MOS Transistors - III - VLSI Circuits & Systems Video Lecture - Computer Science Engineering (CSE)

hello and welcome to today's lecture on
MOS transistors this is the fourth
lecture on the low-power VLSI circuits
and systems course and third lecture on
Mo's transistors in the last lecture we
have discussed how the operation of a
MOS transistor can be visualized with
the help of a very interesting model
known as fluid model and fluid model as
we have seen can be applied to any
charge control devices including MOS
transistors where we have seen how the
most nster operates and how the the
current flow through the channel between
the source and drain can be controlled
with the help of gate voltage and also
the drain voltage and we have seen that
there are three modes of operation one
is accumulation mode another is the
linear mode and third one is the
saturation mode and based on that the
characteristic of MOS transistors can be
divided into three regions as you have
seen cut off cut off region linear
region linear or active or you know that
that partially charged control device
that means partially deeply that that is
actually called when the inversion
region is partial weak inversion region
and then the strong inversion region is
represented by saturation region rather
weak strong inversion mode is
represented by saturation region and
today I shall discuss the electrical
characteristics of MOS transistors and
here is the agenda of today's lecture
after a brief introduction I shall
consider shall so once again the
structural view of a MOS transistor and
which I shall use to explain the
operation and also derive expression for
most of the the the drain current and
particularly which is known as the
electrical characteristics of MOS
transistor then I shall discuss some
about some of the important parameters
like threshold voltage and another
related subject related topic is body
effect and we shall see how this how the
substrate voltage can be changed to
change the threshold voltage with the
help of body effect then another
important phenomenon is known as channel
channel length modulation I shall
discuss about that and finally I shall
discuss about transistor
transconductance so here is the
introduction as I have already mentioned
the drain current IDs depends on two
variable parameters the gate to source
voltage vgs and drain to source voltage
of course here i did not mention about
another important electrical parameter
that is your body bias that is your body
to drain the body to source voltage
rather source to body voltage that also
affects the operation and with the help
of which the threshold voltage can be
controlled so essentially three voltages
gate to source voltage drain to source
voltage and the source to body voltage
these three voltages will control the
drain current and as I have already
mentioned there are three regions of
operation the cutoff region and this is
represented by the accumulation mode
where there is no effective flow of
current between the source of drain
source and drain and we have already
seen in this mode actually the gate
voltage is less than the threshold
voltage and there is no channel
formation so there is no inversion layer
so there is no channel through which
current can flow so in this particular
region there is no flow of current no
effective flow of current between the
source and drain Here I am telling
effective flow the reason for that is
there is various
all amount of current that flows through
the device but for all practical
purposes we can neglect it that's why
you are calling it effective flow of
current I mean the no effective flow of
current but very small amount of current
flows as we shall see that is
represented by the sub threshold leakage
current later on we shall discuss about
it in more detail then we shall consider
the non-saturated region this is the so
called active linear or weak inversion
region as I mentioned and here the drain
current is dependent on both the gate
voltage and then voltages so the as we
shall see the expression for drain
current will be dependent on both gate
and drain voltages then the third region
which is represented by the saturation
region this is the so called strong
inversion region where the drain current
is dependent on the drain to source
voltage
what depends but does not depend on the
here there is a mistake does not depend
on the gate voltage sorry there is it is
independent of the drain to source
voltage but depends on the gate voltage
that means in the saturation region the
drain current slowly depends on the gate
voltage it does not depend as you change
the drain voltage as we shall see so
these are the three different regions
now I have already shown you this
structural view in the last lecture and
particularly there are three important
physical parameters I am calling it
physical because when you do when you do
the fabrication of a MOS transistor then
