A Symmetric Unified Transport and Charge Model for MOSFETs from Diffusive to Ballistic Regimes

This paper proposes a symmetric, unified compact model for MOSFETs that bridges the gap between drift-diffusion and ballistic transport regimes by self-consistently integrating current, charge, and quantum capacitance through a physically motivated high-field scattering framework.

The Gist

Imagine you are trying to design a perfect model to predict how traffic flows through a city.

In a big, sprawling city with many stoplights and intersections, traffic moves slowly and predictably. This is like "Drift-Diffusion" (DD)—the way electrons move in large, old-fashioned transistors. They bump into things, slow down, and follow a steady, predictable flow.

But imagine a tiny, futuristic "micro-city" where there are no stoplights, and the streets are so short that a car can zoom from one end to the other without ever hitting a brake. This is "Ballistic Transport" (BT)—the way electrons move in the ultra-small transistors used in modern smartphones.

The Problem:
For years, engineers have had two different "maps" for these two scenarios. One map works for the big city (DD), and one map works for the micro-city (BT). But as technology shrinks, our "cities" are becoming hybrids—partly crowded and slow, partly wide-open and fast. Using the wrong map leads to massive errors in designing the chips that power our world.

The Solution (The Paper's Contribution):
This paper introduces a "Unified Transport (UT) Model." Think of it as a single, "smart" GPS that automatically switches its logic depending on how short the road is.

Here is how the author’s "Smart GPS" works using three clever tricks:

1. The "Speed Limit" vs. "The Sprint" (The Current Model)

In a big city, your speed is limited by how many red lights you hit (velocity saturation). In a micro-city, your speed is limited by how fast you can physically launch out of your driveway (thermal velocity).

• The Old Way: Previous models tried to blend these by averaging the speeds, which is like saying, "If I drive 20mph in the city and 100mph on the highway, my average speed is 60mph." It doesn't actually describe how the car behaves!
• The New Way: This model uses a "scattering length" (a physical measurement of how much "stuff" is in the way). It treats the speed limit and the launch speed as two different physical rules that merge smoothly. It’s like a car that knows: "If the road is long, I'll hit the brakes; if the road is short, I'll just floor it."

2. The "Crowd Density" Problem (The Charge Model)

When people are walking through a crowded subway station (DD), they are packed tightly together. But if they are sprinting through an empty hallway (BT), they spread out.

• Standard models assumed the "crowd" (the electrical charge) stayed the same regardless of how fast people were running.
• This paper realizes that speed changes the crowd. When electrons start "sprinting" (ballistic transport), the total amount of charge in the channel actually changes. The author created a new formula to account for this "thinning out" of the crowd, making the model much more accurate for tiny chips.

3. The "Smooth Transition" (Symmetry)

If you are driving a car and you flip from "Drive" to "Reverse," you don't want the car to teleport or jerk violently; you want a smooth transition.

• Many previous models had "glitches" (mathematical discontinuities) when the voltage changed direction.
• This model is "Symmetric." It’s like a perfectly balanced seesaw; whether electricity flows from left-to-right or right-to-left, the math remains smooth and continuous. This is vital for engineers who need to simulate complex circuits without the computer crashing.

Why does this matter to you?

As we try to make computers, AI, and smartphones faster and more efficient, we are building transistors that are smaller than a single strand of DNA. This paper provides the "Master Map" that engineers need to design those tiny, lightning-fast components accurately, ensuring that the next generation of technology actually works the way we predict it will.

Technical Summary

Technical Summary: A Symmetric Unified Transport and Charge Model for MOSFETs

1. Problem Statement

As semiconductor technology scales into the sub-20-nm regime (e.g., FinFET, GAAFET), carrier transport in MOSFETs transitions from the classical Drift-Diffusion (DD) regime to the Ballistic Transport (BT) regime. Existing compact models face several critical challenges in bridging these two regimes:

• Incompatibility: Many quasi-ballistic models (like the Virtual Source model) are incompatible with industry-standard DD models, making integration difficult.
• Physical Inconsistency: Some unified models use Matthiessen’s rule to combine average charge velocity with thermal velocity, which is physically unsound because thermal velocity limits the source injection rather than the average charge.
• Charge Modeling Gaps: Standard models often fail to account for the reduction in channel charge that occurs in the quasi-ballistic regime.
• Symmetry Issues: Most source-referenced models suffer from discontinuities or lack of smoothness at $V_{DS} = 0$, failing essential DC and AC symmetry tests.

2. Methodology

The paper proposes a Symmetric Unified Transport (UT) model that avoids channel segmentation and instead uses a continuous mathematical framework. The model is divided into three core components:

• A. Charge Density Model: It employs a dual-approach. For the DD limit, it uses the Gradual Channel Approximation (GCA). For the ballistic limit, it utilizes the Landauer approach to calculate the top-of-the-barrier charge ($q_{top}$), accounting for carrier injection from both source and drain via a transmission rate ($T$).
• B. Current Model: The model calculates two distinct velocities: the DD velocity ($v_{dd}$) and the BT velocity ($v_{bt}$).
  • It uses a physically motivated high-field scattering length ($\lambda_{sat}$) to allow for velocity overshoot, meaning $v_{sat}$ is length-dependent rather than a constant.
  • It applies Matthiessen’s rule to combine $v_{dd}$ and $v_{bt}$ at the Virtual Source (VS) to derive an effective velocity ($v_{eff}$), ensuring the thermal injection limit is respected.
• C. Total Charge Model: To bridge the gap between the local-equilibrium (DD) and non-equilibrium (BT) charge distributions, the author develops a physical charge model where the total charge is a weighted sum of $q_{dd}$ and $q_{bt}$. This accounts for the observed reduction in capacitance in quasi-ballistic devices.

3. Key Contributions

• Unified Framework: A single, continuous formulation that smoothly transitions from the square-law behavior of long-channel devices to the thermal-velocity-limited behavior of ballistic devices.
• Physical Soundness: Unlike previous models, this approach applies velocity limits to the injection velocity at the virtual source, which is physically accurate.
• Symmetry and Continuity: The introduction of specific smoothing functions ensures the model is continuous and passes both DC and AC symmetry tests, making it suitable for high-accuracy circuit simulation.
• Advanced Charge Modeling: A new method to calculate intrinsic gate and drain charges that incorporates ballistic effects, capturing the transition in capacitance.

4. Results and Validation

The model was validated through two primary methods:

• Theoretical Validation: Simulations showed that at $L = 2\mu m$, the model perfectly matches the DD square-law. At $L = 30nm$, it captures velocity overshoot. At $L = 1nm$, it converges to the ballistic limit.
• Experimental Validation:
  • ETSOI (IBM Process): The model successfully fitted experimental data for channel lengths ranging from $980nm$ down to $30nm$ using a single set of universal physical parameters (e.g., $\lambda, v_{th0}$).
  • FinFET (Intel 3): The model was applied to advanced FinFET technology, successfully capturing high on-currents by utilizing a degenerate thermal velocity model.
• Symmetry Tests: The model passed standard DC and AC symmetry tests (Fig. 9), a significant improvement over existing source-referenced models.

5. Significance

This work provides a robust, physically consistent, and industry-ready compact model for next-generation transistors. By bridging the gap between DD and BT regimes without requiring complex channel segmentation, it enables more accurate and computationally efficient SPICE-level simulations for advanced nodes (FinFET, GAAFET). Its ability to maintain symmetry and capture both IV and CV characteristics makes it a highly practical tool for both device physicists and circuit designers.