To the best of my knowledge, all solid rocket boosters so far use a propellant that is a solid mixture of both the fuel and the oxidiser parts of the reaction. However, given that SRBs are generally used in-atmosphere during the earliest parts of a launch (hence the "booster" appellation), one could potentially get a lot more oomph out of a given-sized SRB by only packing the fuel, and grabbing the oxidiser from the ambient atmosphere as you go. Since the SRB is moving at hypersonic speeds near the end of its burn, it'd pretty much be restricted to a ramjet-style intake, necessitating that the propellant immediately surrounding the central bore still contain oxidiser in order to start getting the SRB up to speed; once it's going fast enough, though, it wouldn't need any extra internal oxidiser, allowing the booster to either be made smaller and lighter while still delivering the same thrust for the same amount of time, or to have a longer and/or more powerful burn than a conventional SRB of the same size.
Here's a (very simple) sketch of what I have in mind:
So why don't airbreathing SRBs exist?