data points match the corresponding circles better for tubes with large θ than for tubes with small θ, implying that their fitting parameters suit better for large θ tubes. All of the above examples show cases with the paired transition energies Mii+ and Mii−, for which the (n,m) can be determined based on a single bilayer plot. In the case when unpaired transition energies are provided, one can still make appropriate (n,m) assignments based on several bilayer plots. For example, ref 37 reports M22+ = 2.09 eV and M33− = 2.56 eV for an air-suspended SWNTs. We can accurately assign it to (30,9) by introducing the two oblique guiding lines shown in Figure 3a,b. Sii Difference and the Bilayer Plots for S-SWNTs. For SSWNTs, many previous literatures have reported that the Sii’s are also subject to environmental changes.11,36 For example, systematic Sii comparison between ssDNA-wrapped and SDSwrapped SWNTs shows differences of ∼20 meV in S11 and S22 and ∼40 meV in S33 and S44.32 We observe differences of ∼30 meV in S11 and ∼20 meV in S22 between ionic liquid-dispersed and SDS-wrapped SWNTs.42 Therefore, the Sii differences between the neighboring Sii’s, for example, S22 − S11 and S44 − S33, are less sensitive to the environmental changes than the value of Sii itself. Because the Rayleigh scattering and resonant Raman spectroscopies for S-SWNTs usually involve the S33 and S44 pair, and the photoluminescence excitation (PLE) map usually gives information on the S11 and S22 pair, we hereby show examples of the bilayer plots using S44 − S33 vs (S44 + S33)/2 for air-suspended S-SWNTs (Figure 4a) and S22 − S11 vs (S22 + S11)/2 for ssDNA-dispersed S-SWNTs (Figure 4b). For air-suspended S-SWNTs, the reported equation in ref 37 is given by Sii =

α(i ) β (i ) γ (i ) − 2 + 2 cos(3θ ) dt dt dt

Sii − Sjj =

α(i) − α(j) β(i) − β(j) γ(i) − γ(j) cos(3θ) − + dt d t2 d t2 (10)

and (Sii + Sjj)/2 =

α(i) + α(j) β(i) + β(j) γ(i) + γ(j) − + cos(3θ) 2d t 2d t2 2d t2

(11)

Therefore, Sii − Sjj γ(i) − γ(j)

=

(Sii + Sjj)/2 [γ(i) + γ(j)]/2

+ f (d t )

(12)

and Sii − Sjj =

γ(i) − γ(j) Sii + Sjj + g (d t ) [γ(i) + γ(j)]/2 2

(13)

where f(dt) and g(dt) are functions depending only on dt. As ωRBM is linearly dependent on 1/dt, each ωRBM can be converted into a dt value, and a corresponding g(dt) value can be calculated. Therefore, the ωRBM can be represented by a series of parallel (slope = γ(i) − γ(j) ) but unequally spaced [γ(i) + γ(j)] / 2

(intercept = g(dt)) oblique lines in the Sii − Sjj vs (Sii + Sjj)/2 plot after some mathematical calculations, as shown in Figure 4. We use the sets of fitting parameters reported in refs 29 and 37 for the S33 and S44 transitions of air-suspended S-SWNTs, in which A = 228 nm cm−1 and B = 0. We calculate α = 4.286 eV nm, β = 1.230 eV nm2, γ = 0.412 eV nm2 (MOD1) and γ = −0.412 eV nm2 (MOD2) for the S33 transitions and α = 5.380 eV nm, β = 1.922 eV nm2, γ = −0.644 eV nm2 (MOD1) and γ = 0.644 eV nm2 (MOD2) for the S44 transitions. In Figure 4a, the upper branches correspond to MOD2 tubes, and lower branches are MOD1 tubes. For each 2n + m family, θ decreases from left to right. Because the negative S44 − S33 values are also available, the laser resonance windows are represented by Xshaped regions instead of V-shaped regions. The chiralities in the negative-sloped wing (slope = −2) of the X-shaped region are in resonance with S44 and positive-slope wing (slope = +2) with S33, in a manner similar to those described previously in eq 7. We plot two data points with experimental S33, S44, and ωRBM values reported from refs 29 and 37 in Figure 4a. The data

