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Low-temperature specific heat of single-wall carbon nanotubes
Physica B 316–317 (2002) 468–471
Low-temperature specific heat of single-wall carbon nanotubes J.C. Lasjauniasa,*, K. Biljakovica,b, Z. Benesc, J.E. Fischerc a
Universit!e Joseph Fourier–CNRS, Cen. Recherches sur les Tres Basses Temp., CRTBT, B.P. 166, 38042 Grenoble Cedex 9, France b Institute of Physics, P.O.B. 304, HR-10001 Zagreb, Croatia c University of Pennsylvania, Philadelphia, PA 19104-6272, USA
Abstract Previous low-temperature heat capacity measurements down to 2 K give evidence for 1-D quantum confinement of SWNT vibrational modes (J. Hone et al., Science 289 (2000) 1730). We extended the data on the same sample down to 0.1 K. A good fit is obtained up to 4 K with Cp ¼ Cn T 2 þ 0:043T 0:62 þ 0:035T 3 mJ=gK; where the T 2 term is from nuclear hyperfine interactions in ferromagnetic impurities and the T 3 term originates from intertube coupling in agreement with Hone et al. The origin of the T 0:62 term is unknown. Comparison with the vibrational density of states from inelastic neutron scattering is made. r 2002 Elsevier Science B.V. All rights reserved. PACS: 63.22.+m; 65.80.+n Keywords: Thermal properties; Nanostructure; Lattice dynamics
1. Introduction
2. Results and discussion
The vibrational density of states (DOS) of single-wall carbon nanotubes (SWNT) is of great interest, and has been probed by heat capacity [1] and inelastic neutron scattering [2]. Theory predicts either 1-D, 3-D, or intermediate effective dimensionalities at very low energy [3], depending on the magnitude of inter-tube coupling. Here we extend the heat capacity measurements down to the milliK range in order to provide new information on the low-energy (Eo2 meV) vibrational DOS.
Sample preparation is described in Ref. [1]. It consists of ropes of SWNT, with an average number of (40710) tubes within a rope. Fortyfive milligrams of such sample was used under special outgassing conditions in order to avoid air or water contamination [1,4]. He4 exchange gas was introduced in the sample chamber at either 300 or 77 K at a pressure of 6–7 102 mbar (at 77 K). Once the sample has reached 4 K, we removed the He by heating to 25–30 K [1] under dynamic secondary vacuum. Data were collected between 0.1 and 7 K by a transient heat-pulse technique [5], always under secondary vacuum of B1.5 106 mbar. Run A was performed after this procedure (open circles in Fig. 1). The role of adsorbed He was tested in Run B (solid squares) by introducing
*Corresponding author. Fax: +33-4-76-87-50-60. E-mail address: [email protected] (J.C. Lasjaunias).
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 4 5 - 8
J.C. Lasjaunias et al. / Physica B 316–317 (2002) 468–471
469
limit of 2 K. On the other hand, we find a strongly superlinear increase from 2 to 4–5 K while the Hone et al. data are nearly linear in this interval. Finally, there appears to be a break in slope at B4.5 K to more nearly linear behavior. Fits for the 3 runs are shown in Fig. 2 as solid lines. The negative exponent term is a nuclear hyperfine contribution originating from residual ferromagnetic catalyst particles, the fitted amplitude corresponding to 0.7–1.0 at% Co in agreement with the sample analysis [1]. After subtracting this extrinsic T 2 term, the remainder can be fit by Cp Cnuc ¼ 1:76T 1:30 mJ=gK for run B and Cp Cnuc ¼ 0:07T þ 0:24T 1:85 mJ=gK for run A, both for the entire 0.1–6 K range. Run C was treated in more detail; the best fit Cp Cnuc ¼ 0:043T 0:62 þ 0:035T 3 over the range 0.1–4.5 K. The process leading to this fit is shown in Fig. 3. Here the T 3 term would be the usual 3-D lattice contribution while the sublinear contribution is
Fig. 1. Measured specific heat (log–log plot) of the SWNT sample, for runs A, B, C described in the text. The role of He4 adsorption is maximal in run B and minimal in run C. Powerlaw straight lines are illustrative only.
