Hello Mouad! Thanks for the help, I produced a curve with exchange bias (strange the magnetization goes from 0 to 1 in the Graph1.png )... now I want to vary the number and grain size of the AFM layer, so I understood the Voronoi tessellation produces a vortex in the magnetization, as I associate the command "ext_makegrains (grainSize, maxRegion, randomSeed)" in my case, being that to generate the hysteresis with exchange bias I just used the command "m=vortex" and "relax()", grateful for the help.

Follow the code ( mag.mx3 )!!!

Hello!

See the result that appears in the graph !!

Try to simulate the attached file "mag_01.mx3"

How can I add a spins disorder in the ferromagnetic/antiferromagnetic interface that has Exchange Bias? I want to simulate an interfacial disorder of an FM/AFM bilayer, how can I implement this model in a program like Maple, Matlab or other magnetic simulation program (Mumax...)? Note: I can already simulate magnetization curves that have Exchange Bias.

Reference: https://www.researchgate.net/post/How_can_I_add_a_spins_disorder_in_the_ferromagnetic_antiferromagnetic_interface_that_has_Exchange_Bias

Dear all,

Sadly it is true (and I regret it) that an example of an uncompensated AFM interface (paper 'Modelling exchange bias with MuMax3') was not included in the supplementary material of the paper 'modelling compensated antiferromagnetic interfaces in MuMax3'.

Modelling exchange bias with MuMax3:

https://doi.org/10.1088/0022-3727/49/43/435001

Modelling compensated antiferromagnetic interfaces with MuMax3:

https://doi.org/10.1088/1361-6463/aa8601

However, for reasons of transparency, please find an example code below.

When only dealing with an uncompensated AFM interface, it suffices in MuMax3 to replace the AFM by one layer with thickness equal to the total thickness of the AFM.

The exchange coupling between the FM and the AFM layer can be rescaled (see the papers) to a proper value and can locally be varied by dividing the AFM layer into grains using the Voronoi tesselation. The saturation magnetisation of the AFM layer should be set equal to that of the ferromagnet and demagnetisation energy and the external field should be switched off in the AFM layer. The anisotropy constant K_AFM of the AFM should be set to its real value.

If necessary, one can take into account the external field on the uncompensated AFM interface by appropriately rescaling the value of the applied field. (remember one should put Msat_AFM always equal to Msat_FM, see the supplementary material of the paper 'Modelling compensated antiferromagnetic interfaces with MuMax3'). Please use the relax() function. Limitations of this micromagnetic model is described in the 2 cited papers.

In this model, you obtain exchange bias in the case K_AFM*t_AFM > J_eb with K_AFM the anisotropy constant of the AFM, t_AFM the total thickness of the AFM and J_eb  the surface energy density determined by the exchange interaction at the FM/AFM interface. This situation amounts to pinned AFM grains.

In case K_AFM*t_AFM < J_eb, the AFM grains switch together with the FM layer and so do not lead to exchange bias, but to an enhanced coercivity. In a real system, both types are present and so this model can explain exchange bias (shift of the loop) due to the pinned AFM grains and the enhanced coercivity due to the rotatable AFM grains. This is the case e.g. in the paper 'Modelling exchange bias with MuMax3' when reproducing the experimental hysteresis loop of the Co/CoO bilayer.

For the example script below, we have:

K_AFM = 1.0e5 J/m^3

t_AFM = 3.5e-9 m

J_eb = 0.1e-3 (also labelled as J_I in the paper 'Modelling compensated antiferromagnetic interfaces with MuMax3)

For these values, the theoretical model shows that the bias field should be 35.4 mT. As K_AFM*t_AFM > J_eb, there is no coercivity in the hysteresis loop. Remember that demagnetization energy of the FM is only switched off in this example, in order to be able to compare it to theoretical predictions. (macrospin model) There is no reason to do this in a real system. In the AFM, demagnetization energy should always be switched off. See the function NoDemagSpins.SetRegion(0, 1) in the example code.

When using this model, please cite the corresponding papers.

