Level A - Analytical Model

Level A implements the classical Shockley MOSFET equations with smooth subthreshold transition. It provides fast, analytical solutions ideal for understanding basic MOSFET behavior and quick parameter exploration.

Drain Current Model

Long-Channel Current Equations

The classical long-channel MOSFET model is derived from gradual channel approximation:

Linear Region ( ):

Saturation Region ( ):

Where:

• : Electron mobility (cm²/V·s)
• : Gate oxide capacitance (F/cm²)
• : Channel width
• : Channel length
• : Threshold voltage

Output Characteristics

The following chart shows ideal long-channel output characteristics:

Notice how:

• Current increases linearly with in the linear region
• Current saturates when (pinch-off)
• In saturation, current is independent of (ideal case)

Subthreshold Region

Below threshold ( ), current is dominated by diffusion:

Where:

• : Subthreshold swing factor (typically 1.3-1.5)
• : Thermal voltage (~26 mV at 300K)
• : Pre-exponential current factor

Subthreshold Swing

The subthreshold swing mV/decade determines how sharply the device turns off. Smaller SS means better switching efficiency.

Transfer Characteristics

The transfer curve shows both linear and logarithmic current scales:

Key observations:

• Linear scale (blue): Shows quadratic increase above threshold
• Log scale (red): Shows exponential subthreshold behavior
• Transition region: Smooth interpolation near

Threshold Voltage Calculation

The threshold voltage is calculated from device parameters:

Component Breakdown

Component	Formula	Physical Meaning
Flat-band voltage		Work function difference + fixed charge
Surface potential		Condition for strong inversion
Body effect		Back-bias sensitivity

Fermi Potential

Body Effect Coefficient

Typical values: V^1/2 for modern technologies.

Capacitance Model

Level A includes a high-frequency MOS capacitance model:

Region	Condition	Capacitance
Accumulation
Depletion
Inversion		(HF)

Depletion Capacitance

Where the depletion width is:

Transconductance

The transconductance measures output current sensitivity to gate voltage:

Linear Region:

Saturation Region:

Design Insight

In saturation, , so doubling requires 4× the current. This is a fundamental trade-off in analog circuit design.

Model Assumptions & Limitations

Level A makes several simplifying assumptions:

Assumptions

• Gradual channel approximation (field along channel << field across oxide)
• Constant mobility (no field dependence)
• Abrupt depletion approximation
• Complete ionization of dopants
• No short-channel effects

What's NOT Included

Effect	Impact	Why Missing
Velocity saturation	Reduces	Requires field-dependent mobility
DIBL	Lowers at high	2D electrostatics needed
CLM	Non-zero output conductance	Pinch-off point movement
Mobility degradation	Reduces current	Vertical field effect
Quantum effects	Increases	Carrier confinement

When to Use Level A

✅ Good for:

• Understanding basic MOSFET physics
• Quick parameter exploration
• Educational purposes
• Initial device sizing
• Long-channel devices (L > 1 μm)

❌ Not suitable for:

• Short-channel devices (L < 100 nm)
• Accurate analog design
• Predicting leakage currents
• Process variation analysis

Implementation Notes

The simulator smoothly interpolates between regions to avoid discontinuities:

if Vov <= -3*n*Vt:
    Id = subthreshold only
elif Vov >= 3*n*Vt:
    Id = above-threshold only
else:
    Id = smooth interpolation

This ensures continuous , , and across all bias conditions.