The photon count rate is a standard measurement for SNSPDs where the bias current is increased and the count rate with constant illumination is recorded. This reveals the “intrinsic” detection efficiency ($DE$) of the structure.

The counts show an imperical sigmoidal shape and is fitted with different functions by different groups.

• Sigmoidal Power Law \cite{Baghdadi.Schmidt.ea:2020} where $I_{B0}$ is the inflection point and $p>1$ is changing the steepness of the transition.

$$ DE(I_B) = DE_\mathrm{sat} \left( 1 - \frac{1}{1 + \frac{I_B}{I_{B0}}^p} \right) $$

• »Standard« logistic function with $\sigma$ as width of the transition.

$$ DE(I_B) = DE_\mathrm{sat} \left( 1 - \frac{1}{1 + \exp{\frac{I_B - I_{B0}}{\sigma}}} \right) $$

• Error function based sigmoidal curve \cite{Caloz.Korzh.ea:2017}

$$ DE(I_B) = DE_\mathrm{sat}/2 \left( 1 + \mathrm{erf}~{\frac{I_B - I_{B0}}{\sigma}} \right) $$

Unfortunately this models do not catch the “initial” slope of the PCR as e.g. described here ().