42 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS
spin. The spin of particles can take a value of 0 , 1 / 2 ,, 3 / 2 ,etc. Particles which have inte-
gral spins (in units of) are called bosons, while those that have half-integral spins are called
fermions.
According to thermodynamics, a system with a large number of particles can be described by
macroscopic properties such as temperature, pressure, volume, etc. Under equilibrium condi-
tions (no exchange of net energy with other systems) the system is described by a distribution
function, which gives us the occupation number for any energy level. To find this occupation
we have to minimize the free energyFof the system subject to any constraints from quantum
mechanics (such as the Pauli exclusion principle). The following distribution functions are ob-
tained for equilibrium:
- For fermions such as electrons
f(E)=1
exp[
E−EF
kBT]
+1
Heref(E)is the occupation function;EFis the Fermi energy and its value depends upon particle
density).
In classical physics the occupation function for electrons is
f(E)=1
exp(
E−EF
kBT) (2.4.1)
Note that ifE−EFkBT; i.e.,f(E) 1 , the classical function approaches the quantum
Fermi distribution function.
For completeness we note the distributed function for bosons as well.
- Massless bosons (like photons)
f(E)=1
exp(
E
kBT)
− 1
(2.4.2)
- Bosons with mass (this applies to electron pairs that occur in superconductors)
f(E)=1
exp(
E−μ
kBT)
− 1
(2.4.3)
whereμis an energy determined from the particle density.
- There is a distribution function that proves to be useful in solid state devices. When solving
the Schrodinger equation we can get more than one solution with the same energy. This is the ̈
degeneracygdof a state. Consider a case where a state has a degeneracygiand can, in principle,
be occupied bygdelectrons. However, for dopants and defect levels, when one electron is placed
in the allowed state, the next one cannot be placed because of the Coulombic repulsion. This
happens for some states, such as those states associated with donors or acceptors, traps, etc.