In their article “The quest for room-temperature superconductivity in hydrides” (Physics Today, May 2019, page 52), Warren Pickett and Mikhail Eremets commented that “in the late 1960s William McMillan of Bell Labs extended the [Bardeen-Cooper-Schrieffer] analysis to moderately strong coupling,” which is measured by the electron–phonon coupling constant, λ. According to Pickett and Eremets, the McMillan “equation for Tc was extrapolated beyond its regime of validity to fortify claims that 30 K would be the upper limit for electron–phonon coupling.”
The above comment may not be entirely fair if its subject is the analysis McMillan made in a 1968 article,1 in which he doesn’t mention 30 K as a possible maximum value of Tc but does list 9.2, 22, 28, and 40 K as possible maximums. None of those temperatures are the upper limit of electron–phonon coupling in general. Rather, they are upper limits of Tc in classes of materials represented by lead, niobium, and niobium–tin and vanadium–silicon alloys, and they have not exceeded the regime of validity of the McMillan equation. In particular, McMillan does not exclude higher Tc in other classes, provided that λ does not exceed 2 in his equation.
Specifically, McMillan realizes that Tc from his equation declines when, on average, the phonon frequency becomes either too large or too small and searching for maximum Tc leads to λ = 2. Since in 1968 it was believed that Tc = 7.2 K and λ = 1.3 in Pb, McMillan concludes that Tc may reach 9.2 K in a Pb alloy when λ = 2.8. In that case, Tc was found numerically and therefore was not subject to the λ < 2 limit. Had, say, McMillan found Tc = 203 K with λ = 1.3 from a material in his day, he likely would have concluded that Tc could be higher still in a similar material with λ = 2.8.
In recent work,2 we extended the McMillan equation for 0.6 < λ < 2.67. We found that the original McMillan equation is indeed highly accurate if λ < 2. We also predicted that Tc can reach ~44 K in a beryllium–lead alloy, when the Be to Pb ratio is 0.58 to 0.42 (λ = 1 and Debye temperature is 871 K). Our result may be useful to experimenters because it not only shows that Tc may be high in a class of alloys, but it also gives the exact composition of the alloy, hopefully without extreme pressure.
