Scaling of Resistance and Electron Mean Free Path of Single-Walled
  Carbon Nanotubes
Scaling of Resistance and Electron Mean Free Path of Single-Walled Carbon
Nanotubes
Meninder Purewal1, Byung Hee Hong2, Anirudhh Ravi2, Bhupesh Chandra3, James Hone3, and Philip Kim2
Department of Applied Physics, Columbia University, New York, New York 10027
Department of Physics, Columbia University, New York, New York 10027 and
Department of Mechanical Engineering, Columbia University, New York, New York 10027
We present an experimental investigation on the scaling of resistance in individual single walled
carbon nanotube devices with channel lengths that vary four orders of magnitude on the same
sample. The electron mean free path is obtained from the linear scaling of resistance with length at
various temperatures. The low temperature mean free path is determined by impurity scattering,
while at high temperature the mean free path decreases with increasing temperature, indicating that
it is limited by electron-phonon scattering. An unusually long mean free path at room temperature
has been experimentally confirmed. Exponentially increasing resistance with length at extremely
long length scales suggests anomalous localization effects.
Single walled carbon nanotubes (SWNTs) are 1D con-
ductors that exhibit a rich variety of low dimensional
charge transport phenomena [1], including ballistic con-
duction [2, 3, 4, 5, 6], localization [7] and 1D variable
range hopping [8]. The electron mean free path, Lm, is
one of the important length scales that characterize the
different 1D transport regimes. One method of determin-
ing Lm in SWNTs is to measure ballistic conduction for a
given device channel length. However, this method yields
a lower bound of Lm, and works only at low temperature
[2, 3, 4, 5] or at higher temperature for small length scales
(<60 nm) [6]. Another approach to obtain Lm at room
temperature is to employ scanning probe microscopy to
measure the linear scaling of the channel resistance [9], or
use non-invasive multi-terminal measurements [10]. Due
to the experimental limitations of these approaches, the
characterization of Lm for the same SWNTs over a range
of temperatures is yet to be realized.
Recent advances in the growth of extremely long
SWNTs (>1 mm) [11] now allow for an intensive study
on their intrinsic properties. In this letter, we present
experimental measurements on the scaling behavior of
resistance in individual, millimeter long SWNTs for the
temperature range of 1.6 - 300 K. From the linear scaling
of resistance, the temperature dependent electron mean
free path is calculated for each temperature. Beyond the
linear scaling regime, we observe that the resistance in-
creases exponentially with length, indicating localization
behavior.
Macroscopically long and straight individual SWNTs
were grown on a degenerately doped Si/SiO2 substrate
(tox = 500 nm) using the chemical vapor deposition
method described in Ref.[11]. This was followed by the
fabrication of multiple Pd electrodes with various sepa-
rations (200 nm- 400 µm)(Fig. 1(a)). Pd electrodes were
chosen to create highly transparent SWNT-electrode con-
tacts [4]. The diameters of the SWNTs were measured
by atomic force microscope (AFM). We chose SWNTs
with diameter d less than 2.5 nm to exclude any possi-
bility of including multiwalled nanotubes (MWNT) in
-15 0 15
-30 -15 0
500 m500 m
Vg (V)
Vg (V)
0.8 m
0.8 m
1.5 m
1.5 m
FIG. 1: (a)Optical image showing typical SWNT devices with
multiple Pd electrodes. (Inset) Scanning electron microscope
image of an isolated SWNT contacted with these electrodes.
Room temperature ISD(Vg) of selected channel lengths for (b)
metallic SWNT (M1) and (c) semiconducting SWNT (SC3)
with VSD = 6.4 and 2.7 mV, respectively.
this study. In addition, we confirmed that the high
bias saturation current is < 30 µA for all SWNTs stud-
ied [12], assuring that the samples consisted of single
tubes rather than small bundles or MWNTs. The sub-
strate was used as a gate electrode to tune the chemical
potential of the sample by the application of a gate volt-
age Vg. A small dc source-drain bias voltage (< 10 mV),
VSD, was applied between pairs of consecutive electrodes,
and the two-terminal linear response conductance was
determined from the measured source-drain current ISD.
Fig. 1(b-c) shows the measured ISD as a function of
VSD for selected channel length sections on two repre-
sentative SWNTs. All curves exhibit a ‘gap’ like fea-
ture - a range of Vg where ISD is suppressed. On
the same SWNT, every device (pair of consecutive elec-
http://arxiv.org/abs/0704.0300v2
trodes) shows a similar ISD(Vg) up to a length-dependent
multiplicative factor, once we align the centers of the
gap region for each curve. The similarity of the ISD(Vg)
behavior in different sections for each SWNT sample
indicates that the corresponding ‘gap’ features are de-
rived from the intrinsic electronic structure of the SWNT
rather than the effects of random local variation.
