The quality of greenhouse is a huge variable. For the excavated area, the ratio of depth / surface area probably has a magic number that is just right for whatever latitude. I think a good starting point for figuring this out is looking up the maximum frost depth for your area.
A government entity where I live was promoting half-buried greenhouses for a while. We tried it and abandoned it after the first winter. The south wall cast a long shadow in midwinter, but we get a lot of solar gain so that was a major loss. Also we had to peel away a corner off the top and climb down in to maintain, water o r harvest, which of course isn't good for retaining heat.
If your region has very little solar gain in midwinter it might be worth it, I don't know. At very north latitudes the shadow would be even longer though.
Works at a residential alternative high school in the Himalayas SECMOL.org . "Back home" is Cape Cod, E Coast USA.
Thanks for the replies.
Latitude 34 degrees, north Georgia zone 7b, almost at top of hill heavy red clay soil. According to older maps frost line is 12 inches but I imagine that is rare or out of date, would guess a lot shallower now.