Conductivity Of Semiconductor - Practice - GATE PDF Download

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_n. The other is a positive i.e., hole with mobility µ_p. 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. 
• 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_i² = AT³ 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 ^oC 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_i²
  where n= electron concentration
  p = hole concentration
  n_i² = 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_i² 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_i²
  p_n = n_i²/N_D   
  since (n_n ≈ N_D)
• Mass action law for P-type
  n_p.p_p = n_i²
  n_p = n_i²/ N_A   
  since (p_p ≈ N_A)