Narration script

Unit eleven, the capstone of SOL 101. A semiconductor has a band gap. Photons with less energy pass through. Photons with more energy are absorbed, but their excess energy above the gap turns into heat within trillionths of a second. A p-n junction then sorts the freed electrons and holes to produce voltage and current. Even a perfect cell loses energy to these effects and to unavoidable radiative emission. That sets the Shockley-Queisser limit, about thirty-three point seven percent at best, and about thirty-three point four percent for silicon.

By the end you should be able to explain band gaps and thermalization using a band diagram, describe how a p-n junction separates charge, read an I V curve, compute silicon's two spectral losses, and draw the first energy-loss waterfall from sunlight to an industrial cell.

This is the map for the entire curriculum. Every manufacturing step, texturing, doping, passivation, metallization, encapsulation, exists to remove one of the practical losses between the theoretical limit and a real product.

In a single atom, electrons sit at sharp energy levels. In a crystal with about five times ten to the twenty-two atoms per cubic centimetre, those levels broaden into bands. In a semiconductor, the lower valence band is full of bonding electrons, and the upper conduction band is empty, with a gap between them where no states exist. For silicon at room temperature that gap is one point one two electron-volts.

This band diagram is the picture to remember. A photon with energy at least equal to the gap lifts an electron from the valence band into the conduction band. It leaves behind a hole, a missing bond that behaves like a positive charge. If the photon had extra energy, the electron starts high in the band and quickly tumbles down to the band edge, handing that excess to the crystal as heat. That is thermalization.

So every photon has one of two fates. Below the gap, it is not absorbed, and the whole photon is lost. Above the gap, it makes one electron-hole pair, and everything above the gap energy is lost as heat. Using the real spectrum, we can compute exactly how much each loss costs silicon.

Here it is, computed from the standard spectrum. The pink region, beyond eleven hundred and seven nanometres, is never absorbed: nineteen percent of the power. The orange region is the excess energy of absorbed photons, lost as heat: thirty-two percent. The green region is what remains if every absorbed photon delivers exactly the band gap energy: forty-nine percent. Lowering the gap shrinks the pink but grows the orange; raising it does the opposite. You cannot win both with one gap.

Freeing electrons is not enough; they must be pushed in one direction. Add phosphorus, with five outer electrons, and silicon becomes n-type, with extra electrons. Add boron, with three, and it becomes p-type, with extra holes. Where the two regions meet, carriers diffuse across and recombine, leaving behind fixed charged dopant atoms, which create a built-in electric field in a thin depletion region.

In this diagram the built-in field points from the n side to the p side. Light-generated electrons are swept toward the n side, and holes toward the p side. They leave through the contacts and do work in the external circuit. That is the whole operating principle of a solar cell, in one picture.

To characterise a cell, we sweep its voltage and record the current. That gives the I V curve. The short-circuit current is set by how many photons are collected. The open-circuit voltage is set by how fast carriers recombine, and is always below the band gap. The maximum power point is where voltage times current is largest. The fill factor says how square the curve is. And efficiency is open-circuit voltage times short-circuit current times fill factor, divided by the incoming light power.

Here are the curves, from a model with illustrative parameters typical of a modern industrial cell. The blue line is current versus voltage: nearly flat, then dropping sharply near the open-circuit voltage. The orange line is power, rising to a peak, the maximum power point, at about zero point six two volts and thirty-nine point six milliamps per square centimetre. That is twenty-four point seven milliwatts per square centimetre, or twenty-four point seven percent efficiency, with a fill factor of about zero point eight two. On a full-size cell of three hundred and thirty square centimetres, that is about eight watts at thirteen amps.

In nineteen sixty-one, William Shockley and Hans Queisser asked what the best possible single-junction cell could do. They assumed every photon above the gap is absorbed and makes one pair, that the only recombination is the unavoidable radiative kind, because anything that absorbs must also emit, and that the cell sits at room temperature.

This curve is that limit, computed on this site from the real spectrum. The maximum is thirty-three point seven percent, at a band gap of one point three four electron-volts. Silicon, at one point one two, reaches thirty-three point four. The wiggles in the curve come from the atmospheric absorption bands we saw in unit seven. Silicon is not at the peak, but it is only zero point three percentage points below it. And it is cheap, abundant, stable, non-toxic, and backed by seventy years of process know-how. That is why it makes up about ninety-eight percent of production.

For an ideal silicon cell, our calculation gives an open-circuit voltage of about zero point eight eight volts, well below the one point one two volt gap. That voltage deficit is the entropy cost of emission that we discussed back in unit two.

Real silicon has more limits. Auger recombination, a three-particle process built into silicon, together with its weak absorption, lowers the realistic limit to twenty-nine point four percent. Record cells are around twenty-seven to twenty-eight percent; check the latest Fraunhofer or N R E L chart, because records change often. Mass-production cells are about twenty-five to twenty-six percent, and modules about twenty-three and a half to twenty-four.

Now the payoff, our first energy-loss waterfall. Start at one hundred percent of sunlight. Lose nineteen percent to below-gap photons. Lose thirty-two percent to thermalization. Lose about sixteen percent more to the voltage and fill-factor penalty of an ideal cell. That lands at thirty-three point four, the limit for silicon. Auger and intrinsic silicon physics take about four more points, to twenty-nine point four. Practical losses in record cells take about two more. And industrial cells sit near twenty-five and a half, with modules near twenty-three and a half. Every red step is a loss, and later courses attack them one by one. The two biggest bars, below-gap and thermalization, can only be attacked with tandem cells.

And this is the road ahead: the value chain from quartz and sand, through polysilicon, crystal growth, wafers, cells and modules, with the course that covers each step.

Misconceptions to drop. A lower band gap is not always better. Open-circuit voltage does not equal the band gap. The Shockley-Queisser limit is not a materials problem that research will simply beat; beating it requires changing the rules, with tandems or concentration. And silicon is not used because it is the best absorber; it wins on abundance, stability, cost and manufacturing maturity.

For the job: every process step maps to a loss in this waterfall. Texture and coatings fight reflection. Passivation layers fight recombination and raise voltage. Metallization trades shading against resistance. And I V metrics are the daily language of a cell line, trended by tool and by lot.

Work through the review questions. That completes SOL 101. In SOL 102, we turn from light to matter, and learn the materials science of silicon and the other materials a solar factory runs on.