Two Bounds on the Maximum Phonon-Mediated Superconducting Transition Temperature

TL;DR

This work derives two fundamental bounds on the superconducting transition temperature $T_c$ for conventional phonon-mediated superconductors within Eliashberg theory: a coupling-bound set by the total available electron-phonon interaction, encapsulated in an energy-dependent generalized McMillan-Hopfield parameter ${\widetilde{\eta}}(E)$, and a lattice-stability bound arising from phonon softening under strong electron-phonon coupling. It introduces a local, anisotropy-aware framework using ${\widetilde{\lambda}}\langle{\widetilde{\omega}}^2\rangle$ and an energy-resolved ${\widetilde{\eta}}_i(E)$ computed via supercell methods, enabling rigid-band doping analyses and prediction of maximal $T_c$ without detailed phonon spectra. The authors derive analytical upper and lower bounds on $T_c$ within the strong-coupling regime, incorporating a finite phonon-frequency cutoff and an empirically fitted function $f$, and show how anisotropy can modify these bounds. Applying the framework to covalent metals such as MgB$_2$, Li$_{1-x}$BC, and boron-doped diamond reveals that MgB$_2$ is near its coupling- and stability-limited $T_c$, while hole-doped LiBC and related boron-rich compounds could, in principle, reach substantially higher $T_c$ if synthesis and stability permit. Overall, the paper provides a practical, energy-resolved methodology to assess and guide the search for higher-$T_c$ conventional superconductors within fundamental physical bounds.

Abstract

Two simple bounds on the $T_c$ of conventional, phonon-mediated superconductors are derived within the framework of Eliashberg theory in the strong coupling regime. The first bound is set by the total electron-phonon coupling available within a material given the hypothetical ability to arbitrarily dope the material. This bound is studied by deriving a generalization of the McMillan-Hopfield parameter, $\widetildeη(E)$, which measures the strength of electron-phonon coupling including anisotropy effects and rigid-band doping of the Fermi level to $E$. The second bound is set by the softening of phonons to instability due to strong electron-phonon coupling with electrons at the Fermi level. We apply these bounds to some covalent superconductors including MgB$_2$, where $T_c$ reaches the first bound, and boron-doped diamond, which is far from its bounds.

Figures (10)

• Figure 1: Comparison of $T_c$ upper and lower bounds (assuming $\omega_{max} = 1.5 \sqrt{\langle \omega^2 \rangle}$) with the McMillan formula for an Einstein spectrum of frequency $\omega_{ph}$ with a varying electron-phonon coupling strength $\lambda$ and $\mu^* = 0$.
• Figure 2: Density of states (DOS) (dashed line) and $\widetilde{\eta}(E)$ (solid line) for diamond. This calculation is performed on one unit cell of diamond with a $24\times24\times24$ k-grid and 0.03 Rydberg gaussian smearing.
• Figure 3: Density of states (dashed line) and $\widetilde{\eta}(E)$ (solid line) for silicon in the diamond structure. This calculation is performed on one unit cell with a $24\times24\times24$ k-grid and 0.015 Rydberg gaussian smearing.
• Figure 4: Density of states (dashed line) and $\widetilde{\eta}(E)$ (solid line) for cubic boron nitride. This calculation is performed on one unit cell with a $24\times24\times24$ k-grid and 0.03 Rydberg gaussian smearing. In a single unit cell approximation, $\widetilde{\eta}_i(E)$ is identical for boron and nitrogen.
• Figure 5: Density of states (dashed line) and $\widetilde{\eta}(E)$ (solid line) for graphite. This calculation is performed on one unit cell of AA-stacked graphite with a $24\times24\times12$ k-grid and 0.03 Rydberg gaussian smearing. Only in-plane motions of the carbon atoms are considered; the out of plane motions do not contribute significantly to $\widetilde{\eta}(E)$.