So, now, we're going to think about electric fields in cavities of a dielectric. You may wonder why we are going to solve this problem. As we mentioned before, for gaseous molecules or liquid molecules, we ignore the interactions between individual atoms or molecules. So, we only thought that as pairwise interaction between each molecule in electric field. But as you condense the system to be a solid, like in society, if you have more dense population, then you cannot ignore how other people perceive of you, and that is influencing how you behave. The same applies to materials world as well. Then, the problem gets more complicated. So, here, we're going to show you how physicists have come up with a very simple way to think how the solid case can be solved without doing too much complicated calculations. So, as I showed you, in a dense material, polarization can be large, so the field on an individual atom will be influenced by the polarization of the atoms in his close neighborhood, meaning, others matter. It's not only about us. Others matter. Then, what we have to think about is, what electric field acts on individual atom. If I apply electric field to my system, everybody else will be polarized in response to electric field. But at the same time, my neighbor will also influence myself. So, how do I know that? How can I solve this? So, let's take a look at this fictitious situation where you have sea of individuals in continuum, and now, you're going to carve out a very small cavity where you don't have anybody, and see how electric field will be applied inside that cavity. So, that cavity will be the room where each individual atom will be situated. Now, we have two extreme cases, namely, one, you have cavity that is directed along the electric field and the other one perpendicular to the electric field. Now, we're going to use some tricks that we learned in prior lectures. For the cavity that is aligned along the electric field where you're going to use the circulation of electric field in electrostatics, circulation of electric field is zero. We know that. So, the line integral of electric field along this circle or the loop should be zero. Therefore, we can instantly see electric field inside this cavity should be equal to electric field outside the cavity. No question about it. Now, moving to the cavity that is perpendicular to the electric field, we see that we have to apply different rule, namely, Gauss' law. So, I'm going to ask my teaching assistant, Melodie, what is Gauss' law? So, Gauss' law is, we have the surface integral of the electric field over area, and it's proportional to the charge over epsilon naught. Exactly. Gauss' law states, the divergence of electric field, the flux of electric field is equal to the charge inside that Gauss surface over permitivity. So, when you apply electric field in this cavity, you can imagine you have plus charges near this boundary and minus charges near the upper boundary. Because of that, if you make a Gauss surface like this, you can see you have to have a electric field commensurate with the additional charge you create at the boundary. Therefore, you will have different electric field inside and outside the cavity. So, let's see. Before delving more into this quantity, let's take a break and think about the concept of average field. This is very important. So, let me ask Melodie. In our society, we are interested in mean value. What is the average income of a person living in Korea or living in the United States? So, do you know the average income, annual salary, of people in United States? I want to say it's around 50,000, but I'm not sure. Okay. Let's assume it's $50,000, the average income. Then, if you are earning that average income, does it imply that you are representing the average person? I don't think so. Yes. That's very important. It depends on the distribution. If you have more steeper distribution or non-linear distribution, then getting average income doesn't represent that you have the normal. You can be represented as the median person. So, therefore, in statistics, they also show you median salary as well. Median and mean can be different depending on what kind of distribution you have. The same applies to materials as well. If I say average electric field, which is voltage divided by the distance between two place, that doesn't mean each atom will fill average field. It can be different. It can vary from space to space. In fact, the local field at the point of interest might matter more than the average field. So, that's very important. So, we're going to discuss this line-by-line. If the plates are charged, they will produce an electric field in liquid. But they're also charging the individual atoms, and the total electric field is sum of both of these effects. This true electric field varies very rapidly from point to point in the liquid. It is very high inside the atoms, particularly right next to the nucleus, and relatively small between atoms. So, you can see there's divergence close to the point charge, as we know, and smoother function outside the atoms. If we ignore the fine-grained variations, we can think of an average field E, which is just voltage divided by distance, and we should think of this field as the average over space containing many atoms. Now, you might think that an average atom in an average location would fill this