Answer:
To calculate the probability that an energy state in the bottom of the conduction band is occupied by an electron for various semiconductors, one must use the Fermi-Dirac distribution function. This requires knowing the bandgap values which are not provided. Conceptually, the probability is low at room temperature since semiconductors have a low electron occupancy in the conduction band at such temperatures.
Step-by-step explanation:
The question revolves around determining the probability that an energy state at the bottom of the conduction band of various semiconductors (Si, Ge, GaAs) is occupied by an electron at room temperature (300 K), with the assumption that the Fermi energy level is exactly in the center of the bandgap energy. At T = 0 K, all states below the Fermi energy (ℓ) are filled and all states above are empty.
However, at finite temperatures, we use the Fermi-Dirac distribution function to calculate the occupation probability of energy states.
For semiconductors, the occupation probability of the conduction band can be calculated using the equation:
F(E) = 1 / [exp((E-Eℓ)/(kₖT)) + 1]
Here, E is the energy of the state, Eℓ is the Fermi energy, kₖ is the Boltzmann constant, and T is the temperature. Given that the Fermi level is at the midpoint of the bandgap and the temperature is 300 K, we can substitute these values into the equation to find the occupation probability of the conduction band minimum.
Since specific values for the bandgaps of Si, Ge, and GaAs are not provided, this calculation would typically require those values to be known or provided. As a conceptual answer, at room temperature, since the Fermi energy is in the middle of the bandgap, the actual occupation probability at the bottom of the conduction band for semiconductors with similar bandgap energies would be very low, reflecting the semiconducting property that at low temperatures the conduction band is barely occupied.