I spontaneously developed sebborheic dermatitis outbreaks on my face, scalp, ears, and chest during a prolonged course of 100mg/daily doxycycline while I still healthy. Since decending into health limbo, the SD has never gone away but doesn't seem to get worse either. Nothing seems to kill this microbe, including herbal essential oils, anti-fungals, vinegar, or "no 'poo" and other stratedgies of doing nothing.
The only things that have worked for me to contain the ugly manifestation of SD are hydrocortisone cream and high-dose zinc. The hydrocortisone obviously treats the topical symptoms of skin inflammation and flaking. The zinc is interesting as it seems to reduce SD symptoms everywhere, including the scalp where its impractical to use HC cream.
SD seems to me to be another one of those usually benign micro-organisms that becomes a problem once the immune system stops working correctly. Either the immune system is deficient which lets SD proliferate, or the immune system has priorities that aren't in line with what we want and so we develope embarrasing skin cooties.