300MM WET CHAMBER SIMULATOR METHODOLOGY & PHYSICS
📖 User Guide 🔬 Launch Simulator

Engineering Methodology & Mathematical Modeling

A rigorous technical overview of the depth-integrated, reduced-order hydrodynamic framework (LUB-2D), centrifugal transport kinematics, Marangoni stress formulation, and governing scaling laws implemented in the 300mm Single Wafer Wet Chamber Simulator.

FRAMEWORK: LUB-2D Depth-Integrated Lubrication
SUBSTRATE: 300 mm Single Silicon Wafer (R = 150 mm)
VERSION: Release 14.7 Technical Documentation
📑 Documentation Contents

1. Reduced-Order Physical Philosophy

In semiconductor single-wafer wet processing (wet cleaning, etch back, resist stripping, and deionized water rinsing), chemical distribution and boundary-layer dynamics strongly determine etch uniformity, particle removal efficiency, and chemical consumption efficiency.

Standard industrial Computational Fluid Dynamics (CFD) packages solve the full 3D transient Navier-Stokes equations coupled with Volume-of-Fluid (VOF) surface capturing. While high-fidelity, a 3D transient VOF simulation of a 300mm spinning disk typically requires tens of millions of computational cells and hours or days of high-performance computing (HPC) runtime for a single process second.

The 300mm Wet Chamber Simulator adopts a reduced-order, depth-integrated thin-film hydrodynamic framework. By exploiting the extreme aspect ratio of wet spin processing—where liquid film thickness ($h \sim 10 - 800\,\mu\text{m}$) is orders of magnitude smaller than the wafer radius ($R = 150\,\text{mm}$)—the governing Navier-Stokes equations can be systematically integrated across the film depth. This provides sub-second interactive parameter sweeps, nozzle position optimization, and stability screening suitable for recipe development and engineering education.

Key Distinction: This simulator is an engineering pre-screening and sensitivity analysis tool. It enables rapid comparison of nozzle layouts, RPM sweeps, and chemical blends prior to physical recipe qualification on production tools.

2. Coordinate Systems & Rotating Wafer Transport

Two distinct coordinate systems are required to model single-wafer chemical wet chambers:

Centrifugal Acceleration

The wafer rotates at angular speed $\omega$ determined by the process spin speed (RPM):

$\omega = \frac{2\pi \cdot \text{RPM}}{60} \quad [\text{rad/s}]$

Centrifugal acceleration $a_r(r)$ experienced by liquid elements on the spinning substrate increases linearly with radial distance from the wafer center:

$a_r(r) = \omega^2 r \quad [\text{m/s}^2]$
At 1000 RPM on a 300mm wafer bevel ($r = 0.15\,\text{m}$), $a_r \approx (104.72)^2 \times 0.15 \approx 1,645\,\text{m/s}^2 \approx 168\,g$.

Because centrifugal body forces dwarf gravity ($a_r \gg g$) across the majority of the wafer radius, outward radial flow is dominated by the balance between centrifugal driving acceleration and viscous wall shear stress.

3. Lubrication Approximation (LUB-2D)

Given film thickness $h(r) \ll R$, the characteristic aspect ratio is $\epsilon = H/R \sim 10^{-4} - 10^{-3} \ll 1$. Under standard boundary-layer lubrication scaling (Oron et al., 1997), vertical momentum equations reduce to hydrostatic pressure balance, and radial momentum simplifies to the balance between centrifugal force and viscous shear stress across the vertical coordinate $z$:

$\rho \omega^2 r + \mu \frac{\partial^2 u_r}{\partial z^2} = 0$

Applying boundary conditions:

  1. No-Slip at Substrate: $u_r(z=0) = 0$
  2. Zero Shear at Free Surface: $\frac{\partial u_r}{\partial z}\Big|_{z=h} = 0$ (neglecting initial gas drag)

Integrating twice yields the classical semi-parabolic velocity distribution across the film thickness:

$u_r(z) = \frac{\rho \omega^2 r}{\mu} \left( h z - \frac{1}{2} z^2 \right)$

The maximum surface velocity at $z = h$ is $u_{\text{surf}} = \frac{\rho \omega^2 r h^2}{2\mu}$. Integrating over the depth $z \in [0, h]$ gives the depth-averaged radial volumetric flux per unit azimuthal arc length:

$q_r(r) = \int_0^h u_r(z) dz = \frac{\rho \omega^2 r h^3}{3\mu} \quad [\text{m}^2/\text{s}]$

4. Film Thickness & Scaling Laws

In continuous steady-state single-wafer dispense, total chemical volumetric inflow $Q$ must equal the outward radial flux traversing any concentric circle of radius $r$:

$2\pi r q_r(r) = Q \implies 2\pi r \left( \frac{\rho \omega^2 r h(r)^3}{3\mu} \right) = Q$

Solving explicitly for the local steady-state film thickness $h(r)$ yields the foundational Emslie-Bonner-Peck (1958) rotational thin-film equation:

$h(r) = \left[ \frac{3\mu Q}{2\pi \rho \omega^2} \right]^{1/3} r^{-2/3}$

Key Scaling Principles

Parameter Scaling Dependency Physical Process Interpretation
Angular Velocity ($\omega$) $h \propto \omega^{-2/3}$ Higher spin speed drastically thins the liquid film, decreasing chemical residence time.
Radial Coordinate ($r$) $h \propto r^{-2/3}$ Centrifugal acceleration increases with radius, causing intrinsic thinning toward the wafer bevel.
Fluid Viscosity ($\mu$) $h \propto \mu^{1/3}$ Viscous chemicals (e.g. concentrated SPM or DSP+) form thicker boundary layers.
Volumetric Flow Rate ($Q$) $h \propto Q^{1/3}$ Film thickness exhibits weak cubic-root sensitivity to chemical flow rates.
Radial Transport Velocity ($u_r$) $u_r \propto \omega^{2/3}$ Convective transport velocity increases rapidly with spin speed.

