![\"S-shaped](\"../images/logo_terminal_currents.png\"\n",){kind=logo}
"
]
},
{
"cell_type": "markdown",
"id": "1d2c474a",
"metadata": {},
"source": [
"Supercurrent in a `superscreen.Device` can be classified into one of three categories:\n",
"\n",
"1. **Screening currents**, which are the response of a film to an applied magnetic field.\n",
"2. **Circulating currents**, which are the response of a film to flux trapped in a hole.\n",
"3. **Terminal currents**, which flow into and out of a film through \"transport terminals\" defined on the film boundary.\n",
"\n",
"In this notebook we demonstrate the treatment of **terminal currents** in a `superscreen` model."
]
},
{
"cell_type": "markdown",
"id": "5aff8bea",
"metadata": {},
"source": [
"## Background\n",
"\n",
"We make the following assumptions regarding terminal currents:\n",
"\n",
"1. The current density $|\\vec{J}|$ is uniformly distributed along the terminal.\n",
"2. Along the terminal, the current direction $\\vec{J}/|\\vec{J}|$ is perpendicular to the terminal direction.\n",
"\n",
"The stream function $g(\\vec{r})=g(x, y)$ and 2D supercurrent density $\\vec{J}(x, y)$ are related according to $\\vec{J}=\\vec{\\nabla}\\times(g\\hat{z})=(\\partial g/\\partial y, -\\partial g/\\partial x)$. For some assumed supercurrent distribution $\\vec{J},$ the associated stream function $g$ is given by a line integral,\n",
"\n",
"$$g(\\vec{r})=g(\\vec{r}_0) + \\int_{\\vec{r}_0}^\\vec{r}(\\hat{z}\\times\\vec{J})\\cdot\\mathrm{d}\\vec{\\ell},$$\n",
"\n",
"where $\\hat{z}\\times\\vec{J}=(-J_y, J_x)$ and $\\vec{r}_0$ is some reference position.\n",
"\n",
"The stream function $g$ in a film can be decomposed into $g=g_\\mathrm{screening}+g_\\mathrm{transport},$ where $g_\\mathrm{screening}$ arises from the response to an applied magnetic field or trapped flux, and $g_\\mathrm{transport}$ corresponds to an applied current bias. Below we describe how to set $g_\\mathrm{transport}(\\vec{r})$ for a given current bias configuration."
]
},
{
"cell_type": "markdown",
"id": "e5dfbf85",
"metadata": {},
"source": [
"### Stream function on the boundary\n",
"\n",
"The boundary conditions for terminal currents are as follows:\n",
"\n",
"- The boundary of the film is oriented in a counterclockwise direction.\n",
"- The stream function $g_\\mathrm{transport}(\\vec{r})$ along a source terminal $s$ injecting current $I_s$ changes linearly by a total amount $-I_s,$ and $g_\\mathrm{transport}(\\vec{r})$ for $\\vec{r}$ along the terminal is given by the equation above, where $\\vec{r}_0$ is the start of the terminal.\n",
"- The stream function $g_\\mathrm{transport}(\\vec{r})$ on the boundary between terminals is constant.\n",
"- The sum of all terminal currents must be zero to ensure current conservation."
]
},
{
"cell_type": "markdown",
"id": "39812c6a",
"metadata": {},
"source": [
"### Stream function in the bulk\n",
"\n",
"Once $g_\\mathrm{transport}(\\vec{r}_\\text{boundary})$ has been defined for all $\\vec{r}_\\text{boundary}$ on the film's boundary, we solve for $g_\\mathrm{transport}(\\vec{r}_\\text{bulk})$ inside the film as follows.\n",
"\n",
"1. First we find an \"effective field\" $H_{z,\\,\\text{eff}}(\\vec{r})$ that would generate such a $g_\\mathrm{transport}(\\vec{r}_\\text{boundary})$.\n",
"2. Once $H_{z,\\,\\text{eff}}(\\vec{r})$ is found, $g_\\mathrm{transport}(\\vec{r}_\\text{bulk})$ is given by the response of the film to this applied effective field."
]
},
{
"cell_type": "markdown",
"id": "99d51496",
"metadata": {},
"source": [
"### Stream function in holes\n",
"\n",
"For films with holes, we first solve for $g_\\mathrm{transport}(\\vec{r})$ as described [above](#Stream-function-in-the-bulk) *assuming that there are no holes*. Then, for each hole $h$ we update $g_\\mathrm{transport}(\\vec{r})$ for $\\vec{r}$ inside $h$ to be equal to the average value over the hole area:\n",
"$$g_\\mathrm{transport}(\\vec{r})\\to \\frac{1}{\\mathrm{area}(h)}\\int_h g_\\mathrm{transport}(\\vec{r}')\\,\\mathrm{d}^2r'\\quad\\forall\\:\\vec{r}\\text{ in hole }h$$"
]
},
{
"cell_type": "markdown",
"id": "19b5d1b2",
"metadata": {},
"source": [
"## Examples"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "17e0602f",
"metadata": {},
"outputs": [],
"source": [
"# Automatically install superscreen from GitHub only if running in Google Colab\n",
"if \"google.colab\" in str(get_ipython()):\n",
" %pip install --quiet git+https://github.com/loganbvh/superscreen.git"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "a5a36791",
"metadata": {},
"outputs": [],
"source": [
"%config InlineBackend.figure_formats = {\"retina\", \"png\"}\n",
"%matplotlib inline\n",
"\n",
"import os\n",
"\n",
"os.environ[\"OPENBLAS_NUM_THREADS\"] = \"1\"\n",
"\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"plt.rcParams[\"figure.figsize\"] = (5, 4)\n",
"plt.rcParams[\"font.size\"] = 10\n",
"\n",
"import superscreen as sc\n",
"from superscreen.geometry import box, circle, rotate"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "785f6040",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"| Software | Version |
|---|---|
| SuperScreen | 0.13.0 |
| Numpy | 2.4.4 |
| Numba | 0.65.1 |
| SciPy | 1.17.1 |
| matplotlib | 3.10.9 |
| IPython | 9.13.0 |
| Python | 3.14.4 | packaged by conda-forge | (main, Apr 8 2026, 02:33:53) [Clang 20.1.8 ] |
| OS | posix [darwin] |
| Number of CPUs | Physical: 10, Logical: 10 |
| BLAS Info | Generic |
| Tue Apr 28 20:44:03 2026 EDT | |