If you want to do your PhD with me then that's great! Apply for admission to the program. The deadline will be in December for a start in Fall of the following year. Currently I can accept students admitted to Northwestern's MSE Graduate Program and Applied Physics Graduate Program. Below is some advice.
Do not contact me before you are admitted. Cold emailing a professor is not usually beneficial for top programs. We admit a pool of students and they are matched with a group during the first quarter of their PhD. I have almost no say in admissions decisions, and even if I did, discussing with you beforehand would constitute a conflict of interest. So it may even be detrimental.
Cold emailing me to ask for a PhD position shows that you have not done your research on how admissions actually work. It does not reflect well on you as an independent thinker nor on your interest in our department/Northwestern.
I get dozens of emails a week asking for a PhD position. I do not have time to read them. I certainly do not have time to reply to them.
Make sure your statements respond to the prompts you are given.
Secure your letter writers early.
Recommenders must be senior people who have had some level of supervision over you. If you have done research projects in the past, not having a letter from at least one of those supervisors may be damaging.
A letter from a class instructor is fine. If all your letters are from class instructors then that may be perceived as a weakness.
Make sure your letter writers know you well, can speak about your strengths, ideally with some examples of problems that you solved. It will help them if you give them some bullet points of your main achievements and goals. Even better if you provide them with the rest of your application package.
If you are asked to submit a CV, make sure it is a CV and not a one-page resumé. There is no need to cram all your information onto one page.
It is not usually relevant to put your high school achievements on your CV, unless they were extremely notable. Likewise University entrance exams.
Class ranking, if that information is available to you, is useful.
All lines on your CV should have dates, places, and names of supervisors.
Personal projects are probably not worth putting on your CV unless they had a significant, measurable outcome. Likewise prescribed undergrad lab class experiments.
Include peer reviewed publications as a separate section on your CV. However...
You do not need publications to get in, many of our admitted students have no papers and we routinely reject applicants who have multiple first author papers.