Energy bands, and intrinsic and extrinsic semiconductors
The energy gap between the valence and conduction bands sorts solids into conductors, semiconductors and insulators. Doping turns a pure semiconductor into n-type or p-type.
How energy bands form
In a single atom, electrons sit in sharp, separate energy levels. In a crystal, atoms are packed very close, about 10²³ atoms in every cm³. Their levels would overlap, which Pauli's exclusion principle does not allow, so each level splits. N atoms turn one level into 2N levels. These are so close (about 10⁻²³ eV apart) that together they act as one continuous energy band. The band made from the filled valence levels is the valence band (VB). The band made from the empty higher levels is the conduction band (CB). Electrons in the CB are free to move and carry current.
Classifying solids by their bands
| Property | Conductor | Semiconductor | Insulator |
|---|---|---|---|
| Bands | CB and VB overlap | VB full and CB empty at 0 K, small gap | VB full and CB empty, large gap |
| Energy gap ΔE | Zero | ≤ 3 eV (Si about 1.1 eV) | > 3 eV (diamond about 5.4 eV) |
| Free electrons at room temperature | Very many | A few | Almost none |
| Examples | Copper, aluminium, silver | Silicon, germanium | Diamond, glass, rubber |
| When heated | Resistance rises | Resistance falls | Stays a very poor conductor |
How to draw the three band diagrams
- Draw three panels side by side. Put an upward arrow labelled electron energy on the left.
- Conductor: draw two rectangles that overlap. Label the lower one VB and the upper one CB, and write ΔE = 0.
- Semiconductor: a shaded VB, a narrow gap labelled ΔE ≤ 3 eV, and an empty CB above it.
- Insulator: a shaded VB, a wide gap labelled ΔE > 3 eV, and an empty CB.
- Write conductor, semiconductor and insulator under the three panels.
Intrinsic semiconductor: how electrons and holes form
An **intrinsic semiconductor** is a pure one, such as pure silicon or germanium. Each atom has four valence electrons. It shares one with each of four neighbours in **covalent bonds**.
- 1At 0 K, every electron is held in a bond. There are no free carriers, so the crystal is an insulator.
- 2At room temperature, heat breaks a few bonds. Each freed electron moves up into the conduction band.
- 3The empty place left in the bond is a hole. It acts like a positive charge, equal in size to an electron's charge.
- 4A neighbouring electron can jump into the hole. The hole then appears where that electron was, so the hole moves the opposite way.
- 5Electrons and holes are always made in pairs, so their numbers are equal. Conventional current is taken in the direction the holes move.
- 6★Heating breaks more bonds, so conductivity rises. A semiconductor has a negative temperature coefficient of resistance.
★ key step
Extrinsic semiconductors: n-type and p-type
Doping means replacing a tiny, measured number of atoms, about one in 10⁸, with impurity atoms. The doped crystal is an extrinsic semiconductor, with much lower resistivity. Both types stay electrically neutral: each dopant atom that gives up a carrier is left as a fixed ion of the opposite charge, so the total charge is zero.
Worked example: how much doping helps
Silicon has about 5 × 10²⁸ atoms m⁻³. When pure, it has 1.5 × 10¹⁶ free electrons m⁻³ at 300 K. It is doped with phosphorus at one atom in 10⁸. Find the number of donor electrons per m³ and compare it with the pure value.
- Each phosphorus atom gives one free electron.
- Donor atoms = 5 × 10²⁸ ÷ 10⁸ = 5 × 10²⁰ m⁻³, so there are about 5 × 10²⁰ free electrons m⁻³.
- Ratio = 5 × 10²⁰ ÷ 1.5 × 10¹⁶ ≈ 3.3 × 10⁴.
About 5 × 10²⁰ electrons m⁻³: roughly 33,000 times more than in pure silicon, so the doped crystal conducts far better
Every phone and laptop chip is doped silicon: traces of boron and phosphorus make its p-type and n-type regions. A thermistor in a digital thermometer works because a semiconductor's resistance falls as it warms.
5 marks: “Explain how energy bands are formed in a solid. Using band diagrams, distinguish between a conductor, a semiconductor and an insulator.” Atoms close together: levels split (Pauli's principle); N atoms give 2N very close levels, a band. VB from filled valence levels, CB from empty levels. Conductor: bands overlap, ΔE = 0. Semiconductor: ΔE ≤ 3 eV (Si 1.1 eV). Insulator: ΔE > 3 eV. Three labelled diagrams.
3 marks: “Distinguish between n-type and p-type semiconductors. Why is each of them electrically neutral?” n-type: group V donor (P, As), fifth electron free, electrons are majority carriers. p-type: group III acceptor (B, In), a hole is made, holes are majority carriers. Neutral because the dopant atoms were neutral: each one that gives an electron or a hole is left as a fixed ion of the opposite charge.
2 marks: “What is a hole? Why does the resistance of a pure semiconductor fall when it is heated?” A vacancy left in a covalent bond by a freed electron; it acts like a positive charge. Heat breaks more bonds, making more electron-hole pairs, so conductivity rises and resistance falls.
Zero, small, large: conductor no gap, semiconductor up to 3 eV, insulator above 3 eV. Five gives n, three gives p.