Intrinsic semiconductors and charge carriers
Explain conduction in a pure semiconductor by electron–hole pairs, and why its resistivity falls as its temperature rises.
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ContinueA pure semiconductor, such as a crystal of silicon, is called an intrinsic semiconductor. It has no free electrons of its own: every carrier it has comes from an electron that has gained enough energy to jump the band gap. Each jump makes two charge carriers, not one, and the number of jumps rises steeply with temperature. That is why a thermistor’s resistance falls as it warms.
Electrons and holes
When an electron in the valence band gains energy at least equal to the band gap, it moves up into the conduction band and is free to move through the crystal. It leaves behind an empty state in the valence band, called a hole. The free electron and the hole are created together, as an electron–hole pair, so in an intrinsic semiconductor there are always equal numbers of conduction electrons and holes.
A hole moves because a neighbouring valence electron can move into it, which leaves a new hole where that electron was. In an electric field, valence electrons keep hopping into the hole from one side, so the hole moves the other way. It behaves exactly like a positive charge carrier, of charge + e, even though no positive particle moves.
In an electric field, then:
- conduction electrons drift opposite to the conventional current;
- holes drift in the same direction as the conventional current.
Both movements carry charge in the direction of the conventional current, so both add to the current.
Spot the mistake 1
Which way do the carriers drift?
Student's claim
Choose the drift directions
Show solution step by step
The electrons
Method
Conduction electrons drift to the left.Reason
Conventional current is the direction positive charge would move, and electrons are negative.Working
Answer
Electrons: leftThe holes
Method
Holes drift to the right.Reason
A hole acts as a positive charge, so it moves the way positive charge would.Working
Answer
Holes: rightWhy both add up
Method
Negative charge moving left and positive charge moving right are both a current to the right.Reason
The direction of a current depends on the sign of the charge as well as the direction it moves.Working
Answer
Both carriers add to the same current.
Common mistake
Generation and recombination
A conduction electron that meets a hole can fall back into it. This is recombination: the pair disappears, and the energy is given out. At a steady temperature, pairs are created and destroyed at the same rate, so the number of carriers stays constant. A higher temperature means more pairs are created each second, so the balance settles at a higher number of carriers.
Why resistivity falls as temperature rises
The number of electrons that can cross the band gap depends very strongly on temperature. Near room temperature, a rise of about 10 K roughly doubles the number of carriers in pure silicon. Heating also makes the lattice vibrate more, so carriers collide more often, which on its own would raise the resistivity. But the number of carriers rises so much faster that it wins.
Worked example 2
A thermistor in warm water
Problem
Show full solution
More electrons cross the gap
Method
At the higher temperature, more valence electrons gain enough energy to cross the band gap.Reason
The thermal energy of the electrons increases with temperature.Working
Answer
More electrons reach the conduction band.More carriers of both kinds
Method
Each electron that crosses leaves a hole, so the numbers of conduction electrons and holes both rise.Reason
Carriers are created in electron–hole pairs.Working
Answer
The number density of charge carriers increases.Resistance falls
Method
The resistivity falls, so the resistance falls.Reason
The rise in carrier number density outweighs the extra collisions with the vibrating lattice.Working
Answer
More carriers for the same p.d. give a larger current.
Quick check
Check your understanding 1 Germanium or silicon?
Germanium has a band gap of 0.67 eV and silicon has a band gap of 1.1 eV. Which pure crystal has the lower resistivity at room temperature? Explain.
Show answer
Germanium. With a smaller gap, many more electrons have enough thermal energy to cross it, so pure germanium has a much higher number density of electrons and holes at the same temperature.
Check your understanding 2 A pure semiconductor near absolute zero
Explain why a pure semiconductor behaves almost like an insulator at very low temperatures.
Show answer
At very low temperatures, almost no electrons have enough energy to cross the band gap. The valence band stays full and the conduction band stays empty, so there are almost no electrons or holes to carry a current.
Common mistakes
- Calling a hole a proton or a positive particle. It is an empty state that behaves like a positive charge.
- Saying holes move because positive particles move. A hole moves because valence electrons move into it, one after another.
- Saying electrons and holes drift the same way in a field.
- Explaining the fall in resistivity by “more energy” without saying that more charge carriers are created.
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