SOL 102 › Unit 6

Unit 6 — Thermodynamics: Free Energy, Phase Diagrams, Solidification and Segregation

Big idea. Materials settle into whichever phase has the lowest Gibbs free energy. Phase diagrams map those choices against temperature and composition. When a melt freezes, most impurities prefer to stay in the liquid — the segregation coefficient \(k < 1\) — so crystal growth purifies silicon, while also making dopant levels vary from one end of an ingot to the other (the Scheil equation).

Learning objectives

Why this matters

Czochralski growth, directional solidification, zone refining, aluminium back-surface fields, silver-paste firing and solder joints are all phase-diagram problems. The resistivity spread along an ingot — a daily concern for wafer buyers — is the Scheil equation.

Free energy decides

The Gibbs free energy combines enthalpy \(H\) (bond energy; lower is better bonded) and entropy \(S\) (disorder; higher is favoured at high temperature):

\[G = H - TS\]

In words: at low temperature, strong bonding (low H) wins — solids; at high temperature, disorder (high S) wins — liquids and gases. The melting point is where \(G_\text{solid} = G_\text{liquid}\). For silicon, \(T_m = 1414\) °C (1687 K).

Free energy of solid and liquid vs temperature (schematic). The liquid has higher entropy, so its G falls faster with T; the curves cross at the melting point.
Free energy of solid and liquid vs temperature (schematic). The liquid has higher entropy, so its G falls faster with T; the curves cross at the melting point.

Melting silicon needs its latent heat of fusion, about 50.2 kJ/mol (≈ 1.8 kJ/g; CRC Handbook), plus the heat to warm it from room temperature.

Worked example 1 — minimum energy to melt 1 kg of silicon

Warming 25 → 1414 °C with an average heat capacity of roughly 0.9 J g⁻¹ K⁻¹ (estimate over this range): \(1000 \times 0.9 \times 1389 \approx 1.25\) MJ. Melting: \(1000 \times 1.79 = 1.79\) MJ. Total ≈ 3.0 MJ ≈ 0.85 kWh — a thermodynamic minimum. Real Cz pullers use many times more because of radiation and conduction losses from the hot zone over a multi-day growth run (SOL 301).

Silicon expands on freezing (like water): liquid Si is denser (about 2.57 g/cm³ near the melting point vs 2.33 g/cm³ for the solid; literature values), so solidification increases volume by roughly 10 %. This stresses crucibles in casting and must be handled in crystal-growth design.

Reading a binary phase diagram

A binary phase diagram shows which phases exist at each temperature and composition for a two-element system at equilibrium. - Liquidus: above it, all liquid. - Solidus: below it, all solid. - Two-phase region: between them, liquid + solid coexist with compositions given by the ends of a horizontal tie line. - Eutectic point: the composition with the lowest melting point, where liquid freezes directly into two solids.

Lever rule (fractions in a two-phase region): with overall composition \(C_0\), liquid composition \(C_L\) and solid composition \(C_S\),

\[f_\text{solid} = \frac{C_0 - C_L}{C_S - C_L}\]

In words: the tie line is a see-saw balanced at \(C_0\); the phase on the far side gets the bigger share.

The Al–Si system: why aluminium paste alloys into silicon

Al–Si phase diagram. Eutectic at 577 °C and 12.6 wt% Si; aluminium dissolves up to ~1.65 wt% Si; pure Al melts at 660 °C, Si at 1414 °C. Key points from standard handbooks; curve shapes schematic.
Al–Si phase diagram. Eutectic at 577 °C and 12.6 wt% Si; aluminium dissolves up to ~1.65 wt% Si; pure Al melts at 660 °C, Si at 1414 °C. Key points from standard handbooks; curve shapes schematic.

During firing of a PERC or Al-BSF cell (peak ~750–800 °C for seconds; estimate), the screen-printed aluminium melts and dissolves silicon (moving along the liquidus). On cooling, silicon regrows epitaxially from the melt carrying dissolved aluminium (a p-type dopant), forming a heavily doped p⁺ back-surface field, topped by the Al–Si eutectic. Because firing temperature exceeds 577 °C, Al and Si can react far below either element's melting point — the essence of a eutectic. Silver-paste contacts rely on similar chemistry with glass frit and the Ag–Si system (eutectic around 835 °C; handbook value).

Segregation during freezing

When an alloy or doped melt solidifies slowly, the solid that forms usually has a different impurity concentration than the liquid it grows from. The equilibrium segregation coefficient is

\[k_0 = \frac{C_S}{C_L}\]

For most impurities in silicon \(k_0 \ll 1\): they are rejected into the liquid. Typical values (Trumbore 1960 and later compilations): B ≈ 0.8, P ≈ 0.35, As ≈ 0.3, Ga ≈ 0.008, Al ≈ 0.002, Cu ≈ 4 × 10⁻⁴, Fe ≈ 8 × 10⁻⁶. Oxygen is close to 1 (values debated).