average field. This is not that simple, as we can show by considering what happens if we imagine different-shaped holes in a dielectric. So, let's take a look at this. Why holes in continuum? So, let's think about discrete versus continuum. Which system is easier to understand or calculate? Continuum. Exactly. Continuum is easier to understand, like when you take a picture, you focus on the subject of interest, and the rest of the background, you just defocus. In that way, you can see better what you want to see. The same here. For the background information, we want to assume it is continuum because it's easier to calculate. But for the atom of interest or molecule of interest, we want to use the exact solution, which is discrete. In fact, that's very close to the real solution. So, for myself, it's discrete, continuum for the rest of the people. So, the rest of the people, we think this is continuous media. For our family, it is one person by person. In that way, we can understand what is happening. Now, let's take a look at these people sitting, let's say, in an airplane. If they have space elongated. Along the direction they're sitting, will they feel more comfortable? Yes, I think so. Yes. How about this one? If you make the space wider, but shorter in the direction you sit, would you feel comfortable? No. No, exactly. The same person, depending on how they're sitting, matter. The same as the dipole moment as well. So, as you can see, when you have dipole moment along this cavity versus along this cavity, they will feel different. Now, consider another slot whose large size are perpendicular to E which we just discussed, and we're going to apply Gauss Law as Melody explained to us for polarization charge. As you can see, delta E.S times S is equal to E minus E naught. So, E naught will be the electric field inside the cavity. You can see this flux will be this one, is equal to minus delta P.S over epsilon naught. Why? Because delta P as we mentioned before is the bound charge, the polarization charge. If I multiply by the surface area, that's the charge. E naught is equal to E plus P over epsilon naught. So, you can see the electric field inside this cavity is increased by a factor of P over E epsilon naught. You see that? So, that's different here. You have increase of local electric field when you have cavity lying perpendicular to your electric field. Now, how can we directly measure electric field, as well as displacement? So, it's about a measurement, right? Earlier in the history of physics when it was supposed to be very important to define every quantity by direct experiment, people were delighted to discover that they could define what they meant by E and D in a dielectric without having to crawl around between the atoms, right? The average field E is numerically equal to the field E naught that would be measured in a slot cut parallel to the field and the D field could be measured by finding E naught in a slot cut normal to the field. But nobody ever measures them that way, so it was just one of those philosophical things. Nowadays, we have tools like atomic force microscope where we can make artificial structure, nanostructures, and then we can measure the electric potential. So, we imagine what our ancient scientists couldn't measure, we might be in the position to directly measure electric field and D field. We're going to cover the electric field inside a spherical hole where the situation is similar to most of the liquid, as well as, solid case. I'm going to ask my teaching assistant Melody, in solid case what is the highest form of symmetry? The simple cubic, right? Exactly, simple cubic is the highest form of symmetry which is similar to a spherical hole. The things that we learn here we can apply almost directly to the solid state case as well, but more directly as written here, for most liquids we could expect that an atom finds itself on the average, surrounded by other atoms in what would be a good approximation to a spherical hole. Now, we're going to do a simple math. As you can see, imagine this is the portion of the liquid and now we're going to curve a spherical hole out of it, then you know from superposition rule that we can add the environment with a hole plus the sphere itself. We have learned, in fact, the electric field of a dielectric sphere. So, you can remember where we discussed the uniforms spheres charged with plus and minus, and then displaced by a distance C. I'm going to ask Melody, what was the direction of the electric field inside this sphere? I think it was straight from the bottom to the top, right? Exactly. So, it was like this, straight and uniform no matter where you are. That was from the outside, it was like a dipole. So, you're creating a dipole moment in the plug. So, knowing the electric field of the plug and knowing the electric field when you don't have enough plug, you can just subtract and make an equation to get what would be the field in a spherical hole. So, it's that simple. So, let's do it. So, do you remember this? You had spheres up-and-down, displace and we prove that it's like a sphere with charge density that is dependent on the cosine theta and based on the equation of surface charge due to polarization, one can prove that the surface charge on the dielectric sphere with uniform P in z direction will follow cosine theta distribution. Therefore, the