Thermal Conductivity of a Supported Multi-Walled Carbon Nanotube

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C: Physical Processes in Nanomaterials and Nanostructures

Thermal Conductivity of a Supported Multi-Walled Carbon Nanotube Fabian Könemann, Morten Vollmann, Tino Wagner, Norizzawati Mohd Ghazali, Tomohiro Yamaguchi, Andreas Stemmer, Koji Ishibashi, and Bernd Gotsmann J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00692 • Publication Date (Web): 19 Apr 2019 Downloaded from http://pubs.acs.org on April 20, 2019

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The Journal of Physical Chemistry

Thermal Conductivity of a Supported Multi-Walled Carbon Nanotube Fabian Könemann,∗,† Morten Vollmann,† Tino Wagner,‡ Norizzawati Mohd Ghazali,¶,§ Tomohiro Yamaguchi,¶ Andreas Stemmer,‡ Koji Ishibashi,¶,k and Bernd Gotsmann† †IBM Research - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland ‡Nanotechnology Group, ETH Zürich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland ¶Advanced Device Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan §Malaysia-Japan International Institute of Technology, Universiti Teknologi Malaysia, Jalan Sultan Yahya Petra, Kuala Lumpur 54100, Malaysia kRIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan E-mail: [email protected] Abstract We have extracted temperature-dependent thermal conductivity values from scanning thermal microscopy measurements of a self-heated multi-walled carbon nanotube supported on a silicon substrate. A deliberately introduced segment of amorphous carbon served as integrated nanoheater. Kelvin probe force microscopy was used to supplement the thermometry data with values for the nanotube’s electrical resistivity. This way, both the spatially resolved temperature rise, as well as the Joule heating power density were available for further analysis. A one-dimensional heat diffusion model was fitted to the data to extract values for the thermal conductivity along the nanotube axis and the

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thermal conductance between nanotube and supporting substrate. We found thermal conductivity values that continuously increase from 200 W m−1 K−1 to 400 W m−1 K−1 in a temperature range of 100 K to 400 K above room temperature. The values obtained are about one order of magnitude lower compared to values reported for the freely suspended case. We attribute this observation to increased phonon scattering and quenching of acoustic phonon modes due to the substrate interaction.

Introduction Carbon nanotubes (CNTs) have been recognized for their outstanding transport properties, such as large electrical and thermal conductivity and large ampacity 1,2 . This makes CNTs a candidate material for interconnects in highly integrated electronics 3 . Copper interconnects are currently reaching their limit of miniaturization. Further scaling increases resistivity 4 and failure rates by electromigration 5 . Electromigration is a thermally activated process and therefore self-heating properties are of importance. CNTs could help reduce heating of interconnects, which in modern chips consume up to 50% of the total power 3 and allow for higher current densities than conventional copper interconnects. The thermal conductance of individual CNTs, however, is not easily experimentally accessible. The thermal conductance of freely suspended (unsupported) CNTs is so large that thermal contact resistance tends to dominate measurements in typical experimental geometries 6 . Reported values of thermal conductivity of multi-walled carbon nanotubes (MWCNTs) 6–8 range from 650 W m−1 K−1 to 3000 W m−1 K−1 at room temperature. The high values are related to both the high propagation speeds of phonons along CNTs and the low intrinsic phonon scattering rate 9 . While these large thermal conductance values are certainly encouraging with respect to technological application, it is less clear how much the coupling to a carrying substrate or a surrounding dielectric matrix affects the thermal transport properties of CNTs. It was shown that bundles and sheets of MWCNTs have up to two orders of magnitude reduced 2


