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 $\Gamma$ 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 ($\Delta_j$, where $j = \left(l,r\right)$) 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 Schr\"odinger-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-B\"{u}ttiker} 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.