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)
= G
₀
σk
_{F, s}
²
/4π ∫ Cos (θ
_s
) D
_s
^P(AP)
Sin(θ)dθ
_s
dφ, 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; G
₀
=2e
²
/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 the 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 nm
²
), 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↑
+G
₂
↑+G
_1↓
+G
₂
↓), 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%.