Q.3. Discuss the fermi dirac probability
function.
Ans. Fermi- dirac probability function :-
The thermal
behaviour of electrons in an atom can be explained by fermi-dirac probability
function p (E) given by
.........(i)
Where K is Boltzmann’s
constant, and T the absolute temperature. Fermi-dirac distribution is shown in
figure.
- It illustrates that p (E) = 0 or 1 at T = 0 K. Variation in p (E) with increasing temperature is also shown.
- At higher temperatures, more and more electrons occupy energy greater than fermi energy.
- The kinetic energy of electrons in a metal having maximum energy is equal to its fermi level.
- Only some electrons may attain this level.
Figure - The fermi-direc distribution of free
electrons as a function of temperatures
The fermi energy level of some elements is given in
table. The value of p (E) is 0.5 at cross-over point. This point is invariant
at all temperatures. Thus the fermi energy level can also be defined as the
level at which probability of occupation by an electron is 50%. For E > EF + kT, in eq (i). the exponential term in denominator
becomes too large than unity. Therefore eq(i). may be rewritten as
Q.4. Explain intrinsic semiconductor with
energy diagram.
Ans. Intrinsic
Semiconductors:- Reason of unsuitability of intrinsic semiconductor. We
noticed that the conduction of electrons from valence band to conduction band needs crossing-over the
forbidden gap. The applied electric field required for this conduction is
extremely high in intrinsic semiconductors. For example the forbidden gap in
germanium is 0.7 eV and in silicon 1.1 eV. The distance between locations of
electrons near an ion core and away from it is about 1
. Thus semiconduction of
an electron form valence band to conduction band will take place when a field
gradient of about 0.7 V/1 in Ge and 1.1
V/1 in Si is applied. This comes out to be an
impracticable value of about and respectively.
. Thus semiconduction of
an electron form valence band to conduction band will take place when a field
gradient of about 0.7 V/1 in Ge and 1.1
V/1 in Si is applied. This comes out to be an
impracticable value of about and respectively.
On the other hand, thermal energy at room temperature
can excite limited number of electrons across the energy gap. The number of
electrons crossing-over the forbidden gap may be calculated by fermi-Dirac
probability distribution,
..........(i)
Energy Diagram Of Intrinsic Semiconductor :-
The energy
diagram for an intrinsic semiconductor is shown in figure. It shows that the
fermi level EF lies in the middle of forbidden energy gap Eg. this fermi level is also referred to as intrinsic
energy Ei. The energy gap
Eg is equal to difference in the energies of
valence band Ev and conduction band Ec. The term (E - EF) in eq.(i) is therefore equal to Eg /2. For pure
germanium, (E - EF) is Eg / 2 = 0. 35 eV,
which is 13.5 times higher than kT at room tempereture (= 0.026 eV). Therefore,
as compared to the exponential term in Eq. the value unity is neglected, and
Eq. can be rewritten as
