Notes on superconducting hydrides

Pickett and Eremets reply: X. H. Zheng and J. X. Zheng focus on McMillan’s classic 1968 paper to address the decades-studied but unresolved question of maximum T_c. The last short section of his paper was on issues of maximum T_c. Though he carefully stated that his equation for T_c “was derived for λ ≲ 1,” he nevertheless extrapolated from it to consider possibilities for higher, and maximum, T_c. He recognized that λ and ω (the coupling strength and characteristic frequency) were coupled via the relation λ = η/Mω² in terms of the McMillan–Hopfield electronic stiffness η (more often referred to as the Hopfield parameter), which indicates that λ and ω are strongly intertwined. Within a class of similar materials, it was conjectured, η might be considered to change very little, so one might consider T_c ( η,λ) without explicit dependence on ω. Conversely, one might consider T_c (η,ω). That approximation of constant η has been found, over the years, to be poor in several classes of materials, including hydrides. ¹

But having supposed that, McMillan reported that extrapolation of his equation outside the range of derivation indicated a broad maximum around λ = 2, or ω² = η/2M. Studies conducted a few years after McMillan’s, by Philip Allen and Robert Dynes, ² established rigorous results, but their relevant result here is that the McMillan equation is not accurate around λ = 2 or greater (unlike the claim by Zheng and Zheng). It is widely understood, as pointed out by McMillan and again by Zheng and Zheng, that any “maximum T_c” is material class dependent.

We do not recommend using any T_c equation beyond that of Allen–Dynes to give realistic values of T_c, given the necessary input.

Jim Ho has emphasized the important role that the specific heat c_V(T) continues to play in the understanding of superconducting properties. In 1957, c_V(T) data recorded every 2–3 degrees, ³ and tabulated but not plotted, just missed showing the structure in c_V(T) near 40 K in magnesium diboride that would have led to the discovery of its paradigm-breaking superconductivity. Instead it remained hidden until its discovery ⁴ in 2001. Specific heat is a crucial probe in the understanding of low-temperature superconductivity and of system changes as the superconducting state is entered.

Even in MgB₂, with T_c ~ 40 K, the signal in c_V(T) at T_c is small because the lattice contribution grows so much more rapidly than the electronic contribution. In hydrides at T_c of 200–260 K, the signal relative to the lattice specific heat will be smaller still. Ho suggests that it may still be observable. More to the point, and recognized by Ho, the diamond anvil cells that are necessary to study very high pressure require a cell of size and mass orders of magnitude greater than the sample, so the signal due to the sample is difficult to obtain. Researchers have measured c_V(T) to pressures ⁵ of 10 GPa, but the challenges in extending such measurements to the 200 GPa range are considerable.