Nx := 128

Ny := 128

Nz := 2

Cx := 3.5e-9

Cz := 3.5e-9

setgridsize(Nx, Ny, Nz)

setcellsize(Cx, Cx, Cz)

Aex = 1.3e-11

alpha = 0.9

setgeom(universe())

Msat_FM := 800.0e3

A_FM := 1.3e-11

Ku_AFM := 1.0e5

Msat_AFM := Msat_FM

A_AFM := 1.3e-11

DefRegion(1, Layer(1))

NoDemagSpins.SetRegion(1, 1)

Msat.setRegion(1, Msat_FM)

m.setRegion(1, uniform(1,0,0))

Aex.setRegion(1, A_FM)

J_eb := 0.1e-3

DefRegion(0, Layer(0))

Aex.setRegion(0, A_AFM)

NoDemagSpins.SetRegion(0, 1)

Msat.setRegion(0, Msat_AFM)

m.setRegion(0, uniform(1, 0, 0))

Ku1.setregion(0, Ku_AFM)

anisU.setRegion(0, vector(1, 0, 0))

fc_eb := (2 * A_FM * A_AFM / (Msat_FM * Msat_AFM))/(A_FM/Msat_FM + A_AFM/Msat_AFM)

ext_ScaleExchange(0, 1, J_eb*Cz/(fc_eb*(Msat_AFM+Msat_AFM)))

B_app := 0.0

relax()

save(m)

TableAddVar(B_app, "B_app", "T")

TableAdd(m.Region(1))

for i:=600; i>-601; i--{   

B_app = i*0.0005   

B_ext.setregion(1,vector(i*0.0005,0,0.000001))   

tablesave()   

relax()

}

for i:=-599; i

Many thanks for your contributions Jonas. They have been of great help not only to me but to many researchers in the field who study FM/AFM interfacing exchange. This has been perhaps one of the biggest problems so far, to understand the interfacial behavior of the compensated and uncompensated spins in this process. I simulated the code that you shared and this was the result (hysteresis loop with uncompensated spins.png).

I expected the hysteresis loop to behave similarly to the curves of athermal training reproduction in the IrMn/CoFe bilayers of its article (Figure 17). For they are closer to what is observed experimentally in samples that present Exchange Bias.

For example, samples of NiFe/FeMn bilayers show curves with "almost" the same shape (symmetrical and rounded edges) of an FM sample, only displaced relative to the field. Is it possible, considering the formed grain interface, to approach the experimental characteristic presented by these curves of the example or Figure 17?

Dear Orerci,

In a real system the FM will have anisotropy or some AFM grains will rotate together with the FM (due to a strong coupling and low anisotropy so in this case we would have K_AFM*t_AFM < J_eb, in contrast to the case of exchange bias for which K_AFM*t_AFM > J_eb) and so introduce coercivity in the system. In the example code, the uniform AFM stays almost pinned in the field cooling direction and so acts as a unidirectional anisotropy. So in fact it acts as an exchange spring. The small coercivity you see in the figure, obtained from the example code, is a bit artificial. It results from the fact that the symmetry breaking necessary to switch the FM (thats why there is a z-component in the external field) is still a little bit too low.

Figure 17 which you took from the paper, is the case of a mixed AFM interface, i.e. compensated as well as uncompensated spins are present. A realistic example for a fully uncompensated interface is given in the paper 'Modelling exchange bias with MuMax3' (Co/CoO bilayer) or figure 3.6 of my PhD thesis.

Dear Jonas,

Again, thank you for your clarifications.

A last remark: If you are unsure if the scaling factor of the exchange interaction between the ferromagnet and the antiferromagnet is implemented correctly, first check if the bias field of a simulated hysteresis loop matches with the theoretical predicted bias field for the case of a uniform ferromagnet with no anistropy and demagnetization energy (macrospin) , coupled to a uniform fully uncompensated antiferromagnet, whose anisotropy axis is parallel (or almost parallel as you need to break the symmetry of the system) to the applied external field on the FM.

Also check if the bias field vanishes when K_AFM*t_AFM < J_eb and so is turned into coercivity as the AFM switches together with the FM layer in this case.

The theoretical expression of the bias field in case K_AFM*t_AFM > J_eb can be found at equation (26) of Article Exchange Bias Effect of Ferro-/Antiferromagnetic Heterostructures

Thanks Jonas, I'm reading some articles from Radu Florin, but I believe that simulating an actual uncompensated FM/AFM bicamada is not trivial, due to the innumerable peculiarities that Exchange Bias presents, not only from the interface, but from the materials themselves, regarding thickness, mastery walls, etc ... But come on, hands down !!!

Sir

if I want to model with two FM layers (FM/FM) in mumax with different parameters (magnetic, geometry). How we code this thing in mumax?

Dear Yogesh Kumar,

You have to build two regions on Mumax R1 and R2 with different parameters.

R1 will be a cylinder and R2 will be a cuboid. Something like:

R1: = SetGeom (Cylinder (100e-9, 20e9). Trans (0.0, z));

R2: = SetGeom (Cuboid (400e-9, 100e-9, 20e9). Trans (0.0, -z));

Check the page: http://mumax.github.io/api.html