[Slide 1] Welcome back.
So in the last lecture
we had been talking
about cantilever-based
biosensors.
And I focused primarily
on dynamic-- bio sensing.
That once the biomolecule
lands, the natural resonant frequency 
of the cantilever changes.
And today I'll say
a few more things
about the cantilever
dynamic biosensors.
But then move onto
static bio sensing
where the deflection occurs
but you are not relying
on the oscillation
but rather the degree
of deflection associated
with the cantilever.
And we'll see in
the next lecture
that how the static
deflection can have very large
sensitivity, much larger
than you might expect
from potential metric
or amperometric sensors.
So this is sort of in essence
the preparatory lecture
for the next one.
[Slide 2] Let me tell you one
or two more things
about dynamic biosensing
before we leave this topic.
And then I will move
into static biosensing.
Immediately first emphasize
that static biosensing
on its own doesn't work.
You cannot measure true
static biosensing alone.
The mass of the bacteria, you
always need dynamic biosensing
and I will explain why.
But under specific circumstances
if you design the device
in a particular way, you
will be able to measure it
under static biosensing also.
And that has to do with
nonlinear biosensing.
I'll explain how that
works before I conclude.
[Slide 3] Okay, so you remember the
special feature of the reversal
of frequency at the nanoscale.
This is a very important
thing to remember,
that generally we
always have expected
that as biomolecules sort
of gradually build up,
then the resonant
frequency should go down,
should gradually go down.
And you see that happens,
especially at very
low thicknesses
as the biomolecules are building
up and other part does go
down as one would expect.
But what happens as more
and more biomolecules land
on the sensor surface,
it makes it stiff.
Just like starching your
shirt makes things stiff.
And so therefor what
happens is it begins
to compensate the mass effect.
And at some point
what you will see is
that because this composition
is so perfectly balanced
that you will not see any shift
in the resonant frequency,
even after the biomolecule has
landed on the sensor surface.
And of course if you allow a
longer period of time the shift
in the frequency will
actually become positive.
And I explain to you
the critical frequency
and how to calculate it.
So this is an important
consideration
at nanoscale biosensing
that does not happen
at millimeter scale or even at
micrometers thickness membranes.
All right.
[Slide 4] Now one thing I wanted
to emphasize,
that although I have
focused on the peak,
the resonant frequency, you
must have noticed by now
that this is not a delta
function at the omega naught.
But rather, the red
curve is broadened,
and so is the blue curve.
And this broadening and the peak
position is actually not given
by the simple formula
that I told you about.
This simple formula where
the omega naught is simply
proportional to k divided
by m and the square root.
That's really not
everything in this story.
Remember, I neglected
the-- response
or the damping associated
with this.
And the damping fluid is
of course very different
from the damping in air.
And that can have
an important effect.
That is the reason why
this broadening occurs.
[Slide 5] And this is an important thing,
especially when you are looking
at the experimental data.
That when the damping is
present, the gamma is present,
even if you don't have any
constant external forces.
You give it a ping,
allow it to oscillate.
But depending on whether
it is oscillating in water,
which is high gamma, or
in air which is low gamma,
the response would be
considerably different.
And it turns out that you
can solve this equation.
Instead of just having e to
the power i omega naught t,
by a more damped oscillation
with the sine omega prime
t. This is the new resonant
frequency, the omega prime.
And it turns out that the
solution, which will be given
in the appendix that you
can check in the appendix,
it turns out the omega naught,
the new resonant frequency is
k divided by m square root.
That's fine.
But shifted by this gamma
squared, by 4m squared.
So if your damping
is significant,
you can see the resonant
frequency would be significantly
affected and it will be
shifted at a lower frequency.
So one must account for
this damping when you want
to know what is your
original omega naught
and how far it has shifted
from the original position.
By the way, the gamma with
respect to the omega naught,
original frequency
omega naught, undamped,
is related to the quality
factor Q. therefore
you will often hear
that high quality factor
is very important in order
for accurate cantilever-based
biosensing.
All right, so that's
all I have to say
about dynamic biosensing.
Look at the appendix in order
to see how this variation works.
But that's essentially
the basic story
about nanoscale dynamic sensing
at the nanoscale cantilever.
[Slide 6] Now let's talk about
mass-based sensing,
challenges of mass-based
static sensing.
Now dynamic biosensing requires
that you monitor the frequency
and monitor the deflection
of the laser beam on a ray.
That's generally the question
in AC circuit and others.
So therefore it would
have been much better
if we could just bend it
just like a person standing
on a diving board
at the very end.
If the person is heavy, then
the bending would be more.
If the person is light
the bending will be less.
