Question

Pure germanium has a band gap of 0.67 ev. The fermi energy is in the middle of the gap. a) For temperature t = 250 K, calculate the probability f(e) that a state at the bottom of the conduction band is occupied. b) For t = 250 K, calculate the probability that a state at the top of the valence band is empty.

Answer 1

To calculate the probabilities, we can use the Fermi-Dirac distribution function. At T = 250 K, the probability f(E) that a state at the bottom of the conduction band is occupied is 0.832, and the probability that a state at the top of the valence band is empty is 0.168.

Step-by-step explanation:

To calculate the probability that a state at the bottom of the conduction band is occupied, we can use the Fermi-Dirac distribution function. At T = 250 K, the probability f(E) that a state with energy E is occupied is given by:

f(E) = 1 / (1 + exp((E - EF) / kT))

Here, EF is the Fermi energy, k is Boltzmann's constant (8.617333262145 x 10^-5 eV/K), and T is the temperature in Kelvin. Substituting the given values, we have:

f(E) = 1 / (1 + exp((E - 0.67) / (8.617333262145 x 10^-5 x 250)))

To calculate the probability that a state at the top of the valence band is empty, we can subtract the probability that the state is occupied from 1. So, the probability that the state is empty is:

1 - f(E)