field inside the plug, E plug will be minus sigma naught over 3 epsilon naught which is minus p over 3 epsilon naught. We already know sigma is P.n which is P cosine theta and this is sigma naught cosine theta which is exactly the same as here. So, now we know the electric field in a spherical cavity is E plus P over 3 epsilon naught. We remember that in extreme case of a rectangular slab like this and rectangular slab like this, in this case, we have the same as E, in this case, we have E plus P over epsilon naught. So, we can just infer that sphere should be somewhere in between. So, 1 over 3 epsilon naught is just in between. The spherical hole gives a field one-third of the way between a slot parallel to the field and a slot perpendicular to the field. Now, let's move further. The dielectric constant of liquids. We can now calculate the dielectric constant of liquids based on what we have just solved and in fact, we will learn about the Clausius-Mossotti equation who are the scientists who solve this before we did. So, in a liquid we expect the fuel which will polarize an individual atom is more like E_hole than just E as we just mentioned. The atom inside the hole will feel larger field and the average field. If we use the E_hole for polarizing field then P is equal to N alpha epsilon naught and you have to replace the electric field with the local field which is larger. If you see this, you have P on the left and P on the right. So, you have to rearrange it then polarization will be N alpha over 1 minus N alpha over 3 times epsilon naught E. You can see if N alpha is very small, then you can ignore this and it becomes the equation like in the gas, but if N alpha is large and you cannot ignore it then something else happens. So, remembering that kappa minus one, the dielectric constant, minus one is P over Epsilon naught E, kappa minus one, is N alpha over one minus N alpha over three, and this is the Clausius-Mossoti equation, which give us the dielectric constant kappa of a liquid in terms of alpha, the atomic polarizability. Whenever an alpha is very small, as it is for a gas, kappa minus one is N alpha. So, let's take a look at the table. So you see, for CS_2, the predicted permittivity is 2.76 versus experimental one is 2.64, Oxygen liquid 1.509 versus 1.507. So you can see how close the predicted value is based on the simple approximation. So, our derivation is valid only for electronic polarization in liquids, and it's not right for a polar molecule like water. If we go through the same calculations for water, we get 13.2 for N alpha, which means that the dielectric constant for the liquid is negative, while the observed value of kappa is 80. So, there is some discrepancy. But for most of the liquids that has spherical symmetry, this simple calculation will predict the permittivity to a very good accuracy. Now, let's move on to solid dielectrics. So, let's first discuss electret. So, electret is a dielectric material that has a quasi-permanent electric charge or dipole polarization as depicted on the right side. So, you have shift of minus charge and plus charge in space, and they create dipole moment. One example is wax. So, wax is electret, contains long molecules having a permanent dipole moment like fluoropolymers or polypropylene. But they are different from ferroelectrics that we'll discuss later. They're just electrical analog of a magnet. Complex crystal lattice with permanent internal P, as you can see, each unit cell of the lattice has an identical permanent dipole moment as shown here, and statically they are screened, so we do not normally notice such a P. Because in electrostatics, single charge can move around to screen the dipole moment. So, we don't see that. So, when do we see that, when we break the equilibrium. So for example, if we raise the temperature, heat the materials, then the polarization will change and when they change, you will have in equilibrium, and then additional charge will come or go, and you can just measure the current, and we call that pyroelectricity. Or we can also think about piezoelectricity, when we apply stress to this material, then the dipole moment will change, and due to the change, we have to attract charges or repel charges and that will create current as well, and we call that piezoelectric charge. So, stress-induced P change can also induce current that we can measure. Now, solid dielectrics without permanent polarization, we can only think of electronic polarizability, ionic polarizability, and oriented dipole polarizability. So, these terms will contribute to the dielectric constant of the material. Now, we're going to discuss a special class inside dielectric materials, which is called ferroelectricity, and one prototypical example is Barium Titanate where we have perovskite structures. So, I'm going to ask my teaching assistant, Melody, what is a perovskite structure? Okay, so the perovskite structure is in this figure and we have the A cations here on the corners, and then we have the anions on the faces of the cube, right here, and then finally, we have another cation in the center which is the B cation. Exactly. So, as Melody mentioned, we have a cubic structure or a tetragonal structure, and we are forming oxygen octahedral of which center lies the B cation and titanium cation will rattle up and down, or left and right, or up front and back to create