nanoHUB-U Principles of Nanobiosensors/Lecture 24: Cantilever Sensors Nonlinear Sensing – Flexure FET
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[Slide 1] Welcome back. We are talking about cantilever-based sensors. This is the third lecture on the -- on cantilever sensors. You may remember cantilever sensors we sort of said that it is actually a very simple sensor. A mass lands on a springboard, or a diving board sort of. And it changes the resonant frequency. And that is how it is generally used. Most people, if you tell them that you are thinking about a cantilever sensor they will think that you'd operate it in dynamic mode, that they look at the shift in the resonance frequency when the molecule has arrived. And so that was our first lecture on cantilevers. And then in the second lecture we said that yes, that's good, but it would have been easier if we could detect the deflection after the biomolecule has landed by static shift. And as soon as we put some numbers in we immediately realize that the deflection is so small, you know, couple of hundred maybe femtometer, that it will be impossible to detect by any conventional means. And we said that it would be better if we could somehow weaken the spring so that when the biomolecule lands on the sensor then it causes a big shift, and then we can read it easily. Now there are many ways to read it, but today I'll tell you a special way of reading it by using a transistor, a MOSFET. And we'll call that way of reading it, the whole integrator system, called the flexure FET. And I'll explain the physics associated. Can give tremendous sensitivity unlike that anything we have seen in potentiometric or amperometric sensors.
[Slide 2] So I'll remind you about this notion of weakening the spring by -- and you may remember that we are weakening the spring or making the spring more softer not by changing the material. You know, it is still a silicon beam, still with the same Young's modulus and everything, except now we have used a capacitor and pulled it down close to the critical point. And as I showed last time that the critical point -- at the critical point the spring becomes very weak. So we'll talk about that. And then I will talk a little bit about then the other issues related to flexure FET before I conclude.
[Slide 3] Now very quick reminder that there are three types of sensors, and we are now basically talking about this third sensor. But now I want to specifically look at that picture. You see, when -- last time when I showed you the same picture I deliberately did not talk about the MOSFET that is sitting underneath. I simply focused on the fact that there is this blue cantilever hanging from a spring, and when the yellow biomolecule lands on it it changes the resonant frequency. But you see, what it simultaneously can do is that as it changes the deflection -- static deflection, we can use this MOSFET underneath. Look, this is very similar to the potentiometric sensor on the other side except the gate has been pushed up. And we will use that approach to read the corresponding change in the deflection by using this transistor. So you see, really a hybrid device in that sense.
[Slide 4] All right, so this is how it works. We'll have a simple MOSFET. You'll have the source on the left channel and the drain just like the standard MOSFET. We have a gate dielectric in here, a very thin layer. And then there is an air gap of the thickness Y naught. And on top of it there is a suspended gate. And when the biomolecules -- the red biomolecule land on the suspended gate this is the -- sort of the cantilever, but this time this is fixed on both ends. But that's -- just a way of supporting it. It could be a single cantilever also, a single support -- singly supported cantilever. No matter that once the biomolecules land of course it increases the mass as we know. But at the same time we also know that it can change, the spring constant. And as a result what will happen that if you before the molecule landed if you had pulled it down by this electrostatic coupling then once the biomolecule has landed it can change its position. And that change can be read, and can be -- is deflected and can be read from the current source to drain current, as I will show you in a second. So just to emphasize the essential difference between a potentiometric sensor and a -- this flexure-FET sensor, here it is. In the potentiometric sensor which is on the top the gate the blue gate is directly on top of the channel, the source and drain. And the biomolecules were sort of supported on the gate itself. Now, however, the gate has been lifted up. The gate is suspended, so this blue gate is really like this green suspended gate here. And the bottom dielectric I have not shown here. But that's because it's not essential for this function. So let's see how this particular flexure-FET works.
