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    "# COMSOL transmon capacitance and field\n",
    "\n",
    "::::{admonition} Required extras\n",
    ":class: tip\n",
    "\n",
    "This notebook needs the `comsol` extra (`uv sync --extra comsol`). It installs `MPh`, the Python client for\n",
    "COMSOL, and nothing else: not COMSOL and not a license, so the build and solve cells need a local\n",
    "installation and cannot run on Colab. See the {ref}`extras reference <notebook-extras>`.\n",
    "::::\n",
    "\n",
    "This tutorial walks the path end to end: build a QPDK double-pad\n",
    "transmon, copy it without the Josephson junction layers, extract the metal layout, run a stationary\n",
    "**Electrostatics** solve on a COMSOL sheet model, and read the capacitance and the potential and field the\n",
    "solve stores. The figures are **saved cell outputs** of licensed solves; without a license the COMSOL cells\n",
    "are skipped and the result cells say so.\n",
    "\n",
    "## What is modelled, and what is solved\n",
    "\n",
    "The device is a QPDK {py:func}`~qpdk.cells.transmon.double_pad_transmon_with_bbox`: two pads joined\n",
    "**only** through the Josephson junction, with a SQUID loop at the centre, so the qubit frequency follows\n",
    "from $E_\\text{J}$ and $E_\\text{C} = e^2 / 2C$ {cite:p}`kochChargeinsensitiveQubitDesign2007a`. The sheet\n",
    "model cannot represent the junction overlap or its barrier, so the extraction uses an **EM-only copy** with\n",
    "the junction layers removed: the SQUID loop and its leads are absent from the solved geometry.\n",
    "\n",
    "The solve adds **Electrostatics**, not electromagnetic waves, and returns `es.C11`, the driven pad's\n",
    "capacitance to the grounded rest of the chip, plus `es.intWe`, which agree through\n",
    "$2 W_\\text{e} / V^2 = C_{11}$ {cite:p}`m.pozarMicrowaveEngineering2012`. It is a **quasi-static extraction**,\n",
    "with no Josephson inductance and no resonance solved.\n",
    "\n",
    "::::{only} html\n",
    "```{mermaid}\n",
    "flowchart LR\n",
    "    A[\"Transmon cell\"] --> B[\"EM-only copy\"] --> C[\"Extracted layout\"] --> D[\"Sheet model\"] --> E[\"Electrostatics\"] --> F[\"Mesh, stationary solve\"] --> G[\"C11, intWe, V, normE\"]\n",
    "```\n",
    "::::\n",
    "\n",
    "::::{only} typst or typstpdf\n",
    "The pipeline: transmon cell, EM-only copy with the junction layers removed, extracted layout of pads and\n",
    "ground plane, sheet model with air over silicon, Electrostatics solve driving the left pad, mesh, and the\n",
    "capacitance, energy, and field results.\n",
    "::::\n",
    "\n",
    "::::{admonition} Reading the numbers on this page\n",
    ":class: warning\n",
    "\n",
    "The values come from a licensed electrostatic solve, but they are not a validated device prediction: the\n",
    "SQUID loop and leads are absent, and $C_{11}$ is a **one-terminal** value, the driven pad against the\n",
    "grounded chip, not the two-pad differential capacitance that sets $E_\\text{C}$\n",
    "{cite:p}`blaisCircuitQuantumElectrodynamics2021`. Nothing here is a transmon eigenmode or an $f_{01}$.\n",
    "::::\n",
    "\n",
    "**References:**\n",
    "- QPDK transmon physics and the double-pad cell: {py:func}`~qpdk.cells.transmon.double_pad_transmon_with_bbox`\n",
    "- [COMSOL Electrostatics interface](https://doc.comsol.com/6.3/doc/com.comsol.help.acdc/acdc_ug_electric_fields.07.002.html)\n",
    "- [COMSOL capacitance example](https://doc.comsol.com/6.3/doc/com.comsol.help.models.acdc.capacitor_dc/capacitor_dc.html)\n",
    "- [MPh tutorial](https://mph.readthedocs.io/en/stable/tutorial.html)"
   ]
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   "metadata": {},
   "source": [
    "## Setup and imports"
   ]
  },
  {
   "cell_type": "code",
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   "id": "0a91fb01",
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    "execution": {
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     "iopub.status.busy": "2026-09-30T09:16:17.277174Z",
     "iopub.status.idle": "2026-09-30T09:16:17.279325Z",
     "shell.execute_reply": "2026-09-30T09:16:17.279091Z"
    },
    "tags": [
     "hide-input",
     "hide-output"
    ]
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   "source": [
    "import sys\n",
    "\n",
    "if \"google.colab\" in sys.modules:\n",
    "    import subprocess\n",
    "\n",
    "    print(\"Running in Google Colab. Installing QPDK...\")\n",
    "    subprocess.check_call([\n",
    "        sys.executable,\n",
    "        \"-m\",\n",
    "        \"pip\",\n",
    "        \"install\",\n",
    "        \"-q\",\n",
    "        \"qpdk[comsol] @ git+https://github.com/gdsfactory/quantum-rf-pdk.git\",\n",
    "    ])\n",
    "    print(\n",
    "        \"Note: this installs the Python client only. COMSOL itself and its \"\n",
    "        \"license are not pip-installable, so the build and solve cells below \"\n",
    "        \"cannot run in Colab, and the result cells will report that no exported \"\n",
    "        \"results are present.\"\n",
    "    )"
   ]
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   "id": "927d7bda",
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    "execution": {
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    },
    "tags": [
     "hide-input",
     "hide-output"
    ]
   },
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   "source": [
    "import json\n",
    "import os\n",
    "from pathlib import Path\n",
    "from typing import Any\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.tri as mtri\n",
    "import numpy as np\n",
    "from matplotlib import axes as mpl_axes, font_manager\n",
    "from matplotlib.colors import LogNorm\n",
    "\n",
    "from qpdk import PDK\n",
    "from qpdk.cells.transmon import double_pad_transmon_with_bbox\n",
    "from qpdk.config import PATH\n",
    "from qpdk.simulation import prepare_comsol_layout\n",
    "from qpdk.simulation.comsol.plotting import draw_layout_polygons\n",
    "from qpdk.simulation.comsol.results import (\n",
    "    explain_missing_results,\n",
    "    result_file,\n",
    "    write_json_atomically,\n",
    ")\n",
    "from qpdk.tech import LAYER, LAYER_STACK, get_layer_material_properties\n",
    "\n",
    "try:\n",
    "    import mph\n",
    "\n",
    "    from qpdk.simulation import COMSOL\n",
