Model analysis of temperature dependence of abnormal resistivity of a multiwalled carbon nanotube interconnection
Nanotechnology, Science and Applications
Model analysis of temperature dependence of abnormal resistivity of a multiwalled carbon nanotube interconnection
Yi-chen Yeh 2
Lun-Wei chang 1
hsin-Yuan Miao 0
Szu-Po chen 2
Jhu-Tzang Lue 2
0 Department of electrical engineering, Tunghai University , Taichung , Taiwan
1 institute of electronics engineering, National Tsing hua University , hsinchu , Taiwan
2 Department of Physics
A homemade microwave plasma-enhanced chemical vapor deposition method was used to grow a multiwalled carbon nanotube between two nickel catalyst electrodes. To investigate the transport properties and electron scattering mechanism of this interconnection (of approximately fixed length and fixed diameter), we carried out a model analysis of temperature dependence of resistivity. To explain the abnormal behavior of the negative temperature coefficient of resistivity in our experimental results, we then employed theories, such as hopping conductivity theory and variable range hopping conductivity theory, to describe resistivity in the high- and low-temperature ranges, respectively. Further, the grain boundary scattering model is also provided to fit the entire measured curve of temperature dependence of resistivity.
Extensive research and literature are devoted to understanding
and finding solutions for improving CNT structural defects
and problems. Naeemi and Meindl used equivalent circuit
models to simulate various electron-phonon scattering
mechanisms as a function of temperature and to model the
8 temperature coefficient of resistance (TCR) for different
l--J021u teynpceesd, bdyefceecrttsa,inanddefleecntgsth(ssuocfh
CasNSTtso.n11eR–Wesaisletisvidteyfeisctisnaflnudn31 the Aharonove–Bohm effect), as well as catalysts
remain9o1 ing in the CNTs that cause extra scattering and interface
.19 roughness.12,13 Structure defects of the material, such as
..325 interface roughness and small grain size, also cause variations
312 of the electron transport characteristics.8 Self-heating also
/yb plays a significant role in short-length nanotubes
transport.com ing current under high bias.14 Svizhenko et al reported that
rsse “large diameter nanotubes are preferable because even the
vpe semi-conducting shells can carry current due to small band
od gaps”, and “the conductance decrease with increase in bias
./www l.yno was caused by reflection of incident electrons at crossing
sub/:sp sue bands due to scattering with zone boundary phonons”.15
tth lan In our work, using the concept of phenomenology, we
from rsoe attempt to fit a temperature dependence of resistivity model to
dedoa ropF the experimental high-to-low temperature range (300–25 K),
ln with results from an individual MWCNT (of approximately
dow fixed length and diameter) shown in Figure 1. This
particuitson lar MWCNT was fabricated using a homemade microwave
lica plasma-enhanced chemical vapor deposition (MPECVD)
ppA system.4,16 In order to explain the result of a negative TCR, we
dna took into account the theories of hopping conductivity,
varicne able range hopping conductivity, strong localization (kf l0 ∼ 1,
iceS in the product kf l0, where kf is the quasi-Fermi wave vector, l0
,yog is a finite mean free path, and reflects the degree of the disorder
lhon and localization) and grain boundary scattering.17
If the number of impurities or defects in the crystalline
structure increases, then electron wave function spans only a few
lattices and fades exponentially.18 Therefore, the conductivity
is varied due to electron tunneling between localized states,
and a negative TCR arises from phonon-assisted hopping
conduction. Excitation of electrons to Ec which contributes
to conductivity by hopping can be expressed as strongly
σ = σ min exp − Ec − EF
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Where Ec is the mobility edge and EF is the Fermi energy.
This form of conduction is normally predominant at relatively
high temperatures, or when (Ec−EF) is small (for small value
of Ec−EF, the transport path and the typical hopping distance
are defined, respectively).19
Variable range hopping
Thermally activated, variable range hopping conduction by
electrons in the states near the Fermi energy at low
temperatures18 is described as:
σ ≈ 2e2 R2ν ph N ( EF ) exp − B
Where R is the hopping distance, νph denotes the phonon
energy, B is the coefficient related to the density of state,
and d indicates the dimension. This condition occurs at
sufficiently low temperatures.
