What Determines the Forward Voltage of an LED at low Currents?

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A number of years ago I required a low voltage, extremely low current, voltage regulator.

Normal Zener diodes do not function well at low currents.

I became aware that the forward voltage of red LED's was approximately 1 Volt at even extremely low currents (less than 1 microamp).

My question is (and I have been unable to find an answer), what physical properties determine the forward voltage of LED's at low currents?

Manufacturer's data sheets all indicate that it is approximately 1 Volt at very low (zero?) current.
 

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I understand it comes from the energy in electron volts of the light that the LED emits.
 

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