Physics · Module 25 · the last one
Everything in this course arrives at the same place at once. Standing waves, the Boltzmann distribution, Coulomb's law, drift velocity, the Hall effect, photons, Pauli, tunnelling and the diffraction limit are not nine separate subjects that happen to be useful here. They are the working parts of the object you are holding.
This module is different from the ones before it. Each of those opened a new idea. This one opens almost none: bands, holes and doping are the only genuinely new things in it, and each is built in front of you out of parts you already have. It is an assembly, and the point of it is to watch the parts fit.
So before we start, here is the parts list. Every row is something you already know, and every row is load-bearing. If any one of them were false, the device you are reading this on would not work.
| From | What it contributes |
|---|---|
| Module 7 · standing waves | Confine a wave and only certain states survive. In one atom that gives energy levels; in 1023 atoms it gives bands. |
| Module 22 · Pauli, orbitals | No two electrons in the same state. This is why the levels have to spread out rather than piling up, and why a band can be full. |
| Module 8 · counting states | S, the entropy, is a headcount of the arrangements — W is the number of them — with S = k ln W. The same counting, applied to how a fixed quantity of energy is shared out, gives the Boltzmann factor e−E/kT where E is that shared energy — introduced here, because you will need it in a moment. |
| Module 13 · fields, potential, capacitors | The built-in field of a junction. And the gate of a transistor, which is a capacitor and nothing else. |
| Module 14 · current and drift | I = nAve, where I is the current. Everything a chip does is this equation, switched on and off. |
| Module 15 · F = qvB | F is the sideways force on a carrier moving through a magnetic field, and B is that field. The Hall effect — the experiment that proves the positive carriers in a semiconductor are real and not a bookkeeping trick. |
| Module 21 · photons | E = hf, where E is the photon's energy. An LED is a band gap converted into a colour; a solar cell is the same thing run backwards. |
| Module 23 · tunnelling | How flash memory stores a bit — and the wall that ends the shrinking of transistors. |
| Module 12 · diffraction | The Rayleigh limit — Module 12 gives it as an angle, θ ≈ 1.22 λ/D. Turn that angle into a distance at the wafer and it becomes a limit on feature size. Moore's law is a story about wavelengths. |
Silicon itself is unremarkable: the second most common element in the Earth's crust, the main ingredient of sand and glass, four electrons in its outer shell. What makes it the most important material in the world is a number — a gap of 1.12 electron-volts — and the fact that we learnt how to control it.
Start with one silicon atom on its own. Module 22 tells you what it has: a ladder of sharp energy levels, each holding a fixed number of electrons.
Now bring a second atom close. Their outer orbitals overlap — the standing waves start to occupy the same region of space — and Pauli forbids two electrons from sharing a state. So the shared state cannot simply be the old one twice over. Instead the single level splits into two: one slightly lower in energy (the bonding combination, where the waves reinforce between the atoms) and one slightly higher (the antibonding one, where they cancel).
Bring up a third atom and you get three levels. A hundred atoms, a hundred levels. And in a real crystal there are around 1023 atoms in a cubic centimetre, so you get 1023 levels, spread over a fixed range of a few electron-volts. The spacing between neighbouring levels is then about 10−23 eV, which is so much smaller than anything else in the problem that the ladder has become, for every practical purpose, a continuous band. That argument applies to each rung of the original atomic ladder independently. Every rung spreads into its own band; since the rungs were further apart than each band ends up wide, the bands do not touch. Between them is nothing — no allowed states at all. That gap is the band gap, and it is the single most important number about any solid.
This is a one-dimensional chain with two orbitals per atom, the exactly solvable case. The two starting levels (−8 eV and −4 eV) and the strength of the coupling are chosen to make the effect visible, not measured from silicon — the shape of what happens is right, the numbers are not silicon's.
Two things to do with that bench.
Add atoms. With one atom you have two sharp levels. With five you have two little clusters. With forty-eight the clusters have become solid-looking blocks with a clear empty region between them. Nothing has changed except how many atoms are talking to each other.
