Final answer:
The intrinsic carrier densities in silicon and germanium can be calculated using the band gap energy, temperature, and other parameters. In silicon, the calculation involves determining the effective density of states, while in germanium it differs slightly. The formulas and constants are provided in the detailed answer.
Step-by-step explanation:
The intrinsic carrier density in a semiconductor is the number of charge carriers (electrons or holes) that are thermally generated in the absence of any external stimuli such as doping. It depends on the temperature and the band gap energy of the semiconductor.
In silicon (Si), the band gap energy is 1.12 eV and the intrinsic carrier density at room temperature (300 K) can be calculated as follows:
- Calculate the effective density of states in the conduction band (Nc) and the valence band (Nv) using the formulas:
- Nc = 2(2πmkT/h^2)^3/2 * (2m)^3/2 * (√(Eg/2))^3
- Nv = 2(2πmkT/h^2)^3/2 * (2m)^3/2 * (√(Eg/2))^3
Calculate the intrinsic carrier density (ni) using the formula:
ni = √(Nc * Nv) * exp(-Eg/(2kT))
Substitute the values of the constants: Eg = 1.12 eV, k = 8.617333262145 × 10^-5 eV/K, T = 300 K, m = 9.11 × 10^-31 kg, h = 6.62607015 × 10^-34 J·s.Calculate the intrinsic carrier density using the given values.
For germanium (Ge), the band gap energy is 0.67 eV and the calculation is similar, but with different values for Eg and Nc/Nv.