The Transistor: A Barrier with a Knob

1Keeping last week's promise

Where we stopped: the pn junction builds a real, manufacturable energy barrier — the very object Notes 02 and 03 priced — and Fig 6.1 of Notes 04 teased the fix for its one flaw, a plate above the barrier that moves it by influence through a perfect insulator, with no current through the handle. That teaser has a name: the MOSFET (metal–oxide–semiconductor field-effect transistor — say the long name once, then never again), and it is the switch that every chip you own is made of, several billion times over.

The plan for the two sessions is the plan of these notes. First, what the device looks like when you saw it open (§2). Then how the knob actually grips the barrier, and where "on" begins (§3). Then its three personalities — because the same device behaves like three different components depending on how you treat it (§4). Then the part practicing engineers actually use day to day: two deliberately crude models that replace all the device physics with arithmetic (§5). And finally its mirror twin, which sets up next week (§6). Project 1 is due this week — the closing checklist in §7 is written to get it out the door.

2The device, opened up

The recipe is three moves, all from last week's kitchen. Start with a slab of p-type silicon — spot territory: plenty of empty parking spots, no spare cars. Implant two pockets of n-type into its surface, a short distance apart — two garages, each stuffed with spare cars. Then cover the strip of p territory between them with a film of glass a few atoms thick, and lay a metal plate on the glass. Done. Here it is in cross-section, with its four terminals named the way we will name them all term:

Read the picture with last week's eyes and two facts jump out. First, the device is born off. A car trying to drive from source to drain must leave n territory, cross p territory, and re-enter n territory — two pn borders, back to back. Whichever direction current tries to flow, one of those two junctions is a hill facing the wrong way. No knob, no current: exactly the reverse-biased valve from Notes 04, doubled. And "off" costs nothing to maintain — the barrier holds itself up, the way the junction's built-in hill did.

Second, and this is the sentence to underline: no current can ever flow through the glass. The oxide is quartz-grade insulator — Notes 04's "350 thermal kicks" garage — so the gate is electrically walled off from everything it controls. It can push and pull on the carriers below (charge attracts charge straight through an insulator; that is just electrostatics), but not one car passes through the wall. (Full honesty: at a few atoms thick, a quantum whisper does tunnel through modern oxides — a week-14 story. For every purpose this term: zero.)

Sit with what that buys. A switch whose handle carries no current is a switch that costs almost nothing to hold in either position — you pay to move the knob, not to lean on it. That single property is why the energy story of Notes 03 works at all: a chip can hold billions of bits, and only the bits that are changing send a bill. The whole of week 11 hangs off this sentence.

One more honest note: the names source and drain mean what they say — cars enter the channel from the source and leave at the drain — but the two garages are manufactured identical. Which one plays "source" is decided by the voltages around the device, not by the factory. And the body, the slab itself, is a real fourth terminal; we tie it to a fixed voltage and let it sleep until it briefly matters in week 12.

3Turning the knob

Now put a positive voltage on the plate and watch the territory beneath it. Positive charge attracts electrons — through the glass, no contact needed. The first thing the pull does is shoo the spots away from the surface: the top sliver of p territory empties out, like the depletion zone of Notes 04. Keep turning, and something better happens: actual cars begin to gather under the glass. They do not come through the glass — they cannot — they pour in sideways, from the two garages conveniently parked at either end. (That is why the garages sit exactly flanking the gate: they are the supply depots.)

Past a tipping point, the gathering becomes a crowd: a continuous sheet of cars, a few nanometers thin, running the full width from source to drain. The spot territory under the glass has been electrically converted into a temporary strip of n-type — a channel. The two garages are now connected by a bridge of cars, and the barrier between them is simply gone, for as long as the knob is held up. Let go of the knob and the cars drain back into the garages; the barrier stands back up on its own. This is Notes 04's hill, lowered by a voltage instead of built lower by doping — a barrier with a knob on it.

The tipping point has a name and a number: the threshold voltage, written V_T — typically a few tenths of a volt, around 0.3–0.5 V in modern chips, set at the factory by the doping recipe. Below it, the pull gathers only a few strays. Above it, a real bridge, and the bridge thickens in proportion to how far past the threshold you push — the plate and the channel are two sides of a capacitor, so more knob means more charge means a fatter bridge. That proportionality is the only "formula" the week needs, and it is a sentence, not an equation.

