University of Ljubljana Faculty of Mathematics and Physics

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1 University of Ljubljana Faculty of Mathematics and Physics Oddelek za fiziko Seminar Experimental methods in dielectric spectroscopy Author: David Fabijan Supervisor: doc. dr. Vid Bobnar Ljubljana, May 2012 Abstract Measurements of dielectric properties are among basic experimental techniques in research and characterization of materials. This seminar is an overview of different methods that are used by experimentalists to determine dielectric properties over a wide range of conditions. The main focus is on showing the existence and origin of a frequency dependence of dielectric properties and on different ways to detect the frequency-dependant dielectric response: from the very low frequencies, past the radio and microwave parts of the spectrum, all the way up to optical frequencies.

2 Contents 1 Introduction 2 2 The origins of dielectric response The definition of dielectric properties Origins of polarization Overview of measurement techniques and equipment Low frequency measurements Wien bridge method LCR meter Radio frequency and microwave measurements Impedance analyser Network analyser Optical measurements Practical examples Low frequency liquid crystal measurements Microwave thin film measurements Conclusion 13 References 13 1 Introduction Dielectric materials have a wide range of industrial and scientific applications. A common use of this techniques is in producing new materials with high permittivity. While the high dielectric constant in capacitors is limited by corresponding high losses in such materials, this is not a problem for electromechanical applications. In this electromechanical applications the smaller electrical fields needed to produce the same strain are a good trade off for the larger losses, which would be unacceptable in an electronic device. Dielectric measurements are however used in many other fields as well, perhaps one of the more unusual applications being, the measurements of the dielectric properties of foodstuffs in order to determine their contents [1]. It is of vital importance to be able to measure these properties in an array of different conditions. The state of matter, its temperature and the mechanical pressure acting upon it, as well as the applied frequency of the electrical field and its amplitude are some of the factors that must be accounted for. It seems obvious that fundamentally different methods have to be used to, for instance, measure the dielectric properties of a gas at radio frequencies or, on the other hand, to determine the properties of a solid material in the microwave range. In this seminar I will represent some of the reasons why the dielectric properties change with regards to the before mentioned conditions. The main part will however be dedicated to different methods of measuring these properties, particularly to the different appliances [2] used at different frequency ranges, from the very low up to optical frequencies. In the end some specific examples are shown, specifically measurements of thin films at microwave frequencies and measurements of liquid crystals at low frequencies. 2

3 2 The origins of dielectric response 2.1 The definition of dielectric properties In vacuum the relationship between the electric field (E) and the electric displacement field (D) is expressed as: D = ε 0 E, (1) where ε 0 represents the permittivity of free space, with a value of ε 0 1 µ 0 c F m 1. (2) However, when we talk about the electrical displacement field in a material, we must take into account its electric polarization. Equation 1 must therefore be rewritten as: D = ε 0 E + P, (3) P here stands for electric polarization, the electric dipole momentum contained in a unite volume. From now on our discussion will be limited to homogeneous linear materials. The polarization can in this case be expressed as: P i = j ε 0 χ ij E j, (4) where P i and E j are components of their respective vectors and χ ij are components of the electric susceptibility tensor. For isotropic materials, equation 4 can be further simplified to: P = ε 0 χe. (5) When polarization is defined in this way, we can rewrite equation 3 as: D = ε 0 ε r E. (6) The new symbol ε r is defined as ε r = χ + 1 and will be referred to as the relative dielectric constant of the material. The derivation so far was limited only to a static electric field, a case with little practical applications. Interesting things will actually only begin to take place, if we apply a changing electrical field E(t). Equation 3 still holds, yet we can not assume the same for equation 5. The dipoles in a material actually require some finite time to align themselves in the electric field. That is why the polarization in a time dependent electrical field has to be written as: t P(t) = ε 0 χ(t t )E(t )dt. (7) With the use of Fourier transformation we can however construct a more manageable expression. The result is a frequency dependent polarization, expressed as: P(ω) = ε 0 χ(ω)e(ω). (8) We will now investigate the case of a simple electric field E(t) = E 0 cos(ωt). resulting electrical displacement field is: The D(t) = ε 0 E 0 [ ε (ω) cos(ωt) + ε (ω) sin(ωt) ], (9) and the variables ε (ω) and ε (ω) are the real and imaginary parts of the frequency dependent relative permittivity: ε (ω) = ε (ω) + ıε (ω). (10) 3

