Your green card process will have two steps, the I-140 filing and then the green card application, either in the U.S. with USCIS (known as an I-485 application) or outside the U.S. with a U.S. Consulate (known as an Immigrant Visa application). During the I-140 stage, there is some risk traveling on an F-1 or an E-3 visa (with the F-1 riskier), but this risk is relatively low and can be mitigated. One way to reduce this risk to zero is to switch to O-1 status, which as a PhD grad you probably would qualify for but your employer would need to support this. At the second stage, with an I-485 filing, whether you are in E-3, F-1, or O-1 status, you should expect to be unable to travel for at least 6 months unless you have a family emergency. With an IV application, you will be able to travel on an O-1 visa but there will be some risk traveling on an E-3 or F-1 visa (again, with the F-1 riskier).