Holes, Doping & Conductivity
Two kinds of carrier, and how to choose one.
Lesson at a glanceShow the key points
- Holes act as +e carriers; intrinsic: nₑ = nₕ.
- Donors (P) → n-type; acceptors (Al, B) → p-type.
- σ = e(nₑμₑ + nₕμₕ); nₑnₕ ≈ constant.
- Carriers ∝ T^(3/2)e^(−ΔE/2kT): negative temperature coefficient.
A semiconductor carries current two ways: electrons in the conduction band, and holes in the valence band. Add one impurity atom in a million and you can multiply its conductivity a million times, and choose which carrier does most of the work.
Bonds, broken bonds and holes
Silicon (Z = 14, outer shell 3s² 3p²) has four outer electrons. In the crystal, each atom shares one electron with each of four neighbours, forming four covalent bonds of two electrons each. All these bonding electrons are in the valence band.
A violent enough collision can tear one electron out of its bond. It moves away in a large orbit, hopping from atom to atom: it is now in the conduction band. The broken bond is left one electron short. A neighbouring bond electron can slip into that gap with very little energy (all bond electrons have about the same energy), which breaks its bond instead. The vacancy has moved one step, in the opposite direction to the electron.
In pure silicon every conduction electron leaves one hole, so their numbers are equal. Both drift in a field (electrons against it, holes along it) and both carry current. A pure semiconductor like this is called intrinsic. Germanium has the same crystal structure and behaves the same way.
Doping: n-type and p-type
Replace a few silicon atoms with phosphorus (five outer electrons). Four go into bonds; the fifth circles the phosphorus ion in a wide orbit (about 3 nm). Its energy lies only about 0.045 eV below the conduction band, so heat frees nearly all of them. Electrons become the majority carriers.
Replace a few with aluminium (three outer electrons). One bond is left incomplete. A valence electron needs only about 0.057 eV to move in and complete it, which leaves a hole behind in the valence band. Holes become the majority carriers.
Adding an impurity on purpose is called doping, and a doped semiconductor is extrinsic. The effect is dramatic: phosphorus at one part in a million raises silicon’s conductivity by about a million. On a band diagram, donors add impurity levels just below the conduction band, and acceptors add them just above the valence band.
How many carriers?
Pairs are created by heat and destroyed when an electron falls back into a hole; the two rates balance at a steady density. For pure silicon at 300 K that is about 7 × 10¹⁵ pairs per m³, against about 9 × 10²⁸ free electrons per m³ in copper. When a semiconductor is doped, the product of the electron and hole densities stays roughly constant, so adding many electrons suppresses the holes.
| Material | Electrons per m³ | Holes per m³ |
|---|---|---|
| Copper (metal) | ≈ 9 × 10²⁸ | — |
| Pure silicon | ≈ 7 × 10¹⁵ | ≈ 7 × 10¹⁵ |
| Silicon + phosphorus (1 in 10⁶) | ≈ 5 × 10²² | ≈ 1 × 10⁹ |
| Silicon + aluminium (1 in 10⁶) | ≈ 1 × 10⁹ | ≈ 5 × 10²² |
Conductivity
For a metal, j = nevd; writing the mobility μ = vd/E (drift speed per unit field) gives σ = neμ. A semiconductor has two kinds of carrier with different mobilities, so their contributions add:
An n-type sample has 6 × 10¹⁹ conduction electrons and 2 × 10¹⁸ holes per m³. The mobilities are 0.39 m² V⁻¹ s⁻¹ for electrons and 0.19 m² V⁻¹ s⁻¹ for holes. Find its resistivity.
Temperature
The number of thermally created pairs grows roughly as T3/2e−ΔE/2kT, where ΔE is the gap. Warmer also means more collisions and lower mobility, but the growth in carriers wins by far. So a semiconductor’s resistivity falls as it heats: its temperature coefficient is negative (silicon averages about −0.07 K⁻¹), the opposite of a metal.
A doped crystal has impurity levels 45 meV below the conduction band. (a) Is it n-type or p-type? (b) At what temperature does kT equal 45 meV?
A semiconductor’s band gap equals the energy of a 620 nm photon. Find (a) the minimum energy to make an electron–hole pair, and (b) how many times larger it is than kT at 300 K.
In a p-type crystal the acceptor levels lie 45 meV above the valence band. What is the longest wavelength of light that can create a hole?
Silicon’s gap is 1.1 eV. Taking the number of carriers proportional to e−ΔE/2kT, by what percentage does it grow from 300 K to 310 K?
Pure silicon at 300 K has 7 × 10¹⁵ electron–hole pairs per m³ and 5 × 10²⁸ atoms per m³. Arsenic is added at 1 atom per 10⁸ silicon atoms, and every arsenic atom donates an electron. By what factor does the number of carriers rise?
- 1A broken bond gives a conduction electron and a hole; holes behave as +e carriers.
- 2Intrinsic: electrons = holes. Doping with pentavalent donors gives n-type; with trivalent acceptors, p-type.
- 3Impurity levels sit just below the conduction band (donors) or just above the valence band (acceptors).
- 4σ = e(nₑμₑ + nₕμₕ); nₑnₕ stays about constant under doping.
- 5Carriers ∝ T^(3/2) e^(−ΔE/2kT): semiconductor resistivity falls with temperature.
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