Years ago I purchased frames made out of plastic resin these were approx 12mm dia and I inserted my designs into them,the sizes can vary.

I've seen wooden embroidery hoops used to great effect. They come in different sizes, the smallest I've seen was about 40mm; 12 mm makes a very tiny frame, my wedding ring has a 16mm diameter!

I use to use embroidery hoops as frames for hand stitched items. It would also be a good way to frame machine embroidery items.