Theory of Anisotropic Superexchange in Insulating Cuprates

Spin-orbit corrections to superexchange are calculated using the method of Moriya [T.\ Moriya, {\it Phys.\ Rev.}~{\bf 120} 91, (1960)] for two of the insulating parent compounds of the cuprate superconductors: (1) La$_{2-x}$Nd$_x$CuO$_4$ where the CuO$_6$ octahedra forming each Cu-O layer are tilted in staggered fashion about an axis which depends on $x$ and temperature; and (2) YBa$_2$Cu$_3$O$_{6+x}$ ($x \alt 0.4$) where the Cu-O layers form CuO$_2$-Y-CuO$_2$ bilayers in which the in-plane O$^{2-}$ ions are displaced uniformly towards the Y$^{3+}$ layer. For (1) a simple formula is derived for the weak ferromagnetic moment in each Cu-O layer as a function of the tilting axis and magnitude. For (2) it is shown that the anisotropic corrections to superexchange are different from what has previously been assumed. For the correct spin Hamiltonian a classical N\'eel state in which the Cu spins are lying in the plane is unstable in a single Cu-O layer, but when a bilayer is considered there is a critical value of the interlayer exchange coupling which stabilizes this state. For both cases (1) and (2) spin-wave spectra are calculated and shown to compare favorably with experiment.