Quantum Efficiency and Capture Cross Section of First and Second

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Quantum Efficiency and Capture Cross Section of First and Second Excitonic Transitions of Single-Walled Carbon Nanotubes Measured through Photoconductivity Argyrios Malapanis,† Vasili Perebeinos,‡ Dhiraj Prasad Sinha,† Everett Comfort,† and Ji Ung Lee*,† †

College of Nanoscale Science and Engineering, University at Albany, State University of New York, Albany, New York 12203, United States ‡ IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, New York 10598, United States S Supporting Information *

ABSTRACT: Comparing photoconductivity measurements, using p−n diodes formed along individual single-walled carbon nanotubes (SWNT), with modeling results, allows determination of the quantum efficiency, optical capture cross section, and oscillator strength of the first (E11) and second (E22) excitonic transitions of SWNTs. This is in the infrared region of the spectrum, where little experimental work on SWNT optical absorption has been reported to date. We estimate quantum efficiency (η) ∼1−5% and provide a correlation of η, capture cross section, and oscillator strength for E11 and E22 with nanotube diameter. This study uses the spectral weight of the exciton resonances as the determining parameter in optical measurements. KEYWORDS: Oscillator strength, spectral weight, excitons, photocurrent spectra, carbon nanotubes

S

weight per carbon atom of these excitonic transitions (SC11 or SC22) as a function of tube diameter. We draw our conclusions by comparing the results of a theoretical study, which we briefly outline here, with photocurrent spectroscopy measurements (see Supporting Information, SI). For our experiments we use p−n diodes formed along individual, partially suspended, single-walled carbon nanotubes.22 Our analysis shows that SC11 ≈ 0.8/dt (dt in nm and SC11 in units of 10−17 cm2 eV) while SC22/SC11 ≈ 1 (see SI). For tubes with diameters ranging between 1 and 2 nm, we calculate f C11 ≈ 0.03−0.07 and f C22 ≈ 0.04−0.06 from the modeling data (see SI; f C has a dt dependence similar to SC for the first two exciton subbands; this is expected since the oscillator strength is proportional to the spectral weight23); we estimate η ∼ 1−5% and independent of tube diameter for both exciton peaks, and we find the peak values of the optical capture cross section per atom as σC11 ≈ 1−6 × 10−16 cm2 for the first exciton peak, and σC22 ≈ 0.5−1 × 10−16 cm2 for the second exciton peak (E22), as we outline later. These peak values of the capture cross section per atom are an order of magnitude larger than what has already been reported mostly as a result of photoluminescence experiments.14−17 As we discuss later, however, the peak value of the capture cross section is not a very reliable parameter in measuring optical absorption. Our study shows that the integrated strength of the optical transitions, their spectral weight, can be used as a more

emiconducting single-walled carbon nanotubes (SWNTs) are promising materials for nanoelectronic and nanophotonic applications.1−5 This is due to their quasi onedimensional (1D) character that offers appealing device properties such as reduced carrier scattering,1 enhanced optical absorption,6 and band gaps tunable with a tube diameter1 and electrostatic doping.4 Photoinduced electron−hole pairs, or excitons, with binding energies of several hundred meV, dominate the SWNT optical spectra.2,3,5,7−9 Excitonic transitions in the first, second, and higher subbands are widely used to pinpoint tube diameter and chirality.10−12 Measuring optical absorption and correlating it to structure-dependent factors has been a subject of a number of papers and is fundamental to carbon nanotube photophysics.5,13 Some photoluminescence and other studies have provided initial insights into the optical capture cross section of excitonic transitions for higher subbands, including as a function of tube diameter (dt). Little has been reported, however, on such a relation for the first exciton peak (E11). Furthermore, most reports published to date provide the peak values of the capture cross section per carbon atom without sufficient emphasis on the spectral weight (or integrated oscillator strength) of the optical transitions.14−21 Here we provide such a correlation for the quantum efficiency (η)the probability that excitons will dissociate into electrons and holes that reach the electrodes and contribute to photocurrent, capture cross section per carbon atom (σC), and oscillator strength per carbon atom (f C) of the first and second exciton resonances by analyzing the spectral © 2013 American Chemical Society

