SWNT diameter [nm

Fig. 13. Optical absorption peaks of purified single-wall nanotubes as a function of the synthesis temperature, displayed both on an energy and a diameter axis. The background is subtracted. The dotted, lines indicate the groups of nanotube diameters separated by A d = 0.07 nm (from [46])

Photon energy [eV]

Fig. 13. Optical absorption peaks of purified single-wall nanotubes as a function of the synthesis temperature, displayed both on an energy and a diameter axis. The background is subtracted. The dotted, lines indicate the groups of nanotube diameters separated by A d = 0.07 nm (from [46])

limit for all synthesis conditions studied. This points to the fact that the investigated material consists of nanotubes with a discrete number of diameters grouped around preferred values independent of variations of the process parameters. Furthermore, it was found that the positions of the sub peaks are equidistantly separated on the diameter scale with nearly the same values of A d for both semiconducting and metallic SWNTs. Such an equivalent spacing between nanotube diameters (A d « 0.07nm), common to both semiconducting and metallic SWNTs, can be realized for nanotubes close to the (n, n) armchair geometry. Provided there are no selection rules to favor special helicities, this observation indicates a preferred formation of SWNTs in the vicinity to the armchair configuration. This result certainly needs further investigations, since recent electron-diffraction experiments have revealed a uniform distribution of helicity in the ropes [35], except perhaps on rare occasions where a preferred armchair arrangement was detected [50]. If confirmed, the analysis discussed here clearly shows that the energy of the excited states, in particular the interband excitations between van Hove singularities, first detected by EELS measurements [37] yields valuable information not only on the mean diameter but also on the helicity of SWNT material.

In the following we focus again on the dispersion data of the plasmons of SWNTs which are summarized in Fig. 14, together with similar data on graphite. The dispersions of the n- and the n + a-plasmons of both materials look similar. Graphite is a three-dimensional solid, since the Coulomb interplane interaction is comparable to the intraplanar interaction. As for SWNTs, the individual tubes form bundles having diameters of about 10 nm. Up to 100 SWNTs are located on an hexagonal lattice in a section perpendicular to the bundle, and the distance between the tubes is comparable to the inter-sheet distance in graphite. Such a rope can be described as an effective bulk medium in the spirit of, for instance, the Maxwell-Garnett theory [51]. With such an effective dielectric function at hand, (1) predicts a plasmon dispersion typical of a three-dimensional material. This explains why the quasi two-dimensional graphite system and the quasi one-dimensional SWNT systems show a dispersion like in three-dimensional solids.

On an isolated nanotube, by contrast, theoretical calculations [52,53] predict a plasmon frequency which vanishes with decreasing wave vector, but only for this mode that has full rotational symmetry, whereas in three-dimensions, hw(q) approaches a finite value at zero q. The n- and a-plasmons in an isolated MWNT are also predicted to behave the same way, however, with a crossover from 1D (low q corresponding to a wavelength larger than the nanotube diameter) to 3D (large q) [54]. Strictly speaking, this acoustic behavior applies to free-electron like materials, which the undoped nanotubes are not. In the tight-binding picture, the n-plasmon of a SWNT has a finite energy at q = 0, of the order of 2y0 as mentioned above. This optical-like character of the nanotube plasmon is further reinforced in a bundle [55]. Here, the plasmon frequency depends on both the parallel and perpendicular

Fig. 14. Dispersion of the n-plasmon, the n + a-plasmon (•), and of the excitations due to interband transitions between van Hove singularities (o) of purified single-wall carbon nanotubes. For comparison, the dispersion of the n- and a- plasmons in graphite for momentum transfers parallel to the planes is included (o) (from [37])

Fig. 14. Dispersion of the n-plasmon, the n + a-plasmon (•), and of the excitations due to interband transitions between van Hove singularities (o) of purified single-wall carbon nanotubes. For comparison, the dispersion of the n- and a- plasmons in graphite for momentum transfers parallel to the planes is included (o) (from [37])

components of the transferred wave vector, and it decreases with decreasing \J<i]_ + </jj, as observed experimentally.

