Extracting pinch-off and threshold voltages in quantum transistors

Many times I needed to extract threshold voltages from experimental results. I remember in my PhD days, this was quite a debate, and we could generally agree there is no optimum way of doing it. The problem mostly lied in that you are not exactly sure in which part of the plot the current starts to flow, or the channel is depleted. In room temperature transistors, this was mainly due to the intermediate region of thermal population of the bands. In quantum transistors, we are playing in the low temperature field and most of the time dopants are thought to be frozen.

What is really certain is that all voltages that you compare against will need to be extracted using the same method.

Since what we are interested in is changes in the curvature of the plot, we expect to play around with derivatives a lot. But for this, there is the extra problem that experimental results can be very noisy. Once you take the derivative, you introduce further noise, that goes even worse in the second derivative (see figure below).

Characteristics of a transistor with 2 pinch-off positions in the classical regime. First derivative (blue plot) shows pronounced amount of noise.
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