How about periwinkle (vinca)? It will spread to become a thick mat and choke out most weeds. Here in NH it is virtually evergreen and blooms as soon as the snow melts. When it is not in bloom, it is a lovely dark green. I have it in both deep shade and in sun; it does well in both. The leaves are darker, richer green in the shade and it blooms more profusely in the sun. Good either way!