Correlation of the superconducting critical temperature with spin and orbital excitations in(CaxLa1−x)(Ba1.75−xLa0.25+x)Cu3Oyas measured by resonant inelastic x-ray scattering

Electronic spin and orbital $\left(dd\right)$ excitation spectra of $({\mathrm{Ca}}_x{\mathrm{La}}_{1-x}$ )$\left({\mathrm{Ba}}_{1.75-x}{\mathrm{La}}_{0.25+x}\right){\mathrm{Cu}}_3{\mathrm{O}}_y$ samples are measured by resonant inelastic x-ray scattering (RIXS). In this compound, $T_c$ of samples with identical hole dopings is strongly affected by the Ca/Ba substitution $x$ due to subtle variations in the lattice constants, while crystal symmetry and disorder as measured by linewidths are $x$ independent. We examine two extreme values of $x$ and two extreme values of hole-doping content $y$ corresponding to antiferromagnetic and superconducting states. The $x$ dependence of the spin-mode energies is approximately the same for both the antiferromagnetic and superconducting samples. This clearly demonstrates that RIXS is sensitive to the superexchange $J$ even in doped samples. A positive correlation between $J$ and the maximum of $T_c$ at optimal doping $\left(T_c^{max}\right)$ is observed. We also measured the $x$ dependence of the $d_{xy}\to d_{x^2-y^2}$ and $d_{xz/yz}\to d_{x^2-y^2}$ orbital splittings. We infer that the effect of the unresolved $d_{3z^2-r^2}\to d_{x^2\to y^2}$ excitation on $T_c^{max}$ is much smaller than the effect of $J$. There appears to be dispersion in the $d_{xy}\to d_{x^2-y^2}$ peak of up to 0.05 eV. Our fitting furthermore indicates an asymmetric dispersion for the $d_{xz/yz}\to d_{x^2-y^2}$ excitation. A peak at $\sim 0.8$ eV is also observed and attributed to a $dd$ excitation in the chain layer.