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Usually it is calculated from the diode equation with all the other variables accurately measured (since those are easier)
But I imagine a good pico-ammeter or electrometer could measure it at room temp.
From the diode equation, if you hold the voltage across the diode at n*Vt*(Ln 2) Volts, you will make the diode current equal to the saturation current.
But you can always measure at two different points and calculate it.
If at room temp, Vt = 25 mV and Is is a very strong function of temp so I'm not sure how useful of a measurement it would be.
I just might try this and see if I can measure it directly in one shot at room temp.
So I placed a 1N4148 in a closed box to shield from air currents and carefully wired a pico-ammeter to it making sure my cables & setup would not contribute leakage. When I sourced 26mV in the forward direction, I measured 2.6 nA (somewhat higher than I expected) at a stable temperature of 25C. This should be equal to the Is @ 25C.
A Spice model of a 1N4148 shows Is = 2.68 nA so the measurement seems feasible. Other diodes however may have Is many order of magnitudes lower so it will be more difficult to measure those small current. My electrometer can measure down to 1E-15 A with great pains but that's not a 5 minute test setup.
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