Chemical Kinetics

Semiconductor Manufacturing Process Chemical Kinetics: Mathematics

Introduction

Semiconductor manufacturing relies heavily on chemical kinetics to control thin film deposition, etching, oxidation, and dopant diffusion. This document provides the mathematical framework underlying these processes.

Fundamental Kinetic Concepts

Reaction Rate Expression

The general rate expression for a reaction A + B arrow C is:

r = k[A]^m[B]^n

Where:

• r = reaction rate ((mol)/(m^3 · s))
• k = rate constant
• [A], [B] = concentrations ((mol)/(m^3))
• m, n = reaction orders (empirically determined)

Arrhenius Equation

The temperature dependence of rate constants follows the Arrhenius equation:

k = A exp(-(E_a)/(RT))

Where:

• A = pre-exponential factor (frequency factor)
• E_a = activation energy ((J)/(mol))
• R = universal gas constant (8.314 (J)/(mol · K))
• T = absolute temperature (K)

Linearized Form (for Arrhenius plots):

ln(k) = ln(A) - (E_a)/(R) · (1)/(T)

Chemical Vapor Deposition (CVD)

Overall Rate Model

CVD involves both gas-phase transport and surface reaction. The overall deposition rate is:

R = (C_g)/((1)/(h_g) + (1)/(k_s))

Where:

• R = deposition rate ((mol)/(m^2 · s))
• C_g = gas-phase reactant concentration
• h_g = gas-phase mass transfer coefficient ((m)/(s))
• k_s = surface reaction rate constant ((m)/(s))

Regime Analysis

Surface-Reaction Limited (low temperature, k_s ll h_g):

R ≈ k_s · C_g = A exp(-(E_a)/(RT)) · C_g

Mass-Transport Limited (high temperature, h_g ll k_s):

R ≈ h_g · C_g

Mass Transfer Coefficient

For laminar flow over a flat plate:

h_g = (D_AB)/(L) · 0.664 · Re_L^1/2 · Sc^1/3

Where:

• D_AB = binary diffusion coefficient
• L = characteristic length
• Re_L = (rho v L)/(mu) = Reynolds number
• Sc = (mu)/(rho D_AB) = Schmidt number

Thermal Oxidation: Deal-Grove Model

Governing Equation

The Deal-Grove model describes silicon oxidation (Si + O_2 arrow SiO_2):

x^2 + Ax = B(t + tau)

Where:

• x = oxide thickness (m)
• t = oxidation time (s)
• tau = initial time correction (accounts for native oxide)

Rate Constants

Linear Rate Constant:

(B)/(A) = (k_s C^*)/(N_ox)

Parabolic Rate Constant:

B = (2D_eff C^*)/(N_ox)

Where:

• D_eff = effective diffusion coefficient of oxidant through oxide
• C^* = equilibrium oxidant concentration in oxide
• N_ox = number of oxidant molecules incorporated per unit volume of oxide
• k_s = surface reaction rate constant

Limiting Cases

Thin Oxide Regime (short times, x ll A):

x ≈ (B)/(A)(t + tau)

• Linear growth (surface-reaction controlled)

Thick Oxide Regime (long times, x gg A):

x ≈ sqrt(B · t)

• Parabolic growth (diffusion controlled)

Explicit Solution

Solving the quadratic equation:

x = (A)/(2)[sqrt(1 + (4B(t+tau))/(A^2)) - 1]

Plasma Etching Kinetics

Ion-Enhanced Etching Model

The etch rate combines thermal and ion-assisted components:

R = k_thermal · P · exp(-(E_a)/(RT)) + k_ion · Gamma_ion^alpha · theta

Where:

• k_thermal = thermal etching rate constant
• P = reactive gas partial pressure
• Gamma_ion = ion flux ((ions)/(m^2 · s))
• alpha = ion flux exponent (typically 0.5–1.5)
• theta = surface coverage of reactive species

Sputter Yield Model

Physical sputtering rate:

R_sputter = Y(theta, E) · (Gamma_ion)/(n)

Where:

• Y = sputter yield (atoms removed per incident ion)
• E = ion energy
• theta = ion incidence angle
• n = atomic density of target material

Selectivity

Selectivity between materials A and B:

S = (R_A)/(R_B)

Surface Reaction Kinetics

Langmuir Adsorption Isotherm

For single-species adsorption at equilibrium:

theta = (K · P)/(1 + K · P)

Where:

• theta = fractional surface coverage (0 ≤ theta ≤ 1)
• K = adsorption equilibrium constant
• P = partial pressure

Temperature Dependence of K:

K = K_0 exp((-Delta H_ads)/(RT))

Multi-Species Competitive Adsorption

For species A and B competing for the same sites:

theta_A = (K_A P_A)/(1 + K_A P_A + K_B P_B)

theta_B = (K_B P_B)/(1 + K_A P_A + K_B P_B)

Surface Reaction Rate

Langmuir-Hinshelwood Mechanism (both reactants adsorbed):

r = k_s · theta_A · theta_B = k_s · (K_A P_A · K_B P_B)/((1 + K_A P_A + K_B P_B)^2)

