Calculate domain wall energy density (in unit of J/m^2)

[Zhi Shiuh Lim]

Dear MUMAX community,

Can we use MUMAX3 to output domain wall energy density?

From https://mumax.github.io/api.html,

There is no such function to calculate the domain wall energy density directly.

I notice that "Edens_exch" includes DMI. So I guess it contains exchange energy density and DMI energy density.

On the other hand, I have a set a parameters for thin film that produces hysteretic Skyrmions with magnetic field sweep, that is, if starting magnetic field sweep from negative field saturation towards the positive field direction, then Skyrmions will only be found at the positive small field range but not at the whole negative field range, and vice versa.

Hence, I noticed that the simulated "Edens_exch" is asymmetrical for positive and negative field, but  "Edens_Anis" is symmetrical. Hence, is it valid to calculate domain wall energy density by the following equations at post-simulation data processing:

If H>0, Edens_DMI (H) = Edens_exch (H) - Edens_exch (-H)

If H<0, Edens_DMI (H) = Edens_exch (H) - Edens_exch (H)

Hence, Edens_DMI(H) only has non-zero values at H>0 (asymmetrical).

Likewise:

If H>0, Edens_heisenberg_exch (H) = Edens_exch (-H)

If H<0, Edens_heisenberg_exch (H) = Edens_exch (H)

Hence Edens_heisenberg_exch (H) is symmetrical.

Domain wall energy density (in unit J/m^2) = (4*sqrt(Abs(Edens_heisenberg_exch)*Abs(Edens_Anis))-pi*Abs(Edens_DMI))

*film thickness

Hence the domain wall energy density will be negative at the field range where Skyrmion emerges which also approximately coincides with the range of large topological charge density, but positive at other field range.

Is this method valid? 

But it will fail for cases where Skyrmion emergence is non-hysteretic with field.

Is there a better method?

Thanks!

Regards,

Zhi Shiuh

[Felipe Garcia]

Dear Zhi Shiuh Lim,

The energy of a domain wall depends slightly on the field. Therefore, it is not possible to get it with a procedure like yours. The energy density is a value that depends on the cell.  So it will give you an array of values. If you dump it on the table, it will be averaged. The equation you proposed does not give J/m^2 because the dimensions of the energy density is J/m3.

The best way is to calculate the total energy of a system containing a domain wall and subtract the energy of a system that contains no DW (saturated state). That will give you a value in J. To obtain the value in J/m2, one needs to divide by the lateral surface of the domain, which is normally in PMA systems, that area is the length of the domain wall (obviously not the width) multiplied by the film thickness. One can repeat this for different applied field values.

Best regards,

Felipe

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