[Slide 5] So the basic idea is very simple, you see. Assume that in the beginning you had a capacitor first of all this is the same spring mass system. Remember the standard one that we did in the first lecture. We looked at the oscillation frequency and all. Now in the last lecture I told you that we can put -- if you put a capacitor in the structure and connect it up with the voltage then you can change the air gap. So let's say before you have applied -- you have a certain air gap due to specific voltage. And then when the biomolecule comes let's say it makes the whole thing stiff because you have just added a thin layer on top which has made the structure less compliant. And as a result it will essentially go back. It will be lifted back. And as a result there will be a change in the air gap. Now because of this change in the air gap the current flow that can go from source to drain will be different from the current flow from the source to drain before the molecule was captured. So it's very simple, as you can see, that the spring force, k y naught minus y is being balanced by the electrostatic force. And the solution of this equation for a given voltage V gives me the initial gap yb. So the -- solve this equation. It will be a cubic equation. If we solve it then there will be a corresponding value for y -- a single value for y that will tell us what the initial gap was. Now let's say after the biomolecules have landed the spring constant will change. And the spring constant -- because ka has changed compared to kb, even for the same voltage the solution of the equation will change. And the new ya would be somewhat different. And this difference in the air gap essentially is a function of number of biomolecules you have caught, and as a result will be a direct -- give us a direct readout of the number of biomolecules that has been -- that has arrived and landed on this cantilever.
[Slide 6] So let's do some basic math to find out what the dependence is, that if you have, for example, n number of red biomolecules landing on the cantilevers at first how much would the y change? What is -- in other words, what is the difference between ya minus yb? ya and yb. So the thing you'd do from the last slide, you will cross multiply. If you cross multiply -- remember there was the y squared here. That will get cross multiplied. This is before, and this is after. Now the after -- the expression for the after the biomolecule has landed you can expand it, writing ka as kb, whatever was before plus the delta k, the change. And correspondingly write ya after the gap afterward is yb and the change in delta y. And solve this equation. You see, just expand it, subtract it from the original expression, and if you have solved it out this is a -- one line of algebra. And then realize that you want to bias it close to the pull-in point. If you make that assumption then you will see this middle term will drop away, and the bottom line is that you will get a net change in the deflection which is -- whose square is proportional to the number of biomolecules we just landed. There -- NS is the number of biomolecule which has landed, and you can see it's very different from the linear biosensing. In the linear biosensing the total shift was proportional to NS. Here the square of that is proportional to NS. That's why it's called a nonlinear biosensor. All right, so I now know the difference as a function of number of biomolecules captured. Now how am I going to read it? Well, there are many ways as you can realize. You can bounce off a laser light to see what was the initial gap or what is the final gap. That's one way of doing it. Or you can imbed a piece of electric material; that is another way. But the simplest would be to have a transistor tell you the difference between the air gap. Now how is a transistor going to tell you the difference between the air gap? Here is how it's done.
[Slide 8] So the basic idea is very simple, and we have seen it before, that there are three modes of transistor operation. In one mode when the biomolecules land, let's say on the outside -- I'm talking about classical transistors -- when the biomolecules land on the sensor surface, in that case the number of electrons whatever number of electrons you had before, that number increases. That's called an accumulation. Or whatever number was initial charge, that charge increases. For example, you had 100 electrons before in this space, in the green. After 25 biomolecules have landed, let's say each carrying one charge, the green would be 125. So the total would be 125 electrons. So that's called accumulation. Then there's something called an inversion, that whatever was the original type of the substrate, if that number, essentially the molecules -- electrons get depleted and the inverse type of carriers get -- populate the channel, that's called an inversion. And the final one is called depletion, when a charge carrier comes, the majority carriers are shifted away, pushed away. And in that case the -- and the opposite type of charge carrier just begin to accumulate. This is called depletion. So the transistor bottom line is transistor can operate in any of these three modes. Now let me tell you -- show you how the air gap -- change in the air gap can be sensed by -- through one of these three modes.
