Voronkov Criterion (v/G), Microdefect Control, and Oxygen Thermal History in CZ Growth

01Intrinsic Point Defect Thermodynamics at Solidification

During the solidification of silicon at its melting point ( $1687 K$ ), the crystal lattice incorporates intrinsic point defects: vacancies (V, missing silicon atoms) and self-interstitials (I, extra silicon atoms squeezed between lattice sites).

[ Solidification Interface (T = 1687 K) ]Schematic · source diagram

At high temperatures, both species exist in dynamic equilibrium. As the growing crystal cools below the solidus interface, these point defects recombine ( $V + I ⇌ \emptyset$ ). The species present in higher concentration after recombination dictates the microdefect regime of the entire wafer.

02The Voronkov Criterion: The v/G Ratio

In 1982, V.V. Voronkov established the fundamental mathematical framework governing point defect incorporation in single-crystal silicon. The incorporated defect type is determined by the ratio of pull speed (v, in $mm/min$ ) to axial temperature gradient (G, in $K/mm$ ):

ξ = \frac{v}{G}

Process schematicSchematic · source diagram

The Critical Threshold ( $(v / G)_{crit}$ )

Critical Value: $(v / G)_{crit} \approx 1.3 \times 1 0^{- 3} - 2.2 \times 1 0^{- 3} cm^{2} / K \cdot min$ (depending on exact thermal conditions).
If $v / G > (v / G)_{crit}$ : Vacancy diffusion dominates. Excess vacancies recombine during cooling to form microscopic octahedral voids ( $100 - 300 nm$ in size) known as Crystal-Originated Particles (COPs). When sliced into wafers, COPs create surface pits that destroy transistor gate oxide integrity.
If $v / G < (v / G)_{crit}$ : Self-interstitial diffusion dominates. Excess interstitials aggregate into large dislocation loops and interstitial clusters (A-swirl defects), causing catastrophic electrical leakage across transistor junctions.

The Goal: Perfect "Pure Silicon" ( $P_{v} / P_{i}$ Regions)

By precisely engineering thermal shields inside the furnace to maintain a uniform spatial G(r) across the $300 mm$ ingot radius, crystal pullers operate exactly at $v / G \approx (v / G)_{crit}$ , producing defect-free monocrystalline silicon.

*FIG. 1*A high-resolution photorealistic Transmission Electron Microscope (TEM) visualization style image showing a nanometer-scale octahedral void defect (COP) embedded inside a single-crystal silicon lattice

03Oxygen Incorporation Kinetics and Thermal History

Oxygen is an inevitable chemical species in Czochralski silicon due to the dissolution of the quartz ( $SiO_{2}$ ) crucible wall by molten silicon:

SiO_{2} (solid) + Si (liquid) \to 2 SiO (dissolved in melt)

Dissolved oxygen is transported by melt convection to the growth interface, incorporating interstitially ( $O_{i}$ ) into the silicon lattice at concentration levels between $5 \times 1 0^{17}$ and $1 0^{18} atoms/cm^{3}$ .

[ Interstitial Oxygen (Oi) Dissolved In Lattice ]Schematic · source diagram

Thermal Donors and Oxygen Precipitation Dynamics

Thermal Donor Formation ( $45 0^{\circ} C$ Window): As the ingot slowly cools through the $40 0^{\circ} C - 50 0^{\circ} C$ thermal window, interstitial oxygen atoms aggregate into small clusters ( $SiO_{4}$ ), which act as electrical double-donors. Thermal donors alter the intended electrical resistivity of the wafer and must be destroyed by high-temperature rapid thermal annealing ( $> 65 0^{\circ} C$ for $30 seconds$ ).
Internal Gettering (IG) Engineering: Controlled oxygen precipitation is utilized intentionally for Internal Gettering (IG). High-temperature furnace cycles form microscopic $SiO_{2}$ precipitates in the bulk interior (bulk microdefects, BMDs) while leaving a pristine, defect-free "denuded zone" (DZ) at the top $10 μ m$ surface where transistors are built. The bulk $SiO_{2}$ precipitates act as internal traps, capturing mobile metallic contaminants ( $Fe, Cu, Ni$ ) away from surface transistors.

*FIG. 2*A publication-grade Voronkov v/G defect map diagram

04Mathematical Model of Point Defect Transport

The steady-state radial and axial distribution of vacancies ( $C_{V}$ ) and interstitials ( $C_{I}$ ) is governed by the coupled advection-diffusion-recombination differential equations:

v \frac{\partial C _{V}}{\partial z} = D_{V} (\frac{\partial ^{2} C _{V}}{\partial r ^{2}} + \frac{1}{r} \frac{\partial C _{V}}{\partial r} + \frac{\partial ^{2} C _{V}}{\partial z ^{2}}) - \frac{D _{V} C _{V}}{k _{B} T ^{2}} \nabla T \cdot \nabla E_{V} - k_{I V} (C_{V} C_{I} - C_{V}^{eq} C_{I}^{eq})

v \frac{\partial C _{I}}{\partial z} = D_{I} (\frac{\partial ^{2} C _{I}}{\partial r ^{2}} + \frac{1}{r} \frac{\partial C _{I}}{\partial r} + \frac{\partial ^{2} C _{I}}{\partial z ^{2}}) - \frac{D _{I} C _{I}}{k _{B} T ^{2}} \nabla T \cdot \nabla E_{I} - k_{I V} (C_{V} C_{I} - C_{V}^{eq} C_{I}^{eq})

Where:

v is crystal pull speed.
$D_{V}, D_{I}$ are species diffusion coefficients.
$k_{I V}$ is the vacancy-interstitial recombination rate constant.
$C_{V}^{eq}, C_{I}^{eq}$ are thermal equilibrium defect concentrations.

Numerical finite-element simulation of these equations allows crystal growth engineers to design complex thermal hot-zones that maintain v/G(r) within the defect-free window across the entire $300 mm$ wafer radius.

*FIG. 3*An educational infographic titled 'Oxygen Thermal History & Internal Gettering (IG) in Wafers'

→Prerequisites & continue reading

Prerequisites: Advanced crystal defect thermodynamics, point defect kinetics, heat transfer modeling

Also in Topic 1.3, Monocrystalline Growth: Czochralski & Float Zone:

Tier 1 · Growing the Perfect Single Crystal: Czochralski and …Tier 2 · Czochralski Crystal Pullers and CUSP Magnetic Field …