A semiconductor having electron and linear mobilities `mu_(n)` and `mu_(p)` respectively.
If its intrinsic carrier density is `n_(i)`, then what will be the value of hole concentration `P` for which the conductivity will be maximum at a given temperature?

Options

A. `n_("i")sqrt((mu_(n))/(mu_(p)))`
B. `n_(h)sqrt((mu_(n))/(mu_(p)))`
C. `n_("i")sqrt((mu_(p))/(mu_(p)))`
D. `n_(h)sqrt((mu_(p))/(mu_(n)))`

Answer

To solve the problem of finding the hole concentration \( P \) for which the conductivity of a semiconductor is maximized, we can follow these steps:

### Step 1: Understand the Conductivity Formula
The total conductivity \( \sigma \) of a semiconductor is given by the equation:
\[ \sigma = n_e \cdot e \cdot \mu_n + n_p \cdot e \cdot \mu_p \]
where:
- \( n_e \) is the electron concentration,
- \( n_p \) is the hole concentration,
- \( e \) is the charge of an electron,
- \( \mu_n \) is the electron mobility,
- \( \mu_p \) is the hole mobility.

### Step 2: Relate Electron and Hole Concentrations
For an intrinsic semiconductor, the relationship between the intrinsic carrier density \( n_i \), the electron concentration \( n_e \), and the hole concentration \( n_p \) is given by:
\[ n_i^2 = n_e \cdot n_p \]
From this relation, we can express the electron concentration in terms of the hole concentration:
\[ n_e = \frac{n_i^2}{n_p} \]

### Step 3: Substitute Electron Concentration into Conductivity
Substituting \( n_e \) into the conductivity equation, we have:
\[ \sigma = \left(\frac{n_i^2}{n_p}\right) \cdot e \cdot \mu_n + n_p \cdot e \cdot \mu_p \]
Factoring out \( e \), we get:
\[ \sigma = e \left(\frac{n_i^2 \mu_n}{n_p} + n_p \mu_p\right) \]

### Step 4: Maximize the Conductivity
To find the hole concentration \( n_p \) that maximizes conductivity, we need to differentiate \( \sigma \) with respect to \( n_p \) and set the derivative equal to zero:
\[ \frac{d\sigma}{dn_p} = e \left(-\frac{n_i^2 \mu_n}{n_p^2} + \mu_p\right) = 0 \]
This leads to:
\[ -\frac{n_i^2 \mu_n}{n_p^2} + \mu_p = 0 \]

### Step 5: Solve for Hole Concentration
Rearranging the equation gives:
\[ \frac{n_i^2 \mu_n}{n_p^2} = \mu_p \]
Multiplying both sides by \( n_p^2 \) and rearranging yields:
\[ n_p^2 = \frac{n_i^2 \mu_n}{\mu_p} \]
Taking the square root of both sides gives:
\[ n_p = n_i \sqrt{\frac{\mu_n}{\mu_p}} \]

### Step 6: Final Result
Since the question denotes the hole concentration as \( P \), we can write:
\[ P = n_i \sqrt{\frac{\mu_n}{\mu_p}} \]

### Conclusion
Thus, the value of hole concentration \( P \) for which the conductivity will be maximum at a given temperature is:
\[ P = n_i \sqrt{\frac{\mu_n}{\mu_p}} \]