We use the qualitatively different ISD(Vg) behaviors of
different SWNTs to categorize them as metallic (M-NT)
or semiconducting nanotubes (S-NT). Typical S-NTs
(Fig. 1(c)) exhibit an off current region ISD < 10
when the Fermi energy EF lies n the energy gap [13, 14].
On the other hand, a weaker suppression of ISD(Vg) is
observed in the ‘small gap’ region in M-NTs (Fig. 1(b)).
The ‘small gap’ in M-NTs has been attributed to the
curvature-induced energy gap Eg <100 meV [15], which
is distinguished from the S-NT energy gap, which scales
with diameter as Eg ∼ 1/d (nm) [1]. Among the 11
SWNTs we studied in this letter, we found 4 M-NTs
and 7 S-NTs. Each of these SWNTs exhibit a gap cen-
tered at Vg > 0, indicating their p-doped nature. At
large negative gate voltage (Vg < −20 V), EF lies well
outside of the gap region and ISD(Vg) saturates to I
whose value depends only on the applied VSD and chan-
nel length L of the SWNT section. The two-terminal
resistance of the SWNT section is then obtained from
R(L) = VSD/I
SD. We note that four-terminal resistance
measurements are possible for each section by utilizing
the available multiple electrode configuration. However,
in our experiment, the four terminal measurements yield
essentially similar results to the two terminalR(L), which
prevents separation of the ‘contact’ resistance contribu-
tion from R(L). Such inseparable contact resistance be-
tween SWNT-metal electrodes was reported to be caused
by the invasiveness of metal contacts [16].
We designed many pairs of electrodes with different L
on each SWNT so that the scaling of R(L) can be studied
for a specific sample at a given temperature T . Fig. 2(a)
show R(L) of a representative SWNT measured in the
temperature range of 1.6 - 300 K and with an L range of
200 nm - 50 µm. In these ranges, R(L) increases linearly
and appears to converge to a finite value for small L (in-
set to Fig. 2(a)). We found that this scaling behavior can
be described well by a simple linear dependence with an
offset: R(L) = ρL +Rc, where ρ and Rc are interpreted
as the 1D resistivity and contact resistance, respectively.
The solid lines in Fig. 2(a) are the two parameter line
fits of the data points at a given T value. From these
fits, Rc(T ) and ρ(T ) are obtained as shown in Fig. 2(b)
and Fig. 2(c), respectively. For this sample, Rc remains
fairly constant at ∼8 kΩ and ρ(T ) exhibits typical metal-
lic behavior, i.e. it decreases with T and saturates to a
value ρsat at low temperatures. Similar scaling behavior
of R(L) is observed in other SWNTs, from which both Rc
and ρ(T ) are extracted within the linear scaling regime.
Table I summarizes d, Rc, and ρsat for the 4 M-NTs
200 K
300 K
110 K
1.65 K
L ( m)
T (K) T (K)
0 100 200 300
100 200 300
(b) (c)
0 20 40
L ( m)
FIG. 2: (a) (Inset) R(L) for sample M1 at select temperatures
ranging from 1.6 - 300 K. (Main) A log-log plot highlights the
behaviors at different lengths scaling 3 orders of magnitude.
From the linear fits (solid lines) of these data points, we ob-
tain the 1D resistivity (b) and the contact resistance (c) at
different temperatures. The dashed line in (c) represents RQ.
and 7 S-NTs considered in this study. To understand the
scaling of R(L) in Fig. 2, we begin with the two-terminal
Landauer-Buttiker formula applied to SWNTs [9]. If we
consider 4 low-energy channels in the SWNT, 2 each for
spin and band degeneracy, then the scaling of resistance
is given by R(L) = (h/4e2)(L/Lm + 1) + Rnc, where e
and h are electron charge and Plank constant and Lm
and Rnc are the electron mean free path and the non-
transparent contact resistance, respectively. Note that
we separate out the contribution of Rnc from the total
contact resistance Rc, so that the contact resistance be-
comes the quantum resistance RQ = h/4e
2 when the con-
tacts become fully transparent. From the experimentally
obtained ρ(T ) and Rc, we can deduce Lm = RQ/ρ(T )
and Rnc = Rc − RQ for each of our SWNT samples. In
particular, we note that Rnc <∼ RQ for the majority of
our samples, suggesting that the barrier at the contacts
is very thin and adds only a negligible contribution when
L becomes substantially large.
We now discuss the temperature dependent behavior
of the mean free path. Fig. 3 is the central result of
this letter, showing Lm(T ) of the SWNTs listed in Ta-
ble I. Overall, Lm(T ) exhibits different behaviors in
two regimes separated by Tcr: (i) the high tempera-
ture regime (T > Tcr) where Lm ∼ T
−1 (dashed line
in Fig. 3), which indicates that inelastic scattering be-
TABLE I: Device characteristics for SWNTs used in this study. The character M (SC) is designated for metallic (semiconduct-
ing) SWNTs.
