Not sure. However, I have read that English Ivy is particularly bad.

I don't have a solution. However, mice and rats hang around ground covers for three reasons: 1) as a food source 2) as a nesting source 3) as a cover source where they are running from place to place. English ivy, and all other types of ivy, provide all three. If that same ivy runs up a building, other structure, or tree, it can also provide a climbing vine or conduit.

So what you are looking for is a ground hugging cover without edible seeds, fruit, or berries. Some berries of some vegetation is poisionish enough to be inediable.