• Spin Transport Model :

• Two-terminal Hamiltonian :

• Four-terminal Hamiltonian :

• Contact Self Energy :

• Non-equilibrium Green function :

• Self-Consistent loop :

• Simulation Flow Chart :

• Spin Transfer Torque Calculation :

https://www.worldscientific.com/doi/10.1142/S2010324719500036

Recent study in Ref. 81 showed that the spin-transfer torque (STT) is enhanced by the asymmetry in a graphene lateral spin valve structure. This lateral structure or geometry can be modeled by a four-terminal magnetic tunneling junction (MTJ) as opposite to the conventional two-terminal MTJ. In this paper, using the non-equilibrium Green’s function formalism, we compare the anti-damping components of the STT in a similar non-conventional lateral one dimensional MTJ with that in the conventional MTJ. We find that the lateral geometry renders enhanced anti-damping torques compared with the conventional one, provided that the barrier energy, the scattering length, and the magnetization angle are in a certain parameter region. We also identify this parameter region in the presence of dephasing. The enhancement of the anti-damping torques declines when the scattering region is longer. For the four-terminal MTJ of larger scattering length, the dephasing can expedite the anti-damping torque.

Fig. 1.3 anti damping and field like torque curves versus voltage plots. Here, we select Nox = 4 layer and UI = 3.92 eV for illustrating the bias dependence of the torques for 2- and 4- terminal set up.

https://arxiv.org/abs/0809.4460

https://www.semanticscholar.org/paper/Self-Consistent-Transport-Magnetic-Simulation-and-Mojumder-Augustine/9d3f0a0897e4869a42d867fa9232cbfcd1863ac0

Fig. 4.1 Equilibrium spin resolved energy band profiles for parallel and anti-parallel Fe/MgO/Fe configurations. Green dash line: fermi-energy, red dash line: up-channel and blue dash line: dn-channel.

Fig. 4.2 NEGF simulation of TMR vs. voltage. The parameters used for this calculation are Ef = 2.25 eV, mup = mdn = 0.73 m0 in FM layer, mup = mdn = 0.32 m0 times the 0.2, 0.4, 0.6 0.8 and1 in oxide layer, LFM = 2 layers for left contact and LFM = 2 layers for right contact, Lox = 4, 6, 7, and 8 layers and m0 is the free electron mass.