Is there a monotonic relation between the channel length and threshold Voltage and is it proportional or inverse?
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The short answer is - yes.
The slightly longer answer, it depends on the fabrication details of the transistor and it's targeted use case.
In all cases, below a critical length \$V_{TH}\$ decreases with decreasing length. However once this critical length is met the \$V_{TH}\$ can increase, stay the same or decrease. Here are some pictures.
image 1: flat
image 2: decreasing
image 3: increasing
There is also a correspondance between Gate Oxide thickness, the S/D structure design and this critical length.
answered Oct 5, 2013 at 16:28
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\$\begingroup\$ Does it have any closed form mathematical relation? \$\endgroup\$ Oct 5, 2013 at 20:05
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\$\begingroup\$ I've not seen any, just collections of points. Good question BTW. It might be difficult though as what dominates below critical length is different than what dominates above it. All though I suspect that perhaps Vittoz/Enz charge based transistor models might be able to do this. \$\endgroup\$ Oct 5, 2013 at 20:09
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\$\begingroup\$ would you please include the references to your curves? \$\endgroup\$ Oct 5, 2013 at 20:13
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\$\begingroup\$ The first from the internet, the others are internal and anonymized from different processes, i.e. not available. \$\endgroup\$ Oct 5, 2013 at 20:35
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\$\begingroup\$ please specify the CP full expression in your answer. \$\endgroup\$ Jan 15, 2017 at 20:51