Analytical expression for the spin-transfer torque in a magnetic junction with a ferromagnetic insulator

Analytical expressions for the spin-transfer torque (STT) in magnetic junctions with a ferromagnetic insulator (FI) are derived using the Keldysh formalism. Adopting simple approximations and ballistic transport, both parallel and perpendicular torques are expressed in terms of spin-dependent intersite Green's functions of the insulator. They depend linearly on the bias voltage because of the asymmetry of junctions. The relationship between STT and tunnel magnetoresistance, the difference between the features of the STT obtained in the present and previous works, and effects of the electronic structures of FI and those of impurities on the STT are discussed.

Figure 1

(a) Schematic of the DOS of an ferromagnetic insulator (FI) composed of two types of magnetic atoms $A$ and $B$, (b) a typical structure of the unctions, (c) magnetization alignment of the junctions, and (d) site indices of the junctions with hopping integrals $t$ and $t^{\prime }$ at the $m-n$ and $p-q$ interfaces, respectively.

Figure 2

Energy dependence of up- and down-spin nonlocal Green's functions $\mathrm{Re}\left(g_{16}\right)$ with $k_{\parallel }=(0,0)$ near the energy band gap calculated by using a simple model of an ferromagnetic insulator with six atomic planes. The inset shows the local DOS at an interface layer of the FI, in units of 1/$\left|t\right|$/atom/spin, where $t$ is a hopping integral within the insulator.