# 3.4 Mos regimes

Introducing the Sah equation, and discussing some properties of this equation.

This equation looks a lot like the I-V characteristics of a resistor! ${I}_{d}$ is simply proportional to the drain voltage ${V}_{\mathrm{ds}}$ . The proportionality constant depends on the dimensions of the device, W and L as they intuitivelyshould. The current increases as the transistor gets wider, it decreases as it gets longer. It also depends on ${c}_{\mathrm{ox}}$ and ${\mu }_{s}$ , and on the difference between the gate voltage and the threshold voltage ${V}_{T}$ . Note that if we adjust ${V}_{\mathrm{gs}}$ we can change the slope of the I-V curve. We have made a voltage-controlled resistor!

Caution is advised with this result however, because we have overlooked something quite important. Lets go back to our picture ofthe gate and the batteries involved in the operation of the MOS transistor. Here we have explicitly shown the channel as a black bandand we have introduced a new quantity, ${V}_{c}(x)$ , the voltage along the channel, and a coordinate $x$ , which tells us where we are on the channel relative to the source and drain. Note that once weapply a drain source potential, ${V}_{\mathrm{ds}}$ , the potential in the channel ${V}_{c}(x)$ changes with distance along the channel. At the source end, ${V}_{c}(0)=0$ , as the source is grounded. At the drain end, ${V}_{c}(L)={V}_{\mathrm{ds}}$ . We will define a voltage ${V}_{\mathrm{gc}}$ which is the potential difference between the gate voltage and the voltage in the channel.

${V}_{\mathrm{gc}}(x)\equiv {V}_{\mathrm{gs}}-{V}_{c}(x)$
Thus, ${V}_{\mathrm{gc}}$ goes from ${V}_{\mathrm{gs}}$ at the source end to ${V}_{\mathrm{gs}}-{V}_{\mathrm{ds}}$ at the drain end.

The net charge density in the channel depends upon the potential difference between the gate and the channel at each point along the channel , not just ${V}_{\mathrm{gs}}-{V}_{T}$ . Thus we have to modify the equation of another module to take this into account

${Q}_{\mathrm{chan}}={c}_{\mathrm{ox}}({V}_{\mathrm{gc}}(x)-{V}_{T})={c}_{\mathrm{ox}}({V}_{\mathrm{gs}}-{V}_{c}(x)-{V}_{T})$

This, in turn, modifies the integral relation between ${I}_{d}$ and ${V}_{\mathrm{gs}}$ .

$\int_{0}^{{V}_{\mathrm{ds}}} {\mu }_{s}{c}_{\mathrm{ox}}({V}_{\mathrm{gs}}-{V}_{T}-{V}_{c}(x))W\,d {V}_{c}(x)=\int_{0}^{L} {I}_{d}\,d x$

is only slightly harder to integrate than the one before (Now what is the integral of xdx), and so we get for the drain current

${I}_{d}=\frac{{\mu }_{s}{c}_{\mathrm{ox}}W}{L}(({V}_{\mathrm{gs}}-{V}_{T}){V}_{\mathrm{ds}}-\frac{{V}_{\mathrm{ds}}^{2}}{2})$

This equation is called the Sah Equation after C.T. Sah, who first described the MOS transistor operation thisway back in 1964. It is very important because it describes the basic behavior of the MOS transistor.

Note that for small values of ${V}_{\mathrm{ds}}$ , a previous equation and will give us the same ${I}_{d}-{V}_{\mathrm{ds}}$ behavior, because we can ignore the ${V}_{\mathrm{ds}}^{2}$ term in . This is called the linear regime because we have a straight-line relationship between the drain current and the drain-source voltage. As ${V}_{\mathrm{ds}}$ starts to get larger however, the squared term will begin to kick in and the plot will start tocurve over. Obviously, something is causing the current to drop off as ${V}_{\mathrm{ds}}$ gets larger. This is because the voltage difference between the gate and the channel is becomingless, which means there is less charge in the channel to provide conduction. We can graphically show this by making the channellayer look thinner as we move from the source to the drain. , and in fact, would make us think that if ${V}_{\mathrm{ds}}$ gets large enough, that the drain current ${I}_{d}$ should actually start decreasing again, and maybe even become negative!. This does not seem veryintuitive, so lets take a look in more detail at the place where ${I}_{d}$ becomes a maximum. We can define ${V}_{\mathrm{dsat}}$ as the source-drain voltage where ${I}_{d}$ becomes a maximum. We can find this by taking the derivative of ${I}_{d}$ with respect to ${V}_{\mathrm{ds}}$ and setting the derivative to 0.

$\frac{d {I}_{d}}{d {V}_{\mathrm{ds}}}}=0=\frac{{\mu }_{s}{c}_{\mathrm{ox}}W}{L}({V}_{\mathrm{gs}}-{V}_{T}-{V}_{\mathrm{dsat}})$
On dropping constants:
${V}_{\mathrm{dsat}}={V}_{\mathrm{gs}}-{V}_{T}$
Rearranging this equation gives us a little more insight into what is going on.
${V}_{\mathrm{gs}}-{V}_{\mathrm{dsat}}={V}_{T}={V}_{\mathrm{gc}}(L)$

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!