Chapter 4: Theory of Ferroelectric Capacitance Joe T. Evans, Jr.

Transcription

1 Radiant Technologies, Inc. 2835D Pan American Freeway NE Albuquerque, NM Tel: Fax: Chapter 4: Theory of Ferroelectric Capacitance Joe T. Evans, Jr. The explanation below is aimed at the beginner or nonengineer. It is intended to help the beginner visualize the properties of ferroelectric devices. If you want to delve into the actual physics of the devices, look at the recommended reading list at the end of the lesson. Be forewarned! The mathematics can get intensive but don t be afraid to take a shot at it. Introduction: A ferroelectric capacitor can be defined as a paraelectric capacitor with memory. Remember the paraelectric capacitor plotted in Chapter 3? We will add permanent but switchable dipoles to the paraelectric materials to add memory to the capacitor. The result is a ferroelectric capacitor.

2 High Voltage Paraelectric Capacitor [ 3.3nF 6000V Rating ] Charge (µc) Voltage Figure 4.1 A Paraelectric Capacitor Ferroelectric capacitors are actually very complex beasts on the inside and you will have to refer to the reading list at the end of this chapter to find more scientific explanations of their internal workings. A Ferroelectric Capacitor The polarization vs voltage curve for a ferroelectric capacitor is plotted in Figure 4.2. The sample in Figure 4.2 is a Radiant Type AB capacitor with an area of 10,000 square micrometers and a thickness of 0.26 microns (2600 Ångstroms). The capacitor has the lateral dimensions of 120 micrometers by 80 micrometers. This is the standard capacitor supplied with the RADIANT EDU and it is the largest you can test with the EDU. The EDU is capable of measuring capacitors of smaller areas down to 100 square micrometers. Sample capacitors available from Radiant for the EDU are described in detail in the document Chapter 7 The Ferroelectric Capacitor Structure. 2

3 5 Hysteresis Measurement [ Type AB White ] Charge (nc) Volts Figure 4.2 The Hysteresis of a Standard Radiant AB Ferroelectric Test Capacitor 3

4 The relationship between the ferroelectric capacitor in Figure 4.2 and the paraelectric capacitor of Figure 4.1 can be seen by examining the capacitance vs voltage plot of both devices. Ferroelectric Capacitance vs Voltage [ Type AB WHITE ] microfarads Voltage Figure 4.3 Capacitance vs Voltage for the Capacitor shown in Figure 7 The green dotted line in Figure 4.3 approximates the paraelectric characteristic of the ferroelectric capacitance vs voltage curve. 4

5 The capacitor in Figure 4.2 is said to have memory because, at zero volts, it has two polarization states that do not decay back to zero. The paraelectric capacitor of Figure 4.1 returns to zero polarization at the end of the test. So does a linear capacitor. Not so the ferroelectric capacitor in Figure 4.4, which is a copy of Figure 4.2. At zero volts, the loop has three crossing points of the charge axis, none of them at zero polarization. 5 4 Memory Point B Hysteresis Measurement [ Type AB White ] 3 Charge (nc) Memory Point D Memory Point A 4 Memory Point C Volts Figure 4.4 Memory in the Ferroelectric Capacitor Memory Point A is the starting point of the capacitor after the last test. Since the device remembers, you must take into account what happened to the sample before this test in order to properly interpret what your sample will do during this test. That means that Point A is the last charge state of the capacitor after whatever the previous test did to the capacitor. The previous test could have been a few seconds before this test or it could have been the previous day. In this case, the previous test left the capacitor in the full down condition. Hysteresis loops always move counterclockwise. The orange arrows on Figure 4.4 indicate the direction of the charge response as the test voltage first goes from 0V to 9V, 9V to 0V, 0V to 9V, and finally back to 0V. Memory Point B is the charge state of the capacitor when it passes through 0V while the voltage changes from 9V to 9V. 5