At the end of this section we discuss studies of the plasmons of MWNTs with an inner diameter of 2.5-3 nm and an outer diameter of 10-12 nm, thus consisting of 10-14 layers. In Fig. 15 we show the n-plasmon for various momentum transfers [56,57]. In the inset, the loss function for q = 0.1 A-1 is shown in a larger energy range. For q = 0.1 A-1, the n-plasmon is near 6eV while the n + a-plasmon is near 22 eV. No low-energy excitations are observed due to the larger tube diameter and the corresponding smaller energy of the van Hove singularities. For q > 0.1 A_1, the plasmon energies are intermediate between those of graphite and SWNTs. For small momentum transfers, the n + a-plasmon of the MWNTs shows a decrease in energy, as in the SWNTs. Contrary to the EELS behavior in SWNTs, a strongly decreasing n-plasmon energy is observed in this momentum range, which could be explained by a transition from three-dimensional behavior to one-dimensional behavior with decreasing momentum transfer, as already discussed before.

Recently, EELS spectra of an isolated MWNT have been obtained in a scanning transmission electron microscope, in a well-defined geometry [58,59].

Fig. 15. Plot of the normalized EELS intensity vs energy showing the dispersion of the ^-plasmon (indicated by the arrow) of annealed multi-wall carbon nanotubes for various values of momentum transfers. The inset shows the loss function over a wider energy range (from [56] and [57])

Here an electron beam of nanometer diameter is used in near-perpendicular orientation with respect to the nanotube axis. The EELS spectrum is recorded as a function of the impact parameter and the evolution of the loss peaks is traced against this parameter. If, in the experiments, the n-plasmon remained near 6 eV, insensitive to the impact parameter, the n+<r-plasmon shifted from 27 eV (graphite like) for a near-zero impact parameter to about 23 eV when the beam came closer to a nanotube edge. The interpretation of the shift is that a beam passing through the center of the nanotube probes the inplane component of the dielectric function of graphite, whereas the off-center beam probes the out-of-plane component [44]. The real parts of the corresponding dielectric components of graphite do not vanish at the same energy, and therefore there is a shift of the plasmon position with increasing impact parameter. This is an effect of the graphite dielectric anisotropy mentioned above.

6 Intercalated Single-Wall Nanotubes

Right after the discovery of SWNTs it was clear that one should try to intercalate these materials, in analogy with the well-known examples of Graphite Intercalation Compounds (GICs) and of Ceo. Consequently, a decrease of the resistivity by one order of magnitude has been detected when SWNTs were exposed to potassium (or bromine) vapors [60]. The intercalation of the bundles by K can be followed by electron diffraction [18]. Upon successive intercalation, the first Bragg peak, the so-called rope-lattice peak, characteristic of the nanotube triangular lattice, shifts to lower momentum. This is consistent with an expansion of the inter-nanotube spacings concomitant with intercalation in between the SWNTs in the bundle.

The maximal intercalation can be derived from the intensity of the C 1s and K 2p excitations recorded by EELS [18] (see Fig. 6). A comparison with data of KC6 GICs also shown in Fig. 6 yields the highest concentration of C/K « 7 for SWNTs intercalated to saturation, which is essentially a similar value as for stage I GIC KC8. Like in K-intercalated graphite compounds, the filling of unoccupied n*-states by K 4s electrons cannot clearly be detected in the C 1s spectra (see Fig. 6) [61], since excitonic effects yield deviations of the measured spectral weight compared to the density of states (see Sect. 2).

