# Intermodulation distortion performance enhancement of microwave power amplifiers

код для вставкиСкачатьSIMULATION OF MICROWAVE ABSORPTION IN CONDUCTING THIN FILMS BY Shramana Mishra Submitted to the Department of Physics and Astronomy and the Faculty of the Graduate School of the University of Kansas In partial fulfillment of the requirements for the degree of Master’s of Science ____________________ Chairperson Committee members: __________________ __________________ Date defended: 16th August 2005 UMI Number: 1431757 UMI Microform 1431757 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 The Thesis Committee for Shramana Mishra certifies That this is the approved version of the following thesis: SIMULATION OF MICROWAVE ABSORPTION IN CONDUCTING THIN FILMS Committee: ______________________ Chairperson _______________________ _______________________ Date approved: 29th August 2005 ii Abstract Near-field microwave microscopy has been used as a non-destructive tool for current distribution mapping in high-Tc superconductors (HTS) and other conducting films at room temperature. The focused microwave emitted from the tip of a near-field scanning microwave microprobe (NSMM) is absorbed in the thin film in a small spot of size approximately same as the size of the probe tip. The heat will spread into the adjacent regions in the thin film sample through thermal diffusion. The temperature rise due to local heating will lead to the change in the local resistivity of the sample which results in a voltage response depending on the local current density and the input microwave power. The amount of temperature rise at different regions of the film depends on various sample and environmental parameters like the thermal conductivity and specific heat of the material of the film, thermal conductivity of the substrate, heat transfer coefficient of air etc. The map of voltage response acquired while the microwave probe scans the film surface can provide information about the surface morphology and the non-uniformity of the electromagnetic properties of the film. Hence in order to understand the absorption of microwave radiation in conducting thin films, a theoretical model based on heat diffusion is developed to determine how the different parameters affect the temperature rise and heat diffusion. The temperature profile due to microwave absorption on silver thin film is obtained by solving the heat diffusion equation developed in the theoretical model. The iii temperature profile is related to the voltage response profile which is presented for the one dimensional scan across the width of the film. The variation of the induced voltage due to microwave absorption with time, microwave input power and the thickness of the film is also presented. The voltage profile obtained through the simulation is qualitatively similar to the experimentally obtained profile for the silver micro-bridge. iv Acknowledgements I take this opportunity to thank my advisor Prof. Judy Z. Wu of Department of Physics and Astronomy for her tutelage, guidance and encouragement during the course of my work. I would also like to thank Prof. D. W. McKay and Prof. Jack Shi of Department of Physics and Astronomy for serving in the committee and especially Prof. Jack Shi for the discussions I had with him while developing the simulation model. I would like to thank Jonathan, Dr. R. S. Aga, Jr. and others in thin film lab for helping me with the experimental data and my fellow graduate students in physics G.T.A. office for always being there with a helpful attitude. I am thankful to Department of Physics and Astronomy for supporting me during my stay here. Lastly, I would like to thank my husband for helping me in many ways and my other family members for their moral support. v Table of Contents Page No. Abstract iii Acknowledgements v Chapter 1: Introduction and Background 1 1.1 Introduction 1 1.2 Existing techniques for current mapping 3 1.2.1 Techniques based on magnetic field mapping 1.2.2 ‘Hot-Spot’ techniques 12 1.2.3 Other techniques 17 Microwave mapping 19 1.3 5 Chapter 2: Review of Experimental work 25 2.1 Experimental details 25 2.2 Experimental results 27 2.3 Interpretation of the experimental results 28 2.4 Motivation behind the simulation 29 Chapter 3: Theory of microwave absorption 31 3.1 Microwave absorption in thin films 32 3.2 Heat Diffusion 35 vi Page No. 3.2.1 Heat diffusion equation 36 3.2.2 Mechanisms of heat loss 38 3.2.3 Solution of the heat diffusion equation. 41 3.3 Solution of the continuity in current flux 43 Chapter 4: Results and discussions 48 4.1 Temperature profile 49 4.2 Voltage induced due to microwave 51 4.2.1 Time dependence of V 51 4.2.2 Input power dependence of V 54 4.2.3 Thickness dependence of V 58 4.2.4 4.2.5 4.3 V profile along the width of the film Two dimensional V profile 60 67 Comparison of laser and microwave heating 68 Conclusions 72 References 74 Appendix 78 vii CHAPTER: 1 INTRODUCTION AND BACKGROUND 1.1 INTRODUCTION: Conducting thin films such as high-TC superconductors (HTS) find wide scale applications not only in fundamental research but also in industrial applications like electric power transmission cable, electric motors and generators, power quality tuners and fault current limiting devices. For all the practical application of HTS materials in various electrical applications requires the HTS coated conductors which must be long in length, carry high current and are affordable. The coated conductors which are often called second-generation HTS wires or conductors are comprised of the HTS layer epitaxially grown on a textured buffer layer on a metal tape which gives flexibility and strength to the multilayer conductor1. One of the biggest hurdles to widespread application of HTS coated conductor tape is developing a manufacturing process that will produce long length conductors with minimum mechanical defects. In fact, most coated conductors have only been produced in a laboratory environment with a characteristic area of a few square centimeters. Also for different electrical 1 applications, it is necessary that sufficient critical currents are achieved at sufficient conductor lengths and the coated conductor has to be stable against mechanical loads, current overloads, magnetic fields etc. In HTS coated conductors, uniform critical current density (Jc) greater than 1MA/cm2 at 77 K and self field is required over kilometers of lengths. The transport properties of these coated conductors are strongly influenced by randomly distributed current limiting obstacles such as grain boundaries and mechanical defects. The local Joule heating due to these defects can cause thermal instabilities, which will limit the current-carrying capability of these conductors. Hence, identification and investigation of these defects is very important for the quality control of the current carrying capabilities of these coated conductors. Recently, several techniques have been developedto directly visualize the current flow in the conductors so as to locate these defects. These techniques can be categorized under two categories, techniques involving mapping of magnetic fields like magneto-optical imaging (MOI)2,3,4, Hall probe magnetometry5,6, superconducting quantum interference device (SQUID) microscopy7,8 and magnetic force microscopy (MFM)9,10,11 and the techniques based on local heating of current biased samples like scanning electron12 or laser microscopy13,14,15. Most of the current mapping techniques based on magnetic field mapping involve cooling down the samples to superconducting temperature. There are also some other techniques not based on magnetic