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DIFFUSION CURRENTS

In the previous chapter we saw that apart from the drift current (which is the result of acceleration by the electric field and collisions) there is also the diffusion current. How does this derive from the Boltzmann equation? It derives from the V£f that we have put to zero. Before proceeding further, the reader is advised to reread the caution suggested by the remarks prior to equation 4.17. Since EF enters the probability of occupation of a state, it should be obvious that any spatial variation of EF implies a corresponding variation of the electron density n(r). However, as previously indicated, given that EF characterizes a system at equilibrium and is unique, it is highly debatable that such concepts rest on firm ground. These issues are discussed in the next chapter and will lead us to the methodology of quantum conduction. However, on the assumption of slow variation of the above quantities within large systems, we can cautiously use such concepts.

Taking into account the term VEf and assuming that in equation 4.23 the magnetic field В = 0 and VT = 0, we find that we can follow exactly the same steps of the previous section and write the general expression for the current density, equation 4.28, in the form

where

and и is a unit vector in the direction of the electric field £. Taking into account equation 4.31, the above equation can be put into the form

so we have come to the desired result that in isothermal conditions (VT = 0) we have two types of current densities, a drift current and a diffusion current density a/eVr£f.

All that remains now is to put the second term in equation 4.38 into the well-known form of Fick’s law. Again let us assume a metal with a parabolic conduction band of effective mass m In this case we have from chapter 2, equation 2.72, that

from which we can deduce that

so that 4.38 can be written as where

If, on the other hand, we have a semiconductor, from the exponential relation between electron density and Fermi level we get

where К = Boltzmann’s constant. Then equation 4.40a remains valid but then

Our analysis so far, which was based on the classical Boltzmann equation, has put on a firm basis all of the results of the previous chapter which were based on the mechanics of classical particles without regard to the existence of Bloch states and к space. The Boltzmann equation is also the basis for understanding many solid state phenomena with technological importance such as the Peltier effect. Since our final goal is the quantum analysis of nanoelectronic devices, we leave the description of some of these effects as problems at the end of the chapter. We analyze however two such important effects, the application of a magnetic field or Hall effect and the application of a temperature gradient or the Seebeck effect. Before we do that, we obtain a general expression for the current density in the presence of both an electric field and a thermal gradient.

 
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