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Fluid Dynamics: Mathematical Modeling

Keywords: fluid dynamics, semiconductor fluid dynamics, navier stokes, reynolds number, cfd, wet processing, cmp slurry, gas dynamics

Fluid Dynamics: Mathematical Modeling

1. Overview: Where Fluid Dynamics Matters

Fluid dynamics plays a critical role in numerous semiconductor fabrication steps:

Chemical Vapor Deposition (CVD) — Precursor gas transport and reaction
Spin Coating — Photoresist film formation
Chemical Mechanical Planarization (CMP) — Slurry flow and material removal
Wet Etching/Cleaning — Etchant transport to surfaces
Immersion Lithography — Water flow between lens and wafer
Electrochemical Deposition — Electrolyte flow and ion transport

Each process involves distinct physics, but they share a common mathematical foundation.

2. Fundamental Governing Equations

2.1 Navier-Stokes Framework

The foundation is the incompressible Navier-Stokes equations.

Continuity Equation

abla \cdot \mathbf{u} = 0

Momentum Equation

\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot abla \mathbf{u}\right) = - abla p + \mu abla^2 \mathbf{u} + \mathbf{F}

Where:

$\mathbf{u}$ — Velocity field vector
$p$ — Pressure field
$\rho$ — Fluid density
$\mu$ — Dynamic viscosity
$\mathbf{F}$ — Body forces (gravity, electromagnetic, etc.)

Species Transport Equation

\frac{\partial C_i}{\partial t} + \mathbf{u} \cdot abla C_i = D_i abla^2 C_i + R_i

Where:

$C_i$ — Concentration of species $i$
$D_i$ — Diffusion coefficient of species $i$
$R_i$ — Reaction rate (source/sink term)

Energy Equation

\rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{u} \cdot abla T\right) = k abla^2 T + Q

Where:

$c_p$ — Specific heat capacity
$T$ — Temperature
$k$ — Thermal conductivity
$Q$ — Heat source (reaction heat, Joule heating, etc.)

3. Chemical Vapor Deposition (CVD)

CVD is one of the most mathematically complex processes, coupling gas-phase transport, homogeneous reactions, and heterogeneous surface chemistry.

3.1 Reactor-Scale Transport

In a typical showerhead reactor, gas enters through distributed holes and flows toward a heated wafer. The classic stagnation-point flow solution applies.

Similarity Solution

For axisymmetric flow toward a disk:

u_r = r f'(\eta), \quad u_z = -\sqrt{ u a} \cdot f(\eta)

Where:

$\eta = z\sqrt{a/

u}$ — Similarity variable

$a$ — Strain rate parameter
$

u$ — Kinematic viscosity

This yields the Hiemenz equation :

$$f''' + ff'' - (f')^2 + 1 = 0$$

With boundary conditions:

$f(0) = f'(0) = 0$ (no-slip at surface)
$f'(\infty) = 1$ (far-field condition)

3.2 Key Dimensionless Groups

Damköhler Number

\text{Da} = \frac{k_s L}{D}

Physical meaning: Ratio of surface reaction rate to diffusive transport rate.

Regime	Condition	Implication
Transport-limited	$\text{Da} \gg 1$	Uniformity controlled by flow
Reaction-limited	$\text{Da} \ll 1$	Uniformity controlled by temperature

Péclet Number

\text{Pe} = \frac{UL}{D}

Physical meaning: Ratio of convective to diffusive transport.

Grashof Number

\text{Gr} = \frac{g\beta \Delta T L^3}{ u^2}

Physical meaning: Ratio of buoyancy to viscous forces (important in horizontal reactors).

Where:

$g$ — Gravitational acceleration
$\beta$ — Thermal expansion coefficient
$\Delta T$ — Temperature difference

3.3 Surface Boundary Conditions

The critical coupling between transport and chemistry at the wafer surface:

-D \left.\frac{\partial C}{\partial n}\right|_{\text{surface}} = k_s \cdot f(C, T, \theta)

This is a Robin boundary condition linking diffusive flux to surface kinetics.