4 As the relative dielectric constant for a static field, so is the relative permittivity for the changing field defined as ε (ω) = χ(ω) + 1. Its complex value is a result of the Fourier transformation procedure. The real component can be understood as being the part of the polarization that is in sync with the electrical field, while the imaginary component represent the part of the polarization that is perpendicular to the field. This perpendicular part will manifest itself as heat losses in a device that uses such a material, the heat per unit volume being equal to: Q = ωe2 0 2 ε. (11) It is in fact common to define losses of a dielectric material as the ratio between the real and imaginary part of the permittivity. This measure of losses is usually denoted as tan(δ): tan(δ) = ε ε. (12) 2.2 Origins of polarization Having gained some basic understanding of the concepts, we shall now turn our attention to the actual causes of polarization in a material. There are four main causes of polarization in a material, their effects are schematically represented on figure 1. The first of this effects is the electronic polarization, resulting from the slight redistribution of the negatively charged electrons around their respective atomic nuclei. It has the smallest effect of all the discussed mechanism, but it is the only one that can be found in all materials. A classic example of a material in which this is the exclusive mechanism are noble gasses. The effect of this polarization is strongly dependent on the electron orbitals of the material under investigation. For instance in the before mentioned noble gasses where the shape of the electron cloud is spherically symmetrical, the values will generally be very low. If we take the example of helium it has in its Figure 1: The mechanisms of polarization in a material. [3] liquid phase ε r = In silicon, another material where the electron polarization is the only mechanism ε r = 12, a result of the sp 3 orbital. Electronic polarization will remain nearly constant up to frequencies of about Hz, in that range it will achieve a resonance peak, where it s value can be highly increased. At larger frequencies the movements of the electron cloud can no longer follow the electrical field and the relative permittivity of a material at such high frequencies approaches 1. In materials where chemical bonds have an ionic character, the atomic mechanism of polarization also plays a role. The force of the external electrical field on the positively and negatively charged ions causes them to move slightly further apart, thus creating dipoles. Simple ionic crystals such as NaCl are the characteristic examples of such a material. The effects of atomic polarization will generally be larger then those of electronic polarization. If we take a strongly ionic material such as LiI and measure it s relative permittivity at low frequencies where both the atomic and electronic polarization are at work we get a value of 4

5 ε r = While at infrared frequencies, where the atomic polarization is no longer a factor the value drops to ε r = 3.8. We can observe this change, because the atomic polarization mechanism becomes inefficient at lower frequencies, generally somewhere in the terahertz range. We can intuitively understands this, if we consider that for atomic polarization the electrical field has to move entire atoms, a process that is obviously slow when compared to the movement of the electrons alone as is the case in electronic polarization. A third possible cause of polarization in a material is orientational polarization. It comes into effect when the material includes dipoles that can orientate in an external electrical field. Regular water is a good example of such a material, its polar molecules act as a dipole and the fact that it is in a liquid state allows the dipoles to orient in an external field. In most cases the orientational polarization in a material will be dominant whenever it s present. We can just take water as an example. It s well known that it s relative permittivity is 80, however in ice, where the dipoles can no longer freely rotate in the external field, the relative permittivity at low frequencies is only 4.1. As with all other mechanisms, orientational polarization will also diminish with frequency, the major loss taking place in the microwave or high radio frequency range. While it may seem that there is a pretty limited choice of materials with orientatable dipoles, this is in fact the major source of polarization in ferroelectric materials, which are highly useful in many branches of industry. The last source of polarization in materials occurs when there is some free charge on the boundary of the material itself. This type is only effective at low frequencies and as it is highly dependent on the amount of the before mentioned free charge. As can be seen form figure 1, it is also accompanied by high losses, materials in which this type of polarization is observed are therefore not useful in capacitors. However this is the main source of polarization in some composite materials for electromechanical applications. 3 Overview of measurement techniques and equipment Before we begin exploring the different methods for determining dielectric properties it would be wise to define the useful concept of the device under test (hereafter referred to as DUT). We in fact do not measure the relative permittivity itself, but instead some property of a device, from which we can determine the dielectric properties of the active material in such a device. Most commonly in low frequency measurements such a device is a simple parallel plate capacitor. Under the assumption that the plates of such a capacitor are large in comparison to the distance between them, the real part of relative permittivity of the material filling the space between said plates, can be expressed as: ε r = C d Aε 0, (13) where C is the measured capacitance, d the distance between the plates and A their area. The capacitor is only useful at relatively low frequencies, a statement made obvious by the fact that there is no such thing as an infrared or visible light capacitor. So we have to use some other type of DUT. In the infrared range we usually measure the reflection coefficient, so the DUT has to be some sort of optical element. While in the microwave range some of the most common DUTs are the coaxial cable and cavity resonator. Let us move on to the different techniques used in determining the dielectric properties, starting at the low frequency range. 3.1 Low frequency measurements First we have to to define what is in fact meant by low frequency : For the purpose of this seminar we shall assume as low all frequencies smaller than a few MHz. 5

