Po Pa Pa

where the first term (Ap)c/p0 (denoting classical magnetoresistance) is positive and quadratic at high H, while the second term (Ap)L//p0 due to 3D weak localization and strong electron-electron interaction Js more complicated and is discussed below. The positive quadratic field dependence of (Ap)c/p0 implies that the corresponding magnetoconductivity is negative and quadratic and of the form A07 = —CH2, where the quadratic coefficient C varies as T"3/2 as shown in the inset to Fig. 14.10 [14.16], This positive quadratic term in (Ap)c/p0 may be due to conventional magnetoresistance mechanisms. The second term in Eq. (14.9) (Ap)L //p0 shows a predominantly negative magnetoresistance which is quadratic in H at low H and has an H1'2 dependence at high magnetic field. A T1/2 temperature dependence is found for the magnetoconductivity corresponding to the (Ap)JL //p0 term, and this T dependence has been identified with 3D weak localization effects and carrier-carrier interaction effects. These effects are found to persist to high temperatures, which is attributed to the condition r, <JC r, where t, and t are, respectively, the inelastic and total scattering times. The 3D localization effects arise from disorder, which is associated in part with merohedral disorder and in part with missing or excess K+ at the tetrahedral or octahedral sites, where the K+ ions are normally found in the ideal K3C60 compound. The effect of superconducting fluctuations on the magnetoresistance has been observed in both single crystals and film samples of K3C60 near the superconducting transition Tc. This observation in doped fullerenes is the first time that 3D superconducting fluctuations have been observed in a 3D weakly localized system [14.16,43].

The magnetoresistance of thin K4C70 films [14.26] up to 30 K and in magnetic fields up to 15 tesla is qualitatively similar to that for thin K3C60 films [14.16] and disordered carbon fibers [14.62], and the mechanism for the corresponding magnetoresistance in I^Cyo is identified with 3D weak localization and electron-electron interaction phenomena [14.26]. The electrical resistivity at zero magnetic field for the samples used in the magnetoresistance measurements was well fit by a fluctuation-induced tunneling model for metallic grains separated by poorly conducting material [14.26] (see §14.1.2).

Experimentally, the reported density of states at the Fermi level N(EF) for the M3C60 compounds varies from 1-2 states/eV per C60 per spin [14.95] to over 20 states/eV-CM-spin [14.96]. The theoretical situation also shows a wide range of values, 6.6-12.5 states/eV-C60-spin [14.97] for N(EF) (see §12.7.3), but not as wide a range as for the experiments. An elementary estimate of the electronic density of states at the Fermi level N(EF) can be made by assuming a constant N(E) throughout the tlu band which has 3 electrons per spin state and a bandwidth W, yielding N(EF) = 3/W per spin state per C60 or 6 states/eV-C60-spin using a value of W = 0.5 eV. Detailed calculations by Erwin and Pickett [14.98] yield a value of N(EF) = 6.6 states/eV-C60-spin at a lattice constant of 14.24 A, corresponding to that of K3C60 (see Table 8.3), in good agreement with the rough physical estimate for N(Ef) given above. This approximate relation is also in rough agreement with the estimate N(EF) = 3.5 states/eV-C60-spin that includes the effect of merohedral disorder [14.97]. Other LDA calculations yield values for the density of states of 7.2 and 8.1 states/eV-C60-spin for K3C60 and Rb3C60, respectively [14.99], and a value of 12.5 states/eV-C60-spin for K3C60 [14.13]. [It should be noted that various authors give different normalizations in reporting N(EF) values. These differences must be kept in mind when comparing results between different authors, using various experimental and theoretical techniques.] Theoretical analyses of experimental results have also been helpful in narrowing the range for estimates of N(Ef) for K3C60 and Rb3C60 [14.97],

Photoemission studies are typically used in solid state physics to obtain measurements of the density of states for occupied levels, and inverse pho-

toemission data are used to obtain the corresponding information for the unoccupied states. It is commonly assumed that the final state for an electron in the inverse photoemission is at the LUMO level and the initial electronic state in the photoemission process is at the HOMO level. These assumptions may not be applicable for highly correlated systems such as fullerenes [14.100]. Using the photoemission technique and its conventional interpretation, values of 1-2 states/eV-C60-spin have been obtained [14.95], as mentioned above. It is, however, generally believed that the photoemission estimates for N(EF) for doped fullerenes are low because the spectral weight in the photoemission spectra is smeared out due to strong electron-phonon and electron-plasmon coupling, leading to a reduction in the photoemission intensity at EF [14.101]. The high surface sensitivity of the photoemission technique, the small electron penetration depth, the large lattice constant, and the chemical instability of the K3C60 and Rb3C60 compounds further contribute to the difficulty in obtaining reliable bulk values for the density of states using the photoemission technique.

