Tunnel magnetoresistance in magnetic tunnel junctions with embedded nanoparticles

Abstract

© 2015 IEEE. 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<inf>s</inf>^P(AP) = G<inf>0</inf>σk <inf>F, s</inf>²/4π ∫ Cos (θ<inf>s</inf>) D<inf>s</inf>^P(AP) Sin(θ)dθ<inf>s</inf>d, where Cos(θ<inf>s</inf>) is cosine of incidence angle of the electron trajectory θ<inf>s</inf>, with spin index s=(↑,↓), k<inf>F, s</inf>, is the Fermi wave-vector of the top (bottom) ferromagnetic layers; for simplicity the top and bottom ferromagnetic layers are taken as symmetric; G<inf>0</inf>=2e²/h and σ is area of the tunneling cell. The transmission probability D<inf>s</inf>^P(AP) depends on diameter of NP (d), effective mass m and wave-vector of the electron k<inf>NP</inf> attributing to the quantum state on NP (corresponding to the k-vector of the middle layer in DMTJs [1], and which is affected by applied bias V). Furthermore D<inf>s</inf>^P(AP) depends on Cos(θ<inf>s</inf>), k<inf>F, s</inf>, barriers heights U<inf>1,2</inf> and widths L<inf>1,2</inf>, 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<inf>1↑</inf>+G<inf>2</inf>↑+G<inf>1↓</inf>+G<inf>2</inf>↓), where G<inf>1, s</inf> is dominant conductance with the NP (path I), G<inf>2, s</inf> is conductance of the direct tunneling through the single barrier (path II), and TMR=(G^P-G^AP)/G^AP ×100%.