6 One of the methods that probably deserves to be mentioned, even though it can hardly be named a low frequency measurement, is the measurement of DC capacity. It is a procedure so simple that a decent digital multimeter can perform it. All one needs to do, is to discharge a capacitor trough a well defined resistor. If we measure the voltage over time it will follow the well known equation: V (t) = V 0 e t/rc, (14) knowing the resistor R, this can be used to measure the capacitance C. Determining the relative permittivity from the capacitance can now be done by applying equation 13. Such a measurement has some major drawbacks. For one it gives no information about the frequency dependence or the losses. Also we need to perform at least two measurements, one of them on an equal empty capacitor, in order to determine just how much of the measured capacitance resulted from the inserted dielectric material. G+ C C f V (t) G C ref Figure 2: An electrical scheme of a pulse method electrical circuit for determining the complex relative permittivity of a sample at low frequencies. [4] There does exist a method that alleviates most of this issues. It consists of a slightly more advanced setup shown in figure 2. The two pulse generators G+ and G will send equal but oppositely signed step signals trough the two capacitors C and C ref. The capacitors are equal except for the fact that capacitor C contains a sample of the material, while C ref does not. The amplifier on the right will produce an output voltage V (t), which will equal: V (t) = Q ref (t) Q ( t), (15) C f Q ref (t) and Q ( t) being the charges of the capacitors on the left. From this one can determine the time dependence of the capacitance of our test capacitor (Q = CV ). The last and most interesting step is the application of a Fourier transform: C (ω) = C (ω) ic (ω) = 0 dc(t) e ıωt dt. (16) dt We have thus gained a frequency dependent complex capacitance that can be linearly transformed to a complex relative permittivity. This looks a bit too good to be true, because judging from equation 16, one would say that we have with a single measurement gained data over the whole frequency range. The catch is in the integration limits, because we can not measure time from zero to infinity, but rather from a point close to the initial step function (t 1 ) to a point some time later t 2. Both of this points are limited by the accuracy of our devices. As a consequence our frequency range is limited as well 1/t2 ω 1/t 1. A typical such setup would thus provide measurements only in the range from 10 4 to 10 4 Hz. 6