field mapping, that requires cooling down the samples to superconducting temperatures like terahertz (THz) mapping16,17. In 2 the case of techniques based on local heating, the induced voltage change of a current-biased sample is measured as a function of the beam position. The induced voltage at a constant bias current density is proportional to the resistivity change on the locally heated spot. For HTS such as YBa2Cu3O7 (YBCO), the temperature coefficient of resistivity is around 0.5–0.7 µ cm/K in the normal state and increases by several orders of magnitude near and below the critical temperature. Hence, induced voltage change will be more in low temperature compared to room temperature leading to a high spatial resolution (~1µm13,14), however resolution ~ 60µm is also observed in room temperature15. Microwave, which is an electromagnetic wave, can also be used for current mapping of HTS and other conducting films based on local heating. The larger penetration depth of microwave in conducting thin films compared to laser allows uniform heating through the thickness of the film in contrast to the surface heating by a visible laser. Hence, microwave heating will be more efficient in energy transfer especially in the case of thick films and coated conductors. This motivated us to explore microwave microscopy as an alternative approach for current mapping of HTS samples in room temperature. 1.2 EXISTING TECHNIQUES FOR CURRENT MAPPING: Identification and investigation of the current limiting defects is very important for the quality control of the current carrying capabilities of the coated 3 conductors. Crystalline structure plays a very important role in determining the critical current density (Jc) in HTS thin films and coated conductors. It was observed that randomly oriented polycrystalline HTS materials have Jc < 500A/cm2 whereas, oriented YBCO thin films grown epitaxially on single-crystal oxide substrates, such as SrTiO3 (001), exhibit Jc >1MA/cm2 at 77 K18. This huge difference between randomly oriented HTS polycrystalline and single crystal--like epitaxial films is directly related to the mis-orientation angles at the grain boundaries in polycrystalline materials. Values for Jc across a grain boundary decreases significantly as the mis-orientation angle increases and drops exponentially if the mis-orientation angle exceeds 20−50 19 . In order to achieve high Jc values (~105 to 106A/cm2 at 77 K), the crystallographic orientation of the HTS superconducting wire or tape must have a high degree of both in-plane and out-of-plane grain alignment over the conductor's entire length. X-ray diffraction (XRD) which is commonly used for determination of crystalline quality of thin films can be applied to determine the full-width-at-half-maxima (FWHM) of inplane and out-of-plane mis-orientations in this case2. However, XRD provides the structural information averaged over a segment of the conductors and does not give any information about the distribution of defects with respect to each other. Hence, to have the information about the distribution of the defects, there should be some technique to map the current flow in these films as these defects acts as obstacles to the current flow. 4 Broadly, the current mapping techniques in HTS films can be categorized under two categories: indirect techniques involving the mapping of the magnetic field and the direct techniques based on local heating of a current-biased sample while the voltage change is measured as a function of the beam position (‘hot spot’ technique). 1.2.1 Techniques based on magnetic field mapping: There are many different techniques for current mapping in HTS films which involve magnetic field mapping like magneto-optical imaging (MOI)2,3,4, Hall probe magnetometry5,6, superconducting quantum interference device (SQUID) microscopy7,8 and magnetic force microscopy (MFM)9,10,11. In all these techniques the spatial distribution of magnetic field or force due to magnetic field in superconductor is measured. The current distribution is then obtained from the magnetic field distribution by solving numerically the Biot–Savart law. The penetration of magnetic field in superconductors is different at different temperature range depending on whether it is above or below the critical temperature (Tc). When a superconductor material makes the transition from the normal to superconducting state, it actively excludes magnetic fields from its interior. This is called the Meissner effect. However, the HTS samples are generally Type II materials where a mixed state Meissner effect occurs. In this case, magnetic field is not excluded completely, but is constrained in filaments 5 within the material. These filaments are in the normal state, surrounded by super currents in what is called a vortex state. Hence magnetic field mapping at temperatures less than Tc will give information about the distribution of vortices. The magnetic field mapping can then be de-convoluted to obtain the distribution of the super current density. Hence, magnetic field mapping in superconductors is generally performed at low temperatures in order to create vortices. However, macroscopic defects like mechanical defects will oppose the super current flow in low temperatures also. Similar effect of the mechanical defects on normal current is also expected at room temperature. Hence, the current distribution obtained from the magnetic field mapping at both superconducting temperatures and room temperature should give similar information about the mechanical defects and their distribution. Magneto-optical imaging (MOI) provides a valuable tool for space and time resolved measurements of the magnetic flux density distribution of superconducting thin films. For technological applications such as coated conductors, the investigation of transport currents is of special interest. In coated conductors, mis-orientation at the grain boundaries is the main reason for current suppression. MOI can be used for directly visualizing the current flow in coated conductors2,3,4. It is based on magnetic flux penetration in superconductors. Highresolution MOI is used to map the normal component of the magnetic induction at the surface of the film. The resulting magnetic field profile is then converted into a quantitative map of the critical current density such that in-homogeneities or 6 defects can be located. This technique generally operates at low temperature. The sample has to be cooled down to superconducting temperature for the formation of magnetic vortices and then magnetic field is applied. Fig. 1.1 shows the MO image of YBCO coated conductors at 15K and at the applied magnetic field of 60mT 3. Fig: 1.1: MO image taken across the whole width of the YBCO coated conductor at 15K and applied magnetic field of 60 mT. Bright areas indicate magnetic flux penetration, dark areas flux shielding3. From this figure it is observed that the magnetic flux enters the sample from the edges and propagates preferentially along the grain boundaries and the mechanical defects. Hence the image of magnetic flux penetration will give 7 information about the location of the defects. The spatial resolution obtained is about ~ 1µm 4. For low-magnetic fields, MOI techniques can provide detailed information about the local magnetization of the sample. However, for high fields, the problem of imaging the local magnetic-flux distribution in the background of a large applied magnetic field requires a complementary measurement method with sufficient spatial resolution. Hall-probe magnetometry5,6 can be successfully utilized for this purpose. A spatial resolution 200 nm is