6 Memory Point C is the last point of the measurement. Notice that Point C does not end on Point A, leaving a gap in the loop. The gap is real and is caused by temporary memory that decays away in a few seconds. If we measure the capacitor again immediately after those few seconds, we will find that the loop starts again at Point A. In summary, 9V tests (meaning that the test starts at 0V and goes to 9V first) end at Point C but after the test the capacitor charge then decays to Point A and stays there, waiting for our the test. Here is a very important concept. Inside a linear capacitor, when we go from 0V to 9V and then back to 0V, just as much charge goes back into the capacitor as came out. The capacitor starts and stops at zero charge. In the ferroelectric capacitor, if we start at 0V with the capacitor at Point A, go to 9V, and then back to 0V, the capacitor will end up at Point B. If we stay at 0V instead of going all the way around back to Point C, the capacitor will decay a little bit from Point B to Point D in Figure 4.4 and stay there. More charge comes out of the capacitor when we go to 9V from 0V than goes back in when we return to 0V. Excess charge remains outside of the capacitor, meaning that it has excess charge of the opposite polarity remaining inside. This is the essence of memory, and it is what gives ferroelectric capacitors almost all of their unique and useful properties. Remanent Charge: We have already seen that the charge on a ferroelectric capacitor does not go back to zero at 0V unlike linear and paraelectric capacitors. There is charge inside the capacitor that will not come out on its own. How can that be? In Chapter 1Theory of Linear Capacitance, we learned that the excess electrons we put into a capacitor will repel each other, giving the capacitor a voltage unless it is empty. The ferroelectric capacitor in Figure 4.2 is not empty but yet it still has zero volts on it. The excess charge does not try to come out. Why? The reason is that the ferroelectric material between the plates of the capacitor has a naturally occurring builtin electric field. That electric field pulls in from the circuit just the right amount of excess charge of the opposite polarity to cancel itself at the surface of each plate. Therefore, even though the excess charge on each plate repels itself, it is held in place by the electric field emanating from the ferroelectric material. Below is an explanation that may help you visualize this effect. In the explanation of relative dielectric constant in Chapter 1Theory of Linear Capacitance, it was disclosed that when any material is placed between the plates of a capacitor and the capacitor is then charged, the electric field between the plates causes atoms in the dielectric material to separate slightly into dipoles. This separation goes away when the field is removed. Now let s suppose that some of these dipoles have an 6

7 overcenter lock like the legs of a folding table so that once the dipoles form, they cannot go back to zero. In that case, the electric field they produce will always be present and the capacitor will remain charged up to cancel that remanent electric field. That is the situation in ferroelectric PZT. It has so many atoms crammed into such a small space that they cannot all fit symmetrically. One of the atoms is locked into an up or down position. Two different views of the crystal lattice of PZT, called a Perovskite structure, are shown in the figure below. Lead Titanium (or Zirconium) Oxygen Figure 4.5 One Type of Ferroelectric Crystal Lattice The diagram on the left shows the traditional box view of the Perovskite unit cell. Note two things. 1) The unit cell is a little longer vertically than it is to the sides. 2) The center titanium atom is not centered in the unit cell. (The material we use is PZT with a 20/80 mix. This means that 20% of the unit cells will have a zirconium atom at their center instead of a titanium atom.) The righthand diagram shows how the bonds are actually formed between the atoms of the cell. The bonds between the four center oxygens and the bodycenter atom are tilted upwards because the titanium is offcenter. As well, the electrons of these four bonds tend to hang around the oxygens more than the titanium. This leads to a net upward electric field as shown by the arrow to the left of that diagram. The electric field, the dipole, is always there. If we apply an electric field to this unit cell in the same direction as the natural dipole so positive is against positive, it will push on the dipole of this unit cell. If we push hard enough, we can force the titanium atom to force its way to the other side of the unit cell and cause the natural dipole to switch direction to point downward. Whether the center atom is displaced up or down, the unit cell occupies the same volume and shape in space. The material therefore could have any combination of up or down cells were it not for the fact that the electric field of each dipole pushes or pulls on all of the other unit cells in the 7

8 material. This long range coupling between unit cells is what gives the material its memory. So, according to this model, 1) every unit cell of the PZT has a permanent dipole in it, 2) each dipole will couple with others to form groups called domains, and 3) the domains can be forced to switch directions. The cumulative direction of all of the dipoles in a ferroelectric material is its memory. In real ferroelectric materials, not every unit cell has a dipole but the remaining number of unit cells with dipoles is astronomical. This particular model of perovskite switching is called the rattling titanium model. It is a gross simplification of what really goes on inside the ferroelectric capacitor. In reality, all the atoms move in very complex ways. Material scientists and quantum mechanics still argue over the rules that govern all of this. From our point of view, the actual theory is not important. What is important is the point of view of the tester: how does this permanent dipole affect the charge that comes out of the capacitor when it applies a voltage? A Simple Model for Hysteresis We shall return to the simple vacuum capacitor to gain an understanding of the impact of permanent dipoles on a ferroelectric capacitor. Imagine a capacitor with a vacuum between the plates, no charges on it, and no voltage across it. Negative Charge Positive Charge (a) Uncharged Vacuum Capacitor (b) Insert Dipole and Voltage Develops (c) Connect Wire and Let Charge Flow (d) Charged but No Voltage Figure 4.5 The Dipole and the Vacuum Capacitor 8

