Evolution of the resonance and incommensurate spin fluctuations in superconducting ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$

Polarized and unpolarized neutron triple-axis spectrometry was used to study the dynamical magnetic susceptibility ${\chi }^{″}(\mathbf{q},\omega )$ as a function of energy $\left(ħ\omega \right)$ and wave vector $\left(\mathbf{q}\right)$ in a wide temperature range for the bilayer superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ with oxygen concentrations, x, of 0.45, 0.5, 0.6, 0.7, 0.8, 0.93, and 0.95. The most prominent features in the magnetic spectra include a spin gap in the superconducting state, a pseudogap in the normal state, the much-discussed resonance, and incommensurate spin fluctuations below the resonance. We establish the doping dependence of the spin gap in the superconducting state, the resonance energy, and the incommensurability of the spin fluctuations. The magnitude of the spin gap ${(E}_{\mathrm{sg}})$ up to the optimal doping is proportional to the superconducting transition temperature $T_c$ with $E_{\mathrm{sg}}{/k}_B{T}_c=\mathrm{3.8.}\mathrm{}$ The resonance, which exists exclusively below $T_c$ for highly doped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ with $x=0.93\mathrm{}$ and 0.95, appears above $T_c$ for underdoped compounds with $x<~\mathrm{0.8.}$ The resonance energy ${(E}_r)$ also scales with $k_B{T}_c,$ but saturates at $E_r\approx 40$ meV for x close to 0.93. The incommensurate spin fluctuations at energies below the resonance have structures similar to that of the single-layer superconducting ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4.$ However, there are also important differences. While the incommensurability $\left(\delta \right)$ of the spin fluctuations in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ is proportional to $T_c$ for the entire hole-doping range up to the optimal value, the incommensurability in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ increases with $T_c$ for low-oxygen doping and saturates to $\delta =0.1\mathrm{}$ for $x>~\mathrm{0.6.}$ In addition, the incommensurability decreases with increasing energy close to the resonance. Finally, the incommensurate spin fluctuations appear above $T_c$ in underdoped compounds with $x<~0.6$ but for highly doped materials they are only observed below $T_c.$ We discuss in detail the procedure used for separating the magnetic scattering from phonon and other spurious effects. In the comparison of our experimental results with various microscopic theoretical models, particular emphasis was made to address the similarities and differences in the spin fluctuations of the two most studied superconductors. Finally, we briefly mention recent magnetic-field-dependent studies of the spin fluctuations and discuss their relevance in understanding the microscopic origin of the resonance.