    "except ImportError:\n",
    "    mph = None\n",
    "    COMSOL = None\n",
    "\n",
    "PDK.activate()\n",
    "\n",
    "MPH_AVAILABLE = mph is not None\n",
    "\n",
    "# The checkout stylesheet is absent from installed wheels.\n",
    "for _style in (PATH.docs / \"qpdk.mplstyle\", \"qpdk\"):\n",
    "    try:\n",
    "        plt.style.use(_style)\n",
    "    except OSError:\n",
    "        continue\n",
    "    break\n",
    "\n",
    "# Re-running setup must not wrap Axes.set_title again.\n",
    "if \"Outfit\" in {font.name for font in font_manager.fontManager.ttflist} and not getattr(\n",
    "    mpl_axes.Axes, \"_qpdk_outfit_titles\", False\n",
    "):\n",
    "    _original_set_title = mpl_axes.Axes.set_title\n",
    "\n",
    "    def _qpdk_set_title(self: mpl_axes.Axes, *args: Any, **kwargs: Any) -> Any:\n",
    "        kwargs.setdefault(\"fontfamily\", \"Outfit\")\n",
    "        kwargs.setdefault(\"fontweight\", \"bold\")\n",
    "        return _original_set_title(self, *args, **kwargs)\n",
    "\n",
    "    mpl_axes.Axes.set_title = _qpdk_set_title\n",
    "    mpl_axes.Axes._qpdk_outfit_titles = True\n",
    "\n",
    "# Keep glyphs in the SVG for viewers without the documentation fonts.\n",
    "plt.rcParams[\"svg.fonttype\"] = \"path\"\n",
    "try:\n",
    "    from matplotlib_inline.backend_inline import set_matplotlib_formats\n",
    "except ImportError:\n",
    "    pass\n",
    "else:\n",
    "    set_matplotlib_formats(\"svg\", \"png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19a81ac3",
   "metadata": {},
   "source": [
    "## Build the transmon cell and an EM-only copy\n",
    "\n",
    "`prepare_comsol_layout` rejects geometry on layers it does not model, the junction layers among them. The\n",
    "cell is copied and `JJ_AREA` and `JJ_PATCH` removed, leaving `M1_DRAW` and the `M1_ETCH` mask."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1c4948b6",
   "metadata": {
    "execution": {
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     "iopub.status.busy": "2026-09-30T09:16:19.389262Z",
     "iopub.status.idle": "2026-09-30T09:16:19.404720Z",
     "shell.execute_reply": "2026-09-30T09:16:19.404279Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Component: double_pad_transmon_with_bbox_BE200_PS250_400_PG15_JSsq_c3fdad3b\n",
      "Bounding box (µm): (-457.5,-400;457.5,400)\n",
      "Junction site: center=(0.0, 0.0) µm\n",
      "Layers kept:   [(1, 0), (1, 1)]\n",
      "JJ_AREA/JJ_PATCH removed; M1_DRAW and M1_ETCH remain.\n"
     ]
    }
   ],
   "source": [
    "component = double_pad_transmon_with_bbox(bbox_extension=200.0)\n",
    "\n",
    "junction_port = component.ports[\"junction\"]\n",
    "JUNCTION_CENTER_UM = tuple(junction_port.center)\n",
    "\n",
    "em_component = component.copy()\n",
    "em_component.remove_layers(layers=[LAYER.JJ_AREA, LAYER.JJ_PATCH])\n",
    "\n",
    "print(f\"Component: {component.name}\")\n",
    "print(f\"Bounding box (µm): {component.bbox()}\")\n",
    "print(f\"Junction site: center={JUNCTION_CENTER_UM} µm\")\n",
    "print(f\"Layers kept:   {sorted(em_component.layers)}\")\n",
    "print(\"JJ_AREA/JJ_PATCH removed; M1_DRAW and M1_ETCH remain.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c219b0d",
   "metadata": {},
   "source": [
    "## Extract the COMSOL layout\n",
    "\n",
    "The pads are not transmission-line ports, so no feed ports are requested. The etched moat is a *hole* in\n",
    "the ground-plane polygon; the pads sit inside it as separate polygons."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a8c21bd8",
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     "shell.execute_reply": "2026-09-30T09:16:19.408981Z"
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Metal polygons: 3\n",
      "Feed ports:     ()\n",
      "Prepared bbox (µm): ComsolBoundingBox(xmin=-557.5, ymin=-500.0, xmax=557.5, ymax=500.0)\n"
     ]
    }
   ],
   "source": [
    "layout = prepare_comsol_layout(em_component, feed_ports=None, ground_margin=100.0)\n",
    "\n",
    "print(f\"Metal polygons: {len(layout.polygons)}\")\n",
    "print(f\"Feed ports:     {layout.feed_ports}\")\n",
    "print(f\"Prepared bbox (µm): {layout.bbox}\")\n",
    "\n",
    "# Three conductor faces: the two pads and the ground plane. Each point below\n",
    "# sits well inside its metal region so the face selection resolves to exactly\n",
    "# one face, which the study builder checks before adding the terminals.\n",
    "LEFT_PAD_POINT = (-132.5, 0.0)\n",
    "RIGHT_PAD_POINT = (132.5, 0.0)\n",
    "GROUND_POINT = (-500.0, 0.0)\n",
    "\n",
    "# The study names one face selection per conductor after these tags, so the mesh\n",
    "# sizes and the terminal and ground arguments below can name a face by tag.\n",
    "PAD_L_SELECTION = \"pad_l\"\n",
    "PAD_R_SELECTION = \"pad_r\"\n",
    "GROUND_SELECTION = \"gnd\"\n",
    "CONDUCTORS = (\n",
    "    (PAD_L_SELECTION, LEFT_PAD_POINT),\n",
    "    (PAD_R_SELECTION, RIGHT_PAD_POINT),\n",
    "    (GROUND_SELECTION, GROUND_POINT),\n",
    ")"
   ]
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      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
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     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(6, 6))\n",
    "draw_layout_polygons(ax, layout.polygons)\n",
    "for point, label, offset in (\n",
    "    (LEFT_PAD_POINT, \"left pad\", (-35, 10)),\n",
    "    (RIGHT_PAD_POINT, \"right pad\", (8, 10)),\n",
    "    (GROUND_POINT, \"ground\", (8, 8)),\n",
    "    (JUNCTION_CENTER_UM, \"junction site\", (8, -15)),\n",
    "):\n",
    "    ax.plot(*point, marker=\"x\", color=\"crimson\", markersize=7, zorder=4)\n",
    "    ax.annotate(\n",
    "        label,\n",
    "        point,\n",
    "        textcoords=\"offset points\",\n",
    "        xytext=offset,\n",
    "        fontsize=9,\n",
    "        color=\"crimson\",\n",
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    "ax.set_aspect(\"equal\")\n",
    "ax.set_xlabel(\"x (µm)\")\n",
    "ax.set_ylabel(\"y (µm)\")\n",
    "ax.set_title(\"Pads, ground plane, and the conductor points\")\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c678afc",
   "metadata": {},
   "source": [
    "## Build the COMSOL model and capacitance study\n",
    "\n",
    "Three calls configure the model: {py:meth}`~qpdk.simulation.comsol.model.COMSOL.create_sheet` makes the\n",