Pauli susceptibility (Xp) measurements provide another fairly direct measurement of N(Ef) for simple metals, since x°p neglecting many-body effects is simply x°p = 2fi%N(EF). The results for Xp obtained for K3C60 and Rb3C60 by susceptibility measurements are, respectively, 14 and 19 states/eV-C60-spin [14.102], Many-body enhancement effects calculated by LDA methods suggest correction factors to the magnetic susceptibility determination of N(Ef) so that (Xp/x°p) = [1 - M(EF)]^ where I'1 = 26 states/eV-C60-spin [14.103], Including many-body effects thus leads to N(Ef) values of 9.3 and 11.9 states/eV-C60-spin for K3C60 and Rb3C60, respectively, using the many-body correction factor (Xp/Xp) 1-5 and 1.6 for K3C go and Rb3C60, respectively. ESR measurements of the bulk susceptibility [14.104] of 15.5 states/eV-C60-spin have been reported, and if the many-body correction factor given above is used, a value of 10.3 states/eV-C60-spin is obtained. When these many-body interpretations are considered, the susceptibility measurements overall are in reasonable agreement with calculations of the density of states.

Since thermopower measurements (see §14.12) are sensitive to the bandwidth W, estimates of N(EF) for K3C60 and Rb3C60 can be obtained using the relation N(EF) ~ 3.5/W states/eV-C60-spin given above [14.97]. From analysis of the thermopower measurements of §14.12, estimates for W of 650 meV and 400 meV, respectively, were obtained for K3C60 and Rb3C60, yielding the corresponding rough N{EF) estimate of 5.4 and 8.8 states/eV-C60-spin for these compounds.

Measurements of the low-temperature specific heat (T < 25 K) provide a sensitive tool for measuring the electronic density of states in conventional metals. Specific heat measurements made on K3C60 yield a value of

14 states/eV-Qo-spin, neglecting strong coupling enhancement effects and effective mass enhancement effects [14.102] (see §14.8). Since the magnitude of these many-body effects is not well established, the magnitude of the enhancement factor responsible for the large value of N(EF) is not presently known.

From the Fermi contact term in the nuclear magnetic resonance interaction, a determination of N(EF) can be made from NMR measurements of the temperature dependence of the nuclear spin relaxation rate (see §16.1.7). Using this approach, values of N(EF) — 17 and 22 states/eV-C60-spin were reported for K3C60 and Rb3C60, respectively [14.105]). Recent calculations [14.99] have, however, shown that the NMR determinations of the density of states are likely to be high, because the experimental determinations have attributed all the spin relaxation to the Fermi contact term, while ab initio calculations have shown that the spin dipolar relaxation mechanism dominates over the orbital and Fermi contact mechanisms [14.99]. With this reinterpretation of the NMR measurements, values for N(EF) of 8.7 and 10.9 states/eV-C60-spin are obtained for K3C60 and Rb3C60, respectively.

By bringing theoretical effort to bear on the interpretation of experimental measurements, the range of values for N(EF) for K3C60 and Rb3C60 has been narrowed and brought into rough agreement with theoretical calculations. Based on presently available results, an estimate of N(EF) in the range 6-9 states/eV-C60-spin is reached for K3C60 [14.97], and N(EF) for Rb,C 60 is estimated to be a factor of 1.2-1.6 higher. Although the absolute value of N(Ef) for the M3C60 compounds is not firmly established, the percentage increase in N(Ef) with lattice constant for the fee alkali metal-doped fullerenes is quite well established (see Fig. 14.11).

Fig. 14.11. Calculated Fermi level density of states for K3CM) (solid circles) and for RbjQo (open circles) as a function of lattice constant. The solid line is a parabolic fit to the calculated values [14.106], The percentage increase in the density of states with lattice constant is more accurate than the absolute values given in this figure.

14.6. Pressure Effects

Transport measurements have been used primarily to probe pressure-dependent behavior in the C60 crystalline phases that are stable at atmospheric pressure and to study pressure-induced shifts of the phase boundary of the structural phase transition at Tm (see §7.1.3).

