First-Principles Prediction of Doped Graphane as a High-Temperature Electron-Phonon Superconductor

We predict by first-principles calculations that $p$-doped graphane is an electron-phonon superconductor with a critical temperature above the boiling point of liquid nitrogen. The unique strength of the chemical bonds between carbon atoms and the large density of electronic states at the Fermi energy arising from the reduced dimensionality give rise to a giant Kohn anomaly in the optical phonon dispersions and push the superconducting critical temperature above 90 K. As evidence of graphane was recently reported, and doping of related materials such as graphene, diamond, and carbon nanostructures is well established, superconducting graphane may be feasible.

Figure 1

(a) EDOS per carbon atom in 1D (diamond nanowire), 2D (graphane), and 3D (diamond). (b) EDOS of pristine (solid black line) and 12.5% $p$-doped graphane (dashed red line, $2\times 2$ supercell with one substitutional B). The top of the valence band is set as zero, and $E_F=-0.96\mathrm{eV}$ (green line). The EDOS at $E_F$ is similar in the two models ($0.26\mathrm{\text{states}}/\mathrm{eV}/\mathrm{\text{cell}}$ in rigid band; $0.27\mathrm{\text{states}}/\mathrm{eV}/\mathrm{\text{cell}}$ in supercell). We expect this to hold also at lower doping, where the perturbation to the pristine dispersions is smaller. The supercell calculation has no impurity states in the gap.

Figure 2

Optical phonon dispersions of pristine (solid black line) and 1% $p$-doped graphane (dashed blue lines). The arrows indicate the average Fermi surface diameter, with the Kohn anomalies (see Ref. 25 for mode assignments). Calculations taking B explicitly into account [16] or with nonadiabatic corrections [39] may slightly revise the softening. However, this stands out as a qualitative effect.

Figure 3

(a) PDOS of pristine and $p$-doped graphane. (b) Eliashberg function for several dopings. We use Gaussians of width 2 meV to represent the Dirac deltas in the PDOS and Eliashberg functions. The largest EPC contribution arises from optical phonons, as in diamond [15], but also acoustic phonons couple to holes at the Fermi surface. The hashed region for 1% doping shows the TO stretching contributions to the Eliashberg function. (Inset) Ball-and-stick representations of two TO C-C stretching modes. The arrows indicate the displacement patterns (C atoms gray, H white). The in-plane C-C stretching modes with C and H out-of-phase do not contribute to EPC because, upon softening, the planar modes hybridize so that those at $715{\mathrm{cm}}^{-1}$ have an increased weight on C atoms, the opposite of those at $1257{\mathrm{cm}}^{-1}$.

Figure 4

(a) Total EPC in graphane and ${\mathrm{MgB}}_2$ (solid red line [15]), ${\mathrm{CaC}}_6$ (dashed green line [40]), diamond (solid black line [15]; triangles [16]), and graphene (dashed black line [41]). Note that EPC in graphene is small because the C atoms are $s{p}^2$ and the states are $\pi$ or ${\pi }^{\star }$ combinations of $p_z$, not coupling as much to in-plane C-C stretching [41]. (b) $T_c$ from the modified McMillan formula with Coulomb pseudopotential ${\mu }^{*}=0.13$ [30]. The vertical black line indicates the doping corresponding to the estimated MIT. Below it, our formalism applies to charge transfer or gate-induced doping, while above it also applies to substitutional doping. For comparison, we show $T_c$ of ${\mathrm{MgB}}_2$ (solid red line, $T_c=39\mathrm{K}$ [1]), ${\mathrm{CaC}}_6$ (dashed green line, $T_c=11.5\mathrm{K}$ [42]), and diamond (solid black line, $T_c=4\mathrm{K}$ at $\sim 2.8\%$ B [8]; 5 K at $\sim 3.7\%$ B [43]). Taking explicitly into account a substitutional dopant, such as B, could slightly change the EPCs [16]. However, in B-doped diamond a rigid-band model provides lower EPC and lower bound to $T_c$ [16]. Here we use the isotropic Eliashberg formalism [30]. Anisotropy may increase $T_c$ [10]. (Inset) Ball-and-stick models of B-doped diamond, ${\mathrm{CaC}}_6$, ${\mathrm{MgB}}_2$, and B-doped graphane. C, black; H, blue; B, red; Mg, light green; Ca, green.