Do I need to send e-mail to follow up a PhD interview day?
A polite and courteous thank-you email is never inappropriate. Also, if you've left anything out of your interview day (or promised to follow up on something), it's a good opportunity to do so. 

However, you shouldn't turn this into an opportunity to go overboard and plead or beg for a spot, or oversell yourself. That is unlikely to go over well, and can undo the good job you did on your interview day.