the exchange gas at 8 K, then cooling under static vacuum to 4.2 K. The final pressure was the same as before. Finally, the whole system was warmed to 300 K with the sample chamber under dynamic primary vacuum for one month, following which we performed Run C (filled circles) after pumping to 2 105 mbar for 108 h at 300 K and then introducing He at 77 K and following the same procedure as in Run A. The error bar of individual measurements was of the order or less than 10%, what allows a high accuracy of the obtained fits for the combined results (comparison of A, B, C runs). Fig. 1 shows the addenda-corrected data for the 3 runs described above. In each, Cp decreases smoothly from B6 K, goes through a minimum, and then increases to reach a common 0.1 K limit. Our run C data (best outgassing conditions) are in good agreement with Hone et al. at their lower
Fig. 2. Fits to the data of Fig. 1 (solid curves, see text). Also shown are theoretical behaviors [1,2] of isolated SWNT (linear below 4 K), a bundle of strongly coupled SWNT (quadratic below 4 K), and the estimated electronic contribution if all the tubes were metallic.
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J.C. Lasjaunias et al. / Physica B 316–317 (2002) 468–471
Fig. 3. After subtraction of the hyperfine nuclear term, Cp is the sum of the T 0:62 term plus the 3D vibrational T 3 term defined from 0.3 to 4.5 K.
not understood. Note from Fig. 2 that the Run C data above 2 K lie above both the isolated tube and rope theoretical curves, whereas the Ref. [1] data fell slightly below the isolated tube prediction from 2 to 5 K. We have succeeded in our goal of learning more about the 3-D regime, despite the unanticipated hyperfine contribution from metallic impurities. Since the latter can be fit very well (with the same magnitude for all runs) and then subtracted off, we are able to follow the coupled-tube regime of 3-D T 3 behavior over nearly 4 decades in Cp (Fig. 3) from 0.3 to 4.5 K. The crossover to a weaker power law above 4.5 K is quite evident, suggesting a change in effective dimensionality. In Ref. [1] this is built into the fitting model as the temperature at which acoustical intertube modes are exhausted. These modes are represented as a Debye term with YD a direct measure of intertube coupling strength [1].
Fig. 4. Vibrational density of states of a similar sample measured by inelastic neutron scattering [2]. Note the break in slope at 1.6 meV.
Cp ðTÞ may be calculated from the density of vibrational modes DðEÞ where E is a phonon energy. Inelastic neutron scattering measures a generalized vibrational density of states, or GDOS, closely related but not identical to DðEÞ [2]. Fig. 4 shows the GDOS measured at 100 K from a similar sample [2]. We can identify a slope change from quadratic to linear at 1.6 meV which corresponds to a change in Cp ðTÞ at 4.1 K within the dominant phonon approximation, close to the measured feature at 4.5 K. Furthermore, GðEÞ extrapolates to zero at E ¼ 0; rather than to a finite value expected for isolated tubes [1,2].
3. Conclusion We conclude that the two specific heat experiments are in overall agreement. The present data gives a better estimate of the intertube coupling
J.C. Lasjaunias et al. / Physica B 316–317 (2002) 468–471
energy, 1.7 vs. 1.2 meV in Ref. [1], by extending the data to lower T: Such differences in the coupling energy can be accounted for by differences in tubetube interactions within a rope for different samples. Work at Penn supported by USDOE Grant DEFG02-98ER45701.
471
References [1] J. Hone, et al., Science 289 (2000) 1730. [2] S. Rols, et al., Phys. Rev. Lett. 85 (2000) 5222; S. Rols, Thesis, Universite de Montpellier II, 2000. [3] L.X. Benedict, et al., Solid State Commun. 100 (1996) 177. [4] A. Mizel, et al., Phys. Rev. B 60 (1999) 3264. [5] F. Zougmore, et al., J. Phys. (France) 50 (1989) 1241.
