PRL 104, 247402 (2010)

PHYSICAL REVIEW LETTERS

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Exciton Diffusion in Air-Suspended Single-Walled Carbon Nanotubes
S. Moritsubo,1 T. Murai,1 T. Shimada,1 Y. Murakami,2 S. Chiashi,3 S. Maruyama,3 and Y. K. Kato1,4,*1Institute of Engineering Innovation, The University of Tokyo, Tokyo 113-8656, Japan
2
Global Edge Institute, Tokyo Institute of Technology, Tokyo 152-8550, Japan3
Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan
4
PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan(Received 2 March 2010; published 15 June 2010)
Direct measurements of the diffusion length of excitons in air-suspended single-walled carbon nanotubes are reported. Photoluminescence microscopy is used to identify individual nanotubes and to
determine their lengths and chiral indices. Exciton diffusion length is obtained by comparing thedependence of photoluminescence intensity on the nanotube length to numerical solutions of diffusion
equations. We find that the diffusion length in these clean, as-grown nanotubes is significantly longer thanthose reported for micelle-encapsulated nanotubes.
DOI: 10.1103/PhysRevLett.104.247402

PACS numbers: 78.67.Ch, 71.35.y, 78.55.m

Optical properties of single-walled carbon nanotubes(SWCNTs) are of importance because of their potential
applications in nanoscale photonics and optoelectronics[1], and they exhibit interesting physics that are uniqueto one-dimensional systems. Limited screening of
Coulomb interaction characteristic of SWCNTs causes
electron-hole pairs to form excitons with large binding
energies [2], and these excitons play a central role in
optical processes. There exists an upper limit to the exciton
density in SWCNTs [3,4] caused by exciton-exciton annihilation [5–7]. Since the annihilation rate is determined byexciton diffusion [3,8,9], its elucidation is a key to understanding light emission processes and their efficiencies in
SWCNTs.
Exciton diffusionpis
ﬃﬃﬃﬃﬃﬃﬃtypically characterized by the diffusion length L ¼ D, where D is the diffusion coefficient and  is the exciton lifetime. Stepwise quenching offluorescence [10] has yielded an exciton excursion range
 ¼ 2L ¼ 90 nm, while L ¼ 6 nm has been reported
from time-resolved measurements
pﬃﬃﬃ [9]. Recent near-fieldmeasurement has resulted in 2L ¼ 100 nm [11]. These
measurements have been performed on micelleencapsulated SWCNTs and DNA-wrapped SWCNTs, but
it is expected that the transport properties of excitons are
extremely sensitive to their surrounding environment. The
exciton diffusion length in clean, pristine SWCNTs has thepotential to be considerably longer, but the measurements
done on suspended nanotubes
pﬃﬃﬃ with a chiral index of ð7; 5Þ
turned out to show Ld ¼ 2L ¼ 200 nm [12].
Here we report direct measurements of the exciton
diffusion length L in air-suspended SWCNTs. Individual
nanotubes are identified by photoluminescence (PL) imaging, while their lengths and chiral indices are determinedby excitation spectroscopy and polarization measurements.
With data obtained from 35 individual ð9; 8Þ SWCNTs, weare able to extract L by comparing the dependence of PL
intensity on the nanotube length with numerical solutions
0031-9007=10=104(24)=247402(4)of diffusion equations. We find that L is at least 610 nm,
which is substantially longer than those reported formicelle-encapsulated SWCNTs [9,10]. The apparent diffusion length becomes shorter with higher excitation
powers, consistent with exciton-exciton annihilation
effects.
In order to obtain suspended SWCNTs of various
lengths, trenches with widths ranging from 0.4 to 2:0 m
are prepared on (001) SiO2 =Si substrates. Electron beamlithography and dry etching processes are used to form the2:5 m deep trenches, and an additional electron beamlithography step is performed to define catalyst areas next
to the trenches. Silica supported Co=Mo catalyst suspended in ethanol is spin coated and lifted off. SWCNTs
are directly grown on these substrates by alcohol catalytic
chemical vapor deposition [13]. A scanning electron microscope image of a typical sample is shown in Fig. 1(a).
We note that these suspended as-grown SWCNTs are very
clean and exhibit excellent optical and electrical properties[14–16].
A homebuilt laser-scanning confocal microscope system
is used for the PL measurements. In order to excite
SWCNTs, an output of a wavelength-tunable continuouswave Ti:sapphire laser is focused on the sample with amicroscope objective lens to a spot size of 1 m. PL iscollected through a confocal pinhole corresponding to an
aperture with 3 m diameter at the sample image plane. A
fast steering mirror allows lateral scanning of the laser spot
for acquiring PL images, and the laser polarization can berotated with a half wave plate. PL spectra are collectedwith a single grating spectrometer and a liquid nitrogencooled InGaAs photodiode array. All measurements are
performed at room temperature in air.
Suspended SWCNTs are identified by taking spectrally
resolved PL images. We raster the laser spot across thescan area on the sample and take a PL spectrum at eachposition. By extracting the emission intensity at the desired