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thermal conductivity due to their mutual interaction 8,10 . A similar effect can therefore be expected for CNTs interacting with surrounding materials 11 . It is worth noting that the thermal conductivity of graphene is also strongly reduced if supported by a substrate. The existing methods to measure heat transport along individual CNTs are typically not suitable for supported MWCNTs in which the contribution of the substrate is difficult to separate. Likewise, the thermal coupling between CNTs and a carrier substrate is expected to be important 12 , but experimentally not readily accessible. And, as mentioned above, it is not trivial to experimentally determine the thermal resistance of electrical contacts, typically made from metal, to the CNTs 6 . Here, we address these challenges by measuring the self-heating of an individual supported MWCNT using scanning thermal microscopy (SThM) 13 . Through fitting of the SThM data with suitable models, we extract the thermal conductivity and the thermal coupling strength between MWCNT and an underlying silicon oxide substrate. A similar approach has successfully been used to extract the thermal conductivity of silicon and indium arsenide nanowires 14,15 . To minimize the assumptions needed for the analysis, the combination with Kelvin probe force microscopy 15 is utilized to determine the local Joule dissipation along the MWCNT.

Methods Sample fabrication The MWCNTs are grown by the arc discharge method 20 . They have a length of around 4 µm and a diameter of 10 nm to 50 nm. They are dispersed on a silicon substrate with a native oxide layer. The distance between the contacts is targeted to be around 1 µm. The contacts are patterned by electron beam lithography followed by depositing 40 nm of Palladium in a lift-off process. A damaged section of amorphous carbon has deliberately been introduced with a focused ion beam. Details on the process can be found elsewhere 21 . 3



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Scanning probe thermometry For measuring Joule heating temperature maps, we employ the technique described by Menges et al. 22 . A schematic representation of the technique is shown in Figure 1. This SThM-based technique relies on a micro-cantilever with integrated resistive sensor coupled to a silicon tip. The sensor is used in active-mode, meaning that the applied bias is high enough (Vtot = 2.5 V) to induce significant heating. The result is a temperature rise of ∆Tsensor (out of contact) = 275 K when the tip is out of contact. The measurement is performed in high vacuum (10−6 mbar) at room temperature in a low-noise environment 23 . For taking an image, the tip is brought into contact and raster scanned across the sample. The scan is operated in contact mode of scanning force microscopy. The contact force is monitored and controlled with a laser deflection system. The temperature of the sample is modulated by applying an AC current with a frequency of fmod = 1432 Hz to the MWCNT device. The temperature rise due to Joule heating is then inferred from a simultaneous measurement of the time-averaged sensor signal ∆VDC and the demodulated sensor signal amplitude at 2fmod , denoted as ∆VAC 16,24 :

∆Tsample = ∆Tsensor (out of contact) ×

∆VAC . ∆VDC − ∆VAC

(1)

The influence of the thermal contact resistance between tip and sample has been eliminated from the equations in the derivation of Equation (1). This influence is known to otherwise lead to artifacts and imprecise measurements 22 . The elimination of influences that fluctuations in tip-sample contact resistance have on the measurements is the main motivation for using active mode SThM in combination with a modulated sample temperature. Another advantage is the reduced bandwidth of the lock-in measurement, which leads to an improved signal-to-noise ratio.

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Vtot

50 dB T sample 2fmod ref.

Rs

fmod

I

Lock-in

VAC

VDC

Figure 1: Schematic representation of the SThM thermometry technique applied here. An AC bias with frequency fmod is applied across the MWCNT device and a series resistance Rs . The resulting AC current with amplitude I leads to periodic Joule self-heating with frequency 2fmod . A local spot on the sample surface with temperature Tsample is thermally coupled through the tip to a resistive sensor integrated in the silicon MEMS cantilever. Changes in sensor temperature Tsensor lead to changes of its electrical resistance, which are tracked using a Wheatstone bridge circuit. The DC change in voltage VDC between the legs of the Wheatstone bridge and its AC amplitude VAC at 2fmod then relate the local sample temperature Tsample to the known and constant out-of-contact sensor temperature through Equation (1).

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Kelvin probe force microscopy We obtain the potential profile along the MWCNT by frequency-modulated Kelvin probe force microscopy 25,26 . To measure the sample topography, we perform amplitude-modulated atomic force microscopy (Cypher, Asylum Research) in the net-attractive regime (drive frequency fd ≈ f0 ≈ 300 kHz, free amplitude ≈ 12 nm, amplitude setpoint 90%). Simultaneously to the topography scan, the electrostatic force is modulated by an AC voltage at fm = 2 kHz and 2 V applied to the cantilever (AC160TS-R3, Olympus), and the resulting sidebands at fd ±fm and fd ±2fm due to the electrostatic force gradient are detected by a lock-in amplifier (HF2, Zurich Instruments). The surface potential is then found by adjusting the DC bias to nullify the sidebands at fd ± fm . A custom feedback loop based on a Kalman filter is used to incorporate the information contained in the sidebands fd ± 2fm , thereby minimizing artifacts and topography crosstalk 27 .

