NEBM Simulations

This repository contains all the necessary scripts to run the Nudged Elastic Band method (NEBM) atomistic simulations, using Fidimag.

System

The system under study is an 80 nm X 40 nm X 0.25 nm cobalt nanotrack which is translated into a lattice of 320 X 185 spins which are hexagonally arranged.

Magnetic parameters are:

D     :: VARIABLE meV   DMI constant
J     :: 27.026   meV   Exchange constant
mu_s  :: 0.846    mu_B  Magnetic moment (in Bohr magneton units)
k_u   :: 0.0676   meV   Anisotropy

The simulations are specified for different DMI constants. Folders and scripts have the D parameter in J m**-2 which, in the atomistic simulations, have the following equivalences:

Micromagnetic           Atomistic

D = 26e-4 J m**-2  -->  D = 0.586 meV
D = 28e-4 J m**-2  -->  D = 0.631 meV
D = 30e-4 J m**-2  -->  D = 0.676 meV
D = 32e-4 J m**-2  -->  D = 0.721 meV
D = 34e-4 J m**-2  -->  D = 0.766 meV
D = 36e-4 J m**-2  -->  D = 0.811 meV
D = 38e-4 J m**-2  -->  D = 0.856 meV

Simulations

There are three different NEBM simulations:

1. Skyrmion annihilation at the boundary

2. Skyrmion collapse or skyrmion destruction by a singularity, from linear interpolations as initial state

3. Climbing image NEBM for the NEBM simulations of 2., which results in the skyrmion collapse

For simulation 2., we can obtain either a skyrmion collapse transition or a skyrmion destruction by a singularity. The latest is observed for DMI magnitudes equal or larger than 0.721 meV.

Instructions

To run the simulations, we need:

1. Run the relaxation scripts in sims/relaxation/ to obtain both, the skyrmion and ferromagnetic states as npy files. There is a script for every magnetic configuration and DMI constant. Hence, if we want the simulations for D=0.721 meV, for example, we need to run

   bash stripe_320x185_Co_fm-up_D32e-4_h25e-2nm.sh bash stripe_320x185_Co_sk-down_D32e-4_h25e-2nm.sh

2. With the relaxed states we can now run the NEBM simulations in sims/nebm. Since we have 3 different simulations we have three options:

1. Boundary Annihilation

For this we need to manually generate the initial state for the NEBM. Thus, following the previous example, if we want to run the simulation for D=0.721 meV, we need to

i. Move to sims/nebm/neb_stripe_320x185_Co_sk-down_fm-up_D32e-4_h25e-2nm_GEODESIC

ii. Run the generate_sk-disp_initial_state.sh script. This will create an sk_disp_npys folder with the images for the NEBM initial state

iii. Run the initiate_simulation_skdisp_k1e4.sh script. This will generate the neb_stripe_320x185_Co_sk-down_fm-up_D32e-4_h25e-2nm_GEODESIC_energy.ndt and neb_stripe_320x185_Co_sk-down_fm-up_D32e-4_h25e-2nm_GEODESIC_dYs.ndt files with the data of the energy bands for every iteration of the NEBM.

2. Linear interpolations

For this case, the initial state for the NEBM, which are linear interpolations on the spherical angles that define the spins directions, is automatically generated. Thus it is only necessary to move to the simulation folder, for example, sims/nebm/neb_stripe_320x185_Co_sk-down_fm-up_D32e-4_h25e-2nm_GEODESIC, and run the initiate_simulation_k1e4.sh script. As we mentioned before, some simulations will relax towards a skyrmion collapse and others to the destruction mediated by a singularity.

3. Climbing image NEBM

These simulations are the continuation of the Linear interpolations simulations. Since the climbing image is based on taking the largest energy image in the band and redefining the forces on it, we need to identify this point.

According to our results, we have already identified the climbing images and they are specified on every simulation script (we expect that anyone who runs the simulations will also obtain the same largest energy points). Thus, following the previous examples, if we want to run the simulation for D=0.721 meV, we need to first run the corresponding Skyrmion collapse simulation and then the initiate_simulation_climbing-15_k1e4.sh script, which uses the 15th image of the band as climbing image.

Docker

This repository also contains a Docker file to run a simple example or all the simulations to completely reproduce the data from the paper. We provide in the main folder a Makefile to automatically-easily run them.

When trying to run any of the simulations, the first step of the script is to build a Docker container with Fidimag installed on it, together with all the necessary tools to produce the plots. You can take a look at the Makefile in the docker/ folder for details.

By default, the simulations run with OpenMP using 2 threads.

Example

A simple example produces the data for the case of a DMI constant of D = 0.721 meV (equivalent to D = 3.2 mJ m**-2). This can be called as simply as

make example

These simulations reproduce the three afore mentioned transitions and two kind of plots for every case. One of the plots shows the energy band for the last step of the NEBM and the other is a sequence of snapshots of the NEBM band images, which shows the skyrmion destruction process. Specifically, the snapshots are the magnetisation field coloured according to the out of plane component (z) of the spins. The figures are saved in the nebm_figures folder.

The plot for the energy band when using linear interpolations as initial state is:

And the snapshots look like

Be aware that the system has a significant number of spins, thus the simulations will take a while to finish.

Paper data

To obtain the paper data, running the three different simulations for every DMI value mentioned in the publication, the instruction is just to run

make run_all

This process will probably take a couple of weeks to finish. If you want to use more than 2 threads for OpenMP, just edit the line in docker/Dockerfile which says `ENV OMP_NUM_THREADS=2``.

Every energy band will be plotted in separate files, but if you want to plot all the DMI cases (for a single transition) in a single file, you can call the plot library with `python plot/plot_energy_bands.py --D_list 26 28 32 ...`` for example. See the plot library for more details.