Intrinsic semiconductors: electrons and holes
In pure silicon or germanium, heat breaks a few covalent bonds, making equal numbers of free electrons and holes (nₑ = nₕ = nᵢ). Both carry current.
The structure of silicon and germanium
Silicon (Z = 14) and germanium (Z = 32) each have four valence electrons. In the crystal, every atom shares one electron with each of its four neighbours, forming covalent bonds. At 0 K every bond is complete, so there are no free carriers and the crystal behaves like an insulator.
How electron-hole pairs form and carry current
- 1At room temperature, thermal energy breaks a few covalent bonds.
- 2Each freed electron becomes a free (conduction) electron: it moves up into the conduction band.
- 3The vacancy left in the bond is a hole. It behaves like a charge +e.
- 4An electron from a neighbouring bond can jump into the hole. The hole then appears at the place the electron left, so the hole moves the opposite way.
- 5In an electric field, free electrons drift towards the positive terminal and holes towards the negative terminal. Both make current in the same direction: I = Iₑ + Iₕ.
- 6Free electrons also meet holes and fill them (recombination). At equilibrium, pairs are made and lost at the same rate, so nₑ = nₕ = nᵢ.
How to draw the lattice and band diagrams
- Lattice: draw Si atoms as circles in a square grid. Join each to its four neighbours by double lines (two shared electrons) and write +4 in each circle for the core.
- Show one broken bond: an open circle labelled hole (+e), and a dot with an arrow away from it labelled free electron.
- Band diagram at T = 0 K: a filled valence band, an empty conduction band and the gap Eg.
- Band diagram at T > 0 K: a few dots (electrons) in the conduction band and the same number of open circles (holes) in the valence band.
Effect of temperature
When the temperature rises, more bonds break, so nᵢ increases quickly. More carriers mean higher conductivity and lower resistance. So a semiconductor has a negative temperature coefficient of resistance: its resistance falls as it gets hotter. This is the opposite of a metal.
Worked example: how few carriers there are
Pure silicon at room temperature has nᵢ = 1.5 × 10¹⁶ m⁻³. Silicon has about 5 × 10²⁸ atoms m⁻³. (a) How many free electrons and holes are there in 1 cm³ of silicon? (b) What fraction of the atoms has given a free electron?
- 1 cm³ = 10⁻⁶ m³.
- (a) Free electrons = 1.5 × 10¹⁶ m⁻³ × 10⁻⁶ m³ = 1.5 × 10¹⁰. Holes are the same: 1.5 × 10¹⁰.
- (b) Fraction = 1.5 × 10¹⁶ ÷ 5 × 10²⁸ = 3 × 10⁻¹³.
(a) 1.5 × 10¹⁰ electrons and 1.5 × 10¹⁰ holes; (b) only 3 atoms in every 10¹³, which is why pure silicon conducts poorly
Thermistors in digital thermometers and in the temperature sensor of an air conditioner use semiconductor material whose resistance falls as it warms up. The circuit reads the resistance and works out the temperature.
2 marks: “What is a hole in a semiconductor? How does it move?” A vacancy in a covalent bond left by an electron; it acts as charge +e. A neighbouring electron fills it, so the hole moves the opposite way, towards the negative terminal.
2 marks: “Why does the conductivity of an intrinsic semiconductor increase with temperature?” Heat breaks more covalent bonds, producing more electron-hole pairs. More carriers mean higher conductivity.
3 marks: “Explain how electron-hole pairs are produced in intrinsic silicon, with a two-dimensional lattice diagram and an energy band diagram at T > 0 K.” Thermal energy breaks bonds; each gives a free electron and a hole, so nₑ = nₕ. Lattice with one broken bond; band diagram with equal electrons in CB and holes in VB.
Pure means pairs: in an intrinsic semiconductor every free electron leaves one hole, so nₑ = nₕ.