Langmuir-Hinshelwood Kinetics

R = \frac{k C}{1 + KC}

Features:

First-order at low concentration ($C \ll 1/K$)
Zero-order (saturated) at high concentration ($C \gg 1/K$)

Sticking Coefficient Model

s = s_0 \cdot f(T) \cdot (1 - \theta)

Where:

$s_0$ — Base sticking coefficient
$\theta$ — Surface coverage fraction

3.4 Multi-Scale Challenge

CVD spans enormous length scales:

Scale	Dimension	Physics
Reactor chamber	0.1–1 m	Continuum CFD
Boundary layer	1–10 mm	Convection-diffusion
Surface features	10–100 nm	Ballistic/Knudsen transport
Molecular mean free path	0.1–10 μm	Molecular dynamics

Knudsen Number

\text{Kn} = \frac{\lambda}{L}

Where $\lambda$ is the molecular mean free path.

Regime	Condition	Modeling Approach
Continuum	$\text{Kn} < 0.01$	Navier-Stokes
Slip flow	$0.01 < \text{Kn} < 0.1$	Navier-Stokes + slip BC
Transition	$0.1 < \text{Kn} < 10$	DSMC, Boltzmann
Free molecular	$\text{Kn} > 10$	Ballistic transport

4. Spin Coating

Spin coating deposits thin photoresist films through centrifugal spreading and solvent evaporation.

4.1 Thin Film Lubrication Theory

For a thin viscous layer ($h \ll R$) on a rotating disk, the lubrication approximation applies:

\frac{\partial h}{\partial t} + \frac{1}{r}\frac{\partial}{\partial r}(r h \bar{u}_r) = -E

Where:

$h(r,t)$ — Film thickness
$\bar{u}_r$ — Depth-averaged radial velocity
$E$ — Evaporation rate

4.2 Velocity Profile

Integrating the momentum equation with:

No-slip at substrate ($u_r = 0$ at $z = 0$)
Zero shear at free surface ($\partial u_r / \partial z = 0$ at $z = h$)

Yields:

u_r(z) = \frac{\rho \omega^2 r}{2\mu}(2hz - z^2)

Depth-averaged velocity:

\bar{u}_r = \frac{\rho \omega^2 r h^2}{3\mu}

4.3 Emslie-Bonner-Peck Solution

For a Newtonian fluid without evaporation:

\frac{\partial h}{\partial t} = -\frac{\rho \omega^2}{3\mu} \cdot \frac{1}{r}\frac{\partial (r h^3)}{\partial r}

For uniform initial thickness $h_0$:

h(t) = \frac{h_0}{\sqrt{1 + \dfrac{4\rho \omega^2 h_0^2 t}{3\mu}}}

Asymptotic behavior:

Short time: $h \approx h_0$
Long time: $h \propto t^{-1/2}$

4.4 Non-Newtonian Photoresists

Real photoresists are shear-thinning. Using a power-law model :

\tau = K\left(\frac{\partial u}{\partial z}\right)^n

Where:

$K$ — Consistency index
$n$ — Power-law index ($n < 1$ for shear-thinning)

The governing equation becomes:

\frac{\partial h}{\partial t} = -\frac{n}{2n+1}\left(\frac{\rho \omega^2}{K}\right)^{1/n} \frac{1}{r}\frac{\partial}{\partial r}\left(r h^{(2n+1)/n}\right)

4.5 Evaporation and Marangoni Effects

Coupled Concentration Equation

\frac{\partial(h x_s)}{\partial t} + \frac{1}{r}\frac{\partial}{\partial r}(r h x_s \bar{u}_r) = -\frac{e}{\rho_s}

Where:

$x_s$ — Solvent mass fraction
$e$ — Evaporation mass flux
$\rho_s$ — Solvent density

Marangoni Stress

Surface tension gradients drive Marangoni flows:

\tau_{\text{surface}} = \frac{\partial \sigma}{\partial r} = \frac{d\sigma}{dC}\frac{\partial C}{\partial r}

Marangoni Number

\text{Ma} = \frac{\Delta\sigma \cdot L}{\mu \alpha}

Where $\alpha$ is thermal diffusivity.

