There is different cases regarding PhD subjects, and it vary depending on the country. The normal two cases is doing a master, and by contacting professors you

There is different cases regarding PhD subjects, and it vary depending on the country. The normal two cases is doing a master, and by contacting professors you find opportunities and build with the help of the future PhD director the subject. Second case is there is an open position for a already existing subject. In extremly rare cases, the subject is proposed by (brilliant) students to Prof. but in all cases, you don't work alone and you stand on the shoulders of giants