M1 M2 M3 M4 SC1 SC2 SC3 SC4 SC5 SC6 SC7
d(nm) 2.0 ± .2 1.3 ± .4 1.7 ± .6 1.6 ± .4 1.6 ± .4 1.8 ± .6 1.9 ± .4 2.1 ± .2 2.2 ± .2 2.0 ± .6 2.2 ± .2
Rc(kΩ) 7.9 ± .8 11.5 ± 2.9 8.3 ± 2.5 12.0 ± 4.4 10.2 ± 4.5 14.9 ± 5.7 10.4 ± .9 7.0 ± 2.3 25.4 ± 4.2 6.9 ± 40 21.8 ± 14
ρsat(kΩ/µm) 0.76 ± .02 0.87 ± .02 0.93 ± .01 6.5 ± .08 2.95 ± .05 3.61 ± .05 4.64 ± .01 5.91 ± .12 8.13 ± .31 14.1 ± .19 16.3 ± .13
(µm) 8.56 ± .23 7.65 ± .17 7.07 ± .08 1.00 ± .01 2.24 ± .04 1.83 ± .03 1.40 ± .01 1.10 ± .02 0.80 ± .03 0.47 ± .01 0.40 ± .01
1001 10
T (K)
FIG. 3: (color online) The electron mean free path for the
samples listed in Table I at different temperatures. Most
metallic SWNTs (open circles) saturate at higher values than
that of semiconductors (closed circles). The dashed line rep-
resents T−1 dependence. The insets show scanning gate
microscopy images taken on devices SC2 (upper) and SC7
(lower). Lighter color corresponds less current in the SWNT.
The defects in the SWNT are highlighted by the bright region
(suppressed current) on the SWNT. The scale bar is 500nm.
tween electrons and acoustic phonons is dominant [9, 17]
regardless of chirality [18]; and (ii) the low temperature
regime (T < Tcr) where Lm saturates to the the tube
specific Lsatm . In this low temperature limit, the phonons
freeze out and Lsatm is determined by the temperature in-
dependent elastic scattering with impurities. We believe
the widely spread Lsatm values (0.4-10 µm) in (ii) are a
result of each SWNT sample having a static disorder of
different strengths and densities. We employ scanning
gate microscopy (SGM) [19] to image this static disor-
der. Indeed, the SGM images on S-NTs (insets to Fig. 3)
reveal that the SWNT with a shorter Lsatm shows more
defects. Note also that we have experimentally confirmed
that Lm is generally much higher for M-NTs than that
of S-NTs. This is an indication that the scattering of
electrons is strongly suppressed in M-NTs, as predicted
by Ando et al. [20] and McEuen et al. [21]. In M-NTs
we have experimentally shown that the ballistic electron
0 100 200
1.65 K
110 K
L (µm)
T (K)
0 100 200
L (µm)
101 102
0 100 200 300 400
1.65 K
110 K
L (µm)
0.5 M
T (K)
10050
L (µm)
100 10210110-1
FIG. 4: R(L) in the non-linear regimes for samples (a) SC6
(Lsatm ≈ 460 nm) and (b)M3 (L
m ≈ 7 µm). Note that the
data is magnified in (a) for clarity. The dashed line is an ex-
tension of the linear regime and the solid line is a fit for all
data. Rdev shows the absolute value of the difference between
the actual device resistance and the corresponding linear resis-
tance at 110 K (lower inset a) and 1.65 K (lower inset b).The
non-linearity increases with decreasing temperature, which is
reflected in the value of Lc(upper insets).
conduction is possible for channel lengths up to 8 µm at
low temperature and 0.8 µm even at room temperature.
Finally, we turn our attention to the non-linear scaling
of R(L). Fig. 4 presents R(L) beyond the linear scal-
ing regime of a representative S-NT and M-NT. At ex-
tremely long length scales and low temperatures, R(L)
deviates from the linear dependence extended from the
linear regime (dashed lines in main figure and see also
Rdev = R(L) − Rc − RQL/Lm in lower insets). Since
R(L << Lsatm ) ∼ RQ for all temperatures, we empha-
size here that this non-linear behavior in R(L) is solely
due to increasing electron scattering in the bulk part of
the SWNTs rather than an increasing barrier between
the SWNT and electrodes. In order to experimentally
determine the critical length scale Lc beyond which the
non-linear behaviors is dominant, we use a phenomeno-
logical equation: R(L) = Rc + RQ(L/Lm + e
L/Lc) to
fit the data (solid curves in Fig. 4). While Lc shows a
strong sample dependent behavior, generally we found
Lc >> Lm in all temperature ranges, with the tempera-
ture dependence exhibiting a trend of increasing Lc with
increasing T (upper insets to Fig. 4). This observed be-
havior of Lc(T ) excludes the quantum interference re-
lated to strong localization effects such as Anderson Lo-
calization [7] from the possible scenarios. In particular, in
the high temperature regime (T > Tcr), the phase coher-
ence length Lφ is limited by the phase-breaking electron-
phonon scattering, and thus Lφ ∼ Lm << Lc, inviting
further study to elucidate the observed localization be-
havior beyond the strong localization limit [22, 23].
