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  "id": "atlas-lesson-oxide-to-cell-acceptance",
  "title": "Oxide variation to electrical cell acceptance",
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  "description": "Carry a named material specimen into electrical parameters and separate acceptance gates.",
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  "publishedDate": null,
  "revisionDate": "2026-10-07",
  "author": null,
  "topic": "Devices and logic",
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  "version": "1.3.0",
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      "title": "Oxide variation to electrical cell acceptance",
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      "title": "The mechanism",
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      "title": "Oxide geometry becomes electrical capacitance",
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      "title": "The fixed-charge threshold term",
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      "title": "Geometry and mobility set drive strength",
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      "title": "Engineering example",
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      "title": "Distinctions and tradeoffs",
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      "title": "Specifications and interfaces",
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      "title": "Interfaces",
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      "title": "Check your understanding",
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      "title": "The +20% oxide specimen meets the selected timing and noise limits but violates a 5% thickness tolerance. Is it accepted under all four gates?",
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  "citations": [
    {
      "title": "Explore in Atlas",
      "url": "https://semiaifoundry.com/ai-atlas/#oxide-to-cell-acceptance"
    },
    {
      "title": "MIT 6.012 Fall 2005 — Lecture 8",
      "url": "https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/ba97ec0d4b75ff1049567a48e66b74a0_lec8.pdf"
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    {
      "title": "MIT 6.012 Fall 2005 — Lecture 9",
      "url": "https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/5c5960f14ae37d0a65478c4f3f628d98_lec9.pdf"
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    {
      "title": "MIT 6.012 Fall 2005 — Lecture 10",
      "url": "https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/0498bbf0f0e585507d9cfe5dc3bd54b1_lec10.pdf"
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    {
      "title": "Atlas releases and licensing",
      "url": "https://semiaifoundry.com/ai-atlas/licenses.html"
    },
    {
      "title": "Thermal oxidation and the silicon interface",
      "url": "https://semiaifoundry.com/learn/atlas/thermal-oxidation-interface/"
    },
    {
      "title": "Metal gate and high-k dielectric",
      "url": "https://semiaifoundry.com/learn/atlas/gate-stack/"
    },
    {
      "title": "Metal–oxide–semiconductor field-effect transistor",
      "url": "https://semiaifoundry.com/learn/atlas/mosfet/"
    },
    {
      "title": "Loaded CMOS transfer, noise and transition delay",
      "url": "https://semiaifoundry.com/learn/atlas/loaded-inverter-transition/"
    },
    {
      "title": "Standard-cell library and physical design",
      "url": "https://semiaifoundry.com/learn/atlas/standard-cell/"
    },
    {
      "title": "Complementary CMOS inverter",
      "url": "https://semiaifoundry.com/learn/atlas/inverter/"
    }
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  "content": "Devices and logic · Atlas v1.3.0\n\n# Oxide variation to electrical cell acceptance\n\nCarry a named material specimen into electrical parameters and separate acceptance gates.\n\n[Explore in Atlas](https://semiaifoundry.com/ai-atlas/#oxide-to-cell-acceptance)Declared 2 μm channel, nanometre oxide inputs and one lumped circuit; schematic geometry\n\nLearning objectiveTrace oxide variation through at least three electrical quantities and explain which declared acceptance condition fails.