Theoretical spin transport analysis for a spin pseudovalve-type ${\mathrm{L}}_j$/semiconductor/${\mathrm{L}}_j$ trilayer (with ${\mathrm{L}}_j$ = ferromagnetic)

In this work, a theoretical study of spin transport in a pseudovalve spin (PSV) heterostructure is conducted. For the semiconductor (SC), the conduction band at the $\mathrm{\Gamma }$ point of reciprocal space and spin-orbit coupling (SOC) are considered. For the ferromagnetic (FM) electrodes on the left ($l$) and right ($r$), the internal exchange energy (${\mathrm{\Delta }}_j$, where $j=(l,r)$) and the magnetization normal vector (${\mathbf{n}}_j$) on the barrier plane are taken into account. An analytical expression for the transmission probability as a function of ${\mathbf{n}}_j$ direction was obtained from the {\em Schrödinger-Pauli} equations with the boundary conditions. Furthermore, the tunnel magnetoresistance (TMR) at T $\approx$ 0 K was calculated, depending on the direction of the crystallographic axis favoring the magnetization (${\theta }_m$) of the FM and the thickness of the SC, using the {\em Landauer-Büttiker} formula for a single channel. It is observed that the TMR reaches its maximum value when the ${\mathbf{n}}_l$ direction is parallel to ${\theta }_m$. Applying this physico-mathematical model to the Fe/SC/Fe PSV, with SC as GaAs, GaSb, and InAs, it was found that the {\em Dresselhaus} SOC does not significantly contribute to the TMR.