Semiconductors and doping

Intrinsic semiconductors are chemically pure crystals of silicon (Si) or germanium (Ge). Each atom has four valence electrons and forms four covalent bonds with its neighbours, so at absolute zero every electron is locked in place and the crystal behaves like an insulator. At room temperature, thermal energy is enough to occasionally snap a covalent bond. This releases one free electron into the conduction band and leaves behind a hole — a vacant bonding site that a neighbouring bound electron can hop into, making the vacancy itself appear to drift through the lattice like a mobile positive charge. Because every broken bond produces exactly one electron and one hole together, an intrinsic sample always has equal numbers of each:

where: $n_e$ is the free-electron concentration and $n_h$ is the hole concentration in the intrinsic crystal — the two are generated and destroyed in pairs, so this equality is a direct bookkeeping consequence of how carriers are created, not an independent physical law.

Extrinsic (doped) semiconductors are made by mixing in a tiny, controlled amount of impurity — typically one impurity atom per million to a hundred million host atoms — during crystal growth. This trace addition, called doping, changes the electrical behaviour of the crystal far more dramatically than its minuscule concentration would suggest, because it adds carriers directly rather than waiting on thermal bond-breaking.

• n-type material comes from doping with a pentavalent impurity — an atom with five valence electrons, such as phosphorus, arsenic or antimony. Four electrons form the usual covalent bonds with the four neighbouring host atoms; the fifth has no bond to enter and sits only weakly attached to its parent atom, freed into the conduction band by ordinary thermal energy. Because the impurity atom donates this electron, it is called a donor. Electrons are the majority carriers and holes are the minority carriers.
• p-type material comes from doping with a trivalent impurity — an atom with three valence electrons, such as boron, aluminium or indium. It completes only three of the four bonds a host atom would normally form, leaving one bond incomplete: a hole, present from the moment of doping. Because this atom readily accepts an electron from a neighbouring bond to complete its own, it is called an acceptor. Holes are the majority carriers and electrons are the minority carriers.

In both cases, the doped crystal is still electrically neutral overall. A donor atom, having given up its extra electron, becomes a fixed positive ion locked in the lattice; an acceptor atom, having gained an electron, becomes a fixed negative ion. Neither process creates or destroys net charge — it only redistributes which carrier type is abundant. Doping does not charge the crystal, it biases which carrier dominates conduction.

This topic is purely qualitative on the ISC syllabus — no carrier-density formula is examined — so the reasoning is entirely in the bond picture, and it is worth stating precisely why doping works the way it does.

Step 1. Both silicon and germanium are tetravalent, so an undisturbed crystal is a rigid network of exactly four covalent bonds per atom, with zero free electrons at absolute zero. Substituting one host atom for an impurity atom does not disturb the surrounding lattice geometry — the impurity simply sits in the host atom's site and tries to bond the same way.

Step 2. A pentavalent impurity has one more valence electron than the four bonds around it require. That fifth electron cannot be absorbed into any bond, so it remains attached to its parent atom by a weak electrostatic pull alone — no covalent bond to break, just a shallow binding energy of order $0.01$ to $0.05\ \text{eV}$, far below silicon's $\sim 1.1\ \text{eV}$ band gap. Ordinary thermal energy at room temperature is enough to free it, which is why n-type doping supplies electrons almost independent of temperature-driven bond breaking.

Step 3. A trivalent impurity has one fewer valence electron than four bonds require, so one bond simply cannot form — a hole exists there from the moment the crystal is grown, needing no thermal activation. That hole is filled the instant a bound electron from a neighbouring bond hops across, relocating the hole one lattice site over; repeated hopping makes the hole appear to move freely through the crystal, exactly as if it were a mobile positive charge.

Step 4. The donor atom's nucleus still carries its original positive charge after releasing its fifth electron, so it is left as a fixed positive ion; the acceptor atom gains an electron to complete its bonding, so it is left as a fixed negative ion. These ions are locked in the lattice, but their charge exactly balances the mobile carriers they produced, so the crystal as a whole carries no net charge in either case.

Question: A pure silicon crystal is doped, first with a trace of phosphorus and separately (in another sample) with a trace of indium. For each sample, identify the type of semiconductor formed, name the majority and minority carriers, and state the charge left on the fixed impurity ion.

Step 1. Phosphorus (atomic number 15) has the valence configuration $3s^23p^3$, giving five valence electrons — it is pentavalent. Since silicon is tetravalent, a pentavalent impurity is a donor: four of its electrons form the normal covalent bonds and the fifth becomes a free conduction electron with only a small thermal nudge.

Step 2. The phosphorus-doped sample is therefore n-type, with electrons as majority carriers and holes as minority carriers (the minority holes arise only from ordinary thermal bond-breaking, as in an intrinsic crystal). Having donated its extra electron, the phosphorus atom is left as a fixed positive ion.

Step 3. Indium (atomic number 49) has the valence configuration $5s^25p^1$, giving three valence electrons — it is trivalent. A trivalent impurity in tetravalent silicon can complete only three bonds, leaving one incomplete — a hole present from the start, so indium is an acceptor.

Step 4. The indium-doped sample is therefore p-type, with holes as majority carriers and electrons as minority carriers (again, the minority electrons come only from thermal generation). Having accepted an electron to complete its own bonding, the indium atom is left as a fixed negative ion.

Answer: phosphorus-doped silicon is n-type (electrons majority, holes minority, fixed positive donor ions); indium-doped silicon is p-type (holes majority, electrons minority, fixed negative acceptor ions) — both crystals remain electrically neutral overall.

• Saying a doped semiconductor becomes "charged" overall (net positive for n-type or net negative for p-type) — wrong; fixed ion charge always balances the mobile carriers, leaving the crystal neutral.
• Swapping donor and acceptor: a pentavalent impurity is a donor giving n-type/electrons; a trivalent impurity is an acceptor giving p-type/holes. The valence count, not the element's name, is what to memorise.
• Forgetting that even a heavily n-type sample still has a small population of minority holes (and a p-type sample a small population of minority electrons), from ordinary thermal generation — doping changes the majority carrier concentration, it does not eliminate the minority carrier.
• Treating a hole as a genuine particle like an electron — a hole is a bookkeeping description of the apparent motion of a missing electron through the bonded electron sea; only the electron is a real particle, the hole is a convenient way to track the vacancy it left behind.

Ask the tutor: Want quick-fire practice classifying impurities as donors or acceptors, or a step-by-step walkthrough of the covalent-bond diagram for a new element?