Fermi Dirac statistics

- Assumption not all allowed states are filled

- Electrons are indistinguishable.

- Each state one electron (Pauli exclusion principle)

- Total no. of electrons = fixed

= constant, total energy

Electrons are viewed as indistinguishable "balls" which are placed in allowed state "boxes".

Each box one single ball.

Total energy of system is fixed.

Balls are grouped in rows energy level

no. of boxes in each energy level no. of states of allowed electronic states at a given energy. Fermi function

E F = Fermi energy

k = Boltzman's constant =

For closely spaced levels

Continuous variable E

Fig 6.3

f(E) occupancy factor for electrons at energy E.

g(E) density of states at energy E

1-f(E) occupancy factor for holes at energy E

Distribution of Electron

The distribution of electrons in conduction band

No. of electrons is CB with energy between E and + dE (E > E C )

The distribution of holes in the valence band =

The total carrier concentration in a band: conduction,

The total no. of holes in valence band,

If for CB and for VB

[(Fermi Dirac integral of order 1/2)]

Where, N c = effective density of conduction band states, N v = effective density of valence states.

At room temperature (300k) we have:

Properties of function

gamma function.

here

with a max error of

Thus

From Fig 6 (d) is closely approximated by exp

Thus

Fermi level lies the band gap more than 3kT from either band edge Semiconductor is said to be nondegenerate.