If we could just measure
the bending in the static,
life would have been
much, much, much nicer.
But it turns out
that is difficult.
Let me explain why.
[Slide 7] So assume that in this top
figure the biomolecules are
yet to arrive on the
sensor's surface and captured
by this magenta probe molecules.
And this is a simple silicon
nitride micro cantilever
and could be really thin.
And once the biomolecule
arrives, because of the mass,
also because of the
electrostatic push or repulsion
or because of the
hydration or simply
because the biomolecules
are large,
pushing against each other.
The whole cantilever might bend.
It's just like biometallic
strips.
When you increase the
temperature the whole thing
bends because the
surface stress.
The physics is no different.
Now it could be because of
the mass, surface stress
or the change in
the spring constant.
The whole thing may
deflect in steady state.
The question is,
can you measure it?
Let's see.
[Slide 8] So let's calculate.
We are thinking about steady
state, so life is good.
I don't have any acceleration.
I don't have any damping.
One of the only thing that I have
is the force is being balanced
by the spring.
What is the force?
Well force is mg.
g is a gravitational constant.
All right, so life is simple.
So you can take this
quantity, this balanced force.
Now you can take
log on both sides,
so log k plus log y
is log m plus log g,
and take a differential.
So that makes it
delta k over k naught,
delta y over y naught
and so on so forth.
You remember you see there is no
term called delta g over g naught
because of course hopefully the
gravitational constant doesn't
change significantly when you
change things by a nanometer.
So that is not there.
And so therefore the
net change is equal
to how the mass change due to
mass and change due to spring.
Mass is trying to
make it heavier.
Mass is trying to
make it bigger,
landing on the total spring
and the spring is
sort of pushing back.
And the net difference
gives you the net shift
in delta y. All right,
so that's simple.
By the way, you may remember
that in the resonant
frequency also,
we have this spring competition
between spring constant
and the mass.
And so essentially the reason is
that the essential difference
comes from the same physics.
That will be explained
in a second.
All right, so this looks
like a good strategy.
[Slide 9] So let's take an example.
Let's say you have a
silicon beam again.
Length is 5 micron,
width 1.5 micron.
very thin, density is silicon,
so therefore 2,300 grams.
And let's say you have
some protein molecules,
this prostate specific antigen
marker for prostate cancer.
That comes in, lands
on the sensor surface,
squirts it on the cantilever.
Again I am assuming it about coats the
whole thing uniformly,
length 5 micron, width
about 1.5 micron.
I'm sorry, this would be micron.
Both of them are micron.
H is about 50 nanometer and the
density is somewhat less dense.
So about 1,000 kilograms
per meter cubed.
Let's put it in the
previous equation.
And you have the delta over m.
And once you calculate it you
see the deflection is huge.
Whatever was the
original deflection is
about 100 percent
increase in the deflection.
So you would be very
happy that this is going
to be a very good sensor.
Except that once you
calculate the y itself,
how much deflection
has occurred,
you can see about 40
femtometer which is very small.
There's no technique
that allows you
to measure a distance
that small.
And since it is unmeasurable,
therefore even
with great sensitivity this is
a technique that we cannot use.
So somehow if you
could reduce the k,
if you could somehow reduce
the k. Spring constant,
of course we could increase it.
Now we want to reduce it
by quite a bit, right?
Because you see .0152 which is
already small in the nanoscale.
That didn't do it.
So we may have to reduce
it by a factor of hundred
or thousand before we can
actually measure something.
And so therefore somehow we
have to soften the spring.
Now how are we going
to soften the spring?
This is already small silicon
cantilever hanging there.
How do you soften this spring?
This is something that I'm going
to discuss a little bit later.
[Slide 10] Before I do so, let
me tell you why it is
that we could measure things
in dynamic oscillation,
measure mass.
There is no problem.
But when you try
to do it in static,
we said that oh it is too small.
Why is that?
It turns out that
for the dynamic case,
which was the last lecture,
you remember y was Ae
to the power i omega t
in the real function A equals
sine omega t. So if you get
that acceleration, you will
have omega squared multiplied
by A is the amplitude
of the oscillation.
Now do you remember
what the omega was?
The omega that we
calculated was a couple of gigahertz.
And so by the time
and oscillation was
about a nanometer.
When you multiply these things,
you can see that we are
oscillating at about 20,000 g.
So actually you have
artificially sort
of making the acceleration
significantly larger.
That allowed us to make the
measurement there successfully.
But with the g alone static
deflection just with a single g. We
cannot measure anything.
So that's why this
dynamic sensor worked
but the static sensor
does not work.
We have to do something
about it.
[Slide 11] So let's start by
doing something
about the static sensor.