\n\n## The mechanism\n\nA fabricated layer becomes useful through the electrical behavior it enables. Thickness therefore cannot remain only a geometry annotation in an atlas of computation. INV-01 carries one evaluated oxide value into capacitance per area, the fixed-depletion-charge contribution to threshold, transistor current factor and the next-gate portion of its output load. This coupling prevents changing drive strength while silently holding every related quantity fixed. The nMOS mobility factor and additional threshold shift are explicitly independent perturbations, rather than a claimed prediction from one manufacturing knob. The specimen then faces four different requirements: oxide tolerance, static logic interface, DC noise margin and loaded transition delay. Keeping these gates separate reveals when a material condition is unacceptable even though one electrical metric is favorable, and when nominal geometry still fails a use-dependent timing requirement. The result is a disciplined conceptual bridge from the existing oxidation and device lessons to one executable cell case.\n\n1.  The evaluated thickness is computed from the nominal oxide and a chosen percentage offset; both remain attached to INV-01.\n2.  Permittivity divided by evaluated thickness yields capacitance per area with explicit SI conversion.\n3.  A fixed electrostatic intercept and depletion-charge term give the chosen threshold magnitude; nMOS receives a separately declared shift.\n4.  Mobility and W/L combine with the same Cox to determine drive strength, while next-gate area determines the modeled capacitive load.\n5.  Current balance and a loaded transient produce electrical outputs without changing the original specimen identity.\n6.  Separate tolerance, logic, noise and delay gates determine acceptance under the chosen teaching use.\n\n## Science and assumptions\n\n### Oxide geometry becomes electrical capacitance\n\n$$\nt_{\\mathrm{ox}}=t_0(1+\\delta/100),\\qquad C_{\\mathrm{ox}}=\\frac{\\epsilon_0\\epsilon_r}{t_{\\mathrm{ox}}}\n$$\n\nEvaluated thickness is nominal thickness times one plus its percentage offset; oxide capacitance per area is permittivity divided by thickness.\n\nSymbols and units\n\nt0,tox\n\nNominal and evaluated oxide thickness\n\nm\n\nδ\n\nChosen thickness offset\n\npercent\n\nCox\n\nOxide capacitance per area\n\nF/m²\n\nε0,εr\n\nVacuum permittivity and relative dielectric constant\n\nF/m; dimensionless\n\nThe same evaluated oxide is used in all subsequent device and load calculations. Metrology is not simulated.\n\nAssumptions Planar dielectric with assumed εr=3.9 and ε0=8.854×10⁻¹² F/m; t is converted from nm to m. No high-k multilayer or quantum correction.\n\n### The fixed-charge threshold term\n\n$$\n|V_{Tp}|=V_*+\\frac{Q_d}{C_{\\mathrm{ox}}},\\qquad V_{Tn}=|V_{Tp}|+\\Delta V_{Tn}\n$$\n\nThe threshold magnitude contains a fixed intercept plus depletion charge per area divided by oxide capacitance per area; the nMOS receives an additional declared shift.\n\nSymbols and units\n\nVTn,|VTp|\n\nnMOS threshold and positive pMOS threshold magnitude\n\nV\n\nV\\*\n\nFixed flatband/surface-potential intercept\n\nV\n\nQd\n\nChosen depletion-charge magnitude per area\n\nC/m²\n\nCox\n\nEvaluated oxide capacitance per area\n\nF/m²\n\nΔVTn\n\nIndependent nMOS threshold perturbation\n\nV\n\nIncreasing thickness reduces Cox and raises the fixed-charge contribution to threshold. Both polarities share chosen magnitudes before the independent nMOS shift.\n\nAssumptions V\\*=0.23 V and Qd=4×10⁻⁴ C/m² are specified assumptions with fixed body electrostatics, not values derived from a manufacturing recipe. No variation distribution or impurity profile is solved.\n\n### Geometry and mobility set drive strength\n\n$$\n\\beta_n=\\mu_n C_{\\mathrm{ox}}\\frac{W_n}{L},\\qquad\\beta_p=\\mu_p C_{\\mathrm{ox}}\\frac{W_p}{L}\n$$\n\nCurrent factors multiply mobility, oxide capacitance per area and width-to-length ratio.\n\nSymbols and units\n\nβn,βp\n\nSquare-law current factors\n\nA/V²\n\nμn,μp\n\nCarrier mobility magnitudes\n\nm²/(V s)\n\nWn,Wp,L\n\nChannel widths and common length\n\nm\n\nCox\n\nOxide capacitance per area\n\nF/m²\n\nThe default 450 versus 150 cm²/(V s) mobility magnitudes are balanced by 2 versus 6 μm widths at a common 2 μm length.\n\nAssumptions Constant mobilities, planar long channels; convert cm²/(V s) to m²/(V s) with 10⁻⁴. No mobility degradation or velocity saturation.\n\n### One acceptance conjunction preserves separate causes\n\n$$\nG_{\\mathrm{cell}}=G_{\\mathrm{oxide}}\\land G_{\\mathrm{logic}}\\land G_{\\mathrm{noise}}\\land G_{\\mathrm{timing}}\n$$\n\nThe specimen is accepted only if all four declared gates pass.