Low-temperature specific heat of single-wall carbon nanotubes J.C. Lasjauniasa,*, K. Biljakovica,b, Z. Benesc, J.E. Fischerc a
Universit!e Joseph Fourier–CNRS, Cen. Recherches sur les Tres Basses Temp., CRTBT, B.P. 166, 38042 Grenoble Cedex 9, France b Institute of Physics, P.O.B. 304, HR-10001 Zagreb, Croatia c University of Pennsylvania, Philadelphia, PA 19104-6272, USA
Abstract Previous low-temperature heat capacity measurements down to 2 K give evidence for 1-D quantum confinement of SWNT vibrational modes (J. Hone et al., Science 289 (2000) 1730). We extended the data on the same sample down to 0.1 K. A good fit is obtained up to 4 K with Cp ¼ Cn T 2 þ 0:043T 0:62 þ 0:035T 3 mJ=gK; where the T 2 term is from nuclear hyperfine interactions in ferromagnetic impurities and the T 3 term originates from intertube coupling in agreement with Hone et al. The origin of the T 0:62 term is unknown. Comparison with the vibrational density of states from inelastic neutron scattering is made. r 2002 Elsevier Science B.V. All rights reserved. PACS: 63.22.+m; 65.80.+n Keywords: Thermal properties; Nanostructure; Lattice dynamics
1. Introduction
2. Results and discussion
The vibrational density of states (DOS) of single-wall carbon nanotubes (SWNT) is of great interest, and has been probed by heat capacity [1] and inelastic neutron scattering [2]. Theory predicts either 1-D, 3-D, or intermediate effective dimensionalities at very low energy [3], depending on the magnitude of inter-tube coupling. Here we extend the heat capacity measurements down to the milliK range in order to provide new information on the low-energy (Eo2 meV) vibrational DOS.
Sample preparation is described in Ref. [1]. It consists of ropes of SWNT, with an average number of (40710) tubes within a rope. Fortyfive milligrams of such sample was used under special outgassing conditions in order to avoid air or water contamination [1,4]. He4 exchange gas was introduced in the sample chamber at either 300 or 77 K at a pressure of 6–7 102 mbar (at 77 K). Once the sample has reached 4 K, we removed the He by heating to 25–30 K [1] under dynamic secondary vacuum. Data were collected between 0.1 and 7 K by a transient heat-pulse technique [5], always under secondary vacuum of B1.5 106 mbar. Run A was performed after this procedure (open circles in Fig. 1). The role of adsorbed He was tested in Run B (solid squares) by introducing
*Corresponding author. Fax: +33-4-76-87-50-60. E-mail address: [email protected] (J.C. Lasjaunias).
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 4 5 - 8
J.C. Lasjaunias et al. / Physica B 316–317 (2002) 468–471
469
limit of 2 K. On the other hand, we find a strongly superlinear increase from 2 to 4–5 K while the Hone et al. data are nearly linear in this interval. Finally, there appears to be a break in slope at B4.5 K to more nearly linear behavior. Fits for the 3 runs are shown in Fig. 2 as solid lines. The negative exponent term is a nuclear hyperfine contribution originating from residual ferromagnetic catalyst particles, the fitted amplitude corresponding to 0.7–1.0 at% Co in agreement with the sample analysis [1]. After subtracting this extrinsic T 2 term, the remainder can be fit by Cp Cnuc ¼ 1:76T 1:30 mJ=gK for run B and Cp Cnuc ¼ 0:07T þ 0:24T 1:85 mJ=gK for run A, both for the entire 0.1–6 K range. Run C was treated in more detail; the best fit Cp Cnuc ¼ 0:043T 0:62 þ 0:035T 3 over the range 0.1–4.5 K. The process leading to this fit is shown in Fig. 3. Here the T 3 term would be the usual 3-D lattice contribution while the sublinear contribution is
Fig. 1. Measured specific heat (log–log plot) of the SWNT sample, for runs A, B, C described in the text. The role of He4 adsorption is maximal in run B and minimal in run C. Powerlaw straight lines are illustrative only.
