# Why there is no hole in Sommerfeld's model of free electrons?

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?

• 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. – Jon Custer Jan 10 '17 at 18:15

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.