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type excitons. Electron-electron interactions were treated using the Ohno potential form which is parameterized by an on-site (C-atom) Coulomb strength U = It and a long-range component V — t, where t — 2.0 eV and r0 is an average bond length. Values for t and r0 were optimized for the ion, to fit the optical spectrum. Their calculated results (thick line) are shown in Fig. 13.27 for a bond disorder broadening of 0.2f (0.4 eV). Also shown in Fig. 13.27 is the thin-film K6C60 optical absorption spectrum of Pichler et al. (thin line) [13.97] and the small momentum transfer EELS data by Sohmen and Fink [13.101] for Rb6C60 (dotted line) derived from the electron energy loss function. The EELS data yield a value of e,(0) = 7.1 for the low-frequency dielectric constant, which compares very well to the value 7.2 obtained optically [13.74], As can be seen in Fig. 13.27, the calculated optical spectrum is in good agreement with the EELS-derived result [13.101] and the experimental optical spectrum of Pichler et al. [13.97] for KgCgo in Fig. 13.27 and also the optical spectra for the M6C60 films (M = K, Rb, Cs) in Fig. 13.26. The theoretical results are seen to be in good agreement with experiment up to ~5.0 eV, and the deviations at higher energies have been ascribed [13.148] to the omission of cr-electron excitations in the theoretical model.

13.4.3. Normal State Optical Properties of M3C60

Several optical studies of alkali metal-doped M3C60 (M = K, Rb) have been reported showing these materials to be highly absorbing in the infrared, in contrast to pristine C60, and consistent with the metallic nature

of the transport properties of M3C^y (see §14.1.1). Reflectivity (£%) measurements have been carried out on opaque samples, e.g., pressed powders [13.149-151], doped single crystals [13.152], and thick, vacuum-deposited films [13.153], Transmission (¿F) studies [13.153-155] have also been reported on thinner, vacuum-deposited films. Kramers-Kronig (KK) analyses have been carried out on reflectivity data for M3Cm films to obtain quantitative information on the electronic contribution to the optical conductivity cj-(w). Below ~0.5 eV, the free electron contribution to the optical properties is dominant, and above ~0.5 eV, interband transitions from the partially occupied tlu level to the higher-lying tlg level take place. These contributions to the dielectric function are also sensitively studied by electron energy loss spectroscopy (EELS) (see §17.2.2). In addition, the optical conductivity data show interesting structure in the far IR (w < 400 cm-1). Transmission studies have been reported in the mid-IR region (400-4000 cm"1) and have emphasized the doping-induced changes in the intramolecular IR-active (Flu) phonons (see §11.6.2). These Flu phonons were also observed in single-crystal reflectance studies [13.152], but were not observed in pressed powder samples (3 mm diameter) [13.149,150]. Reflectivity data on M3C60 extending to the far IR region of the spectrum (~ 1 meV = 8 cm-1) are available only for pressed powder samples [13.149,150], though single-crystal optical studies have also been carried out.

In Fig. 13.28 we show the low-temperature, normal state reflectivity (5?) spectra for pressed-powder samples of K3C60 (a) and Rb3C60 (b) [13.151]. The £%(&>) data are plotted on a log frequency scale, covering four decades of frequency (10-105 cm-1). These samples were reported to have been excluded rigorously from oxygen and, furthermore, were found to exhibit superconducting transition temperatures Tc= 19 K (K) and 29 K (Rb) and diamagnetic shielding fractions of ~40-50%. Upon cooling these samples through Tc, large increases were observed in the reflectance below ~100 cm-1, indicative of the onset of superconductivity (see §13.4.4). The frequency-dependent reflectivity data ¿%(a>) in Fig. 13.28 for the normal state of K3C60 and Rb3C60 show a characteristic spectrum of a metal in the "dirty limit," i.e., high 5?(o>) at low frequency, and a broad Drude edge marking the position of the decline in to lower values (~20-30%), the decline in being associated with contributions from electronic interband transitions. The broad width of the Drude edge for both K3C60 and Rb3C60 indicates a short mean free path for the normal state electrons, consistent with transport measurements (see §14.1.2). The Drude edge in the reflectance data of K3C60 reported for doped single crystals [13.152] is also broad and in good agreement with the data shown in Fig. 13.28. (For a comparison between the single crystal and pressed-powder K3C60 reflectivity spectra, see Fig. 5 in Ref.[13.151].)

frequency (cm )

Fig. 13.28. Normal state reflectivity spectra on pressed powders of (a) KjC^ and (b) Rb3C60 at 25 and 40 K, respectively [13.151], frequency (cm )

Fig. 13.28. Normal state reflectivity spectra on pressed powders of (a) KjC^ and (b) Rb3C60 at 25 and 40 K, respectively [13.151],

