McPy - A Micromagnetic Monte Carlo Simulation package

Overview

McPy is a specialized package designed for conducting Monte Carlo (Metropolis-Hastings) simulations in micromagnetics for studying magnetic quasiparticles. This has been developed as part of my Master's thesis to study the thermal stability of Bloch Points, a magnetic quasiparticle pivotal for modern data storage solutions. The package has been developed to allow non-zero temperature simulations in micromagnetics using Metropolis-Hastings.

Integration with Ubermag

McPy seamlessly integrates as an extension of Ubermag, an established micromagnetic simulation framework. Ubermag encompasses a suite of independent Python packages suitable for diverse physics simulations, including Computational Fluid Dynamics. While Ubermag already contains well-maintained energy minimisation solvers, the goal of McPy is to extend the capabilities of Ubermag by adding a non-perturbative approach to energy minimisation that has not been implemented till now.

Key Features

• Developed in Python: Simulation conditions can be easily modified and run using Jupyter Notebook.
• Numba Accelerated: Utilizes the Numba "Just-in-time" compiler for performance comparable to C.
• 3D Finite Difference Micromagnetic Model: Enables continuous magnetization.
• Comprehensive Energy Calculations: Supports:
  • Zeeman energy
  • Uniaxial anisotropy energy
  • Exchange energy
  • Dzyaloshinskii-Moriya (DMI) energy

Why Monte Carlo in micromagnetics?

Here are several papers validating the effectiveness of Monte Carlo in this field

• Simulating anti-skyrmions on a lattice
• The skyrmion lattice phase in three-dimensional chiral magnets from Monte Carlo simulations

Installation:

Requirements

    - Python==3.10.11
    - Ubermag with default oommfc driver: Follow the installation guide at [Ubermag Installation](https://ubermag.github.io/installation.html)
    - NumPy==1.24.3
    - Numba==0.57.0
    - cupy-cuda12x==12.1.0

Installation of the above packages can be achieved by running the following command

conda env create -f environment.yml

Usage:

Visualising a magnetic singularity - Bloch Point

The following example is from Stable and Manuplable Bloch Point. This paper produced results using the Landau–Lifshitz–Gilbert (LLG) equation at absolute zero using a finite element solver. However, we will recreate the results using finite difference Monte Carlo at elevated temperatures.

1. Importing Packages

import discretisedfield as df
import micromagneticmodel as mm
import oommfc as oc
from mcpy.system import MCDriver

2. Defining the System parameters

# Magnetisation
Ms = 3.84e5

# Exchange energy constant
A = 8.78e-12

# System geometry
d = 125e-9
hb = 20e-9
ht = 12.5e-9

# Cell discretisation
cell = (5e-9, 5e-9, 2.5e-9)

# Bilayer disk
D_bloch = {'r1': -1.58e-3, 'r2': 1.58e-3, "r1:r2": 1.58e-9}

subregions = {'r1': df.Region(p1=(-d/2, -d/2, -hb), p2=(d/2, d/2, 0)), 'r2': df.Region(p1=(-d/2, -d/2, 0), p2=(d/2, d/2, ht))}

p1 = (-d/2, -d/2, -hb)
p2 = (d/2, d/2, ht)

3. Creating Mesh and assigning Energy terms

# Creating mesh
mesh = df.Mesh(p1=p1, p2=p2, cell=cell, subregions=subregions)

def Ms_fun(point):
        x, y, z = point
        if x**2 + y**2 < (d/2)**2:
                return Ms
        else:
                return 0

system = mm.System(name='bloch_point')

system.energy = mm.Exchange(A=A) + mm.DMI(D=D_bloch, crystalclass='T')
system.m = df.Field(mesh, dim=3, value=(0, 0, 1), norm=Ms_fun)

4. Visualising the system

system.m.plane('z').mpl()
system.m.plane('x').mpl()

5. Monte Carlo Simulation

# Optinal argument for annealing schedule
schedule={'type': 'FC', 'start_temp': 60, 'end_temp': 0.001, 'steps': 20}

# Defining Monte Carlo driver object
mc = MCDriver(system, schedule_name='bloch_point', schedule=schedule)

# 10 million Monte Carlo iterations
mc.drive(N=10000000)

# Visualising the Bloch Point

system.m.plane('z').mpl()
system.m.plane('x').mpl()

# We can observe a "Head-to-Head" Bloch point at the middle of the disk 

Tests:

To run the automated tests

python -m unittest tests.py

Repository structure

• mcpy: The folder containing the main package
  • system.py: MCDriver and Grid classes to initial the Monte Carlo driver object and Grid object
  • driver.py: Python function to run the Monte Carlo simulations
  • energies
    • numpy_energies.py: Numpy optimised energy calculations
    • numba_energies.py: Numba optimised energy calculations
• tests.py: Unittests
• Notebooks:
  • Instructios.ipynb: Instructions on how to use the module
  • Curie_temperature.ipynb: Curie temperature calculations
  • bloch_point.ipynb: Bloch point simulation

Documentation

You can find the documentation at docs/html/index.html.

Outcomes

The full outcomes and analysis can be found in the final report of this project in reports.

License:

The scripts and documentation in this project are released under the MIT License