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In the beginning…
It seems like LED’s are everywhere.  It is hard to remember a time without them.  The first LED was created by a Russian inventor named Oleg Losev in 1927.  Over the next 3 decades many people studied these devices and the phenomenon of electroluminescence, first discovered in 1907 by H. J. Round. It was not until 1962 that a patent for the first practical LED was filed.
These first LED’s emitted light in the infrared and red portions of the spectrum.  These devices did not give off very much light.  One of the major efforts was to increase the light output.  In 1972, red and red-orange LED’s had been refined to the point that they provided 10x more light than their predecessors. In just 4 more years high brightness LED’s capable of being used for fiber optic telecommunications would be made.
The first blue LED’s were developed by RCA in 1972, but the output was abysmal.  Some blue LED’s were commercially available in1989, but the real breakthrough for blue LED’s came in the mid-1990’s. Three researchers at two different companies were working on high-brightness blue LED technologies based on InGaN and GaN on sapphire.  The three were awarded the Nobel prize for their work in 2014.
Blue LED’s paved the way for the first white LED.  There are several ways to make a white LED. One can use red, blue and green LED’s to mix the light.  You can also use a UV LED and an RGB phosphor or you can use a blue LED with a yellow phosphor.  The latter is the more efficient and the most commonly used method.

Physics: Quantum-something
A diode is created by taking a semiconductor and doping that material with another element.  For example, if silicon, a semiconductor, is doped with antimony it will become an n-type semiconductor.  It has extra electrons.  If you add boron to silicon you can make a p-type semiconductor, it has fewer electrons and more holes.  A diode is created when you put an n-type and a p-type semiconductor together (this is actually done on a single crystal, not something you glue together).
The excess electrons, from the n-type material, in such a device will migrate away from the PN junction toward the P-doped material.  Similarly, the holes will move away from the P-type material and toward the N-type material.  This migration occurs because of the concentration gradient.  A region between the two areas, called the depletion zone or layer, forms that is devoid of mobile carriers.  As the diffusion process continues more and more charges accumulate on either side of the depletion zone.  It is important to note that the electrons and holes themselves combine and are lost to the equation.  But the charged ions they leave behind remain.  This sets up an electric field.  The field is in opposition to the diffusion direction.  The diffusion continues until the electric field is strong enough to stop it.  At this point, there is a potential across the depletion zone, it is called the built-in potential.
When a diode is forward biased, the potential applied to the diode pushes the charge carriers closer together decreasing the size of the depletion zone. The point where appreciable current begins to flow is when the applied potential is equal to the built-in potential.  As the charge carriers, holes and electrons, are pushed closer and closer together they begin to combine.  While from the outside we think of the current as flowing, the actual distance that an electron or hole travels before combining and being neutralized, called the diffusion length, is on the order of a few micrometers.

Here Fermi, Fermi, Fermi
When an electron combines with a hole it moves from a high-energy state to a lower energy state. The holes are unfilled bands in an atom.  As you will remember, the electrons fill discrete energy levels or bands around a nucleus.  The highest band (at 0K) where the electrons are normally present is called the valence band. Above that there is the conduction band.  In the conduction band, electrons can move about in the semiconductor and can be thought of as existing as an ideal gas.  There are discrete levels of energy that an electron can have, independent of its existence in a valence or conduction band. When an electron combines with a hole, it is moving from the conduction band to the valence band.  In order to do so it must release the difference in energy between the two levels, the band gap.  That is the energy difference between the band it occupied in the conduction band and the band it will occupy in the valence band.  Because of the quantization of energy, that quantity cannot be any amount of energy, but it will be a very specific amount determined by the type of material.
Earlier I gave an example of doping of silicon with antimony or boron to make a pn junction.  Silicon does not make a very good LED.  There are two types of semiconductors, relative to the band gaps.  There are indirect and direct band gap semiconductors.  An indirect band gap semiconductor does not release a photon when an electron crosses the band gap.  The extra energy that is liberated is put into the vibrational energy state of the material.  A direct band gap semiconductor does release a photon. For visible light LED’s (not UV or infrared) the energy of the photon must correspond to the energy of visible light.  The color of the light, which is the energy level, is dependent on the band gap energy of the semiconducting material.

Let me out, let me out!
The light, photon, that is released exists inside of the crystal that comprises the PN junction of our LED.  The crystal is transparent to that color of light. So, you would think it would be easy for the photon the exit the crystal.  Most material used to make LED’s have a high index of refraction.  Reaching way back in our memory we remember Snell’s law that describes the relationship between the angles of incidence and refraction when light passes through a boundary between two materials with different indices of refraction. For example, a semiconductor and say, air.  Snell’s law also tells us the critical angle, the angle where the light does not cross the boundary.  This is called total internal reflection.  The angle is small when there is a large difference between the two indices of refraction, such as exists for an LED (3.96 vs. 1.0).  It means that the light must be almost normal to the boundary to exit the crystal, otherwise it will be reflected back inside.  This is the primary cause of inefficiency in an LED, the light just cannot get out and is eventually converted into heat.
Current trends
Gone are the days when an LED was only 3-5mm wide with a warm red glow.  Today LED’s literally span the spectrum, from UV to IR.  The blue LED has revolutionized our ability to make colors and along with advances in power output (measured in lumens per watt), is changing our usage of light.  LED’s are now a viable source of home lighting.  They have over taken the incandescent and the compact florescent bulbs.  They are used in signs, signals and communications.  Their compact nature and long life is allowing lighting in places not possible with conventional incandescent bulbs.

Figure 1, LED array used for a traffic signal.
Final thoughts
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