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Ó 2010 The American Physical Society

PRL 104, 247402 (2010)

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FIG. 1 (color). (a) A scanning electron microscope image of atypical sample. The scale bar is 1 m. (b) PL spectra for a
carbon nanotube (red curve) and Si substrate (black curve). Blueand green shaded areas indicate the 4 meV wide integration
windows that are used to obtain PL images at emission energiesof (c) 0.908 eV and (d) 1.116 eV, respectively. The red dot andthe black dot indicate the positions of the laser spot where the redand black curves in (b) were taken, respectively. The scale barsin (c) and (d) are 1 m. For (b)–(d), excitation energy of
1.653 eV and excitation power of 0.36 mW are used. (e) PL
excitation map for the same nanotube. Excitation power of
1:5–2:5 W is used. The spectra are corrected for the changes
in excitation power with tuning of the excitation energy. Inset isthe laser polarization angle dependence of the PL intensity
showing polarization p ¼ 0:90 for this nanotube. Blue circles
are data and the red line is a fit. Taken at an excitation energy of
1.653 eV and an excitation power of 2:24 W.

emission energy from these PL spectra and replotting it inreal space, PL images can be constructed at any spectral
position. Typical PL spectra from an individual SWCNT
and the substrate are shown in Fig. 1(b). The bright sharp
emission line near 0.9 eV is attributed to PL from asuspended SWCNT, while PL from the Si substrate showsa broad peak around 1.1 eV. PL images at emission energies corresponding to the SWCNT [Fig. 1(c)] and Si substrate [Fig. 1(d)] unambiguously show localized SWCNT