    "air and silicon blocks meeting at $z = 0$, imprints the layout metal on that interface as faces, and\n",
    "assigns materials;\n",
    "{py:meth}`~qpdk.simulation.comsol.model.COMSOL.add_capacitance_study` picks the pad and ground faces from\n",
    "the points above, drives the left pad, grounds the right pad and the ground plane, and adds the mesh and a\n",
    "stationary study; and {py:meth}`~qpdk.simulation.comsol.model.COMSOL.pin_absolute_mesh_sizes` reruns that\n",
    "mesh with absolute element sizes in micrometres, so the near-metal resolution does not move with the\n",
    "domain.\n",
    "\n",
    "The silicon thickness and permittivity are read from the `Substrate` level of the PDK layer stack and its\n",
    "material in `qpdk.tech.material_properties`, so they match every other QPDK simulation. The metal is a\n",
    "zero-thickness sheet, so the M1 film thickness does not enter. The air height and lateral margin only size\n",
    "the simulation domain.\n",
    "\n",
    "`RUN_COMSOL` is `False` by default, so a documentation build without a license skips all of this and the\n",
    "result cells below read saved exports instead. Set it to `True` on a licensed machine to solve and export.\n",
    "The constants are the configuration the saved exports were produced at."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "2545cd6e",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-09-30T09:16:19.629908Z",
     "iopub.status.busy": "2026-09-30T09:16:19.629854Z",
     "iopub.status.idle": "2026-09-30T09:16:19.632308Z",
     "shell.execute_reply": "2026-09-30T09:16:19.631959Z"
    }
   },
   "outputs": [],
   "source": [
    "RUN_COMSOL = False\n",
    "MODEL_DIR = Path.home() / \"comsol_models\"\n",
    "RESULTS_DIR_ENV = \"QPDK_COMSOL_RESULTS_DIR\"\n",
    "# The environment variable wins when it is set and non-empty, so a run without a\n",
    "# license can still read a licensed run's exports.\n",
    "_results_from_environment = os.environ.get(RESULTS_DIR_ENV)\n",
    "RESULTS_DIR: Path | None = (\n",
    "    Path(_results_from_environment).expanduser()\n",
    "    if _results_from_environment\n",
    "    else MODEL_DIR\n",
    "    if RUN_COMSOL\n",
    "    else None\n",
    ")\n",
    "\n",
    "MODEL_PATH = MODEL_DIR / \"comsol_qubit_capacitance.mph\"\n",
    "METRICS_JSON = \"comsol_qubit_metrics.json\"\n",
    "FIELD_TXT = \"comsol_qubit_field.txt\"\n",
    "CORES = 4\n",
    "VOLTAGE_V = 1.0\n",
    "\n",
    "# The substrate thickness and permittivity come from the PDK layer stack; the air\n",
    "# height and lateral margin size the simulation domain, in µm.\n",
    "SUBSTRATE_THICKNESS_UM = LAYER_STACK.layers[\"Substrate\"].thickness\n",
    "SILICON_RELATIVE_PERMITTIVITY = get_layer_material_properties(\"Substrate\")[\n",
    "    \"relative_permittivity\"\n",
    "]\n",
    "AIR_HEIGHT_UM = 1600.0\n",
    "LATERAL_MARGIN_UM = 8000.0\n",
    "# Element size COMSOL's physics-controlled build starts from. The pinned sizes\n",
    "# below replace it, so it only sets the sizing the sequence is materialised with.\n",
    "BASE_MESH_SIZE = 2\n",
    "GLOBAL_HMAX_UM = 1000.0\n",
    "GLOBAL_HMIN_UM = 2.0\n",
    "HGRAD = 1.4\n",
    "HCURVE = 0.5\n",
    "HNARROW = 0.7\n",
    "# Near-metal element sizes (hmax, hmin) in µm, one per conductor face. Each hmin\n",
    "# is a tenth of its hmax.\n",
    "NEAR_METAL = \"0.625/1.25\"\n",
    "PAD_HMAX_UM, PAD_HMIN_UM = 0.625, 0.0625\n",
    "GROUND_HMAX_UM, GROUND_HMIN_UM = 1.25, 0.125\n",
    "\n",
    "if RUN_COMSOL and not MPH_AVAILABLE:\n",
    "    raise RuntimeError(\"RUN_COMSOL needs MPh and a licensed COMSOL installation\")"
   ]
  },
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   "metadata": {},
   "source": [
    "### Solve, save, and export\n",
    "\n",
    "One licensed cell: build, mesh, and solve, then save the model, write `comsol_qubit_metrics.json`, and\n",
    "export $V$ and `es.normE` on a cut plane at $z = 1\\,\\text{µm}$ into `comsol_qubit_field.txt` through\n",
    "COMSOL's Java Data export."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "93d1d16d",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-09-30T09:16:19.633182Z",
     "iopub.status.busy": "2026-09-30T09:16:19.633135Z",
     "iopub.status.idle": "2026-09-30T09:16:19.636185Z",
     "shell.execute_reply": "2026-09-30T09:16:19.635922Z"
    }
   },
   "outputs": [],
   "source": [
    "model = None\n",
    "\n",
    "if RUN_COMSOL and MPH_AVAILABLE:\n",
    "    MODEL_DIR.mkdir(parents=True, exist_ok=True)\n",
    "    client = mph.start(cores=CORES)\n",
    "    model = COMSOL.create_sheet(\n",
    "        client,\n",
    "        layout,\n",
    "        name=\"QPDK Double-Pad Transmon\",\n",
    "        substrate_thickness_um=SUBSTRATE_THICKNESS_UM,\n",
    "        air_height_um=AIR_HEIGHT_UM,\n",
    "        lateral_margin_um=LATERAL_MARGIN_UM,\n",
    "        silicon_relative_permittivity=SILICON_RELATIVE_PERMITTIVITY,\n",
    "    ).add_capacitance_study(\n",
    "        conductors=CONDUCTORS,\n",
    "        terminal=PAD_L_SELECTION,\n",
    "        grounds=(PAD_R_SELECTION, GROUND_SELECTION),\n",
    "        voltage_v=VOLTAGE_V,\n",
    "        mesh_size=BASE_MESH_SIZE,\n",
    "    )\n",
    "    element_count = model.pin_absolute_mesh_sizes(\n",
    "        global_hmax_um=GLOBAL_HMAX_UM,\n",
    "        global_hmin_um=GLOBAL_HMIN_UM,\n",
    "        face_sizes={\n",
    "            PAD_L_SELECTION: (PAD_HMAX_UM, PAD_HMIN_UM),\n",
    "            PAD_R_SELECTION: (PAD_HMAX_UM, PAD_HMIN_UM),\n",
    "            GROUND_SELECTION: (GROUND_HMAX_UM, GROUND_HMIN_UM),\n",
    "        },\n",
    "        hgrad=HGRAD,\n",
    "        hcurve=HCURVE,\n",
    "        hnarrow=HNARROW,\n",
    "    )\n",
    "    model.java.study(\"std1\").run()\n",
    "    for problem in model.problems():\n",
    "        print(f\"The solve reports: {problem}\")\n",
    "\n",
    "    (MODEL_DIR / FIELD_TXT).unlink(missing_ok=True)\n",
    "    model.save(MODEL_PATH)\n",
    "\n",
    "    capacitance_f = float(model.evaluate(\"es.C11\"))\n",
    "    stored_energy_j = float(model.evaluate(\"es.intWe\"))\n",
    "    print(f\"es.C11 = {capacitance_f:.6e} F\")\n",
    "    print(f\"es.intWe = {stored_energy_j:.6e} J\")\n",
    "    print(f\"2*intWe/V^2 = {2.0 * stored_energy_j / VOLTAGE_V**2:.6e} F\")\n",
    "    print(f\"Mesh elements = {element_count}\")\n",
    "    print(f\"Saved model to {MODEL_PATH}\")\n",
