Semiconductor Manufacturing Process: Physics-Based Modeling and Differential Equations
A comprehensive reference for the physics and mathematics governing semiconductor fabrication processes.
1. Thermal Oxidation of Silicon
1.1 Deal-Grove Model
The foundational model for silicon oxidation describes oxide thickness growth through coupled transport and reaction.
Governing Equation:
$$
x^2 + Ax = B(t + \tau)
$$
Parameter Definitions:
- $x$ β oxide thickness
- $A = \frac{2D_{ox}}{k_s}$ β linear rate constant parameter (related to surface reaction)
- $B = \frac{2D_{ox}C^*}{N_1}$ β parabolic rate constant (related to diffusion)
- $D_{ox}$ β oxidant diffusivity through oxide
- $k_s$ β surface reaction rate constant
- $C^*$ β equilibrium oxidant concentration at gas-oxide interface
- $N_1$ β number of oxidant molecules incorporated per unit volume of oxide
- $\tau$ β time shift accounting for initial oxide
1.2 Underlying Diffusion Physics
Steady-state diffusion through the oxide:
$$
\frac{\partial C}{\partial t} = D_{ox}\frac{\partial^2 C}{\partial x^2}
$$
Boundary Conditions:
- Gas-oxide interface (flux from gas phase):
$$
F_1 = h_g(C^* - C_0)
$$
- Si-SiOβ interface (surface reaction):
$$
F_2 = k_s C_i
$$
Steady-state flux through the oxide:
$$
F = \frac{D_{ox}C^*}{1 + \frac{k_s}{h_g} + \frac{k_s x}{D_{ox}}}
$$
1.3 Limiting Growth Regimes
| Regime | Condition | Growth Law | Physical Interpretation |
|--------|-----------|------------|------------------------|
| Linear | Thin oxide ($x \ll A$) | $x \approx \frac{B}{A}(t + \tau)$ | Reaction-limited |
| Parabolic | Thick oxide ($x \gg A$) | $x \approx \sqrt{Bt}$ | Diffusion-limited |
2. Dopant Diffusion
2.1 Fick's Laws of Diffusion
First Law (Flux Equation):
$$
\vec{J} = -D
abla C
$$
Second Law (Mass Conservation / Continuity):
$$
\frac{\partial C}{\partial t} =
abla \cdot (D
abla C)
$$
For constant diffusivity in 1D:
$$
\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}
$$
2.2 Analytical Solutions
Constant Surface Concentration (Predeposition)
Initial condition: $C(x, 0) = 0$
Boundary condition: $C(0, t) = C_s$
$$
C(x,t) = C_s \cdot \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)
$$
where the complementary error function is:
$$
\text{erfc}(z) = 1 - \text{erf}(z) = 1 - \frac{2}{\sqrt{\pi}}\int_0^z e^{-u^2} du
$$
Fixed Dose / Drive-in (Gaussian Distribution)
Initial condition: Delta function at surface with dose $Q$
$$
C(x,t) = \frac{Q}{\sqrt{\pi Dt}} \exp\left(-\frac{x^2}{4Dt}\right)
$$
Key Parameters:
- $Q$ β total dose per unit area (atoms/cmΒ²)
- $\sqrt{Dt}$ β diffusion length
- Peak concentration: $C_{max} = \frac{Q}{\sqrt{\pi Dt}}$
2.3 Concentration-Dependent Diffusion
At high doping concentrations, diffusivity becomes concentration-dependent:
$$
\frac{\partial C}{\partial t} = \frac{\partial}{\partial x}\left[D(C)\frac{\partial C}{\partial x}\right]
$$
Fair-Tsai Model for Diffusivity:
$$
D = D_i + D^-\frac{n}{n_i} + D^+\frac{p}{n_i} + D^{++}\left(\frac{p}{n_i}\right)^2
$$
Parameter Definitions:
- $D_i$ β intrinsic diffusivity (via neutral defects)
