  The next-nearest-neighbour hopping term t' is shown to stabilize the AF state
of the doped Hubbard model with respect to transverse perturbations in the
order- parameter by strongly suppressing the intraband particle-hole processes.
For a fixed sign of t', this stabilization is found to be significantly
different for electron and hole doping, which qualitatively explains the
observed difference in the degree of robustness of the AF state in the
electron-doped (Nd_{2-x}Ce_{x}CuO_{4}) and hole-doped (La_{2-x}Sr_{x}CuO_{4})
cuprates. The t'-U phase diagram is obtained for both signs of the t' term,
showing the different regions of stability and instability of the doped
antiferromagnet. Doping is shown to suppress the t'-induced frustration due to
the competing interaction J'. A study of transverse spin fluctuations in the
metallic AF state reveals that the decay of magnons into particle-hole
excitations yields an interesting low-energy result \Gamma \sim \omega for
magnon damping.
