I'm looking for an intuitive explanation of how a LED works. I have a bachelor level understanding of math (I did take some physics courses as well, but nothing too advanced). I'll explain what I know and what I've figured out (hoping it's correct), and point out where my doubts are.

A LED is built putting side by side two pieces of differently doped semiconductor material, p-type and n-type. On one side, there are electron holes, which basically means that atoms are missing one electron to have their outer shell complete. On the other side, there are free electrons. The idea is that if you have a group of atoms in which each accepts 4 bonds, and you put there one atom that only accepts 3, you get one free electron. The same thing the other way around gives the electron holes.

Now, when a voltage is applied in the right direction, free electrons flow to the part where the holes are. Every time an electron meets a hole, it basically "falls into it" (i.e. it completes the outer shell of the atom having a missing bond), and loses some energy. For this reason, when an electron meets a hole, it emits a photon.

Now to my question. What keeps a LED on? Why doesn't it make a super-short burst of light and then stays off? When an electron meets a hole and falls into a lower energy state, it should stay there, I guess. So it should come a time when all holes are filled up and there's nowhere for the electrons to go. Instead, there must be something that empties the holes for new electrons to come. How does this happen?