Received: March 13, 2013 Revised: June 20, 2013 Published: July 30, 2013 3531

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Figure 1. (a) Scanning electron miscroscopy (SEM) image of single-walled carbon nanotube (SWNT) p−n diode. (b) Device schematic (not drawn to scale). Vg1,2 and Vbg denote the split-gate and back-gate bias, respectively; L is the width of the trench, over which the tube is suspended. (c) First, E11, and (d) second, E22, exciton resonances from 13 devices in photoconductivity experiments reported in the text. The measured photocurrent is normalized to the photon flux and to the number of carbon atoms in the tube’s suspended part, as well as divided by the elementary charge, to give the product of the capture cross section per carbon atom, σC, with quantum efficiency, η. E and λ denote energy and wavelength, respectively.

The quasi-1D character of carbon nanotubes dominates their optical transitions. Light absorption produces strongly correlated electron−hole pairsexcitons.2,3 The geometry of our devices with the suspended part ranging from 0.3 to nearly 3 μm provides a wide range of light absorption without adversely affecting device yield. The asymmetric band structure of our devices, intrinsic to all p−n structures, allows detection of photocurrent from excitons as they dissociate into their constituent electrons and holes. The spectra consist of a series of narrow peaks related to excitonic transitions and characteristic of 1D structures (see Figure 1c,d). As previously demonstrated, our diodes can be used to measure the E11 and E22 peaks, the lowest exciton resonances of the two lowest subbands.22,24 We note that the line widths of the E11 peaks illustrated in Figure 1c are somewhat broader than the line widths extracted from photon-echo studies of nanotubes mixed in polymer.25 In that work, the authors observed an unusually narrow line width due to motional narrowing, a process associated with the disorder in the polymer matrix that reduces exciton−phonon coupling.25 To measure the photocurrent spectra over a broad energy range (0.5−2 eV), we use a quartz tungsten halogen lamp dispersed through a monochromator. The light from the exit slit is then focused on our sample. To measure the photocurrent, we bias the split gates of a device in a p−n or n−p configuration (Vg1 = −Vg2 = +5V or Vg1 = −Vg2 = −5 V, respectively) with drain and source at 0 V and the substrate grounded. To maximize the power output, we set the

accurate gauge in modeling and experimental work on nanophotonics. Figure 1a,b shows a scanning electron miscroscopy (SEM) image and schematic/dimensions of the device we use in our photoconductivity experiments. Details on device fabrication and electrical and optical characterization have been provided elsewhere.22 Briefly, we deposit a 1-μm-thick oxide (SiO2) on a heavily p-doped 300-mm silicon wafer. Using standard deposition, ion implantation, lithography, and etching techniques, we form n-doped polysilicon buried split gates with spacing ranging from 0.5 to 2 μm. We subsequently deposit a thick layer of SiO2 and planarize it through chemicalmechanical polishing to form a 100-nm gate dielectric above the split gates. We then deposit TiN and pattern it to form source (S) and drain (D) contacts. To suspend the nanotubes, we etch a 450-nm-deep trench into the oxide between the split gates. It has been shown that suspending the nanotubes, rather than growing them directly on a substrate, makes the diodes nearly ideal.24 The trench width (L) is about 0.2 μm less than the split-gate spacing. Finally, we grow nanotubes on top of the S and D contacts through catalytic chemical vapor deposition. Synthesizing the nanotubes as the last step in the process results in relatively pristine tubes. We chose to report on diodes that exhibit nearly ideal behavior because they show a strong correlation between the diode leakage current and the first excitonic level, as we have previously described in ref 22, suggesting that we are measuring the intrinsic properties of the nanotubes. 3532

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Table 1. Data Related to the First Exciton Peak Dev.

