Cuprates currently hold the record for the highest temperature superconductivity at ambient pressure, but the microscopic understanding of these materials remains elusive. Here, we utilize nuclear magnetic resonance (NMR) data of planar oxygen and copper from essentially all hole-doped cuprates to provide a universal phenomenology relating the NMR spin shifts, which measure the electronic spin polarization at a given nucleus, with the superconducting dome and maximum critical temperature. There appear to be two separate contributions to the spin shift in planar copper, only one of which is seen at the oxygen site, and we associate them with two different types of carriers. Upon disentangling these two components, their relative size is shown to correlate not only with the doping dependence of the superconducting dome but also with the variation in maximum superconducting critical temperature, $T_{\mathrm{c},\max }$, between different families. One of these components is independent of family and resides in the hybridized planar orbitals of Cu and O. The second component, in contrast, is predominately isotropic and encodes the differences between the families. It is thus related to the charge transfer gap and planar hole sharing. Our findings offer universal insight which should prove useful in the continuing development of a comprehensive theory of the cuprates, as well as an indication of how it may be possible to engineer materials with higher critical temperatures.

Nuclear relaxation, $1/T_1$, is a very robust probe of electronic excitations in superconducting materials above and below the critical temperature of superconductivity, $T_{\mathrm{c}}$. Here, a relaxation phenomenology of hole-doped cuprate superconductors is presented based on the majority of the available literature data from the CuO₂ plane, without assumptions with respect to a hyperfine scenario, form factors, or particular theoretical models. Below a temperature similar to the pseudogap temperature, Heitler–Teller-type relaxation is uncovered universally; i.e., the nuclear spin relaxation above $T_{\mathrm{c}}$ is only determined by the absolute temperature, $1/T_1\propto T$. All materials condense out of this metal at $T_{\mathrm{c}}$, below which relaxation drops even faster, as expected from conventional superconductors, albeit without a Hebel–Slichter peak. It is a ’hidden metal’ in the sense that it has a vanishing uniform response and thus hardly affects the NMR shifts; it is also not seen in planar O relaxation. The hidden metal causes a temperature-independent but material-dependent planar Cu relaxation anisotropy that is strongly correlated with the size of $T_{\mathrm{c}}$. Moreover, the rate measured with the field in the CuO₂ plane is nearly the same for all cuprates: $1/{{}^{63}T}_{1\perp }T\approx 25/$Ks, where $1/{{}^{63}T}_{1\Vert }$ is mainly responsible for the change in anisotropy. Above the hidden metal, the relaxation behavior changes and can be described by an ordinary but renormalized metal, with a reduced Cu relaxation anisotropy. The relaxation phenomenology, which should hold clues to the so-called strange metal, is also discussed in the context of the two spin components previously uncovered in the shifts, as well as the pseudogap and relation to other probes. This new phenomenology should give a better foundation for the understanding of the cuprates.

The electronic properties of high-temperature superconducting cuprates are encoded in NMR data. Without microscopic theory, reliable NMR phenomenologies are in demand. Here we make use of the extensive literature data to develop a different understanding of the cuprates from the shifts of the ${\mathrm{CuO}}_2$ plane. The Cu shift analysis is based only on the symmetry of the two Cu hyperfine couplings, without assumptions about their size. We use an anisotropic $A_{\alpha }$ and isotropic B, as from atomic Cu orbitals, and find two spin components (A- and B-spins) that explain all the shift data. The components differ in size and temperature dependence according to simple rules. Upon doping the cuprates, metallic B-spin appears above a pseudogap temperature, which is shared with the A-spin. Further doping decreases the pseudogap temperature and increases the B-spin, but less so the A-spin. The apparent linear rate of increase in the density of states of the B-spin with doping is nearly threefold above $x=0.20$, where the pseudogap disappears. The pseudogap temperature is a measure of the coupling between A and B, which suppresses the shifts but not nuclear relaxation. Spin-singlet pairing involves A and B according to three simple condensation rates, which will be discussed. The optimal $T_{\mathrm{c}}$ demands a special match between A and B. However, the shifts do not simply predict the highest $T_{\mathrm{c}}$ of all cuprates, in contrast to nuclear relaxation anisotropy and charge sharing between planar Cu and O. Relations to other probes are discussed.