Then squeeze them together. Push the atoms closer and the bands get wider, because the overlap between neighbouring orbitals is stronger and the splitting is bigger. Push far enough and the two bands touch, and the gap closes altogether.
That second observation gives you half the classification of solids, and it is not a metaphor: spacing sets how big the gap is. The other half — whether the band below the gap is full or only partly full — comes next. Go down group 14 of the periodic table — carbon, silicon, germanium, tin — and the atoms get bigger and further apart in the crystal, and the gap shrinks: diamond 5.5 eV, silicon 1.12 eV, germanium 0.66 eV, grey tin 0.08 eV, and lead, at the bottom, is simply a metal. One column of the periodic table walks you from the hardest insulator known to a conductor, by nothing but spacing. The bench runs the other way round, and the difference is worth knowing: in the chain model the gap is what is left of a fixed atomic level separation after the bands have broadened, so squeezing closes it. In group 14 the gap is the bonding–antibonding splitting of the hybrid orbitals, so more overlap makes it bigger — which is why diamond, the most tightly packed of the four, has the largest gap of all. Band broadening and bonding splitting are two different routes to a gap, and this bench shows only the first.
To get a current in silicon, an electron must be lifted from the full lower band (the valence band) across the gap into the empty upper one (the conduction band). It can borrow the energy from heat.
Counting the ways the energy can be shared out tells you how likely that is (this is Module 8's S = k ln W, one step further on): the fraction of particles with energy E or more goes as e−E/kT. Lifting one electron across the gap costs Eg, so the number of pairs created goes as e−Eg/kT. But carriers are also constantly recombining, and in the balance between creation and recombination the number you are left with comes out as the square root of that — which is e−Eg/2kT. Put silicon's numbers in — 1.12 eV against 0.026 eV at room temperature — and the exponent is about −21.7, so the fraction is around 4 × 10−10.
That sounds like nothing. But there are 5 × 1022 atoms in a cubic centimetre of silicon, so you might expect the fraction to multiply straight through. It does not: an electron can only be lifted out of a state near the top of the valence band, and only into one near the bottom of the conduction band, and the effective number of such states is about 2.5 × 1019 per cubic centimetre. 2.5 × 1019 × 4 × 10−10 leaves about 1010 free electrons per cubic centimetre. Small, but not zero — and, crucially, fiercely temperature-dependent, because it sits in an exponent.
Heat a copper wire and its resistance goes up. Module 14 gave you the mechanism — electrons drifting through a lattice are knocked about by the vibrating atoms — and hotter atoms vibrate harder, which is why a light bulb filament draws a surge of current at switch-on and settles down as it comes up to temperature. Now heat a piece of pure silicon.
What happens to its resistance?
It collapses. Warming silicon from 300 K to 400 K cuts its resistance by a factor of several hundred. The two materials behave oppositely because two different things are happening.
In copper, the number of carriers is fixed — one or so per atom, whatever the temperature. Heating only makes the lattice vibrate harder, so electrons are scattered more often and the drift velocity of Module 14 falls. Resistance rises, roughly in proportion to T.
In silicon, that scattering effect is still there and still works the same way. But it is completely swamped by the exponential growth in the number of carriers. Resistance falls off a cliff.
Look at diamond on that bench. Same physics, same equation, one number changed — and the carrier concentration at room temperature comes out at something like 10−27 per cubic centimetre, which is to say that in a cubic kilometre of diamond you would expect to find no free electrons at all. That is what a 5.47 eV gap does inside an exponential.
This exponential sensitivity is also a nuisance. A device whose conductivity changes by hundreds of times when it warms up by a hundred degrees is not something you can build a computer out of. Pure silicon, on its own, is useless for building anything that has to keep working — which is what the rest of this module is about fixing.
The fix is one of the great ideas in engineering, and it is deliberately making the material dirty.
Silicon has four outer electrons, and in the crystal each atom shares one with each of its four neighbours: every electron is spoken for, every bond complete. That is why the valence band is exactly full.
