Magnetic excitations
Magnetic correlations of the hole-doped planes can be extracted from measurements of the magnetic excitations obtained by inelastic neutron scattering. Some examples, measured about the AF wave vector, ${\bf Q}_{\rm AF}$, are shown in Fig. 1 for La$_{2-x}$(Sr,Ba)$_x$CuO$_4$. At the higher energies, one can see that AF-like spin waves survive, though the effective strength of the superexchange is renormalized downwards. Note that the effective bandwidth of the excitations corresponds to $\sim2J_{\rm eff}$.
One change that is apparent in the right-hand panel of Fig.1 is that, below an energy scale $E_{\rm cross}$, the magnetic dispersion is associated with wave vectors split incommensurately about ${\bf Q}_{\rm AF}$. Studies have shown that both the direction and magnitude of the incommensurability are doping dependent.
The nature of the incommensurability has been studied in detail for La$_{2-x}$Sr$_x$CuO$_4$ by neutron scattering [1]. The left-hand side of Fig. 2 summarizes the results. For $x\lesssim0.055$, the system is insulating at low temperature, and elastic scattering is observed at incommensurate peaks split about ${\bf Q}_{\rm AF}$ along a diagonal direction, as indicated by the upper inset in (a); the modulation is uniquely along the orthorhombic $b$ axis. As the doping level is increased to $x\gtrsim0.055$, the system becomes superconducting, the elastic scattering strongly weakens, and the low-energy spin excitations are oriented parallel to the Cu-O bond directions, as shown in the lower-right inset of (a). While the orientation of the peaks rotates with doping, the magnitude $\delta$ of the incommensurability remains very close to $x$ across the transition.