Lecture 27 Electron Gas The model, the partition function and the Fermi function Thermodynamic functions at T = 0 Thermodynamic functions at T > 0

Number of free electrons and number of states The model assumes ideal gas of non-interacting electrons. The only constrain is that no two electrons can occupy the same quantum state, since electrons are fermions Number of states per unit volume between energy = 0 to energy = kT for an ideal gas This number is about 4 orders of magnitude larger than number of electrons is a typical metal. The reason is two fold. (i) The electron density is very high (~ electron per 10Å3) and (ii) electron mass is small, thus G(kT) is small Consequently a classical model of an ideal gas is not operational

Partition Function It is convenient to use grand canonical ensemble The total number of electrons is the sum of number of electrons in single electron quantum states (each quantum state has either zero or one electron) Since the electrons are not interacting

Partition Function - II In term of single energy levels Which is the same as For example with just 2 single electron states the top formula gives which is the same as the bottom formula

Partition Function - III For Nkmax =1 (fermions) Thus the partition function And the logarithm of the partition function

Average occupation number For Nkmax =1 (fermions) Which implies that the average number of electrons in a single electron state is This is called the Fermi function

Fermi function T = 0 T > 0 Nk Nk kT k k

Energy Density of states = 2 times density of state of an ideal gas where is the zero temperature Fermi level

Fermi level (T=0) Number of electrons From which

T > 0 Number of electrons Energy Integrating by parts and expanding around  =  gives

Energy Heat Capacity Nk k kT k The energy ~ T2 can be also seen in the following: Number of excited electrons ~ kT, and the energy of each excited electrons ~ kT, thus the total energy ~ kT x kT ~ (kT)2