Specific heat of aligned multiwalled carbon nanotubes

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INSTITUTE OF PHYSICS PUBLISHING NANOTECHNOLOGY

Nanotechnology 16 (2005) 1490–1494 doi:10.1088/0957-4484/16/9/013

Specific heat of aligned multiwalledcarbon nanotubesCharan Masarapu1, L L Henry2 and Bingqing Wei1,3,4

1 Department of Electrical and Computer Engineering, Louisiana State University,Baton Rouge, LA 70803, USA2 Department of Physics, Southern University, Baton Rouge, LA 70813, USA3 Center for Computation and Technology, Louisiana State University,Baton Rouge, LA 70803, USA

E-mail: weib@ece.lsu.edu

Received 19 April 2005, in final form 18 May 2005Published 29 June 2005Online at stacks.iop.org/Nano/16/1490

AbstractThe specific heat of an aligned bulk multiwalled carbon nanotube samplemade by the chemical vapour deposition method was measured from 250 to1.8 K. The specific heat curve gradually decreased, showingone-dimensional (1D) behaviour down to 40 K. Below 40 K it showed arapid decrease due to the dimensional change from 1D to 3D behaviour,indicated by different temperature dependences at low temperature.Interestingly, a T −2 term was observed below 5 K, suggesting a nuclearhyperfine component due to magnetic impurities which were confirmed tobe present by thermogravimetric analysis and microscopy observations.

1. Introduction

Large scale syntheses of carbon nanotubes (CNTs) [1, 2]have permitted experimental determination of many oftheir physical properties and led to the possibility ofcomparing experimental results with theoretical calculations.Knowledge of the electronic and phonon density of statesof the CNTs at low temperatures is very important fortheir applications in sensors and in quantum computingapplications. A considerable number of theoretical [3–5]and experimental [6, 7] investigations have been done inefforts to explain the phonon structure of single-walled CNTs(SWNTs) at low temperature. However, it is very difficult tonumerically calculate the low energy phonon density of statesof a multiwalled CNT (MWNT) due to its complex structure.Low temperature specific heat measurements can give insightinto the electronic and phonon excitations in CNTs.

Few experiments have been conducted to measure thespecific heat of bulk MWNT samples [8, 9]. Lu et al [8]have indirectly measured the specific heat of millimetre longaligned MWNT bundles from 10 to 300 K by measuringtheir thermal conductivity. They obtained a linear behaviourover the whole temperature range of their experiment andattributed that to the constant phonon density of states excitablein the temperature range. They concluded that the interlayer

4 Author to whom any correspondence should be addressed.

coupling in the sample was very weak. Mizel et al [9] measuredthe specific heat of MWNTs and SWNT ropes. They observedthat the behaviours of the low temperature specific heat forbulk MWNT and graphite were similar. They concluded thatthis is due to the similar structures of the large radius MWNTand graphite. Recently an attempt was made to theoreticallycalculate the low temperature specific heat of a 1–5-layeredMWNT [10] based on force constant dynamical models. Itwas proposed that the low temperature specific heat of a singlelayer was proportional to T 1/2. As the number of layerswas gradually increased from 1 to 5, the T 1/2 dependencechanged to a linear T dependence due to the domination ofthe longitudinal acoustic and twist phonon modes over thetransverse acoustic phonon modes. But the calculated curvesdiffered significantly from the experimental data [8, 9].

We present a detailed analysis of the specific heat of analigned bulk MWNT sample in the temperature range of 250–1.8 K. The specific heat curve shows almost linear dependence(1D behaviour) of the temperature from 250 to 40 K, belowwhich it decreases more rapidly, from linear to a higher ordertemperature dependence, signalling a dimensional crossoverfrom 1D to 3D behaviour as predicted by theory [11]. Since theelectronic contribution to the specific heat at low temperaturesis negligible in MWNTs [9], an ideal curve fit should beT 3 dependent below a temperature of 5 K due to the stronginterlayer coupling caused by the long wavelength phonon

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Figure 1. (a) A typical SEM image of the aligned nanotube sampleused for the specific heat measurements. (b) A TEM image oftrapped Fe nanoparticles. (c) Thermogravimetric analysis indicatesa low percentage of iron impurity in the experimental sample.

modes perpendicular to the tube axis. Interestingly, analysisof experimental results from 1.8 to 5 K gave an additional termof the specific heat that is proportional to the T −2 term, alongwith the T 3 term. We attribute this to the nuclear hyperfineinteractions of ferromagnetic impurities present in the sample.The impurities were inadvertently incorporated in the sampleduring the synthesis process and confirmed by various studies.