9 The initial state of the vacuum capacitor is shown in Figure 4.5(a). The capacitor is empty with no voltage on it. That means that the number of electrons on both sides of the capacitor match the number of protons and they are atop each other. In Figure 4.5(b), we do something rather magical. We insert a dipole inside the gap of the capacitor. This is an imaginary dipole so we can think our way through what happens next. The dipole is fixed in place and does not move or rotate. The two charged ends of the dipole emit electrical field lines that will pass into the plates of the vacuum capacitor. When this happens, the positive field lines will attract electrons to the plate. Electrons will move in the top wire towards the plate until the positive charges left behind in the wire pull back on the electrons in the plate with the same force as the dipole. After charges quit moving, the top plate will have more electrons on it than normal [shown as the red color on the plate in 4.5(b)] and the end of the wire coming from the top plate will be positively charged as shown by the blue color. The opposite situation will occur on the bottom plate since it will have the negative electric field lines contacting it and those field lines will push electrons out of the plate towards the tip of the wire attached to the bottom plate. The capacitor will now have a measurable voltage across it. The voltage was generated when we inserted the dipole into the otherwise uncharged capacitor. An important point here is that the capacitor is still uncharged according to our simple definitions in Chapter 1 even though the capacitor has a voltage across it. The top plate and top wire as well as the bottom plate and bottom wire all have the same number of electrons as protons! A voltage exists because the electrons on both contacts have been displaced from their neutral positions by the dipole. Note: For those of you who are interested in such things, this is exactly how a radio antenna works except that the dipole applying the force is in the transmitting antenna some distance away. The radio receiver just listens to its electrons moving back and forth in its antenna. Before we move onto the next step, let s review 4.5(b) again because it is very important that you correctly visualize what happens. We magically insert a fixed dipole in the middle of the vacuum capacitor. We also magically restrain the dipole so it cannot move or rotate. It is oriented as shown in Figure 4.5(b). The electric field lines from the dipole, positive lines coming out of the top of the arrow and negative lines coming out of the bottom of the tail, apply electrical force to the electrons and protons in the plates across the gap according to the parameter ε o. (Remember that it is a vacuum capacitor so only ε o applies.) The protons and most of the electrons are fixed in place, but some of the electrons in the plates and wires are free to move under the forces exerted by the dipole. Therefore, electrons in both plates change their positions, moving in accordance with the forces from the dipole. When all of the movement is done, the plates and the tips of the wires will be oppositely charged. It is very important to understand that we did not allow 9

10 current to flow between the plates in Figure 4.5(b) so each plate and its attached wire have just as many electrons as protons after the dipole is inserted as before the dipole was inserted. The capacitor has a voltage after the dipole is inserted because the electrons and protons are no longer distributed evenly in the plates and wires due to the electrical force of the dipole. This situation is different than when we charged a vacuum capacitor by forcing a voltage on the capacitor as shown in the presentation Simple Capacitance. In that case, after the capacitor is charged up, one plate and its wire have more electrons than protons while the other plate and its wire have more protons than electrons. This is a crucial difference! Figure 4.6 compares a charge linear capacitor to our mythical capacitorwithdipole in Figure 4.5. #Protons > #Electrons #Protons = #Electrons #Electrons > #Protons #Electrons = #Protons 6(a) A Charged Vacuum Capacitor with No Dipole 6(b) Insert Dipole and Voltage Develops on its own Figure 4.6 Comparison of a Charged Linear Capacitor and the Capacitor in Figure 4.5(b) Look now at Figure 4.5(c) where we connect a wire across the terminals of the capacitor in Figure 4.5(b). Instantly when this pathway is established, the negative charges accumulated at the tip of the wire of the bottom plate of the capacitor in Figure 4.5(b) will repel each other into the connecting wire towards the other plate. On the other terminal of the capacitor, the positively charged area at the tip of its wire is missing some of its electrons and attracts those in the connecting wire. Since the two charged areas at the tips of the wires exactly equal each other but have opposite signs, when everything quits moving those two regions will cancel each other, leaving only the plates with excess charge. Each plate will then have just enough extra charge on it to cancel the electric 10

11 field coming from the dipole between the plates. We can remove the wire from the capacitor terminal, leaving us with the capacitor in Figure 4.5(d). This capacitor has an internal electric field inside but also just enough excess charges on each plate to totally cancel the internal electric field. There will be no voltage between the terminals of the capacitor despite the fact that both plates have excess charges on them. Hence, the capacitor is charged but has no voltage. Let us do one more accounting of electrons. As shown in Figure 4.6(b), the capacitor of Figure 4.5(b) has equal numbers of protons and electrons on both sides of the capacitor despite the fact that the dipole makes the capacitor have a voltage. But, in Figure 4.5(d), the top plate of the capacitor has less electrons than protons while the bottom plate has the opposite population. Even though this situation looks like the charged linear capacitor of Figure 4.6(a) with no dipole, the capacitor in Figure 4.5(d) has no voltage across it! #Protons > #Electrons #Electrons > #Protons #Electrons > #Protons #Protons > #Electrons 7(a) A Charged Vacuum Capacitor with No Dipole 7(b) After letting electrons flow to cancel the dipole, the capacitor of Figure 5(d) has no voltage. Figure 4.7 A Charged Linear Capacitor vs a Charged Capacitor with a Dipole A voltmeter attached across the capacitor in Figure 4.7(b) will verify that the voltage generated by the excess charge on the plates exactly equals the voltage that the dipole generated in the capacitor at step 4.5(b) and show no net voltage between the leads. CONCLUSION: If a permanent dipole is present inside a capacitor, the plates of the capacitor will hold remanent charge to cancel the voltage generated in the capacitor by the electric field of the dipole. 11