5. Multi-Nozzle Spatial Source Model

Industrial 300mm chambers employ multiple dispense arms to counteract natural $r^{-2/3}$ radial thinning. The simulator implements a multi-nozzle source formulation categorized into three functional groups:

Each nozzle arm is parameterized by radial coordinate $r_i$, footprint width $w_i$, and volumetric flow rate $Q_i$. The continuous spatial source term $S(r)$ is represented via Gaussian or smoothed parabolic distributions:

$S(r) = \sum_{i=1}^{N_{\text{nozzles}}} \frac{Q_i}{2\pi r_i \sqrt{2\pi \sigma_i^2}} \exp\left( -\frac{(r - r_i)^2}{2\sigma_i^2} \right)$

Continuity across the wafer surface in the depth-integrated framework satisfies:

$\frac{\partial h}{\partial t} + \frac{1}{r}\frac{\partial}{\partial r}\left( r q_r(r) \right) = S(r) - E(r)$
where $E(r)$ represents localized evaporative mass loss.

6. Temperature & Viscosity Dependencies

Wet chemical properties are highly sensitive to operating temperature ($T$). Viscosity is modeled using the Andrade exponential fitting model:

$\mu(T) = \mu_{25} \exp\left( -B \cdot (T - 25) \right)$

Where $\mu_{25}$ is dynamic viscosity at reference room temperature (25°C), and $B$ is the empirical thermal sensitivity coefficient ($B \approx 0.018 - 0.024\,\text{K}^{-1}$ for typical aqueous mixtures).

When chemicals are dispensed at elevated temperatures (e.g., hot SPM at 130°C or hot DIW at 65°C), dynamic viscosity decreases significantly. This reduction reduces viscous resistance, accelerating outward radial transport ($u_r \propto \mu^{-1}$) and leading to thinner steady-state liquid films that demand higher replenishment flows to prevent dewetting.

7. Solutocapillary & Thermocapillary Marangoni Stresses

Spatial gradients in liquid surface tension exert tangential shear stresses along the free liquid-gas interface. The surface shear stress vector is governed by the Marangoni relationship:

$\tau_M = \nabla_s \sigma = \frac{\partial \sigma}{\partial T} \nabla_s T + \sum_{k} \frac{\partial \sigma}{\partial C_k} \nabla_s C_k$

Two primary mechanisms drive surface tension gradients in semiconductor wet chambers:

  1. Solutocapillary Marangoni Forces: Arise from concentration differences during multi-chemical dispense or rinse transitions. For example, Isopropyl Alcohol (IPA, $\sigma \approx 21.7\,\text{mN/m}$) dispensed near Deionized Water (DIW, $\sigma \approx 72.8\,\text{mN/m}$) produces strong outward pull toward the high-surface-tension water zone.
  2. Thermocapillary Marangoni Forces: Arise from temperature variations ($\frac{\partial \sigma}{\partial T} < 0$ for almost all semiconductor liquids). Warmer zones exhibit lower surface tension, driving liquid toward cooler regions.

The resulting characteristic Marangoni screening velocity is modeled as:

$u_M(r) \approx \frac{h(r)}{2\mu} \nabla_s \sigma(r)$

8. Dimensionless Numbers & Diagnostics

To verify hydrodynamic stability and physical consistency, the simulator computes several key dimensionless criteria:

9. Sanity Checks & Conservation Laws

The simulator continuously executes reproducible sanity checks to maintain physical realism:

  1. Global Mass Balance Check: Total chemical inflow injected across all active nozzles must match the outward radial mass flux passing the 300mm wafer bevel at steady state:
    $\sum_{i=1}^N Q_i = 2\pi R_{\text{wafer}} q_r(R_{\text{wafer}}) + \int_0^{R_{\text{wafer}}} 2\pi r E(r) dr$
  2. Spin-Down Asymptotic Limit: When RPM $\to 0$, centrifugal flux ceases ($q_r \to 0$), and the model relaxes to a capillary puddle governed by gravitational and surface tension equilibrium.
  3. Emslie-Bonner Scaling Match: Single central nozzle dispense under steady spin rigorously converges to the $r^{-2/3}$ profile slope across intermediate radii.

10. Model Scope & Engineering Limitations

⚠️ CRITICAL ENGINEERING NOTICE:

The 300mm Wet Chamber Simulator is a reduced-order 2D lubrication model developed for process screening, comparative sensitivity analysis, recipe parameter scoping, and engineering education. It is not a full 3D transient Navier-Stokes CFD solver.

Users must take the following physical boundaries and assumptions into account:

11. Scientific & Literature References

The physical formulations and mathematical scaling models implemented in this simulator are derived from the following seminal fluid dynamics literature: