Electron-phonon coupling in halogen-doped carbon clathrates from first principles

Electronic structure and electron-phonon coupling of carbon clathrates ${\mathrm{C}}_{40}$, ${\mathrm{C}}_{34}$, and ${\mathrm{C}}_{46}$ doped with interstitial halogen atoms (fluorine, iodine) in the fullerenic cages of the host, have been investigated from first principles. It turns out that, while in ${\mathrm{C}}_{40}$ and ${\mathrm{C}}_{46}$ the halogen is only partially reduced, in ${\mathrm{FC}}_{34}$ an electron is transferred to the guest ion by injecting holes in the valence band of the host. The calculated electron-phonon coupling constant is as large as $\lambda =0.85$. We propose that halogen-doped carbon clathrates might represent another form of tetrahedral carbon, in addition to the recently found B-doped diamond, suitable to turn superconductive at high temperature ($T_c\sim 77\mathrm{K}$ from McMillan formula).

Figure 1

Electronic energy bands of (a) pure ${\mathrm{C}}_{40}$ and (b) ${\mathrm{FC}}_{40}$. The zero of energy is the Fermi level of ${\mathrm{FC}}_{40}$. The electronic (indirect) gap of pure ${\mathrm{C}}_{40}$ is $3.6\mathrm{eV}$. The top of the valence band at the $\Gamma$ point is aligned for the two systems. The band structure is reported along the high symmetry lines of the irreducible Brillouin zone following the notation of Ref. 35. The size of the empty dots in panel (b) represents the fluorinelike character, whose maximum value is 77%, as obtained by projecting the Kohn-Sham orbitals on atomic pseudo-wave-functions (with completeness 0.99). (c) The density of state (DOS) of ${\mathrm{FC}}_{40}$ computed by the tetrahedron method with 793 points in the irreducible Brillouin zone. (d) The DOS projected on the atomic pseudo-wave-functions of all atoms [$N_T\left(E\right)$ thicker line] and of the fluorine atom only [$N_F\left(E\right)$ thinner line]. The projected DOS are obtained from the electronic bands on the same $k$-point grid used for the total DOS, smeared with Gaussian functions $0.03\mathrm{eV}$ wide.

Figure 2

Electronic energy bands of (a) pure ${\mathrm{C}}_{46}$ and (b) ${\mathrm{FC}}_{46}$. The zero of energy is the Fermi level of ${\mathrm{FC}}_{46}$. The electronic (direct) gap of pure ${\mathrm{C}}_{46}$ is $4.1\mathrm{eV}$. The top of the host valence band at the $R$ point is aligned for the two systems. The size of the empty dots represents the fluorinelike character, whose maximum value is 70% (with completeness 0.99). (c) The density of state (DOS) of ${\mathrm{FC}}_{46}$ computed by the tetrahedron method with 726 points in the irreducible Brillouin zone. (d) The DOS projected on the atomic pseudo-wave-functions of all atoms [$N_T\left(E\right)$ thicker line] and of the fluorine atom only [$N_F\left(E\right)$ thinner line]. The projected DOS are obtained from the electronic bands on the same $k$-point grid used for the total DOS, smeared with Gaussian functions $0.03\mathrm{eV}$ wide.

Figure 3

Structure of ${\mathrm{FC}}_{34}$: two cages (${\mathrm{C}}_{20}$ and ${\mathrm{C}}_{28}$) of the clathrate structure are shown. The larger cage contains a fluorine ion at the center (larger and lighter sphere).

Figure 4

Electronic energy bands of (a) pure ${\mathrm{C}}_{34}$ and (b) ${\mathrm{FC}}_{34}$. The zero of energy is the Fermi level of ${\mathrm{FC}}_{34}$. The electronic (indirect) gap of pure ${\mathrm{C}}_{34}$ is $3.9\mathrm{eV}$. The top of the valence band at the $\Gamma$ point is aligned for the two systems. The size of the empty dots represents the fluorinelike character, whose maximum value is 77% (with completeness 0.99). (c) The density of state of ${\mathrm{FC}}_{34}$ computed by the tetrahedron method with 897 points in the irreducible Brillouin zone. (d) The DOS projected on the atomic pseudo-wave-functions of all atoms [$N_T\left(E\right)$ thicker line] and of the fluorine atom only [$N_F\left(E\right)$ thinner line]. The projected DOS are obtained from the electronic bands on the same $k$-point grid used for the total DOS, smeared with Gaussian functions $0.03\mathrm{eV}$ wide.

Figure 5

Superconducting critical temperature of ${\mathrm{FC}}_{34}$ as a function of the Coulomb pseudopotential ${\mu }^{*}$ obtained from the McMillan formula [Eq. (2)] and using $\lambda =0.85$ and ${\omega }_{ln}=1023{\mathrm{cm}}^{-1}$ (see text).

Figure 6

Electronic energy bands of (a) ${\mathrm{C}}_{34}$ and (b) $F_2{\mathrm{C}}_{34}$. The zero of energy is the Fermi level. The top of the valence bands of the host at the $\Gamma$ point is aligned for the two systems. The size of the empty dots represents the size of the projection of the bands on the fluorine orbitals whose maximum value is 90% with completeness 0.99.

Figure 7

Electronic energy bands of ${\mathrm{BC}}_{33}$ with substitutional boron in positions $8a$ (a), $32e$ (b), and $96g$ (c). The zero of energy is the Fermi level. The size of the empty dots represents the size of the projection of the bands on the orbitals of boron and of the neighboring four carbon atoms which contribute to at most 62%, with completeness 0.99.