Universal doping dependence of the ground-state staggered magnetization of cuprate superconductors

Using muon spin rotation we determine the zero-temperature staggered antiferromagnetic order parameter $M_0$ versus hole doping measured from optimum $\Delta p_m$, in the $\left({\text{Ca}}_x{\text{La}}_{1-x}\right)\left({\text{Ba}}_{1.75-x}{\text{La}}_{0.25+x}\right){\text{Cu}}_3{\text{O}}_y$ system. In this system the maximum $T_c$ and the superexchange $J$ vary by 30% between families $\left(x\right)$. $M_0\left(x,\Delta p_m\right)$ is found to be $x$ independent. Using neutron diffraction we also determine the lattice parameters variations for all $x$ and doping. The oxygen buckling angle is found to change with $x$, implying a change in the holes kinetic energy. We discuss the surprising insensitivity of $M_0\left(x,\Delta p_m\right)$ to the kinetic-energy variations in the framework of the $t\text{−}J$ model.