a dipole moment. That dipole moment is causing polarization, as well as, the electronic polarizability that we just discussed. Ferroelectricity is about the property of certain materials that have a spontaneous polarization that can be reversed by the application of an external electric field. So, the term is used in analogy to ferromagnetism, in which a material exhibits a permanent magnetic moment as well. For this prototypical ferroelectric material, Barium Titanate, degree temperatures is 118 degrees Celsius, meaning below which it has ferroelectricity, above which we have no ferroelectricity. So, it's an ordinary dielectrics with enormous kappa, below its dielectric with a permanent dipole moment. Now, the question here is what is the local electric field inside for ferroelectrics? So, in working out the polarization of solid material, we must include fields from the polarization itself, just as we did for the case of a liquid. However, a crystal is not a homogeneous liquid, so we cannot use for the local field what we will get in a spherical hole. The factor one-third becomes slightly different, but not far from one-third because it's simple cubic, as we have just discussed. For simple cubic, let's see if it's just one-third. It is the same. Therefore, we will assume for our preliminary discussion that the factor is one-third for Barium Titanate. So, we can use what we learned for liquid apply it to ferroelectric. Now, let's take a look at the equation again. The Clausius-Mossoti. The kappa minus one is N alpha over one minus N alpha over three. What happens if N alpha is larger than three? The term should become negative. Exactly. So, the term will be negative. Does kappa really go negative? No. It is physically unlikely. Nowadays, we were discussing negative capacitance. So maybe we can create a situation where kappa goes to negative constant, but for most of the material class, we don't have it. Now, let's see what should happen if we gradually increase alpha in a particular crystal and approach N alpha equals three. So, we now assume we cannot increase N alpha larger than three because it will give you negative dielectric constant. So, N alpha equals three is the maximum. Let's see what happens if we increase gradually N alpha up to that point. So, as alpha gets larger, we know the polarization gets bigger making a bigger local field. A bigger local field will polarize each atom more, raising the local fields still more. So, this is like a feedback, positive feedback which will align all of them. So, if the "give" of the atoms is enough, the process keeps going; there is a kind of a feedback that causes the polarization to increase without a limit, assuming that the polarization of each atom increases in proportion to the field. So, it will hit the ceiling and then they will be stuck. We call that polarization catastrophe. The "runaway" condition occurs when N alpha equals to three, because if you put N alpha equals three kappa goes infinity, and infinity is not realistic. So, that's where you have self aligned permanent polarization. So, lattice gets locked in with high self-generated internal polarization. So, that's how we can understand some of the ferroelectrics. So, let's see, what happens in Barrium titanate when N alpha is equal to three. In the case of Barrium titanate, there is in addition to an electronic polarization, also rather a large ionic polarization, presumed to be due to titanium ions that can move a little bit within the cubic lattice as we saw it in the optional tetra. So, we have two terms; the cloud being up and down, titanium atoms rattle up and down. The lattice resists large motion, so after the titanium has gone a little way, it jams up and stops. So, the crystal cell is then left with a permanent dipole moment that comes both from electronic polarization as well as the ionic motion. Now, what happens if we increase temperature so that N alpha becomes less than three. So, we just discussed if N alpha equals three we have stuck situation. Now, if I increase temperature, why does N alpha decrease? Because the N term is dependent on temperature and the alpha. Exactly. So, number of atoms per unit volume, this is N, and if you increase temperature usually, the volume increases, and if the volume increases, the N decreases. So, therefore, you have decrease in N alpha. So then, we can discuss this phenomenon using thermal expansion coefficient which is alpha. So, below the critical temperature, T_c, it is just barely stuck, so it is easy by applying electric field to shift the polarization have a log in a different direction. Above it, we can see that the N alpha will decrease. So, let's derive Curie-Weiss Law based on the knowledge that we just discussed. Let's then analyzed what happens around T _c at critical temperature in more detail. So, we know volume thermal expansion coefficient alpha sub V is defined by one over volume times dV over dT. Let N be the total number of dipole cells and V the volume dipole cells. Also, let's define beta is three times alpha V. Here, beta is a small constant of the same order of magnitude as the thermal expansion coefficients which is about 10 to the minus 5 to 10 to the minus 6 per degree Celsius. So, capital N, which is the density, is number of atoms or dipoles over the volume V, and then, we can replace V with V nought times one plus alpha V delta