{ "cells": [ { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "# Tutorial 03: Uniaxial anisotropy energy term\n", "\n", "Uniaxial anisotropy energy density is computed as\n", "\n", "$$w_\\text{ua} = -K (\\mathbf{m}\\cdot\\mathbf{u})^{2}$$\n", "\n", "where $\\mathbf{m}$ is the normalised ($|\\mathbf{m}|=1$) magnetisation, $K$ is the uniaxial anisotropy constant, and $\\mathbf{u}$ is the anisotropy axis. Uniaxial anisotropy energy term tends to align all magnetic moments parallel or antiparallel to the anisotropy axis.\n", "\n", "In `oommfc`, $\\mathbf{m}$ is a part of the magnetisation field `system.m`. Therefore, only uniaxial anisotropy constant $K$ and axis $\\mathbf{u}$ should be provided as input parameters to uniquely define the energy term. Both $K$ and $\\mathbf{u}$ can be constant in space or spatially varying.\n", "\n", "## Spatially constant $K$ 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", "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": "d9c0cee83bab45169d72ed6429b8d59d", "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 uniaxial anisotropy energy term." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "K = 1e5 # uniaxial anisotropy constant (J/m**3)\n", "u = (0, 0, 1) # uniaxial anisotropy axis\n", "system = mm.System(name=\"uniaxialanisotropy_constant_K_u\")\n", "system.energy = mm.UniaxialAnisotropy(K=K, u=u)" ] }, { "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": "90b26a0e0f224ecc876d1c113873dd17", "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 12:40]... (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 axis (in the $z$-direction)." ] }, { "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": "fe05d96816c5460b952932dfd1968fee", "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 $\\mathbf{K}$\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": "501fee133bb24d698af205104f0a915b", "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 uniaxial anisotropy ($K=0$) in region 1, whereas in region 2 it is $K=10^{5} \\,\\text{Jm}^{-3}$. $\\mathbf{u}$ is in the $z$-direction. `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=\"uniaxialanisotropy_dict_K\")\n", "system.energy = mm.UniaxialAnisotropy(K=K, u=u)\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": "4992e7cb52c743d6b0a6f5baced69f10", "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 12:40]... (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": "004bc862692c46a9802d9d47365841e5", "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", "$\\mathbf{u}(x, y, z) = \\left\\{\n", "\\begin{array}{ll}\n", "(0, 0, 1) & x \\le 0 \\\\\n", "(1, 0, 0) & x > 0 \\\\\n", "\\end{array}\n", "\\right. $\n", "\n", "The value of `u` for the spatially varying anisotropy is set using a Python function." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "def u_fun(pos):\n", " x, y, z = pos\n", " if x <= 0:\n", " return (0, 0, 1)\n", " else:\n", " return (1, 0, 0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The uniaxial anisotropy parameters are" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "K = 1e5\n", "u = df.Field(mesh, nvdim=3, value=u_fun)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The system is" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "system = mm.System(name=\"uniaxialanisotropy_field_u\")\n", "system.energy = mm.UniaxialAnisotropy(K=K, u=u)\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": "577a3ff59bee4b91b8c9e0fc79302f30", "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 12:40]... (0.2 s)\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "7d007637145f471d9d35a66091a4287b", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "md.drive(system)\n", "system.m.k3d.vector(color_field=system.m.z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Higher-order uniaxial anisotropy\n", "\n", "In order to define higher-order anisotropy term:\n", "\n", "$$w_\\text{ua} = -K_{1} (\\mathbf{m}\\cdot\\mathbf{u})^{2} - K_{2} (\\mathbf{m}\\cdot\\mathbf{u})^{4}$$\n", "\n", "anisotropy constants `K1` and `K2` must be passed. Similar to previous example, all parameters (`K1`, `K2`, and `u`) can be spatially varying (dictionary or field)." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:40]... (0.2 s)\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "326622afc96a4a61be0f6faa0d325e45", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Output()" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "K1 = 1e5\n", "K2 = 1e3\n", "u = df.Field(mesh, nvdim=3, value=u_fun)\n", "\n", "system = mm.System(name=\"uniaxialanisotropy_higher_order\")\n", "system.energy = mm.UniaxialAnisotropy(K1=K1, K2=K2, u=u)\n", "system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)\n", "\n", "md.drive(system)\n", "system.m.k3d.vector(color_field=system.m.z)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.12" }, "widgets": { 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