7 3.1.1 Wien bridge method Figure 3: The Wien bridge setup for measuring a capacitor, with variable resistors R 2 and R 4. If we now turn our attention to more direct ways of measuring dielectric properties, the simplest practical method to employ would be the Wien bridge. A simple schematic of such a device is presented in figure 3. As with all Wheatston-type measurement bridges, the desired result is to have no voltage difference flowing between points A and B. This will be achieved when the ratio of impedances on the left side of the bridge equals the ratio on the right side. The resulting formulas for the characteristics of our device would be: C 2 R 4 C x = ( 1 + C 2 2 ω 2 R2) 2, (17) R3 ( ) 1 C2 R x = 2R + ω2 R 2 R 3 2 ω 2. (18) R 4 The important thing now is to figure out what this results mean for our measurement. The C x from equation 17 is quite straight forward, it represents the capacitance of our DUT capacitor, from which we can calculate the relative permittivity. How about the resistance R x? We do not connect in an extra resistor, this resistance in fact represents the losses of our capacitor. The angle δ in the loss parameter tan(δ) which we defined in equation 12 is the angle between the impedance of our DUT (Z x = ( Rx 1 ) 1) + ıωc x and the negative part of the imaginary axes of the impedance plane. Simply put tan(δ) = ωc x /R x. This sort of a bridge can, with appropriately selected components, work over a broad frequency range, from DC up to 300 MHz. Such an instrument can also be quite cheap and can provide quite good accuracies of at least 0.1%. Even if the adjusting of R 2 and R 4 is automatized, the measurements will still be among the more time consuming ones, because covering a wide frequency range will also require the occasional changing of other components. Therefore such devices usually find employment in environments where speed of measurement is not a major factor, for instance in standard setting laboratories LCR meter In most other cases however a faster tool known as the LCR meter is preferable. LCR meters enable us to measure the impedance of a DUT at a wide range of frequencies for 10 Hz all the way up to 110 MHz without any need for human interference in the process. The basis of a LCR meter is the auto-balancing bridge shown in figure 4. 7

8 DUT R OSC1 Null detector OSC2 Figure 4: A simplified circuit diagram of an auto-balancing bridge. [2] In the balanced state the impedance of the DUT at a selected frequency and amplitude on oscillator 1 (OSC1) will equal that of the impedance on the resistor R. To achieve this state the phase and amplitude of the second oscillator are adjusted with the feedback from the null detector. Knowing the resistance of the resistor R and the phase shift and amplitude of both oscillators, it is possible to calculate the impedance of the DUT. Since the difference of amplitudes of both oscillators can only be exact up to a certain limit a LCR meter is able to change the resistor R to a few preselected values so to better match the impedance of the DUT. Common values range from a few Ω to a few MΩ. The operational limits are set by the performance of the null point detector. Lower frequencies then 10 Hz would make the measurements increasingly time consuming. The high frequency limit of 110 MHz is a result of the technical limitations of the null detector circuit. 3.2 Radio frequency and microwave measurements As mentioned before, at frequencies above 110 MHz, other measuring techniques have to be applied. The choices are the radio frequency current-voltage measurement (RF I-V) which is used in impedance analysers and the reflective network analysis method. The first of this two methods can effectively cover the range from 100 MHz to 3 GHz, while the later can work from as low as 10 MHz to over 100 GHz. It may seem that the second option is to be favoured, but in fact it is exactly the opposite. The nature of the network analyser actually makes it much less user friendly Impedance analyser Let s first take a look at the RF I-V method used in impedance analysers. The idea behind its operation is that with the measur- Figure 5: The schematics of the two types of impedance analysers. [2] 8

9 ing of both currents and voltages over our DUT, we should be able to determine its impedance. There are two major types of impedance analysers, for high and low impedance DUTs. Most of the time the actual circuit is combined in a single machine, so it would be better to say that impedance analysers have two modes of operation. A scheme of the two modes can be seen in figure 5. The process of performing a measurement with an impedance analyser does not differ substantially from the one used when measuring with a LCR meter. One does need to be more cautious of possible parasitic effect because of the higher frequencies. The basic procedure of a measurement however remains the same Network analyser The (rather expensive) piece of equipment used to achieve measuring frequencies above 3 GHz is the network analyser. They can be used in a broad frequency range from a few megahertz to well over a hundred gigahertz. The basic schematic of such a device as shown in figure 6, also points at the other issue besides price. What we measure with a network analyser is not the impedance of a device, but rather the reflection of a signal if such a device is connected as a terminator of a wave guide. A two port measurement can be performed Figure 6: The schematics of a single port network analyser. [2] as well, where we measure both the reflected and transmitted signal, which can be useful if our DUT is for example a piece of coaxial cable. In any case the basic problem still remains. While it is possible to determine the impedance of a DUT with the following equation (for a single port measurement [5]): Z = Z S 11 1 S 11, (19) there is a major problem. The accuracy of the measurement is greatest when the impedance DUT closely matches the characteristic impedance (Z 0 ) of the wave guide, which is commonly 50 Ω. As can be seen from figure 7, this fact puts the network analyser in a unfavourable position when compared to the impedance analyser. At high frequencies where this is the only option, it has to be accepted. As will be shown later, on a practical example, this accuracy profile can lead us to some unusual DUTs, which are required to satisfy the 50 Ω condition. 9

