nanoHUB-U Principles of Nanobiosensors/Lecture 3.3: Potentiometric Sensor Charge Screening for Cylindrical Sensors
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[Slide 1] Welcome back. We are talking about different types of biosensors in this set of nine lectures. And right now we are focusing on potentiometric sensors. Amperometric and cantilever sensors will come next. You may remember potentiometric sensors essentially senses the charge and it converts it to potential in the channel. And that potential you can measure as a change in the current in a MOSFET. And so therefore MOSFET is good because it will allow you to integrate on a IC or nanotechnology platform, which is good. Now, if you'll quickly recall where we are, we said that in the beginning, that if you take a simple theory, don't think about it much, then the sensitivity is linearly proportional to
[inaudible] density. You see, that would be great if that could work. But we found out that salt is playing the role of spoiler in our sensors. And therefore, actually, we found a logarithmic dependence, which is far weaker. You see a factor of 100 gets translated one linear factor of 2 increase in sensitivity, which is not very good. So the question, then, you might ask, well we did all this analysis for planar sensors. If we go to nanowire sensors, remember the nanotechnology allows us to create very small sensors. Maybe this straining limit will go away. I will answer that question today and this lecture. And turns out, unfortunately, that that will not really be the case
[Slide 2] So, the outline, again, very quickly on salt and screening. We'll focus on the screening for a cylindrical nanowire sensor. You see, it could be a carbon nanotube. It could be silicon nanowire. Or it could be even a FinFET with a gate exposed. So therefore when I say a cylindrical nanowire, I don't mean a specific technology. I'm talking about the general shape of the channel that allows current conduction. Now, very quickly, after we have looked at the theory, we'll compare it experiment. There have been many experiments done over the last ten years to see whether we are on the correct track in terms of the broad number of variables that people have explored. And then, I want to explain before I conclude some special issues of the small sensors. You know, nanotechnology is good because things are getting small. You can pack more of them. But there are some issues that one has to worry about, even at the fundamental level. I will discuss them before I conclude.
[Slide 3] Very quickly, I'm sure you remember if you were listening to this continuously, remember that we are talking about that DNA getting together, remaining together is difficult because of their strong electronic repulsion. Even for protein. Let's say you have an antibody. Let's say this is an antibody trying to capture a protein molecule. And in that case it's the same thing because as soon as, if you don't put enough salt, the shape, the mutual repulsion will destroy the shape. You cannot capture a protein unless you have the right physiological conditions, high salt concentrations. And we essentially said that therefore we need salt. And the picture will be that we have a DNA or some other biomolecule charge, salt surrounding it, reducing their repulsion. So the question we want to answer, that had this biosensor, had it been cylindrical -- okay, small cross section -- would things be much, much different?
[Slide 4] But there is a reason for this hope, you see? Do you remember that in the Part 1 of this set of lectures we said that a planar sensor has less sensitivity because of the geometry of diffusion? And a nanowire sensor or a nanosphere sensor have much higher sensitivity because it can collect molecules from all sides. And would you expect that because molecules essentially, for
[inaudible] let's say nanosphere sensor. It can sort of surround the whole molecule. Or like for a nanowire sensor, it can surround from all sides. And so therefore, wouldn't you expect a considerably higher sensitivity? Would the shape, again, save you?