Results and discussion Figure 2 shows the results of an SThM scan with an applied modulation current amplitude of I = 349 µA. A CNT diameter value of (37 ± 1) nm can be extracted from the simultaneously obtained topography map (Figure 2a). The Palladium contacts are clearly visible on the left and right sides and have a thickness of 40 nm. Close to the left contact is a carbon particle originating from the CNT growth. It appears to be in direct contact with the nanotube. The location of the damaged section is not apparent from the topography map. The Joule heating map (Figure 2b) however shows a hot spot centered between the contacts, which coincides with the targeted damage position. The hot spot is therefore accredited to be due to increased dissipation at the damaged section. The temperature in this hot spot is about 50 K higher compared to the adjacent defect-free sections of the nanotube. Elevated temperatures are also observed in the regions where the MWCNT is buried by the contacts. This observation has previously been reported for similar MWCNT samples 6


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without a defect 16 . This observation suggests either that charge carriers dissipate all along the CNT and the electrical current density decays on a micrometer length scale, or that the thermal boundary resistance between CNT and metal is large enough to confine heat to within the CNT. (a)

(b)

(c)

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Figure 2: SThM Results: (a) Topography map and (b) simultaneously recorded temperature rise due to Joule heating for a modulation current amplitude of I = 349 µA. Scale bar: 300 nm. (c) Temperature profiles along the nanotube axis (line width of eight pixels at a pixel size of 5 nm) for different values of I. Each line is the average of four individual measurements. Fits of the hyperbolic solution of Equation (2) are shown as thicker lines. Inset: Scaling of median temperature rise with I 2 . The linear trend of temperature against current squared is a signature of the Joule effect. The median temperature rise is defined as the value at x = 1 µm as indicated with circles in the main plot. For further studies, SThM scans at seven different modulation current amplitudes ranging from 187 µA to 524 µA have been performed. For each amplitude, the measurement has been repeated four times. A line profile with an averaging width of eight pixels (pixel size 5 nm) along the tube has been taken on each of the resulting 28 temperature maps. Line profiles for 7



the same modulation amplitude were averaged. The statistical spread in each pixel was found to be on the order of a few percent and below 10% in any case. This is seen as a confirmation of the experiment’s reproducibility. The results are shown in Figure 2c. The influence of the carbon particle is visible as a dip in temperature at approximately 0.5 µm on the x-axis. On the right side of the defect, between 0.8 µm and 1.2 µm, the temperature profile is smooth and suitable for further analysis. We define a median temperature associated with each curve by taking the value measured at 1 µm. This is indicated with purple circles in Figure 2c. Plotting these temperatures against the modulation current amplitude squared yields the linear trend depicted in the inset of Figure 2c. Since dissipated Joule heating power is proportional to the current squared, this result confirms that the observed temperature rise is due to the Joule effect. In general, the temperature distribution is governed by the thermal conductivity of the materials and interfaces involved, the geometry of the device, and the distribution of the heat generation. Consequently, the measured temperature rise alone is not sufficient to determine the nanotube thermal conductivity, and also the local Joule dissipation power is needed. Therefore, we turn to the electrical characterization of the device. For obtaining the MWCNT’s electrical characteristics that can be used to calculate the dissipation power density, KFM measurements were performed on the same sample. During the scans, a DC bias is applied across the MWCNT. The resulting potential maps for bias voltages of 1, 2, and 3 V are shown in Figure 3a. Potential profiles along the tube are shown in Figure 3b. Linear fits yield an average electrical resistivity of ρ = (5.8 ± 0.1) µΩ m, where the diameter extracted from the topography map has been used in the calculation the MWCNT cross-sectional area. An average contact resistance of (10 ± 3) kΩ was extracted from the potential steps at the contacts. Now that both the temperature rise and the nanotube’s electrical resistivity are known, the experiment can be described with the help of the following one-dimensional heat-diffusion