2. Experimental details

The MWNTs were synthesized from a vapour phase mixture ofxylene and ferrocene in a chemical vapour deposition (CVD)tube furnace system [12–14]. Xylene is the carbon source andferrocene is the nanotube nucleation initiator. High resolutionscanning electron microscopy showed aligned MWNTs inbundles, several millimetres in length (figure 1(a)). Theaverage centre-to-centre spacing between adjacent nanotubesis about 50 nm [15]. Analysis of the CNTs using highresolution transmission electron microscopy (TEM) showedthat the diameters of the CNTs were in the range of 20–30 nmwith each tube consisting of 15–25 layers. Catalyst impurityparticles trapped inside the CNT cavity, as shown by the TEMimage in figure 1(b), have been frequently observed and easilyidentified to be face-centred cubic iron, which shows abnormalferromagnetic behaviour [16]. Thermogravimetric analysismeasurements showed that about 1 at.% nanoparticle iron istrapped inside the aligned CNT bundles (figure 1(c)).

The CNT sample used was in the shape of a cylinder,3 mm in diameter and approximately 3 mm long, with amass of 2.2 mg. The measurements of the heat capacity atconstant pressure (Cp) were done using a relaxation techniquein a Quantum Design physical property measurement system(PPMS) with the heat capacity option installed. This pieceof equipment is especially designed to measure the specificheat of materials. To minimize systematic error, the systemwas first calibrated by measuring the specific heat of a pure

copper sample (99.999%) in the temperature range 1.8 K to300 K and compared with the widely accepted data [17]. Thecopper sample was then removed and the specific heat of thecarbon nanotube sample was measured. A parameter called thecoupling strength determines how well the nanotube sampleis attached to the substrate of the sample puck that has thesystem heater and thermometer. If the coupling strength is100% then the temperature of the sample is almost equal to thesystem temperature. At a particular measurement temperatureif the coupling is less than 100% then the system softwareuses a special algorithm to compensate for the error. Eachmeasurement was taken only when the temperature of thesample was completely stabilized. One round of temperaturesweeping from room temperature to 1.8 K took around 10–15 h depending on the number of times the specific heat wasmeasured at a particular set point. In our experiment threemeasurements were taken at each set point temperature andwe believe that the data are reliable with no appreciable errors.

The heat capacity determination can be expressed asCsample = Ctotal − Caddenda. Csample is the heat capacity ofthe sample, Ctotal and Caddenda are the total heat capacity ofthe system during a measurement and the heat capacity due tonon-sample heating (i.e., heating of the sample puck withoutthe sample but with the exact amount of grease—Apiezon N—that will be used to secure the sample to the substrate that hasthe sample heater and thermometer), respectively. Caddenda ismeasured over the entire temperature range to be used for theCsample determination. Following the Caddenda measurementthe sample is placed on the sample substrate and the heatcapacity measurements are performed. The software in thesystem subtracts Caddenda from Ctotal to obtain the Csample at eachtemperature. It then uses the mass to calculate the specific heatof the sample. The pressure during the sample measurementis maintained at the order of 10−5 Torr. The software alsocalculates the Debye temperature for the sample based on thenumber of atoms (1 for carbon) per formula unit that is inputto the system.

3. Results and discussion

A log–log plot of the measured specific heat versus temperatureof the MWNT sample, along with that calculated for two-dimensional graphene and bulk graphite, is shown in figure 2.It can be observed that the specific heats for the three materialsare approximately equal above 80 K and deviate below 80 K.The deviation occurs due to the different low temperatureacoustic phonon densities of states in the materials. Ingraphite the dimensional crossover from 2D to 3D behaviouroccurs in the temperature range of 50–80 K [18, 19]. Theconsequence of this dimensional change is the rapid decreasein the specific heat of graphite below 50 K compared to thatabove 50 K (figure 2). The low temperature specific heatof the 2D graphene sheet is dominated by the out-of-planetransverse acoustic phonon modes and has linear temperaturedependence. For the MWNT sample the specific heat curve ismostly linear from 250 to 40 K. As the temperature decreasesbelow 40 K the long wavelength acoustic phonon modesperpendicular to the tube axis begin to dominate, in additionto the acoustic longitudinal and twist phonon modes, andthe interlayer coupling in the nanotubes becomes significant.