5. Chemical Mechanical Planarization (CMP)

CMP combines chemical etching with mechanical abrasion, mediated by slurry flow between pad and wafer.

5.1 Reynolds Lubrication Equation

For the thin fluid film:

\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} + 12\mu \frac{\partial h}{\partial t}

Terms:

Left side: Pressure-driven (Poiseuille) flow
First term on right: Shear-driven (Couette) flow (wedge effect)
Second term on right: Squeeze film effect

5.2 Slurry as Suspension

CMP slurries contain abrasive particles exhibiting complex rheology.

Shear-Induced Migration (Leighton-Acrivos)

\mathbf{J}_{\text{shear}} = -K_c a^2 \phi abla(\dot{\gamma} \phi) - K_\eta a^2 \dot{\gamma} \phi^2 abla(\ln \eta)

Where:

$a$ — Particle radius
$\phi$ — Particle volume fraction
$\dot{\gamma}$ — Shear rate
$K_c, K_\eta$ — Empirical constants

Physical effect: Particles migrate from high-shear to low-shear regions.

Effective Viscosity (Krieger-Dougherty)

\eta_{\text{eff}} = \eta_0 \left(1 - \frac{\phi}{\phi_m}\right)^{-[\eta]\phi_m}

Where:

$\phi_m$ — Maximum packing fraction (~0.64)
$[\eta]$ — Intrinsic viscosity (~2.5 for spheres)

5.3 Material Removal Models

Classical Preston Equation

\text{MRR} = K_p \cdot p \cdot V

Where:

MRR — Material removal rate
$K_p$ — Preston coefficient
$p$ — Applied pressure
$V$ — Relative velocity

Enhanced Models

\text{MRR} = f(\tau_{\text{shear}}, \phi_{\text{particle}}, k_{\text{chem}}, T)

Incorporating:

Fluid shear stress: $\tau = \mu \left.\dfrac{\partial u}{\partial z}\right|_{\text{surface}}$
Local particle flux
Chemical reaction rate
Temperature-dependent kinetics

5.4 Contact Mechanics

When pad asperities contact wafer:

Greenwood-Williamson Model

p_{\text{contact}} = \frac{4}{3} E^* n \int_d^\infty (z-d)^{3/2} \phi(z) \, dz

Where:

$E^*$ — Effective elastic modulus
$n$ — Asperity density
$\phi(z)$ — Asperity height distribution
$d$ — Separation distance

Force Balance

p_{\text{fluid}} + p_{\text{contact}} = P_{\text{applied}}

6. Wet Etching: Mass Transfer Limited Processes

6.1 Convective-Diffusion Equation

\frac{\partial C}{\partial t} + \mathbf{u} \cdot abla C = D abla^2 C

At the reactive surface (fast reaction limit):

C|_{\text{surface}} = 0

Etch rate:

\text{ER} \propto D \left.\frac{\partial C}{\partial n}\right|_{\text{surface}}

6.2 Rotating Disk Solution (Levich)

For a wafer rotating in etchant:

Velocity Components

u_r = r\omega F(\zeta), \quad u_\theta = r\omega G(\zeta), \quad u_z = \sqrt{ u\omega} H(\zeta)

Where $\zeta = z\sqrt{\omega/ u}$.

Boundary Layer Thickness

\delta = 1.61 D^{1/3} u^{1/6} \omega^{-1/2}

Mass Flux (Levich Equation)

j = 0.62 D^{2/3} u^{-1/6} \omega^{1/2} C_\infty

Key insight : The etch rate is uniform across an infinite disk . This explains why rotating processes achieve excellent uniformity.