- $D^-$ β diffusivity via negatively charged defects
- $D^+$ β diffusivity via singly positive charged defects
- $D^{++}$ β diffusivity via doubly positive charged defects
- $n, p$ β electron and hole concentrations
- $n_i$ β intrinsic carrier concentration
2.4 Point Defect Coupled Diffusion
Modern TCAD uses coupled equations for dopants and point defects (vacancies $V$ and interstitials $I$):
Vacancy Continuity:
$$
\frac{\partial C_V}{\partial t} = D_V
abla^2 C_V - k_{IV}C_V C_I + G_V - \frac{C_V - C_V^*}{\tau_V}
$$
Interstitial Continuity:
$$
\frac{\partial C_I}{\partial t} = D_I
abla^2 C_I - k_{IV}C_V C_I + G_I - \frac{C_I - C_I^*}{\tau_I}
$$
Term Definitions:
- $D_V, D_I$ β diffusion coefficients for vacancies and interstitials
- $k_{IV}$ β recombination rate constant for $V$-$I$ annihilation
- $G_V, G_I$ β generation rates
- $C_V^, C_I^$ β equilibrium concentrations
- $\tau_V, \tau_I$ β lifetimes at sinks (surfaces, dislocations)
Effective Dopant Diffusivity:
$$
D_{eff} = f_I D_I \frac{C_I}{C_I^} + f_V D_V \frac{C_V}{C_V^}
$$
where $f_I$ and $f_V$ are the interstitial and vacancy fractions for the specific dopant species.
3. Ion Implantation
3.1 Range Distribution (LSS Theory)
The implanted dopant profile follows approximately a Gaussian distribution:
$$
C(x) = \frac{\Phi}{\sqrt{2\pi}\Delta R_p} \exp\left[-\frac{(x - R_p)^2}{2\Delta R_p^2}\right]
$$
Parameters:
- $\Phi$ β dose (ions/cmΒ²)
- $R_p$ β projected range (mean implant depth)
- $\Delta R_p$ β straggle (standard deviation of range distribution)
Higher-Order Moments (Pearson IV Distribution):
- $\gamma$ β skewness (asymmetry)
- $\beta$ β kurtosis (peakedness)
3.2 Stopping Power (Energy Loss)
The rate of energy loss as ions traverse the target:
$$
\frac{dE}{dx} = -N[S_n(E) + S_e(E)]
$$
Components:
- $S_n(E)$ β nuclear stopping power (elastic collisions with target nuclei)
- $S_e(E)$ β electronic stopping power (inelastic interactions with electrons)
- $N$ β atomic density of target material (atoms/cmΒ³)
LSS Electronic Stopping (Low Energy):
$$
S_e \propto \sqrt{E}
$$
Nuclear Stopping: Uses screened Coulomb potentials with Thomas-Fermi or ZBL (Ziegler-Biersack-Littmark) universal screening functions.
3.3 Boltzmann Transport Equation
For rigorous treatment (typically solved via Monte Carlo methods):
$$
\frac{\partial f}{\partial t} + \vec{v} \cdot
abla_r f + \frac{\vec{F}}{m} \cdot
abla_v f = \left(\frac{\partial f}{\partial t}\right)_{coll}
$$
Variables:
- $f(\vec{r}, \vec{v}, t)$ β particle distribution function
- $\vec{F}$ β external force
- Right-hand side β collision integral
3.4 Damage Accumulation
Kinchin-Pease Model:
$$
N_d = \frac{E_{damage}}{2E_d}
$$
Parameters:
- $N_d$ β number of displaced atoms
- $E_{damage}$ β energy available for displacement
- $E_d$ β displacement threshold energy ($\approx 15$ eV for silicon)
4. Chemical Vapor Deposition (CVD)
4.1 Coupled Transport Equations
Species Transport (Convection-Diffusion-Reaction):
$$
\frac{\partial C_i}{\partial t} + \vec{u} \cdot
abla C_i = D_i
abla^2 C_i + R_i
$$
Navier-Stokes Equations (Momentum):
$$
