2.2.4 Electrons and holes in semiconductors


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2.2.4 Electrons and holes in semiconductors

As pointed out before, semiconductors distinguish themselves from metals and insulators by the fact that they contain an "almost-empty" conduction band and an "almost-full" valence band. This also means that we will have to deal with the transport of carriers in both bands.

To facilitate the discussion of the transport in the "almost-full" valence band we will introduce the concept of holes in a semiconductor. It is important for the reader to understand that one could deal with only electrons (since these are the only real particles available in a semiconductor) if one is willing to keep track of all the electrons in the "almost-full" valence band.

The concepts of holes is introduced based on the notion that it is a whole lot easier to keep track of the missing particles in an "almost-full" band, rather than keeping track of the actual electrons in that band. We will now first explain the concept of a hole and then point out how the hole concept simplifies the analysis.

Holes are missing electrons. They behave as particles with the same properties as the electrons would have occupying the same states except that they carry a positive charge. This definition is illustrated further with the figure below which presents the simplified energy band diagram in the presence of an electric field.


band1.gif

A uniform electric field is assumed which causes a constant gradient of the conduction and valence band edges as well as a constant gradient of the vacuum level. The gradient of the vacuum level requires some further explaination since the vacuum level is associated with the potential energy of the electrons outside the semiconductor. However the gradient of the vacuum level represents the electric field within the semiconductor.

The electrons in the conduction band are negatively charged particles which therefore move in a direction which opposes the direction of the field. Electrons therefore move down hill in the conduction band. Electrons in the valence band also move in the same direction. The total current due to the electrons in the valence band can therefore be written as:

where V is the volume of the semiconductor, q is the electronic charge and v is the electron velocity. The sum is taken over all occupied or filled states in the valence band. This expression can be reformulated by first taking the sum over all the states in the valence band and subtracting the current due to the electrons which are actually missing in the valence band. This last term therefore represents the sum taken over all the empty states in the valence band, or: The sum over all the states in the valence band has to equal zero since electrons in a completely filled band do not contribute to current, while the remaining term can be written as: which states that the current is due to positively charged particles associated with the empty states in the valence band. We call these particles holes. Keep in mind that there is no real particle associated with a hole, but rather that the combined behavior of all the electrons which occupy states in the valence band is the same as that of positively charge particles associated with the unoccupied states.

The reason the concept of holes simplifies the analysis is that the density of states function of a whole band can be rather complex. However it can be dramatically simplified if only states close to the band edge need to be considered.

As illustrated by the above figure, the holes move in the direction of the field (since they are positively charged particles). They move upward in the energy band diagram similar to air bubbles in a tube filled with water which is closed on each end.


2.2.3 2.2.5

Bart J. Van Zeghbroeck, 1996, 1997