Level C - Numerical Model

Level C provides self-consistent numerical solution of semiconductor device equations using 2D mesh-based simulation. This is the most accurate model but computationally intensive, suitable for research and validation purposes.

Fundamental Equations

The simulator solves the semiconductor device equations self-consistently:

1. Poisson Equation

Relates electrostatic potential to charge density:

Where:

• : Electrostatic potential (V)
• , : Electron and hole concentrations (cm⁻³)
• , : Ionized donor and acceptor concentrations
• : Silicon permittivity

2. Continuity Equations

Conservation of carriers with generation/recombination:

Electrons:

Holes:

For steady-state (DC) analysis:

3. Current Density (Drift-Diffusion)

Using Einstein relation:

Physical Interpretation

• Drift: Carriers move under electric field ( )
• Diffusion: Carriers move from high to low concentration ( )

Mesh Generation

Non-Uniform 2D Mesh

The simulator uses a non-uniform rectangular mesh with refinement in critical regions:

Mesh Refinement Strategy

Region	Typical Spacing	Reason
Si/SiO₂ interface	0.2-0.5 nm	Rapid potential variation
Junction edges	1-2 nm	High field, carrier gradients
Channel center	2-5 nm	Moderate variation
Bulk substrate	5-20 nm	Slow variation

Mesh Quality Requirements

For accurate results:

• Debye length criterion:
• Gradient criterion: Adjacent cells should have < 2× potential difference
• Aspect ratio: Keep cell aspect ratio < 10:1

Numerical Methods

Scharfetter-Gummel Discretization

Standard finite-difference fails for drift-diffusion due to exponential carrier profiles. The Scharfetter-Gummel scheme uses exponential fitting:

Where is the Bernoulli function.

Why Scharfetter-Gummel?

Consider an exponential carrier profile .

Standard FD: → large error

SG scheme: Uses exact solution of drift-diffusion between mesh points → accurate for any

Bernoulli Function Properties

The function is computed carefully to avoid numerical overflow.

Gummel Iteration

Self-consistent solution uses decoupled iteration:

1. Initial guess: ψ from Poisson with n,p from Boltzmann
2. LOOP until converged:
   a. Solve Poisson equation → update ψ
   b. Solve electron continuity → update n
   c. Solve hole continuity → update p
   d. Check: |Δψ|_max < tolerance?
3. Calculate currents from converged solution

Convergence Issues

Gummel iteration may fail to converge for:

• High forward bias (strong injection)
• Very fine mesh (stiff system)
• Poor initial guess

Solutions: Use Newton-Raphson for difficult cases, continuation methods.

Linear Solver: BiCGSTAB

Each Poisson/continuity solve requires solving a sparse linear system .

The BiCGSTAB (Biconjugate Gradient Stabilized) algorithm is used:

• Handles non-symmetric matrices
• Good convergence for device equations
• Memory efficient (no matrix factorization)

Typical settings:

• Tolerance:
• Max iterations: 1000
• Preconditioner: Jacobi (diagonal)

Boundary Conditions

Contact Boundary Conditions

Boundary	Type	Condition
Gate	Dirichlet
Source/Drain (Ohmic)	Dirichlet	, ,
Substrate	Dirichlet	,

Interface Conditions

Oxide-Silicon interface:

Open boundaries (Neumann):

Convergence Behavior

Typical Convergence

| Iteration | | Status | |-----------|---------------------|--------| | 1 | 0.5 V | Initial | | 5 | 0.05 V | Converging | | 10 | 0.001 V | Near solution | | 15 | 1e-6 V | Converged |

Convergence Tips

1. Good initial guess: Start from equilibrium, increment bias slowly
2. Under-relaxation: with
3. Voltage stepping: For large bias, use intermediate steps
4. Mesh refinement: Coarse mesh first, then refine

Computational Cost

Single Bias Point

Mesh Size	Unknowns	Iterations	Time
50×50	7,500	~15	~0.5 s
100×100	30,000	~20	~2 s
200×200	120,000	~25	~8 s

I-V Sweep (25 points)

Mesh Size	Total Time	Per Point
50×50	~10 s	0.4 s
100×100	~45 s	1.8 s
200×200	~4 min	10 s

GPU Acceleration

WebGPU acceleration can provide 10-50× speedup for larger meshes. Currently experimental - see GPU Acceleration Review.

Output Quantities

Level C provides detailed 2D field distributions:

Quantity	Symbol	Unit
Electrostatic potential		V
Electric field	,	V/cm
Electron concentration		cm⁻³
Hole concentration		cm⁻³
Current density	,	A/cm²
Recombination rate		cm⁻³/s

Derived Metrics

• Terminal currents: Integrated from current density at contacts
• Inversion charge:
• Depletion width: From potential profile analysis
• Threshold voltage: Extrapolated from Id-Vgs

Physical Models Included

Carrier Statistics

Boltzmann (default):

Fermi-Dirac (optional):

Recombination Models

Shockley-Read-Hall:

Auger (high injection):

Mobility Models

Doping-dependent:

Field-dependent:

When to Use Level C

✅ Good for:

• Accurate device characterization
• Novel device structure analysis
• 2D field visualization
• Validating compact models
• Research applications
• Understanding internal physics

❌ Not suitable for:

• Quick parameter exploration (too slow)
• Circuit-level simulation
• Large design space sweeps
• Real-time interaction

Comparison: All Levels

Feature	Level A	Level B	Level C
Physics	Analytical	Semi-empirical	Numerical
Accuracy	Low	Medium	High
Speed	~1 ms	~2 ms	~1-10 s
2D fields	❌	❌	✅
SCE	❌	✅	✅ (intrinsic)
Custom structures	❌	❌	✅

Implementation Details

Code Structure

src/physics/levelC/
├── mesh.ts          # Mesh generation
├── poisson.ts       # Poisson solver
├── continuity.ts    # Continuity equations
├── gummel.ts        # Gummel iteration
├── bicgstab.ts      # Linear solver
├── materials.ts     # Material properties
└── worker.ts        # Web Worker interface

Web Worker

Level C runs in a Web Worker to avoid blocking the UI:

• Main thread sends device parameters
• Worker performs simulation
• Results returned via postMessage
• Progress updates for long simulations