CHAPTER 3 EXPERIMENTAL METHODS AND THEIR THEORETICAL BACKGROUND

Transcription

1 38 CHAPTER 3 EXPERIMENTAL METHODS AND THEIR THEORETICAL BACKGROUND Studies on the complex perovskite systems involve the method of synthesis, characterization techniques and physical property measurements. The compositions are synthesized by high temperature solid state reaction technique, characterized using XRD and SEM and the physical properties dielectric characterization, electrical conductivity measurement and Seebeck coefficient measurement are taken. The general principles and methodology employed are presented. 3.1 MATERIALS SYNTHESIS Materials synthesis occupies the central position in the development of advanced ceramics as the experimental methods allow the control of the properties. The dielectric properties of material are found to depend on the sample preparation conditions and the starting materials have become the most critical feature in the formation of homogenous final product after temperature and time control. Careful choice of starting powders should have specified chemical composition without unwanted impurities. The starting materials used are high purity carbonates and oxides. BaCO 3 (99.0% pure Sdfine), SrCO 3 (98+% pure Aldrich), CaCO 3 (99.0% pure Fluka), Sm 2 O 3 (99.0% pure Kemie), Ho 2 O 3 (99.9% pure Acros) and Nb 2 O 5 (99.5% pure Sdfine).

2 39 Samples are prepared by high temperature solid state reaction technique. The main steps involved in the synthesis are given in the flow chart shown in Figure 3.1. Weighing in stoichiometric proportions Dry mixing, wet mixing Calcination Phase confirmation (XRD) Pelletizing Sintering Density measurements Polishing Seebeck coefficient measurements Dielectric measurements Figure 3.1 Flow chart of the important steps involved in the synthesis of the compositions

3 Weighing and Grinding First the powder must be weighed to estimate the exact amount needed. It is highly suggested to sterilize the spatula with acetone between each weighing. All the compounds are placed in an agate mortar and pestle for grinding process. Grinding helps to homogenize the mixture. The powders are first dry mixed and then wet mixed with acetone as medium. Uniform mixing of reagents is necessary for reproducible results Calcinations The mixed powder is placed in an alumina crucible for calcination. Calcine is the chemical process of subjecting absorptive mineral to prolong heating at fairly high temperature with access to air, resulting in removal of water, an increase in hardness, physical stability and absorbent properties of the material. The process converts non-oxide precursors to oxide and react with the precursors to form a phase closer to the final product and it helps to homogenize the material. The calcination temperature and duration should be optimized for the complete reaction of the reagents considered (Wang et al 1994, Wu et al 1995). After calcination single phase formation must be ensured using XRD Pelletizing and Sintering Calcined powders are mixed with 2% polyvinyl alcohol (PVA) which acts as a binder. It reduces brittleness and results in better compactness amongst the granules of the materials. Cylindrical pellets are made using uniaxial isostatic cold press with a help of a hydraulic press. The pellets are heated to C for one hour to evaporate PVA. Sintering involves coordinated change of all grains in a powder compact to allow them change

4 41 themselves in a space filling manner. The grain centers move towards each other thereby reducing the size of the compact and eliminating the pores. The reduction of surface and interface area is the driving force for the process. The sintering is influenced by the particle size, distribution in the particle size and agglomeration of the particles both before and after calcination. Calcination has significant influence on the quality of sintering. Hence one should choose the calcination temperature, duration of calcination carefully to ensure that the reagents react well. High purity chemicals are to be used for proper sintering. This ensures high values of density. Sintering temperature and minimum possible soaking time at the temperature play significant role in determining and reproducibility of the physical properties (Wang et al 1994, Villegas et al 2000) Polishing and Density Measurement Both sides of the sintered pellets are polished well and are made ready for property measurement. Density of the pellets is measured by Archimedes method and from dimensions and weight. The measured density is compared with the calculated X-ray density. Mn Theoretical X-ray density = 3 Na M - n - N - a - Molecular weight Number of molecules per unit cell Avogadro number lattice constant