you will decide the length of the
channel the width of the channel and
thickness of the silicon dioxide these
three are decided at the time of
fabrication so these physical parameters
are will affect the operation of a MOS
transistor and obviously the electrical
characteristics of the most transistor
so we shall express the drain current in
terms of these three physical parameters
l which the length of the channel W the
width of the channel and the thickness
of this silicon die
and as you can see on both sides of this
channel we have got the source I mean
the the the the diffusion regions
heavily diffuse diffusion regions and
here also we have got another diffusion
region that is called less drain so you
have got source and drain and here is
the gate on top of the silicon dioxide
we put polysilicon layer to form the
gate and we can take electrical
connections from source gate and drain
to realize circuits later on we shall
discuss about that coming to the
electrical characteristics I shall try
to derive the expression for drain
current in the same line of the fluid
model we have seen in the fluid model
because of the voltage that we applied
at the gate some charge is induced
so inversion layer is created charge of
opposite polarity in case of n MOS
transistors it is the it is the
electrons which are created and that
charge can flow rather current can flow
between source and drain whenever you
apply some voltage between source and
drain so charge I mean that in weak that
inversion region is created because of
the gate voltage that you apply and then
because of that creation of charge a
conducting path is obtained which is
known as channel between the source and
drain and current will flow when you
apply a voltage you apply an electric
field between the source and drain by
applying a voltage to the drain with
respect to the source so let me write
down the drain current expression so ids
drain to source current
so ids can be represented as charge
induced in the channel
charged in dude induced em de channel
which you can call it icy rather QC and
that can be divided by the electron
transit time electron transit time
transit time is TM so that means here as
you can see not only creation of charge
is important or formation of inversion
layer is not is important it is
necessary that the electron will move
from source to drain as you apply a
voltage electron will form from from one
end source end to the drain end and that
will lead to a constant flow of current
and as a consequence this electron
transit time is important so as you know
that Q that charge that is accumulated
in a parallel plate capacitor is equal
to C into V where C is the capacitance C
is the capacitance and as you have seen
it can be considered as a parallel plate
capacitor and V is the voltage that is
applied across the plate a parallel
plate capacitor and as you know this
capacitance C is equal to epsilon a V
epsilon a area of the plate and the D
thickness of the silicon dioxide layer
so here epsilon is the permittivity of
the insulator permittivity
permittivity of the insulator and this e
will be equal to you know in this case
we shall be using silicon dioxide so yo
X is equal to three point roughly
approximately 3.9 epsilon 0 where
epsilon 0 is the permittivity of the
free space and this we shall use to
represent thee to find out the
expression for the current now this is
the expression for capacitance what
about the electron transit time Tian
Tian will dependent on what what thing
first of all it will depend on two
things number one is speed and second
thing is the distance so we can say that
transit time T n will be equal to me
when into e d s that means mu n is the
mobility of electron mobility of
electron here we are considering n Mo's
type transistor so the charge carrier
our electrons that is why you are
considering mobility of electron and EDS
is the elector is the drain to source
electric field we shall find out what is
the drone then process electric field
EDS it is dependent on what it is
dependent on the voltage that we apply
across the across the drain to source so
the EDS will be equal to VDS by length L
so EDS by L now based on this we can
find out the the electron transition
time so electron transition time will be
equal to first of all let us find out
the velocity of electron tau n tau n is
equal to me when VDS by L and therefore
TN electron transition time this is the
velocity of electron
and TN will be equal to electron
transition time that will be equal to L
by tau n length by tau n that will be
equal to L square mu n by VD s so we
have got the expression for the electron
transition time and it has been found
that typical value typical value of MU n
we when that is the mobility of electron
is equal to 650 centimeter square by
volt and that is at room temperature so
we have got the expression for we have
got the Kappa in the value and the the