\rho\left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot
abla\vec{u}\right) = -
abla p + \mu
abla^2\vec{u} + \rho\vec{g}
$$
Continuity Equation (Incompressible Flow):
$$
abla \cdot \vec{u} = 0
$$
Energy Equation:
$$
\rho c_p\left(\frac{\partial T}{\partial t} + \vec{u} \cdot
abla T\right) = k
abla^2 T + Q_{reaction}
$$
Variable Definitions:
- $C_i$ β concentration of species $i$
- $\vec{u}$ β velocity vector
- $D_i$ β diffusion coefficient of species $i$
- $R_i$ β net reaction rate for species $i$
- $\rho$ β density
- $p$ β pressure
- $\mu$ β dynamic viscosity
- $c_p$ β specific heat at constant pressure
- $k$ β thermal conductivity
- $Q_{reaction}$ β heat of reaction
4.2 Surface Reaction Kinetics
Flux Balance at Wafer Surface:
$$
h_m(C_b - C_s) = k_s C_s
$$
Deposition Rate:
$$
G = \frac{k_s h_m C_b}{k_s + h_m}
$$
Parameters:
- $h_m$ β mass transfer coefficient
- $k_s$ β surface reaction rate constant
- $C_b$ β bulk gas concentration
- $C_s$ β surface concentration
Limiting Cases:
| Regime | Condition | Rate Expression | Control Mechanism |
|--------|-----------|-----------------|-------------------|
| Reaction-limited | $k_s \ll h_m$ | $G \approx k_s C_b$ | Surface chemistry |
| Transport-limited | $k_s \gg h_m$ | $G \approx h_m C_b$ | Mass transfer |
4.3 Step Coverage β Knudsen Diffusion
In high-aspect-ratio features, molecular (Knudsen) flow dominates:
$$
D_K = \frac{d}{3}\sqrt{\frac{8k_B T}{\pi m}}
$$
Parameters:
- $d$ β characteristic feature dimension
- $k_B$ β Boltzmann constant
- $T$ β temperature
- $m$ β molecular mass
Thiele Modulus (Reaction-Diffusion Balance):
$$
\phi = L\sqrt{\frac{k_s}{D_K}}
$$
Interpretation:
- $\phi \ll 1$ β Reaction-limited β Conformal deposition
- $\phi \gg 1$ β Diffusion-limited β Poor step coverage
5. Atomic Layer Deposition (ALD)
5.1 Surface Site Model
Precursor A Adsorption Kinetics:
$$
\frac{d\theta_A}{dt} = s_0 \frac{P_A}{\sqrt{2\pi m_A k_B T}}(1 - \theta_A) - k_{des}\theta_A
$$
Parameters:
- $\theta_A$ β fractional surface coverage of precursor A
- $s_0$ β sticking coefficient
- $P_A$ β partial pressure of precursor A
- $m_A$ β molecular mass of precursor A
- $k_{des}$ β desorption rate constant
5.2 Growth Per Cycle (GPC)
$$
GPC = n_{sites} \cdot \Omega \cdot \theta_A^{sat}
$$
Parameters:
- $n_{sites}$ β surface site density (sites/cmΒ²)
- $\Omega$ β atomic volume (volume per deposited atom)
- $\theta_A^{sat}$ β saturation coverage achieved during half-cycle
6. Plasma Etching
6.1 Plasma Fluid Equations
Electron Continuity:
$$
\frac{\partial n_e}{\partial t} +
abla \cdot \vec{\Gamma}_e = S_{ionization} - S_{recomb}
$$
Ion Continuity:
$$
\frac{\partial n_i}{\partial t} +
abla \cdot \vec{\Gamma}_i = S_{ionization} - S_{recomb}
$$
Drift-Diffusion Flux (Electrons):
$$
\vec{\Gamma}_e = -n_e\mu_e\vec{E} - D_e
abla n_e
$$
Drift-Diffusion Flux (Ions):
$$
\vec{\Gamma}_i = n_i\mu_i\vec{E} - D_i
abla n_i
$$
Poisson's Equation (Self-Consistent Field):
$$
abla^2\phi = -\frac{e}{\varepsilon_0}(n_i - n_e)
$$
Electron Energy Balance:
$$
\frac{\partial}{\partial t}\left(\frac{3}{2}n_e k_B T_e\right) +
