Question

Assume Silicon (bandgap 1.12 eV) at room temperature (300 K) with the Fermi level located exactly in the middle of the bandgap. Answer the following questions.

a) What is the probability that a state located at the bottom of the conduction band is filled?

b) What is the probability that a state located at the top of the valence band is empty?

Answer 1

Concept and theory

Electrons in solid obey Fermi-Dirac statistics.In this type of statistics , one must consider the indistinguishabilty of the electrons, their wave nature, and the Pauli exclusion principle.

The distribution of electrons over a range of allowed energy level at thermal equilibrium is

The fermi distribution function is symmetric about Ef ( fermi level) at all temperature

When we apply F-D statistics to semiconductor. We must recall that F(E) is the probabilty of occupancy of an available state at E. If there is no available state at E.( e.g in the band gap of a semiconductor ), there is no possibility of finding an electron there .

For instrinsic semiconductor like silicon concentration of electrons of holes in the valence band is equal to the concentration of electrons in valence band. Therefore the fermi level lies at the middle of band gap in intrinsic semiconductor.

Answer to part (a) and part (b).

As in intrinsic semiconductor concentration of holes in valence band and electrons are conduction band is equal. Since F(E) is symmetrical about Ef , the electron probabilty tail of F(E) extending into the conduction band is symmetrical with holes probabilty tail (1-f(E)) in the valence band.

This implies that there is an equal chance of finding the electron at the conduction band edge as finding an empty state( hole) at valence band edge.

F( Ec)= 1-F(Ev)

(a) Total probablity sould be 1 so probabilty of finding of electron in the bottom of conduction band F( Ec) = 1/2.

(b) Total probabilty of finding an empty place( hole) in the upper level of valence band ( 1-F(Ev) =1/2