(\[gamma\]) yields the charge current $j_{n-1}^{\rm c}$ if $\frac12{\bm\sigma}$ is replaced by a unit matrix multiplied by the number $e/\hbar$, where $e$ is the electronic charge (negative). All currents flow in the $y$ direction, perpendicular to the layers, and the components of the vector ${\bf  j}$ correspond to transport of $x$, $y$ and $z$ components of spin. The justification of Eq. (\[gamma\]) for ${\bf j}_{n-1}$ relies on an equation of continuity, as pointed out in Section \[quattro\].

To define the present model completely we must supplement the hopping Hamiltonian $H_0$ by specifying the on-site potentials in the various layers. For simplicity we assume the on-site potential for both spins in non-magnetic layers, and for majority spin in ferromagnetic layers, is zero. We assume an infinite exchange splitting in the ferromagnets so that the minority spin potential in these layers is infinite. Thus minority spin electrons are completely excluded from the ferromagnets. Clearly the definition of majority and minority spin relate to spin quantisation in the direction of the local magnetisation. We take $\alpha=0$, so that the magnetisation of the switching magnet is in the z direction and take $\theta=\psi$, where $\psi$ is the angle between the magnetisations.

To describe spin 