Now replace one silicon atom in a million with a phosphorus atom, which has five outer electrons. Four of them go into bonds. The fifth has nowhere to go and is only very loosely held — about 0.045 eV, less than a twentieth of the 1.12 eV it takes to lift an electron across the gap, and only about twice room temperature's kT of 0.026 eV — so at room temperature essentially every one of them is shaken loose. One phosphorus atom, one free electron, and no heat worth mentioning.
Or replace it with boron, which has three. Now one bond is short of an electron. That missing electron is a hole, and here is the thing about holes: a neighbouring electron can slide across to fill it, which completes that bond but leaves a new hole where the electron came from. Do it again and again and the hole moves through the crystal — in the opposite direction to the electrons, and behaving in every respect like a particle carrying a positive charge.
Notice what doping has bought. One phosphorus atom per million silicon atoms gives about 5 × 1016 carriers per cubic centimetre — five million times more than pure silicon has at room temperature. So the thermally generated carriers, and their violent temperature dependence, become an irrelevance: the carrier concentration is now set by how much phosphorus you put in, and that does not change when the chip warms up.
The resistivity also falls by a factor of millions, which is why doping is described as making silicon conduct. But the real prize is control. You choose the number, you choose the sign, and — with a mask — you choose where.
Silicon doped with phosphorus is called n-type, because the carriers it adds are negative. Silicon doped with boron is p-type, because the carriers it adds — the holes — behave as positive. The letters are the sign of the charge that moves.
If one impurity atom per million changes everything, then everything else has to be absent at a far lower level than that. Electronic-grade silicon is refined to what the industry calls eleven nines: 99.999999999% pure, roughly one unwanted foreign atom per hundred billion silicon atoms. It is the purest bulk material human beings make.
Then it is grown as a single crystal: one continuous lattice, with no grain boundaries — no places where one crystal ends and another, tilted differently, begins. It is pulled slowly out of a melt while rotating, as a cylinder 300 mm across and two metres long, called a boule. A modern silicon boule weighs a couple of hundred kilograms and is, atom for atom, one crystal. The wafers your processor is printed on are slices of it, less than a millimetre thick.
Now put a piece of n-type silicon directly against a piece of p-type — not glued, but as one continuous crystal with the doping changed halfway along. What happens next needs nothing but Module 13 and Module 8.
No battery is connected. Nothing is switched on. The crystal just sits there, n-type on one side and p-type on the other.
What is happening at the boundary?
The last ledger box is the point of a diode: at half a volt forward it passes a hundred million times more current than at half a volt backward. Switch the bench to the AC view and you can see what that is for — the sine wave of Module 17 goes in, and only the positive halves come out. That is a half-wave rectifier, and it throws away half the supply. Wire four diodes in a diamond — a bridge — and the negative halves are flipped up instead of discarded, which is what every power supply in your house actually begins with.
The LED. In forward bias, electrons pour into the p-side and fall into holes. That fall releases energy equal to the band gap — and in some materials it comes out as a single photon rather than as heat. Module 21: E = hf, so the colour is fixed by the gap. Gallium arsenide's 1.42 eV gives infrared; gallium nitride's 3.4 eV gives blue, which is why blue LEDs took thirty years longer than red ones and won a Nobel Prize in 2014. White LEDs are blue ones with a coating of phosphor — a material that absorbs blue light and re-emits it at lower energy, as yellow and red. Blue straight through plus yellow and red from the coating adds up to white.
The solar cell. The same junction, run backwards. A photon with more energy than the gap is absorbed and kicks an electron across it, creating an electron–hole pair inside the depletion region. The built-in field then sweeps the two apart before they can recombine — electron to the n-side, hole to the p-side — and that separation of charge is a voltage you can draw current from. The photoelectric effect of Module 21, with the junction's own field doing the collecting.
The photodiode is a solar cell optimised for speed rather than power, and it is what reads the light in a camera sensor, a fibre-optic receiver, and your television remote.
A diode is a valve stuck in one position. What changed the world was a valve with a handle.
