AR is technically a .22 = .223/5.56. Cartridge designed for small bullet (around 35g) driven at fairly high velocity (3000 fps+).
I am not an expert, but .22 does not seem like 55.6 or .223
But I like my .220 swift
The most common .223 (5.56x45) ammo is 55gr Full Metal Jacket.
It is the cheapest and one of the most accurate bullets, but 55 gr is a very light bullet and it does not easily expand upon impact.
It is possible to shoot heavier bullets in 5.56x45 NATO, if you have 1:8 or 1:7 barrel twist. For hunting, I almost exclusively use 70 gr semi spitzer soft point bullets. There are other considerations to apply.