states of the tlu-derived band is found to split into two subbands, the lower subband with eight states and the upper level with four states for most regions of the Brillouin zone [12.75], as discussed further below.

Below 261 K each Qo molecule tries to align with respect to its 12 nearest-neighbors, so that its 12 electron-poor pentagons are located, as much as possible, opposite an electron-rich double bond on the nearest-neighbor Qo (see §7.1.3). Because of the difference in geometry between the 12 pentagons on a regular truncated icosahedron and the 12 nearest-neighbors of an fee lattice, the alignment between the double bonds and pentagons on adjacent Qo molecules cannot be done perfectly, and some misalignment persists down to low temperatures, as discussed in §7.1.4. This issue has been addressed in band calculations, where the highest symmetry compatible with both the icosahedral symmetry of Q0 and the fee lattice (Fm3 or T¿) has been considered [12.5,9]. The effect of orientational disorder on the band structure has been considered by several groups [12.25,93], showing that details in the band structure and density of states are sensitive to orientational disorder [12.94], In particular, it has been shown that orientational disorder, finite resolution effects, and the multiband nature of the bands near EP all lead to a reduction of the predicted dispersion in the angle-resolved photoemission [12.95] consistent with experimental findings [12.96,97].

From the dispersion relations, effective mass estimates for the conduction and valence bands of Qo have been calculated (mle = 1.33m0, m'e - 1.15m0 for the longitudinal and transverse effective electron masses at the Appoint, and m'h = 3.31 m0, m'h — 1.26m0 for the two valence band masses) [12.64], Although one-electron band calculations are not able to explain the details of the optical spectra or the magnitude of the HOMO-LUMO gap (see


Fig. 12.7. Band structure of (a) fee Qo, (b) fee K^C^, and (c) body-centered cubic (bcc) KsQq. The zero energy is either at the top of the valence band or at the Fermi level [12.90], x * i. r WAVE VECTOR


Fig. 12.7. Band structure of (a) fee Qo, (b) fee K^C^, and (c) body-centered cubic (bcc) KsQq. The zero energy is either at the top of the valence band or at the Fermi level [12.90],

§13.1.2), good agreement was obtained between the calculated density of states and photoemission and inverse photoemission measurements (§17.1) [12.7,37,38] with regard to the number of peaks and their relative placements in energy. The good agreement with the photoemission spectra may in part be due to the close relation between the one-electron bands and the molecular levels.

Charge density maps for Qo normal to the (100) direction (see Fig. 12.6), based on an LDA energy band calculation and the Fm3 structure, show almost no charge density at the center of the fullerene (over a diameter of ~4 A) and very little charge density between adjacent Qo molecules, even along the close-packed direction, consistent with the weak coupling between Qo molecules. Even lower charge density is found for crystalline Qo in the direction of the octahedral sites.

Band calculations have placed an octahedral site state at 6.5 eV above the valence band maximum and two levels at 7.0 and 7.2 eV for the tetrahedral site states [12.64], These levels are expected to be significantly perturbed on filling of the states with electrons (see Fig. 12.7). The detailed bandwidths and density of states profiles are sensitive to the details of the structural symmetries and to the disorder [12.75], The narrow bands and weak dispersion found in these calculations of the electronic structure are consistent with the molecular nature of C60.

12.7.3. Band Calculations for Alkali Metal-Doped CM

Local density approximation band calculations have also been extended to treat the electronic energy band structure of doped fullerene solids called fullerides. Most calculations for the doped fullerene compounds also refer to experimental determinations of the lattice constants and structural ge ometry, as was the case for pure C60. For the doped fullerenes, distortions of the icosahedral shell and perturbations of bond lengths and structural geometry due to charge transfer effects may be important in some cases, such as the M3C60 compounds (M = K, Rb) [12.72]. Such effects have been shown to be less important for K^C^ and Rb6C60 [12.67], where the tlu band is completely filled with electrons.

Parallel calculations of the electronic structure for K3C60 and K^Qq [12.65,90] are shown in Fig. 12.7(b) and (c), for the fee and bcc crystal structures, respectively [12.98]. It is shown in this section that the one-electron approach to the electronic structure is capable of explaining a number of key experimental observations. Band calculations show that the cohesive energy between the fullerenes increases upon the addition of an alkali metal. Furthermore, the addition of the alkali metal atom into a tetrahedral site increases the cohesive energy of the fullerene solid more than when the alkali atom is in an octahedral site. Thus alkali metals tend to fill tetrahedral sites before octahedral sites, which in turn have a higher cohesive energy than an endohedral location for an alkali metal atom. These band calculations show that a large charge transfer (close to one electron per alkali metal atom) occurs upon alkali metal addition [12.25,99]. Figure 12.8 shows that the charge transferred to the fullerenes serves to enhance the bonding between fullerenes. The difference between the charge density contours for C60 (Fig. 12.6) and for K3C60 (Fig. 12.8) is striking in terms of the charge coupling between the fullerenes, suggesting that a band picture for K3C60 has merit, and this is also supported by the observation of superconductivity in K3C60. The calculated electronic density of states for M3C60 (M = K, Rb) (see Fig. 12.9) shows a double hump structure with roughly two thirds of the states in the lower-energy hump of Fig. 12.9. Because of the location of the Fermi level on the falling side of the lower-energy hump, it is expected that the various calculations

