A p-n junction has acceptor impurity concentration of `10^(17) cm^(-3)` in the p-side and donor impurity concentration of `10^(16)cm^(-3)` in the n-side. What is the contact potential at the junction? (kT=thermal energy , instrinsic carrier concentration `n_(i)=1.6xx10^(10) cm^(-3)`

Options

A. (a)`((kT)/(e ))ln(4xx10^(12))`
B. (b)`((kT)/(e ))ln(2.5xx10^(23))`
C. (c)`((kT)/(e ))ln(10^(23))`
D. (d)`((kT)/(e ))ln(10^(9))`

Answer

To find the contact potential at the junction of a p-n junction with given impurity concentrations, we can use the formula for contact potential \( V_b \):

\[ V_b = \frac{kT}{q} \ln \left( \frac{N_A N_D}{n_i^2} \right) \]

where:
- \( N_A \) = Acceptor impurity concentration (in the p-side)
- \( N_D \) = Donor impurity concentration (in the n-side)
- \( n_i \) = Intrinsic carrier concentration
- \( k \) = Boltzmann's constant
- \( T \) = Temperature in Kelvin
- \( q \) = Charge of an electron (approximately \( 1.6 \times 10^{-19} \) C)

### Step 1: Identify the given values
- \( N_A = 10^{17} \, \text{cm}^{-3} \)
- \( N_D = 10^{16} \, \text{cm}^{-3} \)
- \( n_i = 1.6 \times 10^{10} \, \text{cm}^{-3} \)
- \( kT \) is given as thermal energy (we will assume a typical room temperature of about 300 K for calculations).

### Step 2: Calculate the product \( N_A N_D \)
\[ N_A N_D = (10^{17}) \times (10^{16}) = 10^{33} \, \text{cm}^{-6} \]

### Step 3: Calculate \( n_i^2 \)
\[ n_i^2 = (1.6 \times 10^{10})^2 = 2.56 \times 10^{20} \, \text{cm}^{-6} \]

### Step 4: Substitute into the logarithm
\[ \frac{N_A N_D}{n_i^2} = \frac{10^{33}}{2.56 \times 10^{20}} = 3.90625 \times 10^{12} \]

### Step 5: Calculate the natural logarithm
\[ \ln(3.90625 \times 10^{12}) \approx \ln(3.90625) + \ln(10^{12}) \approx 1.360 + 27.631 \approx 28.991 \]

### Step 6: Substitute into the contact potential formula
Assuming \( kT \approx 0.0259 \, \text{eV} \) at room temperature:
\[ V_b = \frac{0.0259}{1.6 \times 10^{-19}} \times 28.991 \]

### Step 7: Calculate \( V_b \)
\[ V_b \approx 0.0259 \times 28.991 \approx 0.751 \, \text{V} \]

### Final Answer
The contact potential at the junction is approximately \( 0.751 \, \text{V} \).

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