[Slide 9] So I have turned the transistor around 90 degrees. So you can see the same transistor. There's oxide, source on the bottom, drain on the top now. The green electrons that are going through. Now these are in accumulation. This is the gate on which the biomolecules have landed. It's connected with the voltage VG. And this is the air gap. The air gap is yb. Remember the yb? That was how far away it was before. And of course after the biomolecule has changed -- arrived the air gap will change. So I can simply analyze this structure. This is very easy because you can see, this is just air and oxide. So it's like two capacitors in series. Just two capacitors in a series. So what is the total capacitor of this system? Very simple. When two capacitors are in series then the total capacitor will be C naught. C naught is oxide capacitors and whatever is the air gap capacitors. And the formula comes like this when the two capacitors are in series. And anytime the y changes because the biomolecule has arrived, the capacitors -- the total capacitors will change. Right, very simple. And so therefore, since I've previously told you that the total amount of charge in the MOSFET charge -- there's the green charge -- is equal to whatever amount of charge you have on the mobile gate -- on the suspended gate, which is the blue. So the MOS charge is simply C equivalent -- this is the equivalent capacitance -- multiplied by VG. And so therefore, the current will be whatever charge you have multiplied by the velocity. And the velocity is proportional to the mobility multiplied by the electric field. And electric field is VD over L -- VD is the drain voltage here, divided by L. You see, I don't really have to know all this. Only thing is that any change in I or any change in C will be reflected in I. And so therefore, when the biomolecules land I can read off just by looking at the current how many molecules have landed
[Slide 10] In inversion you know, this is second mode of operation everything is identical. There's no difference. Except that I will need a certain amount of voltage before it begins to invert. And so that voltage is called the threshold voltage. This formula essentially used, but essentially the same thing. The current is again proportional to the amount of charge multiplied by the velocity. Now let me show you how it works when the biomolecules have landed and what the sensitivity you might get.
[Slide 11] So remember, this is my formula. I want to know the change in the current. So therefore, all other things are constant. The change in the current is directly proportional to the change in the total amount of charge in the channel. Now I already know the amount of charge in the channel given a capacitance C, and so therefore, the change in the current is directly proportional to the change in the capacitance. Why? Because VG is constant. I'm not changing VG. Threshold voltage is also constant. So these things don't change. Only thing that change is delta C. And delta C is epsilon naught over yb. That was the initial C. Epislon naught divided by ya, there's the final C. And so therefore, the difference of them will give me the delta C. And if you simplify you will see that the total change in the current is directly proportional to the change in the position of the suspended beam as a function of number of biomolecules captured. Now I have already told you that we have already seen that delta y squared is directly proportional to the number of biomolecule captured, about two slides ago. And so therefore, if you insert it you will see that the change in the current is directly proportional to the square root of the number of biomolecules captured. Now you see this is really a very important result because do you remember in the potentiometric sensors, how were -- what was the dependence? The dependence was logarithmic, remember? There was a log of NS. And you may remember that came from the fact that there was salt solution in here -- there. And the salt was sort of stealing away most of the charge, leaving a very small fraction behind for the transistor. Here we still have salt because without it, you see, this binding would not have occurred. But we are not measuring charge anymore. We are simply measuring the displacement. And that displacement doesn't depend on how many screening you have. And as a result this one doesn't have a log dependence, significantly more sensitive. In fact, you can make it even more sensitive if you operate it in the sub-threshold regime.
[Slide 12] So let me show you how that works. Very simply in this case, in the sub-threshold regime the surface potential psi essentially depends on the capacitors that you have -- again -- once again, air gap and the oxide capacitor and the total amount of voltage that you have applied. So you are simply asking that what is the corresponding voltage at this node when you have a series combination of the capacitors. Now I won't derive it. These things are derived in textbooks. But the only thing I want you to see here -- and by the way, the derivation will be in the appendix -- that this time the charge depends exponentially on the surface potential -- exponentially on this potential at this point. And as a result when there is a small change in the C equivalent capacitors delta psi will change. And once the psi changes the current will change exponentially.