abla \cdot \vec{q}_e = -e\vec{\Gamma}_e \cdot \vec{E} - \sum_j \epsilon_j R_j
$$
6.2 Sheath Physics
Bohm Criterion (Sheath Edge Condition):
$$
u_i \geq u_B = \sqrt{\frac{k_B T_e}{M_i}}
$$
Child-Langmuir Law (Collisionless Sheath Ion Current):
$$
J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{M_i}}\frac{V_0^{3/2}}{d^2}
$$
Parameters:
- $u_i$ β ion velocity at sheath edge
- $u_B$ β Bohm velocity
- $T_e$ β electron temperature
- $M_i$ β ion mass
- $V_0$ β sheath voltage drop
- $d$ β sheath thickness
6.3 Surface Etch Kinetics
Ion-Enhanced Etching Rate:
$$
R_{etch} = Y_i\Gamma_i + Y_n\Gamma_n(1-\theta) + Y_{syn}\Gamma_i\theta
$$
Components:
- $Y_i\Gamma_i$ β physical sputtering contribution
- $Y_n\Gamma_n(1-\theta)$ β spontaneous chemical etching
- $Y_{syn}\Gamma_i\theta$ β ion-enhanced (synergistic) etching
Yield Parameters:
- $Y_i$ β physical sputtering yield
- $Y_n$ β spontaneous chemical etch yield
- $Y_{syn}$ β synergistic yield (ion-enhanced chemistry)
- $\Gamma_i, \Gamma_n$ β ion and neutral fluxes
- $\theta$ β fractional surface coverage of reactive species
Surface Coverage Dynamics:
$$
\frac{d\theta}{dt} = s\Gamma_n(1-\theta) - Y_{syn}\Gamma_i\theta - k_v\theta
$$
Terms:
- $s\Gamma_n(1-\theta)$ β adsorption onto empty sites
- $Y_{syn}\Gamma_i\theta$ β consumption by ion-enhanced reaction
- $k_v\theta$ β thermal desorption/volatilization
7. Lithography
7.1 Aerial Image Formation
Hopkins Formulation (Partially Coherent Imaging):
$$
I(x,y) = \iint TCC(f,g;f',g') \cdot \tilde{M}(f,g) \cdot \tilde{M}^*(f',g') \, df\,dg\,df'\,dg'
$$
Parameters:
- $TCC$ β Transmission Cross Coefficient (encapsulates partial coherence)
- $\tilde{M}(f,g)$ β Fourier transform of mask transmission function
- $f, g$ β spatial frequencies
Rayleigh Resolution Criterion:
$$
Resolution = k_1 \frac{\lambda}{NA}
$$
Depth of Focus:
$$
DOF = k_2 \frac{\lambda}{NA^2}
$$
Parameters:
- $k_1, k_2$ β process-dependent factors
- $\lambda$ β exposure wavelength
- $NA$ β numerical aperture
7.2 Photoresist Exposure β Dill Model
Intensity Attenuation with Photobleaching:
$$
\frac{\partial I}{\partial z} = -\alpha(M)I
$$
where the absorption coefficient depends on PAC concentration:
$$
\alpha = AM + B
$$
Photoactive Compound (PAC) Decomposition:
$$
\frac{\partial M}{\partial t} = -CIM
$$
Dill Parameters:
| Parameter | Description | Units |
|-----------|-------------|-------|
| $A$ | Bleachable absorption coefficient | ΞΌmβ»ΒΉ |
| $B$ | Non-bleachable absorption coefficient | ΞΌmβ»ΒΉ |
| $C$ | Exposure rate constant | cmΒ²/mJ |
| $M$ | Relative PAC concentration | dimensionless (0-1) |
7.3 Chemically Amplified Resists
Photoacid Generation:
$$
\frac{\partial [H^+]}{\partial t} = C \cdot I \cdot [PAG]
$$
Post-Exposure Bake β Acid Diffusion and Reaction:
$$
\frac{\partial [H^+]}{\partial t} = D_{acid}
abla^2[H^+] - k_{loss}[H^+]
$$
Deprotection Reaction (Catalytic Amplification):
$$
\frac{\partial [Protected]}{\partial t} = -k_{cat}[H^+][Protected]
$$
Parameters:
- $[PAG]$ β photoacid generator concentration
- $D_{acid}$ β acid diffusion coefficient