E11 (eV)

E22 (eV)

dt (nm)

(n,m)

σC11·ηexp × 10−17 cm2

fwhm11 (meV)

SC11·ηexp × 10−20 cm2 eV

SC11 model × 10−17 cm2 eV

η11 (%)

σC11 × 10−16 cm2

f C11

1 2 3 4 5 6 7 8 9 10 11 12a 12b

0.565 0.568 0.593 0.618 0.638 0.646 0.667 0.671 0.714 0.851 0.883 0.905 0.929

1.085 1.130 1.139 1.188 1.216 1.223 1.270 1.254 1.400 1.447 1.604 1.661 1.717

1.90 1.88 1.81 1.74 1.71 1.67 1.57 1.56 1.39 1.18 1.15 1.10 1.04

15,13 19,8 18,8 20,4 15,10 17,7 14,9 20,0 14,6 12,5 13,3 10,6 10,5

1.22 1.16 1.34 0.83 0.55 0.89 0.64 0.78 0.23 0.63 0.16 0.20 0.30

7 6 10 8 9 12 11 12 12 29 19 34 35

12.56 8.68 16.76 9.09 6.78 16.33 9.98 12.00 4.88 29.12 3.68 7.45 18.18

0.42 0.43 0.44 0.46 0.47 0.48 0.51 0.51 0.58 0.68 0.70 0.73 0.77

3.0 2.0 3.8 2.0 1.4 3.4 2.0 2.3 0.8 4.3 0.5 1.0 2.4

4.08 5.67 3.52 4.18 3.80 2.62 3.24 3.33 2.68 1.46 2.97 2.00 1.25

0.038 0.039 0.040 0.042 0.043 0.044 0.046 0.047 0.052 0.062 0.063 0.066 0.070

Table 2. Data Related to Second Exciton Peak Dev.

L (nm)

ϕ (deg)

L′ (nm)

N

σC22·ηexp × 10−17 cm2

fwhm22 (meV)

SC22·ηexp × 10−20 cm2 eV

SC22model × 10−17 cm2 eV

η22 (%)

σC22 × 10−16 cm2

f C22

1 2 3 4 5 6 7 8 9 10 11 12a 12b

1800 1800 800 1300 1800 800 1800 800 1800 300 1300 800 800

0 11 44 57 0 17 25 0 29 15 23 59 9

1800 1835 1113 2385 1800 837 1987 800 2058 311 1412 1569 810

400208 403665 235668 485687 360188 163534 365009 146041 334763 42928 190054 201917 98543

0.10 0.09 0.20 0.11 0.13 0.27 0.24 0.31 0.13 0.36 0.33 0.22 0.28

42 40 37 49 47 47 53 42 41 43 66 63 49

5.99 4.86 11.12 8.85 9.98 17.92 21.68 20.05 7.29 23.55 33.00 22.02 14.79

0.46 0.46 0.47 0.49 0.49 0.50 0.52 0.52 0.57 0.63 0.65 0.67 0.69

1.3 1.1 2.3 1.8 2.0 3.6 4.1 3.8 1.3 3.7 5.1 3.3 2.1

0.73 0.83 0.84 0.63 0.66 0.75 0.57 0.81 1.03 0.98 0.64 0.67 1.32

0.042 0.042 0.043 0.044 0.045 0.046 0.048 0.048 0.052 0.058 0.059 0.061 0.063

expected due to a loss in oscillator strength and increased Auger-like nonradiative decay of excitons as a result of the increasing doping level as we ramp up the back-gate bias.26−29 To verify whether our devices were made up of individual SWNT crossing source-drain, and to measure the angle (ϕ) between each nanotube and the direction perpendicular to the trench, we took SEM images of all the devices used in the experiments after completing the measurements. With the exception of one device (Dev. 12a,b in Tables 1 and 2, where we found two nanotubes crossing S−D), we found that all devices were formed along a single SWNT. Measuring ϕ in each case allowed us to calculate the true length of the tube crossing the trench (L′ = L/cos ϕ) and thus to get a fairly accurate estimate of the number of carbon atoms in the tube’s suspended part (N). [We find the number of C atoms per unit length (nL′) from nL′ = 4(n2 + nm + m2)1/2/α√3 = 4πdt/ α2√3,29 and thus N = nL′·L′; where (n,m) are the tube’s chiral indexes, and α = 0.249 nm is the lattice constant for carbon nanotubes, for which the C−C bond length is αC−C = 0.144 nm, slightly larger than graphene’s 0.142 nm.30] The results are shown in Table 2. Since we know that photocurrent is largely produced in the nanotube’s suspended region, N is needed to normalize the experimentally measured parameters to the number of carbon atoms. The photocurrent (IPC) measured in our photoconductivity experimentsproduced as photoexcited electron−hole pairs dissociate into their constituent parts, become free carriers, and reach the electrodesis given by:31