charge that we can obtain by C into V
and we can we have also found out the
electron transition time T now we can
find out the current IDs now to do that
we shall divide the entire region into 3
parts as we have seen number one is
cutoff region when the voltage is less
than the threshold voltage there is no
channel formation so there is no current
so you are not writing an expression for
current for that now we shall write down
the expression for the non-saturated
region non saturated or we also call it
linear or weak inversion region in this
weak inversion region we can find out
the for the current first of all let us
find out the effective voltage that is
being applied between the source and
drain as you have seen we are applying
a voltage VDS between the source and
drain sorry on the gate I'm invidious
between the source and then let me draw
the diagram here a little that will make
it easier to understand so here is your
source and here is your drain this is a
channel then you have got the silicon
dioxide and top of that you have follow
silicon so what you are doing you are
applying a voltage VDS positive voltage
so this is your VDS between the source
and drain and you are also applying a
voltage vgs between source and gate this
is your source drain and is your gate so
vgs now although you are applying a
voltage vgs between the source and gate
the effective voltage across the channel
is not same because this drain voltage
that you are applying is interacting
with the voltage that you apply at the
gate so although you are applying a gate
voltage vgs here the effective effective
voltage in this channel region will be
different for example in here the
voltage you know that the the hole the
voltage VDS will be will be across this
length L and here you can say the
voltage is zero and here the voltage is
VD s so if you take the average value of
0 and VDS this will be equal to VDS by 2
therefore the effective gate voltage
will be equal to vgs that you apply here
minus VT this whole voltage as we know
you the you have to subtract that
threshold voltage because you know the
before that there will be depletion
region you will even require a voltage
VT only after that the inversion region
starts and as a consequence the
effective gate voltage will be vgs
- VT minus V D s by 2 we have taken the
average fell of course it will be it
will be varying after the channel but we
are taking the average value this is the
effective gate voltage now the current
we can derive the expression for current
we can expression for current for that
purpose first we shall find out the QC
this QC charge induced in the channel is
equal to C into V that means C is equal
to the Q C is equal to C into V that is
equal to C what is C value of C epsilon
this is your oxide into into area that
is equal to W into L because we have
seen that parallel plate capacitor has
an area W with W and L is the length so
area is equal to W into L by D this is
the this is the capacitance and your
effective voltage will be equal to vgs
minus VT minus V D s by 2 so this is the
capacitor charge that will be induced in
the channel so this is the charge and
now we know the we already know the
electron tons each time electro what was
the electron tons each time that you
find out this is equal to L square by mu
VDS so this we can substitute that means
IDs IDs is equal to QC by TN that is
equal to Epsilon oxide into W into L by
D into vgs minus VT minus V D s by 2 and
it will be divided by L square
this is the expression for electron
transit time this is the L square TN is
equal to L square by mu and DPS so what
will happen here you will have L Square
and mu will be here QN will be here in
the numerator it will go to the
numerator and here you will have VD s so
this is the this is the expression that
we get this is mu N and this L will
cancel with square so ultimately we get
an expression that is equal to L square
WL by D the current expression that we
get is equal to W mu n this is mu n
epsilon o X this is your permittivity of
electron into D into L and here you get
vgs minus VT minus VDS by 2 into 3 DS
now this is the this is the this is the
current that flows through the electron
so that will be the current that will be
flowing through the electron now you can
express in terms of two other important
parameters one is known as K K is a
constant which depends on the
fabrication so what is the value of K k
is equal to we can express it this
current in terms of K which is equal to
MU and this new N and epsilon of X by D
you know when you fabricate these three
parameters are kept usually constant we
usually change the length this early
changed with the changed but these three
are usually kept constant that's why
this is sometimes is replaced by K that
means if we substitute this value of K
we get IDs is equal to K W by L vgs
minus VT minus VDS by 2 into VDS another
we can also represent it in terms of