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MWNT sample

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Figure 2. Log–log plot of the specific heat versus temperature foran aligned bulk multiwalled carbon nanotube sample, 2D grapheneand graphite. The inset shows the linear plot of the specific heat ofthe MWNT sample. The arrow indicates the dimensional change.

Due to this interlayer coupling the dimensional crossoverin MWNTs is occurring at approximately 35 K, as can beseen from figure 2. A linear plot of the specific heat versustemperature for the MWNT sample is also shown in the insetof figure 2. The change in the slope of the curve at about 35 Kconfirms the dimensional crossover. The dimensional changein the MWNT sample occurs at a lower temperature than thatof graphite (in between 50 and 80 K [18, 19]) indicating that theinterlayer coupling in MWNTs is much weaker as comparedto that in graphite. This might be due to the larger interlayerdistance in MWNTs than in bulk graphite [20]. Since the tubesin the MWNT bundles are weakly coupled, the transition from1D to 3D behaviour [7] is occurring as we move from highertemperature to lower temperature.

To explain the specific heat in detail at very lowtemperatures, it is important to have knowledge of the phonondensity of states in the nanotubes. The detailed phonon densityof states (PDOS) of an individual SWNT has already beendiscussed in the literature [21]. At low temperature only thePDOS contributed by acoustic phonon modes is importantand for SWNTs there are four acoustic phonon modes [7, 9].When the nanotubes get into bundles, the torsional modes ortwist modes on neighbouring tubes couple and become optical,making it effectively a 3D system with three acoustic phononmodes [7]. However, no information was available on thePDOS of MWNTs, plausibly due to its complex structure.But because the diameters of the MWNTs are large with 25–30 nm and 15–25 multiwalls, they effectively are a 3D systemsimilar to that of graphite. So we presume that they haveonly three acoustic phonon modes (two transverse acousticand one longitudinal acoustic) and the effect of the curvaturecontributes to the optical phonon modes.

The specific heat results on the MWNTs obtained by Luet al [8] demonstrate a linear behaviour from 300 down to10 K without any dimensional crossover and closely agreewith the calculated values for graphene sheet below 100 K. Theout-of-plane acoustic phonon modes are strong contributors tothe high specific heat of graphene sheet at low temperatures.However, our data are lower than the calculated specific heat

of graphene (figure 2) and roughly match with those of Mizelet al [9], strongly suggesting that the behaviour of the bulkMWNT sample is very similar to that of graphite.

Figure 3 shows a plot of the specific heat versustemperature below 20 K, showing the regions with specificcurve fits, using short straight lines (for the guidance of thereader). The T 2.8 dependence of the specific heat in thetemperature range of 5–10 K and the T 2.2 dependence in therange 10–20 K are in good agreement with the theoreticalpredictions by Benedict et al [11] that the dimensionality of thephonon density of states of the MWNTs at low temperaturescan be either 1D, 3D or intermediate, depending on the radiusof the tubes and the intertube coupling. Moreover, thisbehaviour is consistent with the theoretical discussion [22, 23]of the low temperature specific heat of graphite, indicatingagain that the specific heat of MWNTs is similar to that ofgraphite.

When the temperature is below 5 K the intertube couplingwill increase and the specific heat should show a strong T 3

dependence due to the dominant acoustic phonon contribution.However, a perfect curve fit is obtained for the data below 5 Kwith C = 0.003T 3 + 0.07T −2 as shown in figure 3(b). This fitis reminiscent of the notion that a general expression for theheat capacity may be comprised of phonon, electron, nuclearetc contributions, with their distinct temperature dependencesin different temperature ranges. The temperature dependencespermit analysis of the data to identify the contributors: forexample, the phonon contribution has a T 3 dependence, theelectronic one has a T dependence and the nuclear term has aT −2 dependence.