grain boundary scattering
Mayadas and Shatzkes were the f irst researchers who
explored the theory of boundary scattering of electrons in
thin, polycrystalline, metal films.20 However, Hoffmann
et al empirically proposed a superior method to take into
account and explain the boundary scattering of electrons.21
The conductivity of thin, polycrystalline films strongly
deviates from the conductivity of its corresponding bulk single
crystalline materials. Hoffmann et al stated that the
reduction of conductivity depends exponentially on the number of
grain boundaries per mean free path (MFP), and all electrons
reflected by the grain boundaries along one mean free path do
not contribute to the resulting current. The DC conductivity
of polycrystalline films can be assumed as:
σ g = ne2l∞ Tg RG ,
lg = l∞ ⋅Tg RG ,
Where l∞/RG is the number of grain boundaries per mean
free path and Tg is the average probability for an electron to
tunnel a single grain boundary, such that the effective mean
free path can be expressed as:
where l∞−1 = lo−1 + lin−1 is due to Matthiessen’s law.22 Here, lo
is the elastic mean free path resulting from acoustic phonon
scattering, while the temperature dependence of inelastic
mean free path lin is expressed as lin−1(T ) ∝ T P, which is
estimated approximately by Bloch-Grüneisen’s law and
then can be assumed as lo/lin(T ) = bTP. If the grain size is rG,
the crystalline proportion inside the film can be denoted as
(rG/RG)3. Therefore, the conductivity can be revised as
σ g =
ne2lo ⋅ 1 r l
mvF 1+ bT P ⋅ ( G )3Tg RG (1+obT P ) .
Furthermore, we can finally obtain the resistivity:
ρ = nmev2Flo (1+ bT P )( RG )3Tg − RG (1+lobT P ) .
Thus, depending on the choice of transmission coefficientT
and the grain size rG, using the temperature dependence of
grain boundary scattering theory can produce a positive or
In this study, for the purpose of modeling an analysis of
the temperature dependence of resistivity, a MWCNT
was fabricated as an interconnector by the MPECVD
method.4,16 Catalyst nickel electrodes were deposited onto
a SiO2 layer which was grown on a silicon wafer by the
traditional thermal oxidation system to prevent current
leakage from the MWCNT device. The gap between the
electrodes was approximately 5–10 µm and benefited from
the wet etching process. The MPECVD procedure was
initially heated and stabilized at 650°C and 10−5 torr; the
CNTs grew slowly with increasing microwave power. A
mixture gas of methane and hydrogen was then introduced
into the deposition chamber through mass flow
controllers. The microstructures of the electrodes and MWCNTs
were examined by scanning electron microscopy, as
shown in Figure 1a. Figure 1b is a cross-sectional image
of another sample of the same batch. The morphology
strongly depended on the growth conditions, especially
temperature and input microwave power. It is obvious
that the MWCNT grew directly between the two electrode
pads with a nearly fixed length of 3 µm and a diameter of
100 nm (even with some bends). Figure 1b clearly shows
that the layered structure has 2.3 nm/per layer
characteristics. The temperature dependence of resistivity in this
individual MWCNT was then measured by the standard
four-point probe method.
Results and discussion
According to the Mooij correlation in disordered metals,
there will be a negative TCR when the resistivity of alloys or
semiconductors is larger than 10–150 µΩ-cm.23 This implies
that a poorly conductive metal always behaves as a negative
TCR material. That is why we presumed there would be a
negative TCR phenomenon in this study.
The original TCR measurement data from the individual
MWCNT with multiple defects formed on the body during
the growth process is shown in Figure 2a. Resistivity would
normally be linear with respect to temperature,24 but for
quantities that vary polynomially or logarithmically with
temperature, it may be possible to calculate a temperature
coefficient that is a useful approximation for a certain range
of temperatures. In order to visualize this behavior for TCR,
Figure 2b shows the logarithmic plot ver. T−1 from Figure 2a.
The dashed lines represent the fitting curves proportional to
T-1 under different slopes at both high temperatures (left side)
and low temperatures (right side).
Clearly, there is a rapid, increasing trend with the decrease
in temperature, especially below 50 K. This represents a
negative TCR (dρ/dT , 0) phenomenon that cannot be
explained by semiclassical theory. Figure 2b seems to imply
that there would need to be two models to handle the variable
trends of TCR with a transient point.
Other researchers have had similar results. Sheng
reported that, for high-resistivity granular disordered
systems, low-field conductivity exp(−A/Tα) with α = 1/2 is
obeyed over large temperature ranges, with possible
crossovers to α = 1/4 at low temperatures and to α . 1/2 at high
temperatures.25 The temperatures at which the crossovers
occur depends on the distribution of grain size. As shown
in Figure 3, Naeemi reported that, for single-wall and
few-wall CNTs, the TCR reaches 1/(T-200K) for lengths
much larger than the electron MFP.11 For MWCNTs with a
large diameter (.20 nm), TCR varies from −1/T to +0.66/
(T-200K) as the length varies from zero to very large values.
There exists a “singular point” around 200 K and a
separate fitting model in two parts. The results were derived by
simulations and assumed the body of the object would be
perfect. However, what is different in a real MWCNT body
which has multiple defects?
In our work, we employed hopping conductivity and
variable range hopping conductivity theories to fit the
experimental results for the two different slopes of Figure 2b.