    "\n",
    "    write_json_atomically(\n",
    "        MODEL_DIR / METRICS_JSON,\n",
    "        {\n",
    "            \"capacitance_f\": capacitance_f,\n",
    "            \"stored_energy_j\": stored_energy_j,\n",
    "            \"voltage_v\": VOLTAGE_V,\n",
    "            \"solver\": \"COMSOL Multiphysics\",\n",
    "            \"study\": \"Electrostatics stationary\",\n",
    "            \"mesh\": \"absolute element sizes, physics-controlled sizing replaced\",\n",
    "            \"element_count\": element_count,\n",
    "            \"near_metal\": NEAR_METAL,\n",
    "            \"lateral_margin_um\": LATERAL_MARGIN_UM,\n",
    "            \"substrate_thickness_um\": SUBSTRATE_THICKNESS_UM,\n",
    "            \"air_height_um\": AIR_HEIGHT_UM,\n",
    "            \"silicon_relative_permittivity\": SILICON_RELATIVE_PERMITTIVITY,\n",
    "        },\n",
    "    )\n",
    "\n",
    "    result = model.java.result()\n",
    "    # Rerunning this cell must not collide with the nodes a previous run left, so\n",
    "    # an existing tag is reused and every setting is written again before export.\n",
    "    datasets = result.dataset()\n",
    "    plane = (\n",
    "        result.dataset(\"cutplane\")\n",
    "        if datasets.hasTag(\"cutplane\")\n",
    "        else datasets.create(\"cutplane\", \"CutPlane\")\n",
    "    )\n",
    "    plane.set(\"planetype\", \"quick\")\n",
    "    plane.set(\"quickplane\", \"xy\")\n",
    "    plane.set(\"quickz\", \"1[um]\")\n",
    "    plane.set(\"data\", \"dset1\")\n",
    "\n",
    "    exports = result.export()\n",
    "    field_export = (\n",
    "        result.export(\"field\")\n",
    "        if exports.hasTag(\"field\")\n",
    "        else exports.create(\"field\", \"Data\")\n",
    "    )\n",
    "    field_export.set(\"data\", \"cutplane\")\n",
    "    field_export.set(\"expr\", [\"V\", \"es.normE\"])\n",
    "    field_path = MODEL_DIR / FIELD_TXT\n",
    "    field_export.set(\"filename\", str(field_path))\n",
    "    field_export.run()\n",
    "    print(f\"Exported V and es.normE to {field_path}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f284f9f6",
   "metadata": {},
   "source": [
    "## Saved capacitance result\n",
    "\n",
    "This cell reads `comsol_qubit_metrics.json` and checks $C_{11}$ against the stored energy through\n",
    "$2 W_\\text{e} / V^2$, the two agreeing to the solver's own precision."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f736a5d2",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-09-30T09:16:19.637002Z",
     "iopub.status.busy": "2026-09-30T09:16:19.636962Z",
     "iopub.status.idle": "2026-09-30T09:16:19.639556Z",
     "shell.execute_reply": "2026-09-30T09:16:19.639270Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C11                    = 122.2183 fF at V = 1 V\n",
      "2 * We / V^2           = 122.2183 fF (relative difference 3.74e-14)\n",
      "mesh                   = 30,901,244 elements, 0.625/1.25 near-metal sizes\n",
      "box                    = margin 8000 µm, silicon 500 µm, air 1600 µm\n"
     ]
    }
   ],
   "source": [
    "metrics_file = result_file(RESULTS_DIR, METRICS_JSON)\n",
    "\n",
    "capacitance_f: float | None = None\n",
    "stored_energy_j: float | None = None\n",
    "voltage_v: float | None = None\n",
    "\n",
    "if metrics_file is None:\n",
    "    print(explain_missing_results(RESULTS_DIR, METRICS_JSON))\n",
    "else:\n",
    "    metrics = json.loads(metrics_file.read_text())\n",
    "    capacitance_f = float(metrics[\"capacitance_f\"])\n",
    "    stored_energy_j = float(metrics[\"stored_energy_j\"])\n",
    "    voltage_v = float(metrics[\"voltage_v\"])\n",
    "\n",
    "    capacitance_from_energy_f = 2.0 * stored_energy_j / voltage_v**2\n",
    "    print(\n",
    "        f\"C11                    = {capacitance_f * 1e15:.4f} fF at V = {voltage_v:g} V\"\n",
    "    )\n",
    "    print(\n",
    "        f\"2 * We / V^2           = {capacitance_from_energy_f * 1e15:.4f} fF \"\n",
    "        f\"(relative difference \"\n",
    "        f\"{abs(capacitance_from_energy_f - capacitance_f) / capacitance_f:.2e})\"\n",
    "    )\n",
    "    print(\n",
    "        f\"mesh                   = {int(metrics['element_count']):,} elements, \"\n",
    "        f\"{metrics['near_metal']} near-metal sizes\"\n",
    "    )\n",
    "    print(\n",
    "        f\"box                    = margin {float(metrics['lateral_margin_um']):g} µm, \"\n",
    "        f\"silicon {float(metrics['substrate_thickness_um']):g} µm, \"\n",
    "        f\"air {float(metrics['air_height_um']):g} µm\"\n",
    "    )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "673aa6a5",
   "metadata": {},
   "source": [
    "## Saved potential and field map\n",
    "\n",
    "The exported field is $V$ and the field norm `es.normE` on the $z = 1\\,\\text{µm}$ plane just above the\n",
    "metal sheet. The export spans the whole box, where the pads are a dot, so the map is a **close-up**:\n",
    "`FIELD_LIMIT_X_UM` and `FIELD_LIMIT_Y_UM` frame both pads, and the remaining nodes are resampled onto a\n",
    "display grid (**display only**: it reads the solved field without re-solving it)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "696a424c",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-09-30T09:16:19.640436Z",
     "iopub.status.busy": "2026-09-30T09:16:19.640389Z",
     "iopub.status.idle": "2026-09-30T09:16:49.639181Z",
     "shell.execute_reply": "2026-09-30T09:16:49.638846Z"
    }
   },
   "outputs": [
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       "z\n",
       "M 57.74908 50.029213 \n",
       "L 58.074808 49.142455 \n",
       "L 58.39722 48.255698 \n",
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       "L 107.211374 239.778352 \n",
       "L 107.614525 239.795342 \n",
       "z\n",
       "M 108.554299 236.248311 \n",
       "L 108.097539 236.217122 \n",
       "L 107.211374 236.179574 \n",
       "L 106.32521 236.116729 \n",
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       "L 60.156542 171.515006 \n",
       "L 60.243021 172.401763 \n",
       "L 60.244658 172.460096 \n",
       "L 60.267746 173.288521 \n",
       "L 60.324015 174.175278 \n",
       "L 60.399511 175.062036 \n",
       "L 60.481972 175.948794 \n",
       "L 60.567961 176.835551 \n",
       "L 60.63706 177.722309 \n",
       "L 60.693258 178.609067 \n",
