First-principles study of superconductivity and Fermi-surface nesting in ultrahard transition metal carbides

Using a recently developed first-principles approach, we show that the variation in the superconducting behavior of group IVb and Vb transition metal carbides is associated with a significant nesting of the Fermi surfaces in group Vb compounds, while their phonon anomalies and density of states at the Fermi level play a minor role. The superconducting pairing arises from the coupling of metal $d$ states to acoustic phonons, and is therefore at variance with the interaction leading to the exceptional mechanical hardness where carbon $p$ states play a substantial role. We provide insight into how to optimize the transition temperature by varying the Fermi surface properties through substitutional doping.

Figure 1

(Color online) Top: Fermi surfaces of (a) $\mathrm{TaC}$ and (b) $\mathrm{HfC}$. Middle: phonon dispersions of (e) $\mathrm{TaC}$ and (f) $\mathrm{HfC}$ (solid lines), together with the experimental data of Ref. 23 (circles). The dashed lines in (d) correspond to the dispersions of $\mathrm{TaC}$ after rescaling the Ta mass to the Hf value. The arrows indicate the wave vectors exhibiting Kohn anomalies. Bottom: Nesting functions of (e) $\mathrm{TaC}$ and (f) $\mathrm{HfC}$ (arbitrary units).

Figure 2

(Color online) Intensity plot of the electron-phonon coupling strength $\lambda \left(\mathbf{q}\right)$ in the $\mathbf{q}=(q_x,q_y,0)$ plane of the Brillouin zone. Top: average electron-phonon coupling strength in $\mathrm{TaC}$ for acoustic (a) and optical (b) modes. Bottom: average coupling strength in $\mathrm{HfC}$ for acoustic (c) and optical (d) modes.

Figure 3

(Color online) (a) Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ for $\mathrm{TaC}$ (solid) and $\mathrm{HfC}$ (dashed). We show the partial contributions to the electron-phonon coupling strength $\lambda$ next to each curve. (b) The corresponding vibrational density of states $F\left(\omega \right)$ in units of modes per meV per unit cell.

Figure 4

(Color online) (a) Wannier-projected density of states for $\mathrm{TaC}$ and $\mathrm{HfC}$: $\mathrm{C}\text{−}2p$ (dashed) and metal-$5d$ (dotted) partial density of states, and total density of states (solid). The vertical lines denote the Fermi levels of $\mathrm{HfC}$ and $\mathrm{TaC}$. (b) Phonon softening vs doping in $\mathrm{HfC}$ (dashed) and $\mathrm{TaC}$ (solid): shown is the doping-dependent frequency of the softest mode for each material. (c) Electron-phonon coupling $\lambda$ as a function of electron concentration for $\mathrm{HfC}$ (dashed) and $\mathrm{TaC}$ (solid).