# 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 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
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!