The high electrical resistivity of C60 is reduced by the application of high pressure [14.107-109], as expected because of the increased interaction between the C60 molecules as the intermolecular distance decreases with increasing pressure. The rapid decrease in electrical resistance with pressure is shown in Fig. 14.12 for C60, C70, and C60I4 pressed powder samples. Since the resistance of C60 was too high to measure accurately at ambient pressure, measurements were made on the less resistive C^ samples, which show a more than five orders of magnitude decrease in resistance with pressures up to 10 GPa [14.109], The temperature dependence of the resistance for C60, C70, and C60I4 could all be fit by the 3D variable-range hopping Mott formula

with the disorder attributed to impurities such as oxygen, lattice defects, stacking faults, and orientational disorder of the molecules with respect to the crystalline axes. The reported pressure dependence of the parameter T0 = [£3/kRN(EF)] could provide valuable information about transport in these samples if independent measurements could be carried out to mea-

Fig. 14.12. Variation of the resistance (plotted on a log scale) at room temperature of fullerenes with pressure: CM (solid squares), C70 (solid circles), and Q0I4 (solid triangles). Note the very steep variation with pressure at low pressures for Qol,, almost reaching metallic values [14.109],

Fig. 14.12. Variation of the resistance (plotted on a log scale) at room temperature of fullerenes with pressure: CM (solid squares), C70 (solid circles), and Q0I4 (solid triangles). Note the very steep variation with pressure at low pressures for Qol,, almost reaching metallic values [14.109],

sure the localization length £ or the density of states at the Fermi level N(Ef). The results shown in Fig. 14.12 further demonstrate an increase in resistance for C60 and C70 at pressures in the 15-20 GPa range, which has been correlated with an irreversible phase transition to a more insulating state [14.109],

Electrical resistance measurements have also been carried out over a much narrower range of temperature and for pressures up to 13 kbar (1 GPa = 9.9 kbar) to study the pressure dependence of the transition temperature T0l associated with the loss of full rotational freedom, yielding a value of 9.9 ± 1 K/kbar [14.110] in good agreement with other measurements of this quantity (see Table 7.1). Two features were identified in the R(T) curve near Tm and the temperature separation between these peaks was found to increase rapidly with increasing pressure. One proposed explanation for this effect is a small difference in the temperature of the rotational alignment of the C60 molecules at the cube edges and cube face centers [14.111], The presence of the two phase transitions is not believed to be due to residual solvent in the sample [14.110], By fitting their R(T) data to an Arrhenius relation as a function of temperature, R(T) = R(0)exp(—Ea/2kBT), values of Ea as a function of pressure were obtained. From these data an estimate of Ea — 2.0 eV at 1 atm was reported [14.110], which is consistent with optical measurements of the band gap in Qo (see §13.3).

As the intermolecular C-C distance decreases upon application of pressure and becomes comparable to the intramolecular C-C distance, an electronic transition might be expected to occur, as discussed in §7.3. However, transport studies in the high pressure phases of C60 [14.28] have not yet been reported.

The dependence of the superconducting parameters on pressure is also interesting and is reviewed in §15.6, where the pressure dependence of Tc for K3C60 and Rb3C60 is discussed in some detail.

Many measurements of the photoconductivity of undoped C60 films have been published [14.2,112-116], and a few measurements have also been reported on undoped C70 films [14.115,117] and potassium-doped C70 films [14.115]. In addition, photoconductivity studies have been carried out on polymers that have been doped with small amounts of fullerenes [14.118— 125]. For example, it has been reported that photoconducting films of polyvinylcarbazole (PVK) doped with fullerenes (a mixture of C60 and C70) show high xerographic performance (see §13.5 and §20.1.2), comparable with that for some of the best photoconductors available commer-

daily [14.113,118]. Thus, practical applications may emerge from the special properties of the photoconductivity of fullerene-doped polymers.

Before reviewing the photoconductive properties of fullerenes in detail, it is important to mention the wide disagreement in the published results on this topic. There are four dominant reasons for these discrepancies. First, the magnitude of the photoconductivity signal is reduced by several orders of magnitude by the presence of oxygen, which acts as a trapping center for carriers; it is therefore important to work with oxygen-free fullerene films for measurements of intrinsic photoconductivity behavior [14.126-128]. Second, the persistent photoconductivity effect described below [14.129-131] implies that photocarrier relaxation times in fullerenes can extend to days. These extremely long relaxation times must be taken into account when using light pulses for photoexcitation and detection. Third, excessive incident light intensities can lead to phototransformation (see §13.3.4) of the fullerene film [14.132], thereby modifying the material during the measurement process. Finally, photoconductivity measurements show sensitivity to the degree of crystallinity and to the concentration of defects. For these four reasons, this review of the photoconducting properties of fullerenes focuses on selected references which have had more success toward making intrinsic measurements.

