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Fermi surfaces of K3C60 in the ordered Pa3 crystal structure. The multiply connected outer surface has two arms at the Brillouin zone boundary along each of the crystal axes kx,ky, and kz, reflecting twofold symmetry [12.5],

Fig. 12.12. Calculated

Fermi surfaces of K3C60 in the ordered Pa3 crystal structure. The multiply connected outer surface has two arms at the Brillouin zone boundary along each of the crystal axes kx,ky, and kz, reflecting twofold symmetry [12.5],

Fig. 12.12. Calculated

Calculations for the band structure of K4C60 have been carried out [12.130], for several different structural models, but in all cases the calculation finds metallic behavior near EF, inconsistent with experimental measurements showing K4C60 to be semiconducting. Further theoretical work is needed to account for the experimental observations, as discussed in §12.7.7.

12.7.4. Electronic Structure of Alkaline Earth-Doped C60

Band calculations using the local density approximation have also been carried out for the alkaline earth compounds, Ca5C60, BagQo, and Sr6C60 [12.36,44], as well as for Ca3C60 [12.44,63]. The results for the electronic structures for these alkaline earth compounds show three important features, as described below. First, at low doping concentrations each alkaline earth atom generally transfers two electrons to the C60 shell. Second, because of the strong hybridization of the 4s Ca levels with the C60 tlg energy levels, as shown in Fig. 12.13, where the energy levels of a [email protected] cluster are presented, the electrons fill the Ca 4s-derived level with two electrons before the tig levels are occupied. With further doping, the 4s alkaline earth level rises above the tlg levels, and only C60-derived levels are filled. Third, the equilibrium position for one Ca in the octahedral site is offcenter, thereby making room for additional calcium ions in an octahedral site. Moreover, different crystal structures are found for these compounds, with Ca5C60 crystallizing in the fee structure, while Ba6C60 and Sr6C60 both crystallize in the bcc structure (see §8.6). For all three compounds Ca5C60,

Fig. 12.13. Electronic energy levels of a CafSC«, cluster, where the Ca atom is located at the center of the CM cage. The energies are measured from the h„ state, the highest occupied state of the Cw cluster. The tlu state, the lowest unoccupied state of the C^, cluster, is now occupied by two electrons. The ag and t,u states derived mainly from Ca 4s and 4p states, respectively, appear at higher energies than the tiu state [12.25,44].

Ba6C60, and Sr6CM, the tlg-hybridized levels are partly occupied in the solid state, and these are the states responsible for the conductivity and superconductivity in these crystals. For the case of CajC^Q, the hybridization between the three Ca orbitals associated with the octahedral site and the adjacent C60 molecules is strong, and at the stoichiometry Ca5C60, the flu level is totally filled, while the (Ca2-" ion hybridized) tlg level of C5(l is half filled [12.44], Experimental evidence for the partial filling of the tlg level comes from both photoemission spectra [12.108,109] and transport measurements [12.30].

Closely related to the electronic dispersion relations £(k) for Ca5C60, discussed above, are the dispersion relations for SreQo and Ba6C60 in the crystalline phase shown in Fig. 12.14(a) and (c), respectively, and both are compared in Fig. 12.14(b) with the corresponding £(k) for a hypothetical bcc form of CA0, calculated in a similar way [12.36]. Of particular interest, both Ba6C60 and Sr6C60 are semimetals, with equal volumes enclosed by their electron and hole Fermi surfaces [12.44,76]. The semimetallic behavior arises from the hybridization of the tXg valence bands with conduction bands of the alkaline earth. Both Sr6C60 and Ba6C60 have strong hybridization between the molecular-derived C60 levels and the alkaline earth (Sr 4d or Ba 5d) states, which for the M6C60 compounds lie lower than the Sr 55 or Ba 6s valence states for the free atoms. This hybridization induces considerable dispersion in the ilg band near the Fermi level and also in higher bands, as seen in Fig. 12.14(a) and (c). Plots of the contour

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Fig. 12.14. Electronic band structure for (a) Sr6Cal, (b) a hypothetical undoped bcc solid Qo, and (c) BagQ,, [12.25]. Energy is measured from the Fermi level at 0 eV in (a) and (c), which is denoted by the horizontal lines. In (b), the energy bands are shifted downward by 2 eV for the sake of easier comparison to (a) and (c) and each bunched band is labeled according to the corresponding state of the CM cluster.

