Diodes - minimum forward conduction current
While most silicon diodes need around 700mV to conduct 'sensible' levels of current, different diodes can have very different saturation currents, and so very different voltage drops at sub-&mu;A levels. For instance the beefy rectifier 1N400x series can conduct a few &mu;A at 100mV.

Some diodes are specified as 'low leakage', which means their saturation current is very low. As part of a study into low leakage protection of amplifier inputs here, I measured the low current voltage of a BAS116 diode (not particularly thoroughly). Several interesting data points are  

  Vf     If
300mV   40pA
450mV   45nA
640mV   16uA    


Note that these are measurements at ambient, not specifications or over temperature, the data sheet specs are much higher. You can rest assured that 1N4148 and 1N400x conduct orders of magnitude more current at the same forward voltages.

At the time (2016), BAS116 was listed as 'until stocks exhausted', so it may have become unavailable since then. However, I expect you can find alternative diodes advertised as 'low leakage'. They are easy enough to test on a bench with a cheap meter and a bit of care. With a 10M input resistance and a 200mV input range, you should be able to resolve 10pA.