obtained in this case. Scanning SQUID microscope (SSM) is a technique capable of imaging a magnetic field distribution in close proximity across the surface of a sample with high sensitivity and with high spatial resolution. It is based on a HTS thin-film SQUID sensor. The spatial resolution and the sensitivity of the microscope are determined by the active area and the sensitivity of the sensor. A SQUID sensor has the highest magnetic flux sensitivity and is able to detect fractions of a flux quantum. However, the active area of commonly used SQUID’s is larger than 10 µm2, limiting the lateral resolution of a SQUID microscope. This technique has been found to be very useful for the observation of current distribution in high TC superconductors like YBCO at low temperature7,8. This technique has higher sensitivity of observing current distribution in superconductors compared to other techniques like MOI and Hall probe as it can resolve single vortex current which cannot be resolved by the other techniques. The current distribution in the 8 superconductor is obtained by measuring the spatial distribution of magnetic field using a SSM. SSM has been effectively used to observe the trapped vortices in YBCO thin films in low temperature7. The films were patterned by photolithography and the ion milling technique into micro-bridges of width ~20–100 µm. The SSM signals were recorded as the two-dimensional distribution map of the z component of the magnetic field7. From these data, the local current distribution, including its magnitude and direction, was derived by solving the Biot–Savart law. The vortices were found to be randomly distributed and the super currents associated with these vortices were found to be circular. The transport super current was observed to flow mainly along the edges of the bridge as observed in the current density profile along the width of the bridge shown in Fig. 1.2. The spatial resolution observed ~10µm 7. Fig: 1.2: Current distribution of YBCO thin films at superconducting temperatures using SQUID microscopy7. 9 All the magnetic field mapping techniques described above have exhibited very high spatial resolution (few microns) but for most of their applications, they have to be operated at low temperatures because the samples must be cooled to superconducting state for the magnetic vortices to form. This makes them inconvenient for routine characterization of large area HTS films and long coated conductors. Hence, it is necessary to develop a current mapping technique which can operate at variable temperatures, preferably room temperature. Magnetic force microscopy (MFM) is one of those techniques, which can operate both at superconducting temperatures10,11 and at room temperature9. In MFM, the sample is probed with a magnetized tip. The tip senses the stray field of a sample. A flexible cantilever, often with an optical sensor of displacement (similar to an atomic force microscopy (AFM) device), is used to measure the magnetic force. The inverse problem is then solved numerically using Biot–Savart law to determine the current density within the sample9,10,11. This technique has been successfully applied in low temperatures for the direct imaging of vortices in YBCO thin films. The arrangement of eight vortices, generated while the YBCO thin film sample was field cooled in the temperature range of 7.7K-74.9K in an external field of 2mT showed a spatial resolution ~ 500nm11. Rous et al9 had successfully applied this method to determine the microscopic current density in a micron-scale, current-carrying Cr/Au line, and obtain images of the current components with spatial resolution ~ 200nm. Fig. 1.3 10 showed the inverted relative current density distribution obtained for a Cr/Au line containing a 45° slanted slit defect9. Fig: 1.3: Current density distribution determined by inversion of MFM phase image a) the total current density, b) the component of the current density perpendicular to the lines edges, and c) the component of the current density parallel to the line edges9. From this figure it is observed that the current crowded in the vicinity of the defect. In the regions far from the defect, the current is uniformly distributed across the line. Hence this technique can be effectively used to identify defects in conducting samples at any temperatures. 11 1.2.2 ‘Hot-Spot’ techniques: All the current mapping techniques using magnetic field mapping had very high spatial resolution however, the only drawback of these techniques are that the current distribution obtained is not through a direct measurement rather it is obtained through the numerical calculation from the distribution of the magnetic field. This problem is eliminated in the direct mapping technique using local heating which are commonly known as ‘hot-spot’ techniques. There are two major hot-spot scanning methods for the investigation of spatial distribution of current in high TC superconductors, namely scanning electron microscopy (SEM) and scanning laser microscopy (SLM). Both of these methods can measure the spatial distribution of current with a resolution of about ~1 µm on samples with characteristic dimensions of the order of millimeters. The principle of current mapping of the above techniques is based on mapping a sample voltage response as a function of the position of a focused electron or laser beam on its surface. The effect of the electron beam or the laser beam is to induce local heating effect resulting in an increase of the sample temperature at the beam position. This will lead to local resistivity change given by r = ( r T) T which in turn will lead to the change in sample voltage given by V = J b rl which can then be recorded as a function of the beam coordinates. Here Jb is the 12 bias current density, ( r T ) is the rate of change of resistivity with temperature and l is the length of the hot spot (spatial resolution). Fig. 1.4 shows the typical resistivity versus temperature (r-T) curve for YBCO film on SrTiO318. Fig: 1.4: A typical resistivity versus temperature (r-T) curve for YBCO thin film on SrTiO3 18. From this figure it is observed that r T increases by several orders of magnitude from the normal state near and below the superconducting transition temperature (TC). Hence, as the current map of HTS using either SEM or SLM is 13 based on local heating, it will be more effective and have maximum resolution near TC. The mechanism of hot-spot technique is very different from the magnetic field mapping technique. However, both can yield similar information about the distribution of mechanical defects. A mechanical defect like scratch or void will obstruct the current flow in a current biased sample and will appear more resistive compared to the other regions of the film. Hence the change in resistivity under local heating will be different in this case compared to other parts of the film. Hence, the voltage induced will be different leading to the identification of the location of the defects. Low temperature SEM (LTSEM) has been effectively used to show the influence of flux trapping in YBCO thin films12. The voltage induced due to local heating by the electron beam is proportional to the critical current density of the superconductor. The image of the critical current density distribution showed the distribution of trapped vortices. The image was obtained with a spatial resolution of about ~ 1µm. SLM can be effectively used in both superconducting temperature13,14 and in room temperature15 for current mapping of superconductors. Sivakov et al13 had used low temperature SLM (LTSLM) to measure the current distribution in BSCCO/Ag tapes. The