Clarification on Interlayer Exchange Prefactor and Mapping to Experimental RKKY Coupling

[SANDHYA RAVICHANDRAN (RA2433002011028)]

Hi sir,

Thank you again for your previous suggestions regarding the implementation of the interlayer exchange coupling in my MuMax3 simulation. Your comments were very helpful.

I have now updated my implementation based on your suggestion:

prefactorZ := Const(AFMAex / (2 * cellsize * cellsize * Ms_eff))

instead of the original MuMax3 tutorial implementation:

prefactorZ := Const((2 * AFMAex) / (cellsize * cellsize * Ms))

My system consists of a Co (1 nm) / Ru (1 nm) / Co (1 nm) synthetic antiferromagnet, where the magnetic layers are separated by a 1 nm Ru spacer. Your explanation that the exchange-coupled magnetic cells are effectively 2 nm apart makes physical sense for this geometry.

After changing the prefactor, I found that using the original tutorial value

AFMAex = -5 × 10⁻13 J/m

resulted in much weaker coupling and the skyrmion no longer exhibited the expected skyrmion → domain-wall pair → skyrmion conversion.

However, increasing the parameter to approximately

AFMAex ≈ -2 × 10⁻12 J/m

restored essentially the same dynamics observed with the original implementation.

I would like to clarify one important point before preparing my manuscript.

In the MuMax3 tutorial, the relation

Aex,RKKY​=2cz​JRKKY​​

is used, so for cz​=1 nm,

• JRKKY​=−1 mJ/m² corresponds to
• Aex,RKKY​=−5×10−13 J/m.

My question is:

1. After modifying the prefactor to account for the 2 nm center-to-center spacing, is it correct to interpret

AFMAex ≈ -2 × 10⁻12 J/m

as representing approximately the same physical interlayer exchange coupling (i.e., about −1 mJ/m²) but with a different normalization?

Or,

2. Does changing the prefactor require a completely new derivation relating AFMAex to the experimental interlayer exchange coupling JRKKY​?

If the latter is true, could you please suggest the correct way to calibrate the custom-field parameter against experimentally reported RKKY coupling values?

I would greatly appreciate your guidance, as I would like to ensure that the coupling parameter used in my simulations has a physically meaningful interpretation.

Thank you very much for your time and assistance.

[Josh Lauzier]

Hi,

The value of Aex chosen in the workshop is already a value with a nonzero thickness, and accounts for that thickness. It was taken from an existing work (cited in the workshop "Performance of synthetic antiferromagnetic racetrack memory: domain wall vs skyrmion" by Tomacello, 2017). That was for CoNi/Ru/CoNi, with the CoNi and Ru thickness being 0.8nm. You may have slightly different values for Co/Ru(1)/Co. The point of the mumax workshop examples was to show that you can use either an implicit spacer, or an explicit spacer method, and it will give the same result. The factor of 1/d^2 changes if you are varying the spacer thickness; in the mumax tutorial they are modelling the same 1nm spacer system, just in two different ways. They are not comparing going from no spacer to adding a spacer. So that Aex value is already normalized for a 1nm spacer in both scripts. So yes, for question 1) one way to think of it as renormalizing the coupling strength. The Aex value -(2e-12)/ (2nm*2nm) is the same effective field strength as -5e-13/(1nm*1nm). In your case, since you're keeping the spacer layer fixed, you don't really have to worry about it that much, it only really matters if you were to vary the spacer thickness.

If you would like to look at the coupling strength in general, the actual behavior for interlayer exchange in real systems is somewhat complicated. You get oscillations in the strength like a sin wave (from negative AFM coupling to positive FM coupling), depending on the thickness of the spacer, multiplied by an overall decay (there is a 1/d^2 factor, which is what I mentioned previously). So it is highly sensitive to the exact thickness.

For details, see the original work of Stuart Parkin:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.67.3598
A theoretical summary paper by Stiles:
https://www.sciencedirect.com/science/article/abs/pii/S0304885399003340

For a newer reference, see:
https://doi.org/10.1103/PhysRevB.104.024422

Within those references, they give the actual equation form for how it varies, which should make it more clear. If you have experimental data, you should try to match it to your system. If not, I don't know the literature well enough to suggest a value for your specific case. Although that last link happens to be Co/Ru(1)/Co, so it may be a useful place to start. (Note also, the exact values will be sensitive to the growth conditions). They have a J for  Co(1)/Ru(1)/Co(1)  of ~ -1 mJ/m^2 (which for J= 2*A_ex/Cz ends up being roughly Aex= -5e-13, as you calculated), but as you can see from e.g. Fig. 3 J changes rapidly even with slight variations, it is extremely sensitive. Hopefully that is more clear.

Best regards,
Josh L.

[Josh Lauzier]

Oh, also, I should mention, it is a good idea to try to keep discussion replies to the same thread, if possible. It makes it much easier to follow the conversation with full context, instead of submitting multiple new ones, which will be disconnected.

[SANDHYA RAVICHANDRAN (RA2433002011028)]

Hi sir ,

Thank you very much for your guidance,

I am simulating SOT-driven domain wall motion in a Pt/Co nanostrip using MuMax3. In my implementation, the field-like torque is defined as

bj=ξSOT​×aj,

where aj is the damping-like torque strength.

I have seen ξSOT​=−2 used in several MuMax3 simulation codes and also in some research papers. I would like to understand how this value is chosen.

Is ξSOT​=−2 a commonly adopted simulation parameter, or should ξSOT​ always be chosen from experimentally measured FLT/DLT ratios for the specific material system (Pt/Co)? 

Thank you .

[Josh Lauzier]

Hi,

ξ is a material parameter, and you should match it to your specific material system. A lot of people often just stick with the factor of -2 out of convenience since it is not trivial to measure, but strictly speaking it is not correct. Depending on how sensitive your system is to field-like torque, it may not matter much. I am not sure for your use-case, I would try checking the literature for a similar system and see if it has been measured before.

Best,
Josh L.