We will consider this as giving -V to the p side and none to the n side. This means the n energy levels remain unchanged and those for p are increased by eV. Due to the application of the external potential difference the system is no longer in dynamical equilibrium and so Fermi-Dirac distributions no longer apply. Despite this, as long as the potential difference applied is relatively low, we can approximate each side as obeying Fermi-Dirac distribution as long as we assign a different Fermi energy to both sides.
Now the energy going from the n side to the p side of the conduction band is eV higher, then only those conduction electrons on the n side with and energy greater than eV are allowed to pass over, so the diffusion current due to electron motion from n to p is reduced. If we call the diffusion current when no external potential difference is present I0 then the diffusion current due to electron motion from n side to p side when there is an external potential difference can be found to be approximately
Apart from being shifted upwards in energy, no change is made to the Fermi-Dirac distribution of the p side, and so the diffusion current due to electron motion from p to n is I0 the same as ever. This means we have a resultant electron motion from p to n and so a resultant current from n to p
and for the holes is the contribution is similar. So if the potential difference is positive, that is the potential for the n side is greater than for the p side, the exponential term will tend to zero and the current tend to the diffusion current. If the potential difference is negative, that is the potential is greater for the p side than for the n side, the exponential term, and so the current, will approach negative infinity.
We see that the diode is a device that only lets current one way. Place it the right way around, with the p-type semiconductor having a higher potential than the n-type and it will conduct easily. This is known as a forward bias p-n junction. If it is the other way round, a reverse bias p-n junction, it will barely conduct at all. Of course the above argument failed to take into account to potential difference dropped along the semiconductor or the wires but it does show us the one way behaviour of the depletion region.
The Light Emitting Diode
If the depletion region has a light shone upon it of a frequency so that it causes valence electrons to move to the conduction band, in effect creating electron hole pairs, the electric field will push the electrons from p-type to n-type and holes from n-type to p-type. If the two ends of the crystal are connected to a circuit this will create a current moving from n-type to p-type. This is the principle on which a solar cell functions.
Driving a current a current through the junction with a battery can result in the opposite effect, conduction electrons decay to the valance band, which can also be thought of as an electron combining with a hole, and light is emitted along the way. As this emits light whilst also working as a diode, such a device is known as a light emitting diode or LED. In the solar cell the current travels from n to p, so it is reasonable to think that in the opposite process, the LED, current must go the other way, from p to n, to create the light. This is true, it is under forward bias that emission occurs.
Not all semiconductors can be used to create LED’s. In a decay from the lowest state in the conduction band to the highest in the valance band both energy and momentum must be conserved. The conservation of energy is satisfied by the emission of light of the correct frequency, but momentum can only be conserved if the wave-vector of the lowest conduction band state is very nearly equal to that of the highest valance band state. Materials that satisfy such a condition are known as direct gap semiconductors and include gallium arsenide, indium antimonide, indium arsenide and gallium arsenide phosphide. The colour of the last of these can be varies over most of the visible spectrum by varying the amount of phosphorus in the alloy.
The LED can also be used to make a laser, although one with low efficiency. To create the laser the decays must induce other decays, causing a cascade of emissions all in phase, known as stimulated emission. This is achieved by making sure there are more electrons in the lowest states of the conduction band than in the highest of the valance band, known as a population inversion. To do this high levels of doping are needed. The laser is of low efficiency as the stimulated emission only occurs in the very centre of the depletion region with lots of spontaneous emissions happening around it. The population inversion therefore takes alot of current to sustain it and produces large amounts of heat.
For my three articles I used the books
Turton, R., The Physics of Solids, Oxford University Press, 2000.
Landshoff, P., Metherell, A. and Rees, G., Essential Quantum Physics, Cambridge University Press, 2001.