Take a piece of p-type silicon. Let phosphorus soak into two patches of its surface, a short distance apart, so that those two patches become n-type; call them the source and the drain. Between them, on top, grow a very thin layer of insulator — silicon dioxide, which is glass, and which happens to grow beautifully on silicon just by heating it in oxygen — and put a metal electrode on that. Call it the gate.
The gate does not touch the silicon. There is no electrical connection to it at all. It is a metal plate separated from silicon by an insulator, which is to say, in the language of Module 13, it is a capacitor.
Switch the bench to the NAND view and work the two input checkboxes. The output is low only when both inputs are high. Now tick both boxes together, as though the two inputs were wired to each other, and watch what the output does: it is the opposite of the input. You have just made a NOT gate out of a NAND.
With NOT in hand, and knowing you can feed one gate's output into another's input, how much of a computer can you build out of copies of this one gate?
NAND is functionally complete. Tie its two inputs together and it is a NOT gate. Follow it with a NOT and it is an AND. Invert both inputs first and it is an OR. From AND, OR and NOT you can build XOR, and from XOR and AND you can build a one-bit adder, and from thirty-two adders you can build a thirty-two-bit adder, and so on all the way up.
Memory too. Wire two NAND gates so that each one's output feeds the other's input and you have a circuit with two stable states, which stays in whichever one you last put it in. That is one bit of memory — a latch — made of two gates, which is eight transistors. Modern static RAM uses six.
So the entire logical content of a computer is: one gate, repeated. A current processor has on the order of 1010 transistors, which is a few billion gates, arranged. Nothing else. No new physics appears anywhere in the stack above this module — everything from here up is architecture, and it is a different subject.
CMOS stands for complementary metal-oxide-semiconductor: metal gate, oxide insulator, semiconductor body — the transistor you just built — and complementary because it comes in two kinds used in pairs. The NAND in the bench uses both: n-channel transistors that switch on when their gate goes high, and p-channel ones that switch on when it goes low. They are wired so that in either steady state, exactly one of the two paths is conducting — either the output is tied to the supply, or it is tied to ground, and never both.
The consequence is that a CMOS gate that is sitting still draws essentially no current at all. It burns power only during the instant it changes state, charging and discharging the gate capacitances of whatever it is driving. That is why a chip's power consumption is roughly proportional to its clock speed, why your laptop gets hot when it is working and cool when it is idle, and why the industry stopped raising clock frequencies around 2005 and started adding cores instead: power goes up with frequency, and there is only so much heat you can pull out of a square centimetre of silicon.
A modern processor has ten billion transistors on a chip the size of a thumbnail. They are not placed one at a time. They are printed — all of them at once, photographically.
The wafer is coated with a light-sensitive polymer. A mask carrying the pattern for one layer is illuminated, and its image is projected — demagnified — onto the wafer through a lens. Where the light lands, the polymer's chemistry changes and it can be washed away; then the exposed silicon underneath is etched, or doped, or has metal deposited on it. Repeat for sixty or so layers and you have a processor.
Which puts the whole industry at the mercy of an equation from Module 12.
A mask is illuminated and its image is projected onto the wafer. The features being printed are now a few nanometres across.
What sets the smallest feature this machine can print — and what would you therefore expect chip-makers to have spent the last fifty years changing?
Diffraction. The same limit that stops an optical microscope resolving a virus stops a lithography machine printing a fine line, and it comes from the same place. Module 12's θmin ≈ 1.22 λ/D, where θ is the diffraction angle, λ is the wavelength, and D is the aperture's width, gives a minimum angular separation; multiply it by the distance to the image and you get a length. Writing the aperture's half-angle as a numerical aperture NA — a measure of how wide a cone of light the lens can gather, at most 1 for a lens in air — turns it into the form the industry uses: the smallest feature is roughly λ divided by twice the NA.
So the history of Moore's law is largely a history of wavelengths. Mercury lamps at 436 nm, then 365. Then excimer lasers at 248 nm, then 193 nm. Then a decade of extraordinary tricks to beat the limit without shortening the wavelength — immersing the last lens in water to raise the numerical aperture, printing the same layer twice with offset masks, correcting the mask shape to pre-compensate for the diffraction it will suffer.
