Effect of dimensionality in polymeric fullerenes and single-wall nanotubes

Physica B 244 (1998) 186 191 Effect of dimensionality in polymeric fullerenes and single-wall nanotubes H. Kuzmany*, B. Burger, M. Fally, A.G. Rinzler, R.E. Smalley Institut fu( r Materialphysik, Universita( t Wien, Boltzmanngasse 5, A-1090 Vienna, Austria Institut fu( r Experimentalphysik, Universita( t Wien, Austria Department of Chemistry and Physics, Rice University, USA Abstract We present new results for two examples where the dimensionality of a solid state system plays an important role. The systems under consideration are AC (A"K, Rb, Cs) and single-wall nanotubes. In the first case we show that a spin-density-wave instability triggered by nearly perfect nesting of a 3-D Fermi-surface is responsible for the transition from a metallic to an insulating state. In the second case the analysis of the radial breathing mode as observed by Raman scattering from single-wall nanotubes is used to determine the types and radial distribution of the as prepared sample of tubes. 1998 Elsevier Science B.V. All rights reserved. Keywords: Fullerenes; Spin-density-wave; Fermi-surface instability; Carbon nanotubes; Raman spectroscopy; Far-infrared spectroscopy 1. Introduction The dimensionality of a solid state is known to be an important characteristic of the system. Only for fully 3-D solids a conventional behavior of the physical properties can be expected. If the dimensionality becomes as low as one, characteristic differences will occur such as a lack of metallicity, a lack of any phase transition, or, more general, a lack of any long-range order. Instead, strong fluctuations will dominate the behavior. Close to 1-D, the divergency of the response function is known to drive several lattice and magnetic instabilities. By a reduction of the dimensionality below one, we understand the approach to material * Corresponding author. Tel.: 43 1 31367 3206; fax: 43 1 310 38 88; e-mail: kuzman@pap.univie.ac.at.; with extremely small extension where finally quantum size effects become important. We report in the following on the influence of the dimensionality for two special systems which have recently attracted considerable attention. Both systems refer to new phases of carbon and are derived from the classical fullerenes. The first system is a polymeric state of C which was obtained from a high-temperature FCC phase of C doped with the alkali metals K or Rb to a stoichiometry 1 : 1. We show that the phase transition from the metallic state at room temperature to an insulating state at low temperatures should not be considered in analogy to the wellknown spin-density-wave transitions in quasi-1-d systems but is rather due to an instability at the 3-D nearly perfect nested Fermi-surface of the half-filled conduction band. 0921-4526/98/$19.00 1998 Elsevier Science B.V. All rights reserved PII S0921-4526(97)00485-7

H. Kuzmany et al. / Physica B 244 (1998) 186 191 187 The other systems under consideration are single-wall carbon nanotubes (SWNT). In these tubes the diameter is so small that quantum size effects become important. The quasi continua for the dispersion relations of electrons and phonons become discrete and the vibrational frequencies and the energies of the electronic states become dependent on the diameter of the tubes. A detailed analysis of the radial breathing mode observed around 180 cm is shown to provide a possibility to analyze the as grown batch of tubes with respect to their type and to their radial extension. In both cases, Raman scattering has proven to be an excellent technique to study the materials. Experimental spectra used for the investigations were recorded either with a Dilor xy-raman system with multichannel detection, and for excitation with various lasers from the red to the blue spectral range, or with a Bruker FT-Raman system for excitation with 1064 nm. In the case of the polymeric fullerenes experiments were carried out on single crystals. The SWNTs were prepared by a two laser desorption process described elsewhere [1]. 2. Polymeric phases of AC (A"K, Rb, Cs) The phases AC have attracted special attention among the various alkali metal fullerides. First discovered by spectroscopic investigations of the doping process at elevated temperatures [2] they undergo a variety of different structural transitions on cooling to room temperature or to lower temperatures. A review of these phenomena is given in Ref. [3]. KC on the one side and RbC and CsC on the other side behave differently with respect to several basic properties. One of them is the stability of the ground state at room temperature. Whereas the ground state for the Rb and Cs compound is an orthorhombic polymeric phase where the C molecules are linked by a cyclobutane ring [4] KC is unstable on cooling and desintegrates into C and K C below about 410 K. Only for fast enough cooling it polymerizes also at the transition temperature. Fig. 1 demonstrates the different behavior of the systems from Raman scattering in the spectral range of the pentagonal pinch mode. All three spectra were recorded at room temperature. Part (a) shows the response after slow cooling of KC. The two lines observed at 1468 and 1447 cm correspond to the phases C and K C. In contrast, parts (b) and (c) of the figure show the response for KC after moderate fast cooling and for RbC. In both cases the pinch mode appears to be split into two components separated by only about 10 cm. The higher-frequency component is assumed to originate from a G mode which was split and became Raman active due to the symmetry-breaking along with the polymerization process. The lower-frequency component originates from the A pinch Fig. 1. Raman response of the pinch mode of KC and RbC at room temperature: (a) KC after slow cooling from its high-temperature phase, (b) KC after moderate fast cooling, and (c) RbC ; (b) and (c) with fitted oscillators.

