An intrinsic semiconductor has a resistivity of 0.50 `Omega` m at room temperature. Find the intrinsic carrier concentration if the mobilities of electrons and holes are `0.39 m^2 V^(-1) s^(-1)` and `0.11 m^2 V^(-1) s^(-1)` respectively

Options

A. `1.2xx10^18 m^(-3)`
B. `2.5xx10^19 m^(-3)`
C. `1.9xx10^20 m^(-3)`
D. `3.1xx10^21 m^(-3)`

Explanation

To find the intrinsic carrier concentration (\(N_i\)) of an intrinsic semiconductor given its resistivity (\(\rho\)) and the mobilities of electrons (\(\mu_n\)) and holes (\(\mu_p\)), we can follow these steps:

### Step 1: Write the formula for resistivity
The resistivity (\(\rho\)) of an intrinsic semiconductor can be expressed as:
\[ \rho = \frac{1}{q \cdot N_i \cdot (\mu_n + \mu_p)} \]
where:
- \(q\) is the charge of an electron (\(1.6 \times 10^{-19} \, \text{C}\)),
- \(N_i\) is the intrinsic carrier concentration,
- \(\mu_n\) is the mobility of electrons,
- \(\mu_p\) is the mobility of holes.

### Step 2: Rearrange the formula to solve for \(N_i\)
To find \(N_i\), we rearrange the formula:
\[ N_i = \frac{1}{\rho \cdot q \cdot (\mu_n + \mu_p)} \]

### Step 3: Substitute the known values
Given:
- \(\rho = 0.50 \, \Omega \cdot m\)
- \(\mu_n = 0.39 \, m^2 \cdot V^{-1} \cdot s^{-1}\)
- \(\mu_p = 0.11 \, m^2 \cdot V^{-1} \cdot s^{-1}\)
- \(q = 1.6 \times 10^{-19} \, C\)

Now, we can substitute these values into the equation:
\[ N_i = \frac{1}{0.50 \cdot (1.6 \times 10^{-19}) \cdot (0.39 + 0.11)} \]

### Step 4: Calculate the sum of mobilities
First, calculate the sum of the mobilities:
\[ \mu_n + \mu_p = 0.39 + 0.11 = 0.50 \, m^2 \cdot V^{-1} \cdot s^{-1} \]

### Step 5: Substitute the sum back into the equation
Now substitute this back into the equation for \(N_i\):
\[ N_i = \frac{1}{0.50 \cdot (1.6 \times 10^{-19}) \cdot 0.50} \]

### Step 6: Calculate \(N_i\)
Now perform the calculation:
\[ N_i = \frac{1}{0.50 \cdot 0.50 \cdot 1.6 \times 10^{-19}} = \frac{1}{0.25 \cdot 1.6 \times 10^{-19}} = \frac{1}{4.0 \times 10^{-20}} = 2.5 \times 10^{19} \, m^{-3} \]

### Final Answer
Thus, the intrinsic carrier concentration \(N_i\) is:
\[ N_i = 2.5 \times 10^{19} \, m^{-3} \]