5 STRUCTURAL AND MICROSTRUCTURAL CHARACTERIZATION X-ray diffraction X-rays are electromagnetic radiation. They are of exactly the same nature as light but with much shorter wavelength. While visible light has a wavelength of the order of Å, X-rays used in diffraction lies in the range of Å. They are found between the gamma and ultraviolet rays in the electromagnetic spectrum. This radiation has been of invaluable importance to solid state physics in two key aspects of research, namely in the finger print identification of crystalline materials and in determining their structure (Anthony 1996, Percharsky 2003). X-rays are produced when electrically charged particles usually electrons, accelerated through a potential >30 KV and therefore of sufficient kinetic energy, are rapidly decelerated by collision with a solid body, e.g a copper target. The electrons hitting the target have sufficient energy to ionize some of the copper 1s electrons, i.e. electrons from the K shell. As this happen, an electron from either of the outer 2p or 3p orbital immediately drops to occupy the vacancy on the 1s level. The energy that is released in this transition appears as X-rays. This is illustrated in Figure 3.2. In the case of copper, the 2p 1s is transition produces K radiation with a wavelength of 1.54Å. The K radiation is the one typically used in diffraction experiments because of its high intensity.

6 43 Figure 3.2 The generation of Cu K X- rays The radiation is produced in an X- ray tube comprising of an electron source (the cathode) which is generally a heated tungsten filament, and an anode to which the copper target is attached. The X- ray tube is evacuated to prevent oxidation of the tungsten filament. Less than one part in a thousand of the energy of the incident electron beam goes into production of X- rays, and the rest gets converted into heat that can easily melt the anode. Therefore, constant cooling of the anode is very important and this is achieved by maintaining a continuous water flow around the latter while the X- ray tube is in operation. Beryllium is used to make up the small windows found on the tube walls through which the X- rays leave. The X- rays emerging from the X- ray tube contain the K as well as radiation of other wavelengths. Since it is best to have a monochromatic beam of X- rays for most diffraction experiments, and the K line is the most intense one, it is desired to filter out all the other wavelengths. In the case of X- rays emitted from a copper target, a Ni foil is used as filter. A schematic view of X- ray powder diffractometer is shown in Figure 3.3. A narrow diffracted peaks and low background with highly monochromatized X- ray beam is made to fall on the sample. The sample is mounted tangent to the focusing circle with the

7 44 scintillation counter tube moving along the circumference of it. It is possible to record the diffracted beam from 2 to 90 o with camera mounted in different mounting positions. The diffractometer is connected to a computer for data collection and analysis. Figure 3.3 Schematic view of X-ray powder diffractometer Depending on the size and shape of the unit cell, the atomic number and position of the different atoms in the structure, different crystalline materials will produce their own distinct diffraction patterns. In a powder sample the lattice planes are oriented in all possible directions, of which some are oriented at the Bragg angle,, to the incident beam, satisfying the Bragg s law, 2d sin = n, where is the angle of incidence, d is the interplanar distance and is the wavelength of the incident beam. Therefore, diffraction occurs for these planes. The diffraction principle and the Bragg s law are explained using Figure 3.4 which shows two X-ray beams, 1 and 2 being reflected from two adjacent planes A and B. For the reflected beams to be in phase, the extra distance XY + YZ, travelled by beam 2 must be equal to a whole number of wavelengths.

8 45 Figure 3.4 Diffraction principle and derivation of Bragg s law Since XY = YZ = d sin XY + YZ = 2d sin and XY + YZ = n where d, the spacing between the two planes and, the angle of incidence / Bragg angle, we obtain the Bragg s law. 2d sin = n (3.1) The diffraction spectra, which are plots of intensity of the Bragg s peak as a function of 2, are used to identify the compounds and to check the phase purity of the samples. Each crystalline solid has unique characteristic X-ray powder pattern which may be used as a finger print for its identification. Compounds of the same structural type do not give rise to the same powder diffraction patterns. This is because they have different unit cell parameters due to their different unit cell size, which make the d spacing and therefore, the peak positions vary. The intensities of the peaks also vary because of the presence of different types of ions with different atomic numbers and hence, different X-ray scattering powers.