The honest aside: below threshold is not off

One aside, and it matters, so it gets its own paragraph. "Below threshold" does not mean zero current — it means the barrier is tall, and Notes 03 taught you exactly what a tall barrier does to a thermal crowd: it holds back all but an exponentially rare few. The leak through an "off" transistor falls by about 10× for every 60–100 mV you push the gate further below threshold — same Boltzmann exponential, new costume. One leaking transistor is laughably little. Ten billion of them, leaking around the clock, is a warm phone in a pocket and a real line item in a datacenter's power bill. We price it in week 11; this is where it comes from.

Sweep slowly from 0 and watch three things move together: the crowd under the glass, the flag on the gauge, and the meter — which is logarithmic, because the honest story spans ten orders of magnitude. Note what the meter does below the flag: it climbs a steady factor of ten for every fixed step of knob. That staircase is Notes 03's exponential, live.

4Three personalities

A bridge alone moves nothing — cars need a reason to cross. So now hold the knob steady and add the second voltage: a pull at the drain, coaxing cars to flow from source to drain through the channel. How much current flows? The answer depends on how hard you pull, and it splits cleanly into three regimes — three personalities of the same device. The household analogy that actually fits is a water tap: the gate is the tap handle, the drain's pull is how hard the far end sucks on the pipe, and current is the flow.

• OFF. Tap closed — knob below the tipping point, barrier up. Pull all you like at the drain; only the §3 leak answers.
• The obedient personality. Tap open, gentle pull: flow follows the pull. Double the pull, double the flow — which is exactly how a resistor behaves, and whose resistance the knob chose. The transistor is, for the moment, a resistor with a control knob.
• The stubborn personality. Keep pulling harder and the obedience stops. Flow levels off and refuses to grow, however hard the drain pulls: the tap is fully committed, and the flow is now set upstream — by the handle — not by the suction downstream. The transistor has become a current source the knob dials.

Why does the obedience stop? Look at the drain end of the bridge, honestly. The bridge exists because the gate's positive pull holds cars under the glass. But the drain is now also positive — that is what pulling means — and at the drain end of the channel the two pulls fight. The harder the drain pulls, the less net grip the gate has on the cars right next to it, so the bridge thins toward the drain like a wedge. Pull hard enough — harder than the knob is past its tipping point — and the bridge pinches off to almost nothing just short of the drain. Cars still cross (they are hurled across the pinched sliver by the strong field there), but how many arrive per second is decided by the wide upstream part of the bridge, which the drain cannot reach. Pulling harder just moves cars faster through a gap that admits no more of them. That is the whole mechanism of stubbornness, and it is drawn live on the bench below.

For the record — and because datasheets and simulators use them — here are the textbook names, once each. You may use the plain names in this course forever.

One honesty footnote on the stubborn region: the flow does not go perfectly flat. As the drain pulls harder, the pinch point creeps slightly toward the source, the wide part of the bridge shortens a little, and the current tilts gently upward. The bench draws that tilt rather than hiding it. Digital designers ignore it; analog designers build careers on exactly how small it is.

Fix the knob and drag the pull from 0 to 1.8 V: the curve rises obediently, then goes stubborn right where the pull passes the knob's excess over threshold — and the wedge above pinches at that same moment. Now change the knob and re-drag: the whole curve rescales and the boundary moves. One device, a family of curves; this family is what ngspice fits when you simulate in Project 2.

5The two models engineers actually use

Here is a professional secret: almost nobody designs with the curves of §4 in hand. Real design runs on deliberately crude caricatures — models that throw away nearly everything and keep the one behavior that answers today's question. For the MOSFET there are exactly two worth owning, one per conducting personality.

Model 1 — the light switch (with a price tag)

For digital work, flatten the whole device to this: knob up → a closed switch with a small resistance; knob down → an open switch. The small resistance is the on-resistance, written R_on — a few thousand ohms for a minimum-size transistor in a modern process — and it is the model's entire memory of all the physics above: the bridge is real, so crossing it is not free. That is the whole model. It is "wrong" in a dozen documentable ways, and it is the model on which weeks 6 through 8 — and every logic gate ever shipped — are built. A chip, viewed through Model 1, is just billions of priced light switches flipping each other.