To discuss the changes of the conduction-band structure upon intercalation, we show in Fig. 16 the loss function of K-intercalated SWNTs for two different concentrations together with that of GIC KC8 [61]. For the latter compound, besides the bound n-plasmon, a charge carrier plasmon due to the filled n* bands is observed at about 2.5 eV. In the case of the interca-

transfer of 0.15 A-1 for (a) GIC KCs; intercalated SWNTs with C/K ratios (b) 7 ± 1 and (c) 16 ± 2. The solid lines represent a fit of a Drude-Lorentz model (from [18])

Fig. 16. Loss function at a momentum

lated SWNTs, the low-energy features cannot be satisfactorily described by a charge carrier plasmon alone. The introduction of an additional interband excitation located at 1.2 eV is necessary. Taking into account the previous discussion on undoped SWNTs, this excitation corresponds to the second transition between the van Hove singularities of the semiconducting SWNTs. In this view, the first transition at 0.6 eV has disappeared by the filling of the lowest unoccupied van Hove singularity due to K 4s electrons transfered to the nanotubes, which lead to a shift of the Fermi level. A filling of these states and a related disappearance of the corresponding interband transitions upon intercalation has been also observed in optical spectroscopy [34,62,63]. Within a Drude-Lorentz model [(4) generalized to include several oscillators], the low-energy features can be explained by a charge carrier plasmon in addition to the remaining interband transitions between the van Hove singularities, and the n-plasmon. The unscreened energy of the charge carrier plasmon in KC7 SWNTs is 1.85eV and 1.2eV in KCi6 [18]. This plasmon is responsible for the peaks observed in the loss function near 0.8 and 0.6 eV, respectively. From the data shown in Fig. 16, an effective mass of the charge carriers has been derived that is about 3.5 times that of the GIC KC8. This effect could be considered to follow naturally from the back-folding of the graphene band structure, which occurs on the wrapping of a graphene sheet to form a SWNT. In addition, the damping of the charge carrier plasmon in the fully intercalated bundles of SWNTs is about twice as large as in fully intercalated graphite. This can be ascribed to a considerable disorder in the nanotube system. From the scattering rates an intrinsic dc conductivity of about 1200 S/cm for the fully intercalated SWNTs could be derived. By comparison, a value of3000 S/cm has been reported from transport measurements in samples doped with K under similar conditions [60].

7 Summary

At present, there exist only a limited number of electron spectroscopy studies on carbon nanotubes. The main reason for that is the non existence of good crystalline materials. Nevertheless, electron spectroscopies have considerably contributed to the understanding of the electronic structure of these new carbon-based n-electron systems. EELS experiments performed on ropes of SWNTs clearly revealed interband transitions between van Hove singularities of the density of states. The intensity of these loss features unambiguously showed that the samples mixed metallic and semiconducting nanotubes in approximate statistical ratio 1:2. The n- and n + a-plasmon structure of the nanotubes was shown to be similar to that of graphite with, however, a few differences among which the indication of quasi-one dimensional dispersion, especially in multi-wall tubes. EELS made it possible to measure the concentration of potassium in intercalated bundles. Charge transfer in potassium-doped nanotubes was also clearly demonstrated from low-energy

EELS spectra. In addition, these methods provided information on the diameter distribution and, indirectly, on the helicity of the nanotubes. Much more detailed studies can be performed in the future when well ordered, oriented and fully dense materials will become available, which even may have a unique helicity.


We are grateful to R. Friedlein, M.S. Golden, M. Knupfer and T. Pichler (Dresden), A. Lucas, P. Rudolf and L. Henrard (Namur) for fruitful discussions.


1. S. Hüfner, Photoelectron Spectroscopy, (Springer, Berlin, Heidelberg 1995) 248, 249, 251, 253

2. B. K. Agarwal, X-Ray Spectroscopy, (Springer, Berlin, Heidelberg 1979) 249

3. J. Lüning, J.-E. Rübensson, C. Ellmers, S. Eisebitt, W. Eberhardt, Appl. Phys. B 56, 13147 (1997) 250

4. For a recent overview, see the issue Appl. Phys. A 65, No.2 (1997) 250

5. J. Fink, Adv. Electron. Electron Phys. 75, 121 (1989) 250, 254, 256

6. D. D. Vvedensky, Topics Appl. Phys. 69, 139 (Springer, Berlin, Heidelberg 1992) 250

7. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons, Springer Tracts Modern Phys. 88 (Springer, Berlin, Heidelberg 1980) 250

