Hole effective mass (me)
"Assumes 3 electrons
"Assumes 3 electrons and Rb^Qo and that K^Q,, and Rb^Qo compounds are not metallic for all x values corresponding to a partially filled LUMO-derived band. Since the splittings of the levels in the LUMO-derived band are small compared with the bandwidth, the conduction band is considered to be degenerate, and the stoichiometry x in an alkali metal-doped compound MxCfi0 corresponds to the filling of the band. Photoemission and optical studies show that the Hubbard U is large and does not change significantly upon variation of the dopant concentration x, indicating that doped fullerenes are strongly correlated systems (see §12.7.7). From the dopant dependence of p we conclude that one-electron theory is insufficient and that many-body electron-electron correlation effects must be important. Further theoretical work is needed to gain an understanding of why the transport behavior is so sensitive to level degeneracies and electron-filling effects and why the half-filled band does not give rise to a metal-insulator transition [14.6],
Another important aspect of the metallic phase is the high value for the pmin that is found experimentally at x = 3. For a given dopant, the doping (or intercalation) conditions and the nature of the fullerene host material both strongly influence the value of the minimum resistivity pmin (or the maximum conductivity <rmax) that is achieved, as well as the magnitude of the superconducting transition temperature Tc and the width of the superconducting transition A Tc for the case of the alkali metal MjC^ (M = K, Rb) compounds. Slower intercalation under well-controlled conditions generally increases Tc, while decreasing A Tc and pmin. The highest Tc and the lowest A Tc and pmin are achieved with single-crystal C60 host materials [see Fig. 14.3(a)], and pmin values of ~ 0.5 x 10_3il-cm have been reported at low T [14.17,18]. Thin-film samples have also been widely studied and tend to have somewhat lower Tc values and larger A Tc widths and higher pmin values [14.16,19,20]. Powder samples have also been studied regarding their transport properties, and they show relatively high pmin values.
One cogent explanation for the high value of pmin arises from merohe-dral disorder which is present because icosahedral symmetry has no fourfold axis, thus giving rise to the random occurrence of the two "standard" orientations for the C60 molecules shown in Fig. 7.6. The conduction bands for a doped C60 crystal arise from the overlap of the LUMO-derived levels of the molecule. The intermolecular hopping matrix element and the electronic states both depend strongly on relative intermolecular orientations. Merohedral disorder significantly reduces the intermolecular hopping matrix element and therefore increases the resistivity. Calculations show that the merohedral disorder itself leads to a large residual resistivity of p0 ~ 300 fiil-cm in the absence of impurities and defects, indicating that merohedral disorder is the dominant scattering mechanism for doped C60 at low temperature [14.21-23], Because of the interaction between the pos-
50 100 150 200 250 300 T(K)
Fig. 14.3. (a)Normalized dc electrical resistivity p(T) of single-crystal KjC^. The inset shows the p(T) behavior near the superconducting transition temperature Tc = 19.8 K [14.15], (b) Temperature dependence of p for two K^Qq films. Sample A has less disorder than sample B. The inset plots the same data as cr(T) us. T1'2 over a smaller range of T (top scale) [14.16]. Samples showing (dp/dT) > 0 tend to have sharp normal-superconducting transitions.
itively charged alkali metal dopants and the negatively charged C60 molecular ions, the energy of the LUMO-derived levels is sensitive to the alkali ion positions. Thus disorder in the position of the alkali ions introduces spatial fluctuations in the LUMO energies. For many MXC60 samples, disorder in the position of the alkali metal M ions commonly occurs because the alkali metal ions are small in size compared to the octahedral sites (see §8.5) and the electrostatic interactions between the alkali metal ions and C60 anions cause ionic displacements from their ideal interstitial positions. Additional ionic displacements can also arise from the presence of neutral ternary species such as ammonia [14.24,25]. The random occupation of octahedral and tetrahedral sites for systems which ideally have site ordering (e.g., Na2KC60, Na2C60, RbC60; see §8.5) is also a source of disorder. Vacancies in the dopant site occupation are another source of disorder. In the case of strong disorder associated with dopant site occupation, carrier localization may occur [see Fig. 14.3(b)], Such effects have been reported in both alkali metal-doped C60 and C70 [14.16,26,27], Calculations show that both the orientational disorder of the C60 molecules and the positional disorder of the alkali metal ions contribute to the large carrier scattering in doped fullerenes [14.21-23], In transport calculations, the hopping matrix is found by evaluating the overlap of LUMO wave functions on adjacent molecules.
The pmin values for Na;tC60 and Cs^Qo in Fig. 14.2 are 0.11 il-cm and 0.20 il-cm, respectively, much higher than pmjn for K_,.C60 and RbxC60. The different behavior for Na3C60 relative to K3C60 and Rb3C60 has been related to the small size of the Na+ ion, thereby allowing some fluctuations in the site positions of the Na+ ions and perhaps occupation by some Na2 molecular ions which provide less charge transfer. For the case of the Cs+ ion, anomalous behavior arises from the large size of the Cs+ ion, which is responsible for the stabilization of a body-centered cubic (bcc) structure at the Cs3Cgo stoichiometry, the face-centered cubic (fee) structure being stable only under externally applied pressure [14.28,29], All compounds and phases in Fig. 14.2 show a local maximum in resistivity at x = 6, corresponding to the filling of the tlu band, if one electron per alkali metal dopant is transferred to the C60 anions.