And then, after twenty years of development and at a cost of billions, extreme ultraviolet light at 13.5 nm. That wavelength is so short that no material is transparent to it, which changes everything about the machine. The "lenses" have to be mirrors. Even the mirrors barely reflect, so each one is coated with dozens of alternating layers a few atoms thick, which reflect by the thin-film interference of Module 12. And the whole beam path has to sit in vacuum, because air absorbs it. The light itself is made by hitting droplets of molten tin with a laser, fifty thousand times a second, to make a plasma that glows at 13.5 nm. There are only a handful of these machines in the world and each costs more than a large aircraft.
All of that to shorten a wavelength, because of a formula about waves going through a gap.
The gate insulator in a modern transistor is about a nanometre thick — five or so atomic layers. Module 23 told you what happens to a barrier that thin: electrons tunnel through it. Gate leakage was becoming a serious power problem by the mid-2000s. The obvious fix — a thicker insulator — was not available, because a thicker capacitor is a weaker one, and the gate only works if it can pull enough charge into the channel. The way out was to change the material: capacitance depends on permittivity as well as thickness (Module 13 again), so a material with a higher permittivity gives the same capacitance from a physically thicker film. Thicker means less tunnelling; the same capacitance means the transistor still switches.
That bought some years. But the trend cannot continue indefinitely, and what stops it is not engineering difficulty. It is physics you already know. A transistor works by keeping electrons where you put them, and Module 23 gave the reason that gets harder and harder: a barrier only blocks an electron if it is thick compared with the distance the electron's wave can leak into it. Shrink everything and the barriers shrink too, and below a few nanometres they stop blocking. Whatever comes next — and there are candidates — will not be a smaller version of this.
That is the end of the course. It seems worth saying plainly what has happened over the twenty-five modules, because from the inside it can look like a list of topics.
You started, in Module 1, by working out where a thrown ball lands. You are finishing by understanding a device that is quantum mechanics made solid. Every part of it is something this course built: the bands are standing waves and the Pauli principle, the carrier count is a Boltzmann factor, the junction is Coulomb's law doing its own assembly, the channel is I = nAve, the printing is diffraction, and the wall at the bottom is tunnelling. A chip is not an application of physics. It is physics, arranged.
Along the way a few things happened repeatedly, and they are more important than any individual result:
What is left out is at least as interesting as what is in. General relativity, which we touched only through GPS. The strong force done properly, and quarks. The weak force, and why the Sun's first step is slow enough for it to burn for ten billion years rather than exploding. Statistical mechanics beyond a first pass. Condensed matter, where superconductivity is waiting. And the honest open questions: what dark matter is, what dark energy is, and how gravity and quantum mechanics are supposed to be reconciled — which nobody knows, and which is not a gap in this course but a gap in the subject.
The syllabus stops here. The subject does not.
| Idea | What it says |
|---|---|
| Bands | Bring N atoms together and each level splits into N. At 1023 atoms the ladder is a continuous band. Between bands, no states at all. |
| Full versus partly full | A full band cannot carry current — nowhere to move to. A partly full band is a metal. A full band with a small gap above it is a semiconductor. |
| Temperature | Carriers ∝ e−Eg/2kT. Silicon's resistance falls steeply with heating; copper's rises. Same physics, opposite mechanism. |
| Doping | One atom in a million of phosphorus or boron sets the carrier concentration and its sign, and swamps the thermal contribution. Control, not just conduction. |
| Holes | Real, in the only sense that matters: the Hall effect measures their sign, and F = qvB from Module 15 is the instrument. |
| The junction | Diffusion and the built-in field reach a standoff, leaving a depletion region and a 0.7 V step. Forward bias floods it exponentially; reverse bias blocks. Diode, LED, solar cell. |
| The transistor | An insulated gate — a capacitor — pulls a channel into existence beneath it. I = nAve, with the gate setting n. It costs nothing to hold on, which is why there can be billions. |
| Limits | Features are printed, so Module 12's diffraction limit governs their size — hence 13.5 nm light. And Module 23's tunnelling is the wall at the bottom. |