\n\nSymbols and units\n\nGcell\n\nOverall acceptance for this declared use\n\nBoolean\n\nGoxide,Glogic,Gnoise,Gtiming\n\nIndependent geometry, static-logic, noise and timing gates\n\nBoolean\n\nA geometry failure, poor static interface, insufficient noise margin and excessive loaded delay remain individually visible. A single success cannot substitute for the other requirements.\n\nAssumptions Only the stated synthetic tolerance and electrical constraints; no statistical yield, process qualification, field reliability or commercial-node signoff.\n\n## Engineering example\n\nAt nominal/evaluated oxide 10 nm, INV-01 has Cox=0.00345306 F/m², total load 65.24896 fF and equal threshold magnitudes 0.3458393 V. A +20% offset yields 12 nm, Cox=0.00287755 F/m² and threshold 0.3690071 V. Its default 50% delay becomes 0.7598888 ns and still meets the chosen 1 ns limit, but the 20% geometry offset fails the independently chosen 5% oxide tolerance. This is one deterministic specimen, not a distribution or a measured wafer yield.\n\n## Distinctions and tradeoffs\n\nA favorable electrical result does not waive a separate material requirement, and nominal material geometry does not establish timing closure.\n\nGeometry, current, capacitance and noise can move together or in opposite directions. The model exposes that coupling while keeping its deliberately ideal physics and chosen acceptance contract explicit.\n\nEvery accelerator is assembled from physical devices whose behavior must satisfy cell-level requirements before higher-level arithmetic and system performance claims are meaningful.\n\n## Specifications and interfaces\n\nSpecimen\n\nINV-01\n\nSame evaluated material/electrical state through all views.\n\nGeometry\n\nL=2 μm; Wn=2 μm; Wp=6 μm\n\nDeclared planar long-channel dimensions; not a process-node label.\n\nSupply\n\n1.2 V\n\nFixed teaching input; no commercial operating point is implied.\n\n### Interfaces\n\nMaterial and process scenario Input\n\nNominal/evaluated oxide, declared nMOS mobility/threshold perturbation and load\n\nCell acceptance evidence Output\n\nCurrent, transfer, noise, delay and independent pass/fail gates\n\n## Check your understanding\n\n### The +20% oxide specimen meets the selected timing and noise limits but violates a 5% thickness tolerance. Is it accepted under all four gates?\n\n1.  Yes, because timing is the only relevant metric\n2.  No, the independently declared geometry gate fails\n3.  Yes, because a digital output was produced\n\nAnswer and reasoning\n\nNo, the independently declared geometry gate fails\n\nThe stated acceptance contract requires all four gates. No metric silently replaces another.\n\n## Evidence and scope\n\nThe physical equations and conventions are primary-source informed. The parameter values, corners, constraints and results are authored teaching choices; no commercial process or human specialist review is represented.\n\nThe process-cell-acceptance-1 studio carries INV-01 through this exact mathematical specimen. It does not run fabrication, TCAD, a SPICE executable or a foundry model.\n\n## Sources\n\n-   [MIT 6.012 Fall 2005 — Lecture 8](https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/ba97ec0d4b75ff1049567a48e66b74a0_lec8.pdf)\n-   [MIT 6.012 Fall 2005 — Lecture 9](https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/5c5960f14ae37d0a65478c4f3f628d98_lec9.pdf)\n-   [MIT 6.012 Fall 2005 — Lecture 10](https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/0498bbf0f0e585507d9cfe5dc3bd54b1_lec10.pdf)\n\n[Published lesson data](https://semiaifoundry.com/ai-atlas/lesson-data/oxide-to-cell-acceptance-0c3831e131a74916.json) · [Atlas releases and licensing](https://semiaifoundry.com/ai-atlas/licenses.html)\n\n## Continue exploring\n\n### Foundations\n\n-   [Thermal oxidation and the silicon interface](https://semiaifoundry.com/learn/atlas/thermal-oxidation-interface/)\n-   [Metal gate and high-k dielectric](https://semiaifoundry.com/learn/atlas/gate-stack/)\n-   [Metal–oxide–semiconductor field-effect transistor](https://semiaifoundry.com/learn/atlas/mosfet/)\n\n### Related lessons\n\n-   [Loaded CMOS transfer, noise and transition delay](https://semiaifoundry.com/learn/atlas/loaded-inverter-transition/)\n-   [Standard-cell library and physical design](https://semiaifoundry.com/learn/atlas/standard-cell/)\n-   [Complementary CMOS inverter](https://semiaifoundry.com/learn/atlas/inverter/)"
}