【Semiconductor】 🧠 08-02. Understanding the Essence of MOSFETs with TCAD — Poisson Equation and Drift–Diffusion

topics: [“Semiconductor”, “TCAD”, “MOSFET”, “Poisson Equation”, “Drift-Diffusion”]

🧭 Introduction — Where Do V–I Characteristics Come From?

The V–I characteristics of a MOSFET are not something that suddenly appear
inside a circuit simulator.

Their true origin lies in the physical phenomena occurring inside the device:

• Electrostatics (potential distribution)
• Carrier transport

In this article, from a TCAD (Technology Computer-Aided Design) perspective, we focus on:

• 🧮 the Poisson equation (which determines potential)
• 🚗 the Drift–Diffusion equations (which determine current)

to explain
👉 why MOSFET $V$–$I$ characteristics take the shape they do.

🔍 What Is TCAD?

TCAD is a simulation methodology that directly solves physical phenomena
inside semiconductor devices using numerical methods.

What you provide includes:

• Device structure
• Doping profiles
• Material parameters
• Bias conditions

And as a result, you can simultaneously and continuously observe:

• Electrostatic potential
• Electric field
• Carrier density
• Current density

📌 If circuit simulation is a world of “observing results,”
TCAD is the world of observing the moment those results are born.

🧮 Poisson Equation — The Origin of Everything

The Poisson equation is the foundation of semiconductor physics, linking:

Charge distribution → Potential distribution

In MOS structures:

• Gate voltage
• Oxide thickness $t_{ox}$
• Substrate doping $N_A$

control surface potential, which in turn governs channel formation.

In short:

Whether a channel forms or not is determined first by electrostatics.

🧱 What Happens in a MOS Structure (Role of Poisson)

As the gate voltage increases, the following sequence occurs in a MOS structure:

1. Gate voltage is applied
2. Surface potential changes through the oxide
3. Carriers accumulate at the surface
4. A channel is formed

All of this is governed by the Poisson equation.

👉 “Potential creates the channel.”

🚗 Drift–Diffusion Equations — What Determines Current

A channel alone does not produce current.
Current is determined by the Drift–Diffusion equations:

• Drift: carrier motion driven by electric fields
• Diffusion: carrier motion driven by concentration gradients

These equations naturally explain:

• Why $I_d$ increases when $V_d$ increases
• Why a linear region appears at low voltages

👉 “Transport equations create $I_d$.”

📈 V–I Characteristics Are a Stack of Physics

With TCAD, the following quantities are observed simultaneously:

• Potential distribution
• Carrier density distribution
• Current density distribution

As a direct consequence of these physical quantities stacking together:

• $V_g$–$I_d$ characteristics
• $V_d$–$I_d$ characteristics

emerge naturally.

🧪 SemiDevKit: An Educational 1D TCAD Playground

SemiDevKit provides a lightweight 1D TCAD environment
specifically designed for educational use.

• Poisson + Drift–Diffusion
• Implemented in Python
• One-to-one correspondence between equations, code, and plots

🔗 TCAD Top Page
https://samizo-aitl.github.io/SemiDevKit/tcad/

📊 Example ①: MOSCAP C–V Characteristics (Result of Poisson)

How does the C–V characteristic of a MOS capacitor change
when oxide thickness $t_{ox}$ is varied?

👉 Differences in potential distribution appear as capacitance.

📊 Example ②: nMOS $V_g$–$I_d$ Characteristics

$n$MOS $V_g$–$I_d$ characteristics with varying oxide thickness.

• Thinner $t_{ox}$ → lower $V_{th}$
• Higher $I_d$ at the same $V_g$

👉 Poisson → Drift–Diffusion → $I_d$

📊 Example ③: nMOS $V_d$–$I_d$ Characteristics

• Low $V_d$: linear region
• High $V_d$: saturation region

👉 Transport equations naturally create operating regions.

🔗 TCAD Is Not the Final Goal

TCAD is extremely powerful, but it has limitations:

• High computational cost
• Too heavy for circuit-level design

This is where BSIM4 comes in.

Compressing the physics observed in TCAD
into a form immediately usable in circuits

That is the role of compact models.

📝 Summary

• MOSFET $V$–$I$ characteristics result from Poisson and Drift–Diffusion equations
• Electrostatics create the channel; transport equations determine current
• TCAD reveals the moment when equations become reality

▶ Next Article

👉 03: What Is BSIM4? — A Compact Model That Translates Physics into Circuits

🧩 The SemiDevKit series continues from
“TCAD → BSIM → SPICE → Reliability.”