       "L 60.776783 179.495824 \n",
       "L 60.832156 180.382582 \n",
       "L 60.876012 181.269339 \n",
       "L 60.919868 182.156097 \n",
       "L 60.997876 183.042855 \n",
       "L 61.104986 183.929612 \n",
       "L 61.130822 184.185813 \n",
       "L 61.195412 184.81637 \n",
       "L 61.276212 185.703127 \n",
       "L 61.36064 186.589885 \n",
       "L 61.438283 187.476643 \n",
       "L 61.515601 188.3634 \n",
       "L 61.593043 189.250158 \n",
       "L 61.681221 190.136915 \n",
       "L 61.809235 191.023673 \n",
       "L 61.937249 191.910431 \n",
       "L 62.016987 192.543605 \n",
       "L 62.045433 192.797188 \n",
       "L 62.101004 193.683946 \n",
       "L 62.187019 194.570704 \n",
       "L 62.326338 195.457461 \n",
       "L 62.448136 196.344219 \n",
       "L 62.573288 197.230976 \n",
       "L 62.701087 198.117734 \n",
       "L 62.833812 199.004492 \n",
       "L 62.903151 199.425886 \n",
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       "L 63.209254 201.664764 \n",
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       "L 63.718153 204.325037 \n",
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       "L 64.404922 207.872068 \n",
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       "L 67.29972 218.513159 \n",
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       "L 70.26566 224.720462 \n",
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       "L 70.901132 225.60722 \n",
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       "L 71.764796 226.578547 \n",
       "L 72.559253 227.380735 \n",
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       "L 73.537125 228.266498 \n",
       "L 73.538462 228.267493 \n",
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       "L 74.871514 229.15425 \n",
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       "L 76.195618 229.926795 \n",
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       "L 78.854112 231.178398 \n",
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       "L 80.473585 231.814523 \n",
       "L 80.62644 231.87172 \n",
       "L 81.512605 232.13788 \n",
       "L 82.398769 232.399554 \n",
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       "L 84.171098 232.871163 \n",
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       "L 92.146578 234.513706 \n",
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       "L 107.211374 236.179574 \n",
       "L 108.097539 236.217122 \n",
       "L 108.554299 236.248311 \n",
       "z\n",
       "M 100.463367 232.701281 \n",
       "L 100.122059 232.666894 \n",
       "L 99.235894 232.596381 \n",
       "L 98.34973 232.536893 \n",
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       "L 76.974801 228.267493 \n",
       "L 76.195618 227.914055 \n",
       "L 75.309454 227.430599 \n",
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       "L 100.463367 232.701281 \n",
       "z\n",
       "M 96.996792 230.041008 \n",
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       "L 95.691236 229.919887 \n",
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       "L 66.751579 159.987157 \n",
       "L 66.771521 160.873914 \n",
       "L 66.796818 161.760672 \n",
       "L 66.835705 162.64743 \n",
       "L 66.85731 163.534187 \n",
       "L 66.874418 164.420945 \n",
       "L 66.884435 165.307702 \n",
       "L 66.918079 166.19446 \n",
       "L 66.955021 167.081218 \n",
       "L 66.983319 167.967975 \n",
       "L 67.015951 168.854733 \n",
       "L 67.052327 169.74149 \n",
       "L 67.082382 170.628248 \n",
       "L 67.109856 171.515006 \n",
       "L 67.135152 172.401763 \n",
       "L 67.152682 173.288521 \n",
       "L 67.179079 174.175278 \n",
       "L 67.218174 175.062036 \n",
       "L 67.252812 175.948794 \n",
       "L 67.262109 176.835551 \n",
       "L 67.313889 177.722309 \n",
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       "L 67.381883 178.609067 \n",
       "L 67.434996 179.495824 \n",
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       "L 67.50371 181.269339 \n",
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       "L 67.599728 183.042855 \n",
       "L 67.626631 183.929612 \n",
       "L 67.682567 184.81637 \n",
       "L 67.747763 185.703127 \n",
       "L 67.78365 186.589885 \n",
       "L 67.838712 187.476643 \n",
       "L 67.900938 188.3634 \n",
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       "L 68.083197 191.023673 \n",
       "L 68.142667 191.910431 \n",
       "L 68.199659 192.797188 \n",
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       "L 68.268832 193.683946 \n",
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       "L 68.453726 196.344219 \n",
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       "L 71.874561 220.286674 \n",
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       "L 73.879203 224.720462 \n",
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       "L 78.79131 227.380735 \n",
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       "L 95.691236 229.919887 \n",
       "L 96.577401 230.005218 \n",
       "L 96.996792 230.041008 \n",
       "z\n",
       "M 100.446623 228.267493 \n",
       "L 100.122059 228.259783 \n",
       "L 99.235894 228.228757 \n",
       "L 98.34973 228.166548 \n",
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       "L 85.943427 227.26988 \n",
       "L 85.057263 227.150919 \n",
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       "L 79.740276 226.386427 \n",
       "L 78.854112 226.21724 \n",
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       "L 74.993577 224.720462 \n",
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       "L 73.086737 220.286674 \n",
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       "L 72.65096 217.93161 \n",
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       "L 100.446623 228.267493 \n",
       "z\n",
       "M 94.754611 226.493977 \n",
       "L 93.918907 226.436698 \n",
       "L 93.032743 226.394981 \n",
       "L 92.146578 226.369233 \n",
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       "z\n",
       "M 116.903611 225.60722 \n",
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       "z\n",
       "M 76.215502 223.833705 \n",
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       "z\n",
       "M 98.075701 223.833705 \n",
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       "L 96.577401 223.749647 \n",