In the hopping conductivity mechanism, the variation
of resistivity at relatively high temperatures is in proportion
to the Boltzman factor, exp(−ε/kT). Hopping conductivity
theory, σ = σminexp(−ε/kT), was used, as shown in Figure 4a,
to display the fitting result curve. The resistivity σmin derived
from the fitting curve is 38 µΩ-cm, and the corresponding
activation energy ε is 4.46 meV, which is quite close to the
energy gap of SWCNT.26 Therefore, as the temperature
increases, the thermal energy will excite an electron from
the valance band to the conduction band, which leads to
However, the TCR at low temperature ranges cannot be
explained by the exponential law above. Instead, when using
strong localization theory (kf l0 ∼ 1), variable range hopping
may prevail at low temperatures, as shown in Figure 4b. The
exact variable range hopping theory that describes the
lowtemperature behavior of resistivity in strongly disordered
systems, where states are localized, can be expressed as:
σ = 2e2 Rav N (ε F )v ph exp(− T 1/ d+1 )
where Rav is the average hopping distance, N(εF) is the
density of state, vph is the coefficient according to the phonon
spectrum, B is the coefficient related to the density of state,
and d indicates the dimension.
100 150 200
Combining these two models, we can cover the
temperature range from high to low (300–25 K), and note that
there exists a singular point around 150 K (which is 50 K
lower than in the work plot of Naeemi in Figure 3).11 This
well-matched result is shown in Figure 4c, and explained
Model: Hopping conductivity
Fitting curve 100 150 200
Model: Variable range hopping
Fitting curve 100 150 200
Model: Hopping conductivity &
Variable range hopping
by taking into account the combined equations of hopping
conductivity and variable range hopping conductivity
theory. The fitting results are shown in Table 1, and are very
consistent with Mott’s predictions for the conductivity of
The meaning of the singular point here was presumed
to be a transient point that governs the conditional change
from inelastic to elastic collisions, ie, a measure of the
temperature above which more modes of phonons in
MWCNTs will be excited. To recognize and describe this
unique property of MWCNTs, we named this singular point
the “Lue point” temperature.
For comparison purposes, the grain boundary scattering
model was also employed to fit for resistivity versus
temperature in our individual MWCNT. The best fitting result
is shown as a solid line in Figure 5, and makes the physical
constraints of equation (
) with p = 1 and Tg , 1.
In general, the value of p depends on the dominant
inelastic scattering mechanism, and is presumed to be
fixed in the 0.5–4.0 temperature range.27 The parameters
obtained by fitting the R-T curves with the least square
Model: Grain boundary
Experimental curve Fitting curve 60 ) 55
errors method are listed in Table 2. The parameter term,
lo/RG, indicates that the electrons are scattered by lo/RG
barriers in one mean free path (lo) where RG is the
interval between grains. According to the scanning electron
microscopy image shown in Figure 1, the diameter of the
MWCNT is smaller than approximately 100 nm, with
about 2.3 nm/per layer, therefore the elastic mean free path
obtained by the fitting result of grain boundary scattering
model is around 3–4 nm.
Parameter b plays a crucial role and affects the
dency of the R-T curve, due to the mean free path’s
/:sp sue l (T ) = lo/(1 + bTp) variance with temperature. According to
h an Matthiessen’s rule, 1/l(T ) = 1/lo+1/lin, where lin is the mean
from rsoe free path due to the inelastic scattering which comes from
dedoa ropF the contribution of the optical phonon, we deduce that the
interaction between the phonon and electron at low
temperatures is small, ie, the inelastic mean free path varying with
temperature causes the negative TCR characteristic in our
study. In addition, the electron transmission probability, Tg,
for electrons to pass a single grain boundary is in the order
of 10−1, as derived from the fitting result.
In this study, we have produced an individual MWCNT
across electrodes by a method of MPECVD and measured
the temperature dependence resistivity of this MWCNT to
investigate its electronic properties.
For comparison, the effect of quantum conf inement
(strong localization) and grain boundary theories were
quoted to explain the anomalous resistivity (a negative TCR
characteristic). With regard to strong localization, the fitting
result can be obtained by combining hopping conductivity
theory in the high-temperature range and variable range
hopping conductivity theory at the low-temperature range.
The most appropriate fitting result is accomplished by using
grain boundary theory. The singular “Lue point”
temperature can be successfully described as a unique property of
MWCNTs. The nanosize of the fabricated MWCNTs ensures
the validity of the grain boundary scattering model and the
temperature-related inelastic mean free path dominates the
R-T curve, causing the negative TCR characteristic.
This work was supported by the National Science Council of
the Republic of China under the contracts
NSC-986-2112M007 and NSC 97-2221-E-029-007-.
The authors report no conflict of interest in this work.
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