6.3 Feature-Scale Transport

In high-aspect-ratio trenches:

Knudsen Diffusion

D_{\text{Kn}} = \frac{d}{3}\sqrt{\frac{8RT}{\pi M}}

Where:

$d$ — Trench width
$M$ — Molecular weight

Concentration Profile in Trench

For a trench of depth $L$ with reactive bottom:

\frac{d^2 C}{dz^2} = 0 \quad \text{(diffusion only)}

With boundary conditions:

$C(0) = C_{\text{top}}$ (top of trench)
$-D\dfrac{dC}{dz}\big|_{z=L} = k_s C(L)$ (reactive bottom)

Solution:

\frac{C(z)}{C_{\text{top}}} = 1 - \frac{z}{L} \cdot \frac{1}{1 + D/(k_s L)}

Thiele Modulus

\phi = L\sqrt{\frac{k_s}{D}}

$\phi \ll 1$: Reaction-limited (uniform etch in feature)
$\phi \gg 1$: Transport-limited (RIE lag)

7. Immersion Lithography

At 193 nm wavelength, water ($n \approx 1.44$) fills the gap between lens and wafer, increasing numerical aperture.

7.1 Free Surface Dynamics

Capillary Number

\text{Ca} = \frac{\mu U}{\sigma}

Where $\sigma$ is surface tension.

$\text{Ca} < \text{Ca}_{\text{crit}} \approx 0.1$: Stable meniscus
$\text{Ca} > \text{Ca}_{\text{crit}}$: Bubble entrainment risk

Young-Laplace Equation

\Delta p = \sigma \kappa = \sigma \left(\frac{1}{R_1} + \frac{1}{R_2}\right)

Where $\kappa$ is the interface curvature.

7.2 Interface Tracking Methods

Level Set Method

\frac{\partial \phi}{\partial t} + \mathbf{u} \cdot abla \phi = 0

Where:

$\phi > 0$: Liquid phase
$\phi < 0$: Gas phase
$\phi = 0$: Interface

Volume of Fluid (VOF)

\frac{\partial \alpha}{\partial t} + abla \cdot (\alpha \mathbf{u}) = 0

Where $\alpha$ is the volume fraction.

7.3 Thermal Management

Light absorption heats the water:

\rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{u} \cdot abla T\right) = k abla^2 T + Q_{\text{abs}}

Refractive Index Sensitivity

\frac{dn}{dT} \approx -1 \times 10^{-4} \text{ K}^{-1}

Temperature variations cause refractive index changes, introducing imaging errors (aberrations).

8. Numerical Methods

8.1 Finite Volume Method (FVM)

The workhorse for semiconductor CFD. Starting from integral form:

\frac{\partial}{\partial t}\int_V \rho \phi \, dV + \oint_S \rho \phi \mathbf{u} \cdot \mathbf{n} \, dS = \oint_S \Gamma abla \phi \cdot \mathbf{n} \, dS + \int_V S_\phi \, dV

Discretization

\frac{(\rho \phi)_P^{n+1} - (\rho \phi)_P^n}{\Delta t} V_P + \sum_f F_f \phi_f = \sum_f \Gamma_f ( abla \phi)_f \cdot \mathbf{A}_f + S_\phi V_P

Where:

$P$ — Cell center
$f$ — Face index
$F_f = \rho \mathbf{u}_f \cdot \mathbf{A}_f$ — Face flux

8.2 Advection Schemes

Scheme	Order	Properties
Upwind	1st	Stable, diffusive
Central	2nd	Unstable for high Pe
QUICK	3rd	Good accuracy, bounded
MUSCL	2nd	TVD, shock-capturing