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       "L 75.485501 154.666611 \n",
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       "L 706.9979 164.420945 \n",
       "L 707.020029 165.307702 \n",
       "L 707.025408 166.19446 \n",
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       "L 707.022317 171.515006 \n",
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       "L 620.707051 244.22913 \n",
       "z\n",
       "M 678.588786 217.641737 \n",
       "L 679.474951 217.747981 \n",
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       "L 702.446852 96.140609 \n",
       "L 702.429482 95.253851 \n",
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       "z\n",
       "M 669.727142 222.079778 \n",
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       "z\n",
       "M 678.466248 217.626401 \n",
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       "L 465.023151 244.22913 \n",
       "z\n",
       "M 557.126912 242.455614 \n",
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       "z\n",
       "M 701.011024 223.833705 \n",
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       "z\n",
       "M 669.112868 222.060189 \n",
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       "z\n",
       "M 704.781444 49.142455 \n",
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       "z\n",
       "M 676.962326 47.36894 \n",
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       "z\n",
       "M 520.851512 184.02414 \n",
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       "L 556.298091 242.402286 \n",
       "L 557.126912 242.455614 \n",
       "z\n",
       "M 578.5042 232.701281 \n",
       "L 578.452202 232.696319 \n",
       "L 577.566038 232.640567 \n",
       "L 576.679873 232.595847 \n",
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       "z\n",
       "M 701.629062 223.911598 \n",
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       "z\n",
       "M 510.217539 215.857578 \n",
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       "L 578.5042 232.701281 \n",
       "z\n",
       "M 576.092869 227.380735 \n",
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       "L 574.907544 227.259994 \n",
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       "L 576.679873 227.438534 \n",
       "z\n",
       "M 493.380414 221.202988 \n",
       "L 494.266578 221.257593 \n",
       "L 495.152743 221.304319 \n",
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       "z\n",
       "M 510.151484 215.852886 \n",
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       "L 576.092869 227.380735 \n",
       "z\n",
       "M 583.541772 225.60722 \n",
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       "L 596.518251 151.119581 \n",
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       "L 596.509581 154.666611 \n",
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       "L 596.5104 157.326884 \n",
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       "L 596.501596 167.967975 \n",
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       "L 584.655353 225.751951 \n",
       "L 583.769189 225.639135 \n",
       "z\n",
       "M 492.824829 221.173432 \n",
       "L 492.49425 221.144868 \n",
       "L 491.608085 221.047186 \n",
       "L 490.721921 220.942321 \n",
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       "L 479.136835 144.02552 \n",
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       "L 477.689844 184.81637 \n",
       "L 477.706401 183.929612 \n",
       "L 477.627157 183.042855 \n",
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       "L 477.591649 181.269339 \n",
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       "L 477.560987 177.722309 \n",
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       "L 477.585472 175.948794 \n",
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       "L 477.595376 172.401763 \n",
       "L 477.476213 171.515006 \n",
       "L 477.613144 170.628248 \n",
       "L 477.46952 169.74149 \n",
       "L 477.563431 168.854733 \n",
       "L 477.503379 167.967975 \n",
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       "L 477.515652 166.19446 \n",
       "L 477.511904 165.307702 \n",
       "L 477.513015 164.420945 \n",
       "L 477.562812 163.534187 \n",
       "L 477.478344 162.64743 \n",
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       "L 477.271562 161.760672 \n",
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       "L 477.329343 163.534187 \n",
       "L 477.366833 164.420945 \n",
       "L 477.365355 165.307702 \n",
       "L 477.36425 166.19446 \n",
       "L 477.280559 167.081218 \n",
       "L 477.373778 167.967975 \n",
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       "L 477.400031 169.74149 \n",
       "L 477.287656 170.628248 \n",
       "L 477.394971 171.515006 \n",
       "L 477.301501 172.401763 \n",
       "L 477.25634 173.288521 \n",
       "L 477.33747 174.175278 \n",
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       "L 477.25048 176.835551 \n",
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       "L 477.277421 183.042855 \n",
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       "L 477.429454 225.133602 \n",
       "L 478.315618 225.304495 \n",
       "L 479.201783 225.298412 \n",
       "L 480.087947 225.222867 \n",
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       "L 481.860276 225.082086 \n",
       "L 482.74644 225.046411 \n",
       "L 483.632605 225.02262 \n",
       "L 484.518769 224.906234 \n",
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       "L 486.291098 224.765593 \n",
       "L 486.981869 224.720462 \n",
       "L 487.177263 224.711393 \n",
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       "L 489.835756 224.625887 \n",
       "L 490.721921 224.620702 \n",
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       "L 492.49425 224.565391 \n",
       "L 493.380414 224.565498 \n",
       "L 494.266578 224.540357 \n",
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       "L 496.038907 224.45671 \n",
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       "L 512.876032 224.274185 \n",
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       "L 514.648361 224.249689 \n",
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       "L 530.599321 224.26081 \n",
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       "L 535.916308 224.268176 \n",