bandwidth high by widening the inlet and outlet slits of the monochromator without affecting the resolution of our instruments. Using this setup, we determine the E11 and E22 peaks of an individual SWNT. This allows us to assign the nanotube diameter and chiral indexes using the Kataura plot.11,22 We find that our assignments are consistent with a recently published atlas of carbon nanotube optical transitions (see SI).12 Unless otherwise indicated, measurements are taken in air at temperature T = 300 K. The incident light is unpolarized. During the experiments, we illuminate the entire nanotube. We have strong evidence, however, that the photocurrent comes largely from the tube’s suspended part. We draw this conclusion by measuring photocurrent as a function of doping density through varying the split gate bias (supported region), while keeping the substrate grounded; and then by doping the nanotube in the suspended region through varying the backgate bias, while keeping the split-gate bias constant. In the first case, we see virtually no change in the intensity or position of the exciton peaks as we double the doping density through the split gates. In the second case, we observe a rapid quenching and small red shift of the first exciton peak; the intensity of E11 diminishes to a small fraction of its original value at a back-gate doping density smaller than that achieved through the split gates (see SI). The stark contrast in the results of these two experiments, conducted in vacuum at T = 150 and 200 K on several devices, shows that photocurrent is largely produced in the tube’s suspended region. The observed quenching is 3533

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Figure 2. (a) First, E11, and second, E22, exciton peaks from device with nanotube diameter dt =1.67 nm. Discrete points are data. The solid line is Lorentzian fit of data; the shaded area under each curve indicates peak’s spectral weight, obtained by integrating a Lorentz profile. Data are normalized as in Figure 1c,d. E and λ denote energy and wavelength, respectively. (b) Spectral weight, SCii, i = 1,2, times quantum efficiency, η, for E11 and E22, show no dependence on dt; their ratio (inset) is ∼1. (c) Quantum efficiency for E11 (solid circles) and E22 (open circles) shows no dependence on dt. However, η does show a correlation with the length of the suspended part of the tube, L′, diminishing with increasing L′ (inset); here the solid line guides the eye. (d) Capture cross section per carbon atom, σCii, i = 1,2, for E11 (solid circles) and E22 (open circles); solid and dashed lines are linear fits of corresponding data, showing that σC22 does not depend on dt, and σC11 scales linearly with dt (σC11 ≈ 2dt , with dt in nm and σC11 in units of 10−16 cm2).

IPC(E) = q·Φ(E) ·σ(E) ·η

estimate the full width half-maximum (fwhm), or line width, of each resonance, as well as each transition’s spectral weight (SCii, where i = 1,2). More accurately, we measure the product of the spectral weight per atom with η (A = ∫ σC(E)·ηdE = η∫ σC(E)dE ≡ η·SC). Tables 1 and 2 show the relevant data, which are also illustrated in Figures 1c,d and 2a−d. To estimate the quantum efficiency, we need to know the absorption cross section, σ. Using the excitonic picture, we find that the first exciton level absorbs nearly 80% of the band− band spectral weight and has similar diameter scaling as that predicted by the single particle picture. In the single particle picture, see ref 33−36 and Supporting Information, the absorption cross section per carbon atom is given by σ(ω), where the frequency of the incident light, ω, is related to the photon energy, E, by E = ℏω, with ℏ Planck’s reduced constant. It is therefore important to understand the single-particle picture. Using this model, outlined in more detail in the Supporting Information, we find that the absorption cross section per carbon atom in the single-particle picture is given by:

(1)

where q is the elementary charge, Φ is the photon flux (measured in photons per unit time and per unit area by calibrating our apparatus with precalibrated photodiodes for the range of photon energy used in our experiments), σ is the optical capture cross section, and η is the quantum efficiency. η gives the probability that the photoexcited electron−hole pairs will dissociate and reach the electrodes, thus contributing to the photocurrent.32 We assume η is independent of energy for the small energy range around the exciton resonances (up to tens of meV). Thus the product of the capture cross section per carbon atom, σC, with quantum efficiency is given by: σC(E) ·η = IPC(E)/(q·Φ(E) ·N )

(2)

