Narration script

Unit seven, diffusion. Atoms in a solid move by random hops. The net result is diffusion, a flow from high concentration to low, described by Fick's laws. Diffusion distance grows with the square root of time, and the diffusion coefficient rises exponentially with temperature. This is how the emitter of a solar cell is formed, with phosphorus diffused a few tenths of a micrometre into silicon at eight to nine hundred degrees. And it is why metals, which diffuse millions of times faster than dopants, can contaminate a whole wafer in seconds at high temperature.

You will learn to explain diffusion as a random walk, interpret Fick's two laws, use the two classic solutions, apply the Arrhenius relation, and compare how fast dopants and metals move in silicon.

Tube diffusion, firing, hydrogenation, gettering, oxidation and the anneals used in TOPCon cells are all diffusion problems. Process engineers tune time and temperature to hit a target junction depth, and contamination engineers worry about metals because they move so fast.

At high temperature, each atom occasionally hops to a neighbouring site, in a random direction. After many hops, the average displacement is zero, because hops go every way equally. But the average of the displacement squared grows linearly with the number of hops. So the typical distance travelled grows with the square root of time.

On the left are six simulated walkers, each wandering randomly. On the right, we averaged three thousand of them, and the mean-square distance climbs in a straight line with the number of hops. That straight line is the signature of diffusion.

Fick's first law says the flux of atoms is proportional to the concentration gradient. In words: atoms flow downhill in concentration, faster for steeper slopes and larger diffusion coefficients. Fick's second law says how the profile changes with time. In words: a dip fills in, and a peak spreads out. The characteristic diffusion length is the square root of the diffusion coefficient times time.

Two classic solutions cover most processes. If the surface is held at a constant concentration by a gas source, as in phosphorus pre-deposition, the profile is a complementary error function. If a fixed dose is deposited and then driven in with no more supply, the profile is a Gaussian. Either way, the junction forms where the diffused dopant concentration equals the wafer's background doping of the opposite type.

These are constant-source profiles after five to sixty minutes, using illustrative values. Each curve pushes deeper with time, and the junction sits where it crosses the red dashed background line. Notice the square root: going from fifteen minutes to sixty, four times longer, only doubles the depth. Real phosphorus emitters have a kink-and-tail shape, because diffusion is enhanced at very high concentrations.

Now temperature. To hop, an atom must get over an energy barrier, so the diffusion coefficient follows the Arrhenius law: a prefactor times e to the minus activation energy over k T. In words: every step up in temperature multiplies the rate by a large factor, and the activation energy sets how sensitive it is.

On an Arrhenius plot, the log of the diffusion coefficient against one over temperature gives straight lines. Boron and phosphorus, with activation energies around three and a half electron-volts, have steep lines far down the chart. Interstitial iron, with an activation energy of only zero point six seven, sits far above. At one thousand degrees, iron diffuses about a hundred million times faster than the dopants.

Boron at one thousand degrees has a diffusion coefficient of about one point five times ten to the minus fourteen square centimetres per second. After one hour, its diffusion length is only about seventy-four nanometres. Dopant diffusion is slow. Microns take hours.

Here is an honest lesson. Using textbook intrinsic values, phosphorus at nine hundred degrees for thirty minutes should only travel about eleven nanometres. Yet industrial emitters reach a few hundred nanometres in similar times. Why? At very high concentrations, phosphorus diffuses much faster through charged point defects. So textbook numbers are a starting point, and real process models are calibrated to measured profiles.

Now iron. At eight hundred degrees, iron's diffusion length after just sixty seconds is about sixty-five micrometres, half a wafer's thickness. Even at two hundred degrees, it moves about five micrometres per hour. So metal picked up anywhere on a wafer spreads everywhere during any heating step. Cleanliness before furnaces is critical, and gettering layers can collect metals from the whole wafer.

Chemical reaction rates follow the same Arrhenius form. You will meet it in thermal oxidation, in alkaline etch rates, where bath temperature is tightly controlled, and in the idea of a thermal budget, the total time-and-temperature exposure, which protects amorphous silicon in heterojunction cells and avoids oxygen precipitation in TOPCon wafers.

Misconceptions. Twice the time does not give twice the depth; you need four times the time. A bit hotter is not a bit faster; near one thousand degrees, fifty degrees more multiplies boron's diffusion by about three. Atoms do diffuse in solids. And dopant diffusion and contamination are not separate problems; it is the same physics, with different diffusion coefficients.

For the job: diffusion recipes set emitter depth and sheet resistance, monitored by four-point probe and depth profiling. Thermal budget distinguishes heterojunction lines from TOPCon lines. And contamination control and gettering are explicitly mentioned in job postings.

Try the review questions. Next unit, the chemistry of silicon itself: from sand to nine-nines polysilicon, and the etching chemistry of the cell line.