PRL 102, 126804 (2009)PHYSICAL REVIEW LETTERS

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27 MARCH 2009

Softening of the Radial Breathing Mode in Metallic Carbon Nanotubes
H. Farhat,1 K. Sasaki,2 M. Kalbac,3 M. Hofmann,4 R. Saito,5 M. S. Dresselhaus,4,6 and J. Kong4
1Department of Materials Science and Engineering, MIT, Cambridge Massachusetts 02139, USA
Department of Quantum Matter, Graduate School of Advanced Sciences of Matter, Hiroshima University,
Higashi-Hiroshima 739-8530, Japan
3J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, V.V.i.,
Dolejskova 3, CZ-18223 Prague 8, Czech Republic
4
Department of Electrical Engineering and Computer Science, MIT, Cambridge Massachusetts 02139, USA
5
Department of Physics, Tohoku University, Sendai, 980-8578, Japan
6
Department of Physics, MIT, Cambridge Massachusetts 02139, USA(Received 21 November 2008; published 25 March 2009)2

The softening of the radial breathing mode (RBM) of metallic single walled carbon nanotubes(m-SWNTs) due to electron-phonon coupling has been studied by observing the Fermi level (F ) dependence of the RBM Raman peak. In situ Raman spectra were obtained from several individual m-SWNTs
while varying F electrochemically. The RBM frequency of an intrinsic m-SWNT is shown to be downshifted relative to highly doped tubes by 2 cm1 . The down-shift is greatest for small diameter andsmall chiral angle SWNTs. Most tubes show no change in RBM linewidth. A comparison is drawn between the RBM and the G band (ALO phonon) with respect to the F dependence of their frequencies andlinewidths.
DOI: 10.1103/PhysRevLett.102.126804

PACS numbers: 78.30.Na, 71.38.k, 78.67.Ch

Electron-phonon (e-ph) interactions govern many aspects of the physical properties of materials. In grapheneand single walled carbon nanotubes (SWNTs), the coupling between electrons and the in-plane C-C stretching
optical phonon modes (G-band phonons) influences thephonon structure [1,2], electrical transport [3], and opticaltransition properties of these materials [4]. Thus, significant effort has been devoted to this subject recently.
Another phonon mode of interest in SWNTs is the isotropic radial deformation of the nanotube called the radial
breathing mode (RBM). This optical phonon is solely theresult of the one-dimensional tubular structure of SWNTs
and its deformations are very different from those of theoptical stretching modes that are common to all othergraphitic materials. e-ph coupling of the RBM is importantbecause it provides a new scattering channel for electrons
that is absent in higher dimensional forms of carbon. Beinga low energy optical phonon, the RBM could be a significant scatterer of low energy electrons such as in electricaltransport at low biases [5].
In metallic SWNTs (m-SWNTs) e-ph coupling can beespecially strong because a wide range of phonon energies
is able to resonantly excite electrons across the linear
electronic bands. Phonons that effectively couple to excitations near the Fermi surface experience lifetime broadening and energy renormalization [1,2]. This phenomenon
is typically investigated by studying how the phonon energy and lifetime evolve as a function of the Fermi energy,F . For the G Raman peak of a m-SWNT, the observed
up-shift and narrowing of the peak with doping has helped
clarify the origin of the metallic G-band line shape and therole of electronic excitations in the softening of the ALO
phonon at F for the neutral m-SWNT [6]. A recent
0031-9007=09=102(12)=126804(4)theoretical treatment of the F dependence of the RBM
predicts a similar, albeit weaker, softening of the RBM
phonon with a significantly larger chiral angle dependence[7]. Because the RBM energy is so much smaller than theG band, the e-ph coupling is expected to be much moresensitive to the fine structure of the electronic bands near
the Dirac point where the valence and conduction bands
touch. An experimental study of the softening of the RBM
frequency in m-SWNTs is therefore important to clarify
the structure-dependent e-ph coupling phenomena associated with the RBM phonon.
In this Letter we present a careful analysis of the frequency !RBM and linewidth RBM of the RBM of individual SWNTs as a function of the electrochemical gating
potential Vg . We observe an increase in frequency when thenanotube is doped with either electrons or holes. Our
experimental results show a diameter (dt ) and chiral angle() dependence of the !RBM softening.
We use a transparent polymer electrolyte (PEO=LiClO4 )to electrochemically dope m-SWNTs. Long, gas flowaligned SWNTs are grown by chemical vapor deposition
and then contacted by a Cr=Au electrode to form theworking electrode of the electrochemical cell. TheSWNTs are typically several hundred microns long andare separated from each other by 20–50 m. A Pt counterelectrode is controlled by a Princeton Applied Research
283 potentiostat to vary the potential (Vg ) of an Ag pseudoreference electrode with respect to the SWNTs [8]. Raman
measurements are made through the transparent electrolyte. The nanotube metallicity is determined by placing!RBM and the excitation energy Elaser on a Kataura plot,and by observing a broadening and shift of the G band [6].
The m-SWNTs under study are either isolated or in small

