Write Hopfion Tutorial

examples/relax-hopfion.ipynb

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+{
+ "cells": [
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "55f0586c",
+   "metadata": {},
+   "source": [
+    "# Hopfion in a nanodisk\n",
+    "\n",
+    "In this tutorial we relax a hopfion in a chiral magnetic nanodisk and calculate its Hopf index. We consider a nanodisk with a radius $r=100 \\, \\text{nm}$, a thickness $t=70 \\, \\text{nm}$, and ends at the top and bottom of thickness $d=10 \\, \\text{nm}$ in which the unit magnetization field $\\boldsymbol{m}$ is frozen along the $z$-direction.\n",
+    "\n",
+    "This tutorial draws on the works [Y. Liu, R. K. Lake and J. Zang. Binding a hopfion in a chiral magnet nanodisk. Phys. Rev. B 98, 174437 (2019)](https://doi.org/10.1103/PhysRevB.98.174437) and [P. Sutcliffe. Hopfions in chiral magnets. J. Phys. A: Math. Theor. 51 375401 (2018)](https://doi.org/10.1088/1751-8121/aad521)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a19ea274",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import oommfc as oc\n",
+    "import discretisedfield as df\n",
+    "import micromagneticmodel as mm\n",
+    "from discretisedfield.tools import hopf_index"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "60719044",
+   "metadata": {},
+   "source": [
+    "We define the region 'cylinder', in which the magnetization evoles, as well as 'top' and 'bottom', the two fixed layers."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d7e56983",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "r = 100e-9\n",
+    "t = 70e-9\n",
+    "d = 10e-9\n",
+    "\n",
+    "region = df.Region(p1=(-r, -r, -d), p2=(r, r, t+d))\n",
+    "subregions = {'bottom': df.Region(p1=(-r, -r, -d), p2=(r, r, 0)),\n",
+    "              'cylinder': df.Region(p1=(-r, -r, 0), p2=(r, r, t)),\n",
+    "              'top': df.Region(p1=(-r, -r, t), p2=(r, r, t+d))}\n",
+    "mesh = df.Mesh(region=region, cell=(2e-9, 2e-9, 2e-9), subregions=subregions)\n",
+    "\n",
+    "mesh.k3d.subregions()"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "cf2c9340",
+   "metadata": {},
+   "source": [
+    "We consider a system with symmetric exchange and DMI energy."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cf8bd5dd",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "system = mm.System(name='hopfion')\n",
+    "\n",
+    "system.energy = (mm.Exchange(A=2.195e-12)\n",
+    "               + mm.DMI(D=3.95e-4, crystalclass='T'))"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "47e2e851",
+   "metadata": {},
+   "source": [
+    "The magnetization is set to be zero outside of the disk geometry."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4bb00e1e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Ms = 3.84e5\n",
+    "\n",
+    "def Ms_fun(pos):\n",
+    "    x, y, z = pos\n",
+    "    if (x**2 + y**2)**0.5 < r:\n",
+    "        return Ms\n",
+    "    else:\n",
+    "        return 0"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "7b213d51",
+   "metadata": {},
+   "source": [
+    "We initialise a hopfion spin texture as a toroidal skyrmion tube with major radius $L=t/2$, skyrmion radius $R=t/4$, and skyrmion domain wall width of $w=t/8$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1552d5fa",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "L = t/2\n",
+    "R = t/4\n",
+    "w = t/8\n",
+    "\n",
+    "def m_init(pos):\n",
+    "    \n",
+    "    x, y, z = pos\n",
+    "\n",
+    "    # Translate to centre of nanodisk\n",
+    "    z -= t/2\n",
+    "\n",
+    "    # Angle around origin in x-y-plane\n",
+    "    psi = np.arctan2(y, x)\n",
+    "\n",
+    "    # x-coordinate in reference plane with a centre coinciding with the skyrmion's centre\n",
+    "    xPrime = (x - L*np.cos(psi)) * np.cos(psi) + (y - L*np.sin(psi)) * np.sin(psi)\n",
+    "\n",
+    "    # Set magnetization in this reference frame\n",
+    "    rho = np.sqrt(xPrime**2 + z**2)\n",
+    "    Phi = np.arctan2(z, xPrime)\n",
+    "    Theta = 2 * np.arctan2(np.sinh(R/w), np.sinh(rho/w))\n",
+    "    myPrime = np.cos(Phi) * np.sin(Theta)\n",
+    "    mz = np.cos(Theta)\n",
+    "    mxPrime = np.sin(Phi) * np.sin(Theta)\n",
+    "\n",
+    "    # Transform magnetization back to global coordinates\n",
+    "    mx = mxPrime*np.cos(psi) - myPrime*np.sin(psi)\n",
+    "    my = mxPrime*np.sin(psi) + myPrime*np.cos(psi)\n",
+    "    \n",
+    "    return (mx, my, mz)\n",
+    "\n",
+    "m = {'bottom': (0, 0, 1), 'cylinder': m_init, 'top': (0, 0, 1)}\n",
+    "system.m = df.Field(mesh, nvdim=3, value=m, norm=Ms_fun)"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "62ea5962",
+   "metadata": {},
+   "source": [
+    "We can image the initialised hopfion texture."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2bacce76",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [],
+   "source": [
+    "system.m.sel('x').mpl()"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "877ebbd9",
+   "metadata": {},
+   "source": [
+    "We can also calculate the Hopf index of this hopfion texture."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cd01d7b7",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(hopf_index(system.m))"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "bc60b0f0",
+   "metadata": {},
+   "source": [
+    "We relax the system with the subregions 'top' and 'bottom' fixed."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cfc1699e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "md = oc.MinDriver()\n",
+    "md.drive(system, fixed_subregions=['bottom', 'top'])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fa219a39",
+   "metadata": {
+    "scrolled": false
+   },
+   "outputs": [],
+   "source": [
+    "system.m.plane('x').mpl()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e5e9b1bf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(hopf_index(system.m))"
+   ]
+  }
+ ],
+ "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",
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+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
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+}