Mechanism for non-Heisenberg-exchange coupling between ferromagnetic layers

We consider a free-electron model of exchange coupling between transition-metal ferromagnetic films separated by a paramagnetic metal spacer. The minority-spin energy bands in the ferromagnets are matched to those of the paramagnetic spacer. The majority-spin electrons experience a repulsive potential arising from the lack of corresponding states in the spacer. The height of the potential barrier is equal to the exchange-energy gap in the ferromagnets. We show how the model, applicable to a trilayer film such as Fe/Cr/Fe or Co/Ru/Co, generates an infinite sum of terms with coefficients ${\mathit{A}}_{12}$,${\mathit{B}}_{12}$, . . . in the coupling energy, extending beyond the Heisenberg-like ${\mathit{A}}_{12}$ term of bilinear coupling between the moments of the ferromagnets. Both the ${\mathit{A}}_{12}$ and non-Heisenberg (biquadratic) ${\mathit{B}}_{12}$ exchange terms oscillate with spacer-layer thickness. We find that the phase of ${\mathit{B}}_{12}$ relative to ${\mathit{A}}_{12}$ shifts with this thickness, so that at certain regions of spacer-layer thickness, the magnitude of ${\mathit{B}}_{12}$ is greater than that of ${\mathit{A}}_{12}$, and ${\mathit{B}}_{12}$ is of the proper sign to cause 90° coupling between the ferromagnets. These special thicknesses are predicted to appear over a broad range of the ratio of Fermi to exchange-gap energies. Thus, we show from our model that biquadratic, or 90°, coupling between the ferromagnets is intrinsic to itinerant-electron exchange across the paramagnetic spacer.