[Slide 5] So let's get started by asking a few simple questions. Let's say this is silicon nanowire just for illustration. And this blue region is oxide coating it, right? There has to be insulator around it, which will separate the fluid from the channel so that this interaction is possible. And we have the receptors, the Y-shaped receptors. It could be antibody. It could be DNA. Just something to cap. And the red biomolecules are coming in. And their interaction between the Y-shaped molecule and the red molecule is being screened by the salt, the blue sodium and the magenta chlorine. So that's the framework. Now, if you took a cut along any of the radial directions starting from the center, you would see that part of the charge -- let's say this is the origin -- part of the charge is within the silicon nanowire, shown here on the left. And I show a small decrease because the charges need a certain distance to decay. We will not focus too much on the distance itself. The width of the accumulation region. So, for the time being, we'll not worry too much about it. This is the oxide, red biomolecules with negative charges, presumably because DNA and the two salt, sodium close by and the chlorine pushed apart. So this is the general configuration. Now, if the radius of the silicon nanowire is a, then and the oxide thickness is t-ox -- you know, that's the oxide thickness -- and the divide length in this particular case is LD. In this case you can wire essentially most of the effect of straining is done by that time. And that's why you can replace it with a sort of a bulk electrode at this point, at this distance. Then life is very simple, you see? At low salt concentration, life is simple. This is how. We know the oxide capacitors for two concentric cylinders is given by a simple formula. We have used it many times before: 2 pi kappa epsilon naught. And log of 1 plus t-ox divided by a. This essentially the ratio of this extra thickness divided by a. And you may recall that when we're thinking about how the particles where cylindrical nanosensor captured biomolecules, we replaced this kappa by d. And we replaced -- and that allowed us to do the diffusion equivalent capacities. But here we are talking about real electrostatic capacitors. So, this is the formula. Remains as is. No change needed. And so the double layer capacitor, essentially it's a capacitor between -- one electrode is here, outside the blue. And the second electrode is a bigger circle. And so therefore, you can again, once again, do the same thing. LD becomes the numerator a plus t-ox becomes the radius of the blue circle. And so therefore you have oxide capacitance. You have double layer capacitance. We know the ratio essentially gives you the fraction of the biomolecule that shows up on the nanowire. And once you have accounted for all these factors, remember the LD is inversely proportional to the salt concentration. Once you have put all these pieces together, right? Let's put them together. And you can easily write a small code to check whether the types of dependency you get. You will get some enhancement. It does offer some advantage compared to your planar sensors because you can choke off the current conduction by surrounding it, surrounding the channel by molecules from one size. And if the diameter is small, then you can -- the choke off will be more effective. And so therefore you will get some advantage. No question. Even in the case when you have a certain amount of salt straining. But, remember that the problem is that life doesn't work at low salt concentration. Life works at high salt concentration, hundred millimolar or so. You remember the unit of molar, right? Molar is Avogadro's Number, 6 times 10 to the power of 23 in a liter of fluid. And so therefore, 100 millimolar is just a fraction, one order of magnitude less. So therefore, very high salt concentration needed. This little theory with low salt concentration will not work. So, what I want to do very quickly is give you the final result at high salt concentration and save the rest for the appendix.
[Slide 6] At high salt concentration, however, once again you have the same biomolecule arriving one at a time through diffusion-limited process or by the droplet-based, you know, the biomolecules are being pulled into the sensor surface. And but this time the double layer capacitance is at high concentration is a little bit scary. But that's all right. People have derived all this. You may see some of your familiar, some of our familiar friends, dielectric constant of water. Remember the LD, which depends on the salt concentration. And if salt concentration is 1 over square root of I -- so LD is 1 over square root of I, so therefore double layer charge increases with salt concentration. Sort of makes sense. Look at that exponential dependence of psi naught. Remember, we said that the charge, when the salt concentration is very high, goes no longer linearly with psi naught, but exponentially in psi naught. You can look at the planar sensor formula. It will be exactly the same and with the extra factor here. And the extra factor has this Bessel functions built into it. But at B is a plus t-ox. This is the circle, or the blue circle, the radius of the blue circle. All right. We have something like this. And we have, once again, the charge for the nanowire. You can look at, this is a cylindrical capacitance multiplied by the psi naught.
[Slide 7] And we recall that this double layer has a lot of terms. But when psi naught is relatively large, in that case it's very easy to approximate it using a much simpler expression. Again, exponentially dependence on the potential almost like a planar sensor. If you compare, you'll see almost the same result in the planar sensor in the last lecture as well. So life gets simplified. If we assume that nanowire is only a small perturbation, most of the biomolecules that's being reflected in the salt. So relate the psi naught with the biomolecule concentration. And once you have done that, and you remember the biomolecule is directly proportional to the number of molecules that has arrived through diffusion and got captured, and that we have something we have already discussed, rho naught 3 minus df over 2. And once you put it in, in the original expression -- you will get the corresponding sensitivity.
