How to contact professors for PhD vacancies?
To supplement the other answers, here are some do's and don'ts for email contact.
The worst thing you can do is make it seem like you're trying to cast a wide net and don't have a clear focus. That's a guaranteed delete.
Therefore,
- Narrow your search to the people whose work you're really interested in
- Read their papers (especially recent ones - I've had people email me about stuff I did 5 year ago - I've moved on :))
- Think about their work. Find something intelligent to say (even a question).
- Email the professor and focus on those questions.
That's most likely to get my (their) attention.
Ultimately, the answer depends upon what admissions system a department uses.
If you need to obtain admission to a group at the same time as (or instead of) the department as a whole, then it is of course absolutely critical that you make contacts ahead of time! In general, I'd even start contacting people well before the time of application. However, make sure that your contact is substantial. When it becomes time to apply, you will have a better shot.
In many departments, you apply for admission to the department as a whole. After your admission, you select an advisor to work for. In such cases, it's not really critical for you to have a direct contact within the department, since it's not necessarily a given that the professor you want to work for sits on the admissions committee. In such cases, you'll have an indirect connection at best. That said, it's still a good idea to have a contact in a department during the admissions process. You don't lose anything by it—unless the professor has a bad reputation amongst his colleagues (and then you might not want to work for him or her, anyways!). By the way, I would caution strongly against going to such a school if there's only one professor at the department you'd be interested in working for. You're taking a very large risk under such circumstances.
Now to answer some of the other issues raised.
Email or other forms of contact. Face-to-face or phone interactions rank above email interactions. There's no doubt about that. However, an email interaction—if actually substantial—can also be viable. However, a quick emil telling someone you're applying and interested in working for them won't really get you anywhere.
Number of people to contact. There is of course no limit to the number of potential advisors you can contact. To some extent, they're competing for you just as much as you're competing for them!
Can I apply on credentials alone? At most top departments (where application is done at the department level), I think it's entirely possible to apply on credentials and recommendations alone. Having the contacts can obviously help, but not having it won't ruin your chances for admissions, either.
Just to add to aeismail's answer:
If you can send an e-mail that demonstrates that
- You are a strong candidate
- You wrote an e-mail specifically for that professor
- You are familiar with some of that professor's work
then it may have an important positive effect. Ideally, you would be able to suggest how things you know or have done could contribute to that professor's research agenda.
Sending a generic e-mail to multiple professors will not help your case and will probably hurt it. I mention this because a large fraction of the e-mails I receive from potential students are obviously part of a mass-mailing.