Any PN junction is an X-ray detector, as long as you don't
package it in an infinite high-Z shell.
Useful, that's another matter. You need substantial gain
to get much signal out of nonlethal levels of X-ray or
gamma.
PIN diodes are used at high dose rate beam lines. They
give tens of mV into 50 ohms at gamma dose rates that
will ruin your day and the city you spend it in. If you are
looking for a small outline element and dentist's office
type dose rates, I'd say cut your losses and pick a new
project (or a lithium battery). Getting a detectable signal
is enough of a design task; powering a chip wants some
orders of magnitude more captured power (which will
go as photovolume, dose rate and the Vf in photovoltaic
mode of your junction). Note that you put volume as a
key requirement. That will lead to some tough choices.
A photovoltaic cell will give you current from X-rays.
You can calculate the volume photocurrent from the
Wirth-Rogers approximation if you know the vertical
construction. You can find this in the literature. The
details touch on That Which Must Not Be Named
(depending on application) and I will not go further
into it here.
You should not expect much current and your circuit
draw should be very, very, very low.