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My question is very naive. I don't understand why there are no holes in Sommerfeld's model of free electrons? Whenever an electron is excited above the Fermi level $E_F=\mu(0)$, there should be a hole. But while calculating the electronic specific heat etc., the hole contribution is not taken into account. In fact, the hole picture is missing in Sommerfeld's model as discussed in books. Why is that? Do I misunderstand the concept of holes?

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    $\begingroup$ In the case of metals, the 'hole' is in the same band as the electron. There is no change in the number of occupied vs unoccupied states in the band, just twiddling of which particular states are occupied. Thus, there is no concept of thermodynamic equilibrium (aka detailed balance) between level occupation in different bands. $\endgroup$ – Jon Custer Jan 10 '17 at 18:15
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First we have to clear up a misconception about holes. The "holes" in the hole picture aren't new and independent objects, they're just a different way of labeling existing occupancy states. We simply say that a state occupied by an electron is not occupied by a hole, and vice versa.

As a result, within each band of states, you can use the hole picture or the electron picture, but not both. If you did count both, many quantities would be off by a factor of $2$, since you'd count each physical state twice.

Next, you could ask why we don't ever use the hole picture in Sommerfeld's model. In solids with nontrivial band structure, the hole picture can be more intuitive for a particular band, if the states in that band near the Fermi energy have negative effective mass. Then switching to the hole picture within this band makes the effective mass positive, which is easier to visualize.

However, in Sommerfeld's free electron theory, this never happens. There is only one band, and its states have constant positive effective mass. So there's no reason to switch to the hole picture, though you could if you wanted to.

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