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0

Figure 3: KFM Results: (a) Potential maps for 1, 2, and 3 V of DC bias applied. About one third of the applied voltage drops across the leads connecting the device. Scale bar: 300 nm. (b) Potential profiles along the nanotube extracted from the three potential maps. Linear fits for the extraction of MWCNT electrical resistivity are indicated. model:

κA ·

∂ 2T − g(T − TA ) = q , ∂x2

(2)

where x denotes the direction along the MWCNT axis, κ is the thermal conductivity along the carbon nanotube in the investigated undamaged section, and g is the thermal conductance per unit length from nanotube to substrate, both of which are assumed to be independent of x. TA is the ambient/substrate temperature and q is the heat generated per unit length due to Joule heating. A denotes the cross-sectional area of the nanotube. We consider the steady state, because the frequency of the modulation (1.4 kHz) is slow compared to the inverse thermal time constant of the nanotube system, which has been measured to be higher than 100 kHz and is expected to lie in the megahertz range. For the analysis, we focus on the areforementioned section between the defect and the right metal electrode, in which the temperature profiles are smooth. Certain properties of the system can be directly extracted from the temperature profiles in this region. Let us assume, in a first case (Figure 4a), that the electrical resistivity of MWCNTs is small such that the contribution of Joule dissipation along this section can be neglected (q ≈ 0). If, in addition, one assumes that the heat transport along the nanotube axis and into the contacts

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(a)

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(b) g

MWCNT Substrate

q

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q

Defect T

x

T

x

x

x

Figure 4: Different solutions of Equation (2) showcased: (a) This solution applies if selfheating in the undamaged sections can be neglected in comparison to power dissipation in the defect region (q ≈ 0) and heat conduction along the MWCNT is so efficient, that conduction into the substrate can be neglected (g ≈ 0). In this case, Equation (2) yields a linear temperature decay from defect region to contact. (b) A nonlinear solution applies if significant self-heating in the undamaged sections (q 6= 0) occurs and/or conduction from nanotube to substrate (g 6= 0) is relevant. is much more efficient than heat transport through the nanotube-substrate contact (g ≈ 0), then a linear temperature decay from the heat source (in this case the defect) to the heat sink (in this case the contact) would be observed. However, the temperature profiles shown in Figure 2 are not linear. They clearly feature some curvature. This indicates that at least one of the the two assumptions, q ≈ 0 and g ≈ 0, does not hold in this sample system. If the model is extended to also account for significant conduction into the substrate (g 6= 0) but still assumes no Joule dissipation in this section (q = 0), then the solution is an exponential decay. This is a non-linear trend, but compared to our experimental observation it has the opposite curvature. Therefore, we can already conclude that Joule dissipation in the undamaged section must be taken into account (q 6= 0). In a first case, one may still assume g ≈ 0, and Equation (2) has the following quadratic solution:

T (x) = −

I 2ρ 2 x + a1 x + a0 , 2κA2

(3)

where ai are constants defined by boundary values, and q = I 2 ρ/A was used, where ρ is the

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electrical resistivity value measured by KFM. If additionally, substrate conduction plays a significant role (g 6= 0), the solution is hyperbolic: cosh(mx) sinh(mx) cosh(mx) q + c1 + c2 T (x) = 1 − cosh(mL/2 g cosh(mL/2) sinh(mL/2) r g with: m = , kA

