(f36)
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:
(f37)
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:
(f38)
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.
Distribution functions
Introduction
The distribution or probability density functions describe the probability with which one can expect particles to occupy the available energy levels in a given system. While the actual derivation belongs in a course on statistical thermodynamics it is of interest to understand the initial assumptions of such derivations and therefore also the applicability of the results.
The derivation starts from the basic notion that any possible distribution of particles over the available energy levels has the same probability as any other possible distribution, which can be distinguished from the first one.
In addition, one takes into account the fact that the total number of particles as well as the total energy of the system has a specific value.
Third, one must acknowledge the different behavior of different particles. Only one Fermion can occupy a given energy level (as described by a unique set of quantum numbers including spin). The number of bosons occupying the same energy levels is unlimited. Fermions and Bosons all "look alike" i.e. they are indistinguishable. Maxwellian particles can be distinguished from each other.
The derivation then yields the most probable distribution of particles by using the Lagrange method of indeterminate constants. One of the Lagrange constants, namely the one associated with the average energy per particle in the distribution, turns out to be a more meaningful physical variable than the total energy. This variable is called the Fermi energy, EF.
An essential assumption in the derivation is that
one is dealing with a very large number of particles. This assumption enables to approximate the factorial terms using the Stirling approximation.
The resulting distributions do have some peculiar characteristics, which are hard to explain. First of all the fact that a probability of occupancy can be obtained independent of whether a particular energy level exists or not. It would seem more acceptable that the distribution function does depend on the density of available states, since it determines where particles can be in the first place.
The fact that the distribution function does not depend on the density of states is due to the assumption that a particular energy level is in thermal equilibrium with a large number of other particles. The nature of these particles does not need to be described further as long as their number is indeed very large. The independence of the density of states is very fortunate since it provides a single distribution function for a wide range of systems.
A plot of the three distribution functions, the Fermi-Dirac distribution, the Maxwell-Boltzmann distribution and the Bose-Einstein distribution is shown in the figure below, where the Fermi energy was set equal to zero.
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