(1) Schottky Vf of (say) 350mV equates to +1dBm RF
signal's crest voltage in a 50-ohm system. That's where
you would pick up anything at all, harvested. This is a
pretty high level for an antenna of any sort to provide
unless you like to do your harvesting real close to the TX.
2) that said, minimum C, minimum Vf, minimum Ileak. The
scrubbing of piece-part attributes I will leave to you.
3) zero-VT RF MOSFET could be a better choice if the
incident power level is low. Perhaps an IC passive mixer
(FET-quad), and mix down to DC (one of two side tones,
2*2.4GHz being the other).
4) Antenna is the big deal, a good LC tank that can "ring
up" the ambient RF until it starts to get taken by the
harvesting network might help you (looked good to me
way back when I was poking at RFID, which combines
low level energy harvesting with "stuff".