simulation Magnetic Force Microscopy (MFM) unit in Mumax

[DongHi Yang]

Hi everyone,

I encountered a question regarding the units in MFM simulations with Mumax. The color bar of the simulated MFM output (e.g., in .ovf files) often displays values around 10^11, whereas experimental MFM results are typically calibrated in degrees ("deg"). I would like to understand the difference between the MFM unit used in Mumax simulations and the unit commonly used in experiments. Are there any established methods to convert the Mumax MFM output values to the experimental unit "deg"?

Thank you very much for your help!

[Josh Lauzier]

Hi,

I don't have a lot of experience with the MFM module, so perhaps someone can answer better. But looking at the source, the official documentation is that it gives back the result in arbitrary units, because it just computes the convolution of the tip with the stray field. From the original mumax3 paper, "The design and verification of mumax3": No attempt is made to reproduce tip fields in absolute terms as our only goal is to produce output proportional to the actual MFM contrast, like shown in Fig.20.

So it should be directly proportional to degrees? The formula for the phase shift in degrees can be found on this paper from Ubermag, Eq 11. If I am reading the mumax paper right (in particular Eq 31), I think mumax may only do the second term in Eq 11, and only the M dot dH_s/dz^2 portion (this can be rewritten as dF/dz).  Also, peeking at the source code, while mumax has both monopole and dipole options in the MFM module, it seems to use the dipole expression for both- for the 'monopole' it just uses a dipole with a tiplength set very far away, hence why it only uses the second term in the phase shift equation.

The numerical values Mumax gives back when you save the MFM data are those in Eq 31, for the derivative of the force.  In order to convert to units of degrees, one needs parameters for the tip including the quality factor Q and the spring constant k. ( In case it is useful: numerically the value of the monopole charge(s) is taken to be 1/u_0, and in the expression for dF/dz, when this is discretized the deltaz is taken to be 1e-9.)

However, there is one final large caveat if you are trying to match experimental data exactly. The formula given in Eq11 has a lot of assumptions that go into simplifying it, and they require making assumptions about things like the magnetization distribution of the tip. Usually MFM ends up being somewhat qualitative because of this, because the signal is sensitive to the exact shape/magnetization of the tip which won't match the simplified assumption, thus to get an exact match requires you to calibrate/characterize each new tip. Exactly quantitatively modeling MFM can end up being quite complicated, it is it's own area of research. You can check the references on the Ubermag paper for more detail. So be aware even if you get a value in units of degrees it may not line up exactly to experiment.

In short: Technically it can be done but realistically it will probably end up qualitative anyway, just scaled differently, so you probably want to treat it as directly proportional already.

Best,
Josh L.

[DongHi Yang]

Dear Josh,

Thank you for your detail explained the MFM in mumax3 and compared it with Ubermag. Your insights are very helpful! 

Now I am more clearly of the MFM in numerical method. Thanks again :)

Best regards,
DongHi