Here are the results of various diode curves that Robert Pease measured.
The slope of the curve is "conductance" with high conductance diodes having a steeper slope. Common transistor base-emitter junctions have an ideal 60 millivolts per decade of current but most silicon diodes are inferior because they are optimized for other things, although even the low leakage diode he tested was not as good as a transistor base-emitter junction. Schottky diodes come in different types with different values of conductance; HP had different categories for low, medium, and high conductance.
He also mentions that red LEDs can have very low forward currents, like 1 picoamp at 0.6 volts forward voltage which is useful in some circuits.