8. Ch. Kittel, Introduction to Solid State Physics, 7th edn. (Wiley, New York 1996) 251

9. P. Rudolf, M. S. Golden, P. A. Brühwiller, J. Electr. Spectrosc. 100, 409 (1999)

10. P. Chen, X. Wü, X. Sun, J. Liü, W. Ji, K. L. Tan, Phys. Rev. Lett. 82, 2548 (1999) 251, 252

11. R. C. Tatar, S. Rabii, Phys. Rev. B 25, 4126 (1982)

N. A. W. Holwarth, S. G. Loüie, S. Rabii, Phys. Rev. B 26, 5382 (1982) 253

12. J. C. Charlier, X. Gonze, J. P. Michenaüd, Phys. Rev. Lett. 43, 4579 (1991) 251, 252

13. W. W. Toy, M. S. Dresselhaüs, G. Dresselhaüs, Phys. Rev. Lett. 15, 4077 (1977)

14. M. S. Golden, M. Knüpfer, J. Fink, J. F. Armbrüster, T. R. Cümmins, H. A. Romberg, M. Roth, M. Sing, M. Schmitt, E. Sohmen, J. Phys. Condens. Matter 7, 8219 (1995) 252, 253, 254

15. J. H. Weaver, J. L. Martins, T. Komeda, Y. Chen, T. R. Ohno, G. H. Kroll, N. Troüllier, R. E. Haüfler, R. E. Smalley, Phys. Rev. Lett. 66, 1741 (1991) 252

16. S. Saito, A. Oshiyama, Phys. Rev. Lett. 66, 2637 (1991) 252

17. S. Eisenbitt, A. Karl, W. Eberhardt, J. E. Fischer, C. Sathe, A. Agni, J. Nord-gren, Appl. Phys. A 67, 89 (1998) 253, 254, 258, 259, 261

18. T. Pichler, M. Sing, M. Knupfer, M. S. Golden, J. Fink, Solid State Commun. 109, 721 (1999) 254, 255, 261, 268, 269

19. E. J. Mele, J. J. Ritsko, Phys. Rev. Lett. 43, 68 (1979) 254

20. E. A. Taft, H. R. Phillip, Phys. Rev. A197, 138 (1965) 254

21. E. Tossati, F. Bassani, Nuovo Cimento B 65, 161 (1970) 255

22. E. Sohmen, J. Fink, W. Kratschmer, Z. Phys. B-Condensed Matter 86, 87 (1992) 256, 257

23. Y. Saito, H. Shinohara A. Ohshita, Japn. J. Appl. Phys. 30, L1068 (1991) 257

24. X. Liu, H. Peisert, T. Pichler, M. Knupfer, M. S. Golden, J. Fink, unpublished 257, 258

25. M. Grioni, J. Voigt, High-resolution photoemission studies of low-dimensional systems, in Electron Spectroscopies Applied to Low-Dimensional Materials, H. Stanberg, H. Huges (Eds.), (Kluwer Academic, Dordrecht 2000) 258

