A semiconductor having electron and hole mobilities μn and μp respectively. If its intrinsic carrier density is ni, then what will be the value of hole concentration P for which the conductivity will be minimum at a given temperature?

Let the hole concentration be nh. Then the electron concentration = ne = ni2/nh
The total conductivity of the semiconductor material = σ = σe + σh = e neμn + e nhμp = e ni2/nh μn + e nhμp
Hence, the condition for minimum conductivity is found by dσ/dnh = 0
⇒ e ni2μn(-1/nh2) + eμp = 0
⇒ nh2 = ni2(μn/μp)
⇒ nh = ni√(μn/μp)
However, the question asks for the condition for minimum conductivity. The conductivity is minimum when the hole concentration is equal to the intrinsic carrier concentration. Therefore there must be a mistake in the question statement or the solution provided above. The condition for minimum conductivity is obtained by setting the derivative of conductivity with respect to hole concentration equal to zero. Solving the resulting equation leads to the value of hole concentration given by the solution provided. In order to check, let us find the second derivative. If it is positive, it signifies the minimum value.

d2σ/dnh2 = 2e ni2μn/nh3 >0

Therefore, the conductivity is minimum when nh = ni√(μn/μp). This solution is incorrect and does not match any of the options. However, the options suggest the correct answer to be ni√μnμp which is incorrect given the analysis above.