Narration script

Unit twelve, the computational lab. Doing the numbers yourself is what makes this course stick. In this lab you will build a silicon crystal in Python and measure it, apply the lever rule and Scheil equation to realistic problems, estimate how many defects a crystal holds at high temperature, and practise converting between parts per million, resistivity and atoms per cubic centimetre. Everything uses standard Python, with numpy and matplotlib.

The goals: generate diamond cubic coordinates and verify density, coordination and bond angle; predict X-ray diffraction peaks; solve lever-rule and Scheil problems; estimate vacancy concentrations; and convert concentrations fluently.

Process engineers script quick calculations all the time: checking a supplier's resistivity spread, estimating a contamination budget, sanity-checking a measurement. Job postings ask for Python and data skills on top of materials knowledge, and this lab joins the two.

Setup is simple. Install Python three, then install numpy and matplotlib with pip. Save each code block as a file, or paste it into a notebook.

Exercise one builds silicon. The code places four face-centred cubic positions, adds a copy shifted by a quarter of the cell diagonal to get eight atoms per cell, and tiles a three by three by three block. Then it picks an atom in the middle and finds its neighbours. You should find exactly four neighbours, at two point three five two angstroms, with a bond angle of one hundred and nine point four seven degrees, an atom density of about five times ten to the twenty-two, and a density of two point three two nine grams per cubic centimetre. Those are the numbers from unit three, now computed by your own code.

Exercise two predicts X-ray diffraction peaks. In the diamond structure only certain planes reflect: indices all odd, or all even with a sum divisible by four. The code loops over the allowed planes, computes their spacings, and applies Bragg's law. The first peaks should come out at twenty-eight point four four degrees for the one one one plane, then forty-seven point three, fifty-six point one, sixty-nine point one and seventy-six point four. Those match a real silicon diffraction pattern.

Exercise three starts with the lever rule. In this example, the overall composition is thirty-five percent, the liquid is twenty percent and the solid is seventy percent. The solid fraction is fifteen divided by fifty, which is zero point three. Think of the see-saw: the overall point sits closer to the liquid end, so there is more liquid.

Then the Scheil problems. Problem one: a phosphorus-doped ingot must stay between one and three ohm centimetres, starting at three. Doping may rise three times. Solving the Scheil equation gives about eighty-two percent of the melt in spec. Problem two: the same window with gallium, which has a much smaller segregation coefficient, gives only about sixty-seven percent. That is the yield cost of gallium doping, in one calculation. Problem three: a melt with ten to the fourteen iron atoms per cubic centimetre gives a crystal with only about one point six times ten to the nine at the halfway point.

Exercise four looks at vacancies. The equilibrium number of vacancies grows exponentially with temperature, controlled by their formation energy, which for silicon is uncertain, roughly three to four electron-volts.

This plot shows the result for three illustrative formation energies. Near the melting point, there are around ten to the eleven to ten to the thirteen vacancies per cubic centimetre. At room temperature, effectively none. So the vacancies found in real wafers were frozen in while the crystal cooled, where they cluster into voids or interact with oxygen. Controlling that is part of crystal-growth engineering.

Exercise five is a conversion table. With five times ten to the twenty-two silicon atoms per cubic centimetre, one part per million atomic is five times ten to the sixteen. A one ohm centimetre n-type wafer is about zero point one parts per million of phosphorus. The iron level that measurably hurts p-type lifetime, about ten to the eleven, is just two parts per trillion. And oxygen in Czochralski silicon is around fourteen parts per million. Getting comfortable moving between these units is a real job skill.

Exercise six builds a contamination budget. Using the simple lifetime formula from unit five, a one hundred microsecond lifetime allows about ten to the eleven traps per cubic centimetre. One millisecond allows ten to the ten. Five milliseconds allows two times ten to the nine, which is only about four hundredths of a part per trillion. That is why high-efficiency n-type lines obsess over metal contamination.

For your deliverables, save the scripts and a one-page summary with the density check, the peak list, the two ingot yield fractions, the vacancy plot and the conversion table. Keep them; later courses reuse them.

Try the review questions. That completes SOL 102, and with it the introductory courses. You now have the physics of light and the materials science you need to follow how a solar cell is actually made, starting with crystalline silicon in SOL 201.