Eley-Rideal Mechanism (one reactant from gas phase):

r = k_s · theta_A · P_B = k_s · (K_A P_A · P_B)/(1 + K_A P_A)

Limiting Behavior

Diffusion Processes

Fick's Laws

First Law (steady-state flux):

J = -D (∂ C)/(∂ x)

Second Law (transient diffusion):

(∂ C)/(∂ t) = D (∂^2 C)/(∂ x^2)

For 3D:

(∂ C)/(∂ t) = D ∇^2 C = D ((∂^2 C)/(∂ x^2) + (∂^2 C)/(∂ y^2) + (∂^2 C)/(∂ z^2))

Concentration-Dependent Diffusion

For dopants where D = D(C):

(∂ C)/(∂ t) = (∂)/(∂ x)[D(C) (∂ C)/(∂ x)]

Analytical Solutions

Constant Surface Concentration (semi-infinite medium):

C(x,t) = C_s · erfc((x)/(2sqrt(Dt)))

Where erfc is the complementary error function:

erfc(z) = 1 - erf(z) = 1 - (2)/(sqrt(pi))∫_0^z e^-u^2 du

Fixed Total Dose (Gaussian profile):

C(x,t) = (Q)/(sqrt(pi D t)) exp(-(x^2)/(4Dt))

Where Q = total dose ((atoms)/(m^2))

Diffusion Coefficient Temperature Dependence

D = D_0 exp(-(E_a)/(kT))

Where k = 8.617 × 10^-5 (eV)/(K) (Boltzmann constant)

Reactor-Scale Modeling

Species Conservation Equation

The convection-diffusion-reaction equation:

(∂ C_i)/(∂ t) + ∇ · (v C_i) = ∇ · (D_i ∇ C_i) + R_i

Expanded form:

(∂ C_i)/(∂ t) + v · ∇ C_i = D_i ∇^2 C_i + R_i

Coupled Equations

Navier-Stokes (momentum):

rho ((∂ v)/(∂ t) + v · ∇ v) = -∇ P + mu ∇^2 v + rho g

Continuity (mass):

(∂ rho)/(∂ t) + ∇ · (rho v) = 0

Energy:

rho c_p ((∂ T)/(∂ t) + v · ∇ T) = k ∇^2 T + Q_rxn

Where Q_rxn = ∑_j (-Delta H_j) r_j is the heat of reaction.

Boundary Conditions

Surface reaction flux:

-D_i (∂ C_i)/(∂ n)|_surface = R_s,i

Inlet conditions:

C_i = C_i,inlet, T = T_inlet, v = v_inlet

Dimensionless Analysis

Damköhler Number

Da = (reaction rate)/(transport rate) = (k_s L)/(D)

Thiele Modulus

For reactions in porous structures:

phi = L sqrt((k)/(D_eff))

Effectiveness Factor:

eta = (tanh(phi))/(phi)

Peclet Number

Pe = (vL)/(D) = (convective transport)/(diffusive transport)

Stanton Number

St = (h)/(rho v c_p) = (heat transfer)/(thermal capacity of flow)

Advanced Modeling Techniques

Microkinetic Modeling

System of coupled ODEs for surface species:

(d theta_i)/(dt) = ∑_j [
u_ij^+ r_j^+ -
u_ij^- r_j^-]

Where:

- theta_i = coverage of species i
-
u_ij = stoichiometric coefficient
- r_j^+, r_j^- = forward and reverse rates of reaction j

Example: Adsorption-Desorption-Reaction:

(d theta_A)/(dt) = k_ads P_A (1-theta_A-theta_B) - k_des theta_A - k_rxn theta_A theta_B

Stochastic Methods

Kinetic Monte Carlo (KMC):

Transition rates:

W_i =
u_i exp(-(E_i)/(kT))

Time step:

Delta t = -(ln(r))/(∑_i W_i)

Where r in (0,1] is a random number.

Master Equation:

(dP_n)/(dt) = ∑_m [W_mn P_m - W_nm P_n]

Multi-Scale Coupling

Computational Methods

Numerical Discretization

Finite Difference (1D diffusion):

(C_i^n+1 - C_i^n)/(Delta t) = D (C_i+1^n - 2C_i^n + C_i-1^n)/((Delta x)^2)

Stability Criterion (explicit method):

(D Delta t)/((Delta x)^2) ≤ (1)/(2)

Operator Splitting

For stiff reaction-diffusion systems:

1. Diffusion step: Solve (∂ C)/(∂ t) = D ∇^2 C for Delta t/2
2. Reaction step: Solve (dC)/(dt) = R(C) for Delta t
3. Diffusion step: Solve (∂ C)/(∂ t) = D ∇^2 C for Delta t/2

Newton-Raphson for Nonlinear Systems

x^(k+1) = x^(k) - J^-1(x^(k)) · F(x^(k))

Where J is the Jacobian matrix:

J_ij = (∂ F_i)/(∂ x_j)

Key Equations Summary

Rate Expressions

Dimensionless Numbers