Digging up part of patio foundation

Hi all, my wife wants to grow a rose up our wall so I’m trying to excavate down to earth under one patio slab. Managed to get the slab itself up pretty easily, but now wondering how to dig out the concrete foundation? Tried unsuccessfully with a standard drill and a sledgehammer… Would an SDS drill do the trick? Think it’s about 15-20cm deep.

Also - might I cause the rest of the patio to cave in? Or even the house itself?!