Correct abbreviation for current PhD student
If this is common in your field, then you should imitate how you see it handled most often in other people's papers.  I don't know of any standard abbreviation for "student" in this context, so I'd imagine you should write it out, but again you should imitate what other people do unless you see a good reason to do otherwise.

If it's not common in your field (and you therefore don't have many examples at hand to compare with), then I'd recommend not including this information.  As MJeffryes points out in a comment, in much of academia papers don't list titles or positions at all.  If that's the case in your field, then identifying the authors by title could look eccentric (or, depending on the title, pompous).