Doping Profile Simulation models dopant distribution resulting from ion implantation and thermal diffusion — predicting 1D/2D/3D dopant concentration profiles that determine junction depth, threshold voltage, and resistance, a core capability of process TCAD essential for transistor design and process optimization.
What Is Doping Profile Simulation?
- Definition: Computational modeling of dopant distribution in semiconductor.
- Inputs: Implant conditions (species, energy, dose, tilt), thermal history.
- Outputs: Dopant concentration vs. position (1D, 2D, or 3D).
- Goal: Predict electrical properties from process conditions.
Why Doping Profile Simulation Matters
- Junction Depth: Determines source/drain, well depths.
- Threshold Voltage: Doping profile controls Vth.
- Resistance: Sheet resistance depends on doping profile.
- Process Optimization: Virtual experiments reduce wafer runs.
- Design-Process Co-Optimization: Link device design to process parameters.
Ion Implantation Modeling
Monte Carlo Simulation:
- Method: Track individual ions through crystal lattice.
- Physics: Binary collision approximation, electronic stopping.
- Advantages: Accurate for channeling, damage, complex geometries.
- Disadvantages: Computationally expensive (millions of ions).
- Use Case: Detailed implant simulation, calibration reference.
Analytical Models:
- Gaussian Distribution: Simple approximation for amorphous targets.
- Formula: N(x) = (Dose / √(2πΔR_p)) · exp(-(x-R_p)² / (2ΔR_p²)).
- Parameters: R_p (projected range), ΔR_p (straggle).
- Advantages: Fast, simple, good for first-order estimates.
- Limitations: Inaccurate for channeling, complex structures.
Pearson IV Distribution:
- Method: Four-moment distribution (mean, variance, skewness, kurtosis).
- Advantages: More accurate than Gaussian, captures asymmetry.
- Parameters: Fit to Monte Carlo or experimental data.
- Use Case: Production TCAD, balance accuracy and speed.
Dual-Pearson:
- Method: Two Pearson distributions for channeled and random components.
- Advantages: Captures channeling effects.
- Use Case: Crystalline silicon implants.
Implantation Parameters
Species:
- Common Dopants: Boron (p-type), Phosphorus, Arsenic, Antimony (n-type).
- Mass Effect: Heavier ions have shorter range.
- Channeling: Lighter ions (B, P) channel more than heavy (As, Sb).
Energy:
- Range: Higher energy → deeper penetration.
- Typical: 1-200 keV for source/drain, 100-1000 keV for wells.
- Scaling: R_p ∝ E^n (n ≈ 1.5-2).
Dose:
- Concentration: Total dopant atoms per area (cm⁻²).
- Typical: 10¹³-10¹⁶ cm⁻² depending on application.
- Peak Concentration: N_peak ≈ Dose / (√(2π) · ΔR_p).
Tilt and Rotation:
- Tilt: Angle from surface normal (typically 7° to avoid channeling).
- Rotation: Azimuthal angle.
- Impact: Reduces channeling, affects profile shape.
Diffusion Modeling
Fick's Laws:
- Fick's First Law: J = -D · ∇C (flux proportional to gradient).
- Fick's Second Law: ∂C/∂t = ∇·(D·∇C) (diffusion equation).
- Solution: Numerical (finite element, finite difference).
Diffusion Mechanisms:
- Vacancy Mechanism: Dopant moves via lattice vacancies.
- Interstitial Mechanism: Dopant moves via interstitial sites.
- Pair Diffusion: Dopant-defect pairs diffuse together.
Concentration-Dependent Diffusion:
- Enhanced Diffusion: D increases at high dopant concentration.
- Mechanism: Excess point defects from high doping.
- Models: Fermi-level dependent diffusion, pair diffusion models.
Transient Enhanced Diffusion (TED):
- Cause: Excess interstitials from implant damage.
- Effect: Temporarily enhanced diffusion during anneal.
- Duration: Minutes to hours depending on damage, temperature.
- Impact: Deeper junctions than expected from equilibrium diffusion.
Activation:
- Process: Dopants move from interstitial to substitutional sites.
