docs/ipynb/02-getting-started-hamiltonian.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Getting started 02: Hamiltonian\n",
    "\n",
    "> Interactive online tutorial:\n",
    "> [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/ubermag/oommfc/master?filepath=docs%2Fipynb%2Findex.ipynb)\n",
    "\n",
    "The energy of a magnetic system can be computed using Hamiltonian, which consists of different energy terms. In addition, the effective field which affects the magnetisation dynamics in the LLG equation, is also computed from the system's Hamiltonian.\n",
    "\n",
    "In this tutorial, we explore different energy terms that can occur in the Hamiltonian.\n",
    "\n",
    "## Zeeman energy\n",
    "\n",
    "The Zeeman energy tends to align all magnetic moments parallel to the external magnetic field $\\mathbf{H}$, so that the energy density $w_\\text{z} = -\\mu_{0}M_\\text{s}\\mathbf{m}\\cdot\\mathbf{H}$ is minimum, where $\\mu_{0}$ is the magnetic constant and $M_\\text{s}$ is the magnetisation saturation.\n",
    "\n",
    "To demonstrate the effect of Zeeman energy we will apply an external magnetic field $H = 10^{6} \\,\\text{A}\\,\\text{m}^{-1}$ in the positive $z$-direction to the one-dimensional array of magnetic moments. For the initial magnetisation configuration, we will take the $(1, 0, 1)$ direction with $M_\\text{s} = 8 \\times 10^{6} \\,\\text{A}\\,\\text{m}^{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import oommfc as oc\n",
    "import discretisedfield as df\n",
    "import micromagneticmodel as mm\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our mesh is a one-dimensional array of discretisation cells."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "7612af8f879b457d948a3513470a269a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Output()"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = (0, 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(p1=p1, p2=p2, cell=cell)\n",
    "\n",
    "mesh.k3d()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "system = mm.System(name='zeeman')  # create the (micromagnetic) system object\n",
    "\n",
    "H = (0, 0, 1e6)  # external magnetic field (A/m)\n",
    "Ms = 8e6  # saturation magnetisation (A/m)\n",
    "system.energy = mm.Zeeman(H=H)  # define system's Hamiltonian\n",
    "system.m = df.Field(mesh, dim=3, value=(1, 0, 1), norm=Ms)  # define initial magnetisation\n",
    "\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After we defined our system, we can minimize its energy using `MinDriver`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (zeeman.mif) ... (1.0 s)\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "md = oc.MinDriver()  # create energy minimisation driver\n",
    "md.drive(system)  # run energy minimisation\n",
    "\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that after applying the magnetic field, all magnetic moments align parallel to $\\mathbf{H}$.\n",
    "\n",
    "### Exercise 1\n",
    "\n",
    "Change the direction of applied field to be in the negative $x$ direction and inspect the minimum magnetisation state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (zeeman.mif) ... (0.9 s)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Uncomment and change the following line\n",
    "# system.hamiltonian.zeeman.H = (1e6, 0, 0)\n",
    "md.drive(system)\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Uniaxial anisotropy\n",
    "\n",
    "Uniaxial anisotropy energy tends to align magnetic moments to be on the anisotropy axis either parallel or antiparallel without a preferred direction. Its energy density is $w_\\text{a} = -K(\\mathbf{m} \\cdot \\mathbf{u})^{2}$, where $K$ is the anisotropy constant and $\\mathbf{u}$ is the anisotropy axis.\n",
    "\n",
    "Let's assume our anisotropy axis is in the $(1, 0, 1)$ direction with $K = 6 \\times 10^{6} \\,\\text{J}\\,\\text{m}^{-3}$ and the initial magnetisation is defined as\n",
    "\n",
    "$$\\mathbf{m} = \\begin{cases} (-1, 0, -0.1) & \\text{for } x \\le 5 \\,\\text{nm} \\\\ (1, 0, 0.1) & \\text{for } x > 5 \\,\\text{nm} \\end{cases}$$\n",
    "\n",
