ABSTRACT
Carbon is known to have several allotropes such as graphite, graphene, diamond, fullerenes, and nanotubes. Each has its own unique characteristics and the application of these materials has been one of the key topics in material science. Although the carbon materials themselves are semiconducting or metallic with low density of states at the Fermi level, they are considered to be good candidates for superconductivity because of high Debye frequency, and thus, it has been attempted to dope carriers in these materials. In fact, several studies show superconductivity in carrier-doped carbon materials such as fullerene compounds [1-3] and intercalated graphite compounds [4,5], where the carrier doping is realized by intercalation of alkali-metal and/or alkaline-earth-metal. On the other hand, diamond was found to become a superconductor by boron doping [6]. In these materials, superconductivity is considered
to be the conventional BCS-type and the transition temperatures are roughly scaled by the Fermi-level density of states as well as the electron-phonon coupling. In addition to these materials, recently, indications of superconductivity in carbon nanotubes (CNTs) have been observed [7-10], and carriers seem to have been introduced by boron doping in some cases [9,10].To study superconductivity in carbon-related materials, density functional theory (DFT) provides a powerful tool to predict geometric and electronic properties from first principles. In fact, DFT approach works fairly well and gives useful predictions for these materials. In this chapter, we show results for alkali-doped fullerene compounds to present how the DFT contributes to this field and discuss the possibility of superconductivity in carrier-doped CNTs. 4.1 Alkali-Doped Fullerene CompoundsThe macroscopic production of C60 fullerenes and the discovery of solid C60 [11] have stimulated a significant amount of work on applications of C60. One of the most interesting application is superconductivity of alkali or alkaline earth-doped fullerene compounds, AxC60 (A = K, Rb, etc.) [1,2]. Although these alkali or alkaline earth fullerides have various lattice structures depending on the kind of and the number of intercalated atoms, face-centered cubic (fcc) A3C60 fullerides have received the most attention because of high transition temperatures until the discovery of A15 Cs3C60 [3]. 4.1.1 Fcc A3C60For the fcc A3C60, the transition temperature TC is known to be a monotonic function of the fcc lattice constant, which is longer for heavier alkali elements and can also be controlled by pressure. In the theoretical viewpoint, it is known that this relation can be understood in terms of the Fermi-level density of states, D(EF), as follows. In electronic structure studies, it has been revealed that the conduction band, which is composed of the lowest unoccupied three-degenerated t1u states of the C60 molecule, is half-filled by three electrons per unit cell which are transferred from alkali valence states as shown in Fig. 4.1. Because the size of C60 is the same in these A3C60, the lattice expansion gives smaller spatial overlap of the
adjacent t1u states and, therefore, a narrower t1u band, which leads to a larger D(EF). On the other hand, the transition temperature is usually estimated using McMillan’s formula [12,13], as TC = − +− + ω λλ µ λlog. exp . ( )*( . ) .1 2 1 04 11 0 62 Here, λ, µ*, and ω log are electron-phonon coupling constant, Coulomb pseudopotential, and typical phonon frequency, respectively. In this system, important phonon modes are considered to be intramolecular modes and using this fact, λ can be expressed as λ = D(EF)V, where V is the electron-phonon coupling parameter and is independent of the lattice constant [14]. Though there is a controversy in estimating µ* [15], several first-principles studies have shown that V is sufficiently large to explain the experimental transition temperature [16-19]. Therefore, a larger lattice constant leads to a larger D(EF), a larger λ, and a higher TC. According to the above discussion, fcc Cs3C60 was considered to have the highest TC, though the highest TC in the fcc A3C60 was 33 K in Cs2RbC60 [2]. Note that in the DFT study, A15 Cs3C60 was found to be more stable than fcc Cs3C60 [20].