9 46 Scintillation detector can be moved in steps of o by means of stepper motor and the diffracted beam can be closely scanned to study peak profiles. When the grain size of a polycrystalline material is very small, say in the nanometer range, the crystals cause broadening of the diffracted beam due to diffraction at angles near to, but not equal to the exact Bragg angle. The crystallite size (D) and the full width at half maximum (FWHM) of the diffracted line (in radian) are related by D = 0.9 / cos p (3.2) where is the wavelength of X-rays used p is the Bragg s angle of the peak in degree. The equation (3.2) is known as the Scherrer formula (Cullity 1967). In the present work PANalytical X pert PRO model with nickel filtered Cu K radiation is used for structural analysis Scanning Electron Microscopy Scanning Electron Microscopy (SEM) is a powerful tool for unveiling important and useful information about the microstructure of solids. SEM is able to produce very detailed 3D images at considerably higher magnifications than can be achieved by an optical microscope (Anthony 1996). The scanning electron microscope uses electrons rather than light waves to create those images. SEM analysis reveals detailed information about ceramic surface texture, grain sizes and shapes and grain boundaries. For an SEM experiment, the sample s surface has to be made conducting and this can be achieved either by applying a coating of graphite paste or by sputtering a thin layer of gold on the surface to be scanned. The sample is then mounted on a holder inside the sample chamber and air is

10 47 pumped out of the chamber in order to create a high vacuum. Once this is achieved, an electron gun placed right on top of the sample emits a high energy electron beam which is focused onto a fine spot on the specimen (Fishbane et al 1996, Skoog et al 1998). This focused beam is then made to scan the surface of the specimen by moving back and forth. As the high energy electrons hit the surface of the specimen, secondary electrons are ejected from a depth of about Å. This is the result of interactions between the energetic electrons from the beam and conduction electrons which are weakly bound in the sample. The secondary electrons then hit a detector which counts the electrons and sends the signals to an amplifier and the final image is created based on the number of electrons which are emitted from the different spots on the sample. A schematic representation of the instrumentation involved in a Scanning Electron Microscope is shown in Figure 3.5. In this work microstructure of all the samples are analyzed using Scanning Electron Microscope JOEL JSM / 6360 model. Figure 3.5 Schematic representation of a Scanning Electron Microscope

11 THERMO GRAVIMETRY / DIFFERENTIAL THERMAL ANALYSIS (TG / DTA) It is important to have the knowledge of thermal stability, chemical stability and the possibility of phase transition if any, of a material. Methods commonly used for this characterization of the materials include thermal analysis techniques, namely, Thermo gravimetry (TG) and Differential Thermal Analysis (DTA) (Brown 2001). When heat is applied to a compound, two events may take place simultaneously, a thermal change and weight loss, but the latter does not always occur. Thermo gravimetry measures the change in weight of the sample as a function of time or temperature, while the differential analysis measures the difference in temperature between an inert reference material and the sample while both are being heated at a constant rate in a suitable atmosphere (Brown 2001). In DTA, two thermocouples of the same nature are connected together, i.e. either the positive or negative legs of both thermocouples are electrically connected, meaning that the net electromotive force (emf) for the two thermocouples is zero at any given temperature. Therefore an emf response will only occur when the temperature of one thermocouple differs from the other. In a DTA experiment the sample is put next to one thermocouple and an inert reference material, usually alumina powder is put next to the other thermocouple, since its heat capacity remains constant even up to its melting point of 1930 o C. As both are heated at an equal heat flow, thermal changes can be followed since each thermocouple produces its own emf as temperature changes. Since no thermal changes take place in the reference, any changes that are detected come from thermal events in the sample. These temperature differences therefore, serve as reaction indicators. A plot of these temperature differences as a function of temperature gives rise

