CONDUCTIVITY OF SEMICONDUCTOR:
In a pure semiconductor, the no. of holes is equal to the no. of electrons. Thermal agitation continue to produce new electron-hole pairs and these electron-hole pairs disappear because of recombination. With each electron hole pair created , two charge carrying particles are formed . One is negative which is a free electron with mobility µn . The other is a positive i.e., hole with mobility µp . The electrons and hole move in opposite direction in an electric field E, but since they are of opposite sign, the current due to each is in the same direction. Hence the total current density J within the intrinsic semiconductor is given by
J = Jn + Jp
=q n µn E + q p µp E
= (n µn + p µp)qE
Where n=no. of electrons / unit volume i.e., concentration of free electrons
P= no. of holes / unit volume i.e., concentration of holes
E=applied electric field strength, V/m
q= charge of electron or hole in Coulombs
Hence, σ is the conductivity of semiconductor which is equal to (n µn + p µp)q. The resistivity of semiconductor is reciprocal of conductivity.
Ρ = 1/ σ
It is evident from the above equation that current density with in a semiconductor is directly proportional to applied electric field E.
For a pure semiconductor, n=p= ni where ni = intrinsic concentration. The value of ni is given by
ni2=AT3 exp (-EGO/KT)
therefore, J= ni ( µn + µp) q E
Hence, conductivity in intrinsic semiconductor is σi= ni ( µn + µp) q
Intrinsic conductivity increases at the rate of 5% per o C for Ge and 7% per o C for Si.
Conductivity in extrinsic semiconductor (N-Type and P-Type):
The conductivity of intrinsic semiconductor is given by σi= ni ( µn + µp) q = (n µn + p µp)q
For N-type , n>>p
Therefore, σ= q n µn
For P-type ,p>>n
Therefore, σ= q p µp
CHARGE DENSITIES IN P-TYPE AND N-TYPE SEMICONDUCTOR:
Mass Action Law:
Under thermal equilibrium for any semiconductor, the product of the no. of holes and the concentration of electrons is constant and is independent of amount of donor and acceptor impurity doping.
where n= electron concentration
p = hole concentration
ni2= intrinsic concentration
In N-type semiconductor as the no. of electrons increase, the no. of holes decreases. Similarly in P-type, as the no. of holes increases the no. of electrons decreases. Thus, the product is constant and is equal to ni2 in case of intrinsic as well as extrinsic semiconductor.
The law of mass action has given the relationship between free electrons concentration and hole concentration. These concentrations are further related by the law of electrical neutrality as explained below.
Law of electrical neutrality:
Semiconductor materials are electrically neutral. According to the law of electrical neutrality, in an electrically neutral material, the magnitude of positive charge concentration is equal to that of negative charge concentration. Let us consider a semiconductor that has ND donor atoms per cubic centimeter and NA acceptor atoms per cubic centimeter i.e., the concentration of donor and acceptor atoms are ND and NA respectively. Therefore, ND positively charged ions per cubic centimeter are contributed by donor atoms and NA negatively charged ions per cubic centimeter are contributed by the acceptor atoms. Let n, p be concentration of free electrons and holes respectively. Then according to the law of neutrality,
ND + p =NA + n ...............................……………eq 1.1
For N-type semiconductor, NA =0 and n>>p. Therefore, ND ≈ n …………….eq 1.2
Hence for N-type semiconductor the free electron concentration is approximately equal to the concentration of donor atoms. In later applications since some confusion may arise as to which type of semiconductor is under consideration at the given moment, the subscript n or p is added for N-type or P-type respectively. Hence eq 1.2 becomes ND ≈ nn
Therefore, current density in N type semicomductor is J = ND µn q E
And conductivity σ= ND µn q
For P-type semiconductor, ND = 0 and p>>n. Therefore NA ≈ p
Or NA ≈ pp
Hence for P-type semiconductor, the hole concentration is approximately equal to the concentration of acceptor atoms.
Current density in N-type semiconductor is J = NA µp q E
And conductivity σ= NA µp q
Mass action law for N-type, nn pn= ni2
pn= ni2/ ND since (nn≈ ND)
Mass action law for P-type, np pp= ni2
np= ni2/ NA since (pp≈ NA)