How is a half door made in Minecraft?

There used to be a bug that resulted in doors being incorrectly destroyed, and leaving half a door remaining. I believe this has been fixed, as per Tom Wijsman's answer. However, if you would like to make half-doors, gates etc., consider the Multi-use Doors Mod, which adds this functionality to the game, as well as other door-related happiness and excitement:

After update 1.7_01, doors would make purple particle effects when hit with any tool, hand, item, or block. This was fixed in 1.7.3.

— Minecraft Wiki - Doors

I believe as part of the door duplication and door bug problems, the lower half bug has been fixed...

1.7.3: Corrected a block duplication bug when using pistons.

1.7.3: Doors no longer create purple particles.

1.7.1: The door duplication bug is back again, this time happening when a piston pushes on the lower half of a door. If it is a normal piston that does it, then the piston can be destroyed and the bottom half of the door works like a normal door.

— Minecraft Wiki - Version history

In 1.7.3, the door duplication bug with Pistons is most likely to have been fixed.

Thus, the ability for the lower half door to be able to exist alone in the world could be gone...

What you're looking for is basically a fence gate. Craft one with two planks vertically stacked and two sticks on each side.