Semiconductor Manufacturing: Ion Implantation Mathematical Modeling

Keywords: implant modeling, ion implantation, doping, dopant diffusion, range straggling, damage

Semiconductor Manufacturing: Ion Implantation Mathematical Modeling

1. Introduction

Ion implantation is a critical process in semiconductor fabrication where dopant ions (B, P, As, Sb) are accelerated and embedded into silicon substrates to precisely control electrical properties.

Key Process Parameters:

• Energy (keV): Controls implant depth ($R_p$)
• Dose (ions/cm²): Controls peak concentration
• Tilt angle (°): Minimizes channeling effects
• Twist angle (°): Avoids major crystal planes
• Beam current (mA): Affects dose rate and wafer heating

2. Foundational Physics: Ion Stopping

When an energetic ion enters a solid, it loses energy through two primary mechanisms.

2.1 Total Stopping Power

Where:

• $N$ = atomic density of target ($\approx 5 \times 10^{22}$ atoms/cm³ for Si)
• $S_n(E)$ = nuclear stopping cross-section (elastic collisions with nuclei)
• $S_e(E)$ = electronic stopping cross-section (inelastic energy loss to electrons)

2.2 Nuclear Stopping: ZBL Universal Potential

The Ziegler-Biersack-Littmark (ZBL) universal screening function:

Where $x = r/a_u$ is the reduced interatomic distance.

Universal screening length:

Where:

• $a_0$ = Bohr radius (0.529 Å)
• $Z_1$ = atomic number of incident ion
• $Z_2$ = atomic number of target atom

2.3 Electronic Stopping

Low energy regime (velocity-proportional, Lindhard-Scharff):

Where:

High energy regime (Bethe-Bloch formula):

Where:

• $m_e$ = electron mass
• $v$ = ion velocity
• $I$ = mean ionization potential of target

3. Range Statistics and Profile Models

3.1 Gaussian Approximation (First Order)

For amorphous targets, the as-implanted profile:

Peak concentration:

3.2 Pearson IV Distribution (Industry Standard)

Real profiles exhibit asymmetry. The Pearson IV distribution uses four statistical moments:

Four Moments:

1. First Moment (Mean): $R_p$ — projected range 2. Second Moment (Variance): $\Delta R_p^2$ — spread 3. Third Moment (Skewness): $\gamma$ — asymmetry

• $\gamma < 0$: tail extends deeper into substrate (light ions: B)
• $\gamma > 0$: tail extends toward surface (heavy ions: As)

4. Fourth Moment (Kurtosis): $\beta$ — peakedness relative to Gaussian

Typical values for Si:

3.3 Dual Pearson Model (Channeling Effects)

For implants into crystalline silicon with channeling tails:

Where:

• $P_{random}(x)$ = Pearson distribution for random (amorphous) stopping
• $P_{channel}(x)$ = Pearson distribution for channeled ions
• $f_{ch}$ = channeling fraction (depends on tilt, beam divergence, surface oxide)

Channeling fraction dependencies:

• Beam divergence: $f_{ch} \downarrow$ as divergence $\uparrow$
• Tilt angle: $f_{ch} \downarrow$ as tilt $\uparrow$ (typically 7° off-axis)
• Surface oxide: $f_{ch} \downarrow$ with screen oxide
• Pre-amorphization: $f_{ch} \approx 0$ with PAI

4. Monte Carlo Simulation (BCA Method)

The Binary Collision Approximation provides the highest accuracy for profile prediction.

4.1 Algorithm Overview

FOR each ion i = 1 to N_ions (typically 10⁵ - 10⁶):
    
    1. Initialize:
       - Energy: E = E₀
       - Position: (x, y, z) = (0, 0, 0)
       - Direction: (cos θ, sin θ cos φ, sin θ sin φ)
    
    2. WHILE E > E_cutoff:
       
       a. Calculate mean free path:
          $\lambda = 1 / (N \cdot \pi \cdot p_{max}^2)$
       
       b. Select random impact parameter:
          $p = p_{max} \cdot \sqrt{\text{random}[0,1]}$
       
       c. Solve scattering integral for deflection angle $\Theta$
       
       d. Calculate energy transfer to target atom:
          $T = T_{max} \cdot \sin^2(\Theta/2)$
       
       e. Update ion energy:
          $E \to E - T - \Delta E_{\text{electronic}}$
       
       f. IF T > E_displacement:
          Create recoil cascade (track secondary)
       
       g. Update position and direction vectors
    
    3. Record final ion position (x_final, y_final, z_final)

END FOR

4. Build histogram of final positions → Dopant profile

4.2 Scattering Integral

The classical scattering integral for deflection angle:

Where:

• $p$ = impact parameter
• $r_{min}$ = distance of closest approach
• $V(r)$ = interatomic potential (e.g., ZBL)
• $E_c$ = center-of-mass energy

Center-of-mass energy:

4.3 Energy Transfer

Maximum energy transfer in elastic collision:

Where $\gamma$ is the kinematic factor:

4.4 Electronic Energy Loss (Continuous)

Along the free flight path:

5. Multi-Layer and Through-Film Implantation

5.1 Screen Oxide Implantation

For implantation through oxide layer of thickness $t_{ox}$:

Range correction:

Straggle correction:

5.2 Moment Matching at Interfaces

For multi-layer structures, use moment conservation:

Where $w_i$ is the weighting factor for layer $i$.

6. Two-Dimensional Profile Modeling

6.1 Lateral Straggle

The lateral distribution follows:

Relationship between straggles:

6.2 Masked Implant with Edge Effects

For a mask opening of width $W$:

6.3 Full 3D Distribution

7. Damage and Defect Modeling

7.1 Kinchin-Pease Model

Number of displaced atoms per incident ion:

Where:

• $E_D$ = damage energy (energy deposited into nuclear collisions)
• $E_d$ = displacement threshold energy ($\approx 15$ eV for Si)

7.2 Modified NRT Model (Norgett-Robinson-Torrens)

The factor 0.8 accounts for forward scattering efficiency.

7.3 Damage Energy Partition

Lindhard partition function:

Where:

7.4 Amorphization Threshold

Critical dose for amorphization:

Typical values:

7.5 Damage Profile

The damage distribution differs from dopant distribution:

Where $R_d < R_p$ (damage peaks shallower than dopant).

8. Process-Relevant Calculations

8.1 Junction Depth

For Gaussian profile meeting background concentration $C_B$:

For asymmetric Pearson profiles:

8.2 Sheet Resistance

With concentration-dependent mobility (Masetti model):

8.3 Threshold Voltage Shift

For channel implant:

Simplified (shallow implant):

8.4 Dose Calculation from Profile

Verification:

Where:

• $I$ = beam current
• $t$ = implant time
• $A$ = implanted area

9. Advanced Effects

9.1 Transient Enhanced Diffusion (TED)

The "+1 Model": Each implanted ion creates approximately one net interstitial.

Enhanced diffusion equation:

Enhanced diffusivity:

Where:

• $D_i$ = intrinsic diffusivity
• $C_I$ = interstitial concentration
• $C_I^*$ = equilibrium interstitial concentration

9.2 Dose Loss Mechanisms

Sputtering yield:

Where:

• $\alpha$ = angular factor ($\approx 0.2$ for light ions, $\approx 0.4$ for heavy ions)
• $U_0$ = surface binding energy ($\approx 4.7$ eV for Si)

Retained dose:

9.3 High Dose Effects

Dose saturation:

Snow-plow effect at very high doses pushes peak toward surface.

9.4 Temperature Effects

Dynamic annealing: Competes with damage accumulation

Where $E_a \approx 0.3$ eV for Si self-interstitial migration.

10. Summary Tables

10.1 Key Scaling Relationships

10.2 Common Implant Parameters in Si

10.3 Simulation Tools Comparison

Quick Reference Formulas

Essential Equations Card

-
┌─────────────────────────────────────────────────────────────────────────────────────────────┐
│  GAUSSIAN PROFILE                                                                           │
│  $C(x) = \Phi/(\sqrt{2\pi} \cdot \Delta R_p) \cdot \exp[-(x-R_p)^2/(2\Delta R_p^2)]$        │
├─────────────────────────────────────────────────────────────────────────────────────────────┤
│  PEAK CONCENTRATION                                                                         │
│  $C_{max} \approx 0.4 \cdot \Phi/\Delta R_p$                                                │
├─────────────────────────────────────────────────────────────────────────────────────────────┤
│  JUNCTION DEPTH                                                                             │
│  $x_j = R_p + \Delta R_p \cdot \sqrt{2 \cdot \ln(C_{max}/C_B)}$                             │
├─────────────────────────────────────────────────────────────────────────────────────────────┤
│  SHEET RESISTANCE                                                                           │
│  $R_s = 1/(q \cdot \int \mu(C) \cdot C(x) dx)$                                              │
├─────────────────────────────────────────────────────────────────────────────────────────────┤
│  DISPLACEMENT DAMAGE                                                                        │
│  $N_d = 0.8 \cdot E_D/(2E_d)$                                                               │
└─────────────────────────────────────────────────────────────────────────────────────────────┘

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