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emission at the trench position. If the nanotube PL position
does not coincide with the underlying trench, we exclude
those nanotubes from further measurements as they may
not be fully suspended. We also exclude nanotubes if theyshow considerably lower emission intensity, as those arelikely to have defects or surface contamination.
Once we find a suspended SWCNT with bright emission,
we perform PL excitation spectroscopy for chirality assignment. As shown in Fig. 1(e), only a single peak isvisible throughout the measurement range of excitationand emission energies, and we determine the chirality ofthis nanotube to be ð9; 8Þ from tabulated data [17,18]. If wefind two or more peaks in the PL excitation spectra, thosenanotubes are rejected since they may be bundled.
Finally, PL intensity is measured as a function of polarization angle of the excitation laser [Fig. 1(e), inset]. We fit
the data to I0 þ I1 sin2 ð’ þ ’0 Þ, where I0 and I1 are unpolarized and polarized PL intensity, respectively, ’ is theexcitation polarization angle, and ’0 is the angle offset.
From the fit parameters, we compute the polarization p ¼
I1 =ðI0 þ I1 Þ and use it as a measure of the straightness ofthe nanotube. Since uncertainties in the nanotube length
caused by bending are undesirable, we limit ourselves tonanotubes with p > 0:5 for the measurement of the exciton
diffusion length. Under the assumption that the nanotube isrelatively straight, the length l of the suspended portion of
the nanotube is given by l ¼ w= sin’0 , where w is thewidth of the trench.
Following such careful selection and characterization
procedures, we have investigated 35 individual SWCNTs
with a chiral index of ð9; 8Þ. We focus on a single chirality
in order to avoid any chirality dependent effects. For eachof these nanotubes, we have collected a series of PL spectra
as a function of excitation power P. These measurements
are done with the laser spot at the center of the nanotube,
the laser polarization adjusted for maximum emission
intensity, and the excitation energy tuned to the peak of
the resonance.
Typical data are shown in Fig. 2(a), and we fit the
nanotube peak with a Lorentzian function in order toextract the peak height, width, and position. We calculate
the peak area from the fit parameters, and use this as themeasure of the PL intensity. It shows a sublinear behavior[Fig. 2(b)], as expected from exciton-exciton annihilation[3,7]. This is observed even at the lowest P that we used,suggesting that such effects might play a role even at very
low excitation powers [4]. There are, however, reports ofextended linear regimes on similar samples [7], and moreinvestigation is necessary to clarify this behavior. We canalso estimate the amount of laser induced heating from thebroadening of the linewidth [Fig. 2(c)] by comparing withprevious temperature dependent measurements [7,19,20].
Heating may also influence the diffusion constant [12], but
we find that the increase in the temperature is less than
30 K in all of the nanotubes under investigation. We also
observe the blueshift of the emission line [Fig. 2(d)], which
may be related to gas adsorption [21,22].

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FIG. 3 (color). (a) Simulated exciton density spatial profile forl ¼ 0:5 m (blue curve), 1:0 m (purple curve), and 2:0 m(red curve). Exciton diffusion length L ¼ 0:3 m is used for
obtaining these curves. (b) Simulated PL intensity as a functionof l for L ¼ 0:0 m (black line), 0:2 m (red line), 0:4 m(blue line), 0:6 m (green line), and 0:8 m (orange line).

FIG. 2. (a) Nanotube emission spectra at excitation powers of
0:47 W (circles), 2:32 W (squares), and 13:7 W (triangles). Data are taken at an excitation energy of 1.553 eV.
Data are offset for clarity and lines are Lorentzian fits to data.
The same nanotube as in Figs. 1(b)–1(e) is used. Panels (b)–(d)show photoluminescence intensity, FWHM of the emission line,
and emission energy as a function of excitation power, respectively.

Since we want to analyze the dependence of the PL
intensity on l to obtain the diffusion length, we havesimulated the exciton density profile based on a steadystate one-dimensional diffusion equation given byD

d2 nðzÞ nðzÞþ ðzÞ ¼ 0;
dz2

where nðzÞ is the exciton density, z is the position on the
nanotube, and ðzÞ is the exciton generation rate. Thisequation does not explicitly contain the exciton-exciton
annihilation term, but to first order approximation, suchan effect can be described by a shorter  within this simple
model. We set the origin to be at the center of the nanotube
and impose boundary conditions to be nðl=2Þ ¼ 0, assuming that the quenching of PL due to interaction with thesubstrate is sufficiently strong. Since the exciton generation rate is proportional to the laser intensity profile, we letðzÞ ¼ 0 expðz2 =2 Þ, where 0 is a proportionality
constant and  ¼ 520 nm is the radius of the laser spot.
The diffusion equation becomesd2 nðzÞ
z2
L2

nðzÞ
þ
N
exp

¼ 0;
dz2
2
where N ¼ 0  is a constant.
We numerically solve this equation to obtain nðzÞ, andthe results for L ¼ 0:3 m are plotted in Fig. 3(a). For lshorter or comparable to L, the exciton density profile isnearly parabolic, indicating that the majority of excitons
diffuse to the unsuspended regions before recombining.
Since the nonradiative recombination within the unsus-