[Slide 8] You see, you equate these two. This is the biomolecule. This is the double layer charge, allowing you to calculate psi naught. Once you have psi naught, the total number of charges, total amount of charges within the nanowire is directly proportional to this psi naught. So that has been put in here. And once you have done that -- this is the psi naught. And once you have done that you will get the full expression. So what happens for a cylindrical nanowire is exactly the same result as a planar nanowire -- a planar sensor. Except the C1, C2, C3 is slightly different. The P factors are slightly different. But the analyte density, the logarithmic dependence on time, logarithmic dependence on salt concentration, all of them remains exactly the same. Now, you might be surprised. Why is it that I have a planar sensor, the geometry was very different. And this is a cylindrical sensor. And cylindrical sensor, shouldn't it be very different? At least in low salt concentrations it's very different? What is happening at high salt concentration that the results are getting so close? On the top two figures it gives you the answers. You see, this central electrode is a nanowire itself. The blue is the oxide surrounding it. Now, in case of a low salt concentration, we have a picture, a figure, which is on the right. And you can see the depletion rate, the double layer thickness, LD, is much bigger for low salt concentration. So therefore, there is an effect here. But at high salt concentration the entire depletion double layer becomes sort of much, much smaller. Because even that smaller region you have enough salt to neutralize the -- almost neutralize the biomolecules. So, this is as if a person, you see, when we walk around an arc, we are sort of hugging close to the ground. Still, therefore, the arc almost looks like a flat land. We don't see the big curvature of the arc. This is the same thing. That essentially point by point it looks for the biomolecules here as if it's a planar surface. No wonder, therefore, that the results are quite comparable.
[Slide 9] So, at the end, let's check it out to see whether these results make any sense. After all, the experiments should be able to tell us whether these results that we got is really helpful or not. Let's start with the logarithmic dependence with analyte density. You may remember this particular plot I showed to introduce the positive result that linear dependence -- a linear axis on y and logarithmic axis on density. And you do see that the theory essentially predicts the c is same logarithmic dependence on rho naught, which is very good because that's what you started with. We have to get that puzzle right before we can do anything with biosensors, potentiometric biosensors. What about salt concentration? Again, look at this. As you are increasing the salt concentration, remember this 10 to the minus 1 molar is 100 millimolar, physiological condition. And if you go lower -- of course, you cannot go too low because then the biomolecules will not bind, right? Remember, too much repulsion. But you can go, let's say, a factor of 5 or 10 or so. And you expect a logarithmic dependence, y-axis linear, x-axis logarithmic. Sure enough, the theory tells you essentially the same thing. What about logarithmic dependence on time? Remember from the optical experiments we saw that the time, essentially arrival, is more or less linear. Yeah. But, it turns out that a logarithmic dependence on time is expected. Experiment on the left, or the blue, a different analyte concentration. And time dependence on the right. You see the theory, once again. If you look at these regions, where the experiments can probe it, at very low concentration you cannot probe it because of the noise. But then, essentially, you will see the same logarithmic dependence. The pH sensitivity also works out. But this is something I have not discussed. In the next lecture we'll be talking about pH, but just as an aside. I just wanted to show you that if you change the pH, again, you expect a linear dependence on pH. Sure enough, the theory says exactly the same thing. Look, all we did was to take a simple capacitor, a simple oxide capacitor and then made two capacitors in parallel, one side salt, one side electrostatics, one side general conduction. And these two little capacitors, with a little bit of, a few lines of algebra, it has combined, has been able to interpret experiments from tens of different laboratories. That, I would say, is pretty impressive.
