The Distribution Coefficient

The key equation governing zone refining: $k = \frac{C_s}{C_l}$

Where:

• $C_s$ = concentration of impurity in solid phase
• $C_l$ = concentration of impurity in liquid phase
• For zone refining to work: k < 1 (impurity prefers liquid)

How Multiple Passes Improve Purity

After n passes of the molten zone: $C_n = C_0 \cdot k^n$

Where C0 = initial impurity concentration.

Example:

• k = 0.1 (typical for Si zone refining)
• After 3 passes: C3 = C0 × (0.1)^{3} = 0.001 × C0 (1000× improvement)
• After 5 passes: C5 = C0 × (0.1)^{5} = 0.00001 × C0 (100,000× improvement)

This explains why semiconductor-grade Si (>99.9999%) is achievable — starting from 99.9% Si, even 3 passes gives ppb level impurities.

Physical Setup

The impure Si ingot sits in a quartz boat. A narrow RF (radio-frequency) heating coil moves slowly from one end to the other, melting a small zone. As it moves, the solid crystallizing behind the zone is purer, and the molten zone (with impurities) moves forward. Impurities accumulate at the final end, which is cut off.

Why Si and Ge, Not Cu?

• Si and Ge are metalloids requiring purity at ppb level for semiconductor function
• Their melting points (Si: 1414°C, Ge: 938°C) are accessible
• They don't readily dissolve in ionic solutions, ruling out electrolytic refining
• Most impurity atoms have k < 1 in Si/Ge melts