8.3 Pressure-Velocity Coupling

SIMPLE Algorithm

1. Guess pressure field $p^$ 2. Solve momentum for $\mathbf{u}^$ 3. Solve pressure correction: $ abla \cdot (D abla p') = abla \cdot \mathbf{u}^$ 4. Correct: $p = p^ + \alpha_p p'$, $\mathbf{u} = \mathbf{u}^* - D abla p'$ 5. Iterate until convergence

8.4 Moving Boundary Problems

For etching/deposition where geometry evolves:

Arbitrary Lagrangian-Eulerian (ALE)

\left.\frac{\partial \phi}{\partial t}\right|_{\chi} + (\mathbf{u} - \mathbf{u}_{\text{mesh}}) \cdot abla \phi = \text{RHS}

Where $\mathbf{u}_{\text{mesh}}$ is mesh velocity.

Level Set Velocity Extension

\frac{\partial d}{\partial \tau} + \text{sign}(\phi)(| abla d| - 1) = 0

Reinitializes the level set to a signed distance function.

8.5 Stiff Chemistry

CVD with multiple reactions has time scales from ns (gas reactions) to s (deposition).

Operator Splitting

1. Solve transport: $\dfrac{\partial C}{\partial t} + \mathbf{u} \cdot abla C = D abla^2 C$ 2. Solve chemistry: $\dfrac{dC}{dt} = R(C)$ (using stiff ODE solver)

Implicit Methods

For stiff systems:

\mathbf{C}^{n+1} = \mathbf{C}^n + \Delta t \cdot \mathbf{R}(\mathbf{C}^{n+1})

Requires Newton iteration with Jacobian $\partial R_i / \partial C_j$.

9. Dimensionless

Group	Definition	Physical Meaning
Reynolds (Re)	$\dfrac{\rho UL}{\mu}$	Inertia / Viscosity
Péclet (Pe)	$\dfrac{UL}{D}$	Convection / Diffusion
Damköhler (Da)	$\dfrac{k_s L}{D}$	Reaction / Transport
Knudsen (Kn)	$\dfrac{\lambda}{L}$	Mean free path / Length
Capillary (Ca)	$\dfrac{\mu U}{\sigma}$	Viscous / Surface tension
Marangoni (Ma)	$\dfrac{\Delta\sigma \cdot L}{\mu \alpha}$	Marangoni / Viscous
Grashof (Gr)

u^2}$ | Buoyancy / Viscous |

Schmidt (Sc)

u}{D}$ | Momentum / Mass diffusivity |

Sherwood (Sh)	$\dfrac{k_m L}{D}$	Convective / Diffusive mass transfer
Thiele ($\phi$)	$L\sqrt{\dfrac{k_s}{D}}$	Reaction / Diffusion in pores

10. Current Research Frontiers

10.1 Machine Learning Integration

Surrogate models replacing expensive CFD for real-time process control
Physics-informed neural networks (PINNs) for solving PDEs
Digital twins for predictive maintenance and optimization

10.2 Atomic Layer Processes (ALD/ALE)

Highly transient, surface-reaction-dominated
Requires time-dependent modeling of pulse/purge cycles
Surface coverage evolution:

\frac{d\theta}{dt} = k_{\text{ads}} C (1-\theta) - k_{\text{des}} \theta

10.3 Extreme Aspect Ratios

3D NAND with aspect ratios > 100
Transition to molecular flow ($\text{Kn} > 0.1$)
Transmission probability methods :

P = \frac{1}{1 + 3L/(8r)}

10.4 EUV-Related Flows

Hydrogen buffer gas flow for debris mitigation
Tin droplet dynamics in source
Molecular outgassing and mask contamination

10.5 Plasma-Flow Coupling

Low-pressure plasma processes require multi-physics:

abla \cdot \mathbf{J}_e = S_e - R_e \quad \text{(electron continuity)}

abla \cdot \mathbf{J}_i = S_i - R_i \quad \text{(ion continuity)}

abla \cdot (\epsilon abla \phi) = -e(n_i - n_e) \quad \text{(Poisson)}

Coupled to neutral gas Navier-Stokes equations.

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