Semiconductors

The entire digital world — every phone, every laptop, every server humming in a data centre — runs on a material that is, on paper, a mediocrity: it is neither a good conductor nor a good insulator. Pure silicon at room temperature carries almost no current, yet it is not diamond-like in its refusal either. It sits precisely on the fence. And that fence-sitting is the whole point. Because a semiconductor is poised so delicately between conducting and not conducting, we can tip it either way — and we can do it with breathtaking control. Stir in one impurity atom for every million of silicon, a mere pinch, and the conductivity leaps by a factor of a billion.

That tunability is what makes silicon the most important material of the modern age. This page picks up directly from band theory: a semiconductor is simply an insulator with a small gap. We will see how heat lifts a few electrons across that gap, leaving positively charged holes behind; how deliberately adding impurities — doping — floods the material with carriers of one sign; and how joining a piece flooded with electrons to a piece flooded with holes makes a p–n junction, the one-way valve for current that is the seed of the diode, the transistor, and every logic gate built from them.

An insulator with a small gap

Recall the band-theory verdict: a full valence band beneath an empty conduction band, with a forbidden gap E_g between them, means no conduction — a full band carries no current. An insulator like diamond has a huge gap (\sim 5.5\ \text{eV}) and stays dead. A semiconductor has exactly the same band arrangement but a modest gap, roughly 1\ \text{eV}:

Why does a small gap change everything? Because at temperature T the thermal energy scale is k_B T \approx 0.025\ \text{eV} at room temperature. That is far too small to lift an electron across diamond's 5.5\ \text{eV} gap — the chance is astronomically tiny. But across silicon's 1.1\ \text{eV} gap the odds, while still small, are large enough that a measurable trickle of electrons is thermally promoted from the valence band into the empty conduction band. Each promoted electron is now free to roam an almost-empty band and carry current. A pure (intrinsic) semiconductor conducts a little because heat alone jostles a few electrons across the gap.

Holes: the empty seat that carries charge

Promoting an electron does two things, not one. It puts an electron into the conduction band — obviously a carrier — but it also leaves behind an empty state in the valence band. That empty state is called a hole, and it is every bit as important as the electron.

Here is the beautiful bookkeeping. A valence band with one electron missing is almost full. When a field pushes the sea of remaining valence electrons, they shuffle along, and the empty seat moves the opposite way — exactly as a bubble rises while the water around it falls. Rather than track 10^{23} electrons minus one, we track the single empty seat and give it the properties it appears to have: a positive charge +e and its own effective mass. A hole is a genuine mobile positive carrier for every purpose of current flow. So thermal excitation in a semiconductor creates carriers in pairs: one negative electron up in the conduction band, one positive hole down in the valence band.

The intrinsic material therefore has equal numbers of the two: n = p = n_i, where n is the electron concentration, p the hole concentration, and n_i the intrinsic carrier concentration.

How the carriers explode with temperature

A careful count of how many electrons are thermally promoted (integrating the Fermi–Dirac occupation against the band densities of states) gives the intrinsic carrier concentration:

n_i \propto T^{3/2}\,\exp\!\left(-\frac{E_g}{2k_B T}\right).

The T^{3/2} in front is a gentle rise, but it is the exponential that dominates and makes semiconductors so dramatic. Two features are worth burning into memory. First, the factor of 2 in the denominator: it is E_g/2, not E_g, because the Fermi level sits near the middle of the gap, so a carrier only has to climb half the gap on average. Second, the ferocious sensitivity: because E_g and T sit inside an exponential, a small change in either sends n_i flying over orders of magnitude. Warm the crystal and the carrier count rockets; widen the gap and it collapses. Drag the gap slider and watch the curve's whole scale transform.

This exponential is exactly why a pure semiconductor is a lousy, temperature-sensitive conductor — and why we almost never use one pure. To make a reliable device we need a carrier concentration we set ourselves, one that does not swing wildly with the weather. That is what doping delivers.

Doping: choosing the carriers by hand

Silicon sits in group IV: each atom shares four electrons with its neighbours, and every one is locked into a bond. Now replace a single silicon atom with an impurity from a neighbouring column of the periodic table.

The carrier you added in overwhelming excess is the majority carrier (electrons in n-type, holes in p-type); the other, still present in tiny numbers from thermal pair-creation, is the minority carrier. Because typical doping adds far more carriers than heat ever could, doping sets the conductivity almost independently of temperature — and lets us dial it up or down by choosing how much impurity to add.

The law of mass action

Doping floods the crystal with one kind of carrier — but it does not change the fundamental balance between electrons and holes as much as you might guess. Electrons and holes are constantly created in pairs and constantly recombining, and in equilibrium a remarkable relation holds no matter how you dope:

This is the semiconductor engineer's most-used equation. Suppose we dope silicon n-type with a donor concentration N_D large compared with n_i. Then essentially every donor gives up its electron, so n \approx N_D, and the minority hole concentration follows immediately:

p = \frac{n_i^2}{n} \approx \frac{n_i^2}{N_D}.

Because n_i is tiny and N_D is large, the minority concentration is minute — heavy doping doesn't just raise the majority carriers, it actively suppresses the minority ones. That lopsidedness is what makes a p–n junction work.

The p–n junction: a one-way valve for current

Join a piece of p-type silicon to a piece of n-type silicon and something wonderful happens at the boundary. Electrons from the electron-rich n-side spill across into the hole-rich p-side and recombine; holes spill the other way. This leaves behind the fixed, ionised dopant atoms — positive donor ions stranded on the n-side, negative acceptor ions on the p-side — creating a carrier-starved depletion region and, with it, a built-in potential that opposes further diffusion. Equilibrium is a standoff.