And the trick would
be the following.
We'll add a capacitor.
Remember previously it was
just a spring cantilever.
Electrical things were
not around sort of.
Now we add a capacitor.
So the capacitor has two plates.
One is a bottom plate and
the top plate would be this
cantilever beam.
And we'll apply a voltage.
Let's see how the property
of the sensor changes as a result.
What it will do effectively
is it will change the k,
reduce the k significantly.
That's what's going to happen.
So again, you will have
the force balance equation
exactly the same as before.
You have that acceleration
term, damping term, spring term.
Equal to the external force,
any force that you apply.
And then because I
have used a capacitor,
there will be an
electrical force associated
with the capacitor.
Now in steady state of course
the time-dependent terms
are 0.
The first and second terms are
0 so I have this spring.
External force and
the electrical force.
How do you calculate the
electrical force in a capacitor?
Again, from the college-level
physics courses you may have
noticed this, that the energy
of the capacitor
is half CV squared.
Everybody knows that.
Now how do you get
force from the energy?
You take a derivative.
Derivative of the
energy with respect
to the change in position.
That's the force.
So if you take a derivative,
it will be half V squared dCdy.
By the way, the negative
sign comes from the fact
that y is measured
vertically down.
That's why.
And the V squared is a constant
because here look
at this battery.
That battery is always connected
to the constant voltage.
The voltage is not changing.
But once the cantilever
has come down a little bit,
the gap has changed and so
therefore the C has changed.
The capacitor has changed.
And so dCdy.
If it were a parallel
plate capacitor,
what is the formula
for the  parallel plate capacitor?
Epsilon naught A over y. And so dCdy
simply will be epsilon A naught
over y squared.
So you can put it in.
You can immediately get the
capacitors, the force associated
with this additional capacitor.
[Slide 12] Now let's think about a
person who does not know
that you have added a capacitor.
They are still thinking that
the same old spring mass system
shown here in B. What
I want to tell you,
that that person will think
that the spring has the
effective spring constant K
effective, which is different
from the physical
k that you have.
Because we have added
this extra capacitor.
This person doesn't know
that there is this capacitor
business going on in here.
So this is the effective
spring constant.
And let me tell you how that
spring constant becomes very,
very small once you put
in a capacitor like this.
All right.
So remember, in equilibrium the
spring constant is being directly
balanced by the electrostatic
force associated
with the capacitor.
And you remember why this one
is over y squared, because dCdy
in a parallel plate
configuration
and this one over y squared.
Now you see, for somebody who
is in the B configuration,
the person may think
that he had an original spring
and there is some extra
force that is coming
from this capacitor, which
he really doesn't know about.
For him, he sees the
total external force.
And essentially the
sum of these two.
And if you do that, then the
corresponding spring constant
would be this dF over dy.
And the spring constant
will be of course the k.
That's fine.
With a negative sign.
That's fine.
But there will be a y
cubed dependence in here
because there is
a y squared here.
And first of all, the
spring now looks weaker,
considerably weaker.
This is our original
k minus this quantity.
The higher the V, the
weaker the spring.
And in fact you can
make the V so large
that the spring constant
may disappear completely.
In that case, remember I will
have a huge change once the
biomolecule comes in.
There will be huge deflection
because I have effectively made
this spring constant disappear.
Now this particular expression
2V squared, you can rewrite it
by using the first
expression here.
This is how it works.
So V squared epsilon A 
over 2 y squared.
So you can essentially take
this quantity, put it in here
and then you will
immediately see
that will give you two-thirds ky
naught minus y. And once you sort
of put it in you will
get the final expression.
So what does it mean?
It says that as soon as you
keep increasing the bias, the voltage V,
the effective spring constant
will gradually go down.
And eventually when
it's two-thirds y naught,
at that point the
spring constant will
essentially vanish.
The spring would have
softened considerably.
[Slide 13] Now what's special about
this two-thirds y naught?
Well in order to understand
it you have to understand
that this spring mass
system in the presence
of a capacitor is a
highly nonlinear system.
It is such that if
you apply a voltage,
the spring will come down,
oscillate and then stabilize.
Put a little bit more voltage, comes
down, oscillates and stops.
But if you put beyond the
sudden voltage as you get closer
to this two-thirds y naught,
it can then snap shut
beyond this point.
It becomes highly nonlinear
capacitor in this system.
And if you do a sort of
simulation you will see
that for one voltage
the black line is sort
of oscillation stabilized
at a given distance.
y naught is about less
than 3 micron.
A little bit more
voltage, red stabilizes.
Blue stabilizes.
But apply a little
bit more voltage
and it doesn't stabilize at all.
