Foreign key in main table or supplementary table?
Suggested that previous comment be posted as 'answer'.  Here 'tis.

Even if a justifiable one-to-one, I'd still treat the "primary" table as if it were a parent, and the "detail" as if it were one of  potentially multiple children. Thus the FK would go in the 'details' table. More appropriately named 'additional_student_details'.