12 Multiple Dipole Model: What happens if we put two dipoles in our capacitor of Figure 4.5? The dipoles do not have to face the same direction, so there are several combinations. Both can be up, both can be down, or both can be in opposite directions. Figure 4.8 below addresses only two of the combinations, as the other two are antisymmetrical with the ones in Figure 4.8. #Electrons = #Protons #Electrons > #Protons #Protons = #Electrons #Protons > #Electrons 8(a) Opposing dipoles cause charge separation but no excess charge and no voltage. 8(b) Dipoles in the same direction result in excess charge on the plates but no voltage. Figure 4.8 Combinations of Two Dipoles In both Figures 4.8(a) and 4.8(b), the capacitors have zero volts across them. One [4.8(b)] has excess charge on the plates while the other [4.8(a)] does not. Therefore, the net polarization of a ferroelectric capacitor cannot be measured externally with a voltmeter even though it can range from zero to the sum of all dipoles if they all point in the same direction. CONCLUSION: The static polarization state of a ferroelectric capacitor cannot be determined by measuring the voltage across the plates of the capacitor. Under the condition that no external voltage is being applied to the capacitor, the voltage created by the internal dipoles will always exactly be canceled by the voltage of the charge redistributed on the plates and the capacitor will have no measureable voltage across it. 12

13 CONCLUSION: When a ferroelectric capacitor is switched to saturation in one direction by the application of an external voltage (something we will discuss in the next chapter), the total charge that comes out will be determined by the relative number of up and down dipoles in the material prior to the measurement. Only those dipoles oriented in the same direction of the as the applied voltage will switch to oppose the applied field. The others will not. How Many Dipoles Are There? We will close this chapter with some population calculations. The area and thickness of the Radiant PZT AB White capacitors are 10,000 square micrometers and 0.26 micrometers respectively. So, the capacitor has an approximate volume of Capacitor volume in cubic micrometers = 10,000µ 2 x 0.26µ = 2,600 cubic micrometers (or µ 3 ). The average perovskite unit cell in the PZT is approximately 4 Ångstroms on a side, which is µ. The average volume of the unit cell is then Unit cell volume in cubic micrometers = µ x µ x µ = 64 x µ 3 It is certainly not true that every unit cell in the capacitor is without a defect and has a dipole. However, we can gain insight into the approximate number of dipoles in the capacitors you will test with the RADIANT EDU by dividing the volume of the capacitor by the volume of the unit cell. Number of dipoles in a test capacitor = 2.6 x 10 3 µ 3 / 64 x µ 3 = 40,625,000,000,000 That s a lot of dipoles! Even though each dipole is incredibly tiny and probably very weak, together they add up to a significant amount of remanent charge on the plates. Switching: When the capacitors are first made, they must start life with all 40,625,000,000,000,000 dipoles aligned with half up and half down. This creates the condition of the capacitor we examined in Figure 4.8(a) where there is no need for excess charge on the plates of the capacitor. The positive and negative ends of the dipoles facing each plate require 13

14 equal numbers of protons and electrons to cancel their fields. Applying a voltage to the capacitor after it is made, we can force all of the dipoles up or down together to leave remanent charge on the plates as in Figure 4.8(b). The switching of ferroelectric capacitors is the subject of the next chapter. Before proceeding to that chapter, you may want to review this chapter again. It is critical that you grasp the effect of an internal dipole on the charge distribution on the capacitor s plates. The second critical concept is that there are a specific number of positive and negative charges on the plates to cancel a specific number of dipoles. If we can count the number of charges making up the remanent charge on the plates, then we will know the total strength of the dipoles inside the capacitor. Recommended Textbooks: (Recommendations courtesy of Dr. Susan TrollierMcKinstry, Pennsylvania State University, and Dr. Ramamoorthy Ramesh, University of California, Berkeley. Note: Dr. Ramamoorthy Ramesh is a good friend of mine and he does not think anyone can spell or pronounce his first name so he goes by Ramesh!) A. J. Moulson and J. M. Herbert, Electroceramics: Materials, Properties, Applications, Chapman and Hall, London, New York, 1990 B. Jaffe, W. R. Cook, H. Jaffe, Piezoelectric Ceramics, Academic Press, London, New York, 1971 M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford,