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STATISTICS OF ELECTRON OCCUPANCY, THE PAULI PRINCIPLE AND THE FERMI – DIRAC DISTRIBUTION

AND THE FERMI - DIRAC DISTRIBUTION_

If we put N electrons in a quantum well, or in any system for that matter, how are they going to be distributed among the energy levels Ekx,ky,kzy. The answer to this question will come in two stages: first by analyzing the so-called Pauli principle and then by analyzing the effect of the temperature T. We begin with the first. There are two kinds of particles in nature, fermions and bosons. Electrons are fermions which have the property that no two electrons in the same system can have the same quantum numbers. The quantum numbers in the 3-dimensional well are the kx, ky, kz. In other systems, atoms for example, they may be different, but still each Ч'(г) will be characterized by three indices corresponding to the three dimensions of the system.

These three indices are not enough to characterize the electron—a fourth related to spin is required. We assume that the reader is familiar with the concept of spin. If not, an introduction is given in Appendix A. For the purposes of this section it is enough to know the following: the electron is assumed to have an inherent magnetic moment Цв = eh /(2m), with m the mass of an electron which can point in nominally “up” or “down” directions. Therefore, given a state with quantum numbers kxt ky, kz, we can put two electrons in it, one with “up” and the other with “down” spin. This does not violate the Pauli principle because they will differ in a fourth quantum number, a magnetic one which determines the direction of its magnetic moment. This is quantitatively discussed in Appendix A.

So at T = 0 it is easy to see what the distribution of the electrons among the energy levelswill be—they will fill in ascending order (i.e. increasing energy) the levels E[kx>ky,kz) in pairs of spin “up” and “down” until N72 levels have been filled up. If the system contains many electrons with their energy levels very close to each other, the highest occupied state is called “Fermi energy” and is usually symbolized as EF.

But what is the effect of a finite temperature T > 0? We know that electrons, like any particle, can gain energy KT from the environment, (K is Boltzmann’s constant), and jump to a higher empty or singly occupied state. Then the Pauli principle must still be observed. What is the rule for such transitions in an equilibrium state? It is given by the Fermi-Dirac probability of occupation of any state P(E)

where EF does not necessarily have the previous meaning of the highest occupied energy level but it is just a variable to be determined by the total number of electrons, as we will find out.

The graph of the Fermi-Dirac distribution function is shown in figure 1.6. We observe that at T = 0 all the states with E < EF have P(E)= 1 and all the states with E > EF have P(E) = 0. So why does EF does not have the meaning it has in the paragraph above? The answer is that EF is not necessarily an allowed state. In metals, where there is a continuum of states, this distinction hardly makes any difference, but it does in semiconductors.

A careful examination of figure 1.6 shows that at T>0 all the electrons that occupy the states with E> EF have come from the states a few KT below EF. In fact the whole

The Fermi-Dirac distribution function/(£) which gives the probability of occupation of a state

FIGURE 1.6 The Fermi-Dirac distribution function/(£) which gives the probability of occupation of a state: /(£) is 1 for £ < EF and zero for £ > EF at T = 0.

distribution P{E) is unaltered at E «: Ее and E » All the changes have occurred a few KT below and above Ep. It is important to remember this as we shall see that all conduction mechanisms occur when electrons change state near Ep.

 
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