Model 2 — the current source

For the stubborn region, the caricature is: a current source whose value the knob sets, indifferent to the drain. One sentence on why anyone cares: a current that ignores what the output is doing is the raw material of amplification, which is why analog designers park their transistors in the stubborn region and live there. We will borrow this model exactly once, in week 14's tour of CMOS analog computing; digital design almost never needs it.

The discipline worth naming: choose the model by the question, and know which caricature you are holding. "Is this gate fast enough?" is a Model 1 question. "How much does this amplify?" is Model 2. "Why is my sleeping chip warm?" is neither — that one needs the §3 leak, which both models deliberately forget. Half of engineering judgment is remembering what your model chose not to know.

6Two flavors: mirror twins

Everything so far used cars — electrons — as the carriers. Now run the whole story through a mirror. Swap every ingredient for its opposite: put two p-type garages (stuffed with spare spots) into an n-type slab, same glass, same plate. The result is a transistor whose bridge is made of spots instead of cars, and whose knob works upside down: you turn it on by pulling the gate down (a low gate attracts the positive spots up under the glass), and off by pushing it up. Every sentence of §§2–5 survives the mirroring word for word — just swap cars↔spots and up↔down.

The two flavors are called nMOS (n-type bridge, cars) and pMOS (p-type bridge, spots), and the reason a factory bothers making both is a single sentence that is all of next week: put one twin of each in series between the supply and ground, drive both knobs from the same input, and you get the CMOS inverter — a logic gate that burns almost nothing while holding still, and the most manufactured object in human history. Well over 10²² MOSFETs have been built to date — comfortably more than there are grains of sand on Earth — and essentially all of them were built to stand in pairs like that. Notes 06 opens the pair up.

7Work before Session 9

Session 9 is Monday, October 5, and it builds the CMOS inverter out of this week's twins — nothing new is needed beyond these notes. The bigger event is that Project 1 is due this week, on the date in the specification. You have carried this model since week 2; what remains is finishing, not inventing.

• Review these notes. The memory test: draw Fig 3.1's three frames — knob off, tipping point, bridge — and narrate them aloud in garage language. Then say, in one sentence each, what the obedient and stubborn personalities act like and which model (§5) belongs to each.
• Ten minutes per bench. On Bench 3.1, find the gate voltage where the current first reaches 1 µA, then step the knob down in equal steps below threshold and confirm the meter drops by roughly a factor of ten per step. On Bench 4.1, for two different knob settings, find the pull where obedient turns stubborn — and check that the boundary moved with the knob, as §4 says it must.

Optional, for the curious: pull up the datasheet of the 2N7000 — a discrete MOSFET you can buy for pocket change — find its threshold voltage and on-resistance, and compare both with the benches. The numbers differ (it is a big, rugged device), but the personalities are identical.

8Exercises

Not graded, as always. Conceptual on purpose — the garage and the tap answer all three. Bring your answers, or your stuck points, to Session 9.

1. The glass paradox. A friend objects: "If no current can ever cross the oxide, the gate is not connected to anything — how can it possibly do anything?" Resolve this with the garage picture. Then take it one step further: given that the handle carries no standing current, what does flipping a MOSFET actually cost energy-wise — and when is that cost paid? (Notes 03 has the vocabulary.)
2. The two-junction lock. With the knob off, use Notes 04's junction picture to explain why no current flows from source to drain in either direction — identify which of the two pn borders does the blocking for each direction. Then say precisely what the channel changes about that picture, and why the barrier stands back up by itself the moment the knob is released.
3. The stubborn tap. In your own words: once the bridge is pinched at the drain end, why does pulling harder at the drain buy almost no extra current — where is the bottleneck, and who controls it? Then the judgment question: why can a digital designer flatten this entire drama into "a switch with an on-resistance," while an analog designer cannot? (Hint: ask what question each one is trying to answer, per §5.)

§Symbols and constants