Tribology & Luo-Dornfeld Abrasive Mechanics: Micro-Scale Material Removal

01Executive Summary

While Preston's equation describes CMP behavior at a macro-scale continuum level, it fails to explain micro-scale phenomena such as removal rate saturation at high pressures, non-linear abrasive concentration effects, or particle size dependence. To overcome these limitations, Luo and Dornfeld developed a fundamental tribological model based on elastic-plastic contact mechanics, pad asperity statistics, and active abrasive particle counts. This deep-dive monograph examines the Luo-Dornfeld model, solid-augmented chemical removal kinetics, and sub-micron abrasive contact physics in oxide and polysilicon CMP.

02Limitations of the Macro-Scale Preston Model

Preston's empirical equation ( $MRR = K_{P} P V$ ) assumes a continuous, direct contact between two ideal flat surfaces. In reality:

Contact occurs only at the discrete tips of rough polyurethane pad asperities.
Material removal is mediated by discrete abrasive nanoparticles ( $10 - 100 nm$ ) suspended in a liquid boundary layer.
At elevated pressures, the removal rate deviates from linearity due to pad deformation, abrasive saturation, and slurry hydrodynamics.

03The Luo-Dornfeld Model Formulations

Luo and Dornfeld (2001) modeled material removal by integrating plastic indentation mechanics of individual abrasive particles with Greenwood-Williamson statistics for pad asperity heights.

Micro-Scale Abrasive Contact Mechanics:Schematic · source diagram

2.1 Indentation Depth of a Single Particle

Assuming elastic deformation of the pad and plastic deformation of the wafer's soft hydrated surface layer, the indentation depth $δ$ of an abrasive particle of diameter $x_{ab}$ into the wafer surface is:

δ = C_{1} \cdot x_{ab} \cdot (\frac{P _{a s p}}{H _{w}})^{2/3}

Where:

$P_{a s p}$ is the local contact pressure exerted by a single pad asperity.
$H_{w}$ is the hardness of the hydrated wafer surface layer.
$C_{1}$ is a dimensionless constant incorporating the elastic moduli of the pad ( $E_{p}$ ) and particle ( $E_{a}$ ).

2.2 Active Particle Density

Not all particles in the slurry contribute to polishing. Only particles trapped between pad asperities and the wafer surface (active particles $N_{a c t}$ ) cut material:

N_{a c t} = C_{2} \cdot D_{s} \cdot x_{ab} \cdot A_{re a l}

Where $D_{s}$ is the volumetric particle concentration in the slurry, and $A_{re a l}$ is the real contact area between pad asperities and the wafer ( $A_{re a l} ≪ A_{n o mina l}$ ).

2.3 Comprehensive Luo-Dornfeld MRR Equation

Combining indentation volume per particle, relative velocity V, and active particle density yields the complete Luo-Dornfeld removal rate expression:

MRR = C_{L D} \cdot x_{ab} \cdot D_{s}^{1/3} \cdot (\frac{E _{p}}{H _{w}}) \cdot P^{1/2} \cdot V + MRR_{c h e m}

Where:

$P^{1/2}$ captures the non-linear pressure saturation regime caused by pad elastic deformation.
$MRR_{c h e m}$ represents pure chemical dissolution in the absence of mechanical contact.

04Solid-Augmented Removal Dynamics in Ceria vs. Silica Slurries

The chemical nature of the abrasive particle fundamentally changes the removal mechanism:

Colloidal Silica ( $SiO_{2}$ ): Mechanical Abrasion Dominant

Colloidal silica particles rely on mechanical shear to sweep away hydrated surface layers ( $Si(OH)_{4}$ ). The interaction is non-selective, polishing silica, silicon nitride ( $Si_{3} N_{4}$ ), and polysilicon at similar rates depending on slurry pH.

Ceria ( $CeO_{2}$ ): Chemical Tooth Mechanism

Ceria particles exhibit a unique "chemical tooth" effect through chemical bonding between surface cerium atoms and silicon oxide molecules:

\equiv Ce-OH + \equiv Si-OH ⇌\equiv Ce-O-Si \equiv + H_{2} O

This formation of temporary $Ce-O-Si$ bonds allows ceria abrasives to tear away entire silica molecular clusters with extremely low mechanical force, yielding exceptionally high oxide removal rates ( $> 500 nm/min$ ) while stopping completely on silicon nitride stopping layers ( $SiO_{2} : Si_{3} N_{4}$ selectivity >100:1).

05Quantitative Model Comparison

Model Feature	Preston Model (1927)	Scire / Cook Model (1990)	Luo-Dornfeld Model (2001)
Primary Physics	Macro Continuum Kinematics	Glass Hertzian Contact	Elastic-Plastic Pad Asperity Contact
Pressure Dependence	Linear ( $P^{1}$ )	Linear ( $P^{1}$ )	Non-linear ( $P^{1/2}$ to $P^{2/3}$ )
Particle Size Sensitivity	None ( $K_{P}$ constant)	Inverse proportional ( $1/ x_{ab}$ )	Proportional to indentation ( $x_{ab} \cdot D_{s}^{1/3}$ )
Pad Roughness Factor	Ignored	Ignored	Incorporates Greenwood-Williamson distribution
Predictive Accuracy	Baseline trend analysis	Polishing optics	High-precision sub-10nm CMP node design

→Prerequisites & continue reading

Prerequisites: ["p03-t3.1-t2-prestonian-kinematics-cmp-tool-mechanics"]

Also in Topic 3.1, Oxide & Polysilicon CMP:

Tier 1 · Oxide and Polysilicon CMP Fundamentals: Achieving Gl…Tier 2 · Prestonian Kinematics & CMP Tool Architecture: Press…