nanoHUB-U Principles of Nanobiosensors/Lecture 3.4: Potentiometric Sensors ISFET as a pH-Meter
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[Slide 1] Welcome back. We are discussing potentiometric biosensors. We have already seen that how potentiometric biosensors respond to charges from the biomolecule. And there was a puzzle or a surprise that the dependence was logarithmic, other than we originally expected a linear dependence on analyte concentration, and very quickly figured out that salt was really the trouble maker. Salt is necessary for the ions to stay together, the DNA to stay together, protein to stay together, but at the same time you have to give the taxes to the salt which is that they took away a significant amount of charge, leaving behind a very small fraction for the MOSFET. And that's where the logarithmic dependence came from. Now we are moving on to a second non-ideal effect, and this non-ideal effect is good in some ways because it will allow us to measure pH of a solution. And in fact, measurement of the pH is very important for a wide variety of applications. But -- and as we will see on this lecture this will be a good thing, this non-ideal thing would be a good thing because it will allow us to measure something that we really care about. In the next lecture when we go there, when we discuss the measurement of biomolecules we will find that this non-ideality is something that we wish to get rid of. And how to get rid of that? Well, we'll learn that in this lecture as well.
[Slide 2] I will begin by a reminder about pH. PH is potential of hydrogen. Now unless you really liked your chemistry in your college you may not have remembered everything about pH, so I'll give a quick reminder. And then we'll talk about how this pH gets reflected on ISFET and do a quantitative calculation using something called the surface binding model. You'll see it's pretty simple and will give us the total amount of charge as a function of pH on the sensor surface. And what we will find through this analysis today is the fundamental limit for these sensors. And these sensors are called ion sensitive field effect transistors because they are sensitive to the protons. And pH is the potential of hydrogen, which is potential of proton. So it, therefore, has the name ion sensitive field effect transistor. And after we discuss the fundamental limit we will conclude.
[Slide 3] So what is a pH of a solution? pH is that in any given solution whatever is the number of protons that you have, H plus, take a log based on 10 with a negative sign because this number is a negative -- this is a number smaller than one, so once you take a log and the minus that will give you a positive number. For example, if the proton concentration is 10 to the power 7 molar, its a 100 nanomolar, in that case once you put it in that formula it will be -- the answer will be seven. So in that case one would say, "A solution containing so many molars of proton has a pH of seven." Now pH of seven sits nicely sort of in the middle. In water in particular it turns out that the pH only ranges from zero to 14. You cannot go more than 14 or less than zero. I'll explain it a little bit, that why it has to be 14, not 17, not 25. Within that scale, below seven things are acidic, so anytime if you have your orange juice, and if it looks particularly -- tastes particularly tart, then we are on the acidic side. And from seven onward it will be alkaline or base. And so therefore, it is often important, as it was the case for Beckman, remember the original tech billionaire who made a lot of money by integrating transistor technology, vacuum tube technology, with this potentiometric sensor. Well, in that -- in his time the main question was there are different types of orange juices, and people couldn't really tell their acidity apart. And so he used the sensor to essentially tell that which type of orange juices are more acidic and people would like it, and thereby sell more of them. Now in case of orange juice, of course, the pH can change quite a bit. And so therefore, even a crude sensor would be of immense use. In case of a human being the pH is controlled within .1, and so therefore, in that case a little change in the pH beyond this limit can be very dangerous to our health. So measuring pH, which is essentially the measure of the amount of hydrogen atoms -- hydrogen -- the protons that we have in the solution is an extremely important problem. So how does it work? How does a simple MOSFET which is in all of -- in our -- on our iPhones, and we carry billions of them in your pocket, how does it all in a sudden can sense the orange juice -- the acidity of orange juice? How does it do so? Actually it turns out to be pretty simple. You see, if you have a silicon and an oxide -- remember I have removed the gate, and the gate is now -- has become this reference electrode. And here I have the fluid on top of the oxide, and -- which has water in it. Now one thing that is very important in the discussion is to realize that any time you have an oxide, there -- this oxide is not cleanly terminated. These oxides have dangling bonds -- silicon dioxide, so the oxides have dangling bonds, the oxygen atoms. And they're terminated by hydrogen. So there are all these OH groups, like receptors. Remember for the biomolecule we had a -- we had to explicitly plant receptors so that the other DNA could come and attach to it. Here we get the receptors for free. Because as soon as you put the oxide in water all this
[inaudible], dangling bonds, they pick up hydrogen, and they become OH sites. Now where does hydrogen come from? Well, it turns out that a small fraction of water when you put it in, they can be associated into hydrogen and OH ions. And some of those hydrogen can essentially come and sit on the sensor surface. Now of course you see in this case things are charged neutral, so there is nothing that the silicon channel can fill unless, of course, what happened? That this hydrogen essentially had a dynamic reaction with the surface. And sometimes -- and it can proton it -- it can add a proton or take away a proton, and thereby make the entire surface charged. You see, if a proton
[inaudible] the surface or one
[inaudible] group, that bond will become positive; the other can become negative. And so overall there will be a charge that can be felt by the sensor surface. And that is cleanly determined by simply the hydrogen concentration of the potential of hydrogen in the solution. And so the MOSFET underneath, you see what will happen? Let's say for a given pH, given amount of hydrogen -- given amount of protons, let's say pH of seven. You had a particular value of current. Now as soon as you change the pH to a slightly different value, by let's say putting orange juices in here, then this ratio of the positive and negative charges will change because this reaction will be sent off balance. And as a result the corresponding charge will change. The MOSFET will carry a different current, and it will be shifted. And from the shift you could say that, "What is the corresponding pH for this solution?" We'll get into more details of this in a minute, but for the time being let's -- this is the basic idea.
[Slide 4] Now let's get into this calculation of our surface here. It'll turn out to be very simple, but in case you have forgotten, you know, the undergraduate chemistry let me remind you of some basic, basic facts, and then we'll very quickly get to the final answer.
[Slide 5] In water, as I mentioned, because of this exchange of charges there will be -- it -- this chain reaction, protonation and deprotonation, there's always a certain amount of charge on the sensor surface. So let's think about water pure, without the sensor. You know, the water may actually dissociate in the proton and OH groups. I can go back and forth. Of course, will break apart and come together. But it turns out that only a very small fraction of water does so. So if you wrote a rate equation about forward dissociation and reverse process you will see that the product of the hydrogen and OH with respect to the water molecule itself is 10 to the power minus 14. This minus 14 -- this is an experimental measured quantity. This minus 14 defines the range of the pH, right, because all the solutions essentially are in water. Therefore, this defines the range. If you had some other fluid in the background your pH range will not be from zero to 14 anymore. Now how many hydrogen and OH group do you have? Remember, every time one water molecule breaks you have one hydrogen and one OH, so their number must always be equal. And you can immediately solve the problem as follows. This H2O is not the original number of H2 that you started with. But the H2O, that is sort of the water that has been -- that is determined by this kinetic balance. So if you call the x naught, unbroken number of water molecules in the beginning, this is the one after the dynamic equilibrium has been reached. And y plus y -- 2y, one y for hydrogen, another one for OH. And you remember x is equal to y. Solve this equation. That tells you that the concentration of hydrogen is 10 to the power minus 7. What is the pH then? All you have to do is to take a log 10 at the minus sign. So pH of neutral water with nothing in it, no impurities is 7 because of these constraints. Good, what about OH for a pure water -- potential of OH atoms -- ions? It is again, you can take a log of the OH. In this particular case everything is nicely balanced. Both hydrogen and OH, they are equal. So the potential of OH is also 7. And when you sum them up sum is 14, reflecting the original dissociation rate in water. All right, so this is the basic thing in case you had forgotten.
[Slide 6] Now how do you control the pH of a solution? Very simple, let's say you put a little bit of acid -- of .04 molar. So it's mostly water still. A small amount of acid has been added. So of course then it will be ionized between the proton and the rest of the ions. Almost looks like a water dissociating into the two ions. Again, you solve it exactly the same way. This is the number of unbroken acid molecule that you started with. This is the one -- x is the after dynamic equilibrium, the value of unbroken acid molecules that you have left. h and y are the two species. And remember since it's the acid most of the ions will break -- most of the ions will -- most of these atoms will be ionized. So x after a while will be very small. It's almost like sodium chloride. Remember the sodium chloride all broke apart after we put them in water. Very similar, there's no x left. And h is equal to y. So once you solve for it you will get the solution for the pH for this particular case. And it turns out that it is simply equal to the original value of the acid itself, the amount of acid that you added. And here the pH is 1.4. Lower the pH, more acidic it is, so that is something to remember. And you can do the same thing for a base, like NaOH. There's the .028 molar. Go through the same calculation. There it's ionized. Remember, most of them -- because it's a strong base most of them will be ionized, so there will not be any left in the solution after dynamic equilibrium has been reached. And sodium is N; OH is O. They must be equal to each other in number. And so when you put it in x naught will equal -- will be equal to N and will be equal to the OH atoms. And you can correspondingly calculate that -- take a log of O to find the pOH; 1.55 would be the pOH, potential of OH. What is the potential of hydrogen with such an alkaline solution? It is 14 minus 1.55 because the sum -- they must sum up to 14 for when things are dissolved in water. And so the number will be 12.45. Higher the pH number, more alkaline it is.
