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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

\frac{dE}{dx} = N \left[ S_n(E) + S_e(E) \right]

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:

\phi(x) = 0.1818 e^{-3.2x} + 0.5099 e^{-0.9423x} + 0.2802 e^{-0.4028x} + 0.02817 e^{-0.2016x}

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

Universal screening length:

a_u = \frac{0.8854 \, a_0}{Z_1^{0.23} + Z_2^{0.23}}

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):

S_e = k_e \sqrt{E}

Where:

k_e = \frac{1.212 \, Z_1^{7/6} \, Z_2}{(Z_1^{2/3} + Z_2^{2/3})^{3/2} \, M_1^{1/2}}

High energy regime (Bethe-Bloch formula):

S_e = \frac{4\pi Z_1^2 e^4 N Z_2}{m_e v^2} \ln\left(\frac{2 m_e v^2}{I}\right)

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:

C(x) = \frac{\Phi}{\sqrt{2\pi} \, \Delta R_p} \exp\left[ -\frac{(x - R_p)^2}{2 \Delta R_p^2} \right]

Symbol	Definition	Units
$\Phi$	Implant dose	ions/cm²
$R_p$	Projected range (mean depth)	nm or cm
$\Delta R_p$	Range straggle (standard deviation)	nm or cm

Peak concentration:

C_{max} = \frac{\Phi}{\sqrt{2\pi} \, \Delta R_p} \approx \frac{0.4 \, \Phi}{\Delta R_p}

3.2 Pearson IV Distribution (Industry Standard)

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

f(x) = K \left[ 1 + \left( \frac{x - \lambda}{a} \right)^2 \right]^{-m} \exp\left[ - u \arctan\left( \frac{x - \lambda}{a} \right) \right]

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:

Dopant	Skewness ($\gamma$)	Kurtosis ($\beta$)
Boron (B)	-0.5 to +0.5	2.5 to 4.0
Phosphorus (P)	-0.3 to +0.3	2.5 to 3.5
Arsenic (As)	+0.5 to +1.5	3.0 to 5.0
Antimony (Sb)	+0.8 to +2.0	3.5 to 6.0

3.3 Dual Pearson Model (Channeling Effects)

For implants into crystalline silicon with channeling tails:

C(x) = (1 - f_{ch}) \cdot P_{random}(x) + f_{ch} \cdot P_{channel}(x)

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:

\Theta = \pi - 2p \int_{r_{min}}^{\infty} \frac{dr}{r^2 \sqrt{1 - \frac{V(r)}{E_c} - \frac{p^2}{r^2}}}

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:

E_c = \frac{M_2}{M_1 + M_2} E

4.3 Energy Transfer

Maximum energy transfer in elastic collision:

T_{max} = \frac{4 M_1 M_2}{(M_1 + M_2)^2} \cdot E = \gamma \cdot E

Where $\gamma$ is the kinematic factor:

Ion → Si	$M_1$ (amu)	$\gamma$
B → Si	11	0.702
P → Si	31	0.968
As → Si	75	0.746

4.4 Electronic Energy Loss (Continuous)

Along the free flight path:

\Delta E_{electronic} = \int_0^{\lambda} S_e(E) \, dx \approx S_e(E) \cdot \lambda

5. Multi-Layer and Through-Film Implantation

5.1 Screen Oxide Implantation

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

Range correction:

R_p^{eff} = R_p^{Si} - t_{ox} \left( \frac{R_p^{Si} - R_p^{ox}}{R_p^{ox}} \right)

Straggle correction:

(\Delta R_p^{eff})^2 = (\Delta R_p^{Si})^2 - t_{ox} \left( \frac{(\Delta R_p^{Si})^2 - (\Delta R_p^{ox})^2}{R_p^{ox}} \right)

5.2 Moment Matching at Interfaces

For multi-layer structures, use moment conservation:

\langle x^n \rangle_{total} = \sum_i \langle x^n \rangle_i \cdot w_i

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

6. Two-Dimensional Profile Modeling

6.1 Lateral Straggle

The lateral distribution follows:

C(x, y) = C(x) \cdot \frac{1}{\sqrt{2\pi} \, \Delta R_\perp} \exp\left[ -\frac{y^2}{2 \Delta R_\perp^2} \right]

Relationship between straggles:

\Delta R_\perp \approx (0.7 \text{ to } 1.0) \times \Delta R_p

6.2 Masked Implant with Edge Effects

For a mask opening of width $W$:

C(x, y) = C(x) \cdot \frac{1}{2} \left[ \text{erf}\left( \frac{y + W/2}{\sqrt{2} \, \Delta R_\perp} \right) - \text{erf}\left( \frac{y - W/2}{\sqrt{2} \, \Delta R_\perp} \right) \right]

6.3 Full 3D Distribution

C(x, y, z) = \frac{\Phi}{(2\pi)^{3/2} \Delta R_p \, \Delta R_\perp^2} \exp\left[ -\frac{(x - R_p)^2}{2 \Delta R_p^2} - \frac{y^2 + z^2}{2 \Delta R_\perp^2} \right]

7. Damage and Defect Modeling

7.1 Kinchin-Pease Model

Number of displaced atoms per incident ion:

N_d = \begin{cases} 0 & \text{if } E_D < E_d \\ 1 & \text{if } E_d < E_D < 2E_d \\ \displaystyle\frac{E_D}{2E_d} & \text{if } E_D > 2E_d \end{cases}

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)

N_d = \frac{0.8 \, E_D}{2 E_d}

The factor 0.8 accounts for forward scattering efficiency.