- Electrical Activity: Only substitutional dopants are electrically active.
- Incomplete Activation: Some dopants remain inactive (clusters, precipitates).
Clustering:
- High Concentration: Dopants form clusters at high concentration.
- Boron-Interstitial Clusters (BICs): Common in boron doping.
- Impact: Reduces electrical activation, affects diffusion.
Thermal Budget
Annealing Conditions:
- Temperature: 800-1100°C typical for activation anneal.
- Time: Seconds (RTA) to hours (furnace anneal).
- Ambient: Inert (N₂, Ar) or oxidizing (O₂).
Rapid Thermal Anneal (RTA):
- Duration: 1-60 seconds at high temperature.
- Advantage: Minimal diffusion, good activation.
- Use Case: Shallow junctions, advanced nodes.
Furnace Anneal:
- Duration: Minutes to hours.
- Advantage: Uniform, well-controlled.
- Disadvantage: More diffusion than RTA.
Spike Anneal:
- Duration: <1 second at peak temperature.
- Advantage: Minimal diffusion, ultra-shallow junctions.
- Challenge: Requires precise temperature control.
Simulation Workflow
Step 1: Define Structure:
- Geometry: 1D, 2D, or 3D simulation domain.
- Materials: Silicon substrate, oxide, nitride layers.
- Mesh: Discretization for numerical solution.
Step 2: Implantation:
- Specify Conditions: Species, energy, dose, tilt, rotation.
- Run Implant Simulation: Monte Carlo or analytical.
- Result: As-implanted dopant profile.
Step 3: Thermal Processing:
- Specify Anneal: Temperature vs. time profile.
- Run Diffusion Simulation: Solve diffusion equations.
- Result: Annealed dopant profile.
Step 4: Activation:
- Model: Compute electrically active dopant concentration.
- Clustering: Account for inactive dopants.
- Result: Active doping profile.
Step 5: Validation:
- Compare to SIMS: Secondary Ion Mass Spectrometry for concentration profile.
- Compare to Electrical: Sheet resistance, junction depth from electrical tests.
- Calibrate: Adjust model parameters if needed.
Output Metrics
Junction Depth (x_j):
- Definition: Depth where dopant concentration equals background.
- Typical: 10-100nm for source/drain, 100-1000nm for wells.
- Impact: Determines short-channel effects, leakage.
Sheet Resistance (R_s):
- Formula: R_s = 1 / (q · ∫ μ(x) · N_active(x) dx).
- Units: Ω/square.
- Impact: Determines contact resistance, RC delay.
Peak Concentration:
- Location: Depth of maximum dopant concentration.
- Value: Maximum concentration (cm⁻³).
- Impact: Affects tunneling, breakdown voltage.
Dose Retention:
- Definition: Fraction of implanted dose remaining after anneal.
- Loss Mechanisms: Outdiffusion, segregation to oxide.
- Typical: 70-95% retention.
Applications
Source/Drain Engineering:
- Shallow Junctions: Low energy implants, minimal anneal.
- Low Resistance: High dose, good activation.
- Abruptness: Steep profiles for short-channel control.
Well Formation:
- Deep Junctions: High energy implants, longer anneals.
- Retrograde Wells: Peak concentration below surface.
- Latch-Up Prevention: Proper well doping prevents parasitic thyristors.
Threshold Voltage Adjustment:
- Channel Implants: Low dose implants to adjust Vth.
- Halo/Pocket Implants: Angled implants for short-channel control.
- Optimization: Balance Vth, short-channel effects, variability.
Tools & Software
- Synopsys Sentaurus Process: Comprehensive process simulation.
- Silvaco Athena: Process simulation with implant and diffusion.
- Crosslight CSUPREM: Process simulator.
- UT-MARLOWE: Monte Carlo implant simulator.
Doping Profile Simulation is a core TCAD capability — by accurately predicting how ion implantation and thermal processing create dopant distributions, it enables virtual process optimization, reduces experimental iterations, and provides critical insights for transistor design and manufacturing at advanced technology nodes.