T. Using the binomial expansion and approximation, we can put that to the denominator with different sign here. So, N over V times one minus alpha V delta T. Then this becomes N nought and this becomes N nought alpha V delta T. From this relationship, we can say N nought minus N nought beta delta T over three. So, this is the equation we can easily derive. Let's say T_c is the critical temperature at which N alpha is exactly three when polarization is locked, and let N nought be the density of dipole cells at critical temperature. Then, N alpha equals N nought alpha minus N nought alpha beta Delta T over three. N alpha nought because it's at critical temperature is three. N alpha nought here is also three. So therefore, we only have beta delta T and that's three minus beta times T minus T_c. If I put this into our Clausius-Mossotti Equation, then I can prove that kappa minus one is equal to the right term. If I rearrange that with an assumption, that this difference is very small, smaller than one, then kappa minus one is nine over beta T minus T_c. You remember, Curie Law was one over T. Curie-Weiss Law is one over T minus T_c, just if you have a defined Curie temperature then it's shifted and we will see if that's the case. So, here's an example of Curie-Weiss Law from this journal. As you can see, as the temperature decreases toward T_c, you see it goes up like this. It follows one over T minus T_c relationship. Okay. Now, let's take a look from the structural point of view what happens when T is less than T_c. We have just discussed about the situation when T is larger than T_c and we just derived the Curie-Weiss law based on linear thermal coefficient. So, if we imagine lattice of unit cells like the one on the right side as we just discussed, we have perovskite structure. It is possible to pick out chains of ions along vertical lines. So, we're going to think about the set of ions making a chain, and think about the first nearest neighbors, second nearest neighbors, and third nearest neighbors, and so on and so forth. To our surprise, you will see also in our society, the people who influence most is our neighbors. It's not people faraway, it's people around us and that's the same here in materials. With first nearest neighbors, second nearest neighbors, you can account for most of the behavior. We're going to prove it. One of them consists of alternating oxygen and titanium ions and you can see whether you caught in particular crystallographic orientation, in this case, 001 or 110, you will see different cross-section. There are other lines made up of either barium or oxygen ions, but the spacing along these lines are greater. All right. So, you can make this very simple schematic based on what you see across different crystallographic lines. So, let's see simplification and increase in complexity, that's our strategy. To understand a complicated phenomena, we just start from very simple case and then we add complexity to reach to the reality. So, we make a simple model to imitate the situation by imagining a series of chains of ions. Along what we call the main chain, the separation of the ion is a, a is the lattice, the cubic lattice parameter which is half the lattice constant. The lateral distance between identical chain is 2a. So, this is the full lattice. There are less dense chain in between which we will ignore for the moment. So, we're going to ignore that and see if it makes sense. To make the analysis a little easier, we will also suppose that all the ions in the main chain are identical. It is not a serious simplification because all the important effects will still appear, this is one of the tricks of theoretical physics and one does a different problem because it's the easier to figure out the first time. Then when one understand how the thing works, it is time to put all in the complications. All right. So, let's take a look at the electric field at each atom along the chain. This is one chain that we just picked. You can see we use different colors for different ions along the chain and see how that matches the number in the series. Using mathematics, we can calculate the electric field along the chain which is P over Epsilon nought times 0.383 over a to the power of three. So, at any given atom, the dipoles at equal distances above and below it give fields in the same direction, so for the whole chain we get this and you see, it has the same direction. So, they are helping each other, they are helping to create dipole moment along the same direction. You also see the magnitude here. You can see the field at the distance r from a dipole in a direction along it's given axis is like this. We already saw this in a prior lecture. So, let me ask my teaching assistants Melodie what happens to the neighboring chains? What does the ions here, what kind of field do they impose on the neighboring chain? So, I think because the electric field here is going in the same direction, then these molecules and the atoms are here are going to be aligned in a different direction. Exactly, they will feel electric field that is opposite to their polarization, therefore, they're being suppressed. So, you can see along the chain there helping, but beside change they are opposing, suppressing. But because the electric field is also a strong function of the distance, this opposing field is weaker than supporting field of