10 Figure 7: Comparison of measurement sensitivity for network and impedance analysis. [2] 3.3 Optical measurements Measurements in the terahertz range are outside the scope of this seminar. Therefore there will be a slight jump in the frequency scale up to the optical frequencies. Here we need the ideas that where discussed in the previous section. An optical capacitor is an unheard of device, but reflection of an optical signal is something quite common. So that is what we will be working with. What we usually measure is the reflection index of a given material, which we shall for the rest of this subsection refer to as R (not to be confused with resistance that is denoted with the same symbol throughout the rest of the text). But ratio of the incoming and reflected signal does not provide us with enough information. We need to get the phase shift as well. As in fact we only measure the amplitude of the reflection, the phase shift is obtained by elipsometry or more commonly by the use of the Kramers-Kronig-dispersion relation: φ(ω) = ω ln R(ω ) ln R(ω) π ω 2 ω 2 dω. (20) 0 Once both the amplitude and the phase angle of reflection have been determined the next step is to get the complex reflection index of the measured sample. Maxwell s equations provide a simple way of achieving this: n = n + ık, (21) R = (n 1)2 + k 2 (n + 1) 2 + k 2, (22) tan φ = 2k 1 n 2 k 2. (23) If we now further assume that our sample has no noticeable magnetic properties (a valid assumption in practically all cases of interest), we can use the equation ε = n to get the real and imaginary parts of the relative permittivity as: 4 Practical examples ε = n 2 k 2, (24) ε = 2nk. (25) In the real world, setting up a measurement is not necessary a straight forward process. For simple non-ferroelectric ceramic materials it may just include applying metallic electrodes to the sample, inserting it into an appropriate fixture and measuring it at a few frequencies. For the measurements of thin film materials one usually starts with a lengthy 10

11 lithography process to manufacture the miniature DUT. This is followed by a set of precise network analyser measurements, from which the final results can often only be obtained by comparing them to numerical simulations. In the rest of this seminar we will look at two cases of such real world measurements. One will be the before mentioned thin film measurements in the microwave range, while the other will be low frequency measurements of liquid crystals. 4.1 Low frequency liquid crystal measurements The device of choice for low frequency measurements is a LCR meter. The real question is how the sample should be prepared. For solid materials the answer is straight forward, a piece of the material to be measured is shaped into a thin layer (for ceramics discs with a diameter of around 1 cm and a thickness of a few millimetres are common), with electrodes deposited directly onto the material. This is done so that there is no additional layers between the electrode and the material. In liquid crystals or in fact in any liquid such a procedure is clearly impossible. A solution is to employ the use of a holding cell, such as the one schematically represented in figure 8. It constitutes of 5 major components: 1. the substrate, which provides the rigid structure that holds the sample, 2. the dielectric spacers that are used to keep the substrate plates at a fixed distance, 3. the thin metal electrodes that act as the plates of our DUT capacitor, 4. the alignment layer, used to align the liquid crystals (in a cell used for regular liquids this layer would be left out) 5. the liquid crystal sample. It is of course necessary to measure both an empty cell and a cell with the inserted sample to deembed the properties of the sample from the gathered data. The measurement with the LCR meter is itself pretty straight forward. The only thing one has to be careful of is the amplitude of the applied Figure 8: A holding cell for liquid crystal measurements with for parallel (a) and perpendicularly (b) aligned molecules. AC signal. For most materials the common choice is 1 V. A liquid crystal at such voltages might already undergo a Frederiks transition so smaller voltages of 0.1 V are commonly used. 4.2 Microwave thin film measurements Another interesting problem is the measurement of thin films at microwave frequencies. A thin film is a layer of material, typically thinner then 1 µm. Such a thin layer can not be freely transported, it is always bound to the substrate on which it was manufactured. A problem that commonly occurs is that the structure of the thin film is strongly dependent on the substrate material. If it is possible to get the desired thin film structure by growing it on a metallic substrate the measurements are quite simple. All one has to do is to 11

