# Carriers

 Density of States An Analogy The Fermi Function Where are they?

### The Parable of the Empty Seat

An analogy that can be applied to carriers in a semiconductor can be found in a football stadium.  No matter if anyone is playing or what day it is, the stadium has the same number of seats in a static arrangement.  This is the stadium's capacity, and we can call this the "Density of Seats".  Think of the rows of seats as being different energy levels.

Now imagine a game is being played and some people are in the stands.  Because people like to be close to the action, they tend to sit low in the stands.  The lower seats, the rows lower in energy, are more likely to be filled.  Electrons in a semiconductor are similar; they tend to "prefer" to occupy lower energy electronic states.  Even so, one can always find some electrons at the higher energy states just as you will find some people choose seats in the higher rows of the stadium, even when some closer seats are available.

Once the game has started and the people are settled, take a snapshot of where people are seated.  Wait 15 minutes and take another.  Chances are some people will have moved around since the first snapshot.  Some may have left, some may have come in, some may have just switched seats.

Now imagine that the stadium was sold out so all the seats are filled.  Can anyone move?  The answer is no, because in order to move, there must be an empty seat somewhere in the stadium.  If a few people didn't show up, someone would be able to move to one of those seats, leaving an empty seat behind.  Someone can move into this empty seat and leave another empty seat behind, and so on and so forth.  This also happens with electrons.  Electrons need an empty electronic state nearby in order to be able to move.

Two electrons (people) cannot occupy the same state (seat) at the same time.  For an electron to move, it must have an empty state nearby.  Otherwise it will not move, no matter how much energy is applied.  When a semiconductor is at zero Kelvin, all the states in the valence band are full and all the states in the conduction band are empty.  There is no conduction (movement) of electrons in the conduction band because it doesn't have any electrons.  The valence band is full of electrons, but there aren't any empty states to where an electron can move.  Therefore, at zero Kelvin a semiconductor does not conduct electricity.

If this analogy really modeled a semiconductor in equilibrium:

1. The number of people in the stadium would essentially remain unchanged, but the people would continuously be moving.
2. The average number of people would be a function of temperature, their number increasing with increasing temperature (more people show up on a warm fall day than a cold winter day).
3. Even though people are continually moving, their distribution in the rows would always be similar and would be determinable mathematically with something like the Fermi function.

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