(9) 10515


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ACS Nano

coordinates need to be taken into consideration, which greatly simplifies the situation. Less data overlap is observed in the bilayer plots than in the Kataura plot, owing to the facts that the data variation in ΔEii is smaller than in Eii and that only one data point is necessary for a pair of Eii’s in the bilayer plots, but two data points are required in the Kataura plot. The only disadvantage of the bilayer plots is that each bilayer plot contains the information on only a pair of Eii’s. When multiple Eii’s of both M- and S-SWNTs are required, one needs refer to a series of bilayer plots instead of only one Kataura plot. In general, the bilayer plots offer not only all the information provided in the Kataura plot but also additional information about the degree of interactions between the SWNT and its local environments. We strongly encourage the researchers in the nanotube community to utilize the bilayer plots for all (n,m)-related studies, especially for accurate and convenient (n,m) determination. We have also launched an (n,m) assignment program on our group Web site (www.chem.pku. edu.cn/cnt_assign) to assist the researchers in accurately and conveniently determining the (n,m) of various SWNT samples under complex environments based on the bilayer plots.

points agree with the corresponding symbols of (23,17) and (16,11). For SDS-dispersed S-SWNTs, the S11 and S22 values are collected by PLE map and fitted using the equations reported in ref 10 as γ 1eV + 12 cos(3θ ) S11 = 0.1270 + 0.8606d t dt (14) and S22 =

γ 1eV + 22 cos(3θ ) 0.1174 + 0.4644d t dt

(15)

where γ1 = 0.04575 eV nm and γ2 = −0.1829 eV nm for MOD1 tubes and γ1 = −0.08802 eV nm2 and γ2 = 0.1705 eV nm2 for MOD2 tubes. A = 223.5 nm cm−1 and B = 12.5 cm−1 are reported with a C−C bond distance of 0.144 nm for SDSdispersed SWNTs,10 whereas a C−C bond distance of 0.142 nm is used in all the other plots. For ssDNA-dispersed SWNTs, we construct the bilayer plot (Figure 4b) for the S11 and S22 transitions by utilizing the reported parameters of A = 218 nm cm−1 and B = 18.3 cm−1 27 and by redshifts of 20 meV from the above values for SDSdispersed SWNTs. Note that some kinks in the ωRBM lines are present in Figure 4b but not in Figure 4a. This is because the slopes for the ωRBM lines happen to be the same for both MOD1 and MOD2 tubes in Figure 4a but not the same in Figure 4b due to different fitting equations and parameters. The positions of those kinks can be determined by solving the corresponding equations with the same ωRBM values. We plot the reported data points of S11 and S22 in ref 32 as red squares and the reported ωRBM in ref 27 as blue diamonds. The good match between the data points and the symbols in Figure 4b leads to straightforward and convenient assignments based on the bilayer plot. Bilayer Plots for Various SWNT Samples. We provide a series of bilayer plots in the Supporting Information for various SWNT samples, including air-suspended SWNTs, SWNTs on silicon substrates, SWNT arrays on quartz substrates, “supergrowth” SWNTs, SDS-dispersed SWNTs, and ssDNAdispersed SWNTs. Different fitting parameters, which are listed in detail, should be used for samples in different environments. Note that in order to obtain correct information about the degree of interactions between the SWNTs and the environments, the appropriate link between Eii average and ωRBM is important. 2

2

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b05860. The Web site of an (n,m) assignment program, and a series of bilayer plots applicable for various SWNT samples under complex environments, including airsuspended SWNTs, SWNTs on silicon substrates, SWNT arrays on quartz substrates, “super-growth” SWNTs, SDS-dispersed SWNTs, and ssDNA-dispersed SWNTs (PDF)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Juan Yang: 0000-0001-5502-9351 Yan Li: 0000-0002-3828-8340 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS We thank Prof. Zhongfan Liu and Prof. Shigeo Maruyama for encouraging us to propose the bilayer plots. We thank Prof. Kaihui Liu and Prof. Rong Xiang for helpful discussion. This research is financially supported by Ministry of Science and Technology of China (2016YFA0201904), National Natural Science Foundation of China (grants 21631002 and U1632119).

CONCLUSIONS The Kataura plot is an Eii−dt or Eii−ωRBM plot, which can provide multiple Eii’s of both M- and S-SWNTs. The significant disadvantage in determining the (n,m) based on the Kataura plot is that both Eii and the ωRBM−dt relation are strongly affected by interactions with the environments. Therefore, one has to simultaneously consider the shifts in both vertical and horizontal coordinates. In addition, some data points overlap with each other in the Kataura plot, especially for the zigzag and near zigzag chiralities, which is also an disadvantage for accurate (n,m) determination. Our bilayer plots use ΔEii vs Eii average in one layer and ΔEii vs ωRBM in the other layer, where ΔEii represents Mii splitting for M-SWNTs and Sii difference for SSWNTs. Because ΔEii is less sensitive to the environmental changes than the value of Eii itself, the shifts in the vertical coordinate are insignificant, and only the shifts in the horizontal

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Bilayer Plots for Accurately Determining the Chirality of Single-Walled

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