- $k_{loss}$ β acid loss rate (neutralization, evaporation)
- $k_{cat}$ β catalytic deprotection rate constant
7.4 Development Rate β Mack Model
$$
R = R_{max}\frac{(a+1)(1-M)^n}{a + (1-M)^n} + R_{min}
$$
Parameters:
- $R_{max}$ β maximum development rate (fully exposed)
- $R_{min}$ β minimum development rate (unexposed)
- $a$ β selectivity parameter
- $n$ β contrast parameter
- $M$ β normalized PAC concentration after exposure
8. Epitaxy
8.1 Burton-Cabrera-Frank (BCF) Theory
Adatom Diffusion on Terraces:
$$
\frac{\partial n}{\partial t} = D_s
abla^2 n + F - \frac{n}{\tau}
$$
Parameters:
- $n$ β adatom density on terrace
- $D_s$ β surface diffusion coefficient
- $F$ β deposition flux (atoms/cmΒ²Β·s)
- $\tau$ β adatom lifetime before desorption
Step Velocity:
$$
v_{step} = \Omega D_s\left[\left(\frac{\partial n}{\partial x}\right)_+ - \left(\frac{\partial n}{\partial x}\right)_-\right]
$$
Steady-State Solution for Step Flow:
$$
v_{step} = \frac{2D_s \lambda_s F}{l} \cdot \tanh\left(\frac{l}{2\lambda_s}\right)
$$
Parameters:
- $\Omega$ β atomic volume
- $\lambda_s = \sqrt{D_s \tau}$ β surface diffusion length
- $l$ β terrace width
8.2 Rate Equations for Island Nucleation
Monomer (Single Adatom) Density:
$$
\frac{dn_1}{dt} = F - 2\sigma_1 D_s n_1^2 - \sum_{j>1}\sigma_j D_s n_1 n_j - \frac{n_1}{\tau}
$$
Cluster of Size $j$:
$$
\frac{dn_j}{dt} = \sigma_{j-1}D_s n_1 n_{j-1} - \sigma_j D_s n_1 n_j
$$
Parameters:
- $n_j$ β density of clusters containing $j$ atoms
- $\sigma_j$ β capture cross-section for clusters of size $j$
9. Chemical Mechanical Polishing (CMP)
9.1 Preston Equation
$$
MRR = K_p \cdot P \cdot V
$$
Parameters:
- $MRR$ β material removal rate (nm/min)
- $K_p$ β Preston coefficient (material/process dependent)
- $P$ β applied pressure
- $V$ β relative velocity between pad and wafer
9.2 Contact Mechanics β Greenwood-Williamson Model
Real Contact Area:
$$
A_r = \pi \eta A_n R_p \int_d^\infty (z-d)\phi(z)dz
$$
Parameters:
- $\eta$ β asperity density
- $A_n$ β nominal contact area
- $R_p$ β asperity radius
- $d$ β separation distance
- $\phi(z)$ β asperity height distribution
9.3 Slurry Hydrodynamics β Reynolds Equation
$$
\frac{\partial}{\partial x}\left(h^3\frac{\partial p}{\partial x}\right) + \frac{\partial}{\partial y}\left(h^3\frac{\partial p}{\partial y}\right) = 6\mu U\frac{\partial h}{\partial x}
$$
Parameters:
- $h$ β film thickness
- $p$ β pressure
- $\mu$ β dynamic viscosity
- $U$ β sliding velocity
10. Thin Film Stress
10.1 Stoney Equation
Film Stress from Wafer Curvature:
$$
\sigma_f = \frac{E_s h_s^2}{6(1-
u_s)h_f R}
$$
Parameters:
- $\sigma_f$ β film stress
- $E_s$ β substrate Young's modulus
- $
u_s$ β substrate Poisson's ratio
- $h_s$ β substrate thickness
- $h_f$ β film thickness
- $R$ β radius of curvature
10.2 Thermal Stress
$$
\sigma_{th} = \frac{E_f}{1-
u_f}(\alpha_s - \alpha_f)\Delta T
$$
Parameters:
- $E_f$ β film Young's modulus
- $
u_f$ β film Poisson's ratio
- $\alpha_s, \alpha_f$ β thermal expansion coefficients (substrate, film)
- $\Delta T$ β temperature change from deposition
