Calculation of Charge Centroid in FETs

• For small fin thickness (DG-like structures) or channel diameter (CG-like structures),

there’s notable structural and electrical confinement.

• This is also true for the triangular potential well confinement for III-V structures.

• This results in quantum mechanical effects such aso Threshold voltage shift due to band gap wideningo Shift of the charge centroid away from the interface

• The charge centroid location is dependent on the potential profile and hence on the

applied biases

Understanding the centroid shift:• This centroid shift can be understood by considering the electron wave functions

in the potential well.• Consider a triangular potential well as shown:

Understanding the centroid shift:• This confinement gives rise to electron wave functions () for the first two energy subbands as

shown.

• gives the probability density, the maxima of which (roughly) corresponds to the centroid for that energy level.

• Charge density profile for a CG FET is shown

S. Venugopalan et. al., IEEE TED, April 2013

How do you model that?

For a triangular potential well:

The potential profile can be given as

The Schrödinger’s equation becomes

which gives , where is the airy function

A. Dasgupta et. al., IEEE Proceedings of ICEE, 2014A. Dasgupta et. al., IEEE TED, submitted 2014.

A plot of the Airy function is shown.

We can get the location of the centroid from the maxima of as

A. Dasgupta et. al., IEEE Proceedings of ICEE, 2014A. Dasgupta et. al., IEEE TED, submitted 2014.Image source: Nextnano

For rectangular wells:

• For rectangular potential wells, the wave function would be nearly sinusoidal.

Potential Function is given by

V(x)=

Schrodinger Wave equation for well region is

Solution of this equation is

=

we can get the location of the centroid from the maxima of as

= ; m=1,3,5… 2n-1

In this case the centroid will be located at o 0.5L for the first subband levelo0.25L,0.75L for the second , and so on…

• Centroid shift adds to the effective insulator thickness• Affects capacitance behaviour

Effects of centroid shift:

A. Dasgupta et. al., IEEE TED, submitted 2014.