Any undoped or doped semiconductor has two types of free carriers, electrons and holes. Under equilibrium conditions, i. e. without external stimuli such as light or current, the law of mass action teaches that the product of the electron and hole concentrations is, at a given temperature, a constant, i. e.

n0 P0 = П2

where n0 and p0 are the equilibrium electron and hole concentrations and ni is the intrinsic carrier concentration. The validity of the law of mass action is limited to non-degenerately doped semiconductors (see, for example, Schubert, 1993).

Excess carriers in semiconductors can be generated either by absorption of light or by an injection current. The total carrier concentration is then given by the sum of equilibrium and excess carrier concentrations, i. e.

n = щ + An and p = P0 + A p (2.2)

where An and Ap are the excess electron and hole concentrations, respectively.

Next, we consider recombination of carriers. The band diagram of a semiconductor with
electrons and holes is shown in Fig. 2.1. We are interested in the rate at which the carrier concentration decreases and denote the recombination rate as R. Consider a free electron in the conduction band. The probability that the electron recombines with a hole is proportional to the hole concentration, that is, R <x p. The number of recombination events will also be proportional to the concentration of electrons, as indicated in Fig. 2.1. Thus the recombination rate is proportional to the product of electron and hole concentrations, that is, R x n p. Using a proportionality constant, the recombination rate per unit time per unit volume can be written as

This equation is the bimolecular rate equation and the proportionality constant B is called the bimolecular recombination coefficient. It has typical values of 10-11—10-9 cm3/s for direct-gap III-V semiconductors. The bimolecular recombination coefficient will be calculated in a subsequent section using the van Roosbroeck-Shockley model.

Fig. 2.1. Illustration of elec - tron-hole recombination. The number of recombination events per unit time per unit volume is proportional to the product of electron and hole concentra­tions, i. e. Rocnp.