[Slide 10] Now, before I wrap up, I want to tell you a little bit more. Because I have been cheating a little bit, but it is important for me to come clean now as I go to other aspects of this experiment. One thing I wanted to mention, this importance of salt. And I have already mentioned to you that the salt get distributed. If it's a positive charge, negative charges come close. Positive charges in the fluid, and salt gets pushed apart. You see, in here I assumed as if the original biomolecule doesn't get perturbed. The salt is simply coming in and sort of taking away some part of the charge. Not true, as I explain to you later. So there is an additional correction that one has to think about. And let me explain where that comes from.
[Slide 11] You remember that when we looked into the MOSFET and there was salt on this side and channel on the left side. Current was going in. The total amount of charge in the MOSFET was a fraction of the biomolecule charge because of this charge partitioning. Now, we assume that this is unaffected. The biomolecules, if you caught 500 biomolecules, let's say, 500 DNAs. And each DNA, let's say, contained 10 units of charge. We assume that there'd be 5,000 units of charge. Salt will eat away, let's say, 90% of it so that the MOSFET will get 10% of it. That is what we said. But we said that the 5,000 will not change. What I want to tell you, that's not true either. It turns out that this salt will also come and reduce this number for a very simple reason.
[Slide 12] You see, when we looked at the biomolecule, I intentionally flattened it. Pushed it back, flattened it. And therefore, it looked as if I can always see the biomolecule. But in reality, the biomolecule may be often standing upright on the sensor surface. So here I show in vertical. And so what will happen, that only the fraction of the biomolecule which is inside the double layer will be seen by the MOSFET. The rest of this thing is as if it's lost in a fog, fog of the salt. And so only a fraction would be available. The total charge is not -- doesn't get to the sensor to begin with. And so therefore, only a fraction of the molecules, this LD will not only determine the capacitance of the double layer, but also the fraction of the biomolecule that can infringe from this side. Because beyond this point, anything that is available is actually lost to the sensor itself. And so if we assume hybridization efficiency of 1, meaning that everybody who is coming in is getting hybridized, there's something called a Manning coefficient of theta. For the time being, don't worry about the specific numbers. I'll explain them later. Important point is the turn, per turn of the DNA is .34 nanometers per turn. Then you can see the total biomolecule charge in the number of biomolecules, DNA, that arrived. But only a fraction of them, LD over d, is the fraction that can be seen by the sensor. And then, of course, there's another correction that we'll discuss later. This reduction in total amount of biomolecule charge is another bad effect associated with screening. You see, screening is killing us in many ways. Reducing the total amount of charge itself, and then whatever charge I am left with, it's also taking a large fraction of it. Stealing a large fraction of it away. Leaving something for the smaller part for the MOSFET. So potentiometric sensors does have some important screening effects to contend with.
[Slide 13] Now, the second thing that is very important is that I have often drawn it like a flattened charge. But you realize that the original charge is a DNA molecule, which are standing like clamps, blades of grass, at different points. And what does -- how should I be able to make this approximation? It turns out in this particular case, water saves us. Because water is a high dielectric constant material. So instead of the fill line directly trying to go to the channel, in fact, it's preferable for the fill lines to stay in the water a little bit and then go down. That is much more preferable. So the effective charge seen by the channel, although the original charge isn't discrete, the effective charge seen by the channel is actually flattened out.
[Slide 14] And here is a simulation that clarifies the story that if you have a charge, this is the nanowire. And if you had a rate charge sitting in the middle, then there is a big dip. Because the charge is very localized. But only if it is in the air. But if it is in the -- if it is in water, in that case, the whole thing sort of spreads out. And the carrier concentration that are induced for each one effectively it spreads out over a micron or so, a least a tenth of a micron or so. Although the original charge may be just a few nanometers. And once you have a few of these molecules, it's perfectly fine to assume that as far as the channel conduction is concerned, that these molecules are, biomolecules, are as if homogenous and flatly laid out.