12 49 to a curve with peaks that either increase (exothermic) above or decrease (endothermic) below the baseline. If evaporation of water (or any type of solvents) occurs, an endothermic peak appears on the DTA curve because the process absorbs heat, causing the temperature of the thermocouple next to the sample to go down. If a structural change that tends to release heat occurs in the sample, then an exothermic peak will be observed. Schematic representation of a typical DTA setup and a DTA curve are shown in Figure 3.6. Thermo gravimetric analysis measures the change in the weight of a sample as a function of increasing temperature. The apparatus consists of an analytical balance, which has a weight - change detector on one side of it. The equipment also has a temperature controlled furnace, which houses a crucible containing the sample. A change in sample weight is detected and the current in the drive coil changes to counter the displacement of the beam. The change in current is proportional to the change in the weight of the sample. This change is plotted as a function of temperature. In the present work thermal analysis is carried out using Simultaneous TG / DTA, SDTQ 600 V 8.2 build 100 up to a maximum temperature of 1000 o C at the heating rate of 20 o C / min in air.

13 50 Figure 3.6 (a) Instrumentation for Differential Thermal Analysis (DTA) (b) DTA curve showing exothermic and endothermic peaks

14 DIELECTRIC CHARACTERIZATION In order to characterize the microstructure and electrical properties of electroceramics, techniques that can probe or distinguish between the different microstructural regions of a ceramic are required. One of the methods available to measure the electrical properties of ceramics is impedance spectroscopy (Ross Macdonald 1987). This technique has gained a lot of importance because it has proved to be very powerful for unravelling the complexities of electroceramics. The frequency dependence of the different parts of an electroceramic allows the overall electrical properties of the material to be separated, thus making them easier to be studied and/or modified. In impedance spectroscopy, an alternating voltage is applied across a sample and a standard resistor, which are in series, and the in and out of phase components of the voltage across the sample are measured. These components are then divided by the magnitude of the current to give the resistive and reactive components of the impedance. The measurements are repeated as a stepwise function of frequency which typically ranges from 10-2 to 10 7 HZ, and the different regions of the material are characterized according to their electrical relaxation times or time constants, by a resistance and a capacitance, usually placed in parallel. The principle of dielectric measurement of a material is shown in Figure 3.7. Figure 3.7 Principle of dielectric measurement The sample usually in the form of a disc is sputtered with a layer of gold on each side and the electrical wires (Gold) are attached using silver paste. As shown in Figure 3.7 a sinusoidal voltage V (t) with a fixed frequency /2 is applied to the sample by means of the generator, and this

15 52 produces polarization in the sample. This voltage causes a current I (t) to flow through the sample at the same frequency. V (t) = V o sin t I (t) = I o sin ( t + ) Generally, there is a phase shift between the resulting current and the applied voltage, which is known as the phase angle. This is graphically represented in Figure 3.8. By measuring the complex impedance Z* of the sample and by separating it into its real and imaginary parts, Z and Z, the capacitance, C and conductance, G can be calculated as follows: Z* = Z + Z V / I = V o sin t / I o sin ( t + ) = 1 1 (3.3) G C The dielectric constant of the sample, which is the real part of the dielectric permittivity, can then be obtained from the impedance measurement. Figure 3.8 Relationship of amplitude and phase difference between applied voltage and resulting current

16 53 The permittivity, of a dielectric material is related to its capacitance C. The application of a voltage V, to the sample and the flow of a current, I through it causes the accumulation of a given amount of charges Q, on the opposite plates of the capacitor, producing an electric field between the plates. The presence of dielectric material between the parallel plates of the capacitor causes a reduction in the effective field E. This is because the polarization of the material produces an electric field, E Polarization, which opposes the field of the charges on the plates, E o. This is illustrated in Figure 3.9. Therefore E = E o + E Polarization = / o (3.4) where - Charge per unit area o - - Permittivity of free space and The dielectric constant of the material E = V/d; Q = CV and = Q / A, V = Ed = Q/C Therefore, C = Q / Ed = Q o / d C = ' d (3.5) where C - Capacitance A - Area of the plates and d - Distance separating the plates (or) thickness of the dielectric material.