tapes were cut into bridges which were typically 0.5 mm wide and 10–15 µm thick. A Helium-Neon laser is focused directly on the bridge. For temperatures T greater than TC, the major contribution to the induced voltage 14 response due to laser absorption is proportional to r T . For T < TC and the bias current Ib > Ic (critical current), there is an additional contribution to the LTSLM response, which can be related to photo-induced flux creep, presence of superconducting-normal (S-N) transition, etc. The ratio between the two components in the response is expected to be different for different sample regions, depending on the local mechanism of resistivity. The voltage response was found to be distributed in-homogeneously over the sample surface. This is due to the non uniform distribution of the super current. The map showed wellseparated spots with low critical current density. These spots had dimensions that are of the order of the grain size (several µm). Thus LTSLM imaging can be used to visualize the grain boundaries in HTS. Klein et al15 have used SLM in room temperature (RTSLM) to image transport current distributions in SmBa2Cu3O7 (Sm123) and YBCO thin films on LaAlO3 (LAO) substrate. The samples are photo-lithographically patterned into micro-bridges. A Helium-Neon laser is used as a light source. For the YBCO micro-bridge, a void of size 60µm×60µm is created at the centre of the microbridge17. The RTSLM image of the YBCO micro-bridge with the void in the centre is shown in Fig. 1.5. From this image it is observed that there is larger induced voltage response at the edges of the bridge and around the void. This indicates that the temperature rise due to local heating is larger near the edges of the bridge and around the void. The void could be easily observed in the image clearly indicating a spatial resolution of less than 60µm. 15 Fig: 1.5: RTSLM image of the YBCO micro-bridge with a void of size 60µm×60µm at the centre15. Thus RTSLM could be effectively used to map current distributions in HTS thin films. This current mapping can then be effectively used to detect the defects that create disturbances to the current flow in room temperature. The only drawback for this system is the small penetration depth of laser in superconductors. Penetration depth of the laser is about ~10nm in YBCO. Hence for the films of larger thickness, as the laser light does not penetrate through the whole thickness, long pulse width for the input radiation is used in order to allow the heat to diffuse through the whole film thickness. 16 1.2.3 Other techniques: There can be some other techniques for current mapping in HTS films which neither involve magnetic field mapping or local heating. Terahertz (THz) mapping16,17 is one such technique which operate at superconducting temperatures. The principle used here is that the THz beam can be emitted from high-TC superconductors due to optical super current modulation16, 17. A modelocked Ti:sapphire laser operating at a repetition rate of 82 MHz and producing 50 femto-second pulses is focused by an objective lens onto a YBCO thin film mounted on a cold finger of a closed-cycle helium cryostat16. The temperature of the system is lowered to the superconducting temperatures. The photo-excitation is absorbed and breaks the Cooper pairs and produces high energy quasi-particles and thereby producing high energy electromagnetic pulses. The decrease in the number of the Cooper pair results in the decrease in the super-current. The amplitude of THz radiation emitted from the opposite side is proportional to the local current density of the area of the film that is optically excited and hence can be used to map the current density in the superconductor16. For the YBCO thin film patterned into a structure consisting of a number of voids, it is observed that the super-current flows in such a way so as to avoid the voids. Hence from the current map, the voids or the defects can be easily located. The spatial resolution is limited by the laser spot size and is found to be ~ 25µm in this case16. 17 Table 1.1 summarizes the spatial resolution obtained in all the different mapping techniques discussed above. TABLE: 1.1: Comparison of spatial resolution in different mapping techniques. SPATIAL No. TECHNIQUE MATERIAL TEMP (K) RESOLUTION Magneto-optical YBCO 15 K ~1 µm YBCO 77 K 0.2 µm YBCO 77 K ~ 10 µm YBCO 7.7-74.9K ~ 0.5µm Cr/Au RT ~ 0.2µm YBCO 16.5 K ~ 25 µm YBCO 35 K ~ 1 µm Scanning Laser BSCCO/Ag 102 K ~ 10 µm Microscopy YBCO RT < 60µm 1. Imaging (MOI) Hall probe 2. Magnetometry SQUID 3. Microscopy Magnetic force 4. Microscopy (MFM) Terahertz (THz) 5. Mapping Scanning Electron 6. Microscopy 7. 18 1.3 MICROWAVE MAPPING: The microwave radiation is an electromagnetic radiation operating in the frequency range 300MHz-30GHz20. It has wide scale application in various industrial and medical fields like transmission and reception of information for communication purposes, processing of food, pasteurization of vegetables, drying of paper or textiles, thermal treatment of pharmaceutical products, vulcanization of rubber, etc. Microwave can also be used for rapid thermal processing (RTP) like dopant activation, silicide formation and post metal annealing for the fabrication of devices21. In majority of RTP techniques, optical heating of substrate is used. In most cases, it is difficult to maintain the uniformity of substrate temperature. This problem can be solved to some extent by using microwave heating because for microwave, there is volumetric heating of wafer instead of surface absorption and thermal diffusion. Most of the heating applications, such as microwave ovens etc, involves large samples and operate with a microwave power of hundreds of watts. However in the case of conducting thin films, the local microwave properties can be studied using the near-field techniques. A near field scanning microwave microprobe (NSMM) can be used to direct the microwave to a local spot on the sample and various properties can be measured as a function of the beam coordinates. Standard optical microscopy is not capable of obtaining a transverse resolution better than approximately half a wavelength of light ( /2) due to the 19 diffraction limit, also termed the Rayleigh or Abbe limit. In the case of near-field microscopy, as both the tip-to-sample separation and the tip aperture are a small fraction of the wavelength (i.e, << ), the spatial resolution is given approximately by the tip diameter. Resolutions significantly better than the best far-field microscope can be obtained. The local microwave properties of superconducting thin films are probed by using the microwave near-field scanning technique22,23,24 at variable temperatures. The NSMM consist of a high quality factor Q (Q ~ 1000), quarter wave coaxial resonator with an STM-like niobium or tungsten tip fixed to the centre conductor. The near-field interaction between the tip and the scanned surface shifts the resonant frequency (f0) and changes the quality factor (Q) of the resonator. The shift in f0 strongly relates to tip-sample distance and the variation of the dielectric property of the sample, while the Q of the resonator changes very sensitively with the variation of the local surface resistance of the sample under the probe. Therefore, by monitoring the variation of f0 and Q as the tip scans over the surface of a sample, one can measure the local microwave properties of the sample22,23,24. In the experiment, the thin film sample is mounted on the lowtemperature sample stage and the dc resistivity of the film is measured by standard four-probe transport measurement through evaporated