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II: Theory of Conduction

Simple Classical Theory of Conduction

EXTERNAL VOLTAGES AND FERMI LEVELS

In this chapter we will give an account of the elementary or standard theory of classical conduction, i.e. the theory which is based on the continuity and the drift-diffusion equations. As such, the chapter will deal with the relation between applied voltages and observed currents in both semiconductor materials and devices. A note on notation is mandatory at this stage. In the previous chapters we used the capital letter V for the potential energy of an electron in the Schroedinger equation. In this and subsequent chapters, we would like to reserve this symbol for the quantity voltage or potential, so that the potential energy of an electron corresponding to an arbitrary potential V(x) will be U(x)= -eV(x) = -1.6xlO-19(C)V(x). We note that the symbol e is always considered positive.

Imagine a homogeneous piece of metal onto which an external voltage difference V is applied, see figure 3.1a. This voltage will create a potential energy U(x) inside the metal

and a ID electric field £, which, being equal to £ = ————, will give the rate of change of

e dx

the potential energy U(x). If the applied electric fields are small compared to the internal electric fields due to the crystalline potential energy Vcr, then we can always assume the following two statements:a) that in a small length element dx, U(x) may be considered constant, and b) at the same time within dx, we have enough atoms to form a crystal with all the properties of a crystal that we have discussed so far. Under these circumstances, we can think that the top of the band which is at EF follows U(x) under the action of the applied V, see figure 3.1b. Therefore under the action of V (or £) the electrons will slide down from the higher energies to the lower energies.

Furthermore, the Fermi level EF will no longer be constant but the difference in EF at the two ends of the metal piece will be equal to eV. If we call Ef and EF the Fermi levels at the two ends (left and right) of the metal then we will have Bending of the conduction band of a metal with distance x when an external voltage V is applied to it. The external voltage produces an additional internal potential energy U(x) in the metal

FIGURE 3.1 Bending of the conduction band of a metal with distance x when an external voltage V is applied to it. The external voltage produces an additional internal potential energy U(x) in the metal.

The situation described above has certain underlying assumptions over and above all those just described. First and foremost, the Fermi level is a concept or quantity valid only at equilibrium and pertaining to the whole of the system under consideration. It is a space- independent quantity, unique for any system. One can talk about the Fermi level variation in space if the deviation from equilibrium is weak and the system is large. In principle, one has to solve the Boltzmann equation which gives the probability of occupation of a state in terms of both the position vector r and the wavevector k. We will do that in the next chapter. Flowever, equation 3.1 is always regarded as valid in the sense that the applied electrodes are big enough to provide two thermodynamic reservoirs, one at each end, that keep the Fermi levels constant.

In semiconductors there are further complications due to the existence of two types of carriers, electrons and holes, and the sensitivity of their concentrations on the Fermilevel. The well-known relationship

does not hold away from equilibrium. As we will see, the majority carriers change marginally, while the minority carriers drastically. The concentration of the latter is calculated by means of the continuity equations. However as more of a mathematical device and less as a physical concept, we define the quasi Fermi levels for electrons and holes separately Ep„ and Epp, when usually electrons or holes are each majority carriers respectively, by extending the validity of equations 2.62 and 2.63, i.e. by writing

We emphasize again that equations 3.2a and 3.2b are mere definitions of Ep„, Epp. Essentially Ep„ and Epp constitute a device by which the functional form for n and p at equilibrium is preserved at non-equilibrium. Multiplying 3.2a by 3.2b we get

If we have thermodynamic equilibrium, then there is only one Fermi level, Ep„ = Epp and we get back to the well known law of mass action.

 
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