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Ó 2009 The American Physical Society

Intensity [a.u]140 145 150 155 160 165 140 145 150 155 160 165a)

2εF

Eeh

εF0V

0V

-0.2

-10
Gate Voltage (V)13152.521521151.5-101
Gate Voltage (V)-101Gate Voltage (V)0-1a)

ε (eV)
F00.2b)70

158050156030ω LO1540d)c)

1010

19486193-1FIG. 1 (color online). (a) Waterfall plots of the RBM spectra ofa m-SWNT at several positive (left-hand panel) and negative(right-hand panel) values of gate potential Vg . To evaluate the!RBM down-shift we use the vertical red lines as fiduciary marks
indicating the peak frequency at Vg ¼ 0. The inset of (a) is aschematic band structure of a m-SWNT illustrating the allowede-h transitions. (b) Fitted frequency !RBM and (c) FWHMlinewidth RBM values versus Vg . Error bars denote 95% confidence interval. Solid curves guide the eye. The inset of (b)shows the fitted !RBM ðVg Þ for a s-SWNT.

-0.2

0.2-1

4157-1c)158

5γRBM (cm-1)-1

ωRBM (cm )

153

159

ωRBM (cm )1.5V

ωG- (cm )

1600
-1.5V

ε (eV)
F0-1

Raman Shift (cm )γG- (cm )-1-1

Raman Shift (cm ) values [7]. These m-SWNTs all exhibit a similar characteristic !RBM (Vg ) behavior. Meanwhile, the RBM peak
for a semiconducting SWNT (s-SWNT) shows no appreciable change in !RBM (Vg ) as a function of Vg , as, forexample, shown in the inset of Fig. 1(b). It is thus clear thatthe behavior in Fig. 1(b) is an effect specific to m-SWNTs.
The behavior of !RBM for m-SWNTs can be understoodby considering that, like the G-band phonons, the RBM isalso an optical phonon capable of exciting verticalelectron-hole (e-h) pairs across the linear k valence andconduction bands of the m-SWNT near the K (or K0 ) point.
If the e-ph matrix element for such transitions is nonzero,these scattering events contribute to renormalizing thephonon energy and decreasing its lifetime. As illustrated
in the inset of Fig. 1(a), the Pauli exclusion principle limits
the available e-h pair transitions to those satisfying Ee-h >2jF j, thus giving rise to the Vg dependence of !RBM . As
F is shifted away from the Dirac point, the number ofexcitations contributing to the dressed RBM phonon isreduced and !RBM approaches the frequency of the bare
RBM phonon [7].
In Fig. 2 we compare the F dependence of the G peak(ALO phonon) to !RBM of a m-SWNT which we tentatively
assign as a (13,4) SWNT (!RBM ¼ 193 cm1 , Elaser ¼
1:91 eV). We see a striking resemblance between the Vg
dependence of !G and that of !RBM , with both peaks
displaying a minimum in frequency around Vg ¼ 0.
However, the RBM behavior of the two peaks is quite
different with the RBM peak showing no noticeable broadening in contrast to the 60 cm1 broadening of the G
peak. The lack of broadening for the RBM indicates anegligible resonant decay of the RBM into an e-h pair.

γRBM (cm )bundles of two or three SWNTs. However, only one SWNT
contributes to the signal for these measurements.
Figure 1(a) shows the RBM spectrum of a m-SWNT atseveral values of Vg . Here, a positive (negative) Vg corresponds to electron (hole) doping. A subtle up-shift of2 cm1 for both () polarities of Vg is observed.
Changes in !RBM are more evident after fitting the peaks,
whereby the fit gives both !RBM and the FWHM linewidthRBM versus Vg , shown in Figs. 1(b) and 1(c), respectively.
We use a Voigt profile to deconvolute the instrumental
broadening from the Lorentzian linewidth of the RBM
peak. The !RBM of the m-SWNT shown in Fig. 1(b) makes
an almost symmetric ‘‘V’’ shape about Vg ¼ 0. Previous
studies suggest that environmental effects such as thevan der Waals interaction of the SWNT with the substrateor surrounding solution may modify !RBM [9]. The latter
effect presumably only contributes to a constant background with respect to Vg . The large drop in the peakintensity observed with increasing Vg may be caused bya change in the resonance condition. Because of the loss inpeak intensity with increasing Vg , we are only able to follow !RBM (Vg ) for those m-SWNTs that have strong signals, which biases our sample set towards tubes with small

