How is an LED constructed, is it possible to build my own
Yes you can! And no specialistic knowledge is required.
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Building a DIY LED from SiC
Michael Lippert writes:

The green glow of the DIY LED! At the contact point of pin and crystal greenish light is emitted.
The setup is very simple. I took a crystal of SiC and attached a clamp to it to supply it with about 20 Volts. It is important that the plus clamp is used. It is further advisable to limit the current to something around 30 mA. This will prevent the crystal from heating up too much, as the diode forward voltage will be only on the order of 9 Volts @ 30 mA. If you have no current limiter on your power supply or want to use a battery, use 12 Volts as fixed voltage and be careful not to overheat your setup. The negative lead is attached to a wire which holds a regular pin from your sowing supplies. After applying the voltage the pin should be moved across the surface forming a cat’s whisker detector. Every now and then a tiny light appears at the contact point.

Here's a photograph of the operational LED:

Requires: Silicon carbide (sandpaper etc), pin, wire, power supply.