# ⚛  L1 Principle — Drift-Diffusion Semiconductor Transport (TCAD)

**ID:** `L1-488` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Semiconductor Physics — *Device transport*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** device IV curves and field profiles
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554130

---

## 🧠 1. Introduction

**Drift-Diffusion Semiconductor Transport (TCAD)** is a **nonlinear inverse problem** whose unknown lives in **device IV curves and field profiles** space, within the **Device transport** sub-domain of **Semiconductor Physics**.

Measurements consist of electrons collected by an electron detector via a **terminal IV CV** sensing mechanism.

The forward operator applies, in order: E · poisson operator; E · continuity n operator; E · continuity p operator; a fixed-point or gradient iteration on the unknown; O · terminal IV operator.

Observations are corrupted by additive Gaussian noise. Well-posed elliptic-parabolic system with Dirichlet/mixed BC at contacts; inversion for (mu, tau) from I-V is moderately ill-conditioned (mobility-lifetime trade-off).

## ⚙ 2. Forward Model

Physical chain: **x** → E · poisson → E · continuity n → E · continuity p → O · terminal IV → **y** (detector).

```
y = `O.terminal_IV` `E.continuity_p` `E.continuity_n` `E.poisson` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.poisson` | E · poisson operator |
| `E.continuity_n` | E · continuity n operator |
| `E.continuity_p` | E · continuity p operator |
| `O.terminal_IV` | O · terminal iv operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Semiconductor Physics |
| Sub domain | Device transport |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | device_IV_curves_and_field_profiles |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Well-posed elliptic-parabolic system with Dirichlet/mixed BC at contacts; inversion for (mu, tau) from I-V is moderately ill-conditioned (mobility-lifetime trade-off).

| Metric | Value |
|---|---|
| Metric | IV_log_relative_error |
| Secondary | Vth_error_mV |

## 📏 5. Operating Range (Ω)

**Center problem class:** `device_parameter_inversion` · **Forward operator:** `drift_diffusion_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 300 |
| Mesh n | N | 64 |
| N a cm3 | — | 100000000000000000 |
| N d cm3 | — | 1000000000000000000 |
| T ox nm | nm | 2 |
| L channel nm | nm | 50 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 200 – 500 |
| Mesh n | N | 32 – 512 |
| N a cm3 | — | 1000000000000000.0 – 100000000000000000000 |
| N d cm3 | — | 1000000000000000.0 – 100000000000000000000 |
| T ox nm | nm | 0.5 – 20 |
| L channel nm | nm | 5 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** log10(I_D) error <= 0.03

| Metric | Range |
|---|---|
| Iv log relative error | 0.005 – 0.3 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **IV_log_relative_error**, with κ = `800` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xac30a5d9507b3f0c2be5058d1f997da330ca02cfaa2350bc5059acf1447878e8`
- **Chain tx hash:** `0x1aa1ba71a065c90c7d6a0380ac73b50f77f82db64b73d56b2000e17973fa0476`
- **Chain block:** `41554130`

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## File Mapping

This bundle consists of: `L1-488.md`, `L1-488.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-488.md` | Source of truth — edit this | Human or LLM |
| `L1-488.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-488`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._