their own. So, even with single chain consideration, we can get a very good estimate of the dielectric constant inside field electrics. So, it is not too hard to show that if our model were like a completely cubic crystal, that is, if the next identical lines for only the distance AOA, the number 0.383 will change to 0.333 or one over three. So, it's a very small change. In other words, if the next lines were at the distance a, they would contribute only minus 0.050 unit to our sum. So, that's a very small negative influence. right? However, the next main chain is at the distance 2a and the field from a period structured dies off exponentially with the distance. Therefore, these lines contribute much less than 0.05 and we can just ignore all the other chains. So, we just focus on the chains where they help each other. Okay. So, let's think about runaway process for a single chain. So, the single chain will represent the collective behavior of all crystals in perovskite structure. So, it is necessary to find out what polarizability alpha is needed to make the runaway process work. Suppose that the induced moment p of each atom of the chain is proportional to the field on it, p is equal to alpha epsilon the electric field along the chain and the electric field along the chain is 0.383 over a to the power of 3 p over epsilon and here we have two solutions, right? If I want to solve these two equations, either E and p be both zero, which is trivial solution or alpha equals eight to the power three over 0.383 with E and p both finite. Thus, if alpha is as large as eight to the power three over 0.383, a permanent polarization sustained by its own field will set in. This critical quality must be reached for barium titanate at critical temperature. This is very simple. Right? So, let's discuss about polarizability at a critical temperature for barium titanate. So, for barium titanate, the spacing is the a is two times ten to the minus a eight centimeter. Som I'm going to ask my teaching assistant Melody, what is this number in angstrom or nanometer? Two angstrom. Is two angstrom. So, half of the lattice is two angstrom. So, fool lattice is four angstrom. Then, if we put this number in, we must expect alpha to be 21.8 times 10 to the minus 24 cubic centimeter. Let's compare this with the known electronic alphas of the individual atoms. Okay? For oxygen, alpha is equal to 30.2 times 10 to the minus 24 cubic centimeter and for titanium, alpha is 2.4 times 10 to the minus 24. All along the chain, we had equal number of titanium in oxygen. So, if we took the average out of those two, we've got 16.3 times 10 to the minus 24 cubic centimeter which is very close, right? But, it's not high enough to give a prominent polarization. So, you see, most of the polarization comes from electronic polarizability of those two, but we're missing something, right? So, there is also some ionic polarization due to the motion of the titanium ion as we mentioned. All we need is an ionic polarizability of 9.2 times 10 to the minus 24th cubic centimeter, or a more precise computation shows that actually 11.9 times 10 to minus 24 cubic centimeters needed. To understand the properties of the barium titanate, we have to assume that such an ionic polarizability exists. Okay? So, let's see. So, we have just discussed based on the simple approximation also knowing only the ions along the main chain influenced the whole collective behavior for electric crystals. We also understand at critical temperature, we have not only electronic polarizability but also ionic polarizability. But, there are some remaining complexities as shown here. Why the titanium ion in barium titanate should have that much ionic polarizability is not known. Furthermore, why at a lower temperature, it polarizes along the cube diagonal in the face diagonal equally well it's not clear. Okay? Now, we're going to discuss Antiferroelectric versus Ferroelectric. So, Ferroelectric has a hysteresis loop that shows well-defined square loop like this. This is polarization, this is electric field. Antiferroelectric, shows a double loop that is depicted here. So, at zero field, it has zero moment while for ferroelectric, at zero field, you either have plus p or minus p. You will see what kind of structures are responsible for different behavior. So, returning to our simple model, we see that the field from one chain would tend to polarize neighboring chain in the opposite direction. So, in that case, if I do that like in this case, then there is no net permanent polarization. So, at zero field, you will have zero as we see in our polarization states I drew and we call it Antiferroelectric. But, barium titanate is ferroelectric and is close to the arrangement shown to the right picture. Where the opposing field is mitigated by the less dense chain and then because of this, another interaction, the next chain will have the same direction. So, the oxygen titanium chains are all polarize in the same direction because there are intermediate chains of atoms in between. This intermediate chains creates smaller electric field that will have small influence to the net polarization. So, you can have non-zero net polarization at zero electric field and that's the ferroelectric case. So, with that, we'd like to thank you for your attention and we are going to wrap up our whole lecture. Thank you very much. Bye bye.
