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**Conductivity of Semiconductors**

**➢ Conductivity in Intrinsic Semiconductor**

- In a pure semiconductor, the
**number of holes is equal to the number of electrons**. Thermal agitation continues to produce new electron-hole pairs and these electron-hole pairs disappear because of**recombination.**

Conductivity in Intrinsic Semiconductors

- With each electron-hole pair created, two charge-carrying particles are formed. One is negative which is a free electron with mobility
**µ**The other is a positive i.e., hole with mobility_{n}.**µ**The electrons and hole move in the opposite direction in an electric field E, but since they are of opposite sign, the current due to each is in the same direction._{p}. - Hence the
**total current density J**within the intrinsic semiconductor is given by:

J = J_{n}+ J_{p}

= q.n.µ_{n}.E + q.p.µ_{p}.E

= (n.µ_{n}+ p.µ_{p})qE

= σ.E

where n=no. of electrons/unit volume i.e., the concentration of free electrons

p= no. of holes/unit volume i.e., the 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 µ_{n}+ p µ_{p})q. The**resistivity of semiconductor is reciprocal of conductivity.**

Ρ = 1/σ - It is evident from the above equation that current density within a semiconductor is directly proportional to applied electric field E.
- For a pure semiconductor,
**n = p = n**_{i}

where n_{i}= intrinsic concentration.

The value of n_{i}is given by:

n_{i}^{2 }= AT^{3 }exp (-E_{GO}/KT)

therefore, J = n_{i}(µ_{n}+ µ_{p})qE - Hence, conductivity in an intrinsic semiconductor is:
**σ**_{i }= n_{i}(µ_{n}+ µ_{p})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:

σ_{i }= n_{i}(µ_{n}+ µ_{p})q

= (n.µ_{n}+ p.µ_{p})q - For
**N-type**, n>>p

Therefore, σ = q.n.µ_{n} - For
**P-type**,p>>n

Therefore, σ = q.p.µ_{p}

Extrinsic Semiconductors

**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 the amount of donor and acceptor impurity doping**.

n.p = n_{i}^{2}

where n= electron concentration

p = hole concentration

n_{i}^{2 }= 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 n_{i}^{2}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 N
_{D}donor atoms per cubic centimetre and N_{A}acceptor atoms per cubic centimetre i.e., the concentration of donor and acceptor atoms are N_{D}and N_{A,}respectively. - Therefore, N
_{D}positively charged ions per cubic centimetre are contributed by donor atoms and N_{A}negatively charged ions per cubic centimetre are contributed by the acceptor atoms. Let n, p be concentration of free electrons and holes respectively. Then according to the law of neutrality,

N_{D}+ p = N_{A}+ n ...eq 1.1

For N-type semiconductor, N_{A }= 0 and n>>p.

Therefore, N_{D}≈ 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
**N**_{D}≈ n_{n}. - Therefore, the current density in N-type semiconductor is
**J = N**_{D}.µ_{n}.q.E

and conductivity**σ = N**_{D}.µ_{n}.q - For P-type semiconductor,

N_{D }= 0 and p>>n.

Therefore N_{A}≈ p or**N**_{A}≈ p_{p} - 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 = N_{A}.µ_{p}.q.E

And conductivity σ = N_{A}.µ_{p}.q **Mass action law for N-type**

n_{n}.p_{n }= n_{i}^{2}

p_{n }= n_{i}^{2}/N_{D }

since (n_{n }≈ N_{D})**Mass action law for P-type**

n_{p}.p_{p }= n_{i}^{2}

n_{p }= n_{i}^{2}/ N_{A}

since (p_{p }≈ N_{A})

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