26. M. Bockrath, D. H. Cobden, J. Lu. A. G. Rinzler, R. E. Smalley, L. Balents, P. L. Mc Enen, Nature 397, 598 (1999)

Z. Yao, H. W.Ch. Postma, L. Balents, C. Dekker, Nature 402, 273 (1999) 258

27. C. Schönenberger, A. Bachtold, C. Strunk, J. P. Salvetat, L. Forro, Appl. Phys. A 69, 283 (1999) 258, 261

28. N. Hamada, S. I. Sawada, A. Oshiyama, Phys. Rev. Lett. 68, 1579 (1992) 260

29. R. Saito, M. Fujita, G. Dresselhaus, M. S. Dresselhaus, Appl. Phys. Lett. 60, 2204 (1992) 260

30. J. C. Charlier, Ph. Lambin, Phys. Rev. Lett. 57, R15037 (1998) 260

31. J. W. G. Wildoöer, L. C. Venema, A. G. Rinzler, R. E. Smalley, C. Dekker, Nature 391, 59 (1998) 260, 263

32. T. W. Odom, J. L. Huang, Ph. Kim, Ch. M. Lieber, Nature 391, 62 (1998) 260, 263

33. A. M. Rao, E. Richter, S. Bandow, B. Chase, P. C. Eklund, K. A. Williams, S. Fang, K. R. Subbaswamy, M. Menon, A. Thess, R. E. Smalley, G. Dresselhaus, M. S. Dresselhaus, Science 275, 187 (1997) 260

34. R. Saito H. Kataura, chapter in this volume 260, 269

35. L. Henrard, A. Loiseau, C. Journet, P. Bernier, Eur. Phys. J. B 13, 661 (2000) 260, 265

36. S. Rols, E. Anglaret, J. L. Sauvajol, G. Coddens, A. J. Dianoux, Appl. Phys. A 69, 591 (1999) 260

37. T. Pichler, M. Knupfer, M. S. Golden, J. Fink, A. Rinzler, R. E. Smalley, Phys. Rev. Lett. 80, 4729 (1998) 261, 262, 265, 266

38. R. Kuzno, M. Teranchi, M. Tanaka, Y. Saito, Jpn. J. Appl. Phys. 33, L1316 (1994) 262

39. L. Henrard Ph. Lambin, J. Phys. B. 29, 5127 (1996) 262

40. A. A. Lucas, L. Henrard, Ph. Lambin, Phys. Rev. Lett. 49, 2888 (1994) 262

41. T. Stöckli, J. M. Bonard, A. Chatelain, Z. L. Wang, P. Stadelmann, Phys. Rev. Lett. 57, 15599 (1998) 262

42. M. F. Lin, K. W. K. Shung, Phys. Rev. Lett. 50, 17744 (1994) 263

43. S. Tasaki, K. Maekawa, T. Yamabe, Phys. Rev. Lett. 57, 9301 (1998) 263

44. O. Stephan, P. M. Ajayan, C. Colliex, F. Cyroy-Lackmann, E. Sandre, Phys. Rev. Lett. 53, 13824 (1996) 263, 267

45. H. Kataura, Y. Kumazawa, Y. Haniwa, I. Umezu, S. Suzuki, Y. Ohtsuka, Y. Achiba, Synth. Met. 103, 2555 (1999) 263

46. O. Jost, A. A. Gorbunov, W. Pompe, T. Pichler, R. Friedlein, M. Knupfer, M. Reibold, H.-D. Bauer, L. Dunsch, M. S. Golden, J. Fink, Appl. Phys. Lett. 75, 2217 (1999) 263, 264

47. M. S. Dresselhaus, G. Dresselhaus, P. C. Eklund, Science of Fullerenes and Carbon Nanotubes (Academic, San Diego 1996) 263

48. R. Saito, G. Dresselhaus, M. S. Dresselhaus, Phys. Rev. Lett. 61, 2981 (2000) 263

49. C. H. Olk, J. P. Heremans, J. Mater. Res. 9, 259 (1994) 263

50. L. C. Qin, S. Iijima, H. Kataura, Y. Maniwa, S. Suzuki, Y. Achiba, Chem. Phys. Lett. 268, 101 (1997) 265

51. F. J. Garcia-Vidal, J. M. Pitarke, J. B. Pendry, Phys. Rev. Lett. 78, 4289 (1997) 265

52. M. F. Lin, K. W. K. Shung, Phys. Rev. Lett. 47, 6617 (1993) 265

53. S. Das Sarma, E. H. Hwang, Phys. Rev. Lett. 54, 1936 (1996) 265

54. C. Yannouleas, E. G. Bogachek, U. Landman, Phys. Rev. Lett. 53, 10225 (1996) 265

55. M. F. Lin, D. S. Chuu, Phys. Rev. Lett. 57, 10183 (1998) 265

56. R. Friedlein, T. Pichler, M. Knupfer, M. S. Golden, K, Mukhopadhyay, T. Sugai, H. Shinohara, J. Fink, in Electronic Properties of Novel Materials - Science and Technology of Molecular Nanostructures, AIP Conf. Proc. 486, 351 (1999) 266, 267