Fig. 12.8. Contour maps of the valence electron densities of

K^Qo on the (110) plane. Solid circles denote the tetrahedral and octahedral sites at which K atoms are located. The highest-density contour is 0.90 electrons/A3, and each contour represents half or twice the density of the adjacent contour

Fig. 12.8. Contour maps of the valence electron densities of

K^Qo on the (110) plane. Solid circles denote the tetrahedral and octahedral sites at which K atoms are located. The highest-density contour is 0.90 electrons/A3, and each contour represents half or twice the density of the adjacent contour

Fig. 12.9. Calculated density of states for unidirectionally ordered fee Cm. The position of the Fermi level for M^C^ with n = 1,2, and 3 is indicated, where M denotes an alkali metal such as K or Rb [12.75].

of N(Ef) would give relatively high values for N(EF) for M3C60 compounds (6.6-9.8 states/eV C60 spin) [12.66,68,75,100-102], as well as a fairly large spread in values of N(EF). It should be mentioned that the oft-quoted N(EF) = 12.5 states/eV C60 spin [12.25] is somewhat larger than that for other calculations [12.5,72]. Despite the wide spread of values for N(EF) among the various calculations for K3C60, the ratio of N(EF) for Rb,C 60, and K3C60, shows a much smaller spread (1.14-1.27) [12.72], Another source of unreliability in the density of states calculations N(EF) stems from the common neglect of issues relating to merohedral disorder, insofar as the calculations are generally carried out for unidirectionally oriented fullerene molecules [12.72], Tight binding calculations directed at exploring the effect of merohedral disorder on N(EF), however, find little change in the value of N(EF), although the density of states curve N(Ef) is significantly broadened by the effect of merohedral disorder [12.93,103,104], in contrast to GW quasiparticle calculations which show a decrease in N(EF) with merohedral disorder, but little broadening of the -derived band [12.9], Reasons why the tight-binding model works as well as it does for the M3C60 compounds have been attributed to a very weak contribution to the LUMO wave functions and energy eigenvalues by the alkali metal $ orbitals and to the lack of overlap of the LUMO tiu band with higher-lying bands [12.72,75], It is further found that the density of states is insensitive to the dopant (whether K or Rb), except for differences in the lattice constant between I^Qo and Rb3C6o (see §14.5).

In the case of K^o [see Fig. 12.7(c)], the Fermi level is located between the tiu(fiu) and the ilg(/lg)-derived excited state levels, giving rise to a narrow gap semiconductor, in agreement with experiment. Although the connection to the molecular C60 levels is similar for £(k) in the C6(l and

Energy eV

M6C60 crystalline phases, some differences arise because of the bcc crystal structure for the M6C60 compounds (K^Qq, Rb6C60 and Cs6C60). Some interesting features about the M6C60 bands are the calculated reduced band gap (0.8 eV) between the hu and -derived bands and between the tlu and tlg-derived levels (0.3-0.5 eV), as the Fermi level for K^Qq and other M6C60 compounds lies between the tlu and tlg energy bands [12.65,90]. Some calculations of E(k) for MgQ,, give an indirect band gap [12.65], while others [12.90] suggest a direct gap at the P point of the bcc Brillouin zone. In both cases, the bands are broadened versions of the molecular levels, with little hybridization of the carbon-derived bands with the alkali metal levels. Some relaxation of the K ions in KgQo has been suggested, with lattice distortions that do not lower the lattice symmetry [12.67].

The most interesting case for the alkali metal-doped C60 occurs for MjQo, which corresponds to a half-filled band and a maximum in the conductivity cr versus composition x curve for M^Qq (see §14.1.1). The band calculations further show that the Fermi surface for K3CM has both electron and hole orbits [12.81,105] and that the superconducting transition temperature Tc depends approximately linearly on the density of states for alkali metal-doped C60 (see §15.2) [12.106]. To show the relative stability of various KXC60 stoichiometrics, the calculated total energy is plotted as a function of lattice constant in Fig. 12.10. The figure shows the relative stabilities of KXC60, with K3C60 being the most stable (largest binding energy). As we will see below, the situation for the alkaline earth dopants is different because of the filling of the flg bands and the greater hybridization of the alkaline earth levels with the £(k) for the fullerene bands near EF.

Figure 12.11 shows the density of states for Na2CsC60 as an example of an M3C60 compound where the two small sodium atoms are in tetrahe-dral sites, while the larger Cs atom is in an octahedral site, as determined from structural measurements [12.107], This figure is typical of LDA density of states calculations for M3C60 compounds, showing narrow bands with some structure in the density of states N(E). The magnitude of N(EF) = 22.4 states/eV-C60 (including both spin orientations) [12.25,81,106] is large compared to other calculations, N(EF) = 13.2 states/eV/C60 [12.66], indicating a strong sensitivity of N(EF) to the details of the band calculation. The Fermi surface for K3C60 is shown in Fig. 12.12 [12.66], where two in-equivalent sheets are seen. One sheet is a closed spheroidal surface with protrusions in the (111) directions, while the second sheet is a multiply connected surface, reflecting contact of the Fermi surface with the Brillouin zone boundary and showing two (rather than four) necks along the (100) directions, indicative of the (Pa3) lower-symmetry space group.

Fig. 12.10. Calculated total energy of ^Cffl as a function of lattice constant a, normalized to the experimental lattice constant aa of pristine C^. The total energies are measured from the sum of the energies of the isolated Cm cluster and the isolated x potassium atoms. The total energy is the negative of the cohesive energy [12.25].

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