transport in the structure we adopt the generalised Landauer approach of Waintal [*et. al.*]{} [@4]. Thus the structure is placed between two reservoirs, one on the left and one on the right, with electron distributions characterised by Fermi functions $f(\omega-\mu_{\rm L})$, $f(\omega-\mu_{\rm R})$ respectively. The system is then subject to a bias voltage $V_{\rm
b}$ given by $eV_{\rm b}=\mu_{\rm L}-\mu_{\rm R}$, the difference between the chemical potentials. We discuss the ballistic limit where scattering occurs only at interfaces, the effect of impurities being negligible. We label the atomic planes so that $n=0$ corresponds to the last atomic plane of the polarising magnet. the planes of the spacer layer correspond to $n=1,2\, \cdot\cdot\,N $ and $n=N+1$ is the first plane of the switching magnet.

Consider first an electron incident from the left with wave-function $|k,maj\rangle$, where $k>0$, which corresponds to a Bloch wave $|k\rangle = \sum_{n}{\rm e}^{{\rm i}kna}|{\bf
k}_{\parallel}n\rangle$ with majority spin in the polarising magnet. In this notation the label ${\bf k}_{\parallel}$ is suppressed. The particle is partially reflected by the structure and finally emerges as a partially transmitted wave in the lead, with spin $\uparrow$ corresponding to majority spin in the switching magnet. Thus the wave-function is of the 