Surrogate keys as Foreign Keys
You could just add the name of the student in Table B instead of Student_ID but that would not be normalized and you would be duplicating data, also if you don´t have constraints it may be possible for you to add students that are not registered in Table A. Name column for sure will use more bytes than Student_ID. Lookup will be efficient since you´re using Student_ID as clustered index in Table A. If you make queries filtering by that column your search would be fast, SQL Engine will use an index seek operation in the execution plan and even if the table is small SQL will put the entire table in cache (RAM) so it won´t have to go and read from disk. For a transactional app your design is fine, for a data-warehouse you may use a denormalized design where duplicating data is allowed.