silver electrodes on the sample. The surface resistance of YBCO is compared with silver22,23. The surface resistance can be calculated from Q value and the dc resistance obtained from electrical transport measurement. At room temperature, the surface 20 resistance of the YBCO thin film is found to be much higher than that of the silver film, while at 81 K, the surface resistance of the YBCO thin film is slightly less than that of the silver film. Fig. 1.6 shows the two-dimensional image of Q values for YBCO and silver thin film at 81K22. Q Fig: 1.6: Two-dimensional image of Q values for YBCO and silver thin film at 81K22. In this image some small regions with higher surface resistance in YBCO are observed which may be due to the in-homogeneity of the thin film or the existence of some defects22,23. Takeuchi et al24 had used microwave Q scan for two-dimensional mapping of patterned YBCO thin films. Films were patterned into arrays of 100µm×100µm square patches using photolithography. The twodimensional Q map was able to resolve clearly the square patches however, the 21 resolution is much better when the temperature is lowered to the superconducting temperature of 80K. This shows that the mapping of microwave properties of superconductors can lead to the identification of current limiting defects in the superconductor. NSMM can also be used for localized heating of the sample similar to heating by laser or electron beam. Low power near field microwave has been used for localized heating of soft matter like egg-white, albumin, plant leafs, and raw meat 25. The sample is mounted on an XYZ stage and is brought to the distance of 10–200µm from the probe in order to operate in the near-field regime. The probe is based on a cylindrical metal-coated dielectric resonator with a narrow slot in the apex. The temperature distribution upon localized microwave heating is highly inhomogeneous, the maximum temperature being achieved in the area just beneath the probe. By scanning the sample with regulated power and the irradiation controlled microwave heating of the sample was obtained. Microwave localized heating using NSSM can also be used like laser to map the current density in conducting thin films. The advantage of microwave over laser is that microwave penetration depth is much larger than that of laser. For YBCO, microwave has a penetration depth of ~ 20µm whereas for laser it is only ~10nm (or 0.01µm). Hence microwave heating has greater efficiency than laser because microwave is uniformly absorbed throughout the thickness of the film whereas laser is absorbed in the uppermost layer of the sample. Localized microwave heating has been used to map current distributions in silver thin films 22 in room temperature26. A silver micro-bridge ~ 200 µm wide and ~ 0.1 µm-thick on a glass substrate was patterned into an array of voids with dimension of 20µm×20 µm. Fig. 1.7a shows the optical microscope image of the bridge with an array of voids. The region enclosed by the dotted line was imaged with the NSMM and Fig. 1.7b shows the resonant frequency f0 scan. Fig. 1.7c and Fig 1.7d shows the comparison between the f0 and induced voltage images ( V) taken on a smaller scan area of 100×40 µm2 26. o p tic a l m ic r o s c o p e im a g e Ib (b) Ib (c) m id d le v o id (d) (a) 10 um Fig: 1.7: Silver micro-bridge with an array of voids showing a) the optical microscope image, b) 3-D and c) 2-D image of the resonant frequency scan and d) induced voltage scan26. 23 From this figure it is observed that the resonant frequency scan was able to resolve clearly the voids, however the boundary of the voids shown in the V image is not very clear even though the voids could be easily observed. It is also observed that the V response around the void is higher, similar to that observed in RTSLM. This suggests that near field microwave heating can be effectively used for the identification of defects in conducting thin films in room temperature. Hence it is very important to understand the mechanism of microwave heating and extend this technique for the mapping of the current density in high TC superconductors. This technique if successfully used may provide a convenient and practical alternative for quality control of commercial HTS films and coated conductors. 24 CHAPTER: 2 REVIEW OF EXPERIMENTAL WORK The near field scanning microwave microprobe (NSMM) has been integrated with the standard current voltage (I-V) characterization for mapping of electrical current density of conducting thin films in room temperature. The focused microwave emitted from the tip of the NSMM has been used to change the local resistance of the film which results in voltage response depending on the local current density and the input microwave power. The map of voltage response acquired during scanning over the whole film can then give important information about the surface morphology and the non uniformity of the electromagnetic properties of the film. 2.1 EXPERIMENTAL DETAILS: The experimental setup and the microwave probe are given in Fig. 2.1. The experimental setup shows the sample on a motorized XYZ stage and the microwave micro-probe. The microwave probe is also shown separately. The details of the experimental setup and the probe are given elsewhere26,27. 25 Microwave Probe Fig: 2.1: The experimental setup and the microwave microprobe. A silver thin film of thickness of ~ 0.1µm is patterned into a micro-bridge by the photolithography technique. The bridge has a dimension of 0.5cm×0.05cm. The schematic of the experimental setup is given in Fig. 2.2. Fig: 2.2: Schematic of the experimental setup. 26 The patterned silver bridge is mounted on the XYZ motorized stage and biased with pulse DC current at 50Hz. It is connected by a network of resistors to form the Wheatstone bridge. NSMM which is used to direct microwave at a local spot on the film and collect response from it is mounted on top of the silver micro bridge. NSMM has a tip of diameter ~ 20µm. Microwave frequency used is 2GHz. The microwave is applied to the NSMM as a pulse of pulse width 0.7 sec. This is the shortest pulse width achievable in this set up. Short pulse width is used in order to minimize the heat diffusion due to microwave absorption. The microwave induced voltage response ( V) is measured via the Wheatstone bridge using lock-in technique. V is the difference between the peak voltage at the end of the microwave pulse and the sample voltage without microwave. The voltage response is collected as the sample is moved along the XYZ stage. 