Empirical rule to reconcile Bardeen–Cooper–Schrieffer theory with electron–phonon interaction in normal state

Abstract

We introduce a simple empirical rule wherein the pairing interaction in superconductors is cancelled when normal and umklapp phonon scattering coexist. Superconductivity then arises solely from the residual umklapp contribution. As a result the deduced electron-phonon interactions in niobium, tantalum, lead and aluminum become virtually identical in the normal and superconducting states. Transition temperatures calculated under the rule are accurate within a few per cent when compared with experimental data. Features of the Matthias relations are also explained. The high T c so far predicted for metallic hydrogen is probably overly optimistic.

BCS theory has to be overhauled: Reassurance from numerical survival rate

“…While doing so we retain the exact strength of the electronphonon interaction found in the normal state without change. Previously, in contrast, we merely demonstrated its consistency with the interaction strength in the superconducting state [3,4].…”

“…However, uncomfortable signs of strain, both conceptual and numerical, have kept emerging to question the validity of the theory [2]. The problem conceptually is that the theory doubly and contradictorily defines electron pair occupancy probabilities when the electrons from normal and umklapp scattering processes compete for the same destination state [3,4]. Numerically, attempts to calculate superconducting properties with the BCS theory from first principles have always been to varying degrees less than successful.…”

“…Now the numerical value of T c is reduced when reasonable values of ρ(T ) are implemented. It turns out that the introduction of the empirical S(q) settles the BCS numerical issue in the cases of lead, aluminium, niobium and tantalum with high accuracy [3,4] as it does too in tungsten, iridium, molybdenum and vanadium [19,20].…”

Comparison of Heine–Abarenkov and alternative pseudopotentials for electron–phonon interaction in aluminium, lead, lithium and calcium

“…Should we adopt h(k ) or should we adopt u(k )? That is the dilemma underlying the current issue [25].…”

“…In the original BCS theory the above dilemma manifests itself through a ground state electron wavefunction that cannot be normalised [25]. Keeping this in mind we have devised an electron-pairing rule in the BCS theory that allows a consistent pseudopotential to describe the electron-phonon interaction in both the normal and superconducting states [21].…”