{ "cells": [ { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "# Tutorial 05: Cubic anisotropy energy term\n", "\n", "Cubic anisotropy energy density is computed as\n", "\n", "$$w_{ca} = -K_{1} [(\\mathbf{m} \\cdot\n", " \\mathbf{u}_{1})^{2}(\\mathbf{m} \\cdot\n", " \\mathbf{u}_{2})^{2} + (\\mathbf{m} \\cdot\n", " \\mathbf{u}_{2})^{2}(\\mathbf{m} \\cdot\n", " \\mathbf{u}_{3})^{2} + (\\mathbf{m} \\cdot\n", " \\mathbf{u}_{1})^{2}(\\mathbf{m} \\cdot\n", " \\mathbf{u}_{3})^{2}]$$\n", "\n", "where $\\mathbf{m}$ is the normalised ($|\\mathbf{m}|=1$) magnetisation, $K_{1}$ is the cubic anisotropy constant, and $\\mathbf{u}_{1}$ and $\\mathbf{u}_{2}$ are the anisotropy axes. Cubic anisotropy energy term tends to align all magnetic moments parallel or antiparallel to one of the three anisotropy axes.\n", "\n", "In `oommfc`, $\\mathbf{m}$ is a part of the magnetisation field `system.m`. Therefore, only cubic anisotropy constant $K_{1}$ and axes $\\mathbf{u}_{1}$ and $\\mathbf{u}_{2}$ should be provided as input parameters to uniquely define the energy term. All parameters can be constant in space or spatially varying.\n", "\n", "## Spatially constant $K_{1}$ and $\\mathbf{u}$\n", "\n", "Let us start by assembling a simple simple simulation where neither $K$ nor $\\mathbf{u}$ vary in space. The sample is a \"one-dimensional\" chain of magnetic moments." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import discretisedfield as df\n", "import micromagneticmodel as mm\n", "\n", "import oommfc as oc\n", "\n", "p1 = (-10e-9, 0, 0)\n", "p2 = (10e-9, 1e-9, 1e-9)\n", "cell = (1e-9, 1e-9, 1e-9)\n", "\n", "region = df.Region(p1=p1, p2=p2)\n", "mesh = df.Mesh(region=region, cell=cell)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The mesh is" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "fbb48161494b483ea0a509bb4b0b3352", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mesh.k3d()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The system has a Hamiltonian, which consists of only cubic anisotropy energy term." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "K = 1e5 # cubic anisotropy constant (J/m**3)\n", "u1 = (0, 0, 1) # cubic anisotropy axis\n", "u2 = (0, 1, 0) # cubic anisotropy axis\n", "system = mm.System(name=\"cubicanisotropy_constant_K_u\")\n", "system.energy = mm.CubicAnisotropy(K=K, u1=u1, u2=u2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We are going to minimise the system's energy using `oommfc.MinDriver` later. Therefore, we do not have to define the system's dynamics equation. Finally, we need to define the system's magnetisation (`system.m`). We are going to make it random with $M_\\text{s}=8\\times10^{5} \\,\\text{Am}^{-1}$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "import random\n", "\n", "import discretisedfield as df\n", "\n", "Ms = 8e5 # saturation magnetisation (A/m)\n", "\n", "\n", "def m_fun(pos):\n", " return [2 * random.random() - 1 for i in range(3)]\n", "\n", "\n", "system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The magnetisation, we set is" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "a858f54c8e7946c98c99d55bcc20695a", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "system.m.k3d.vector(color_field=system.m.z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we can minimise the system's energy by using `oommfc.MinDriver`." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:33]... (0.4 s)\n" ] } ], "source": [ "md = oc.MinDriver()\n", "md.drive(system)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We expect that now all magnetic moments are aligned parallel or antiparallel to the anisotropy axes." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "2c3f177da06f480a92ff770db6f7aa4d", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "system.m.k3d.vector(color_field=system.m.z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spatially varying $\\mathbf{K}_{1}$\n", "\n", "There are two different ways how a parameter can be made spatially varying, by using:\n", "1. Dictionary\n", "2. `discretisedfield.Field`\n", "\n", "### Dictionary\n", "\n", "In order to define a parameter using a dictionary, regions must be defined in the mesh. Regions are defined as a dictionary, whose keys are the strings and values are `discretisedfield.Region` objects, which take two corner points of the region as input parameters. " ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "p1 = (-10e-9, 0, 0)\n", "p2 = (10e-9, 1e-9, 1e-9)\n", "cell = (1e-9, 1e-9, 1e-9)\n", "subregions = {\n", " \"region1\": df.Region(p1=(-10e-9, 0, 0), p2=(0, 1e-9, 1e-9)),\n", " \"region2\": df.Region(p1=(0, 0, 0), p2=(10e-9, 1e-9, 1e-9)),\n", "}\n", "region = df.Region(p1=p1, p2=p2)\n", "mesh = df.Mesh(region=region, cell=cell, subregions=subregions)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "30d2759ce6e94350b0f9fef4a55b50a7", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mesh.k3d.subregions()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us say that there is no cubic anisotropy ($K=0$) in region 1, whereas in region 2 it is $K=10^{5} \\,\\text{Jm}^{-3}$. `K` is now defined as a dictionary:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "K = {\"region1\": 0, \"region2\": 1e5}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The system object is" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "system = mm.System(name=\"cubicanisotropy_dict_K\")\n", "system.energy = mm.CubicAnisotropy(K=K, u1=u1, u2=u2)\n", "system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Its magnetisation is" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "23b0040c293249548acb2f1635c4a9e3", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "system.m.k3d.vector(color_field=system.m.z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "After we minimise the energy" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:33]... (0.2 s)\n" ] } ], "source": [ "md.drive(system)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The magnetisation is as we expected." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "b901092885584546b39931e21d97c255", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "system.m.k3d.vector(color_field=system.m.z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### `discretisedfield.Field`\n", "\n", "Let us define the spatailly varying uniaxial anisotropy, so that\n", "\n", "$K(x, y, z) = \\left\\{\n", "\\begin{array}{ll}\n", "0 & x \\le 0 \\\\\n", "1e5 & x > 0 \\\\\n", "\\end{array}\n", "\\right. $\n", "\n", "The value of `K` for the spatially varying anisotropy is set using a Python function." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "def K_fun(pos):\n", " x, y, z = pos\n", " if x <= 0:\n", " return 0\n", " else:\n", " return 1e5" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The uniaxial anisotropy parameters are" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "K = df.Field(mesh, nvdim=1, value=K_fun)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The system is" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "system = mm.System(name=\"cubicanisotropy_field_u\")\n", "system.energy = mm.CubicAnisotropy(K=K, u1=u1, u2=u2)\n", "system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and its magnetisation is" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "8aaf2f77527f4bb196eaac8272434738", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "system.m.k3d.vector(color_field=system.m.z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "After the energy minimisation, the magnetisation is:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:33]... 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