[Slide 15] Third. And this is unfortunately negative effect associated with doping. Remember that this is a very small nanowire. Nanotechnology gives you nanowires which may have 10, 20, 30 nanometer radius. Very, very small. Carbon nanotubes maybe even smaller, right? In that case, of course, in order to get high sensitivity, we had the same before, from geometry of electrostatics discussion, that it's good to have lower density. Why? Because if you have lower density, your background current, your original current will be small. And therefore any change that is brought about by the DNA molecule will be larger, right? So therefore, your sensitivity would be larger. But there is a problem with the low density, you see? Because when you have low density, some of the sensors may not have any molecules to begin with. Because it's a discrete number. Some may have two. Some may have five. And so there'll be large variability from one sensor to the next because you cannot control doping at that low a level for every sensor in a uniform way. As a result, what happens that there is big distribution of the sensor response, not a single value associated with a given diameter. Even for a given diameter there's a big distribution. You may have an average. But then, for every sensor in that ensemble, let's say you 500, every sensor will respond slightly differently to the captured biomolecules, shown here in blue. But, since you do not know the original red. If you don't know, let's say, if you haven't captured it in some way, then what will happen, you'll not be able to sort of distinguish the capture before and after events based on these plots alone. So, therefore, it's very important to work with relatively high density. Otherwise you have to save the initial values for each sensor and then look at the differential response itself. So, nanotechnology comes with some prices that has to be paid in order so that we can work with very small sensors.
[Slide 16] Finally, I want to emphasize one point. I have been talking all about sensors. But you may recall that I mentioned there's a mystery box in the sensor configuration which is very important. Let me quickly tell you what the mystery box is. This mystery box is a reference electrode. Remember I had to take the gate out and put it as a reference electrode within the fluid. Now, this reference electrode can come in two forms. It can have an insulator around it. We'll call that non-Faradic. And in that case, only a fraction of the voltage that you apply on the gate will show up on the channel by this fraction. Because essentially there's one insulator here, another insulator here. So two capacitors separated by this fluid conductor. Or, you can have a Faradic conductor, which allows easy exchange of charges from the electrode to the sensor surface. And this resistance essentially reflects that charge can come through this radian relatively quickly. It is extremely important for potentiometric sensors that we use the blue Faradic sensor, his Ohmic contact. If we do not use this one, this type of sensor, or if you omit it completely, all together, then the results can be completely wrong. There is much debate in the literature regarding this. So therefore, I hope you will always remember your reference electrode when you do any experiment or write any paper.
[Slide 17] Let me conclude on this lecture. We have, in this lecture we have talked about potentiometric sensors and how it responds to charges. It's a camera for charge. At low concentrations we saw that nanowire screening is -- could be different from planar sensors. But only at low concentration, unfortunately. For binding to
[inaudible] between DNA or to protein, to keep its shape, you need high concentration. And in that case, essentially you get no difference. The constants are slightly different, but not significantly. Now, the doping cannot be too low because it looks like you can have high sensitivity. But then at the expense of significant statistical variation. So doping of the sensors should be good, relatively high. And one thing I want to emphasize from the very beginning, that if you have water versus if you dissolve things in oil, they have dramatically different response because oil has low dielectric constant. So many solvent are dissociated. So depending on the experiment you wish to do, then, one has to use -- the responses would actually be very different. So the fluid is not a bystander. Fluid is an active participant of nanobiosensors, as you might expect, equally important as the MOSFET itself. And finally, I emphasized the importance of Faradic electrode so that electrons or the charges can come in and out relatively quickly. If this is missed, then most of the results will not be reliable. It's very important. And I hope that you always remember to include it in any discussion of biosensors, potentiometric biosensors. Next up, I'll talk about another problem of biosensing, which has to do with the exposed surface. I said the the exposed surface without the biomolecules look clean. Not really. But fortunately, this unclean surface made a professor from Cal Tech, which is Beckman, a billionaire. And so I will tell you because he spent about 10 years in Bell Labs. And when he came back, he was immediately able to see the opportunities of an imperfect surface in being able to make big money. Next lecture we'll be talking about these same sensors, but now as a pH meter.