As you saw in previous chapters, conductivity of a semiconductor highly depends on the carrier density. Qualitatively, we have seen how changing temperature or doping can affect the density of electrons and holes. Let’s look at the densities of electrons and holes more carefully and see what determines their values.
The valence and conduction bands contain a vast number of allowed energy levels. However, these energy states are quantized and there are a finite number of them in any given energy range. Hover over the vallance and conduction band to zoom in.
Can you calculate the number of energy states per unit energy (density of states) at various energy levels (e.g. around 0.7eV, 0.9eV, or -0.68 eV)?
Can you calculate roughly how many states exist in the energy range of 0.6eV-0.7eV?
Can you plot density of states versus energy.
Change the volume and see how density of states changes.
Volume (cm3) :
Each energy state can accommodate two electrons, one with spin up and one with spin down. At zero temperature, all vallance band energy levels are full (have two electrons) and all conduction band energy levels are empty (no electrons). At non-zero temperature, however, some electrons in the vallance band get excited and go to the conduction band.
Can you calculate the probability of each state being occupied (Fermi Function)? What about the probability of having a hole?
Can you calculate roughly how many electrons occupy the energy states in the energy range of 0.6eV-0.7eV?
At what energy value the number of electrons in the zoomed box is the largest? What about holes?
Change the temperature and see how the probabilities change.
Temperature (K):
Dope the lattice with donor atoms and repeat what you did before. What happens to the fermi energy level?
Temperature (K):
Donor Density: (pair/cm3)
Dope the lattice with acceptor atoms and repeat what you did before. What happens to the fermi energy level?
Temperature (K):
Acceptor Density: (pair/cm3)
The earlier cases, the temperature did not go below 100K. Let’s see what happens if we lower the temperature to very low values.
Change the temperature to observe the behavior of the added dopants. Please wait a few more seconds between each change to ensure best results.
Temperature (K):
Donor Density: (pair/cm3)
If the temperature is very low, not all dopants get ionized. In other words, the fifth electrons of many donor atoms won’t have enough energy to break free.
We consider the hole to be a positively charged carrier, and it moves freely in the lattice similar to the free electron.
The motion of these two carriers is responsible for conduction in semiconductors.