Xf_STT

Download

The Xf_STT class was created to enable simulation of the injection of multiple spin currents into a ferromagnet in OOMMF.

The spin torque equation derived by Slonczewski [1] and modified by Xiao [2] is written as (as implemented in OOMMF):

\frac{\partial\textbf{m}}{\partial t} = -|\gamma|\textbf{m}\times\textbf{H}_\text{EFF}+\alpha\textbf{m}\times\frac{\partial\textbf{m}}{\partial t}+\textbf{m}\times\textbf{STT}
\textbf{m}\times\textbf{STT} = \left|\gamma\right|\beta \left( \epsilon \left( \textbf{m}\times \textbf{m}_{p} \times \textbf{m} \right) - \epsilon'\left(\textbf{m}\times\textbf{m}_{p}\right) \right)
\textbf{STT} = \left|\gamma\right|\beta \left( \epsilon \left( \textbf{m}_{p} \times \textbf{m} \right) - \epsilon'\textbf{m}_{p} \right)
\textbf{H}_\text{STT} = \beta \left( \epsilon \left( \textbf{m} \times \textbf{m}_{p} \right) + \epsilon'\textbf{m}_{p} \right)

According to the OOMMF user guide:

\textbf{m}=\text{reduced magnetization,} \frac{\textbf{M}}{M_{sat}}
\gamma=\text{Gilbert gyromagnetic ratio}
\beta=\left|\frac{\hbar}{\mu_{0}e}\right|\frac{J}{tM_{sat}}
\textbf{m}_{p}=\text{(unit) electron polarization direction}

\gamma has a default value of -2.211\times{}10^{5}~\text{m/A}\cdot\text{s}. In the definition of \beta, \hbar is the reduced Planck’s constant, e is the electronic charge in C, J is the current density exerting spin-torque in \text{A/m}^{2}, t is the thickness of the free layer in meters in the direction which the current density is flowing, and M_{sat} is the saturation magnetization in \text{A/m}. Note that \beta may be rewritten as,

\beta=\left|\frac{\hbar}{\mu_{0}e}\right|\frac{I}{M_{sat}Vol}

where I is the current flowing homogeneously into the magnetic cell in A, and Vol is the volume of the magnetic cell in \text{m}^{3}.

\epsilon and \epsilon' gives the in-plane and out-of-plane spin-torque terms, respectively. As implemented in OOMMF,

\epsilon=\frac{q_{+}}{A_{+}+A_{-}\left(\textbf{m}\cdot\textbf{m}_{p}\right)}+\frac{q_{-}}{A_{+}-A_{-}\left(\textbf{m}\cdot\textbf{m}_{p}\right)}
q_{\pm}=P_{fixed}\Lambda_{fixed}^2\sqrt{\frac{\Lambda_{free}^2+1}{\Lambda_{fixed}^2+1}}\pm P_{free}\Lambda_{free}^2\sqrt{\frac{\Lambda_{fixed}^2-1}{\Lambda_{free}^2-1}}
A_{\pm}=\sqrt{\left(\Lambda_{fixed}^2\pm 1\right)\left(\Lambda_{free}^2\pm 1\right)}

In the case where P_{fixed}=P_{free} and \Lambda_{fixed}=\Lambda_{free}, \epsilon reduces to

\epsilon=\frac{P\Lambda^2}{\left(\Lambda^2+1\right)+\left(\Lambda^2-1\right)\left(\textbf{m}\cdot\textbf{m}_{p}\right)}

The Specify block for the Xf_STT class has the form

\text{Specify Xf\_STT:}\textit{name}~\{
    \text{P}~~~~\textit{polarization}
    \text{P\_fixed}~~~~\textit{p\_fixed\_layer}
    \text{P\_free}~~~~\textit{p\_free\_layer}
    \text{Lambda}~~~~\Lambda
    \text{Lambda\_fixed}~~~~\Lambda\textit{\_fixed\_layer}
    \text{Lambda\_free}~~~~\Lambda\textit{\_free\_layer}
    \text{eps\_prime}~~~~\textit{ep}
    \text{J}~~~~\textit{current\_density}
    \text{J\_direction}~~~~\textit{current\_flow\_direction}
    \text{J\_profile}~~~~\textit{Jprofile\_script}
    \text{J\_profile\_args}~~~~\textit{Jprofile\_script\_args}
    \text{mp}~~~~\textit{p\_direction}
    \text{propagate\_mp}~~~~\textit{prop\_mp}
\}

The spin-transfer torque term does not actually change the energy in this implementation in OOMMF, and its implementation as a derivation from Oxs_Energy class is technically illegal. However, the spin-transfer torque term as an equivalent H-field in OOMMF using the Oxs_Energy class gives the flexibility to model the injection of multiple spin currents into one region in the ferromagnet.

The default values for P and Lambda are 0.4 and 2, respectively. If preferred, values for the fixed and free layers may be instead specified separately, through P_fixed, P_free, Lambda_fixed, and Lambda_free. Otherwise P_fixed = P_free = P and
Lambda_fixed = Lambda_free = Lambda. Lambda must be larger than or equal to 1; set Lambda=1 to remove the dependence of \epsilon on \textbf{m}\cdot\textbf{m}_{p}. If you want non-zero \epsilon', it is set directly as eps_prime.

Current density J and unit polarization direction mp are required. The units on J are \text{A/m}^{2}. Positive J produces torque that tends to align \textbf{m} towards \textbf{m}_p.

Simulation of domain-wall dynamics under current-induced spin-torque is enabled by setting propogate_mp to 1. The setting propogate_mp is 0 (disabled) by default. When propogate_mp is enabled, mp is actually \Delta_{x}\times\frac{\partial\textbf{m}}{\partial x} , where x is the flow direction and \Delta_{x} is the cell dimension in that direction. The flow direction may be set by setting J_direction as one of six options:
-z, +z, -y, +y, -x, +x. The default is -z. The direction changes the mp used to calculate the spin torque at each cell site.

Parameters J, mp, P, Lambda, and eps_prime may all be varied pointwise (by specifying them as Oxs_ScalarField objects), but are fixed with respect to time. However, J can be multiplied by a time varying “profile,” to model current rise times, pulses, etc. Use the J_profile and J_profile_args options to enable this feature. The Jprofile_script should be a Tcl script that returns a single scalar. Jprofile_script_args should be a subset of {stage stage_time total_time }, to specify arguments appended to Jprofile_script on each time step. Default is the entire set, in the order as listed.

The Xf_STT module provides the scalar output “average J,” and the vector field outputs “Field” (which is |\gamma|\beta\left[\epsilon\left(\textbf{m}_{p}\times\textbf{m}\right)-\epsilon'\textbf{m}_{p}\right]) and “J*mp.”

2 comments on “Xf_STT
  1. Xiangjun Xing says:

    Dear Xuanyao,
    What is the difference between Xf_STT and Oxs_SpinXferEvolve? It seems that the former ties together the functions of the latter and the Anv_SpinTEvolve evolver.

    Xiangjun, Cheers>

    • xfong says:

      Xf_STT allows you to to include the effect of spin-transfer torque into any evolver class you want, since Xf_STT returns spin-transfer torque as an Oxs_Energy class. This allows you to simulate structures in which spin current may be injected into the same region from several different interfaces, each with a polarization factor and lambda.

Leave a Reply

Your email address will not be published. Required fields are marked *

*

Recent Posts

Recent Comments

Archives

Categories

Skip to toolbar