And so we are talking
about biasing the sensor
in the blue curve,
close to the blue curve,
where the sensitivity
is the greatest.
So how does it work?
Why is there transition like
that at two-thirds y naught?
In order to understand that,
you have to plug the y,
the displacement, as a
function of the force.
Remember the spring force
goes as k y naught minus y,
which is the red line.
As you pull the spring
more and more away from y naught,
the force you require
increases linearly.
So of course you
cannot go below y naught
because the two places
would be together.
So that's the maximum point.
It increases linearly
with spring constant.
Now what happens because of the
electrostatic force goes as 1
over y squared so in the
beginning the spring force is
significantly larger than
the electrostatic force.
So nothing much happens.
So this is by the way, this y
square comes because of the fact
as I told you before dCdy will
have a y squared dependent.
That's why this F electric
force has a 1
over y squared dependence.
So it stabilizes at
a particular point.
The spring constant has
reduced a little bit
but not significantly.
Now if you increase the
voltage a little bit more,
then you can see the difference
between the electrostatic
and the spring has
narrowed considerably.
So the spring is getting weaker
and at one point essentially
they will be tangential
to each other and this
electrical force will be
always larger.
This blue line will be
always larger than the red
and the spring will not
be able to hold it back.
It will snap.
And so we want to operate
very close to this point
because at this point the
spring is the weakest.
Any small vibration can
cause a significant change
in the displacement.
So you can find out what this
position is simply by noting
that at this position, at
this critical position we have
to be a little bit below that.
The forces are equal,
the tangents are equal.
And if you equate this to
when you will do it as part
of the homework, you'll find
that two-third y naught the
spring is the weakest.
And at that point you actually
have the greatest sensitivity.
Now you can look at it in a
slightly different way also
in terms of total energy.
[Slide 14] This is the spring energy,
ky naught minus y squared
divided by 2.
Electrical energy
is half CV squared.
If we don't have any voltage,
then essentially the
electrical force has no effect.
As you increase the voltage
then you have the pink
and the green curve.
And if you put them
together, you will see
that in the beginning you have
the total system has equilibrium
at y naught.
As you apply more and more
voltage, the spring gets weaker
because the curvature gives
you the spring constant.
The curvature is the second
derivative of this with respect
to y. That gives me
the spring constant.
The curvature becomes
small and at some point
on the green curve essentially
there is no stable point.
It goes and clamps shut.
So we want to operate close
to the weaker spring position.
And that is something I
will explain in detail
in the next lecture, how we
can use this spring weakening,
spring softening in order
to have highly sensitive
cantilever-based biosensor.
[Slide 15] So let me conclude then.
So I told you about
the importance
of dynamic biosensing.
In the last lecture we
simplified things a little bit.
I explained the frequency
reversal and all,
but didn't really emphasize
the importance of damping,
fluid damping, and
broadening and quality factor.
I pointed that out
this time around.
In fact, this is why it's
difficult to go below a picogram
in fluid, because of
the damping issue.
In fact, if you didn't have
damping, if you had a vacuum,
an order of 100 zeptograms
is easily measurable.
And so therefore this
is an important thing.
Damping is very important.
It's not a secondary effect
especially at nanoscale.
Now I explained to you pure mass
sensing is difficult to measure.
Why? Because the gravitational
constant is so weak.
You know, for us
it's very strong.
We can walk around and
we stick to the earth.
That's not a problem.
But think about a little
virus or a bacteria.
They don't care about the
gravitational constant.
In their lives, gravitation
plays no role whatsoever.
So in that case, either one has
to do this dynamic biosensing,
which makes the acceleration
thousands of times larger.
Or have to weaken the
spring in such a way
so that even the small mass can
cause significant deflection.
And one of the ways
of spring-softening,
you can always have a new
material that you can try
to find, very difficult, is
to simply add a capacitor.
And if you add a capacitor
for the biomolecule sitting
on the cantilever surface,
the biomolecule doesn't know
that you have added a
capacitor underneath.
It doesn't know that.
It feels like the overall
spring has weakened somehow.
And when it's landing on it,
the deflection is significant.
And that is the physics
I explained
in the last three
or four slides.
It turns out that physics of
this transition, that transition
of spring weakening,
is that at the heart
of this phase transition,
how water become solid
and the solid ice becomes water.
This phase transition and
this physics are equivalent,
and the great sensitivity
that you have
at the phase transition is
exactly what you are using here
to measure the mass of
a virus or a bacteria.
So that's it.
Let me end here.
In the next lecture I will show
how the static bio deflection
and transistor-based sensing
can be combined together
to greatly enhance sensitivity.
These are nonlinear biosensing.
But until that point, take care.
