Contacting a Professor about a possible Ph.D.
I formulated my mail as a "letter of motivation" - Just in the same way you would write an application in Germany. You could start with something like 


  "Thank you for the nice discussion about the open PhD position in your
  research group at the XYZ conference in October 2017..."


Then, you should mention that you had a very close look to his/her research and that this perfectly fits your interests because of "this" and "that".

In any case I would already attach the CV with all certificates and required documents.