In conclusion, we determine the length dependent re-
sistance for SWNTs with channel lengths ranged 200 nm
- 400 µm. From the scaling behavior we evaluate the
electron mean free path and localization length of the
SWNT for a range of temperatures. While the low tem-
perature mean free path is determined by the impurity
scattering, an unusually long mean free path is demon-
strated at room temperature, even with the dominant
electron-phonon scattering.
We thank I. Aleiner, B. Altshuler, and P. Jarillo-
Herrero for helpful discussions. This work is supported
by the NSF NIRT(ECS 0507111), CAREER (DMR-
0349232), NSEC (CHE-0117752), and the New York
State Office of Science, Technology, and Academic Re-
search (NYSTAR).
[1] R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physical
Properties of Carbon Nanotubes (Imperial College Press,
London 1998).
[2] J. Kong, E. Yenilmez, T.W. Tombler, W. Kim, H. Dai,
R.B. Laughlin, L. Liu, C.S. Jayanthi, and S.Y. Wu, Phys.
Rev. Lett. 87, 106801 (2001).
[3] W. Liang, M. Bockrath, D. Bozovic, J.H. Hafner, M.
Tinkham, and H. Park, Nature 411, 665 (2001).
[4] D. Mann, A. Javey, J. Kong, Q. Wang, and H. Dai, Nano
Lett. 3, 1541 (2003).
[5] A. Javey, J. Guo, Q. Wang, M. Lundstrom and H. Dai,
Nature, 424, 654 (2003).
[6] A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M.
Lundstrom, and H. Dai, Phys. Rev. Lett. 92, 106804
(2004).
[7] C. Gomez-Navarro, P.J. de Pablo, J. Gomez-Herrero, B.
Biel, F.J. Garcia-Vidal, A. Rubio and F. Flores, Nat.
Mater. 4, 534 (2005).
[8] B. Gao, D.C. Glattli, B. Placais and A. Bachtold, Phys.
Rev. B 74, 085410 (2006).
[9] J. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, H. Us-
tunel, S. Braig, T.A. Arias, P.W. Brouwer and P.L.
McEuen, Nano Lett. 4, 517 (2004).
[10] B. Gao, Y.F. Chen, M.S. Fuhrer, D.C. Glattli, and A.
Bachtold, Phys. Rev. Lett. 95, 196802 (2005).
[11] B.H. Hong, J.Y. Lee, T. Beetz, Y. Zhu, P. Kim, and K.S.
Kim, J. Am. Chem. Soc. 127, 15336 (2005).
[12] Z. Yao, C.L. Kane, and C. Dekker, Phys. Rev. Lett. 84,
2941 (2000).
[13] S.J. Tans, A.R.M. Verschueren, and C. Dekker, Nature
393, 49 (1998).
[14] J. Appenzeller, J. Knoch, V. Derycke, R. Martel, S. Wind
and Ph. Avouris, Phys. Rev. Lett. 89, 126801 (2002).
[15] C. Zhou, J. Kong, and H. Dai, Phys. Rev. Lett. 84, 5604
(2000).
[16] A. Bezryadin, A. R. M. Verschueren, S. J. Tans, and C.
Dekker, Phys. Rev. Lett. 80, 4036 (1998).
[17] V. Perebeinos, J. Tersoff, and Ph. Avouris, Phys. Rev.
Lett. 94, 086802 (2005).
[18] X. Zhou, J. Park, S. Huang, J. Liu, and P. L. McEuen,
Phys. Rev. Lett. 95, 146805 (2005).
[19] A. Bachtold, M.S. Fuhrer, S. Plyasunov, M. Forero, E.H.
Anderson, A. Zettl, and P.L. McEuen, Phys. Rev. Lett.
84, 6082 (2000).
[20] T. Ando and T. Nakanishi, Jpn. J. Appl. Phys. 67, 1704
(1998).
[21] P.L. McEuen, M. Bockrath, D.H. Cobden, Y. G. Yoon,
and S.G. Louie, Phys. Rev. Lett. 83, 5098 (1999).
[22] F. Triozon, S. Roche, A. Rubio, and D. Mayou, Phys.
Rev. B 69, 121410(R) (2004).
[23] R. Avriller, S. Latil, F. Triozon, X. Balse, and S. Roche,
Phys. Rev. B 74, 121406(R) (2006).
