Model analysis of temperature dependence of abnormal resistivity of a multiwalled carbon nanotube interconnection

Nanotechnology, Science and Applications, Jul 2010

Model analysis of temperature dependence of abnormal resistivity of a multiwalled carbon nanotube interconnection Yi-Chen Yeh1, Lun-Wei Chang2, Hsin-Yuan Miao3, Szu-Po Chen1, Jhu-Tzang Lue11Department of Physics and 2Institute of Electronics Engineering, National Tsing Hua University, Hsinchu, Taiwan; 3Department of Electrical Engineering, Tunghai University, Taichung, TaiwanAbstract: 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.Keywords: multiwalled carbon nanotube, resistivity, hopping conductivity, temperature dependence

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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 c e t o n a N Theory hopping conductivity 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 localized states: σ = σ min exp  − Ec − EF    kT  submit your manuscript | www.dovepress.com Dovepress A B 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   T1/ d+1  ( 2 ) ( 1 ) 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: l∞ σ g = ne2l∞ Tg RG , mvF l∞ 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: 8 1 0 2 l u J 3 1 n o 9 1 1 . 9 5 . 2 3 . 3 1 2 y b / m o c . s s e r p e v o d ./www l.yno 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 ) . RG Furthermore, we can finally obtain the resistivity: ρ = nmev2Flo (1+ bT P )( RG )3Tg − RG (1+lobT P ) . rG 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 negative TCR. Experimental design In this study, for the purpose of modeling an analysis of the temperature dependence of resistivity, a MWCNT ( 3 ) ( 4 ) ( 5 ) ( 6 ) 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 A 60 −1/T 0.66(T-200) 1/(T-200) 1.2 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 high conductivity. 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: B 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. 8 1 0 2 l u J 3 1 n o 9 1 1 . 9 5 . 2 3 . 3 1 2 y b / m o c . s s e r p e v o d ./www l.yno /:sp sue tt l h a n from rsoe dedoa ropF l n w o d s n o it a c li p p A d n a e c n e i c S , y g o l o n h c e t o n a N B 60 C 60 ) 55 m c (µΩ50 y t i itv45 s i s e R40 35 0 ) 55 m c (µΩ50 y t i itv45 s i s e R40 35 0 ) 55 m c (µΩ50 y t i itv45 s i s e R40 35 0 50 50 50 100 150 200 Temperature (K) 250 300 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 Experimental curve Fitting curve 100 150 200 Temperature (K) 250 300 Model: Variable range hopping Experimental curve Fitting curve 100 150 200 Temperature (K) 250 300 Model: Hopping conductivity & Variable range hopping Experimental curve Fitting curve Fitting curve 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 amorphous materials.18 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 ( 6 ) 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 ) 55 250 300 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 ten 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 tt l 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. Conclusion 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 42 temperature-related inelastic mean free path dominates the R-T curve, causing the negative TCR characteristic. Acknowledgments 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-. Disclosure The authors report no conflict of interest in this work. 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Yi-Chen Yeh, Lun-Wei Chang, Hsin-Yuan Miao, Szu-Po Chen, Jhu-Tzang Lue. Model analysis of temperature dependence of abnormal resistivity of a multiwalled carbon nanotube interconnection, Nanotechnology, Science and Applications, 2010, 37-43, DOI: 10.2147/NSA.S11696