the exchange gas at 8 K, then cooling under static vacuum to 4.2 K. The final pressure was the same as before. Finally, the whole system was warmed to 300 K with the sample chamber under dynamic primary vacuum for one month, following which we performed Run C (filled circles) after pumping to 2 105 mbar for 108 h at 300 K and then introducing He at 77 K and following the same procedure as in Run A. The error bar of individual measurements was of the order or less than 10%, what allows a high accuracy of the obtained fits for the combined results (comparison of A, B, C runs). Fig. 1 shows the addenda-corrected data for the 3 runs described above. In each, Cp decreases smoothly from B6 K, goes through a minimum, and then increases to reach a common 0.1 K limit. Our run C data (best outgassing conditions) are in good agreement with Hone et al. at their lower
Fig. 2. Fits to the data of Fig. 1 (solid curves, see text). Also shown are theoretical behaviors [1,2] of isolated SWNT (linear below 4 K), a bundle of strongly coupled SWNT (quadratic below 4 K), and the estimated electronic contribution if all the tubes were metallic.
470
J.C. Lasjaunias et al. / Physica B 316–317 (2002) 468–471
Fig. 3. After subtraction of the hyperfine nuclear term, Cp is the sum of the T 0:62 term plus the 3D vibrational T 3 term defined from 0.3 to 4.5 K.
not understood. Note from Fig. 2 that the Run C data above 2 K lie above both the isolated tube and rope theoretical curves, whereas the Ref. [1] data fell slightly below the isolated tube prediction from 2 to 5 K. We have succeeded in our goal of learning more about the 3-D regime, despite the unanticipated hyperfine contribution from metallic impurities. Since the latter can be fit very well (with the same magnitude for all runs) and then subtracted off, we are able to follow the coupled-tube regime of 3-D T 3 behavior over nearly 4 decades in Cp (Fig. 3) from 0.3 to 4.5 K. The crossover to a weaker power law above 4.5 K is quite evident, suggesting a change in effective dimensionality. In Ref. [1] this is built into the fitting model as the temperature at which acoustical intertube modes are exhausted. These modes are represented as a Debye term with YD a direct measure of intertube coupling strength [1].
Fig. 4. Vibrational density of states of a similar sample measured by inelastic neutron scattering [2]. Note the break in slope at 1.6 meV.
Cp ðTÞ may be calculated from the density of vibrational modes DðEÞ where E is a phonon energy. Inelastic neutron scattering measures a generalized vibrational density of states, or GDOS, closely related but not identical to DðEÞ [2]. Fig. 4 shows the GDOS measured at 100 K from a similar sample [2]. We can identify a slope change from quadratic to linear at 1.6 meV which corresponds to a change in Cp ðTÞ at 4.1 K within the dominant phonon approximation, close to the measured feature at 4.5 K. Furthermore, GðEÞ extrapolates to zero at E ¼ 0; rather than to a finite value expected for isolated tubes [1,2].
3. Conclusion We conclude that the two specific heat experiments are in overall agreement. The present data gives a better estimate of the intertube coupling
J.C. Lasjaunias et al. / Physica B 316–317 (2002) 468–471
energy, 1.7 vs. 1.2 meV in Ref. [1], by extending the data to lower T: Such differences in the coupling energy can be accounted for by differences in tubetube interactions within a rope for different samples. Work at Penn supported by USDOE Grant DEFG02-98ER45701.
471
References [1] J. Hone, et al., Science 289 (2000) 1730. [2] S. Rols, et al., Phys. Rev. Lett. 85 (2000) 5222; S. Rols, Thesis, Universite de Montpellier II, 2000. [3] L.X. Benedict, et al., Solid State Commun. 100 (1996) 177. [4] A. Mizel, et al., Phys. Rev. B 60 (1999) 3264. [5] F. Zougmore, et al., J. Phys. (France) 50 (1989) 1241.