This is the parameter we use to measure each exciton peak, as Figure 1c,d shows. All values are adjusted for irradiation with unpolarized light, which means the values of the optical capture cross section have been multiplied by a factor of 2, since only light parallel to the nanotube axis contributes to photocurrent.14,21 Fitting the data for each exciton peak to a Lorentzian profile and integrating the area (A) under the peak allows us to 3534

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Nano Letters σC(ω) =

Letter

σ(ω) 4 3 αt = σ0 N π dt ℏ2ω 2

∑ i

Δi 2 ℏ2ω 2 − Δi 2 (3)

where N is the number of carbon atoms under light illumination and producing photocurrent; the band index i = 1, 2, 3, etc. labels all subbands with band gaps Δi = μΔ1 = μ[(2tα)/√3dt], with Δ1 the fundamental band gap and μ = 1, 2, 4, 5, 7, 8, etc. (an integer which is not a multiple of 3); t = 3.1 eV is the nearest neighbor hopping integral in the tight-binding approximation; and σ0 = (q2/4ε0cℏ)(√3α2/4) = 0.61 × 10−17 cm2 the absorption cross section of graphene (with ε0 the permittivity of free space and c the speed of light).37,38 Note that in the limit of infinitely large tube diameter, summation in eq 3 can be replaced by an integral and an interband absorption cross section in pristine graphene of σC(ω) = σ0 is obtained. To calculate the absorption cross section per atom in a carbon nanotube under light illumination, however, we need to use the excitonic picture, not the single-particle picture outlined above. Therefore, we solve the Bethe-Salpeter equation39 on the basis of the tight-binding wave functions to calculate absorption.9 The results are outlined in the Supporting Information. The integrated oscillator strength, or spectral weight, of the excitonic transitions is expected to be dependent on the dielectric constant of the tube’s environment.9 For a typical value of the dielectric constant appropriate for modeling carbon nanotubes in our experimental conditions, we find that dependence of the spectral weight on tube diameter can be captured by the empirical scaling relations (see SI): (4)

Figure 3. Data from model (see SI). (a) Spectral weight, or integrated oscillator strength, per carbon atom for the first, SC11 (solid circles), and second, SC22 (open circles), exciton peaks as a function of tube diameter, dt. The solid line is fit for SC11 data: SC11 ≈ 0.8/dt; and dashed line for SC22 data: SC22 ≈ 0.73/(dt0.68), where dt is in nm and SCii in units of 10−17 cm2 eV. (b) Ratio of spectral weight of second to first exciton peak is ∼1. (c) Oscillator strength per carbon atom, f Cii, i = 1,2, for E11 (solid circles) and E22 (open circles). The solid line is fit for f C11 data:f C11 ≈ 7.3/dt; and dashed line for f C22 data: f C22 ≈ 6.7/ (dt0.68), where dt is in nm and f Cii is in units of 10−2. Inset: f Cii scales almost linearly with P/dt, where P = 1 for E11 (solid squares), and P = 2 for E22 (open rhombi)a scaling similar to that in ref 40.

where dt is in nm and SCii in units of 10−17 cm2 eV. Figure 3a illustrates the relevant data. The ratio of the spectral weights of the second to the first exciton peaks is: (SC22/SC11 ≈ 1 in the diameter range of interest here (dt = 1−2 nm), as Figure 3b shows. Our modeling is consistent with a recent ab initio calculation providing an explicit formula for the optical oscillator strength per atom of excitonic transitions in SWNTs (see also Figure 3c, inset).40 Using eq 3, we can also estimate the maximum integrated oscillator strength if the spectral weight from the single-particle picture were to be transferred to the exciton completely: SC,maxii = σ0(4√3/π)(αt/ dt) ≈ (1.0/dt), which is independent of the band index i. We can now compare our experimental results with the calculations from the theoretical model described above, and outlined in more detail in the Supporting Information, to obtain an estimate of the quantum efficiency, ηii ≈ (SCii·η)exp/ (SCii)mod, the peak value of the capture cross section per carbon atom, σCii, and the dimensionless oscillator strength per carbon atom, f Cii. As the results in Tables 1 and 2 show, the data for the quantum efficiency estimated in this fashion show no diameter dependence but have a large scatter (see also Figure 2c). The latter is likely due to the relative inhomogeneity of the tubes in the devices used in our experiments. As previously reported, even a small amount of adsorbates on the tube’s suspended part could result in trap states in the band gap, affecting the electrical and optical properties of the devices; uneven adsorbate coverage could cause inhomegeneity in optical behavior from device to device.41 Adsorbates can act as charge impurities that serve as localization centers for bright excitons in single-walled carbon nanotubes.42 However, we see a striking