Now apply a voltage. Forward bias (positive on the p-side) lowers the built-in barrier, and carriers flood across: current flows easily, rising exponentially with voltage. Reverse bias (positive on the n-side) raises the barrier and widens the depletion region, and almost nothing gets through — just a tiny saturation current. The junction conducts one way and blocks the other. It rectifies. The current–voltage law is the Shockley diode equation, where V_T = k_B T/e \approx 0.026\ \text{V} is the thermal voltage:

I = I_s\left(e^{V/V_T} - 1\right).

The curve tells the whole story: flat and near-zero for reverse (negative) voltage, then a sharp turn-on knee where forward current explodes. That asymmetry is the diode. Adjust the saturation current and watch the knee shift.

Stack two junctions back to back — n–p–n or p–n–p — and a small current at the middle terminal controls a large current through the whole sandwich: that is the transistor, the amplifier and switch from which all of digital logic is assembled. Every idea on this page, scaled down to a few nanometres and repeated billions of times on a fingernail of silicon, is a microprocessor.

Worked examples

Example 1 — minority carriers by mass action. Silicon at room temperature has n_i \approx 1.0\times 10^{10}\ \text{cm}^{-3}. It is doped n-type with N_D = 1.0\times 10^{16}\ \text{cm}^{-3} donors. Find the majority and minority concentrations. Essentially every donor ionises, so the majority electron concentration is

n \approx N_D = 1.0\times 10^{16}\ \text{cm}^{-3}.

The minority hole concentration follows from n\,p = n_i^2:

p = \frac{n_i^2}{n} = \frac{(1.0\times 10^{10})^2}{1.0\times 10^{16}} = \frac{1.0\times 10^{20}}{1.0\times 10^{16}} = 1.0\times 10^{4}\ \text{cm}^{-3}.

The electrons outnumber the holes by a factor of 10^{12} — doping has made this an overwhelmingly electron-dominated material.

Example 2 — the exponential sensitivity to the gap. Two materials at the same temperature have gaps differing by \Delta E_g = 0.6\ \text{eV}. Ignoring the slowly varying prefactor, the ratio of their intrinsic carrier concentrations is \exp\!\big(\Delta E_g/(2k_B T)\big). With k_B T = 0.025\ \text{eV} the exponent is 0.6/(2\times 0.025) = 12, so

\frac{n_i(\text{small gap})}{n_i(\text{large gap})} \approx e^{12} \approx 1.6\times 10^{5}.

A gap difference smaller than an electron-volt changes the carrier population by a hundred thousand times — which is why germanium (0.67\ \text{eV}) is intrinsically far more conductive than gallium arsenide (1.4\ \text{eV}).

Example 3 — recovering n_i. A sample is measured to have n = 4\times 10^{16}\ \text{cm}^{-3} electrons and p = 2.5\times 10^{3}\ \text{cm}^{-3} holes in equilibrium. What is the intrinsic concentration? By mass action n_i = \sqrt{n\,p}:

n_i = \sqrt{(4\times 10^{16})(2.5\times 10^{3})} = \sqrt{1.0\times 10^{20}} = 1.0\times 10^{10}\ \text{cm}^{-3}.

Consistent with room-temperature silicon — as it should be, since n_i depends only on temperature and gap, never on the doping.

Watch out — false, and this is the misconception that trips up almost everyone meeting semiconductors. A hole is not a particle. It is the absence of an electron in an otherwise nearly-full valence band — an empty seat in a packed theatre. There is no little positive object sitting in the silicon. What happens is that the vast sea of remaining valence electrons shifts under a field, and the empty seat appears to drift the opposite way. Tracking that one empty seat, and assigning it a charge +e and an effective mass, is simply a wonderfully economical way to describe the collective motion of 10^{23} electrons-minus-one. The hole behaves exactly like a positive mobile charge — so we are entitled to call it one — but underneath it is missing-electron bookkeeping, not a new particle.

A second trap hides in the name "n-type". It is tempting to think n-type silicon is negatively charged because it is "full of electrons". It is not — n-type material is perfectly electrically neutral. Every mobile electron the donors released left behind a fixed, positive donor ion locked into the lattice, and those stranded positive ions balance the mobile negative electrons charge-for-charge. "n-type" tells you the sign of the mobile carriers (negative), not the net charge of the material. The same goes for p-type: neutral overall, with positive holes balanced by fixed negative acceptor ions. Confuse "carriers are negative" with "the block is charged" and half of junction physics stops making sense.

It comes down to the barrier and which way you push. At the junction, diffusion has already carved out a depletion region and built up an internal potential barrier that mobile carriers must climb. When you forward bias — pushing positive voltage onto the p-side — you lower that barrier. Suddenly holes from the p-side and electrons from the n-side can pour across in enormous numbers, and because the barrier height enters the carrier statistics exponentially, the current shoots up exponentially with voltage: the e^{V/V_T} term takes over.

Reverse bias does the opposite. It raises the barrier and pulls carriers away from the junction, widening the depletion region. Now the only current is the tiny drift of thermally generated minority carriers — the saturation current I_s, often billionths of an amp — and it barely grows however hard you push. So the same device that is a near short-circuit forwards is a near open-circuit backwards. That built-in asymmetry is rectification, and it is the reason a diode can turn alternating current into direct current, protect circuits from reversed batteries, and — stacked into transistors — switch the ones and zeros of every computer.