What to write in email to potential PhD advisor?
If you have a specific potential advisor at a university, start by sending a brief (and polite, of course) email. Ask whether she would be willing to take a new student and say a few words about your background (what you did in your bachelor/master, where and with whom). Thus you can show that your background is compatible with the advisor's research. If you manage to add a few words about how you liked something specific about the advisor's recent work, that will certainly be a big plus. End by thanking the supervisor for her time and say that you are ready to provide more info if necessary. The advisor can then help you get through the formalities if she sees a potential in you.