Fig. 12.14. Electronic band structure for (a) Sr6Cal, (b) a hypothetical undoped bcc solid Qo, and (c) BagQ,, [12.25]. Energy is measured from the Fermi level at 0 eV in (a) and (c), which is denoted by the horizontal lines. In (b), the energy bands are shifted downward by 2 eV for the sake of easier comparison to (a) and (c) and each bunched band is labeled according to the corresponding state of the CM cluster.

maps for Sr6C60 and Ba6C60 show considerable electron charge distribution around the metal sites, indicative of incomplete charge transfer to the C60 and consistent with the strong hybridization between the metal d levels and the t]g level discussed above [12.36]. The semimetallic nature of Ba6C60 can be seen by noting the hole pockets at the N point and the electron pockets at the H point [see Fig. 12.14(c)] [12.36,76]. The density of states at the Fermi level N(EF) is calculated to be lower in Sr6C60 (11.6 states/eV-C50) and Ba6C60 (4.3 states/eV-C60), as compared with that for many of the M, Qq alkali metal compounds, although their superconducting Tc values are 7 K for Ba6C60 and 4 K for Sr6C60 (see §15.1). In contrast to the behavior of the M3C60 alkali metal compounds, there is no clear relation between Tc and N(EF) for the alkaline earth-derived superconducting compounds. Since EP for Sr6C60 and Ba6C60 lies in a broad, strongly hybridized band, the Coulomb repulsion between the superconducting carriers is expected to be smaller than for the M3C60 compounds [12.36], but ¡jl* for Sr6C60 is expected to be larger than /u,* for Ba6C60 based on the more uniform charge distribution calculated for Ba6C60, thereby perhaps accounting for the higher Tc in Ba6C60, despite its lower N(EF) value relative to Sr6C60 [12.130].

One-electron band calculations have significantly influenced progress by experimentalists by providing insights and explanations for observed phenomena, by making predictions for new phenomena, and by suggesting systems of theoretical interest that experimentalists may be able to study, such as alkaline earth-alkali metal alloy compounds (e.g., Ca2CsC60) for study of hybridization effects and their potential as superconductors [12.25]. The calculated density of states for Ca2CsC60 (with doubly charged Ca2+ in each of the tetrahedral sites and Cs in the octahedral site) gives a relatively high value for N(EF), suggesting Ca2CsC60 as an attractive candidate for superconductivity, and since the t^u band in this case could be filled with five out of six electrons, this would be expected to be a hole-type superconductor. Other suggested candidates are the M3M3C60 fee structure where M is an alkaline earth and M' is an alkali metal whereby ~9 electrons would be transferred to C60 to half fill the tlg band, with the alkaline earth ions going into a distorted tetrahedral site surrounded by four pentagon-oriented C60 clusters and the alkali metal going into distorted tetrahedral sites surrounded by four hexagon-oriented C60 clusters. Also, it has been suggested that the study of intermetallic Ba3Sr3C60 might prove interesting with regard to charge transfer issues [12.130].

12.7.5. Band Calculations for Other Doped Fullerenes

One promising band structure calculation is for an intercalation compound of graphite in which a (111) layer of C60 or K4C60 is introduced between the graphite layers to stoichiometrics C60C32 and K4C60C32 [12.110]. In the case of C60C32, a small charge transfer is found from the graphite layer to the C60 intercalate because of the curvature of the C60 molecules which admixes some s state character into the tt orbitals and thus lowers the energy of the C^-derived tt state. The resulting charge transfer stabilizes the Ceo in a symmetric position with respect to the surrounding graphite layers. The coupling between the C60 and graphite layers in this case is weak, so that the energy bands of the isolated constituents are only weakly perturbed. Since all the atoms of this structure are carbon, this graphite intercalation compound is expected to be stable under ambient conditions.

