i took a year out (personnal reasons) after finishing my master and now i am looking for phds, by the time i obtain one, i amy have over two years out with no working experience, how can i explain this on my cv and during the interview?

Yes, good question. I'm in the same situation. Some advice would be very welcome here.

after a long search for proposed phds, i found none relevant to my interests. I've been recomended to contact directly those who may become my supervisor; how should this contact be: email? what shall I write, especially to explain this past year since I graduated, where I hardly did mcuh productive? i'd be greatful ofr your help.

If you choose to contact supervisors by email make sure your message refers to specifically to their research, as they receive many unsolicited PhD enquiries and can spot the mail merged ones a mile off.

If you took a year out for personal reasons or because your decision to look for a PhD was taken a couple of months after you graduated then it's no big deal. If you're asked, explain it, don't make too many excuses or blame other people. On your application mention the things you did do, don't spend time explaining why you were doing nothing. If you really have done nothing and for no good reason, then start doing something. Try and get some lab or other research experience by contacting people at relevant local institutions (universities, libraries, hospitals, museums, R&D companies) and volunteering to work for free.

What's PHD study like?