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       "L 582.883025 225.55777 \n",
       "L 583.541772 225.60722 \n",
       "z\n",
       "M 586.938856 224.720462 \n",
       "L 586.427682 224.663508 \n",
       "L 585.541518 224.541352 \n",
       "L 584.655353 224.42854 \n",
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       "L 587.236449 212.305856 \n",
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       "L 587.302206 205.211795 \n",
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       "L 587.304277 203.43828 \n",
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       "L 587.314228 201.664764 \n",
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       "L 587.341072 198.117734 \n",
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       "L 587.332408 196.344219 \n",
       "L 587.32706 195.457461 \n",
       "L 587.341225 194.570704 \n",
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       "L 587.343547 192.797188 \n",
       "L 587.345112 191.910431 \n",
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       "L 587.345451 190.136915 \n",
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       "L 587.351808 187.476643 \n",
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       "L 587.356291 174.175278 \n",
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       "L 587.353012 172.401763 \n",
       "L 587.357202 171.515006 \n",
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       "L 587.35922 169.74149 \n",
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       "L 587.359018 167.967975 \n",
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       "L 587.367735 166.19446 \n",
       "L 587.366526 165.307702 \n",
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       "L 587.363786 163.534187 \n",
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       "L 587.367648 161.760672 \n",
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       "L 587.366339 159.100399 \n",
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       "L 587.366429 157.326884 \n",
       "L 587.390167 156.440126 \n",
       "L 587.371553 155.553369 \n",
       "L 587.341073 154.666611 \n",
       "L 587.358283 153.779853 \n",
       "L 587.362134 152.893096 \n",
       "L 587.366036 152.006338 \n",
       "L 587.359975 151.119581 \n",
       "L 587.375005 150.232823 \n",
       "L 587.3732 149.346065 \n",
       "L 587.365812 148.459308 \n",
       "L 587.370242 147.57255 \n",
       "L 587.373907 146.685793 \n",
       "L 587.383376 145.799035 \n",
       "L 587.370492 144.912277 \n",
       "L 587.376469 144.02552 \n",
       "L 587.371971 143.138762 \n",
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       "L 587.372935 140.478489 \n",
       "L 587.365661 139.591732 \n",
       "L 587.368652 138.704974 \n",
       "L 587.36477 137.818216 \n",
       "L 587.364857 136.931459 \n",
       "L 587.359586 136.044701 \n",
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       "L 587.367768 134.271186 \n",
       "L 587.362715 133.384428 \n",
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       "L 587.362414 131.610913 \n",
       "L 587.361351 130.724156 \n",
       "L 587.369402 129.837398 \n",
       "L 587.353374 128.95064 \n",
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       "L 587.365557 115.649276 \n",
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       "L 587.36173 113.875761 \n",
       "L 587.358269 112.989003 \n",
       "L 587.375578 112.102246 \n",
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       "L 587.371282 110.328731 \n",
       "L 587.351435 109.441973 \n",
       "L 587.372198 108.555215 \n",
       "L 587.364772 107.668458 \n",
       "L 587.356027 106.7817 \n",
       "L 587.35112 105.894943 \n",
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       "L 587.362812 104.121427 \n",
       "L 587.355147 103.23467 \n",
       "L 587.357369 102.347912 \n",
       "L 587.358874 101.461154 \n",
       "L 587.362342 100.574397 \n",
       "L 587.355087 99.687639 \n",
       "L 587.356674 98.800882 \n",
       "L 587.364539 97.914124 \n",
       "L 587.362095 97.027366 \n",
       "L 587.356522 96.140609 \n",
       "L 587.36939 95.253851 \n",
       "L 587.349167 94.367094 \n",
       "L 587.364269 93.480336 \n",
       "L 587.35246 92.593578 \n",
       "L 587.346146 91.706821 \n",
       "L 587.345487 90.820063 \n",
       "L 587.352058 89.933306 \n",
       "L 587.344224 89.046548 \n",
       "L 587.345575 88.15979 \n",
       "L 587.350942 87.273033 \n",
       "L 587.344576 86.386275 \n",
       "L 587.341564 85.499517 \n",
       "L 587.341957 84.61276 \n",
       "L 587.329428 83.726002 \n",
       "L 587.356402 82.839245 \n",
       "L 587.344739 81.952487 \n",
       "L 587.344556 81.065729 \n",
       "L 587.35784 80.178972 \n",
       "L 587.332283 79.292214 \n",
       "L 587.313847 78.592305 \n",
       "L 587.308244 78.405457 \n",
       "L 587.313847 78.247213 \n",
       "L 587.333659 77.518699 \n",
       "L 587.338628 76.631941 \n",
       "L 587.321743 75.745184 \n",
       "L 587.323008 74.858426 \n",
       "L 587.328875 73.971669 \n",
       "L 587.319018 73.084911 \n",
       "L 587.319298 72.198153 \n",
       "L 587.323339 71.311396 \n",
       "L 587.330402 70.424638 \n",
       "L 587.329506 69.53788 \n",
       "L 587.313847 69.103215 \n",
       "L 587.293868 68.651123 \n",
       "L 587.303825 67.764365 \n",
       "L 587.311508 66.877608 \n",
       "L 587.292728 65.99085 \n",
       "L 587.291674 65.104092 \n",
       "L 587.282351 64.217335 \n",
       "L 587.268663 63.330577 \n",
       "L 587.268717 62.44382 \n",
       "L 587.27591 61.557062 \n",
       "L 587.265738 60.670304 \n",
       "L 587.25591 59.783547 \n",
       "L 587.243115 58.896789 \n",
       "L 587.241983 58.010032 \n",
       "L 587.227642 57.123274 \n",
       "L 587.185303 56.236516 \n",
       "L 587.176048 55.349759 \n",
       "L 587.166028 54.463001 \n",
       "L 587.131531 53.576244 \n",
       "L 587.081268 52.689486 \n",
       "L 587.013677 51.802728 \n",
       "L 586.946074 50.915971 \n",
       "L 586.816337 50.029213 \n",
       "L 586.597157 49.142455 \n",
       "L 586.427682 48.816835 \n",
       "L 585.774579 48.255698 \n",
       "L 585.541518 48.168037 \n",
       "L 584.655353 47.97822 \n",
       "L 583.769189 47.76332 \n",
       "L 582.883025 47.666181 \n",
       "L 581.99686 47.573103 \n",