The Scheil equation

Assume (1) no diffusion in the solid, (2) a perfectly mixed liquid, (3) local equilibrium at the interface. Then as fraction \(f_s\) of the original melt solidifies, the solid forming has concentration

\[C_S = k_0 C_0 (1 - f_s)^{k_0 - 1}\]

In words: the first solid is cleaner than the melt by factor \(k_0\); as rejected impurity accumulates in the shrinking liquid, each new layer of solid gets dirtier; the last bit to freeze holds most of the impurities.

Scheil curves for boron, phosphorus, gallium and iron in silicon. Iron stays near 10⁻⁵ of the melt concentration for most of the ingot (purification!), while gallium's very low k makes its concentration rise steeply toward the tail. Values: typical literature k₀.
Scheil curves for boron, phosphorus, gallium and iron in silicon. Iron stays near 10⁻⁵ of the melt concentration for most of the ingot (purification!), while gallium's very low k makes its concentration rise steeply toward the tail. Values: typical literature k₀.

Worked example 2 — resistivity along an n-type ingot

Phosphorus, \(k_0 = 0.35\). At the seed end (\(f_s = 0\)): \(C_S = 0.35\,C_0\). At 80 % solidified: \(C_S = 0.35\,C_0 (0.2)^{-0.65} = 0.35\,C_0 \times 2.85\). Doping rises 2.85× → resistivity falls to about 1/2.85 of the seed-end value (mobility changes slightly). So an ingot that starts at 3 Ω·cm ends near 1 Ω·cm — a wide distribution that wafer and cell makers must manage (e.g., continuous Cz with melt replenishment, SOL 301).

Worked example 3 — why boron was convenient and gallium is hard

Boron (\(k_0 = 0.8\)): at \(f_s = 0.8\), \((0.2)^{-0.2} = 1.38\) → only 38 % increase. Gallium (\(k_0 = 0.008\)): \((0.2)^{-0.992} = 4.9\) → nearly 5× increase. That is the price p-type makers paid for switching from B to Ga to avoid B–O LID (Unit 5).

Worked example 4 — purification of iron

Iron \(k_0 = 8\times10^{-6}\): a melt with 10¹⁵ Fe/cm³ grows a seed-end crystal with \(8\times10^{9}\) cm⁻³ — a 125,000-fold purification in one step. Even at 90 % solidified, \((0.1)^{-1} \approx 10\) → \(8\times10^{10}\) cm⁻³. Tail ends ("tops and tails") are cut off and recycled carefully because they concentrate impurities.

Zone refining and directional solidification

Zone refining passes a molten zone repeatedly along a bar, sweeping impurities to one end each pass (the float-zone process uses a single RF-heated zone). Directional solidification of cast ingots purifies by the same Scheil physics; upgraded metallurgical-grade (UMG) silicon routes rely on it, with limits for B and P whose \(k_0\) is closer to 1.

Common misconceptions

Misconceptions

  • "A eutectic is a special compound." It is a composition where liquid freezes into two solid phases at the lowest melting point.
  • "Crystal growth only shapes silicon." It also purifies (k ≪ 1 for metals) and sets doping profiles.
  • "Doping is uniform along an ingot." It follows Scheil: it increases toward the tail for k < 1.
  • "Phase diagrams show what happens in fast processes." They show equilibrium; fast firing and quenching can trap non-equilibrium states.

What this means for manufacturing and the job

Key equations

Equation In words
\(G = H - TS\) phase stability: bonding vs disorder
\(f_s = (C_0 - C_L)/(C_S - C_L)\) lever rule
\(k_0 = C_S/C_L\) segregation coefficient
\(C_S = k_0 C_0 (1-f_s)^{k_0-1}\) Scheil: concentration along a directionally frozen ingot

Review questions

1. At 700 °C, is an Al-rich alloy with 12.6 wt% Si solid or liquid?

Liquid — it is above the 577 °C eutectic temperature at the eutectic composition.

2. Phosphorus k₀ = 0.35. Ratio of dopant concentration at 50 % solidified to the seed end?

(1 − 0.5)^(0.35−1) = 0.5^(−0.65) = 1.57.

3. Why are ingot tails often scrapped or recycled carefully?

The last-to-freeze melt is enriched in impurities with k < 1 (metals, and excess dopant), giving poor lifetime and out-of-spec resistivity.

4. Using the lever rule: C₀ = 30 %, C_L = 20 %, C_S = 60 %. Fraction solid?

(30 − 20)/(60 − 20) = 0.25.

5. Why does silicon's expansion on freezing matter for casting?

The ~10 % volume increase stresses the crucible and the solidifying ingot; designs must allow for it to avoid cracking.

Sources

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