153.5 b)week ending
27 MARCH 2009PHYSICAL REVIEW LETTERS

PRL 102, 126804 (2009)

01Gate Voltage (V)-101Gate Voltage (V)FIG. 2 (color online). Comparing the G and RBM peaks
versus Vg for a (13,4) m-SWNT (!RBM  193 cm1 , Elaser ¼
1:91 eV). (a) the G frequency and (b) G linewidth versus Vg(bottom axis) and F (top axis). The Vg dependence of the G isused to estimate the gating efficiency,  ¼ 0:24 eV=V. The box
in (b) gives the T ¼ 0 energy range within which the LO phononcan excite real e-h pairs. (c) !RBM and (d) RBM for the same mSWNT.

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27 MARCH 2009PHYSICAL REVIEW LETTERS

To understand why the RBM is down-shifted but notbroadened, it is instructive to draw comparisons to the e-phcoupling of the optical phonons that contribute to the Gband of graphene, m-SWNTs and s-SWNTs, which haverecently been studied in great detail [6,10–13]. Inm-SWNTs, the LO phonon (with energy 0.2 eV) is able
to create real and virtual e-h pairs across the linear electronic bands, resulting in the strong broadening and downshift in frequency characteristic of the metallic G peak asshown in Fig. 2. Similarly, in graphene, the G-band phonons are both broadened and down-shifted because they
couple to e-h excitations near the Dirac point. In the case ofs-SWNTs, the optical phonons do not have sufficient energy to excite e-h pair excitations across the large electronic energy gap. However, these optical phonons can stillcreate virtual excitations which contribute to the downshift of the phonon frequency. Since there is no decay into
real states, there is no linewidth broadening and the frequency shift is modest compared to that of m-SWNTs, asrecently verified in experiments [13,14].
Since @!RBM is a small fraction of the LO=TO phonon
energies, the e-h excitations for the RBM occur muchcloser to the Dirac point. On this energy scale a very smallenergy gap, such as the curvature-induced minigap, becomes significant. The latter, given by Egap ¼ ðA=d2t Þ cosð3Þ, is greatest for zigzag and absent for armchair
m-SWNTs. The value of A is about 60 meV based on an
extended tight-binding model [7]. From the perspectiveof the RBM phonon, the electronic bands of small diameterzigzag m-SWNTs with Egap > @!RBM appear semiconductinglike. Only the armchair SWNTs have truly metallic
bands when close to the Dirac point. Indeed, for the (13,4)
SWNT shown in Fig. 2, Egap ¼ 32 meV (dt ¼ 1:2 nm,
 ¼ 13 ) exceeds the @!RBM (24 meV) and hence nolinewidth broadening is expected.
In Fig. 3 we show !RBM for the (13,4) nanotube calculated from the effective mass theory in [7], which also
predicts no change in RBM as a function of F and a!RBM behavior qualitatively similar to what we observein our measurements. For a quantitative comparison, weconverted the Vg scale of Fig. 2 to an energy scale (bottomaxis) using a gate efficiency of  ¼ 0:24 ðeV=VÞ. The value is determined by fitting the broadening window ofthe G peak to G ¼ o þ EPC ðVg Þ, as shown inFig. 2(b). Here, EPC , the linewidth due to e-ph coupling,is given by Eq. (29) of [1]. The Vg independent contributions to the linewidth are included in o which is taken as10 cm1 based on our experimental results. Comparing
Fig. 2(c) with Fig. 3 within the same F range, we seethat the experimental shift of !RBM is approximately afactor of 2 smaller than that predicted.
It is important to know in what dt range the softening of!RBM becomes significant. The down-shift in !RBM isgreatest in smaller dt m-SWNTs as seen in Fig. 4, whichplots the observed frequency down-shift relative to Vg ¼V0 þ 1V (black) and relative to Vg ¼ V0  1V [gray (red)]

196
195
194
193-0.4-0.2

00.2

0.4

FIG. 3 (color online). Calculated !RBM versus F for a (13,4)
m-SWNT. For this m-SWNT the e-ph coupling contribution tothe RBM is zero. Black points !0 give the bare !RBM and gray(red) points give the corrected !RBM [7].