17 54 Figure 3.9 Reduction of effective electric field between the plates of a capacitor by a dielectric material Ferroelectric and relaxor ferroelectric materials cause a considerable lowering of the effective electric field, E, between the capacitor plates. Ferroelectrics have their polarization oriented in different directions, which gives rise to the formation of domains, and within a given domain, all dipoles are oriented in the same direction. The application of an external field then causes domain walls to move as all the dipoles begin to align themselves in the same direction. This leads to a large increase in polarization which results in an electric field, E polarization, opposing the applied electric field E o. Owing to these non-linear effects, large values of static dielectric constant are observed in these materials. When subjected to an alternating field, a point is reached when at certain frequencies the dipoles which are responsible for polarization can no longer keep up with the oscillation of the electric field. The reorientation of the dipoles and the reversal of the field therefore become out of phase causing a dielectric relaxation, i.e. a decrease in the real part of permittivity Impedance Spectroscopy Impedance Spectroscopy is a simple and powerful tool to study the electrical and dielectric properties of a compacted insulating, ionically conducting and semi conducting materials and their interfaces.

18 55 The electrochemical processes represent a complex combination of various individual circuit elements, such as a resistor (R), a capacitor (C) and an inductor (L), and the contribution of any given component depend on the frequency of measurement. Complex plane analysis is a mathematical technique in which the circuit elements are determined from the data obtained over a wide range of frequencies (Ross Macdonald 1987). It involves the plotting of the real and imaginary parts of complex electrical quantities such as (i) (ii) (iii) (iv) Z* - complex impedance Y* - complex admittance * - complex permittivity and M* - complex modulus The plots are the characteristics of particular circuits and consist of a combination of semicircles and straight lines. Intercept with real axis and frequency values corresponding to the maximum of Y-coordinate lead to simple component values. Impedance spectroscopy is a powerful method for studying many electrical properties of materials and their interfaces with electronically conducting electrodes. It may be used to investigate the dynamics of bound or mobile charges in the bulk or interfacial regions of any kind of solid or liquid materials like ionic, semiconducting, mixed electronic-ionic and insulators (dielectrics). There are three different types of electrical stimuli which are used in Impedance Spectroscopy. First in transient measurement a step function of voltage {V (t) = V o for t > 0, V (t) = 0 for t < 0} may be applied at t = 0 to the system and the resulting time-varying current I (t) is measured. The ratio V o / I (t), often called the indicial impedance or the time-varying

19 56 resistance, measures the impedance resulting from the step function voltage perturbation at the electrochemical interface. This quantity, although easily defined is not the usual impedance referred to in Impedance Spectroscopy. The second technique in Impedance Spectroscopy is to apply a signal V (t) composed of random (white) noise to the interface and to measure the resulting current. Again, one generally Fourier transforms the results to pass into the frequency domain and obtain impedance. This approach offers the advantage of fast data collection because only one signal is applied to the interface for a short time. This technique has the disadvantages of requiring a true white noise and Fourier analysis, which is computationally difficult and time - consuming. The third approach, the most common and standard one, is to measure impedance directly in the frequency domain by applying a singlefrequency voltage to the interface and measuring the phase shift and amplitude of real and imaginary parts, of the resulting current at that frequency. Commercially available instruments measure the impedance as a function of frequency automatically in the frequency range of about 10-2 to 10 7 Hz and they are interfaced with laboratory microcomputers. The advantages of this approach are the availability of these instruments and the ease with which one can use them, as well as the fact that the experimentalist can control the frequency range to examine the domain of his interest. It is useful and not surprising that modern advances in electronic automation have included Impedance Spectroscopy. Sophisticated automatic experimental equipment have been developed to measure and analyze the frequency response of a small amplitude a.c signal between about 10-4 and 10 6 Hz, interfacing its results to computers and their peripherals.

