Occupancy of impurity states in semi-conductors

[capandbells]

Homework Statement

Calculate the average number of electrons (the occupation probability) in a localized state of impurity in a semiconductor at a finite temperature T. The impurity state in a semiconductor can be empty or it can be occupied by only one electron.

The Attempt at a Solution

Is this as easy as it seems? The partition sum is just 1 + exp(-(E - μ)/kT). The average occupancy is exp(-(E - μ)/kT) / [1 + exp(-(E - μ)/kT)] = 1 / [exp((E - μ)/kT) + 1], the FD distribution. Is there anything more to this?

 

[anathema]

Yes, calculating the average number of electrons in a localized state of impurity in a semiconductor at a finite temperature T is relatively straightforward. As you correctly pointed out, the partition sum for this system is 1 + exp(-(E - μ)/kT), where E is the energy of the impurity state, μ is the chemical potential, and k is the Boltzmann constant. However, it is important to note that this partition sum assumes that the impurity state can either be empty or occupied by only one electron.

To calculate the average occupancy, we need to take into account the fact that there can be multiple impurity states with different energies. In this case, the partition sum becomes a sum over all possible states, and the average occupancy is given by the sum of exp(-(Ei - μ)/kT) / [1 + exp(-(Ei - μ)/kT)] over all states i. This is known as the Fermi-Dirac distribution, and it accurately describes the distribution of electrons in a system at a finite temperature.

Furthermore, the average occupancy also depends on the temperature and the chemical potential. As the temperature increases, more states become thermally accessible and the average occupancy increases. Similarly, as the chemical potential increases, more states become energetically accessible and the average occupancy increases.

In summary, while the calculation of the average number of electrons in a localized state of impurity in a semiconductor at a finite temperature may seem simple, it is important to consider all factors such as multiple impurity states, temperature, and chemical potential in order to accurately determine the average occupancy. I hope this helps clarify any confusion and please let me know if you have any further questions.