2.2 EXPERIMENTAL RESULTS: Fig. 2.3 gives the V line scan across the width of the micro-bridge for different bias currents in the range 5mA-45mA. Microwave input power is 1W. From this figure it is observed that as the bias current Ib is raised, the whole V curve shifts upwards. This is due to the linear dependence of Ohm’s law. It is also observed that the V on Ib from V is higher at the edge of the film compared to its centre and is fairly uniform between the two edges. The deviation 27 of the V curves from the general trend for higher Ib values (45mA) is due to the additional sample heating by the bias current Ib. 1.6 5mA 15mA 25mA 35mA 45mA 1.4 1.2 V(µV) 1.0 0.8 0.6 0.4 0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Width(mm) Fig 2.3: V line scans across the width of the micro-bridge for different bias currents. Microwave input power is 1W. 2.3 INTERPRETATION OF THE EXPERIMENTAL RESULTS: For a uniform film without any defects, it can be assumed that there is no spatial variation in resistivity. When microwave is absorbed by the film, there will be local heating leading to local resistivity change. In the absence of spatial variation of resistivity, the change in resistivity due to microwave absorption 28 should be uniform everywhere on the bridge. As the microwave induced voltage response ( V) is linearly proportional to the change in resistivity from Ohm’s law, it also should also be uniform everywhere. However, experimentally observed higher V at the edge of the film compared to its centre suggests that the change in resistivity at the edge due to microwave absorption is more than that of the centre. This suggests that the temperature rise due to absorption of microwave is more at the edge than at the centre of the film. 2.4 THE MOTIVATION BEHIND SIMULATION: The atoms at the centre of the film are surrounded by similar atoms on all sides, whereas the atoms at the edge of the film are surrounded by similar atoms only on one side and the other side is exposed to air. Hence, when the microwave probe is at the centre of the film, there is uniform heat diffusion due to thermal conduction through the neighboring atoms, but when the probe is at the edge, there is heat loss to the air by convection on one side and diffusion through the film on the other side. This will lead to different temperature rise near the edge of the film compared to the centre of the film. For a fixed microwave power, the amount of temperature rise will depend on various sample and environmental factors like the thermal conductivity and specific heat of the material of the film, thermal conductivity of the substrate, heat transfer coefficient of air etc. Hence in order to understand the absorption of microwave radiation in thin films, it is very 29 important to develop a theoretical model to determine how the different parameters mentioned above affects the temperature rise. This model can also help to explain the microwave mapping for identification of electrical current limiting defects in thin films. Also, quantitative current mapping (J-mapping) in conducting thin films remains challenging. Hence, before it is reached, a full calibration involving both experimental and theoretical modeling is necessary. Thus the combination of both experimental and theoretical modeling will help to establish microwave mapping as a simple and non-destructive technique for identification of defects in conducting thin films in room temperature. 30 CHAPTER: 3 THEORY OF MICROWAVE ABSORPTION Microwave heating finds wide scale application in various industrial and medical fields. Most of the heating applications, such as microwave ovens etc, involves large samples and operate with a microwave power of hundreds of watts. However, using near-field techniques, more localized heating can be achieved in the case of conducting thin films. A near field scanning microwave microprobe (NSMM) can be used to direct the microwave to a local spot on the sample. The microwave will be absorbed in that small spot of size approximately same as the size of the probe tip. The heat will spread into the adjacent regions in the thin film sample through thermal diffusion. The temperature rise due to local heating will lead to the change in the local resistivity of the film which in turn will results in voltage response depending on the local current density and the input microwave power. The temperature rise at the local spot will depend on various sample and environmental parameters like the thermal conductivity and specific heat of the material of the film, thermal conductivity of the substrate, heat transfer coefficient of air etc. Hence in order to understand the absorption of microwave radiation in thin films, it is very important to develop a theoretical model to determine how the different parameters affect the temperature rise. 31 3.1 MICROWAVE ABSORPTION IN THIN FILMS: The schematic of the model is given in Fig. 3.1. The thin film sample has the lateral dimension of 2 x l (length) and y l (width) and of thickness z l . It is grown on a glass plate which acts as a substrate. The microwave beam falls on a pre-specified location (at the edge or at the center as indicated by the cylindrical shape in the figure). As shown in the figure, Y-axis runs through the centre of the film-substrate system. The film is initially at a temperature Tair, which is also the temperature of the surroundings. Fig. 3.1: The schematic of the model to study heat diffusion. 32 Before solving the unsteady state heat flow equation, it is very important to determine the amount of microwave power absorbed by the system. The microwave radiation is in the frequency range 300MHz-30GHz 20. The advantage of using microwave radiation compared to other optical radiation is that it will have much larger penetration depth. The penetration depth ( ) of an electromagnetic wave in a conducting film depends on the electrical conductivity ( ) of the film and the frequency of the radiation ( ) and is given as 20: = 2 --------- (3.1) 2µ 0 For silver which is used in the experiment, ~ 61.0 x106 -1 m-1 20 . For microwave frequency of 2GHz as used in the experiment, the penetration depth of microwave in silver is found to be ~1.41 µm. The films have thickness ~ 0.1µm which is much less than the penetration depth of microwave in silver. When a microwave is incident on a thin film, the energy can be reflected, transmitted or absorbed depending on the material properties as shown in Fig. 3.2. Fig. 3.2: Thin film model to study the absorption of electromagnetic wave. 33 The fraction of microwave power that is absorbed (A) can be calculated as: A=1 R2 T2 ----------- (3.2) Here R is the reflection coefficient and T is the transmission coefficient. For a microwave radiation of frequency 2GHz, wavelength ( ) is about ~ 0.15m. This is much larger than penetration depth and thickness z l of the film. Hence thin film approximation ( z l < << ) can be applied to this system. Under this approximation, the fraction of incident power absorbed by the film is given as28: A= Here, 2 --------------- (3.3) (1 + ) 2 z 2 = l and s = s 2 = 2 0 . Thus, the amount of microwave µ0 power absorbed by thin film depends on its penetration depth which in turn depends on the sample properties. A has maximum value of 0.5, occurring when z l = s as observed from Equation 3.3. Hence, a maximum of 50% of the incident power can be absorbed by the film. Fig. 3.3 shows the plot Equation 3.3 as shown by Bosman et al28. From this figure, it can be observed that the amount of power absorbed by the film falls drastically when the ratio of z l and s ( ) is much away from 1 and insignificant power is absorbed when the ratio tends to 104 or 10-4. In the case of silver with ~ 61.0×106 -1 m-1 and thickness ~ 0.1µm, is found to be ~1193.3 (~103). The corresponding value of A is found to be ~ 34 1.67×10-3. Hence, a very small fraction of microwave power is absorbed by the system. Fig. 3.3: The fraction of incident power absorbed by the thin film as a function of its thickness28. 3.2 HEAT DIFFUSION: When the microwave energy is absorbed in the thin film, the local temperature of the film will change with time. This change can be evaluated by solving the unsteady state heat flow equation. The energy that is absorbed in the film is responsible for the rise in temperature. Once the heat energy is absorbed, it could spread into the adjacent regions near the point of incidence by thermal 35 conduction. The heat energy could also flow out of the film through convection and radiation into surrounding environment and by thermal conduction into the substrate. Hence all these factors have to be taken into consideration for developing the mathematical model for heat flow. 3.2.1 Heat diffusion equation: Unsteady state heat flow equations were solved for the model given in Fig.1. Assuming the solid film to be isotropic with thermal conductivity K, specific heat Cp, and the density , application of the first law of thermodynamics to this system (a control volume as shown in Fig. 3.4) yields: {Rate of input of heat} – {Rate of output of heat} = {Rate of accumulation} D G zl A H C F x dy B E y dx Fig. 3.4: The volume element to study heat diffusion. 