In preparing a typical photoconductivity sample, thin (~2000 Á) fullerene films are deposited on glass substrates with partially precoated silver electrodes, and on top of the fullerene films, semitransparent aluminum electrodes (~ 200 A thick) are deposited. Light illumination is through these top electrodes and into the fullerene film, allowing both photoconductivity and dark conductivity measurements to be carried out, when a voltage (e.g., 1 V) is applied to the top electrodes, while the bottom electrodes are grounded [14.115]. To measure intrinsic photoconducting properties, attention must be given to address the four critical issues described above.

Oxygen-free C60 has a very high quantum efficiency (~55%), which is defined as the number of photogenerated carriers per absorbed photon [14.114,116], This high quantum efficiency, which is attributed to the long lifetimes of excited carriers in Cm, can be reduced by several orders of magnitude by oxygen uptake. Electrons are easily trapped at the interstitial octahedral sites where the oxygen molecules collect [14.133], giving rise to shallow traps for electron-hole recombination [14.116], In this review, we emphasize reported results on nominally oxygen-free samples.

In general, both photoexcited electrons and holes should contribute to the photoconductivity. However, for most of the common photoconductors the lifetime for one of the carrier types tends to be much greater, so that the photoconductivity is often dominated by a single carrier. Experiments such as the Dember effect indicate that electrons are the dominant photocarri-

ers in C60 films [14.128,134]. The Dember effect arises from the different mobilities of electrons and holes produced by light, so that a net excess of positive charge is observed at the illuminated electrode relative to the dark electrode which has a net negative charge; this charge separation gives rise to the Dember voltage which has been measured in C60 films [14.128].

The carrier generation can be explained in terms of the following relations:

for a monomolecular (or unimolecular) process, and for a bimolecular process [14.135], where An is the density of photoexcited excess carriers, n0 the density of thermal carriers, Sn the capture coefficient by recombination, and g the carrier generation rate. Because the quantum efficiency tj is defined by where np is the absorbed photon density which is proportional to g, then 17 becomes constant in a monomolecular process and proportional to npl/2 in a bimolecular process. Thus we see that photocarrier generation enhances photoconduction, while recombination processes inhibit photoconductivity.

The photoconductivity shows a wide range of rise times and decay times ranging from hundreds of picoseconds to days, with different mechanisms involved as the time scale is varied. As the different time scales are probed with various experimental techniques, different photocarrier excitation mechanisms come into play.

Figure 14.13 summarizes the transient photoconductivity response on the picosecond scale in pristine and oxygenated C60 films measured at a photon energy of ha> = 2.0 eV at room temperature [14.136]. Of particular significance is the very large magnitude of the transient photoconductivity, which is reported to be as much as nine orders of magnitude greater than the steady-state photoconductivity [14.137]. The transient photoconductivity of a pristine C60 film (open squares □ in Fig. 14.13) consists of short- and longer-lived components with relaxation times of 693 ps and 7.2 ns, respectively. From the temperature-dependent photoconductivity studies, it was shown [14.136] that these two components are dominated by two distinct transport mechanisms. The carrier relaxation times, as well as the magnitudes of the photo and dark conductivities, decrease dramatically with oxygen exposure, indicating that oxygen in the C60 film creates efficient deep traps for the photocarriers, which diminish the probability of

Fig. 14.13. The time-resolved transient photocurrent (T =300 K, ha) = 2.0 eV) in a pristine C^ film and in the film at various levels of oxygen content. The transient photocurrent data for the oxygen-free sample are represented by the □ points, and each sample was characterized by Id its dark current: □ Id = 3.6 nA; o Ij = 5.67 pA; A Id = 0.24 pA; 0 Id < 0.01 pA, where 1 pA = 10'2A. The solid lines are the best fit of the experimental data to a double-exponential function mentioned in the text. The inset shows the normalized transient photocurrent of pristine at different laser intensities: 2.7 x 1014 (a), 1.1 x 1015 (•), and 5.4 x 1015 photons/cm2 (■) [14.136],

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