20 General a.c circuit theory The applied a.c. voltage and the measured current in an electrical network are given by V (t) = V o e j t (3.6) I (t) = I o e j ( t+ ) (3.7) j 1 where is the phase angle, V o and I o are the maximum amplitude of the voltage and current respectively. The impedance Z ( ) of the circuit is given by Z ( ) = Z e -j = Z cos - j Z sin = Z r j Z i (3.8) where Z r and Z i are the real and imaginary parts of the complex impedance. The relationships between various quantities are given by 2 2 Z Zr Z i 1/2 (3.9) tan = Z i / Z r (3.10) For an RC circuit shown in Figure 3.10 (a), 1/ Z = 1/ R + j C (3.11) Simplifying this equation, Z = R / (1+ R C )- j R C / (1+ R C ) (3.12)

21 58 Thus Z r =R/(1+ω R C ) (3.13) and Z i = R C / (1+ ω R C ) (3.14) Eliminating and C and rearrange these equations, one gets, (Z -R/2) +Z = R /4 (3.15) r i which is the equation of a circle of radius R/2 with centre at (R/2,0). The typical impedance spectrum in the complex impedance plane corresponding to the circuit of Figure 3.10(a) is shown in Figure 3.10(b). (a) (b) Figure 3.10 (a) A common RC circuits (b) its equivalent complex impedance plane plot.

22 Constant phase elements A main feature of the solid electrolyte is the noticeable flattening of the semi-circle and tilting of the spike compared with the idealized shape obtained from an equivalent circuit composed of series and parallel combinations of resistors and capacitors. The equivalent circuit shown in Figure 3.11 is the simplest arrangement of resistors and capacitors that can account for the physical features present in the material, namely the bulk electrolyte resistance, the charge transfer capacitance and the geometric capacitance that remains even in the presence of the electrolyte. A new type of circuit element, the constant phase angle element or simply constant phase element (CPE) is needed to account for the semi-circle flattening and spike tilting in the complex impedance (CI) plane. It may be thought of conceptually as a leaky capacitor, the physical origin of which perhaps is related to the presence of crystalline non-conducting region interwoven with conducting amorphous material within sherulites. Its impedance is given by -P Z CPE = k(j ),0 p 1 (3.16) When p = 0, Z is independent of frequency and k is just the resistance R, whereas when -1 p =1, Z =1/j k (the constant k -1 now corresponds to the capacitance C. When p is between 0 and 1, the CPE acts in a way intermediate between a resistor and a capacitor. The Z CPE is incorporated into an expression so that it can be used to model equivalent circuits to interpret the experimental complex impedance plots. The above equation can be recast using Euler s formula from which, we have j e = cos + jsin, (3.17)

23 60 j p 1/2 p p/ 2 ( 1) ( 1) (3.18) When =, Eq. (3.17) yields i e = cos( ) + jsin ( )=-1 (3.19) Figure 3.11 An equivalent circuit for constant phase element and its impedance spectrum (Solid line: spectrum for equivalent circuit, Dashed line: A typical experimental spectrum from an ceramic electrolyte.) Combining Eqs. (3.18) and (3.19), we arrive at -p j -p/2 -jp j = (e ) = e so that, substitution into Eq. (3.17) yields Z = k cos(p / 2) - jsin (p / 2) / CPE p The cosine term contributes to Z and the sine term Z. The use of the series CPE term tilts the spike and parallel CPE terms depress the semicircle as shown in Figure 3.12.

24 61 Figure 3.12 Depression of semicircle and tilting of spike caused by replacing capacitors by a constant phase element Automated impedance analyzer Generally, this class of ac analyzer operates with a so called autobalance bridge. The desired signal (comprising both ac and dc components) is applied to the unknown impedance, as shown in Figure The current follower effectively constrains all the current flowing through the unknown impedance i r to flow through the resistor R r, presenting a virtual ground at the terminal marked Low. For this condition the impedance can be measured as Z unknown = R r (e i / e r ) (3.20) The complex ratio (e i /e r ) is measured in a manner very similar to that of Barbarian-Cole bridge circuits, in which the in-phase and the quadrature fractions of the input signal are summed with the unknown output signal (current) until the result is zero. The oscillator that produces the input perturbation signal e i, also outputs in phase and quadrature (90 o out of phase)