57. R. Friedlein, PhD thesis, TU Dresden (2000) 266, 267

58. L. Henrard, O. Stephan, C. Colliex, Synt. Met. 103, 2502 (1999) 266

59. M. Kociak, L. Henrard, O. Stephan, K. Suenaga, C. Colliex, Phys. Rev. Lett. 61, 13936 (2000) 266

60. R. S. Lee, H. J. Kim, J. E. Fischer, A. Thess, R. E. Smalley, Nature 388, 255 (1997) 267, 269

61. J. J. Ritsko, Phys. Rev. Lett. 25, 6452 (1982) 268

62. P. Petit, C. Mathis, C. Journet, P. Bernier, Chem. Phys. Lett. 305, 370 (1999) 269

63. S. Kazaoui, N. Minami, R. Jacquemin, H. Kataura, Y. Achiba, Phys. Rev. Lett. 60, 13339 (1999) 269

Phonons and Thermal Properties of Carbon Nanotubes

James Hone

Department of Physics, University of Pennsylvania Philadelphia, PA 19104-6317, USA [email protected]

Abstract. The thermal properties of carbon nanotubes display a wide range of behaviors which are related both to their graphitic nature and their unique structure and size. The specific heat of individual nanotubes should be similar to that of two-dimensional graphene at high temperatures, with the effects of phonon quantization becoming apparent at lower temperatures. Inter-tube coupling in SWNT ropes, and interlayer coupling in MWNTs, should cause their low-temperature specific heat to resemble that of three-dimensional graphite. Experimental data on SWNTs show relatively weak inter-tube coupling, and are in good agreement with theoretical models. The specific heat of MWNTs has not been examined theoretically in detail. Experimental results on MWNTs show a temperature dependent specific heat which is consistent with weak inter-layer coupling, although different measurements show slightly different temperature dependences. The thermal conductivity of both SWNTs and MWNTs should reflect the on-tube phonon structure, regardless of tube-tube coupling. Measurements of the thermal conductivity of bulk samples show graphite-like behavior for MWNTs but quite different behavior for SWNTs, specifically a linear temperature dependence at low T which is consistent with one-dimensional phonons. The room-temperature thermal conductivity of highly aligned SWNT samples is over 200 W/mK, and the thermal conductivity of individual nanotubes is likely to be higher still.

1 Specific Heat

Because nanotubes are derived from graphene sheets, we first examine the specific heat C of a single such sheet, and how C changes when many such sheets are combined to form solid graphite. We then in Sect. 1.2 consider the specific heat of an isolated nanotube [1], and the effects of bundling tubes into crystalline ropes and multi-walled tubes (Sect. 1.3). Theoretical models are then compared to experimental results (Sect. 1.4).

1.1 Specific Heat of 2-D Graphene and 3-D Graphite

In general, the specific heat C consists of phonon Cph and electron Ce contributions, but for 3-D graphite, graphene and carbon nanotubes, the dominant contribution to the specific heat comes from the phonons. The phonon contribution is obtained by integrating over the phonon density of states with

M. S. Dresselhaus, G. Dresselhaus, Ph. Avouris (Eds.): Carbon Nanotubes, Topics Appl. Phys. 80, 273-286 (2001) © Springer-Verlag Berlin Heidelberg 2001

a convolution factor that reflects the energy and occupation of each phonon state:

where p(w) is the phonon density of states and wmax is the highest phonon energy of the material. For nonzero temperatures, the convolution factor is 1 at w = 0, and decreases smoothly to a value of ~ 0.1 at hw = kBT/6, so that the specific heat rises with T as more phonon states are occupied. Because p(w) is in general a complicated function of w, the specific heat, at least at moderate temperatures, cannot be calculated analytically.

