Here are three possibilities:
1) device characterization engineer: spend your day taking measurement of silicon and optimizing parameter sets for simulation with eg BSIM3; you might also design and lay out test chips

2) process development engineer: work on improving devices: change doping profiles, process recipe, mask layouts, etc. to change the electrical behavior of devices; or figure out how to add new devices (eg add BJTs to CMOS process)

3) spice programmer: add new equations to a spice simulator to allow simulation of new device characteristics or new features of advanced cmos processes

There are other possibilities, too, I expect.  I spend some of my time doing 3.

I have a PhD, but it's not required for (3).  What *is* required is a maniacal attention to details and the ability to get equations -- and especially derivatives -- rigorously correct.

It doesn't hurt to have a PhD, and it would help to have a reasonable understanding of device physics, like what the diode I(v) equation is, etc.