correlation between the quantum efficiency and the length L′ of the suspended segment of the nanotube (see Figure 2c, inset). η tends to decrease as L′ increases. This suggests that the longer the suspended part of the tube the larger the possibility for exciton quenching to occur, diminishing quantum efficiency. The experimentally obtained products of spectral weight with quantum efficiency are approximately the same for the first and second exciton peakstheir ratio being approximately 1, as the inset of Figure 2b showssuggesting a similar dissociation mechanism for E11 and E22; the scattering in the experimental data of this ratio (three outliers are much larger than 1) is likely due to the scattering of η. A recent photoconductivity study with devices similar in structure to ours reported quantum efficiencies of ∼12% and 9% for the first and second exciton peaks, respectively.21 The authors of that study, however, used the geometric, rather than the optical, capture cross section. (This may be valid for evaluating the technical merits of these diodes for applications; under those circumstances, the geometric area is the relevant parameter for evaluating efficiency.) If we were to use a similar parameter (the geometric capture cross section = L′ × dt normalized to the number of carbon atoms) we would obtain quantum efficiencies up to 15%. Figure 2d illustrates the dependence on tube diameter of the peak values of the capture cross section per carbon atom, σC, for the first and second exciton peaks, estimated by dividing the measured peak values of σ C ·η (Figure 1c,d) by the corresponding quantum efficiency, η (Tables 1 and 2). As Figure 2d illustrates, σC22 shows no dependence on dt , but σC11 increases linearly with diameter: σC11 ≈ 2dt , with dt in nm and

SC11 ≈ 0.8/d t

and

SC 22 ≈ 0.73/(d t 0.68)

3535

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σC11 in units of 10−16 cm2. This result is of limited significance, however, since we find that the peak value of the capture cross section per atom is not a very reliable parameter in measuring optical absorption. The reason for this conclusion is the following. If we were to use the peak values of σC11 published to date (∼1 × 10−17 cm2) for individual SWNTs with dt in a range similar to ours (0.8−2 nm),14−17 and compare them to our measured value of σC11·η, we would get quantum efficiencies of ∼100% in our devices and oscillator strengths 1−2 orders of magnitude smaller than those predicted by theory and confirmed experimentally.14,40 Moreover, an efficiency of ∼100% would be difficult to justify, particularly for the E11 excitons, since their large binding energies2,3,5 would require a very efficient dissociation mechanism, which has yet to be identified. We thus conclude that extrinsic effects play a role in the estimation of the peak value of σC11. This argument is also supported by the oscillator strength per atom we calculate for the E11 excitons, f C11, which shows the opposite trend in its scaling with tube diameter than σC11 does, as we demonstrate below. The dimensionless oscillator strength per carbon atom is given by:23 fC =

4ε0mc hq

2

∫ σC(E)dE ≡

4ε0mc hq2

SC

(5)

where m is the electron mass; h Planck’s constant and the other parameters have been defined earlier. The prefactor to the integral in eq 5 is 0.0911 × 10−17 cm−2 eV−1; we use it to calculate the oscillator strength from the modeling results (eq 4) and the experimentally assigned dt (Tables 1 and 2). As Figure 3c shows, illustrating the results of our theoretical model, the oscillator strength per atom thus obtained scales with tube diameter the same way as the spectral weight per atom does: f C11 ≈ 7.3/dt and f C22 ≈ (6.7/dt0.68) where dt is in nm and f Cii in units of 10−2. This is expected, because the oscillator strength is proportional to the spectral weight, as eq 5 shows. We finally need to discuss the exciton dissociation mechanism for the lowest subband. Photoexcited electron− hole pairs are produced largely in the tube’s suspended region and diffuse to the edges of the trench, where the longitudinal (parallel to the tube’s axis) electric field (FII) increases by at least an order of magnitude compared to the rest of the tube’s suspended portion, as finite element analysis of our device shows (see SI); the value of FII at the trench edges, however, is not high enough to dissociate the excitons by field ionization a conclusion consistent with our experimental results. As we noted earlier, the intensity and position of the first exciton peak do not change as we double the split-gate bias and thus the corresponding electric field (see SI), leading us to exclude exciton dissociation through field ionization. The exciton dissociation rate due to field ionization has been shown to increase exponentially with the longitudinal electric field.32 It is likely that excitons dissociate near the trench edges, because their binding energy is reduced to tens of meV or less, there, due to band gap renormalization.4,22,43,44 Thermal energy at room temperature (∼25 meV) could provide enough boost in those areas of the tube for exciton dissociation in the first subband. A thermal study of several of the devices used in our experiments, with a range of diameters, shows a weak dependence on temperature for E11 (see Figure 4a), and no such dependence for E22, which is consistent with this picture.