7.3 Damage Energy Partition

Lindhard partition function:

E_D = \frac{E_0}{1 + k \cdot g(\varepsilon)}

Where:

k = 0.1337 \, Z_1^{1/6} \left( \frac{Z_1}{Z_2} \right)^{1/2}

\varepsilon = \frac{32.53 \, M_2 \, E_0}{Z_1 Z_2 (M_1 + M_2)(Z_1^{0.23} + Z_2^{0.23})}

7.4 Amorphization Threshold

Critical dose for amorphization:

\Phi_c \approx \frac{N_0}{N_d \cdot \sigma_{damage}}

Typical values:

Ion	Critical Dose (cm⁻²)
B⁺	$\sim 10^{15}$
P⁺	$\sim 5 \times 10^{14}$
As⁺	$\sim 10^{14}$
Sb⁺	$\sim 5 \times 10^{13}$

7.5 Damage Profile

The damage distribution differs from dopant distribution:

D(x) = \frac{\Phi \cdot N_d(E)}{\sqrt{2\pi} \, \Delta R_d} \exp\left[ -\frac{(x - R_d)^2}{2 \Delta R_d^2} \right]

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

8. Process-Relevant Calculations

8.1 Junction Depth

For Gaussian profile meeting background concentration $C_B$:

x_j = R_p + \Delta R_p \sqrt{2 \ln\left( \frac{C_{max}}{C_B} \right)}

For asymmetric Pearson profiles:

x_j = R_p + \Delta R_p \left[ \gamma + \sqrt{\gamma^2 + 2 \ln\left( \frac{C_{max}}{C_B} \right)} \right]

8.2 Sheet Resistance

R_s = \frac{1}{q \displaystyle\int_0^{x_j} \mu(C(x)) \cdot C(x) \, dx}

With concentration-dependent mobility (Masetti model):

\mu(C) = \mu_{min} + \frac{\mu_0}{1 + (C/C_r)^\alpha} - \frac{\mu_1}{1 + (C_s/C)^\beta}

Parameter	Electrons	Holes
$\mu_{min}$	52.2	44.9
$\mu_0$	1417	470.5
$C_r$	$9.68 \times 10^{16}$	$2.23 \times 10^{17}$
$\alpha$	0.68	0.719

8.3 Threshold Voltage Shift

For channel implant:

\Delta V_T = \frac{q}{\varepsilon_{ox}} \int_0^{x_{max}} C(x) \cdot x \, dx

Simplified (shallow implant):

\Delta V_T \approx \frac{q \, \Phi \, R_p}{\varepsilon_{ox}}

8.4 Dose Calculation from Profile

\Phi = \int_0^{\infty} C(x) \, dx

Verification:

\Phi_{measured} = \frac{I \cdot t}{q \cdot A}

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:

\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} \left[ D^* \frac{\partial C}{\partial x} \right]

Enhanced diffusivity:

D^* = D_i \cdot \left( 1 + \frac{C_I}{C_I^*} \right)

Where:

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

9.2 Dose Loss Mechanisms

Sputtering yield:

Y = \frac{0.042 \, \alpha \, S_n(E_0)}{U_0}

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:

\Phi_{retained} = \Phi_{implanted} \cdot (1 - \eta_{sputter} - \eta_{backscatter})

9.3 High Dose Effects

Dose saturation:

C_{max}^{sat} = \frac{N_0}{\sqrt{2\pi} \, \Delta R_p}

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

9.4 Temperature Effects

Dynamic annealing: Competes with damage accumulation

\Phi_c(T) = \Phi_c(0) \exp\left( \frac{E_a}{k_B T} \right)

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

10. Summary Tables

10.1 Key Scaling Relationships

Parameter	Scaling with Energy
Projected Range	$R_p \propto E^n$ where $n \approx 0.5 - 0.8$
Range Straggle	$\Delta R_p \approx 0.4 R_p$ (light ions) to $0.2 R_p$ (heavy ions)
Lateral Straggle	$\Delta R_\perp \approx 0.7 - 1.0 \times \Delta R_p$
Damage Energy	$E_D/E_0$ increases with ion mass

10.2 Common Implant Parameters in Si

Dopant	Type	Energy (keV)	$R_p$ (nm)	$\Delta R_p$ (nm)
B	p	10	35	14
B	p	50	160	52
P	n	30	40	15
P	n	100	120	40
As	n	50	35	12
As	n	150	95	28

10.3 Simulation Tools Comparison

Approach	Speed	Accuracy	Primary Use
Analytical (Gaussian)	★★★★★	★★☆☆☆	Quick estimates
Pearson IV Tables	★★★★☆	★★★☆☆	Process simulation
Monte Carlo (SRIM/TRIM)	★★☆☆☆	★★★★☆	Profile calibration
Molecular Dynamics	★☆☆☆☆	★★★★★	Damage cascade studies

Quick Reference Formulas

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