A Kramers-Kronig (KK) integral transform of the ifl(w) data in Fig. 13.28 was performed to determine the optical conductivity cr(w) = cTx{<a) + ia2(w) for K3C60 (a) and RbjQo (b) [13.151]. The results thus obtained for the real part, tr^w), which is associated with optical absorption, are plotted in Fig. 13.29 on a log frequency scale. The solid and dash-dotted curves represent, respectively, the experimental ax(io)

frequency (cm")

Fig. 13.29. Normal state optical conductivity for (a) KjC«, and (b) RbjC«, at 25 and 40 K, respectively, as evaluated from a Kramers-Kronig analysis of the Sk(o>) reflectivity data in Fig. 13.28. The phenomenological fits to Eq. (13.16) are the dashed-dotted curves, and values for the fitting parameters are summarized in Table 13.8 [13.151],

frequency (cm")

Fig. 13.29. Normal state optical conductivity for (a) KjC«, and (b) RbjC«, at 25 and 40 K, respectively, as evaluated from a Kramers-Kronig analysis of the Sk(o>) reflectivity data in Fig. 13.28. The phenomenological fits to Eq. (13.16) are the dashed-dotted curves, and values for the fitting parameters are summarized in Table 13.8 [13.151], obtained from the KK analysis and that calculated on the basis of a model discussed below. At w = 0, the cr,(0) data match the dc transport value, consistent with the low-frequency data extension in Fig. 13.29 [13.151]. Above 400 cm"1, the ctx{u>) data in Fig. 13.29 for Rb3C60 and K3C60 are similar. Below 400 cm-1, the Rb3C60 data exhibit a more noticeable dip at ~100 cm"1, or equivalently, the peak at ~400 cm-1 is more pronounced in the Rb3C60 data. This peak is reminiscent of the so-called "mid-IR" band exhibiting maxima in the range 2000-4000 cm"1 in high Tc cuprate materials [13.156]. IR-active, Flw-derived phonons would appear as narrow peaks riding on the electronic background. They have been observed in films and crystals in the range w < 1700 cm-1 [13.153-155], but these peaks are apparently too weak to be detected in the powder samples of Fig. 13.29 [13.151]. Finally, the experimentally observed structure above ~3000 cm-1 is quite similar for both compounds and therefore must be identified with interband transitions between C60-derived states, rather than with "charge transfer" excitations involving initial or final states (but not both) associated with the alkali metal ions.

The model calculation results for crj(w) in Fig. 13.29 are based on the dielectric function given by

where ex is the core dielectric constant associated primarily with higher energy interband transitions, and vp is the electronic plasma frequency. In the notation of Eq. (13.16) the optical conductivity is, in general, related to the dielectric function by

The first term in the brackets of Eq. (13.16) is a harmonic oscillator function which describes the experimentally observed "mid-IR" absorption band, and the second term is the Drude expression for the dielectric response of conduction electrons. As written, the oscillator strength fG (or spectral weight) of the mid-IR band is borrowed from the Drude term. The electrons described by Eq. (13.16) behave approximately as free electrons for v < yD and as bound electrons for v > yD. An alternative, and more conventional approach is to decouple the two terms of Eq. (13.16) in brackets, i.e., to set fG = 0 in the second term [13.151],

As can be seen from Fig. 13.29, the model dielectric function given by Eq. (13.16) leads to a reasonably good description of the experimental arx((o) data below ~1000 cm-1, and the values for the parameters in Eq. (13.16) which correspond to the dash-dotted curves in Fig. 13.29 are summarized in Table 13.8. In this table the plasma frequency vp is found from v2 — (ne2)/vmb, where mb is the band mass and n is the free electron concentration. Using Eq. (13.16) and choosing a value for n — 4.1 x 1021 cm-3, consistent with three electrons in the conduction band per C^ molecule, a band mass of mb ~ 4m0 is obtained, where m0 is the free electron mass. This value is in reasonable agreement with that obtained by

Table 13.8

Parameters for the phenomenological fit to the optical data for KjC^ and RbjQo using Eq. (13.16): the total plasma frequency vp and the damping yD for the Drude term, the resonance frequency vG and the damping yG for the harmonic oscillator, the high-frequency contribution to the dielectric function and fG the weight factor between the Drude and harmonic oscillator terms [13.151].

Parameters for the phenomenological fit to the optical data for KjC^ and RbjQo using Eq. (13.16): the total plasma frequency vp and the damping yD for the Drude term, the resonance frequency vG and the damping yG for the harmonic oscillator, the high-frequency contribution to the dielectric function and fG the weight factor between the Drude and harmonic oscillator terms [13.151].

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