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In my experience, you don't just e-mail someone and say "will you be my advisor." You certainly don't do that before you start the PhD, though I do encourage you to start getting to know faculty during your first year. By the time you are formally choosing your committee, you should know your potential advisor well enough that there is no need for awkward, formal e-mails.
Approaching someone as a potential advisor is just one of many steps in developing a relationship with faculty. You meet faculty by attending seminars and taking classes. You should make a point of talking to people who you might want to work with. You can have informal conversations at the end of a seminar: "I was interested in the point you made about serial correlation. Can you describe it to me in a little more detail?" You can go to office hours; if you do, it helps to have a few specific questions rather than just "tell me about your research." You can send an e-mail asking to discuss a potential research idea; again, you should have some specifics in mind, and ask for advice about something concrete. Many schools, including U-Mich, have organized summer RA programs for students who have finished first year, and asking about a summer RA position can be a natural way to broach the topic of working with someone.
These are all low-key, casual interactions that faculty have with students all the time. They can, and should, start as early as your first year. By second year, you should know the professors in your fields fairly well, and they should certainly know who you are. You will probably meet with your field faculty at some point during your second year to discuss the term papers that are typically assigned in field courses. Take advantage of the meeting and follow up, including after the paper is finished.
In short, you should have a lot of interaction with many faculty members before you reach the stage of formally choosing an advisor. The initial contact is low key and casual, and it gives you a sense of how to best interact with the various professors at your school. By the time you are actually choosing a committee, you will know the potential members well enough that you won't need a form e-mail from TM to approach them!
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