what is known as the gate capacitance cg
what is the hell of CG CG is equal to
this is your e o X into WL by D that we
have already seen that is a capacitance
this this is the we have already seen
epsilon a by D here is control W by L so
we can also express ideas in terms of
this capacitance so in terms of K
constant in terms of CG so IDs we can
write as IDs is equal to CG mu n by L
square into vgs minus VT - okay VDS by 2
into VDS okay so this is the current
that we get whenever the circuit is in
the linear region it may you may note
that we are calling it linear region but
it is not exactly linear why it is not
exactly linear because you can see if we
assume that VDS by 2 is too small
compared to vgs minus VT only then it
becomes proportional to VDS that means
if we neglect this part only then it
becomes linear and that is a reason why
the this the linear region is not
exactly linear but it approximately
linear particularly when the drain
voltage is very close to 0 only then it
is linear so that means VDS is
negligible compared to VG's minus VT
then it is linear okay so we have
derived the expression for
the nonlinear region now let us consider
the saturated region so we can say in
the second part is your saturated region
so in the saturated region what is the
key difference between the linear region
and saturated region one important
difference is in the saturated region
the current is independent of the same
voltage and when it happens when the
voltage drop across the channel is equal
to the effective gas voltage then if the
drain voltage is such which is equal to
VDS is equal to VG's minus VT that means
this occurs when VDS is equal to vgs
minus VT this is the effective gate
voltage which is equal to the den
voltage beyond which beyond this point
the drain current is independent of VDS
that means this is the point from where
we can consider we can say that
saturated region has started below that
it is linear or non-saturated region so
what we can do we can obtain the
expression for drain current in the
saturated region by substituting VDS is
equal to VG's minus VT so we can say
that ids is equal to kW by l into vgs
minus VT minus vgs by 2 into here you
can substitute this sorry this is d s
VDS minus sorry v GS minus VT we are
substituting this vgs minus ax t so this
is equal to kW by l into vgs minus VT
square and here it becomes here also it
will be
- VT so it will be equal to vgs minus VT
square by two okay so this will be equal
to K WL by 2 because hop this if we VG's
minus VT square minus vgs minus VT
square by 2 will become vgs minus VT
square by 2 so this will be equal this 2
we have taken here so k WL KW by 2 l
into vgs minus VT square so here indeed
we find an expression where it is
independent of the drain voltage so what
it is there is a it is a there is a
squirrel or dependence on vgs minus VT
that means this saturation current will
increase at the square law as the gate
voltage is increased so that is
proportional to vgs minus VT square so
this is the expression for drain current
and also you can express it in terms of
the capacitance that will be equal to CG
mu n by 2 L square into vgs minus VT
square so this is the expression for
that in terms of CG gate capacitance and
of course you can you can represent it
with the help of another parameter there
is another parameter which is known as
CE o x co x and CG these two are
different
the Co X is the you need gate
capacitance that means capacitance of
unit area so what is the relationship
between CG and co x cg will be equal to
Co X into W that area I have to multiply
W into L so Co X into W into L if we
substitute it here we can write it in
this way this L it will become at L
square this L will cancel out so it will
be
muin will be there in any case then w
will be there and it will be Co X by 2
into L and here it will be vgs minus VT
square so this is in terms of the unit
gate capacitance co x okay so we have
seen the expression for saturation
current now let us represent the n MOS
electrical characteristics and moss
characteristic let us now plot that
saturation and non saturation current we
have seen that we can divide into three
regions let's assume it varies from 1 2
3 4 5 1 volt 2 volt 3 volt 4 volt and
fiber and this IDs usually varies in the
range of in the milliampere region so it
is milli ampere so 0 to let's say you
know it is 0.3 0.3 0 mph
so usually it will vary in this region
now as you have seen there are three
regions IDs is equal to 0 for vgs less
than VT we can write V TN if we consider
it n MOS transistor so which part will
represent that this line that means this
line will correspond to this particular
line will correspond to this region that
means irrespective of the drain voltage
current is 0 ids is 0 so this represents
the cutoff region car top
region so this this is your cut abrasion
now nonlinear linear region IDs is equal
to we have already seen the expression