[Slide 7] All right, so we are ready. We are ready to actually bring back our MOSFET because we have been -- just been talking about the fluid itself, that if you have a little bit of fluid, put orange juice in it, changes-- changes pH conditions. Or put some other alkaline material in, changes pH. But now the MOSFET must be able to respond to it. Respond, how would we do that? It turns out that if you are in air then of course there is no discussion about any hydronium -- any proton because nothing will be dissociated in air. And everything is charged neutral, so the transistor doesn't know of anything. It is happy because this surface is charged neutral. Now as soon as you put things in water of course the hydrogen atoms -- hydrogen ions, the protons will protonate and deprotonate the surface and innate charge will be generated. Let's calculate what this value is. So now because of this innate charge, although the original surface in air was completely charge-free, now it will develop a charge. Turns out there is a very simple calculation. We'll be able to do it relatively quickly.
[Slide 8] Here is my surface. Deprotonation means the hydrogen has gone away, the proton has gone away. Protonation means hydrogen has come up, and you had one OH, and it has become OH2 plus. So positive with protonation because positive with deprotonation things become negative. Well, the reactions are simple. SiOH breaking into SiO and H plus. You know the rate equations; this balance equation's unknown. The amount of hydrogen that you have in the surface, that dictates how many protons would go away. And that rate constant's unknown and measured, called Kb. And similarly the protonation expressions are also known because SiOH2 can get broken into a proton, an SiOH -- return to its original SiOH. Corresponding number is also measured and is called Ka. So if you know these two numbers all I need to know is how many positive charges, how many negative charges, and my total charge is just completed. So I can calculate the total charge and see how -- what is the charge that is sitting on the sensor surface given a certain proton concentration, certain pH of the solution. Now one thing that we'll be returning to is that bulk concentration and surface concentration are not exactly the same. I'll explain in a second why that is the case.
[Slide 9] So here is a very simple calculation. I already know from the previous expression all I need -- this H plus is already known. SiOH, as soon as the surface or the oxide is specified SiOH concentrations are known. I know Kb, so I know how many negative charges on the surface. Just put a few numbers in the calculator, I'd know how many of the negative charges there would be on the surface. Similarly, after protonation you can calculate how many positive charges you have on the surface because H plus will come from pH. This is already known, is a property of the oxide itself. Aluminum oxide may have a different value, compared to silicon dioxide, compared to hafnium oxide, for different insulators. And K is also a material parameter that is measured. And so therefore, once you remember that a total number after protonation and deprotonation must be equal N naught, is the value in the air. And so therefore, you can simply substitute. Once you substitute you have this -- all these hydrogen atoms climbing from pH. Substitute two lines of algebra. Bottom line is now the charge, and the surface is completely specified. These are coming from the pH. These are measured quantity. So as soon as you change your pH the Q will change. And immediately the channel underneath will know that the pH has changed. And you will be able to say by how much has the pH changed? Now often for the sake of simplicity people write not Ka, but the potential of Ka sort of as a logarithmic quantity. And the small -- a small p up front, and so many times that simplifies the formula. It's something good to know in calculating various quantities associated with pH sensing. Generally these values are measured, minus 2. So Ka -- capital Ka is .01, and Kb is 10 to the power minus 6. So this is -- I'm sorry. This is about 100. Ka is about 100, and Kb is 10 to the power minus 6. So from these values one can immediately calculate the entire expression here. So all the information unknown in that sense to the
[inaudible] order, and therefore we can calculate the charge for any given pH.
[Slide 10] Now it's very interesting that this charge depends on the pH value itself, and there is a point where this can become zero. So there is a pH value where the numerator can become zero -- equal to zero. So although the surface has a lot of dangling bonds, OH groups, at some point there our positive and negative will nicely balance. So for a MOSFET sitting underneath it will appear the charge -- the surface -- oxide surface has just become charge neutral. And that point is called point of zero charge, or isoelectric point. And you can immediately see by setting it equal to zero you have a condition, and it becomes H squared. And we -- if you substitute -- if I substitute these two things in here, take a log on both sides, the point of zero charge will be the potential for Ka and pKb divided by 2, so the average of those two values. So in the previous case, for example, we had 6 and minus 2. So the point of zero charge would be 2. So in that case it will turn out that at that point the charges will disappear. It's a defective surface, but at that pH the positive and negative, they balance each other, no further charges on the sensor surface, effectively.
[Slide 11] All right, now I cheated a little bit. Let me tell you a little bit more and then I will wrap up, in terms of there's one more thing you have to do. Do you remember that when we are talking about biomolecules first we had the biomolecules, but then we had to worry about this salt screening it away. The same thing will happen here. Assume that you have a certain pH, and as a result on the sensor surface you have these positive and negative charges. And overall let's say you have a surface which has a positive charge. Now as soon as that happens this positive charge will kick all the hydrogen atoms away from it because those are positive, and they'll push it away. And we're trying to bring all the OH groups close to the surface. So therefore, you have to have a self-consistent -- consistency requirement before this charge is fully calculated. And here is how it's done. Remember this total charge on the sensor surface is the blue. The important thing is this subscript S, you see. This subscript says that it's not the bulk solution that we are interested in, but whatever hydrogen atoms are available at the surface is what is important. And that will itself depend on what the charge itself is. So there is a self-consistency that we have to worry about. And this self-consistency we can again remember that this is a relatively large concentration. So therefore, just as before we did it for the salt, we have this double-layer charge, which is the charge in here, exponentially related to the potential psi naught, not of the surface. You remember that the total charge -- the total charge from the surface must be balanced through the double layer and the
[inaudible]. And the double layer charge is approximately equal to Q and is given by this relationship. And once you know psi naught -- and by the way, this Q is now not the biomolecule charge. This is the charge of the charge surface. Once you know psi naught you put this value of psi naught in. With respect to the bulk solution, therefore, it may either increase or decrease, depending on the sine of psi naught. And then you put it back in there, over here. Do a new calculation for Q and go through this process a few times until -- generally once is good enough -- until you have a full solution. Now often people utilize -- write this expression. The sensitivity of change in Q with respect to change in the surface concentration is a parameter called intrinsic buffer capacity, this ability to screen little change in the surface charges -- little change in pH concentrations close to the surface is called a buffer capacity. I just wanted to quickly point that out.
[Slide 12] So in general this is how people often view these problem -- self-consistency problem. They will start by saying the surface concentration is different from the bulk concentration because the potential close to the surface is different because of the accumulation of these surface charges. If you take a log, the surface potential -- pH of the surface will be slightly different from the bulk pH through this potential. And if you wanted to take a derivative that how the surface potential changes with respect to the surface pH all you would do is in the numerator write this particular expression that relates the bulk concentration with the surface potential. And once you have this quantity you can write it as a product of two factors, essentially DQ cancelling out. But these are not DQ. This is essentially the double-layer capacitors that we already know. And this quantity is the buffer capacity. So therefore, you could relate the change in the potential with respect to the bulk pH concentration -- no longer surface pH -- bulk pH concentration. And you can show that this would be about 60 millivolt, 2/3 kT, divided by a factor. It's always going to be less than 60 millivolt per pH for the bulk surface concentration. Now you remember this is very important, because this psi naught at the end is going to determine how much charge you have in the channel, and correspondingly how sensitively the MOSFET responds to these charges on the surface.
[Slide 13] Then the last slide I want to explain to you what does this 59 millivolts but decayed come from? You see, we just said that you cannot exceed it based on the previous slide. But let me now explain why is it that you cannot explain it -- you cannot exceed it, before I conclude.
[Slide 14] You see, the Nersnst definition of an ion sensitive FET is-- works something like this. You remember the total drain current -- excess drain current is proportional to the amount of surface charge you have. Protonated, deprotonated. Plus and minus. And whatever the difference is, the bottom line is that difference is a function of pH. There's something we already know. Now the surface concentration is related to the bulk concentration in terms of the potential of the surface itself and as well as the fluid gate. Because this fluid gate, or the reference gate, can move the whole potential up and down, and thereby can change the surface concentration. And this hydrogen atom concentration, or the pH, you can write it as 2.3 in terms of pH because pH was based on log 10 if you wanted to do it in terms of log E. So therefore, you have to pick up a 2.3 factor up there. Now let's consider two different cases. Let's say you have increased the pH of a solution, as a result you have a current ID1, and for pH1 there'll be a current ID1. For pH2 there'll be a current ID2. Let's say you have two different currents. The question one should be asking is that how should you change the fluid gate voltage -- this voltage -- so that these two currents become equal to each other? Because then you would know actually how much pH change had actually occurred. So the recipe's very simple, you add -- simply add some acid or change its pH, the current changes. And you change the reference gate potential until this current returns back to its original value. And you note down what is the change in the reference gate potential, that will essentially tell you what is the change in the surface potential -- what is the change in the bulk pH concentration. So this is what I mean, making it return back to its original value ensures that the surface potentials are the same, or the potential at this point are exactly the same. And that implies that, therefore, you should equate these two quantities, and once you do that you will see that delta VG, or delta change in the fluid gate, is essentially 2/3 -- 2.3 KT over Q 1/2 delta pH. pH1 minus pH2. And so if you wanted to know the limit, the limit is 59 millivolt per pH. It will always be lower. You cannot do it any better than 59 millivolt per pH. Now therefore, this pH change, the maximum shift you can have is essentially 59 millivolt. Now you see immediately there is a problem. Do you see the problem? You see, it's one thing about orange juice, that where the pH can change maybe one or two, and therefore, you may have 100 millivolt shift in the gate voltage. And it's easy to measure 100 millivolt shift. But think about a human being. In a human being it was 7.35 to 7.45; .1 pH -- change in pH. And that would only mean that this has 6 millivolt per range -- entire range in physiological condition. And often the noise is bigger than that, so to use it for proving our pH control is somewhat more problematic. Orange juice is are much easier.
[Slide 15] So to let me conclude, so in this lecture I explained that pH sensor is one of the simplest type of biosensors. Biosensors in the sense that pH is an indicator -- a small-molecule indicator of the health of an individual or a biological system. Remember that we had three types of molecules I said we'll be talking about; right? Small molecules like glucose. You can view proton as a small molecule also. And then there were DNA as polymers, and there are viruses. So we have just talked about a small molecule detector using field effect transistors. Now the way this pH sensitivity comes from is because there is a dangling bond on the oxide surface. Oxide has a tetrahedral structure, and so the bonds stick up in the air. And therefore, it can interact with the fluid itself. Now think about it, if you had graphene would you expect any pH sensitivity? I hope you'd say the answer is no because graphene is a two-dimensional material. And essentially all the bonds are essentially satisfied. And so therefore, it's difficult for it to interact with the outside -- it's a self-satisfied bond, so therefore, difficult to interact with the fluid around it. And so therefore, pH sensitivity of graphene would be very small, unlike silicon dioxide or hafnium dioxide, or aluminum oxide, for example. The maximum pH sensitivity of 59 millivolts per the pH is a concern, as I said. That this is not too big, but this is a fundamental limit. You cannot do -- in semi-classical device you cannot do something better than this. Now you may remember that a seminal number occurs for transistors. Transistors also have 59 millivolts per decayed limit. And in fact it turns out that these two things are intimately connected, both comes from the relationship Bozeman distribution of carriers at a -- as a function of potential. Both has same underlying physics here. And finally, the first pH meter that made Beckman a very rich man, it measured the acidity of various types of orange juices, as I mentioned. But this is different types of potentiometric sensors, not the field effect register we are talking about because field effect register was not invented back in 1920s. What he used was a combination of an electrode, potentiosensic -- or potentiometric electrode that can measure the potential difference and amplified it by vacuum tubes, something that he learned when he was at Bell Labs ten years before. So you can see that the physics is very similar. There's a marrying of the technology of electronics with biosensing even at that time. But this is much more sophisticated because now we are talking about a MOSFET in doing both the sensing as well as the amplifying, not two different pieces where you have an electrode to measure the pH and a vacuum tube to amplify it. This integration and an ability to make things small actually makes the potential of these techniques really significant as we'll see later on.