pended region efficiently removes excitons, the densityof excitons stays low compared to longer nanotubes. As
the nanotube length gets longer, the exciton density increases until it saturates when the nanotube length becomes long enough compared to 2ð þ LÞ. In such asituation, most of the excitons recombine before theydiffuse out to the unsuspended part, so that l does notplay a role.
In order to compute how the PL intensity changes with l,
we integrate the exciton density profile to obtain the total
number of excitons. We simulate the PL intensity for arange of nanotube length and a series of diffusion lengths,
and plot them in Fig. 3(b). They scale as l3 for small lbecause of the nearly parabolic profile of nðzÞ, then transition to linear behavior when l becomes comparable to 2L,
and finally saturate at very long l. If the diffusion length isvery short, the saturation of the PL intensity is expected for
nanotube length longer than the laser spot size. As the
diffusion length gets longer, the transition to the linear
behavior shifts to longer nanotube length, and saturation
would not be observed.
Now we compare the l dependence of the measured PL
intensity to the simulation [Figs. 4(a)–4(c)]. We perform
least-squares fits to the experimental data by looking for
optimum values for L and N. At all excitation powers, wefind reasonable agreement between the data and the simulation. Although the experimental data show some dispersion from the fit, it is expected that there are some tube-totube variations in the PL intensity. Such inhomogeneities
have been observed in PL imaging of very long SWCNTs[23] and can result from changes in the exciton lifetime 
induced by gas adsorption, contamination, or defects.
Since we observe that the PL intensity varies by 20%for nanotubes of similar lengths, lifetime may also vary bycomparable amounts. Taking such uncertainties in  as theerror in the determination of the diffusion length, we plot L
and N as a function of P in Fig. 4(d).
The apparent diffusion length decreases with increasing
P, which can be qualitatively explained by a reduction of 
caused by exciton-exciton annihilation. The sublinear P
dependence of N is also consistent with this interpretation.

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FIG. 4. Panels (a)–(c) show measured PL intensity as a function of nanotube length for P ¼ 0:47, 2.32, and 13:7 W,
respectively. The solid lines represent the best fits, giving L ¼
610, 430, and 280 nm for (a)–(c), respectively. Polarization
dependence of the detection efficiency has been corrected.
(d) Excitation power dependence of exciton diffusion length.
Inset shows N as a function of the excitation power in arbitrary
units.

However, the z and l dependences of  is not accounted forin our simple model, so it may not be accurate at highexcitation powers. A more rigorous modeling is required toclarify the effects of exciton-exciton annihilation on L.
Nevertheless, such effects should be small at lower powers.
We find L ¼ 610 nm for the lowest P, and an extrapolationof the data down to P ¼ 0 W suggests even longer L.
The observed diffusion length is much longer compared tomicelle-encapsulated SWCNTs [9,10]. It is possible that
surfactants cause additional exciton scattering, and sonication may introduce defects in those samples. Long diffusion lengths are favorable for fabricating single photon
sources from SWCNTs [24], because such a device willneed to have a length less than the exciton diffusion length
to ensure annihilation of excess excitons.
The diffusion length can give us some insight to transport properties of excitons. From L ¼ 610 nm, we obtainD ¼ 44 cm2 s1 for  ¼ 85 ps [4]. Using the Einstein
relation kT ¼ eD, where  is the exciton mobility, k is
the Boltzmann constant, and e is the electronic charge, wefind  ¼ 1:7  103 cm2 V1 s1 . This is comparable toreported values of carrier mobilities in SWCNTs [25–27], although different scattering mechanisms and effective masses should be considered in general.
We acknowledge support from KAKENHI, ISTF,
Murata Science Foundation, Research Foundation for
Opto-Science and Technology, Center for Nano
Lithography & Analysis at The University of Tokyo, andPhoton Frontier Network Program of MEXT, Japan.

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*Corresponding author.
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