we can write it KW by l into vgs minus
VT minus vgs VDS by 2 into VDS this part
as we have seen is separated by a line I
mean this corresponds to VDS is equal to
vgs minus VT that means this part is
your linear region so obviously there
will be different curves for different
gate voltages so this is this
corresponds to V let's assume vgs is
equal to VT plus 0.5 volt let's assume
this corresponds to VT plus 1 volt 1.0
volt and this will correspond to let us
assume this is VT plus 1.5 volt okay so
the in this part the current is constant
this part current is constant so this is
the saturation region so this is the
linear region this is the cutoff region
and this is the saturation region
saturated region and in the saturated
region the current expression is given
by and okay this is for this
non-saturated region is for V G s is
greater than equal to V T and V D s is
less than vgs minus VT
that means VDS has to be less than vgs
- PT in this region and VJs has to be
greater than 50 so for these two we call
it linear region this is cutoff this is
your cutoff region car top this
corresponds to cut opinion this
corresponds to linear non saturated or
weak inversion region and this
corresponds the third region this is
your one two three IDs is equal to we
have seen the expression is kW by to l
into vgs minus VT square and this
happens when vgs is greater than equal
to VT and also VDS has to be greater
than VG's minus VT so this is your
saturated region this corresponds to
saturated so we have seen we have
derived the expression for drain current
for to regionals linear and saturated
regions and we have defined the regions
based on the gen faulty and gate voltage
here we have not taken into account the
body voltage with respect to the source
we have assumed that the body is
connected to the source that means if we
consider this diagram this simple
diagram we can say this body is
connected to ground
that means this is connected to ground
source is connected to ground and body
is also connected to time and usually
this is done you know by having a region
this is called p plus and this p plus is
actually used to connect the substrate
to ground or to some other voltage later
on we shall see how we can really apply
this voltage
to have different assault polities so
for the time being let us assume that
this drain and the source to substrate
voltage is zero okay now we have
considered the N Mo's depletion type
transistor layer we have n Mo's
enhancement type transistor let us now
focus on n Mo's depletion type
transistor and as we know in case of n
Mo's depletion type transistor a channel
formation is done by implanting suitable
type impurity in the channel region so a
conducting layer already exists and as a
result you don't require a gate voltage
to to form a channel so current can flow
even when the gate voltage is zero and
you can you can stop the flow of current
by applying a negative gate voltage and
that is precisely shown in this diagram
you can see here even when the gate
voltage is zero there there is a flow of
current and if you want to stop it you
have to apply negative voltage that is
minus 1 minus 1 minus 1 volt minus 2
volt and so on otherwise there is no
difference in the characteristic curve I
mean it is very similar to the N Mo's
depletion type transistor I mean
enhancement type transistor except that
these gate voltages are different for
different currents that means normally
in case in case of enhancement type
transistor this will start when the gate
voltage is slightly more than the
threshold voltage but here as you can
see this is minus 2 volt so in case of
depletion mode of transistors only
difference that we will get is that for
different gate voltages you will get
different curves that means you have to
apply a negative voltage to stop the
flow of current and if you apply
positive gate voltage then current will
increase and obviously as we know that
it will increase following square law
vgs minus VT square
o so we have seen the characteristic
curve for n MOS type transistors both
enhancement type and depletion type what
about the
pti power transistors actually that can
be drawn say p MOS transistors we shall
not derive expression for that I shall
simply show you the characteristic curve
how it will look like so you have got
four quadrants you can see here you have
got four quadrants this is your IDs and
this is your vgs this is positive and
his negative for n MOS transistors the
car base on this side on the other hand
for few MOS transistors the car will be
on the negative side that means you will
be applying your the pole then the
voltage that drain voltage sorry this is
your V D s VD s so here it will be
negative here also VD s but here it is
say 0 1 2 here it is minus 1 - 2 - 3 - 4
- 5 and so on that means you will apply
a negative gate voltage a tip voltage to
the drain with respect to source and
current will flow in the opposite
