Question

You are given an intrinsic silicon at “some” temperature. It is known that the Fermi-Dirac probability for electrons at the conduction band edge is Fe(EC)=1∗10−5. Answer the following questions:

Hint: Even though it looks like a calculation question, it is not. You should be able to “see” the answer directly even without using the formula of the Fermi-Dirac function.

1.What is the Fermi-Dirac probability for electrons at the valence band edge Fe(EV)?

2.What is the Fermi-Dirac probability for holes at the valence band edge Fh(EV)?

3.What is the Fermi-Dirac probability for electrons at the intrinsic Fermi level Fe(Ei)?

4.When temperature increases, state whether Fe(EC), Fe(EV) and Fe(Ei) will increase, decrease or remain unchanged?

a.Fe(EC)

i.increases ii. decreases iii.remains unchanged .

b.Fe(EV)
i.increases ii. decreases iii.remains unchanged

c. Fe(Ei)
i.increases ii. decreases iii.remains unchanged  

5.Doping is introduced to the silicon so that the Fermi-Dirac probability for electrons at the conduction band edge becomes Fe(EC)=1∗10−3 at the initial temperature. Is the silicon after doping n-type or p-type?

i. n-type    ii. p-type

6.Under the new condition with the introduced doping, will the Fermi-Dirac probability for electrons at the valence band edge Fe(EV) and intrinsic Fermi0-level Fe(Ei) increase, decrease or remain unchanged compared with the initial values?
a.Fe(EV)
increases  decreases remains unchanged   

b.Fe(Ei)
  increases         decreases remains unchanged

Answer 1

Electrons are fermions (spin 1/2 particles). So the distribution of electrons in different energy levels are governed by Fermi-Dirac distribution. The probability that an energy level with energy E is filled by an electron is given by the Fermi Dirac Distribution function, F(E).

where, E_F is the energy of fermi level in which the probability of electron occupancy is for all temperatures greater than absolute zero, K_{B is the Boltzmann constant and T is the temperature.}

The vacancy of an electron is called a hole.

Therefore the sum of the probabilities of finding an electron and a hole in an energy level is unity.

i.e., the probability of finding a hole in an energy level with energy E,

For an intrinsic semiconductor the number of electrons in the conduction band edge is equal to the number of holes in the valance band edge so their probabilities are equal. i.e., the probability of finding an electron in the conduction band edge is equal to the probability of finding an hole in the valance bandedge.

1. The Fermi-Dirac probability for electrons at the valence band edge

2. The Fermi-Dirac probability for holes at the valence band edge

3. From definition, the Fermi-Dirac probability for electrons at the intrinsic Fermi level

4.

a. Increases

Since the conduction band edge is above the fermi level as temperature increases Fe(EC) increases

b. Decreases

Since valance band edge is below the fermi level as temperature increases Fe(EV) decreases

c. Remains unchanged

For all temperature greater than absolute zero Fe(Ei) = 0.5

5. n-type

Fermi-Dirac probability for electrons at the conduction band edge is increased upon doping so the material become n-type

6.

a. Remains unchanged

b. Remains unchanged

Even though the Fermi-Dirac probability for electrons at intrinsic Fermi0-level Fe(Ei) remains unchanged, the position of fermi level gets shifted near to the conduction band edge