For MWNTs in the temperature range of our study, if theyare free of any impurities then there should be a dominantT 3 dependence of the specific heat since the electroniccontribution is negligible [24] and the specific heat shouldapproach zero at the lowest temperatures following a T 3

dependence. For our sample, as the temperature is loweredthe data follow a T 3 dependence; however, it can be seen thatbelow 3 K the data deviate from a pure T 3 behaviour (seefigure 3(b)) and more closely follow a fit that includes a T −2

term. It is well known that the heat capacity associated withthe nuclear hyperfine magnetic field for magnetic atoms hassuch a T −2 temperature dependence [6, 25]. The error barsin figure 3(b) show the system error in measuring the specificheat of the sample. Even if the data are assumed to be shiftedin this error region, the fit is closer to C = 0.003T 3 +0.07T −2,instead of the only T 3 fit, showing that this deviation does notcome from the system error but does indeed come from theimpurities in the sample. This leads to the conclusion thatthere are magnetic impurities in the sample, thus causing theT −2 term indicative of nuclear hyperfine interactions. Thisconclusion is supported by the finding mentioned earlier thatferromagnetic iron nanoparticle impurities were inadvertentlyincorporated in the sample (figure 1) during the CVD synthesisprocess. In this case, a return to a rise of the experimental curveat even lower temperatures is expected due to the dominantrole of the nuclear interaction (the constant for the T −2 part ismuch greater than that for the T 3 part) in this temperatureregion, but it is beyond the temperature limitation of ourmeasurement equipment. To our knowledge, this T −2 termin the specific heat measurements of MWNTs has not been

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Figure 3. (a) Specific heat versus temperature plot for a bulkmultiwalled carbon nanotube sample with curve fittings from 20 to1.8 K. The solid straight lines are for the guidance of the reader,showing the T dependence of the specific heat. (b) Specific heatversus temperature plot for the same sample below 5 K, along withthe error bars showing the curve fitting. The fitting is improved whenthe T −2 term is added. This indicates the presence of impurities inthe sample, consistent with TEM and TGA measurements.

(This figure is in colour only in the electronic version)

reported previously. The non-observation may be due to thefact that the measurements by Lu et al [8] were conductedonly down to 10 K, and, although Mizel et al [9] measuredthe specific heat down to 1 K, the MWNT samples in theirexperiment were fabricated using an arc-discharge method.

We can also observe the temperature dependence ofthe specific heat components by plotting the specific heatdivided by the different powers of the temperature versus thetemperature T . Since the specific heat contribution of thehyperfine term, because it has 1/T 2 dependence, becomessignificant only at the lowest temperatures, and the electronicspecific heat contribution, which has a linear temperaturedependence, is negligible in these materials, then plotting C/Tversus T , C/T 2 versus T and C/T 3 versus T , respectively,will provide explicit demonstrations of the features of thecontributors. This is done in figures 4(a)–(c). In figure 4(a),the dimensional crossover can be easily observed, and it canbe seen from figures 4(b) and (c) that the heat capacity has anupturn and increases quite rapidly below 3 K showing the effect

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Figure 4. Plots of the specific heat divided by various powers of Tversus T , showing more clearly the dimensional change: (a) C/T asa function of T , and the effect of the impurities in the sample,(b) C/T 2 as a function of T and (c) C/T 3 as a function of T .

of the T −2 term. For an ideal Debye solid, the C/T 3 versusT plot results in a horizontal line at a value of C/T 3 equal tothe coefficient of the T 3 term up to about one tenth of the lowtemperature limit of the Debye temperature [26]. Our resultplotted in figure 4(c) demonstrates that the electronic specificheat of the MWNT sample is negligible when compared tothe phonon specific heat at low temperature due to the quitesmall deviation of the curve from the value of C/T 3 = 0.003.This indicates that MWNTs behave similarly to an ideal Debyesolid.

4. Conclusions

We conclude that the specific heat of the aligned bulkMWNT sample is similar to that of graphite. However, the

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dimensional change in MWNTs from 1D to 3D occurs at alower temperature (∼35 K) than that of graphite (∼50 K from2D to 3D) due to the weak interlayer coupling in MWNTs.The specific heat data deviate from a pure T 3 behaviour atlow temperatures and the fit requires an additional T −2 termbelow 5 K. This additional term is due to the nuclear hyperfineinteractions of the ferromagnetic impurities in the sample anddoes not show up for a pure MWNT. The contribution of theelectrons to the specific heat is negligible, even at very lowtemperatures.

Acknowledgments

The authors are grateful for the financial support fromLouisiana Board of Regents under award numbers LEQSF(2005-08)-RD-A-13 and LEQSF(2005-08)-RD-B-05, andLSU Council on Research.

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