directions and these are four different
different gate voltages and more
negative they correspond to more
negative gate voltages and just like
your n MOS transistors here also there
is a and there are three regions cut off
so just like your n most in most
transistor here also you have got linear
region this is the saturation region
and this line obviously will represent
the cutoff region this is the cut
opinion for a MOS transistor this is the
cutoff region for n MOS transistor so on
that means this this this the the
characteristic curve for P Mo's
transistor will be on this quadrant both
for enhancement type and depletion type
okay now let us focus on the threshold
voltage this threshold voltage has been
found to be one of one of the most
crucial parameter which controls the
operation of a MOS transistors MOS
transistor so here you can see this
corresponds to n Mo's enhancement type
transistor you have to apply a threshold
voltage
I mean VT n to start the flow of current
and characteristic curve will be
somewhat like this that this is the
drain current will increase and as you
know this essentially represents the the
drain current in the saturation mode
that means in the saturation mode
although I have drawn it linearly but it
will be quadratic as we have seen this
current is proportional to IDs is
proportional to vgs minus VT square so
it increases at this rate so far as the
depletion type transistor is concerned
carb is somewhat similar but it will be
shifted towards the towards the negative
quadrant as you can see you have to
apply a minus VTN a threshold voltage
which is the minus VTN in this case to
stop the flow of current and if you
increase the ND i mean negative to a
less negative to zero and then positive
then the drain current in disease
following the same nature of cod and of
course the the amount of current will be
dependent on the gate voltage so you can
see the these two essentially represent
the
again the drain current with respect to
get to source voltage for enhancement
type transistor and for depletion type
transistor similarly you can consider
the gate to source voltage for P Mo's
type transistors this corresponds to the
characteristic curve for P Mo's type
transistor here you can see the
variation of drain current with respect
to the gate voltage gate to source
voltage and a negative voltage is
required to for the flow of current and
you can see the curve is somewhat
similar but inverted in this case
because in there it is in a negative
quadrant similarly this is this discard
correspond to corresponds to the P Mo's
depletion type transistor and here as
you can see you have to apply a positive
voltage to stop the flow of current and
as you as you make it smaller and
smaller and when it becomes zero this
side is the gate to source voltage so
when it becomes zero still there is a
flow of current and as you make it more
and more negative the current increases
in case of P Mo's depletion type
transistor so these four these two and
these two represent the the variation of
get gen current with respect to the
variation of gate to source voltage and
you can see the threshold voltage
actually plays a place yes I act
somewhat like the cutting voltage of no
cutting voltage of bipolar Junction
transistors as you know in case of
bipolar Junction transistors flow of
current is dependent you know you
require a cutting voltage for example
for silicon the best base voltage should
be more than 0.6 0.7 volt that is known
as cutting voltage here also here also
it is somewhat similar to that but we
call it threshold voltage not cutting
voltage okay
now one one very important as I told
threshold voltage plays a very important
role and on what factors or parameter it
defends you can really derive an
expression for threshold voltage in
terms of various parameters but here I
have simply taken the expression without
and I'm not going to derive it and as
you can see VT the threshold voltage is
equal to VT 0 plus gamma into root 2 5
minus 2 Phi plus VT PSB absolute value
of that and minus absolute value of 2
Phi V here this gamma is known as
substrate bias come coefficient Phi
substrate bias coefficient as you can
see this vs b vs b is actually the
source to body voltage that you are
applying source to body voltage that
will be applying that means if we
consider a curve here you can apply a
voltage between the source and body that
means this particular source and this
body you can apply positive voltage to
the body with respect to the source and
that will actually increase the
threshold voltage as we shall see so by
changing the body bias substrate body
bias the threshold voltage can be
controlled and in this expression you
have got vs B that's why it is called
substrate bias coefficient gamma and
gamma is equal to root 2 Q Q is the
charge of lectern epsilon si which is
the permittivity of electron and na na
is the carrier concentration in the in