Even more interesting is the case of K^QqC^ [Fig. 12.15(a)], where potassium atoms go into interstitial positions between the C60 and the surrounding graphene sheets. The shortest K-K distance in this model calculation is 5.7 A and the shortest C-K distance to C60 is 3.1 A and to the graphene plane is 3.3 A. In this case the calculations show that complete charge transfer takes place from the K to both the C60 and the graphene sheets. From Fig. 12.15(b), it is seen that the resulting intercalation compound is predicted to be metallic, with a high density of states at the Fermi level [N(EF) = 24.9 states/eV], which is similar in magnitude to that calculated for K3C60 using the same technique. The coupling to the graphite layers gives rise to a graphite conduction band with a large bandwidth that serves to reduce the Coulomb repulsion energy /jl*. The large N(Ef) and small are favorable for high-rc superconductivity. In the graphene layers electrons are attractively coupled by vibrations associated with molecular-type C60 vibrations. A rough cohesive energy calculation indicates that C60C32 should be stabilized by a very weak attractive van

Large open spheres and small decorated spheres denote Cm clusters and K ions, respectively. (b) Electronic band structure of the K-doped C^,-graphite cointercalation compound, KjCa^j, with a c-axis repeat distance c = 12.530 A. The energy is measured from the Fermi level of the system, which is at the anticrossing point of the CM tlu conduction band and the graphite conduction band [12.110].

der Waals interaction, while K4Q0C32 is further stabilized by electrostatic forces [12.110], If K4C60C32 were found to be superconducting, a whole class of materials, obtained by varying either the dopant (K) or the number of graphene sheets (stage) would become of interest.

Band calculations for halogen doping of CM have also been carried out, showing a cohesive energy of ~2-3 eV for solid C60Br, and indicating that bromine doping should be possible [12.111], Experimental observation of the formation of brominated C60 supports these conclusions (see §10.5). However, because of the high electronegativity of C60, it is not likely that holes would be created in the //„-derived C60 HOMO level, also in agreement with experiment [12.112]. The calculations for Br on a tetrahedral site or on an octahedral site both show that the Fermi level lies in the Br 4p state, which lies between the hlu HOMO and tXu LUMO levels, thereby indicating that no charge transfer occurs, and a large charge accumulation takes place near the Br site [12.113]. Similar results were obtained for C60C1 [12.130].

A local density approximation calculation for a BC59 molecule has been carried out [12.113] showing the fivefold degenerate hu band to bunch into four lower-lying bands and one higher-lying band which is half empty, thereby indicating hole doping with EF ~ 0.2 eV below the top of the valence band (see Fig. 12.16). The possibility of formation of substitutionally doped BC59 has been demonstrated experimentally in mass spectra [12.114],

level calculated for solid BC59 [12.113].

but this species has been neither isolated nor purified, so that no measurements on the physical properties are yet available. The results of the band calculation suggest that BC59 might be an interesting candidate for a hole-doped fullerene system.

12.7.6. Band Calculations for Other Fullerenes

Regarding smaller fullerenes, the electronic structure for a solid constructed from a C28 molecule with Td symmetry has been calculated [12.115,116], and the electronic structure in this case was found to be semiconducting. Other calculations have also been reported for small fullerenes [12.25].

With measurements now becoming available for higher fullerenes, including NMR experiments on Q4 [12.117], photoemission spectroscopy results on C76 and C84 [12.118-120], scanning tunneling microscopy on C76, Cgo, C82, and C84 [12.121,122], and photoabsorption spectroscopy on C76 and C84 [12.117,123], there is great interest in determining the electronic band structure and density of states for such higher-mass fullerenes. There is particular interest in identifying the most probable isomers for each C„c; such information would also provide insights into the growth model for fullerenes.