36 The heat flow equation is assumed two dimensional. In Fig. 3.4, the rate at which heat flows in the direction of x across a plane of area (dy)(zl) normal to the axis x and passing through the mid point of the volume element is: q xx = K T dy z l x ------------ (3.4) The rate of heat input across plane area ABCD is then: Q in xx = q xx q xx dx = x 2 K T T dx K dy z l + x x x 2 ----- (3.5) Similarly, the rate of heat output across plane area EFGH will be: Q out xx = q xx + q xx dx = x 2 K T x x T dx dy z l x 2 K ----- (3.6) Hence, from Equations 3.5 and 3.6, the net heat flow in to volume element in the x direction will be: in Q net xx = Q xx Q out xx = q xx dx = x x K T dx dy z l ------ (3.7) x Similar expression can be obtained for q yy in y-direction. The difference between the rate of heat input and the rate of heat output is: Q net = x K T T K + x y y dx dy z l ------ (3.8) The rate of heat accumulation is given as: ! C p dx dy z l T t 37 ------ (3.9) Equating the Equations 3.8 and 3.9, the following equation of heat diffusion is obtained: !C p T = K t 2 T + x2 2 T y2 ------- (3.10) 3.2.2 Mechanisms of heat loss: There are different mechanisms of heat loss. Heat can be lost through convective heat transfer from the surface of the solid to adjoining air. There can be heat loss by radiation from the surface of the film. There can also be some heat loss by conduction through the bottom glass plate. i) Convective Loss: There can be heat transfer from the surface of the film of temperature T to the surrounding temperature Tair by convection. The heat transfer rate is written as: Qc = h(T Tair )dxdy -------------- (3.11) Here h is the convective heat transfer coefficient of air. The value of h is dependent on the geometry of the sample, type of media, gas or liquid, temperature and the flow properties such as velocity, viscosity etc. Generally for air, it ranges from ~ 5-40 W/m2K 29 depending on the velocity of the flowing air. 38 ii) Radiative Loss: Heat can be lost from the surface of the film through radiation obeying the Stefan-Boltzman’ law: Qr = Here, s (T 4 4 Tair )dxdy is the surface emissivity of the film and --------------- (3.12) s is the Stefan-Boltzman constant. Radiative heat loss plays a very important role especially in the high temperature range in determining the temperature profile. It has been shown by Zhang et al30 that for microwave heating of a silicon wafer, the radiative heat loss is the most important factor counter-balancing the heat coming into and out of the wafer resulting in a stable temperature profile. iii) Loss through the substrate: The silver film is formed on the glass substrate whose thermal conductivity (~1W/mK 31 ) is two orders of magnitude less than the conductivity of the film (~ 417.1 W/mK 29,31). Hence, the cross diffusion of heat from one edge of the film to other through substrate will be negligible. However, there can be some loss of heat through the substrate. The substrate has a heat capacity which will allow it to hold some heat. As the thermal conductivity of glass substrate is ~ 102 times less than that of silver film and the thickness of the substrate (~1.0mm) is ~105 times more than that of the film, the fall in substrate temperature will be less compared to the fall in film temperature when the microwave power source is 39 switched off. Hence, when the temperature of the film reduces to a value close to Tair, the substrate temperature will be higher leading to heat flow into the film from substrate through the film-substrate interface. This will result in tailing in the time dependent response. This tailing becomes more pronounced only when the temperature of the film drops significantly and is close to Tair. All experimental measurements were done after 0.7 sec, just before the microwave is turned off. By that time, the temperature profile is substantially established or it is in the steady state. At this stage, there can be steady state heat loss due to conduction through the substrate. If the thermal conductivity of the substrate is Ks and thickness is z s , then the conductive heat loss through the substrate will be: Q sub = K s (T Tair ) dxdy zs ----------- (3.13) Thus it can be concluded that the heat capacity of the substrate will mostly affect the tailing of the time dependence of V and will not affect the voltage profile when the temperature profile is substantially established. However, at steady state, there can be some amount heat loss due to conduction through the substrate. This loss can be significant if the thickness of the film is very small compared to the penetration depth. 40 3.2.3 Solution of heat diffusion equation: Taking into account the different forms of heat losses, listed above, the governing heat diffusion equation can be written by combining Equations 3.103.12: !C p T =K t 2T x2 + 2T y2 h {T zl Tair } zl { s T4 4 Tair } ------ (3.14) The microwave heat is introduced in one point in the film, depending on the location of the microwave source. The right hand side of the above equation is discretized over the grid points with heat source added in the equation for one of the grid points. With the heat source added, the above equation is written as: !C p T =K t 2 T x2 + 2 h {T zl T y2 Tair } s zl {T 4 } 4 Tair + Pnet - (3.15) 2z l x y Here, Pnet = A . power where A is the fraction of microwave power absorbed as given by Equation 3.3. x and y represent the grid size. The Equation 3.15 does not take into account the conductive heat loss through the substrate. Taking the heat loss through substrate into account, the Equation 3.15 will be modified as: !C p 2 2 T T T =K + 2 t x y2 1 zl h+ f Ks {T Tair } zs s zl {T 4 } 4 Tair + Pnet - (3.16) 2zl x y Here f is the fraction of heat loss through the substrate. The thin film sample is divided into two symmetric halves for the purpose of solving heat diffusion equation. The axis of symmetry runs through the center of the slab along 41 y-axis as shown in Fig. 3.1. The heat equations are solved for the right half of the film assuming the condition of symmetry at the center. The mirror image of the temperature profile for the other half of the film is extracted from the solution. The initial condition is given as: At t = 0 $ T = Tair . At the boundaries x = x l , y = 0, y = y l there is heat transfer to air. The boundary conditions are given as: T =0 x T x = xl $ K = h(T x x =0$ For x, ----------- (3.17) Tair ) T = h(T Tair ) y T = h(T Tair ) y = yl $ K y y=0$ K For y, ------------ (3.18) The Equation 3.16 is first discretized in space using central finite difference formulation. This resulted in coupled ordinary differential equations (ODE), one for each grid point. Following dimensionless variables are also introduced: %= T , Tair xd = x , xl yd = y yl For a point which is ‘i’ grids away from the y-axis and ‘j’ grids away from the x-axis, the Equation 3.16 becomes: 42 d% i , j dt = K &,% i +1, j 2% i , j + % i + !C p &* xl2 ( xd )2 f K s (% i , j 1) h+ zs !C p zl 1, j + 3 4 sTair (% i , j % i , j+1 2% i , j + % i , j 1 )& yl2 ( yd )2 1) !C p zl ( &' Pnet + 2!C p zl xl yl Tair ( xd )( yd ) - (3.19) These coupled ordinary differential equations (ODEs) are solved using LSODE package available from Lawrence Livermore National Laboratory (free download-able version). The solver is based on GEAR algorithm (a sophisticated predictor-corrector scheme, ideal for solving stiff ODEs). The package consists of a FORTRAN program, where the equation is introduced in one of the subroutines. The main program contains the initial condition and the time at which the output is sought. The program decides the size of the time step depending on the stiffness of the problem. 