When I read the headline of the question I was going to suggest growing moss! Not what you want, obviously. Have you tried a grass mixture for shade? If that doesn't work and you don't like moss, how about gravel or bark?

Moss makes a perfect, beautiful and maintainance-free lawn in areas of deep shade - think of the glowing emerald moss gardens of Japan.

All grasses, and Chamomile too for that matter, require sun to thrive.

The best option, otherwise, might be to incorporate the shady area into a bed and plant with suitable shade loving plants, of which there are very many.

Lidl have all sorts of gardening stuff this week - including grass seed for shade. If you can't tolerate your moss I think Ilex's suggestion is a good one. You could click on "S" at the bottom of the page and go to the "Shade" section for ideas.