(4) (5)

where again, the ci are constants defined by boundary values and L is the length of the segment under consideration 17 . This is illustrated in Figure 4b. The extended models can still be fitted to the measured temperature profiles, but now also the local Joule power dissipation in the undamaged section is needed as input for the model. An upper limit for radiative heat losses is given by black body radiation. Assuming a homogeneous temperature rise of 400 K along the CNT and treating it as a black body emitter, we would expect radiative heat losses on the order of 1 nW with a Wien wavelength of about 4 µm. This is eight orders of magnitude smaller than the power that is electrically dissipated in the device at I = 500 µA. Therefore, we can neglect radiation in our analysis. This should be justified even if we take into account potential enhancement of radiative losses by several orders of magnitude above the black body limit due to the fact that device dimensions are smaller than Wien’s wavelength. We fitted both the quadratic solution Equation (3) and the hyperbolic solution Equation (4) to the experimental data. The results for the hyperbolic model are drawn with thicker lines in Figure 2c. The resulting temperature-dependent thermal conductivity values are shown in Figure 5a, where the previously defined median temperature values are again used as reference. The heat conduction per unit length obtained from the hyperbolic fit is shown in Figure 5b. The values obtained for g in the hyperbolic fit are small, which is also apparent from the similar fitting results when using the quadratic and the hyperbolic equations. In the limit g → 0, the two solutions become the same. Nevertheless, we can extract values also for the nanotube-to-substrate conduction per unit length. This is in agreement with the 11



(a)

(b)

Figure 5: Analysis results: (a) Thermal conductivity along MWCNT in undamaged section. (b) Thermal conductance into substrate per unit length. observation of a temperature rise in the substrate around the self-heated nanotube. The thermal conductivity values are about one order of magnitude lower than reported values for free-standing MWCNTs 8 . This is likely due to the nanotube’s interaction with the substrate and can either be explained by quenching of acoustic phonon modes 10 , which are as a result no longer available for heat conduction, or increased phonon scattering rates. Increased phonon scattering rates may also stem from a high defect density in the crystal structure. To test this, we performed transmission electron microscopy on an MWCNT fabricated by the same process (see supporting information). No crystalline defects could be observed. Therefore, we do not expect poor crystallinity to be the origin of the observed deviation from values reported for the free standing case. Figure 5a shows an increase of κ in the investigated temperature range. This is in contrast to temperature-dependent measurements of freely suspended CNTs, in which the thermal conductivity has been found to only increase up to room temperature and then saturate 6 or even decrease again. The reduction of thermal conductivity with increasing temperature typically can be attributed to umklapp and three-phonon scattering processes above room temperature 8,19 . A constant or slightly increasing thermal conductivity for increasing temperature above room temperature, however, has been observed for strongly

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confined phonon conductors like silicon nanowires, in which the boundary scattering does not increase with temperature like umpklapp scattering does.

Conclusion We have extracted thermal conductivity data from scanning probe thermometry measurements of self-heated multi-walled carbon nanotubes fully supported on a silicon substrate using an SThM thermometry technique. The method allows thermal transport measurements in systems with large contact resistance and for cases with substrate support. We report continuously increasing thermal conductivities from 200 W m−1 K−1 to 400 W m−1 K−1 in a temperature range of 100 K to 400 K above room temperature. These values are about one order of magnitude lower compared to theoretical predictions and previously reported measurements for freely suspended MWCNTs.

Acknowledgement The authors thank Ute Drechsler, Steffen Reidt, and Anel Zulji for technical assistance, Andreas Schenk, Guillermo Villanueva, Kirsten Moselund and Walter Riess for continuous support, Seiji Akita of Osaka Prefecture University for providing us with the MWCNTs, and Bissera Ivanova for helping us with editing the manuscript. This work has received funding in the course of the H2020 projects “CONNECT” under the grant agreement № 688612 and “QuIET” under the grant agreement № 767187. Koji Ishibashi (KI) acknowledges a partial support by an MEXT Grant-in-Aid for Scientific Research on Innovative Areas “Science of Hybrid Quantum Systems” (№ 15H05867). Supporting information. Transmission electron micrograph of an MWCNT that was arc-discharge grown in the same process as the MWCNT discussed here. Transmission electron micrograph of an MWCNT grown by chemical vapor deposition for comparison.

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Kelvin probe force microscopy for local characterisation of active nanoelectronic devices. Beilstein Journal of Nanotechnology 2015, 6, 2193–2206.

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1

2 V, 179 µA

3 V, 289 µA


0

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Thermal Conductivity of a Supported Multi-Walled Carbon Nanotube

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