Electron-Phonon Interaction via Electronic and Lattice Wannier Functions: Superconductivity in Boron-Doped Diamond Reexamined

We present a first-principles technique for investigating the electron-phonon interaction with millions of $k$ points in the Brillouin zone, which exploits the spatial localization of electronic and lattice Wannier functions. We demonstrate the effectiveness of our technique by elucidating the phonon mechanism responsible for superconductivity in boron-doped diamond. Our calculated phonon self-energy and Eliashberg spectral function show that superconductivity cannot be explained without taking into account the finite-wave-vector Fourier components of the vibrational modes introduced by boron, as well as the breaking of the diamond crystal periodicity induced by doping.

Figure 1

Phonon dispersions along the high-symmetry directions of the diamond BZ: VC model (red dashed line) and SC model (blue solid line, from the local maxima of the spectrum in the inset). The modes with $q<2k_F$ are shaded. Inset: intensity map obtained by unfolding the SC vibrational eigenmodes in the BZ of bulk diamond.

Figure 2

Intensity map $\Gamma (\mathbf{q},\omega )$ of the phonon full widths calculated as $\Gamma (\mathbf{q},\omega )=\sum _{\nu }{\Gamma }_{\mathbf{q}\nu }\delta (\omega -{\omega }_{\mathbf{q}\nu })$. Top: two-dimensional map of the vibrational eigenmodes with full widths $>1\mathrm{meV}$ (blue disks; the red lines are the phonon dispersions from Fig. 1). Bottom: three-dimensional view of $\Gamma (\mathbf{q},\omega )$ vs $\mathbf{q}$ and $\omega$ (Gaussian broadening of 1 meV). The left (right) panels refer to the VC (SC) model.

Figure 3

Eliashberg function calculated for the SC model (blue solid line), the ${\mathrm{SC}}_{\Gamma }$ model (red dashed line), and the VC model (black dotted line). Inset: cumulative integral of the Eliashberg function ${\lambda }^{<}(\omega )\equiv 2{\int }_0^{\omega }{\alpha }^{2}F({\omega }^{\prime })/{\omega }^{\prime }d{\omega }^{\prime }$ showing the contribution of the acoustic modes to $\lambda$ (about 10% and 20% for the VC and the SC model, respectively).