       "L 581.110696 47.436417 \n",
       "L 580.740698 47.36894 \n",
       "L 581.110696 47.297449 \n",
       "L 581.99686 47.159167 \n",
       "L 582.883025 47.067436 \n",
       "L 583.769189 46.971458 \n",
       "L 584.655353 46.788491 \n",
       "L 585.541518 46.672689 \n",
       "L 586.427682 46.545078 \n",
       "L 587.074612 46.482183 \n",
       "L 587.313847 46.450459 \n",
       "L 587.565994 46.482183 \n",
       "L 588.200011 46.524951 \n",
       "L 588.977919 47.36894 \n",
       "L 589.082266 48.255698 \n",
       "L 589.086176 48.485408 \n",
       "L 589.118561 49.142455 \n",
       "L 589.086176 49.551147 \n",
       "L 589.068644 50.029213 \n",
       "L 589.083091 50.915971 \n",
       "L 589.061569 51.802728 \n",
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       "z\n",
       "M 477.10816 223.833705 \n",
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       "z\n",
       "M 593.988414 223.833705 \n",
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       "L 588.977919 47.36894 \n",
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       "L 585.541518 224.541352 \n",
       "L 586.427682 224.663508 \n",
       "L 586.938856 224.720462 \n",
       "z\n",
       "M 477.429454 224.197669 \n",
       "L 478.315618 224.265763 \n",
       "L 479.201783 223.93765 \n",
       "L 479.922796 223.833705 \n",
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       "L 477.10816 223.833705 \n",
       "z\n",
       "M 594.403163 224.221385 \n",
       "L 595.289327 224.220779 \n",
       "L 596.175491 223.838198 \n",
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       "L 595.019523 90.820063 \n",
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       "L 595.055427 88.15979 \n",
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       "L 595.0409 83.726002 \n",
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       "L 595.035901 81.952487 \n",
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       "L 595.062402 79.292214 \n",
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       "L 595.009358 76.631941 \n",
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       "L 595.028906 70.424638 \n",
       "L 595.030127 69.53788 \n",
       "L 594.966549 68.651123 \n",
       "L 595.028239 67.764365 \n",
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       "L 594.972684 56.236516 \n",
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       "<Figure size 1200x400 with 4 Axes>"
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    "FIELD_LIMIT_X_UM = 300.0\n",
    "FIELD_LIMIT_Y_UM = 250.0\n",
    "FIELD_GRID_X = 300\n",
    "FIELD_GRID_Y = 250\n",
    "FIELD_LEVELS = 20\n",
    "\n",
    "field_file = result_file(RESULTS_DIR, FIELD_TXT)\n",
    "\n",
    "if field_file is None:\n",
    "    print(explain_missing_results(RESULTS_DIR, FIELD_TXT))\n",
    "else:\n",
    "    field = np.loadtxt(field_file, comments=\"%\")\n",
    "    field_x, field_y = field[:, 0], field[:, 1]\n",
    "    in_view = (np.abs(field_x) <= FIELD_LIMIT_X_UM) & (\n",
    "        np.abs(field_y) <= FIELD_LIMIT_Y_UM\n",
    "    )\n",
    "    if in_view.sum() < 3:\n",
    "        raise ValueError(\n",
    "            f\"No field nodes lie within ±{FIELD_LIMIT_X_UM:g} by \"\n",
    "            f\"±{FIELD_LIMIT_Y_UM:g} µm, so there is nothing to draw a close-up from\"\n",
    "        )\n",
    "    triangulation = mtri.Triangulation(field_x[in_view], field_y[in_view])\n",
    "    grid_x = np.linspace(-FIELD_LIMIT_X_UM, FIELD_LIMIT_X_UM, FIELD_GRID_X)\n",
    "    grid_y = np.linspace(-FIELD_LIMIT_Y_UM, FIELD_LIMIT_Y_UM, FIELD_GRID_Y)\n",
    "    grid_xx, grid_yy = np.meshgrid(grid_x, grid_y)\n",
    "    field_v = mtri.LinearTriInterpolator(triangulation, field[in_view, 3])(\n",
    "        grid_xx, grid_yy\n",
    "    ).filled(np.nan)\n",
    "    field_e = mtri.LinearTriInterpolator(triangulation, field[in_view, 4])(\n",
    "        grid_xx, grid_yy\n",
    "    ).filled(np.nan)\n",
    "    field_e_max = float(np.nanmax(field_e))\n",
    "\n",
    "    voltage_label = f\"at V = {voltage_v:g} V, \" if voltage_v is not None else \"\"\n",
    "\n",
    "    fig, axes = plt.subplots(1, 2, figsize=(12, 4))\n",
    "    potential = axes[0].contourf(\n",
    "        grid_x, grid_y, field_v, levels=FIELD_LEVELS, cmap=\"viridis\"\n",
    "    )\n",
    "    axes[0].set_title(f\"Electric potential {voltage_label}z = 1 µm\")\n",
    "    fig.colorbar(potential, ax=axes[0], label=r\"$V$ (V)\")\n",
    "\n",
    "    norm_e = axes[1].contourf(\n",
    "        grid_x,\n",
    "        grid_y,\n",
    "        field_e,\n",
    "        levels=np.geomspace(1.0, field_e_max, FIELD_LEVELS),\n",
    "        norm=LogNorm(vmin=1.0, vmax=field_e_max),\n",
    "        cmap=\"inferno\",\n",
    "        extend=\"min\",\n",
    "    )\n",
    "    axes[1].set_title(f\"Electric field norm {voltage_label}z = 1 µm\")\n",
    "    fig.colorbar(norm_e, ax=axes[1], label=r\"$|\\mathbf{E}|$ (V/m)\")\n",
    "\n",
    "    for axis in axes:\n",
    "        axis.set_aspect(\"equal\")\n",
    "        axis.set_xlabel(\"x (µm)\")\n",
    "        axis.set_ylabel(\"y (µm)\")\n",
    "    plt.tight_layout()\n",
    "    plt.show()"
   ]
  },
  {
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   "id": "64871d61",
   "metadata": {},
   "source": [
    "The potential holds across the driven pad and falls through the gap to the grounded pad; the field\n",
    "concentrates in that gap and at the pad edges, where the pad capacitance mainly lives."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "552e0157",
   "metadata": {},
   "source": [
    "## Limitations and next steps\n",
    "\n",
    "The solve is electrostatic and the EM-only copy omits the SQUID loop and its leads, so $C_{11}$ is a\n",
    "**one-terminal** capacitance to the grounded chip rather than the two-pad charging capacitance, and it is\n",
    "not a transmon eigenfrequency or $f_{01}$. The finite zero-charge outer walls of the box act at any mesh,\n",
    "so the value is not a converged or validated device number. The natural next step is the full two-pad\n",
    "capacitance matrix feeding the QPDK Hamiltonian workflow\n",
    "({doc}`/notebooks/scqubits_parameter_calculation`).\n",
    "\n",
    "## References\n",
    "\n",
    "```{bibliography}\n",
    ":filter: docname in docnames\n",
    "```"
   ]
  }
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