Posted by Steve Simon on February 27, 2013, 9:37 pm, in reply to "Holes, free electrons and conductivity"
Very good question. Yes, the number of electrons in the conduction band will equal the number of holes in the valence band. Naively you might think that this would result in a perfect cancellation and you would get no Hall effect at all. However, it turns out that it is not quite so simple. The cancellation is only perfect if the mobility of electrons is the same as the mobility for holes. (Mobility is e tau/m, closely related to the conductivity up to a factor of e and density). Generally the mobilities for the two species will not be the same so there will not be cancellation. (There is an exercise in the book, 17.9, which asks you to work this out in detail, but it is a * problem and it doesn't give you any hint). Here is a hint to explain how you don't get cancellation. First, consider the simple drude conductivity for electrons and holes seperately. sigma_{electron} = n e^2 tau_e / m_e sigma_{hole} = p e^2 tau_h / m_h where n and p are the electron and hole densities (assumed to be equal to each other here), tau_e and tau_h are the electron and hole scattering times, and m_e and m_h are the electron and hole effective masses. Now suppose you apply a voltage across the sample. Some of the current is carried by each species, but the current carried by the electron versus the hole are not the same. Similarly, if you force a current into the sample, you will discover that some of this will be carried by the electrons, and some will be carried by the holes, but it will not be divided equally. In particular, let us suppose that the conductivity (mobility) of the holes is much greater than that of the electrons. Then, when you push a current through a system, almost all the current will be carried by the holes. As a result, when you apply a magnetic field, you will discover a positive hall coefficient. If on the other hand, the mobility of the electrons were greater, then you would discover a negative hall coefficient. Does this make sense? 
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