Narration script

Unit six. Materials settle into whichever phase has the lowest Gibbs free energy, and phase diagrams map those choices against temperature and composition. When a melt freezes, most impurities prefer to stay in the liquid. We say their segregation coefficient is less than one. So crystal growth purifies silicon, but it also makes dopant levels vary from one end of an ingot to the other, following the Scheil equation.

By the end, you can explain Gibbs free energy, read a binary phase diagram including the lever rule, explain the aluminium-silicon eutectic and why it matters for contacts, and use the Scheil equation to predict concentrations along an ingot.

Crystal growth, ingot casting, aluminium back-surface fields, silver paste firing and solder joints are all phase-diagram problems. And the resistivity spread along an ingot, a daily concern for wafer buyers, is the Scheil equation.

Gibbs free energy combines enthalpy, which is about bond energy, and entropy, which is about disorder. G equals H minus T times S. In words: at low temperature, strong bonding wins, and you get solids. At high temperature, disorder wins, and you get liquids and gases.

This schematic shows the free energy of the solid and the liquid as temperature rises. The liquid has more entropy, so its curve falls faster. Where the two curves cross is the melting point. For silicon, that is fourteen hundred and fourteen degrees Celsius.

How much energy does it take to melt a kilogram of silicon? Warming it from room temperature takes roughly one and a quarter megajoules, using an estimated average heat capacity. Melting it takes about one point eight more. Total, about three megajoules, or zero point eight five kilowatt-hours. That is the thermodynamic minimum. Real crystal pullers use many times more, because the hot zone loses heat for days during a growth run. And one odd fact: silicon expands by about ten percent when it freezes, like water, which stresses crucibles and must be designed around.

A binary phase diagram shows which phases exist at each temperature and composition. Above the liquidus line, everything is liquid. Below the solidus line, everything is solid. Between them, liquid and solid coexist. The eutectic point is the composition with the lowest melting point, where liquid freezes directly into two solids. And the lever rule gives how much of each phase you have: picture the tie line as a see-saw balanced at the overall composition. The phase on the far side gets the bigger share.

Here is the aluminium-silicon phase diagram, with key points from standard handbooks. Pure aluminium melts at six hundred and sixty degrees, pure silicon at fourteen hundred and fourteen. But at twelve point six percent silicon, the eutectic, the mixture melts at only five hundred and seventy-seven degrees.

That matters for cells. During firing, printed aluminium paste melts and dissolves some silicon. On cooling, silicon regrows from the melt carrying dissolved aluminium, a p-type dopant, forming a heavily doped back-surface field. Because firing exceeds five hundred and seventy-seven degrees, aluminium and silicon can react far below either one's own melting point. That is the essence of a eutectic. Silver paste contacts rely on related chemistry.

Now segregation. When a doped melt freezes slowly, the solid usually has a different impurity concentration from the liquid. The ratio of solid to liquid concentration is the segregation coefficient, k. For most impurities in silicon, k is much less than one: they are pushed into the liquid. Boron is about zero point eight. Phosphorus, about zero point three five. Gallium, about zero point zero zero eight. And iron, about eight millionths.

The Scheil equation describes freezing with no diffusion in the solid and a well-mixed liquid. In words: the first solid is cleaner than the melt by the factor k. As rejected impurity builds up in the shrinking liquid, each new layer of solid gets dirtier. And the last bit to freeze holds most of the impurities.

These curves show it for four elements, on a log scale. Boron, with k near one, stays almost flat. Phosphorus rises moderately toward the tail. Gallium starts very low and climbs steeply. And iron stays around a hundred thousand times below the melt concentration for most of the ingot. That is purification, built into the physics of freezing.

A practical example. In a phosphorus-doped ingot, by the time eighty percent has solidified, the doping has risen about two point eight five times. So an ingot that starts at three ohm centimetres ends near one. Wafer and cell makers must manage that spread, for example with continuous Czochralski growth that keeps topping up the melt.

Compare boron and gallium at the same point. Boron rises only thirty-eight percent. Gallium rises nearly five times. That was the price p-type makers paid when they switched from boron to gallium to escape the boron-oxygen degradation of unit five.

And iron. A melt with ten to the fifteen iron atoms per cubic centimetre grows a crystal with only about eight times ten to the nine at the seed end, a hundred and twenty-five thousand fold purification in one step. The tail ends concentrate impurities, so they are cut off and recycled carefully.

Zone refining passes a molten zone repeatedly along a bar, sweeping impurities to one end each time; the float-zone process uses a single zone. Directional solidification of cast ingots purifies by the same physics, though it struggles with boron and phosphorus, whose k values are closer to one.

Misconceptions. A eutectic is not a special compound; it is a lowest-melting composition. Crystal growth does not just shape silicon; it purifies it and sets its doping profile. Doping is not uniform along an ingot. And phase diagrams show equilibrium, not what happens in fast processes like firing.

For the job: ingot resistivity specs and dopant choices come from segregation. Firing profiles are phase-diagram engineering. Solder alloys are eutectic systems chosen for their melting temperatures. And crystal-growth roles at SpaceX and Tesla mention melt and segregation control. This is the foundation.

Try the review questions, especially a Scheil calculation. Next unit: diffusion, how atoms move through solids.