Using a tight-binding and an LDA approach, the electronic structure and density of states for several of the C76, Cg2 and C84 isomers have been calculated [12.124,125]. For C76, the electronic structure and density of states (DOS) of the D2 isomer have been calculated using the tight binding model [12.126,127] and by a local density approximation [12.128], yielding good agreement with the experimental photoemission spectrum [12.118], including consideration of both occupied and unoccupied bands. The DOS of Td C76 has also been calculated [12.129]. For C84, tight binding [12.125,130]

and LDA [12.131] electronic structure and density of states calculations have been reported on four D2 candidates for the major C84 isomer, as well as on the minor Dld Cg4 isomer. The density of states for three possible candidates for the major C2 isomer of Cg2 has also been calculated [12.129]. Comparison has also been made between measured properties (such as photoemission and inverse photoemission spectra [12.118,132,133]) and model calculations, using both the LDA and tight-binding methods for finding distributions N(E) in the density of states. Such studies have already provided sensitive identifications for the most likely isomers for C76, C82, and C84 [12.25]. As more experimental data on higher fullerenes become available we can expect more detailed calculations to be made on the various isomers of interest.

In the limit that the number of carbon atoms on a fullerene becomes very large, the electronic level picture for such a large molecule approaches that of a graphene sheet where each carbon atom is in an sp2 a bonding configuration. In this limit, the ir electron states are separated into an occupied bonding it band and an empty antibonding 7r band with no band gap separating these two bands, and electron-electron correlation effects are not expected to be of importance.

12.7.7. Many-Body Approach to Solid C60

Many-body treatments of the electronic structure of crystalline C60 have been strongly stimulated by measurements of the luminescence and absorption phenomena associated with the optical absorption edge in solution and in the solid phase (see §13.2 and §13.3). On this basis, considerable experimental evidence has been demonstrated for the presence of strong electron-electron intramolecular Coulomb interactions [12.73,134-138]. In addition, experimental determinations of the magnitude of the HOMO-LUMO band gap through photoemission studies have significantly exceeded LDA calculations, and furthermore experimental angle-resolved photoemission data show very little dispersion.

These observations have stimulated theoretical work in two main directions [12.33,139-147]. Along one approach, efforts have been made to correct the LDA underestimation of the energy gap in solid C60 by properly treating the electron excitations with an ab initio quasiparticle approach. Work along these lines has been done by Shirley and Louie [12.9,89]. In their GW approach, the Dyson equation is solved within a Greens ftinc-tion (G) formalism using a dynamically screened Coulomb interaction W to obtain the quasiparticle energy spectrum. The unperturbed LDA wave function and eigenvalues used as a basis set are in good agreement with other LDA calculations [12.64,99,100,148]. A large increase in the calcu lated energy gap (from 1.04 to 2.15 eV) is found using this GW quasi-particle approach yielding better agreement with photoemission data (see Fig. 12.17), as measured by the energy differences for the onsets of the photoemission and inverse photoemission spectra (see §17.1.1). Reasonably good agreement was also obtained between the quasiparticle GW calculations (3.0 eV) and the peak-to-peak energy in the photoemission and inverse photoemission spectra [12.9]. Finally, the GW quasiparticle technique was used to calculate the angle-resolved photoemission spectra along TM and TK for the hu, tiu,'t2u, tXg, and t2g bands, thereby yielding an explanation for the very weak dispersion found experimentally for these energy bands [12.96,149]. This calculational technique has also been very useful for predicting the effect of merohedral disorder on the observed density of states (see Fig. 12.18) and on the angle-resolved photoemission spectra [12.9], In Fig. 12.18 we see a comparison between the density of states calculated by the quasiparticle GW technique for an orientationally ordered Fm3 structure (the highest symmetry compatible with icosahedra on a cubic fee lattice), an ordered version of the Pa3 structure (which is observed experimentally in C60 below Tm), and the merohedrally disordered Pa3 structure. A comparison of these densities of states shows that merohedral disorder washes out structure in the tlu density of states without

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