shown in Fig. \[fig1\], where [**p**]{} and [**m**]{} are unit vectors in the direction of the magnetisations. This models the layered structure of the pillars used in experiments but the atomic planes shown are considered to be unbounded instead of having the finite cross-section of the pillar. This means that there is translational symmetry in the $x$ and $z$ directions. The structure consists of a thick (semi-infinite) left magnetic layer (polarising magnet), a non-magnetic metallic spacer layer, a thin second magnet (switching magnet) and a semi-infinite non-magnetic lead. In the simplest model we assume the atoms form a simple cubic lattice, with lattice constant $a$, and we adopt a one-band tight-binding model with hopping Hamiltonian

$$H_0=t\sum_{{\bf k}_{\parallel}\sigma}
\sum_{n}c^{\dagger}_{k_{\parallel} n\sigma} c_{k_{\parallel}
n-1\sigma}\,\, +\,\, {\rm h. c.}. \label{hamiltonian}$$

Here $c^{\dagger}_{k_{\parallel} n\sigma}$ creates an electron on plane $n$ with two-dimensional wave-vector ${\bf k}_{\parallel}$ and spin $\sigma$, and $t$ is the nearest-neighbour hopping integral.

In the tight-binding description the operator for spin angular momentum current between planes $n-1$ and $n$, which we require to calculate spin-transfer torque, is given by $${\bf j}_{n-1}=-\frac{{\rm i}t}2\sum_{{\bf
    k}_{\parallel}}\left(c^{\dagger}_{k_{\parallel},n,\sigma\uparrow},
c^{\dagger}_{k_{\parallel},n,\downarrow}\right){\bm\sigma}
\left(c_{k_{\parallel} ,n-1,\uparrow}, c_{k_{\parallel}
,n-1,\downarrow} \right)^{\dagger} +\,\, {\rm h. c.}. \label{gamma}$$ Here ${\bm\sigma}=\left(\sigma_x,\sigma_y,\sigma_z\right)$ where the components are Pauli matrices. Eq. 