the in the substrate so na is the
carrier concentration of the substrate
and Co Co X is the unit efficient unit
the capacitance of the gate region so
this is the expression for gamma so it
will depend on these parameters
and five is the Fermi band potential
which is equal to KT by Q K is the
Kelvin constant and T is the temperature
in absolute value of temperature not in
degree or Fahrenheit and is the absolute
value of temperature and Q's the charge
of electron and ni is the carrier
concentration of the intrinsic silicon
so intense axilla Khan will have some
carrier consideration this is the Ni and
na is the carrier concentration of the
substrate as I have already told so
based on that you can find out the
threshold voltage by substituting
different parameter values for a
particular fabrication technology
obviously so far as this part is
concerned Q epsilon si K these are
constants on the other hand the Co X is
the unit gets gate capacitance or these
carrier concentrations you can will vary
from process signal of one process
technology to the other process
technology and as a consequence the
threshold voltage will be different now
let us take up an example let us see the
calc we can later you can calculate the
threshold voltage and you can see how
exactly the threshold voltage varies as
you vary the substrate bias YB is equal
to you can substitute different
parameter values this ni is equal to 1
point 4 5 into 10 to the power 10
centimeter
I mean per centimeter cube and na is
equal to 10 to power 16 so 0.26 and
actually this KT by Q as you know is 20
26 millivolt that is the KT by Q value
and this is the ratio of the ni by na
obviously the intrinsic silicon has much
layer lower carrier concentration
compared to the carrier concentration of
this substrate that means as you know
substrate is uniformly
I adopt to have more carrier
concentration compared to the intrinsic
substrate and Phi B value is calculated
see this is the value of Co x5 e is
minus 0.35 and Co X is equal to epsilon
X by T o X and is the value 7.0 3 into
10 to power minus 8 farad per centimeter
square and you can substitute the
various values to get gamma gamma is
equal to becomes 0.8 to here the value
of the charge of electron the
permittivity of silicon dioxide na these
are all substituted and Co X these are
all substituted and by substituting
various values you get V T is equal to
roughly point 0.4 assuming that point
four is the threshold voltage whenever
the body bias is 0 that means source is
connected to the body and this part is
is quill this is the point 8 2 this is
your gamma into root 0.7 plus vs V we
have to take the absolute value and
minus root point 7 now you can actually
very vsb and to get different from
different value of threshold voltage so
threshold voltage VT is not constant
with respect to the voltage difference
between the substrate and the source of
the electrons as you can see using that
expression how the threshold voltage can
change as you as you change the
substrate body bias so you can see here
when the substrate bias is zero as you
have seen the threshold voltage is point
four volt but as you increase the
substrate bias say from zero to one goal
to 2 volt to 3 volt it can it can reach
say one point six so for a large range
point four to one point six a large
change in the threshold voltage that can
take place as you vary the
the substrate via switch and threshold
voltage will change and later on we
shall use this and see how the threshold
voltage can be controlled to reduce what
is known as leakage current because the
leakage current is heavily dependent on
the threshold voltage and later on we
shall see instead of this positive bias
we shall usually apply negative body
bias to low to - to increase the same I
mean with respect this is called the
reverse body bias we shall do that to
increase the threshold voltage such that
the leakage current is lowered that
means larger the threshold voltage
smaller is the leakage current smaller
the threshold voltage larger is the link
leakage current later on we shall
discuss more about it
so this is the body effect now we shall
consider another very important
parameter that is your channel length
rather phenomenon channel length
modulation here in our expression for
drain current we have assumed that this
L is constant fixed it does not really
change but in reality you will see that
this L is dependent on the drain voltage
however this dependence is not visible
until the channel length is small that
means it is it becomes predominant in
short channel devices when the channel
length is say 0.25 micron 0.25 micron
micro millimeter then this variation is
so small that we don't really bother
about it but whenever you are
considering say 60 nanometer nanometer
or say 45 nanometer technology instead
of 250 nanometer technology then this
channel and variation becomes visible or
it cannot be neglected in number first
