Research statement
Research statement is a narrative that lays out your research program. I believe that it is useful to start it earlier than the last year of the Ph.D. because writing one can provide a good overview of where you’re going. I think it’s useful to think about from the early stage of PhD. Richard McElreath has a template for a short PhD research proposal.
There is a github repo that assembled successful job application statements. Some other research statements: one by Lars Backstrom and another by Scott Aaronson.