Summary of Chapter 3: Energy Bands and Charge Carriers in Semiconductors

Excess Carriers in Semiconductors

Note: These notes were written when ECE 4339 used another text, and so the notation for excess carriers is different from what we are using now. What we are calling n and p is here indicated as n and p.

Basic Ideas

When we shine light on a semiconductor (photoconductor, solar cell) or drive a current through it, the concentration of electrons and holes changes from its equilibrium value, and the semiconductor is no longer in equilibrium. There may be more carriers than at equilibrium, or fewer. The difference from equilibrium is called the excess, which can be positive or negative (fewer carriers than at equilibrium). We define the total carrier concentrations as

Terminology: A carrier (or charge carrier) is an electron or a hole.

At equilibrium, the generation rate g and recombination rate r for electrons and holes are equal, so there are no net carriers generated, on average. When there is more than the equilibrium value of electrons or holes, the probability of recombination (i.e., the recombination rate) increases; when there are fewer, the generation rate (the probability of generation) increases. In either case, the number of carriers tends to return to the equilibrium value.

Excess carriers, and in particular excess minority carriers, are crucial to the operation of both optical and electronic devices. Therefore, we need to know something about their properties and about the equations that govern them. This is the subject of Chapter 4.

Carrier Lifetime

If we have an excess of carriers, say electrons, in a semiconductor, we can ask how their number decreases with time due to increased recombination. This problem was set up for the simple case of an excess electron density generated by a flash of light on a semiconductor. When the light goes off, the excess electron density decreases exponentially in time.

Under the assumption that n = p, and that n and p are small compared to the majority carrier density, we have

The solution to this equation is

where r is a proportionality constant, n(t) is the excess electron density as a function of time, n is the excess electron density at t = 0 (we would have to know this or have some way to calculate it separately), and n = (rpo)-1 is the minority carrier lifetime.

Looking at these equations, and defining the net recombination rate R = r g, we have

Ideas:

We are interested in examining minority carriers, so we set up the equation for excess electrons, n, and stipulated that the material was p-type (po >> no).

The approximation n, p