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Resistance, material and temperature

Electron Fermi velocity and lattice phonons

Free electrons in a metal come with a range of velocities up to around the 'Fermi' velocity.  The Fermi velocity is typically a million metres per second or so.  It's different for different metals and depends on the way the electrons are arranged around an atom.

The ions that make up the lattice of a metal vibrate.  This vibration can be quite ordered and wave-like because the ions are fairly neatly arranged and are indirectly connected to each other.  These lattice waves are called phonons and they're described by quantum theory.

It's the relationship between the free electrons and the phonons that plays an important part in electrical resistance.

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Scattering of free electrons by phonons

When a free electron interacts with a phonon it gets scattered.  A typical free electron might be scattered 50 trillion times a second.

This means that whatever direction it was going in to begin with, it ends up moving off in some other direction.  When a metal isn't part of an electric circuit the electrons move in every direction equally.  Scattering events change these directions all the time.

Electrons get scattered more often the faster they move because they come across phonons more frequently.  Certain frequencies of phonons will also tend to scatter electrons more readily.

Resistance and scattering of free electrons

When a piece of metal is part of an electric circuit then the free electrons move very quickly in all directions with lots of velocities up to around the Fermi velocity.  The electric field superimposes a very tiny extra drift velocity, perhaps 1 millimetre a minute, in the direction of electron flow on top of this random motion.

However scattering events tend to disrupt this drift.  The more often free electrons are scattered the harder it is for them to progress round the circuit and the higher the resistance of the piece of metal.

Resistance and type of metal

The Fermi velocity is different for different metals.  Silver, copper and and aluminium have Fermi velocities of 1.39, 1.57 and 2.03 million metres per second, respectively.  The best conductor in this list is silver, then copper then aluminium.  So low Fermi velocity means less frequent scattering and lower resistance.

However this is by no means the whole story.  Potassium has a very low Fermi velocity but is not a brilliant conductor.  Just as important is how easy it is for electrons to become free in the first place.

In addition, different metals have different characteristic phonon frequencies and this also has to be taken into account.  The crystal lattice structure of a metal tends to have faults in it and these faults also cause scattering.

On the whole it's messy and you can see why it's not easy to predict which metals will be good conductors from first principles.

Resistance and temperature

The higher the temperature the faster the free electrons in a metal move.  This means they tend to come across phonons more often and be scattered more frequently.  This is why increasing temperature increases the resistance of a metal wire.

back to Lesson 7: Resistance and Ohm's Law