25 62 Figure 3.13 (a) Impedance analyzer and (b) Direct measurement of impedance using an impedance analyzer (Ross Macdonald 1987) reference signal that are proportional in amplitude to e i. These are fed to a summing circuit and summed with the unknown current until the current to the detector, i d, is zero. Under these conditions the low-potential terminal is at ground voltage, and Z i = R r /(a+jb) (3.21)

26 63 Thus, the unknown impedance can be determined directly from the value of the range resistor, R r, and the attenuation factors a and b imposed by the null detector to achieve the null condition. The advantages of this method are relatively high speed and high precision. Being null method, the effects of stray capacitances are somewhat reduced, although, unlike in a true bridge, currents do flow through the unknown impedance at the null condition. The intrinsic disadvantage of this method is because of its two terminal nature. The dc potential cannot be applied to the electrode of interest with respect to a suitable reference electrode. The potential e i across the specimen varies during the balance procedure and hence the measurements give potential information. Solartron 1253 impedance/gain phase analyzer and 1287 dielectric interface is used in the frequency range of 100Hz to 5MHz and for temperature range of 300K to 700K to study the electrical transport of the samples in this work. 3.5 MEASUREMENT OF SEEBECK COEFFICIENT Seebeck coefficient or thermoelectric power (S) is a bulk property of the material like electrical resistivity and is dependent on the nature, number and mobility of the charge carriers. The sign of the seebeck coefficient will determine whether electrons or holes make dominant contribution to the conduction process (positive for holes and negative for electrons). The phase transitions and changes in the mechanism of electrical conduction may also affect the Seebeck coefficient of a material by way of change in magnitude and or sign. The true Seebeck coefficient of a material is defined by S = ( E / T) T 0 and

27 64 E (E E ) / T T T H C H C (3.22) where E H and T H are the electric potential and temperature at the hot end, E c and T c the corresponding values at the cold end. Since the charges flow from higher to lower thermal gradient, the cold end of the sample becomes positive or negative according to the nature of the effective charge carriers accumulated on it. Thus the cold is positive for holes (p type) and negative for electrons (n type). The condition for obtaining high degree of reproducibility is that the surface of the specimen in contact with the metal against which Seebeck coefficient is measured be exactly parallel and finely polished (Parker 1978). In the present study Seebeck coefficients are measured on polycrystalline samples in pellet form by a compact unit (Subramanian 1989) shown in Figure The unit machined out of brass has copper electrodes isolated by concentric teflon insulator. The temperature gradient is produced by electrical heating of one electrode using non inductive winding of nichrome wire with asbestos thread insulation. The entire unit can be placed in an outer heater to study Seebeck effect as a function of temperature for a constant temperature gradient. The temperature range for measurements is 300K to 473 K. A screw head arrangement provided the pressure contact. Chromel - alumel thermocouple is used and the leads are kept in a small hole drilled in the copper block very close to the sample. Two separate leads are brazed to the blocks and are used for thermoemf measurements. The thermoemf voltage across the specimen is measured by a very sensitive

28 65 S 1 S 2 Leads for thermoelectric measurements H 1 H 2 Heater coils C, C Copper blocks C1 Copper rod S Screw head T Teflon X Sample I Insulating cover Figure 3.14 Cell for thermoelectric measurements micro voltmeter. The thermoelectric power is measured for a temperature difference of T = 20K across the sample thickness of 2-3mm. The Seebeck coefficient(s) and the carrier concentration (n) is related by the equation, 7 3 K P S= 2 nh ln 3 e P 5 2(2 mkt ) 2 2 (3.23) For an ionic lattice P = ½

29 66 3 K nh S= e 3 ln 2(2 ) 3 mkt 2 The minus sign for the value of S are due to the assumption that the charge carriers are electrons. For holes the Seebeck coefficient is positive. The Seebeck coefficient decreases with increasing carrier concentration (Jain 1967).