we shall show you why this variation or
take place you can see the channel
region is not uniform throughout the
channel the reason for that is here the
drain voltage is 0 here the den voltage
is high and because of the interaction
between the gate voltage and drain
voltage the dense the the thickness of
the channel region is more near the
source and it is very narrow near d.j
and as we further increase the drain
voltage what happens at some point of
time the very close to the drain there
is no carrier that means it becomes
pinch off if we increase the drain
voltage further you will find that
that that pincher point moves towards
source so here there is small region
where there is no charge carrier
so essentially channel length is now
becoming smaller than the the physical
length of the device so effective
channel length is smaller and this is
the that's why we call it channel length
modulation essentially it is dependent
on the drain voltage how do you take
into account we can take into account by
substituting L effective which is equal
to L minus Li Delta L so n minus Delta L
is actually we can represent it by this
we can represent it by this one minus
this is the expression where we have
substituted effective channel length and
we have got this expression and this and
this variation is proportional to VDS
minus VD set essentially when the
saturation starts that is your vgs minus
VT and this is the VDS so this change in
the channel length is proportional to
the drain voltage when the drain voltage
is large only then this reduction occurs
and this can be represented this change
1 minus Delta L by L can be represented
by 1 minus lambda VDS where lambda is
the is known as channel length
modulation coefficient so we can now
express ideas in terms in terms of this
lambda and it becomes like this mu n Co
X by 2
WN by L n into vgs minus VT square into
1 plus lambda VDS so you can see here it
is no longer 1 it is here normally it is
VDS so but it is 1 plus lambda VDS so it
varies with drain voltage and this can
be depicted pictorially with the help of
this diagram as you can see in the
saturation region the although I have
drawn it in a little X exaggerated form
it will not be the slope will not be so
high but you can see the drain current
is varying even in the saturation region
because of channel-length modulation
effect and you can see here it is
becoming the it is no longer parallel
but it is there is a point where it is
touching and this is how the
characteristic can be represented okay
now another very important parameter
that is known as transistor
transconductance for bipolar Junction
transistors which is a current control
device it is represented by gain of the
transistor beta the collector current by
base current on the other hand in case
of MOS transistors it is a voltage
control device and similar parameter is
known as transconductance that means
rate of change of drain current by rate
of change of gate voltage so here we
have seen the the drain current is
controlled by the gate voltage so this
transconductance is represented by delta
ids change or change your drain current
by change of gate voltage Delta VDS
keeping the drain to source voltage
constant so that we can derive as we
know ids is equal to QC by TSB so delta
ids is equal to delta QC by TS d and
this is the TS d that expression we have
already derived i'm not going into the
details of that so delta ID delta ids is
equal to MU n cg s by L square into VDS
into delta vgs and GM is equal to delta
ids by delta vgs as a as you have seen
which is equal to c g into mu n into VDS
by l square and finally by substituting
VDS is equal to VG's minus VT we get G M
is equal to MU n epsilon AI NS into
epsilon zero w by D into l into vgs
minus VT so you can see the the gain i
am there which is represented by a
transistor transconductance
Mord again it is better we can see it is
dependent on the mobility of the device
width of the device thickness of the
silicon dioxide length of the channel
and the effective gate voltage which is
equal to
- VT so this is the parameter and more
we can how we can increase the value of
GM is clear from this expression which
parameter we can clearly that means if
the width is more more current will flow
so that's why GM will be more if the
thickness is small of the silicon
dioxide gate voltage you will have more
control and obviously GM will be large
similarly the length is small more
current will flow so GM will be increase
so okay so here we can summarize we have
discussed about three regions of the
operation of MOS transistor we have
derived the expression of threshold
voltage and explained what is known as
body effect and we have also considered
channel length modulation phenomenon and
then explained what is known as
transistor transconductance so with this
let us stop here in the next lecture we
shall discuss how a transistor can be
used as a switch thank you