188 H. Kuzmany et al. / Physica B 244 (1998) 186 191 mode of the charged polymer. It is downshifted by about 7 cm from the position for the neutral linear polymer [5]. The small line at the highfrequency edge of the line profile originates from a small amount (about 5%) undoped and unpolymerized C. The downshift of the pinch mode as compared to this line is slightly lower for KC than for RbC. The values from the computer fit are 17 and 18 cm for the former and the latter, respectively. This difference is consistent with a slightly lower degree of ionization for KC as compared to RbC and thus, with a slightly lower carrier concentration for the former. This difference is consistent with the different magnitude of the spin susceptibility observed for KC and RbC or CsC [6]. Another intriguing difference between the K compound and the Rb compound is their behavior at low temperatures. Whereas RbC undergoes a phase transition to an insulating spin-densitywave state, KC remains metallic down to any low-temperature checked so far [6, 7]. In analogy to well-known results from quasi-1-d systems, this phenomenon was linked with an assumed quasi-1- D state for RbC and CsC whereas KC was believed to remain 3-D. This interpretation is problematic since all three compounds are isostructural and all calculations of the electronic structure reveal a fully 3-D band structure [8]. Thus, the analogy to the quasi-1-d systems does not apply. In a recent theoretical analysis we found that an instability of the system will also occur for a 3-D Fermi-surface if the conduction band has tight binding character and is exactly half-filled. The tight binding character for this band was proved for the AC phases from ab initio calculations [8]. Also half-filling is well established since the threefold degenerate t band is split into three nondegenerate bands. Since the splitting is strong enough the lowest band is exactly half-filled. The 3-D tight binding band is described by ε(k)"!2t cos(ak )!2t [cos(bk )#cos(ck )]!μ. (1) The corresponding Fermi-surface for an exactly half-filled band is perfectly nested for the vector q"(π/a, π/b, π/c). Under these conditions the system is unstable towards a transition to a spindensity-wave state for an arbitrary small particle particle interaction λ. Any deviation from the half filling, from the tight binding character or from the 3-D structure needs a finite value for λ to trigger the instability at a particular temperature ¹. The conditions for the triggering can be evaluated from the renormalized density density response function. χ(q, ω)" χ (q, ω) 1!λχ (q, ω), (2) where χ (q, ω) is the free particle density density response function. The conditions for the divergency of χ(q, ω) allow to determine the temperature for the phase transition as a function of the anisotropy and of the band filling for a given value of λ. Fig. 2 shows an example for λ"272 mev. The xy-plane is a phase diagram for the transition to a spin-density-wave state. The dashed line is the border between the disordered metallic and the spin-ordered insulating state. The z-direction gives ¹ for the transition. Whereas the conventional quasi-1-d systems are located in the upper right corner of the xy-plane the AC phases are located in the lower left corner of the plane. Thus the instability of the latter is triggered from the 3-D Fig. 2. Three-dimensional phase diagram for the stability of a system with a partially filled tight binding band versus a transition to a spin-density-wave state. The dashed line in the xy-plane is the phase boundary as a function of the bandfilling and of the anisotropy. The hatched area is the region for the well-known quasi-1d systems. The full and the open circle label possible positions for RbC and KC.