[Slide 13] And in fact you can immediately see this. See, here the change in the delta psi, where is it coming from? Because the capacitors changed -- the total amount of capacitors changed. And since it's on top of an exponential that will give you -- and the change in the current, which is exponentially sensitive. In the appendix I show you that this delta psi, yes, depends on a combination of -- it has a functional dependence on the number of biomolecules. These constants are known. These are all -- all these these things are known. For example, this is the height of -- Ht is the height of each biomolecule. At is the area of each biomolecule. A is the cross-sectional area of the cantilever. So on and so forth. But the bottom line is the sensitivity which is the current before and current after is exponentially sensitive to the number of biomolecules. And you can see how this works. So you can see as you apply a certain voltage on the top left, in the beginning lets say y naught was 100. You pull it down and begin to operate about 9 volts or so let's say you begin to operate the device. Once the biomolecule has landed then you get the red curve because it has gotten stiffer. And there is a significant shift, let's say about 10 nanometer shift, which is easily measurable. And that change is about 8 to 10 nanometers once you calculate the current. Remember, this is the y axis, exponentially dependent. It will go from let's say 10 nano-amps to essentially less than a pico-amp, huge change in the current. And so therefore if you take the ratio between the red and the blue you will see that you can have hundreds of times enhancement in sensitivity. Not log but rather exponential dependence.
[Slide 14] And these things have been plotted. For example, in here you see the classical, standard mechanical sensors that we talked about on the first lecture of cantilevers. This blue curve is a dependence. If you think about potentiometric sensors, then correspondingly these would be the red or the green curves. These are all approximately sensitivities on the order of 1 or so. You know, 1, 2, 3, less than 1, something like this. But the flexure-FET in the sub-threshold can give you very high sensitivity. Now of course one has to look at signal-to-noise ratio and other things that we'll discuss in the selectivity lectures, which is coming down the road. But for the time being if you just look at the sensitivity, the flexure-FET operated in this mode -- in this high-bid mode where you have combined the cantilever with the modern transistor and put them together, that creates the nonlinear biosensors that is exponentially more sensitive than anything that has been around. All right.
[Slide 15] So let me conclude then. So the key idea of this nonlinear biosensing is the notion of spring softening, making the spring weak by biasing it close to the critical point which is for the parallel-plate structures 2/3 y naught. y naught is the original gap. Now in the sub-threshold regime the sensitivity is maximum. Now it's true that I didn't really explain what a sub-threshold is, accumulation, inversion in great detail. For that you need to understand transistor physics a little bit more. But just accept -- if you just accept that these formulas are correct then you will see that exponential sensitivity is possible. Even at accumulation or inversion the sensitivity is considerably better compared to a potentiometric sensor. Now the key of this enhanced response is lack of -- absence of salt screening. Not that the salt screening isn't there. Salt is there, screening is there. It is just that in this sensor mode you measure displacement, not the charge. And as a result even though screening is there, fully present just like in any other case, you don't care about it. And this sensor, essentially, is fully oblivious of that -- of the salt solution. But it doesn't mean that the cantilever sensor despite its sensitivity. First the biomolecules have to come. So everything that we have learned about settling time, everything about selectivity issues, those are still there for all sensors. It is that among the three this one doesn't need any reference electrode, can be miniaturized, easily integrated to nanoelectronic structures, nanoelectronic transistors. And so therefore, this has a number of advantages which makes it very attractive. So you will see a large number of papers in various journals and other books on this cantilever-based sensing. So next time I hope when you read them you will not be afraid. This ends our set of lectures associated with different types of biosensors, right, potentiometric, amperometric, and cantilever. In the next set of lectures we'll take up the issue of selectivity, ability to differentiate between molecules, the right one and the wrong one. And until that time take care.