M3 is a Matlab based micromagnetics code, written by C.K.A. Mewes & T.Mewes.
If you are interested in using M3 just sent us an email at cmewes@mint.ua.edu or tmewes@mint.ua.edu!

It uses a Fast Fourier transform (FFT) method to calculate the magnetostatic interaction field, utilizing Newell's formulation of the demagnetizing tensor [1] at short distances and a dipole approximation for the far field.

For the exchange interaction three different formulations are currently implemented: a six neighbor method, a twelve neighbor method and a twenty-six neighbor method. The six and twelve neighbor methods can use either Dirichlet or Neumann boundary conditions, the twenty-six neighbor method uses Neumann boundary conditions [2]. Interlayer exchange coupling is implemented using modified boundary conditions for the layers next to the interlayer.

M3 can assign arbitrary anisotropies (unidirectional, uniaxial, cubic, etc.) to the simulation volume. The applied field can be static or varying as a function of time. M3 includes spin-transfer torque [3] based on the extensions of Berger's original work [4] by Zhang and Li [5].

M3 includes various high level visualization tools together with functions to convert between cgs and SI units. M3 can take advantage of the parallel computing capabilities of Matlab when executed on a computer cluster.

Do you have questions or suggestions? Please take a look at the M3-Forum.

Problems solved using M3:

µMAG Standard Problem #4

For standard problem #4 the dynamics of a 500 x 125 x 3 nm thick rectangular parallelepiped are calculated. The parameters of the material are as follows:

A = 1.3x10-11J/m (1.3x10-6erg/cm)
Ms = 800 kA/m (800 emu/cc)

= 0.02

= 2.21x105 m/As (1.7595x107 Hz/Oe)

no anisotropy

The initial state is a S-state, which can be obtained by reducing a field along the [1 1 1] direction slowly to zero. Starting from this state a field is applied at t=0 and the dynamics calculated for the next couple of nanoseconds. According to the specification of problem #4 two slightly different switching fields are applied.

For a 2.5 nm discretization in the x- and y-direction and a 1.5 nm discretization in the z-direction, which corresponds to a total of 20000 elements, M3 calculated the following results:



Field 1:

Field 2:


t=0.15[ns]

Reversal movie


t=0.14[ns]

Reversal movie

Average magnetization components:

Average magnetization components:


Reversal of an elliptical disc

This is a slight modification of standard problem #4, which uses an elliptical disc rather than a rectangular parallelepiped, while all other parameters remain the same. For a 5 nm discretization in the x- and y-direction and a 1.5 nm discretization in the z-direction M3 calculated the following results:


Field 1:

Field 2:


t=0.21[ns]

Reversal movie


t=0.13[ns]

Reversal movie

Average magnetization components:


Note: the average includes unoccupied cells.

Average magnetization components:



Proposed Spin-Transfer Torque standard problem

This problem has been proposed in 2009 by Najafi et al. [6] as a possible standard problem for micromagnetic calculations including spin-transfer torque. The problem uses a cuboid with dimensions of 100 x 100 x 10 nm in the x,y,z directions respectively and the following material parameters:

A = 1.3x10-11J/m (1.3x10-6erg/cm)
Ms = 800 kA/m (800 emu/cc)

= 1 for relaxation of the initial configuration and= 0.1 for the calculation including spin-transfer torque

= 2.211x105 m/As (1.7595x107 Hz/Oe)

After the initial relaxation without spin-tranfer torque into a vortex state a homogeneous fully spin polarized current with a magnitude of 1012 A/m2 (2.9979x1017[statC/cm2]) in the x-direction is turned on at t=0, i.e. the electrons flow from right to left.


Initial configuration (no current):




Movie of vortex motion under spin-transfer torque

Final configuration (electrons flowing from right to left):

Time dependence of average magnetization components:



Ferromagnetic resonance calculations:

M3 can be used to calculate ferromagnetic resonance in magnetic structures, the examples below show calculations for structures for which analytical results are available.


Ferromagnetic resonance calculated for a 2nm thin film of Permalloy with the field applied in-the plane of the film (red symbols) and perpendicular to the film plane (blue symbols). The solid lines represent the analytic result using Kittel's formula.


In-plane ferromagnetic resonance for two coupled films of different thickness and saturation magnetization and an interlayer exchange coupling of 0.4 erg/cm2. The solid lines are analytical results based on [7].


Interlayer exchange coupling:

M3 implements both bilinear (J1) and biquadratic (J2) interlayer exchange coupling using modified boundary conditions at the interlayer interfaces. Below the phase diagram as a function of J1 and J2 is shown for the case of identical films with no anisotropy, see for example [8].



Other contributions:

2009: REU summer student A. Montgomery developed and tested scripts to investigate µMAG Standard Problem #3 using M3.



References:

[1]: A.J. Newell, W. Williams, D.J. Dunlop, J. Geophys. Res. 98, 9551 (1993).

[2]: M.J. Donahue, D.G. Porter, Physica B 343, 177 (2004).

[3]: J. Slonczewski, J. Mag. Mag. Mat. 159, 1 (1996), J. Mag. Mag. Mat. 247, 324 (2002).

[4]: L. Berger, Phys. Rev. B 54, 9353 (1996).

[5]: S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).

[6]: M. Najafi et al., J. Appl. Phys. 105, 113914 (2009).

[7]: A. Layadi, J.O. Artman, J.Magn.Magn.Mater 92, 143 (1990).

[8]: S.O. Demokritov, J. Phys. D: Appl. Phys. 31, 925 (1998).



Acknowledgements:

This material is based upon work supported by the National Science Foundation under Grant No. 0804243. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

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