Upper limit for superconducting transition temperature in electron-phonon superconductors: very strong coupling

TL;DR

This work analyzes the maximum superconducting transition temperature $T_c$ achievable by electron-phonon mechanisms within the Eliashberg-McMillan framework under very strong coupling. It shows that in the strong-coupling limit, $T_c$ scales as $T_c \approx 0.182\sqrt{\lambda\langle\Omega^2\rangle}$, replacing the exponential BCS behavior with a square-root dependence, while phonon renormalization and lattice stability impose practical caps (e.g., $T_c \lesssim 0.13\Omega_0$ for Einstein phonons). The paper introduces a material-bound upper limit $T_c^{\star}=0.18\sqrt{\dfrac{N(0)\langle I^2\rangle}{M}}$, highlighting a fundamental-constants-based ceiling that still depends on electronic structure. It also discusses potential limitations from Fröhlich instabilities and specific-heat considerations, and critically assesses metallic hydrogen via the jellium model, showing that weak-coupling pictures underestimate $T_c$ and that strong coupling with phonon softening is essential to reach high $T_c$ values observed in hydrides. Overall, Eliashberg theory remains robust for conventional electron-phonon superconductors, and the results provide practical guidance for estimating maximal $T_c$ and guiding experimental searches, particularly in hydrides and metallic hydrogen.

Abstract

We present a brief review of some recent work on the problem of highest achievable temperature of superconducting transition $T_c$ in electron-phonon systems. The discovery of record-breaking values of $T_c$ in quite a number of hydrides under high pressure was an impressive demonstration of capabilities of electron-phonon mechanism of Cooper pairing. This lead to an increased interest on possible limitations of Eliashberg-McMillan theory as the main theory of superconductivity in a system of electrons and phonons. We shall consider some basic conclusions following from this theory and present some remarks on the limit of very strong electron-phonon coupling. We shall discuss possible limitations on the value of the coupling constant related to possible lattice and specific heat instability and conclude that within the stable metallic phase the effective pairing constant may acquire very large values. We discuss some bounds for $T_c$ derived in the strong coupling limit and propose an elementary estimate of an upper limit for $T_c$, expressed via combination of fundamental physical constants. Finally we also briefly discuss some pessimistic estimates for $T_c$ of metallic hydrogen obtained in ``jellium'' model.

Figures (4)

• Figure 1: Temperature of superconducting transition in Einstein model of phonon spectrum in units of $T_c/\Omega_0$ as a function of pairing constant $\lambda$AD: 1 -- lower bound (\ref{['TcAD']}), 2 -- numerically exact solution of the full system of equations AD. McMillan expression for $T_c$McM is shown by dashed line (for the case of $\mu^{\star}$=0).
• Figure 2: Dyson equation for the full ("dressed") phonon Green's function.
• Figure 3: Experimental values of the temperature of superconducting transition for conventional superconductors dependence on their Debye temperature $\Theta_D$EKS.
• Figure 4: Superconducting transition temperature of metallic hydrogen in "jellium" model calculated from the weak -- coupling BCS expression. At the insert: the same calculated from McMillan formula for intermediate coupling.