Cuprate superconductors have the highest critical temperatures (T_c) at ambient pressure, and they exhibit rich physical behaviors due to strong electron correlation. Researchers have been trying hard to identify the key driving force for the Cooper pairing, among a variety of intertwined degrees of freedom. An effective approach in this regard is to find an energy scale that is directly related to T_c – this is the reason why the isotope effect in conventional superconductors has been so important for establishing the BCS mechanism, as the atomic mass is directly linked to the phonon energies, which in turn control T_c.

While magnetic interactions are widely believed to be crucial for Cooper pairing in the cuprates, establishing a correspondence between their energy and T_c has remained an experimental challenge. The strength of magnetic interactions is reflected by the energy spectrum of magnetic excitations, known as paramagnons in cuprates that are doped away from their antiferromagnetic “parent” state to become superconductors. The doping changes T_c, but it does not change much the paramagnon spectrum, so when we say “a correspondence with T_c”, we actually mean a correspondence with the maximal T_c at optimal doping. Establishing such a correspondence thus requires comparing different cuprates, but when the chemical compositions are different, how do we know that a change in (the optimal) T_c is related to a change in the paramagnon energy, and not to other factors? The strategy taken here by the collaborating team is to compare cuprates with high but sufficiently different T_c, and they did it within one cuprate “family” to minimize changes in the other factors. They chose to study the mercury family of cuprates, which has very high T_c’s on the order of 100 kelvins, even though it is also infamous for difficulty in single-crystal growth. Over the years, Prof. Yuan Li’s group have achieved significant improvements in the crystal size and quality (Physical Review Materials 2, 123401 (2018)), which has finally enabled the present study.

Using high-resolution inelastic photon scattering, including resonant inelastic x-ray scattering (RIXS) and Raman spectroscopy, Profs. Yuan Li and Yingying Peng’s groups from ICQM, together with collaborators from Germany and the United Kingdom, have determined and compared the paramagnon spectra in two of the mercury-family member compounds (HgBa₂CuO_4+x, Hg1201; and HgBa₂CaCu₂O_6+x, Hg1212) for the first time. The key finding is that the paramagnon energies are different by 20-30% between the two systems (Fig. 1). This result is in favor of a magnetic pairing mechanism, because the two systems’ optimal T_c’s are different by just 30%.

They further extended the comparison to additional cuprates and revealed an approximate linear relation between the optimal T_c and paramagnon energy (Fig. 2). There are still several low-T_c outliers in the plot, but specific reasons detrimental to T_c (such as disorder and small electron hopping range) can be found for each of them. The full story has been published on June 7^th, 2022 in Nature Communications (link to article https://doi.org/10.1038/s41467-022-30918-z ).

The work reveals a simple and near-proportional relation between paramagnon energy and T_c based on direct spectroscopic measurements of high-quality samples. The result encourages researchers to think deep about magnetic pairing mechanisms (or, to reflect on alternative mechanisms to encompass the newly discovered relation). Dr. Lichen Wang (currently a Humboldt Fellow at MPI-FKF, Germany) and Ph.D. student Guanhong He from the ICQM are the first authors of this work. Profs. Yuan Li and Yingying Peng are the corresponding authors. The RIXS experiment was performed at beamline I21, Diamond Light Source, UK. The work at Peking University was financially supported by the NSFC and MOST of China.

Figure 1. (a-d) RIXS data measured on Hg1201 at in-plane momentum transfers Q_// quoted in reciprocal lattice units. Shaded peaks are fitted paramagnon signals. (e-h) Same as (a-d), but for Hg1212, the characteristic energy of which is higher than Hg1201 by 20-30%.

Figure 2. T_c of various cuprates at optimal or the best available doping, plotted versus J extracted from measurements of (para)magnons. J is the nearest-neighbor antiferromagnetic interaction (inset).