Question
In a non-degenerate bulk semiconductor with electron density n = 1016 cm−3, the value of (EC − EFn) = 200 meV, where EC and EFn denote the bottom of the conduction band and electron Fermi level energy, respectively. Assume the thermal voltage as 26 meV and the intrinsic carrier concentration as 1010 cm−3.
For n = 0.5 × 1016 cm−3, the closest approximation of the value of (EC − EFn) is:
- 226 meV
- 174 meV
- 218 meV
- 182 meV
Step-by-Step Solution
Step 1: Write the Carrier Concentration Equation
For a non-degenerate semiconductor, the electron concentration is given by:
n = ni exp[(EFn − Ei) / kT]
Alternatively,
EC − EFn = kT ln(NC / n)
Since the effective density of states NC remains constant at a fixed temperature, we can compare two operating conditions directly.
Step 2: Use the Difference Formula
The change in the Fermi level position is:
(EC − EFn)2 −
(EC − EFn)1
= kT ln(n1 / n2)
Step 3: Substitute the Given Values
| Electron concentration (n₁) | 1016 cm−3 |
|---|---|
| Electron concentration (n₂) | 0.5 × 1016 cm−3 |
| Initial value of (EC − EFn) | 200 meV |
| Thermal voltage (kT) | 26 meV |
Step 4: Calculate the Energy Shift
ΔE = 26 × ln(1016 / (0.5 × 1016))
Since,
1016 / (0.5 × 1016) = 2
Therefore,
ΔE = 26 × ln(2)
Using,
ln(2) = 0.693
ΔE = 26 × 0.693 = 18.02 meV
Step 5: Find the New Value of (EC − EFn)
Add the energy shift to the original value:
(EC − EFn) = 200 + 18
(EC − EFn) = 218 meV
Step 6: Physical Explanation
The electron concentration decreases from 1016 to 0.5 × 1016 cm−3. As the electron density decreases, the Fermi level moves farther away from the conduction band. Therefore, (EC − EFn) increases.
Final Answer
(EC − EFn) = 218 meV
Correct Option: (C) 218 meV