12 construct an appropriate top electrode and make a connection to the bottom metallic layer and we obtain a capacitor that we can measure. Figure 9: The design of a planar interdigital capacitor with 7 fingers as vied from the top (left), and its cross section (right). When it is required that the substrate is a specific non-conductive material (alumina being a common example) another type of device is required. Microstrip lines are a possible choice, but the one that will be described in more details is the planar interdigital capacitor. An example of such a capacitor is shown in figure 9. It consists of two electrodes deposited on top of the sample. The shield of the coaxial cable leading to the network analyser is connected at two symmetrical points to the outer (G for ground) electrode, while it s center is connected to the central (S for source) electrode. The capacitor is formed between the fingers of both electrodes with the majority of the electrical field density being contained in the thin film layer. To achieve a good measurement the impedance of the DUT has to be close to Z 0 = 50 Ω. At sample thickness s of less then a micrometer and with frequencies above one gigahertz this implies devices that are a few hundred micrometers in size. Lithographical manufacture of such a device is not a major problem nowadays however, a problem arises when we want to connect this microscopic device to our measuring equipment. For this task a probe station with microwave probe tips is required. A microwave probe tip is a device that connects on one end to the coaxial cable of a network analyses and on the other is shaped into tips of appropriate dimensions (pitches from 50 to a few hundred micro meters are common). A probe station is an elaborate mechanical setup that allows for precise positioning of said tips, with accuracy of a few micro meters. A vital part of a micro probe setup is also the accompanying calibration standard without which the measurement from the probe tips would be garbled beyond recognition. Once the DUT has been manufactured and connected with a calibrated probe, the reflection coefficient S 11 is measured. Using the equation 19, the impedance of the DUT is then calculated. Now we are faced with one final obstacle. The requirements of the material have forced us to construct quite an uncommon capacitor. As a consequence we can no longer use equation 13 to determine the relative permittivity and losses. While there are some closed analytical formulas form determining a capacitance of a planar capacitor, they usually carry with them an inaccuracy of around 10%. The only viable solution we are left with are numerical simulations. So the final part of such a measurement consists of finely tuning the permittivity and losses of a simulated system until the simulation results correspond with the measurement. 12

13 5 Conclusion This seminar has hopefully given the reader some sense of what dielectric measurements are about, and a brief overview of the most common experimental techniques. There is of course a much larger range of techniques out there for all the different materials and circumstances under which they are to be measured. A reader wishing to perform such measurements should however if at all possible consult with an expert in the field before doing so. Because although the basic principles are not exceedingly difficult to understand, there is a whole array of issues that a first time experimentalist can stumble upon. Finally, the last figure 10 sums up the devices and experimental techniques (many of them described in the seminar) that are used in a typical dielectric spectroscopy laboratory. Figure 10: A representation of the devices and techniques used at the Dielectric Spectroscopy Laboratory at the Institute for Physics of the University of Augsburg, Germany [?] References [1] Hu Lizhi, K. Toyoda, and I. Ihara. Dielectric properties of edible oils and fatty acids as a function of frequency, temperature, moisture and composition. Journal of Food Engineering, 88(2): , [2] Agilent. Agilent impedance measurement handbook a guide to measurement technology and techniques. Measurement, (3), [3] Derek D. Hass. Dielectric sensing of ceramic particle suspensions. Master s thesis, University of Virginia, [4] F. I. Mopsik. Precision time-domain dielectric spectrometer. Review of Scientific Instruments, 55(1):79 87,

14 [5] S A Arcone. A numerical study of dielectric measurements using single-reflection timedomain reflectometry. Journal of Physics E: Scientific Instruments, 19(6):448,