    "with $M_\\text{s} = 8 \\times 10^{6} \\,\\text{A}\\,\\text{m}^{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "system = mm.System(name='uniaxial_anisotropy')\n",
    "system.energy = mm.UniaxialAnisotropy(K=6e6, u=(1, 0, 1))\n",
    "\n",
    "def m_initial(pos):\n",
    "    x, y, z = pos\n",
    "    if x <= 5e-9:\n",
    "        return (-1, 0, -0.1)\n",
    "    else:\n",
    "        return (1, 0, 0.1)\n",
    "\n",
    "system.m = df.Field(mesh, dim=3, value=m_initial, norm=Ms)\n",
    "\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now relax the magnetisation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (uniaxial_anisotropy.mif) ... (0.9 s)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "md.drive(system)\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that in the relaxed state, all magnetic moments are aligned with the aniostropy axis. The initial configuration affects which orientation along the anisotropy direction is chosen. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exchange energy\n",
    "\n",
    "The Zeeman and uniaxial anisotropy are *local* energy terms; they only depend on the magnetisation at a given location. In contrast, the exchange energy represents a short range interaction and tends to align all spins parallel to each other without a preferential direction. \n",
    "\n",
    "The exchange energy density is $w_\\text{ex} = A[(\\nabla m_{x})^{2} + (\\nabla m_{y})^{2} + (\\nabla m_{z})^{2}]$, where $A$ is the exchange energy constant. To demonstrate how it affects the magnetisation configuration, we will use $A = 8 \\,\\text{pJ}\\,\\text{m}^{-1}$ on a non-uniform magnetisation configuration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "system = mm.System(name='exchange')\n",
    "system.energy = mm.Exchange(A=8e-12)\n",
    "\n",
    "def m_initial(pos):\n",
    "    x, y, z = pos\n",
    "    if x <= 5e-9:\n",
    "        return (0, 0, 1)\n",
    "    else:\n",
    "        return (1, 0, 0)\n",
    "    \n",
    "system.m = df.Field(mesh, dim=3, value=m_initial, norm=Ms)\n",
    "\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (exchange.mif) ... (1.0 s)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "md.drive(system)\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dzyaloshinkii-Moriya energy\n",
    "\n",
    "While the exchange interaction aims to align (neighbouring) magnetic moments parallel to each other, the Dzyaloshinskii-Moriya (DM) energy wants to align them perpendicular to each other.\n",
    "\n",
    "The (interfacial) DMI energy density is $w_\\text{dmi} = D[m_{z}\\nabla\\cdot\\mathbf{m} - (\\mathbf{m}\\cdot\\nabla)m_{z}]$ (crystallographic class $C_\\text{nv}$), with $D$ being the DM constant. Again, we demonstrate its effect by starting from a uniform configuration, with $D = 3\\times10^{3} \\,\\text{J}\\,\\text{m}^{-2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (dmi.mif) ... (0.9 s)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "system = mm.System(name='dmi')\n",
    "system.energy = mm.DMI(crystalclass='Cnv', D=3e-3)\n",
    "system.m = df.Field(mesh, dim=3, value=(0, 0, 1), norm=Ms)\n",
    "\n",
    "md.drive(system)\n",
    "\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exchange and Zeeman energies\n",
    "\n",
    "So far, we investigated the effect of individual energy terms in the Hamiltonian. Now, we look at how multiple energies in the Hamiltoninan contribute to the final magnetisation configuration. We start with the simplest example where only exchange and Zeeman energies are present. As we showed before, exchange energy wants to align all magnetic moments parallel to each other without a preferential direction, while Zeeman energy wants them to be parallel to the external magnetic field. Therefore, we can expect that the minimum energy configuration is going to be the uniform state in the direction of an applied field $\\mathbf{H}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "system = mm.System(name='exchange_and_zeeman')\n",
    "\n",
    "# We can add multiple interactions using the 'plus' operator\n",
    "system.energy = mm.Exchange(A=8e-12) + mm.Zeeman(H=(0, 0, -1e6))\n",
    "\n",
    "def m_initial(pos):\n",
    "    x, y, z = pos\n",
    "    if x <= 5e-9:\n",
    "        return (0, 0, 1)\n",
    "    else:\n",
    "        return (1, 0, 0)\n",
    "    \n",
    "system.m = df.Field(mesh, dim=3, value=m_initial, norm=Ms)\n",
    "\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (exchange_and_zeeman.mif) ... (0.9 s)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "md.drive(system)\n",
    "system.m.plane('y').mpl()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both energies are minimal in the above relaxed state: the magnetisation is uniform (minimising the exchange energy) and is aligned with the external field (minimising the Zeeman energy)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Competition of DMI and exchange\n",
    "\n",
    "Here, we study the competition of the exchange and (interfacial) DMI energy.\n",
    "\n",
    "### Exercise 2\n",
    "\n",
    "Relax a one-dimensional chain of magnetic moments of length $L = 20 \\,\\text{nm}$ with discretisation cell size of $(d, d, d)$, with $d = 1 \\,\\text{nm}$. The Hamiltonian should consist of:\n",
    "\n",
    "- exchange energy with $A=1 \\times 10^{-11} \\,\\text{J}\\,\\text{m}^{-1}$, and\n",
    "- Dzyaloshinskii-Moriya energy with $D = 4\\pi A/L \\approx 6.28 \\times 10^{-3} \\,\\text{J}\\,\\text{m}^{-2}$.\n",
    "\n",
    "The magnetisation saturation is $M_\\mathrm{s} = 8 \\times 10^{6} \\,\\text{A}\\,\\text{m}^{-1}$.\n",
    "Initialise the system with any uniform state. \n",
    "\n",
    "Questions:\n",
    "- What is the relaxed state? \n",
    "- Double the value of DMI constant to be $D = 12.56 \\times 10^{3} \\,\\text{J}\\,\\text{m}^{-2}$. What has changed?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "54df454c55e846f2b2920146b4dacb7c",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Output()"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "system = mm.System(name='exchange_and_DMI')\n",
    "\n",
    "A = 1e-11  # exchange energy constant (J/m)\n",
    "D = 6.28e-3  # DMI energy constant (J/m**2)\n",
    "Ms = 8e6 # Saturation magnetisation (A/m)\n",
    "\n",
    "region = df.Region(p1=(0, 0, 0), p2=(20e-9, 1e-9, 1e-9))\n",
    "mesh = df.Mesh(region=region, cell=(1e-9, 1e-9, 1e-9))\n",
    "m_initial = (0, 1, 1)    # uniform initial magnetisation\n",
    "system.m = df.Field(mesh, dim=3, value=m_initial, norm=Ms)\n",
    "system.m.plane('y').k3d_vectors()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (exchange_and_DMI.mif) ... (1.0 s)\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5a048a18136a419283e7d53a0dde54c8",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Output()"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "system.energy = mm.Exchange(A=A) + mm.DMI(D=D, crystalclass='Cnv')\n",
    "\n",
    "md = oc.MinDriver()  # create energy minimisation driver\n",
    "md.drive(system)\n",
    "\n",
    "system.m.plane('y').k3d_vectors(color_field=system.m.z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020/03/09 10:48: Running OOMMF (exchange_and_DMI.mif) ... (1.0 s)\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "dc60b299ad4c4062bca791a93cee07e8",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Output()"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Uncomment and modify the value of D in the following line to double it\n",
    "# system.hamiltonian.dmi.D = D\n",
    "md.drive(system)\n",
    "system.m.plane('y').k3d_vectors(color_field=system.m.z)"
   ]
  }
 ],
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   "display_name": "Python 3",
   "language": "python",
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     "3007bc9e903d46478dd40a498bfe655d": {
      "model_module": "k3d",
      "model_module_version": "2.7.4",
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      "state": {
       "_backend_version": "2.7.4",
       "_dom_classes": [],
       "_view_count": null,
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       "auto_rendering": true,
       "axes": [
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       "axes_helper": 1,
       "background_color": 16777215,
       "camera": [
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