Figure 4. (a) Activation energy, Ea, data (solid squares) of first, E11, exciton peak of four devices with a range of nanotube diameters (dt = 1.10, 1.18, 1.57, 1.67 nm) show there is a weak dependence of exciton dissociation on temperature. Inset: Photocurrent (IPC) spectra of device with dt = 1.57 nm at temperature T varying between 200 and 350 K. IPC is normalized to the photon flux here. Plotting the peak IPC value as a function of 1/T allows the extraction of Ea; since IPC ∝ exp(−Ea/kBT), where kB is Boltzman’s constant, the slope of ln(IPC) vs 1/T is a measure of the activation energy. (b) Full-width at halfmaximum, fwhm, of E11 (solid circles), and E22 (open circles), exciton peaks show a pronounced dependence on energy, E. (c) Spectral weight per atom times quantum efficiency, SC·η, of E11 as a function of T for same device as in (a) inset. Discrete points are data; the solid line is the exponential fit of data.

As Figure 4b shows, the behavior of E22 line widths we observe experimentally (at T = 300 K) demonstrates a pronounced energy dependence, which agrees quite well with that reported in ref 45. As that study showed, the decay mechanism of E22 excitons is well understood, with excitons of the second subband decaying into the continuum of the E11 bright exciton and into the doubly degenerate dark E11 state with the emission of optical phonons.45 Our experimental data show a similar energy dependence of the E11 line width as that 3536

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of E22 broadening (see Figure 4b), suggesting a similar decay mechanism for E11 as E22. In pristine tubes, however, there are no states 160 meV below E11 to activate the optical phonon decay channel, which is available for E22.46 The splitting between the dark states and E11 is significantly smaller than the optical phonon energy.5,46,47 Therefore, the existence of localized states48−50 below E11 may provide the density of final states for the optical phonon decay path. Alternatively, the diameter dependence of the Coulomb impurity potential42 may give rise to the inhomogeneous diameter-dependent line width broadening. More work is needed, however, to identify the nature of the localized states and the role of impurities on the E11 line width. Figure 4c shows a very similar scaling with temperature of the spectral weight per atom times quantum efficiency for the E11 in one of the devices used in our thermal study. The displayed data are typical of the results in the other devices. This set of data shows a strong increase of SC11·η with T, which suggests that the quantum efficiency increases rapidly with T. This indicates that exciton diffusion,51−53 and thus the exciton dissociation rate, increases substantially with T. To summarize, we have demonstrated that modeling of the photocurrent spectra of single-walled carbon nanotubes shows that the spectral weight per carbon atom of the first exciton peak is inversely proportional to tube diameter, and a similar scaling holds for the second exciton peak. Comparing the modeling results with photoconductivity measurements using SWNT p−n diodes allowed us to estimate the quantum efficiency, optical capture cross section per atom, and oscillator strength per atom of the first and second exciton transitions in carbon nanotubes with a range of chiralities, and to provide scaling of these parameters with tube diameter in the range dt = 1−2 nm. Our study also suggests a route to enhance quantum efficiency by scaling the suspended region of the nanotube appropriately.

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ASSOCIATED CONTENT

S Supporting Information *

Effect of doping through split gates vs back gate, modeling photoconductivity in carbon nanotubes, confirming excitonic nature of photoconductivity peaks, and finite element analysis. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A Naval Research Laboratory grant (N001731216008) through a Defense Threat Reduction Agency Military Interdepartmental Purchase Request (MIPR no. HDTRA 135701) supported this work.

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REFERENCES

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Nano Letters

Letter

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