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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.

enter image description here image 1: flat

enter image description here image 2: decreasing

enter image description here image 3: increasing

There is also a correspondance between Gate Oxide thickness, the S/D structure design and this critical length.

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  • \$\begingroup\$ Does it have any closed form mathematical relation? \$\endgroup\$ Commented Oct 5, 2013 at 20:05
  • \$\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\$ Commented Oct 5, 2013 at 20:09
  • \$\begingroup\$ would you please include the references to your curves? \$\endgroup\$ Commented Oct 5, 2013 at 20:13
  • \$\begingroup\$ The first from the internet, the others are internal and anonymized from different processes, i.e. not available. \$\endgroup\$ Commented Oct 5, 2013 at 20:35
  • \$\begingroup\$ please specify the CP full expression in your answer. \$\endgroup\$ Commented Jan 15, 2017 at 20:51

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