Evolution of Magnetic Properties of Lightly Doped Copper Oxides

We study how doping destroys the AF order in the layered cuprates within the framework of the charge--transfer insulator concept. We use the criterion of stability of the AF background to show that the stability problem is one of the main issues in any correspondence between results for the $t-J$ model and, say, the three--band model for the lightly--doped layered oxides. Provided a phenomenological conduction band is chosen to satisfy the criterion of stability, a detailed picture of how dopants influence the spin wave spectrum at $T=0$ is presented. The critical concentration $x_c$ for the destruction of the AF long range order is due to the Cherenkov effect when the Fermi velocity first exceeds the spin wave velocity. We then discuss the overall spectrum of spin excitations and find that the spin wave attenuation for $x < x_c$, $T=0$ due to Landau damping appears in the range of magnon momenta $k(x) = 2 m^* s \pm \alpha \sqrt{x}$. We also argue that in the presence of superconductivity, the Cherenkov effect is eliminated due to the gap in the spectrum. This may restore the role of the AF fluctuations as the main source of dissipation at the lowest temperatures.