Tunnel magnetoresistance in magnetic tunnel junctions with embedded nanoparticles

We present a theoretical simulation to calculate the tunnel magnetoresistance (TMR) in magnetic tunnel junction with embedded nano-particles (npMTJ). The simulation is done in the range of coherent electron tunneling model through the insulating layer with embedded magnetic and non-magnetic nano-particles (NPs). We consider two conduction channels in parallel within one MTJ cell, in which one is through double barriers with NP (path I in Fig. 1) and another is through a single barrier (path II). The model allows us to reproduce the TMR dependencies at low temperatures of the experimental results for npMTJs [2-4] having in-plane magnetic anisotropy. In our model we can reproduce the anomalous bias-dependence of TMR and enhanced TMR with magnetic and non-magnetic NPs. We found that the electron transport through NPs is similar to coherent one for double barrier magnetic tunnel junction (DMTJ) [1]; therefore, we take into account all transmitting electron trajectories and the spin-dependent momentum conservation law in a similar way as for DMTJs. The formula of the conductance for parallel (P) and anti-parallel (AP) magnetic configurations is presented as following: G_s^P(AP) = G₀σk _{F, s}²/4π ∫ Cos (θ_s) D_s^P(AP) Sin(θ)dθ_sdφ, where Cos(θ_s) is cosine of incidence angle of the electron trajectory θ_s, with spin index s=(↑,↓), k_{F, s}, is the Fermi wave-vector of the top (bottom) ferromagnetic layers; for simplicity the top and bottom ferromagnetic layers are taken as symmetric; G₀=2e²/h and σ is area of the tunneling cell. The transmission probability D_s^P(AP) depends on diameter of NP (d), effective mass m and wave-vector of the electron k_NP attributing to the quantum state on NP (corresponding to th- k-vector of the middle layer in DMTJs [1], and which is affected by applied bias V). Furthermore D_s^P(AP) depends on Cos(θ_s), k_{F, s}, barriers heights U_1,2 and widths L_1,2, respectively. The exact quantum mechanical solution for symmetric DMTJ was found in Ref.[1]. Here we employ parallel circuit connection of the tunneling unit cells, where each cell contains one NP with the average d less than 3 nm per unit cell´s area (σ =20 nm²), while tunnel junction itself has surface area S and consists of N cells (N=S/σ). The total conductance of the junction is G = Nx (G_1↑+G₂↑+G_1↓+G₂↓), where G_{1, s} is dominant conductance with the NP (path I), G_{2, s} is conductance of the direct tunneling through the single barrier (path II), and TMR=(G^P-G^AP)/G^AP ×100%.