At low temperature (T ^ ®d), however, the temperature dependence of the specific heat is in general much simpler. In this regime, the upper bound in (1) can be taken as infinity, and p(w) is dominated by acoustic phonon modes, i.e., those with w ^ 0 as k ^ 0. If we consider a single acoustic mode in d dimensions that obeys a dispersion relation w <x ka, then from (1) it follows that:

Thus the low-temperature specific heat contains information about both the dimensionality of the system and the phonon dispersion.

A single graphene sheet is a 2-D system with three acoustic modes, two having a very high sound velocity and linear dispersion [a longitudinal (LA) mode, with -y=24km/s, and an in-plane transverse (TA) mode, with v=18km/s] and a third out-of-plane transverse (ZA) mode that is described by a parabolic dispersion relation, w = 5k2, with 5 ~ 6 x 10~7 m2/s [2,3]. From (2), we see that the specific heat from the in-plane modes should display a T2 temperature dependence, while that of the out-of plane mode should be linear in T. Equation (1) can be evaluated separately for each mode; the contribution from the ZA mode dominates that of the in-plane modes below room temperature.

The phonon contributions to the specific heat can be compared to the expected electronic specific heat of a graphene layer. The unusual linear k dependence of the electronic structure E(k) of a single graphene sheet at EF (see Fig. 2 of [4]) produces a low-temperature electronic specific heat that is quadratic in temperature, rather than the linear dependence found for typical metals [5]. Benedict et al. [1] show that, for the in-plane modes in a graphene sheet,

The specific heat of the out-of plane mode is even higher, and thus phonons dominate the specific heat, even at low T, and all the way to T = 0.

Combining weakly interacting graphene sheets in a correlated stacking arrangement to form solid graphite introduces dispersion along the c-axis, as the system becomes three-dimensional. Since the c-axis phonons have very low frequencies, thermal energies of ~50K are sufficient to occupy all ZA phonon states, so that for T > 50 K the specific heat of 3-D graphite is essentially the same as that of 2-D graphene. The crossover between 2-D and interplanar coupled behavior is identified as a maximum in a plot of Cph vs. T2 [6,7]. We will see below that this type of dimensional crossover also exists in bundles of SWNTs. The electronic specific heat is also significantly changed in going from 2-D graphene to 3-D graphite: bulk graphite has a small but nonzero density of states at the Fermi energy N(EF) due to c-axis dispersion of the electronic states. Therefore 3-D graphite displays a small linear Ce(T), while Cph has no such term. For 3-D graphite, the phonon contribution remains dominant above ~1K [8,9].

1.2 Specific Heat of Nanotubes

Figure 1 shows the low-energy phonon dispersion relations for an isolated (10,10) nanotube. Rolling a graphene sheet into a nanotube has two major effects on the phonon dispersion. First, the two-dimensional band-structure

Fig. 1. Low-energy phonon dispersion relations for a (10,10) nanotube. There are four acoustic modes: two degenerate TA modes (v = 9km/s), a 'twist' mode (v = 15km/s), and one TA mode (v = 24km/s) [3]. The inset shows the low-energy phonon density of states of the nanotube (solid line) and that of graphite (dashed line) and graphene (dot-dashed line). The nanotube phonon DOS is constant below 2.5 meV, then increases stepwise as higher subbands enter; there is a 1-D singularity at each subband edge

Figure 1 shows the low-energy phonon dispersion relations for an isolated (10,10) nanotube. Rolling a graphene sheet into a nanotube has two major effects on the phonon dispersion. First, the two-dimensional band-structure

0 0

Post a comment