Fermi-surface of the half-filled tight binding band. This interpretation is fully consistent with the 3-D electronic structure of AC. A possible position for RbC is indicated as full circle. A slightly lower band filling (i.e. a slightly lower carrier concentration) and/or a lower anisotropy (i.e. a higher value for t /t ) for KC as compared to Rb or CsC is enough to render the potassium compound in an area of the phase diagram where it remains stable for all temperatures (open circle). With respect to their position in the phase diagram of Fig. 2 the AC compounds are unique materials so far. H. Kuzmany et al. / Physica B 244 (1998) 186 191 189 3. Single-wall carbon nanotubes Since the discovery of the nanotubes by Iijima in 1991 [9] this material has attracted considerable attention. Nanotubes consist of a set of concentric cylinders of graphene sheets. Typical diameters of the tubes are 2 20 nm depending on the number of concentric sheets. Each cylinder can be characterized by two numbers (n, m) which determine the way the graphene sheet was rolled up [10]. Cylinders with n"m are labeled as armchair, those with either n"0 or m"0 are labelled as zigzag. All other tubes are chiral. If the tubes consist of only one cylinder (single-wall nanotubes) their diameter is in general less than 2 nm. In this case line positions in the Raman spectra can depend on the energy of the exciting laser as it was demonstrated recently by Rao et al. [11]. This shift of Raman lines with changing energy of the exciting laser is well known for conjugated polymers [12] and graphitic materials. It has been called a photoselective resonance scattering or a dispersion of Raman lines. Whereas in the polymers the C C stretch mode around 1500 cm exhibits the strongest dispersion, the effect is most significant in the SWNTs for the radial breathing mode. Fig. 3 shows the shift of the lines for the two materials. For graphitic materials or for multiwall nanotubes the dispersion effect is known to occur for a defectinduced line (D-line) around 1340 cm. From simple considerations one would expect that the vibrational modes shift continuously upwards with increasing energy of the laser. This is Fig. 3. Dispersion of the radial breathing mode for single-wall nanotubes (a) and for the C C stretch mode in polyacetylene (b) after excitation with different laser energies. indeed so for the line in polyacetylene and for the D-line in the MWNTs as shown in Fig. 4. SWNTs behave differently as the peak position for the radial breathing mode shifts upwards first and decreases for the highest laser energies used. By looking in more detail at the Raman response from the radial breathing mode in the SWNTs a well-expressed fine structure can be resolved. An example is given in Fig. 5 which shows a blown up version of the line after excitation with λ"476 nm. The fine structure is assumed to originate from contributions of SWNTs of different type and with different radii. The full-drawn line through the experimental points is the sum of eight oscillators used for the fit. During the fitting process the line width was kept constant at 7$1cm. The described procedure of fitting the observed line shape with oscillators was performed for excitations with six different laser lines, extending from 586 to 457.9 nm. As a result 13 different lines could be identified and were assigned to mode frequencies calculated for the different nanotubes [10, 13]. The strongest line was observed for 185 cm for excitation with 2.5 ev in very good agreement with a mode frequency of 185 cm calculated for the radial breathing mode of the (9, 9) armchair tube. The next strongest line was found at 177 cm which corresponds to the (16, 0) zigzag tube. Other relevant but weaker lines were observed at 166 and

190 H. Kuzmany et al. / Physica B 244 (1998) 186 191 Fig. 4. Relative line shift for single-wall nanotubes (SWNT), multi wall nanotubes (MWNT), and polyacetylene ((CH) ) versus energy of the exciting laser. The numbers assigned to the three different graphs label the frequency for the excitation with lowest energy. Fig. 5. Fine structure of the Raman response for the radial breathing mode in as grown samples of single wall nanotubes after excitation with λ"476 nm. 163 cm which corresponds to (10, 10) and (17, 0) tubes. The assignment of the lines is taken from the calculated results for the mode frequencies. This means it scales with this calculation. The diameter for the (9, 9) and for the (16, 0) tubes are 1.24 and 1.28 nm, respectively, in reasonably good agreement with a diameter of 1.38 nm derived from X-ray analysis [1]. All lines identified above exhibited a characteristic dependence on the laser energy. They showed a maximum for a characteristic energy and became very small or even disappear for energies shifted away from the maximum. In some cases also a minimum was observed for the intensity. Comparing the energy dependence of the excitation with calculated joint density of states allowed to determine the optical transition used for the resonance scattering. Since the transition energies scale with the nearest-neighbor overlap integral γ between two carbon atoms the latter could be determined from a fit between the observed intensities and the calculated joint density of states. The value obtained was 2 ev as compared to 2.5 ev calculated for the single graphene sheet. 4. Conclusions In conclusion, it was shown that new carbon materials derived from the fullerenes exhibit structures where the dimensionality plays an important role. For the polymeric phases of AC an explanation for the instability at low temperatures was given which is consistent with the 3-D nature of the electronic structure. In the case of SWNTs quantum size effects were shown to be the reason for the unusual dispersion of the Raman lines. The experiments allowed to determine the components of the

H. Kuzmany et al. / Physica B 244 (1998) 186 191 191 as grown tube material with respect to type and diameter of the tubes. Acknowledgements The work was supported by the FFWF in Austria, project P11943. References [1] A. Thess et al., Science 273 (1996) 483. [2] J. Winter, H. Kuzmany, Solid State Commun. 84 (1992) 935. [3] H. Kuzmany, J. Winter, in: P. Eklund, and A.M. Rao (Eds.), Fullerene Polymers and Fullerene Polymer Composits Springer, Berlin, in press. [4] S. Pekker et al., Solid State Commun. 90 (1994) 349. [5] J. Winter et al., Phys. Rev. B 54 (1997) 1. [6] Bommeli et al., Phys. Rev. B 51 (1995) 14794. [7] O. Chauvet et al., Phys. Rev. Lett. 72 (1994) 2721. [8] S.C. Erwin, G.V. Krishna, E.J. Mele, Phys. Rev. B 51 (1995) 7345. [9] S. Iijima, Nature 354 (1991) 56. [10] R.A. Jishi et al., J. Phys. Soc. Japan 63 (1994) 2252. [11] A. Rao et al., Science 275 (1997) 187. [12] H. Kuzmany, Pure Appl. Chem. 57 (1985) 235. [13] M.S. Dresselhaus, G. Dresselhaus, P. Eklund, Science of Fullerenes and Carbon Nanotubes, Ch. 19, Academic Press, San Diego 1996.