Since the alkali metal-doped fullerenes are chemically unstable in air and are reactive with other chemical species, great care needs to be exercised in sample handling and in the execution of transport measurements. In general, the electrodes are attached prior to alkali metal doping, and the doping procedure is carried out under the constraints of maintaining the integrity of the electrodes. To minimize the effects of phase separation on the transport measurements of the maximum conductivity phase, great care must be exercised to prepare the M3C5Q stoichiometry accurately in the transport samples and to ensure that the samples are well annealed.
Because of the small magnitude of the mean free path I (~2-31 A) in the M3C60 compounds and because I tends to be comparable to the lattice constant a (14 A) and to the superconducting coherence length £0 (~25 A), the effects of crystal defects and grain boundaries are very important, so that film and single-crystal samples tend to exhibit somewhat different detailed transport behavior, especially regarding the magnitude of pmin and I and the temperature dependence of p(T), as discussed in §14.1.2. A theoretical estimate has been given for the mean free path I of 18 A in N^Qo, considering the effect of strong orientational fluctuations [14.30].
Studies of the temperature dependence of the resistivity of polycrystalline alkali metal-doped M^Qq samples in the normal state show that conduction is by a thermally activated hopping process except for a small range of x near 3 where the conduction is metallic [14.4,15,31], as discussed in §14.1.1. As indicated by the arrows in Fig. 14.1, the magnitude of the activation energy Ea increases as x deviates further and further from the resistivity minimum at x = 3. For example, Ea has a magnitude of 0.12 eV for x = 1 in KjC60 [14.5]. In contrast, the temperature dependence of p(T) reported for Na^Qo showed a metallic dependence in the higher temperature range (200-500 K) [14.32], which does not seem consistent with photoemission spectra [14.33,34] and susceptibility measurements [14.35]. Further work is needed to clarify the behavior of p(T) for the Na^C60 compounds.
In the metallic regime, results for p(T) for a superconducting single-crystal K3C60 sample [see Fig. 14.3(a)] show a T dependence of p(T) with a positive slope [14.15,31,36,37], A linear temperature dependence for p(T) is found over a wide temperature range, when the p(T) measurements are made on a sample at constant volume [14.38], Closer to Tc, some authors have also reported a T2 dependence for p(T) [14.8,31], and closer examination shows that the T2 dependence arises from measurements at constant pressure [14.38], At Tc the mean free path I is comparable to the C60-C60 intermolecular distance, and at room temperature I is smaller than the C60-C60 intermolecular distance for most film samples, implying relatively easy carrier transport on a singular molecular shell but strong scattering upon transfer to an adjacent shell.
The linear T dependence for the alkali metal-doped fullerenes is much weaker than that for the high Tc-cuprates [14.39]. There are presently significant differences in the functional form of p(T) and the magnitudes of p at various temperatures from one research group to another for the same nominal single-crystal M3C60 stoichiometry, depending on measurement conditions and crystalline quality [14.15,31,36,37], Larger differences in the functional dependence of p(T) are found when comparisons are made between reported single-crystal data and data on polycrystalline films and powder samples [14.5,16,19,36,40,41]. Disordered films often show a negative slope for p(T) above Tc [14.16], as discussed further below in connection with localization effects. Measurements of p(T) on K3C60 and Rb3C60 thin films with a grain size of ~1 /¿m and a sharp superconducting transition show a linear T dependence in the temperature range 280-500 K [14.8], rather similar to the trends observed for p(T) for single-crystal samples.
Microwave determinations of p(T) at 60 GHz on pressed powder samples have also been reported [14.42], showing a low value of p(T+) ~ 0.5 x 10 3 fl-cm, a functional form for p(T) that emphasizes the T2 dependence and a much larger difference between p(Tc+) and p(300 K) than shown in Fig. 14.3. Here 7+ = Tc + e is used to denote temperatures just above the superconducting transition temperature for the superconducting-normal state transition.
Attempts to fit the observed p(T) of single-crystal samples in the normal state with theoretical models [14.18,43] indicate that electron-phonon scattering is the dominant scattering mechanism near room temperature and that the lower-frequency intramolecular vibrations play an important role in the carrier scattering. Analysis of infrared measurements [14.44] implies a value of ~ 0.4 x 10~3 il-cm for p(T+), in good agreement with the microwave result and recent single-crystal dc p(T) experiments [14.17,18],
Film samples with small grain size (<1 /xm) tend to exhibit a negative temperature coefficient of p(T) above Tc [see Fig. 14.3(b)] while single-crystal samples exhibit a positive dp(T)/dT above Tc [see Fig. 14.3(a)], The observed temperature dependence of the resistivity of films of M^Qq (for 0 < x < 6) has been interpreted by some authors in terms of a granular conductor with grain sizes in the 60-80 A range [14.45] with hopping conduction between grains. This granularity strongly affects the superconducting properties of the M3C60 films, as well as their properties in the normal state [14.45]. For more crystalline samples, merohedral disorder is the dominant scattering mechanism at low-temperature, as discussed above.
The normal state of K3C60 is highly resistive, and if the Drude model is used to describe the transport properties, the mean free path is given by
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