Spin transport analysis for a spin pseudovalve-type L_l/SC/L_r trilayer for L = {FeCr, Fe, Co, NiFe, Ni} and SC = {GaSb, InSb, InAs, GaAs, ZnSe}

In this work, we present a theoretical study of spin transport in a trilayer pseudospin-valve (PSV) heterostructure composed of electrode (L_l)/insulator/electrode (L_r). The insulating layer corrresponds to a semiconductor (SC) with a zinc-blende crystal structure from the III-V (GaSb, InSb, InAs, and GaAs) or the II-VI (ZnSe), while the electrodes are ferromagnetic materials L_j = {FeCr, Fe, Co, NiFe, Ni}. This combination yields 125 possible PSV configurations. The theoretical model implemented is based on the approach proposed by J. C. Slonczewski. In our approach, the exchange splitting in the ferromagnetic materials and the spin-orbit coupling (SOC) of the Dresselhaus and Rashba types in the semiconductors are included, allowing control of the wave vector associated with the spin states. The tunnel magnetoresistance (TMR) is calculated at low temperature as a function of the semiconductor thickness, parameterized with respect to the crystallographic axis that favors the magnetization direction in the ferromagnetic electrodes, within the Landauer–Büttiker formalism in the single-channel regime. The results show that the TMR reaches its maximum value independently of the relative orientation between the magnetization vector and the crystallographic direction. The most efficient configuration corresponds to Fe_{90}Cr_{10}/GaSb/Fe_{90}Cr_{10}, with a TMR value of 83.60%. Furthermore, the Dresselhaus SOC contributes more significantly to the TMR than the Rashba SOC. Finally, the TMR varies when the electrodes L_j are permuted, due to differences in their Fermi energies. The obtained results are compared with previous studies reported in the literature based on alternative theoretical frameworks or assumptions, showing good agreement.

💡 Research Summary

In this paper the authors present a comprehensive theoretical investigation of spin‑dependent transport in a pseudospin‑valve (PSV) trilayer of the form L_l/SC/L_r, where the ferromagnetic electrodes L_j are chosen from the set {Fe₉₀Cr₁₀, Fe, Co, NiFe, Ni} and the insulating barrier is a zinc‑blende semiconductor (SC) taken from {GaSb, InSb, InAs, GaAs, ZnSe}. By combining the five electrode materials with the five semiconductors, a total of 125 distinct PSV configurations are examined.

The theoretical framework is based on the Slonczewski spin‑torque model. In the ferromagnetic regions the Hamiltonian includes a free‑electron kinetic term and an exchange‑splitting term Δ_xs that separates majority and minority spin bands. In the semiconductor barrier the Hamiltonian incorporates both Dresselhaus (β) and Rashba (α) spin‑orbit coupling (SOC) terms appropriate for zinc‑blende crystals. The magnetization direction of each electrode is parameterized by angles θ and φ, allowing the authors to treat arbitrary relative orientations between the two electrodes.

The Schrödinger‑Pauli equation is solved piecewise in the three regions. Plane‑wave solutions are multiplied by spinors that diagonalize the respective Hamiltonians. Continuity of the wavefunction and of the mass‑weighted derivative at the two interfaces yields analytic expressions for the reflection, transmission, and evanescent coefficients. The transmission probability T_σ(k_∥,θ) (σ = ↑, ↓) is expressed in terms of complex parameters ρ_ε and λ_ε that contain the barrier height, effective masses, and SOC strengths. The authors consider a rectangular barrier of thickness a = 1–6 nm and an in‑plane wavevector k_∥ ranging from 0 to 1 nm⁻¹, effectively working in a single‑channel regime at T ≈ 0 K.

Conductance is calculated using the Landauer‑Büttiker formula G = (e²/ħ)∫T(k_∥,θ) dk_∥, with separate evaluations for the parallel (θ = 0) and antiparallel (θ = π) magnetic configurations. Tunnel magnetoresistance (TMR) is then defined as (G_P – G_AP)/G_AP. Material parameters—Fermi wavevectors k_F↑, k_F↓, exchange splittings Δ_xs for each electrode, effective masses, band gaps, and SOC constants β and α for each semiconductor—are taken from experimental data and recent first‑principles calculations. The authors also interpolate or estimate values where direct measurements are unavailable (e.g., Cr‑doped Fe).

Key findings are:

1. Maximum TMR: The highest TMR obtained is 83.60 % for the symmetric structure Fe₉₀Cr₁₀/GaSb/Fe₉₀Cr₁₀ with a barrier thickness a ≈ 1.92 nm. This configuration benefits from the large exchange splitting of Fe₉₀Cr₁₀ (Δ_xs ≈ 4.22 eV) and the relatively high Dresselhaus coefficient of GaSb (β ≈ 0.30 eV·Å³).

2. SOC Contributions: Dresselhaus SOC consistently yields a larger enhancement of TMR than Rashba SOC. Materials with larger β (GaSb, InSb) show the most pronounced SOC‑induced increase, while the Rashba term α contributes only marginally.

3. Orientation Independence: The TMR is essentially independent of the crystallographic direction that defines the easy axis of the ferromagnets. Whether the magnetization aligns along