I often get email from students who want to join my lab. I welcome these requests, and I ignore most of them.
Why? Because most of them are not helpful. They don't tell me what I want to know. I'm writing this advice to save us both time and frustration, in the hope that this won't happen to you.
Mark Corner (my PhD advisor) posted some blunt but solid advice here. You should read it.
When a student contacts me, I initially want to know three things.
I want to know what you can do. You can tell me where you went to school, what classes you took, and what languages you know, but that isn't really what I care about. I really want to know what you can do. Can write solid code, design great circuits, solve difficult problems, think creatively, or write coherently? Specifics are good, especially if they relate to the work that I do. Show me what you've done.
I want to know why you want to work with me. Some students send form emails to a large number of professors. These emails waste my time. I am not interested in students who just want to get into graduate school. I want students who are excited about the kind of research that I do, and have thought about how their interests fit with my research program. The more you know about the work I do and how it fits in with your interests, the more likely I will be interested in talking with you about it. Again, specific is better than vague.
I want to know what you want to do. I know you likely don't have a specific project in mind. I know you're probably new to research. You're probably a bit uncertain about how exciting or novel your ideas are. Still, I want to know how you think. What problems have you thought about? What problems excite you? What makes you the right person to tackle these problems? Getting a PhD is a very self-driven process, and I'm not interested in students who are going to sit around waiting to be told what to do. What you actually do will probably differ from your current plan, but I want to see that you are thinking about it and maybe have the beginnings of a plan.
I look forward to talking with you.