as a function of 1=dt for several m-SWNTs, where V0 is thegate voltage where the minimum in frequency occurs. Thee-ph matrix element and hence the frequency shift !RBMat a constant Vg is expected to be linear in 1=dt [7]. The
experimental data increase monotonically versus 1=dt withsome variation that we attribute to the expected  dependence of !RBM [7]. Note that the shift on the negative gateside is in most cases greater than it is on the positive side.
We attribute the mild asymmetry with respect to the sign ofVg to the C-C bond softening (stiffening) due to charging
the lattices with electrons (holes). There may also be adifference in the gating efficiency for Vg .
To further explore the  dependence of !RBM we havemeasured the Vg dependence of three consecutivem-SWNTs from the 2n þ m ¼ 24 family, namely, (11,2),(10,4), and (9,6) as shown in Fig. 5. For a given Vg , themagnitude of the measured !RBM decreases from the(11,2) to the (9,6) nanotube. This is in agreement withFIG. 4 (color online). The down-shift !RBM in !RBM vs
1=dt . Red circles indicate the shift relative to Vg ¼ V0  1V(hole doping) and black squares indicate the shift relative toVg ¼ V0 þ 1V (electron doping). Solid lines represent linear fits
to the experimental data points.

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PRL 102, 126804 (2009)

0.2

654
32322(9,6)238c) -0.2

10.2d) -0.2

0.2

76237

54236(10,4)30.2f) -0.2

0.2

244

765243

4(11,2)242

γ RBM (cm-1)

231e) -0.2
ωRBM (cm-1)b)-0.2-101
Gate Voltage (V)γ RBM (cm-1)

ωRBM (cm-1)

ωRBM (cm-1)

233a)

ε F (eV)0.2

γ RBM (cm-1)

ε F (eV)-0.2

3-101
Gate Voltage (V)

2FIG. 5 (color online). Fitted !RBM and RBM for threem-SWNTs from family 2n þ m ¼ 24. (a),(b) (9,6); (c),(d) (10,4); (e),(f) (11,2). The curvature-induced energy gaps
for the three m-SWNTs are 19.3, 41.7, and 60.5 meV, respectively. The energy scale (top axis) is estimated for each mSWNT as in Fig. 2. Solid red lines are curves to guide the eye.

the predicted  dependence of the down-shift in !RBM [7].
The e-ph matrix element, given approximately by
he-hjHe-ph jRBMi ¼ gðjsz j=dt Þ cosð3Þ, where g is theoff-site e-ph coupling constant and sz is the radial displacement, is greatest for the zigzag m-SWNT and approaches
zero for the armchair m-SWNT [7]. Consequently thezigzag RBM is expected to exhibit the largest frequency
shift but no broadening, while the armchair RBM should
show neither a change in !RBM nor in RBM . An intermediate chiral m-SWNT could be expected to exhibit botha frequency shift and a broadening. Within the 2n þ m ¼
24 family, only the (9,6) and (8,8) tubes have a RBM
phonon which is greater in energy than the curvatureinduced band gap. As shown in Ref. [7], the (9,6) tube is
expected to exhibit a broadening  ¼ 2 cm1 and anegligible frequency shift at 300 K. However, we observea small frequency shift and no change in linewidth. On theother hand, the (10,4) SWNT is not expected to exhibit any
broadening, but we observe a possible small broadening of ¼ 1 cm1 . Besides these deviations, the overall measured dt and  dependences of the !RBM down-shift are ingood qualitative agreement with Ref. [7].
It should be noted that the energy gaps for the (9,6) and(10,4) nanotubes are only 7 meV below and 15 meV above
their RBM phonon energies, respectively. The !RBM andRBM of m-SWNTs are predicted to depend strongly on theposition of the cutting line with respect to the K point, as
well as on the size of the energy gap with respect to theweek ending
27 MARCH 2009@!RBM . Small perturbations to the lattice such as strain[15], the displacement of the ALO phonon [16,17], andbreaking of mirror symmetry [18] could sufficiently
change the electronic structure to alter the F dependenceof the RBM peak.
In summary, we have experimentally confirmed that
there is a mild softening of the !RBM of m-SWNTs which
can be removed by both positive and negative doping. The
magnitude of the down-shift scales with 1=dt , and is largest
for small . The measured !RBM shifts agree qualitatively
with the curvature-dependent softening in Ref. [7]. No
gate-induced change in RBM is found for most of theSWNTs that we studied, as is expected [7] for small (dt )SWNTs. Our result shows that chirality dependent corrections to !RBM are required to better assign the dt and (n; m)
indices for small dt m-SWNTs, and that environmental
doping can be responsible for some of the variability inthe observed !RBM . This work also implies that the role of
the RBM in electrical transport and other phonon assisted
relaxation processes warrants further investigation.
This work is mainly supported by the Materials,Structure and Devices (MSD) Center, one of the five programs in the focus center research program (FCRP), aSemiconductor Research Corporation program. Raman
measurements were carried out in the George R. Harrison
Spectroscopy Laboratory supported by NSF-CHE 0111370
and NIH-RR02594. M. S. D. was supported by NSF/DMR
07-04197 and R. S. by MEXT Grants (No. 20241023 andNo. 16076201).

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