11. Electromigration (Reliability)
11.1 Black's Equation (Empirical MTTF)
$$
MTTF = A \cdot j^{-n} \cdot \exp\left(\frac{E_a}{k_B T}\right)
$$
Parameters:
- $MTTF$ β mean time to failure
- $j$ β current density
- $n$ β current density exponent (typically 1-2)
- $E_a$ β activation energy
- $A$ β material/geometry constant
11.2 Drift-Diffusion Model
$$
\frac{\partial C}{\partial t} =
abla \cdot \left[D\left(
abla C - C\frac{Z^*e\rho \vec{j}}{k_B T}\right)\right]
$$
Parameters:
- $C$ β atomic concentration
- $D$ β diffusion coefficient
- $Z^*$ β effective charge number (wind force parameter)
- $\rho$ β electrical resistivity
- $\vec{j}$ β current density vector
11.3 Stress Evolution β Korhonen Model
$$
\frac{\partial \sigma}{\partial t} = \frac{\partial}{\partial x}\left[\frac{D_a B\Omega}{k_B T}\left(\frac{\partial\sigma}{\partial x} + \frac{Z^*e\rho j}{\Omega}\right)\right]
$$
Parameters:
- $\sigma$ β hydrostatic stress
- $D_a$ β atomic diffusivity
- $B$ β effective bulk modulus
- $\Omega$ β atomic volume
12. Numerical Solution Methods
12.1 Common Numerical Techniques
| Method | Application | Strengths |
|--------|-------------|-----------|
| Finite Difference (FDM) | Regular grids, 1D/2D problems | Simple implementation, efficient |
| Finite Element (FEM) | Complex geometries, stress analysis | Flexible meshing, boundary conditions |
| Monte Carlo | Ion implantation, plasma kinetics | Statistical accuracy, handles randomness |
| Level Set | Topography evolution (etch/deposition) | Handles topology changes |
| Kinetic Monte Carlo (KMC) | Atomic-scale diffusion, nucleation | Captures rare events, atomic detail |
12.2 Discretization Examples
Explicit Forward Euler (1D Diffusion):
$$
C_i^{n+1} = C_i^n + \frac{D\Delta t}{(\Delta x)^2}\left(C_{i+1}^n - 2C_i^n + C_{i-1}^n\right)
$$
Stability Criterion:
$$
\frac{D\Delta t}{(\Delta x)^2} \leq \frac{1}{2}
$$
Implicit Backward Euler:
$$
C_i^{n+1} - \frac{D\Delta t}{(\Delta x)^2}\left(C_{i+1}^{n+1} - 2C_i^{n+1} + C_{i-1}^{n+1}\right) = C_i^n
$$
12.3 Major TCAD Software Tools
- Synopsys Sentaurus β comprehensive process and device simulation
- Silvaco ATHENA/ATLAS β process and device modeling
- COMSOL Multiphysics β general multiphysics platform
- SRIM/TRIM β ion implantation Monte Carlo
- PROLITH β lithography simulation
Processes and Governing Equations
| Process | Primary Physics | Key Equation |
|---------|-----------------|--------------|
| Oxidation | Diffusion + Reaction | $x^2 + Ax = Bt$ |
| Diffusion | Mass Transport | $\frac{\partial C}{\partial t} = D
abla^2 C$ |
| Implantation | Ballistic + Stopping | $\frac{dE}{dx} = -N(S_n + S_e)$ |
| CVD | Transport + Kinetics | Navier-Stokes + Species |
| ALD | Self-limiting Adsorption | Langmuir kinetics |
| Plasma Etch | Plasma + Surface | Poisson + Drift-Diffusion |
| Lithography | Wave Optics + Chemistry | Dill ABC model |
| Epitaxy | Surface Diffusion | BCF theory |
| CMP | Tribology + Chemistry | Preston equation |
| Stress | Elasticity | Stoney equation |
| Electromigration | Mass transport under current | Korhonen model |