3.3 SOLUTION OF CONTINUITY IN CURRENT FLUX: Once a temperature profile is obtained by solving the heat diffusion equation for the entire film, the resistivity (r) at each location is computed based on the following relationship: r = r0 + - (T Tair ) --------- (3.20) Here r0 is the coefficient of resistivity and - is the temperature coefficient of resistivity. Hence, the resistivity at any spatial point r ( x , y ) can be calculated 43 using the Equation 3.20. This is then used to determine the voltage distribution. This is because the induced voltage due to microwave absorption ( V) at a constant bias current density (Jb) is related to the resistivity change ( r) on the hot spot and is given by V = J b rl ----------- (3.21) Here l is the length of the hot spot (spatial resolution). r due to temperature change can be estimated from Equation 3.20 and is given as r= r T T ----------- (3.22) Thus, high temperature coefficient of resistivity ( r = - ) and high T heating efficiency ( T) are both necessary to achieve high spatial resolution. The schematic of the model to determine the voltage distribution is given in Fig. 3.5. Fig. 3.5: The schematic of the model to determine the voltage distribution in the thin film. 44 Voltage distribution in the thin film is obtained by solving the continuity equation of the current flux32: . .J = 0 ----------- (3.23) Here J is the current flux given by: ) ) J = Jxx + Jyy Or, J= V ( x, y) ) V ( x, y) ) x+ y r ( x, y ) x r ( x, y ) y --------- (3.24) The equation is solved for the same control volume given in Fig. 3.4. Within this volume assuming r ( x , y ) is constant, Equation 3.23 can be written as: J= V ( x, y) ) V ( x, y ) ) x+ y x r( x, y) y r( x, y) ---------- (3.25) Substituting Equation 3.25 into Equation 3.23, the following equation is obtained, which is used to calculate the voltage distribution: 2 2 V ( x, y) V ( x, y) =0 + x 2 r( x, y) y 2 r( x, y) ------- (3.26) The bias current Ib flows into and out of the film in the x-direction as shown in the Fig. 3.5. There is no flow of current out of the film in the y direction. The boundary conditions are given as: y = 0$ For all x, V =0 y V y = yl $ =0 y 45 --------- (3.27) V r ( 0, y ) = I b x y z l l r (2 x , y ) V l x = 2x $ I = l b x y z l l x = 0$V = 0 & For all y, --------- (3.28) Equation 3.26 was discretized in the central difference form along the same grids that was earlier used to solve the heat diffusion equation. Gauss-Siedel iterative technique was used to solve the voltage distribution. The Gauss-Siedel technique provides estimate the ratio of voltage to resistivity (V/r) at a grid point as a function of the average of V/r values of the four adjacent points as shown in Equation 3.29. Vi , j ri , j = 1 ,&Vi + 1, j Vi 1, j Vi , j + 1 Vi , j 1 )& + + + + ( 4 &* ri + 1, j ri 1, j ri , j + 1 ri , j 1 &' ---------- (3.29) The Gauss-Siedel sweep begins from the right edge and move towards left edge as shown in Fig. 3.5. Iterations in sweep from right to left were continued until a steady voltage profile is reached. Since the voltage at the right edge of the film is zero, the voltage drop across the whole length of the film is the voltage at the left edge of the film (averaged over all y even though the variation is minimal). The voltage drop before the microwave was switched on has to be subtracted from the total voltage rise to obtain the voltage rise induced due to microwave. 46 The entire analysis, presented above, is with power being added only in one grid point. For a complete one-dimensional or two-dimensional scan, a sequence of computations was performed with source term added to all the grids, one at a time. 47 CHAPTER: 4 RESULTS AND DISCUSSIONS The microwave induced voltage response for a thin film of silver of dimension 0.5cm×0.5cm and thickness of ~ 0.1µm was simulated. Microwave heating causes the temperature rise to varying extent in different portions of the film. This change in temperature profile of the film causes the local resisivity to increase which in turn results in voltage drop across the film. The map of voltage response acquired while the microwave probe scans the film surface can provide information about the surface morphology and the non-uniformity of the electromagnetic properties of the film. This is known as the ‘hot-spot’ technique. In this chapter, first the temperature profile due to microwave absorption on silver thin film will be presented. The temperature profile will be related to the voltage response profile which will be presented for the one dimensional scan across the width of the film and two dimensional scan for the area of the film. The variation of the input voltage with time, input microwave power and thickness of the film will also be presented. Finally, the voltage profile obtained through the simulation will be compared with the experimental results of the silver micro-bridge. 48 4.1 TEMPERATURE PROFILE: The temperature profile due to microwave heating of silver film was calculated by solving the differential equation (Equation 3.15) given in Chapter 3. The parameters used for silver are given bellow: ! (Density) =10.5x103Kg/m3 31 C p (Specific Heat) =234.0 J/KgK 31 29,31 K (Thermal Conductivity) =417.1 W/mK r0 (Coefficient of Resistivity at 200C) = 1.59x10-8 m 20,33 h (Heat Transfer Coefficient of air)=5.6 W/m2K (Air is assumed to be stationary, v air = 0 in this case) 29 (Surface Emissivity) =0.03 34 -12 s (Stefan-Boltzman Constant)=5.6x10 Js-1cm-2K-4 31 Tair = 293K (200C) The microwave probe has a tip diameter of ~ 200µm. The grid size is taken to same as the tip diameter. The microwave pulse has duration of 0.7 sec and the measurement is made just after the pulse ends. For the incident power of 1W, the temperature profile when the microwave is incident at the centre and edge is given in Fig. 4.1. 49 b) a) Fig: 4.1: The temperature profile when the microwave is incident at the a) edge and b) centre of the silver film. Input microwave power is 1W and the tip diameter is ~ 200µm. Thickness of the film ~ 0.1µm From the above figure, it is observed that the temperature rises by ~ 400C when the probe is at the edge and by ~ 250C when the probe is in the centre. When the probe is at the centre, the groups of atoms that are heated are surrounded by similar atoms on all sides. Hence, there is uniform heat diffusion due to conduction through the neighboring atoms, but when the probe is at the edge, the atoms that are heated are surrounded by similar atoms only on one side and the other side is exposed to air. Hence, there is heat loss to the air by convection on one side whereas there is diffusion through the film on the other 50 side. The heat loss to air is substantially less than the heat diffusion through similar atoms. This will lead to higher temperature near the edge of the film compared to the centre of the film. Thus, the thermal conductivity of the film and the heat transfer coefficient of air play an important role to determine the temperature profile due to microwave absorption. 4.2 VOLTAGE INDUCED DUE TO MICROWAVE: The temperature profile due to microwave absorption can lead to the local resistivity change as given in Equation 3.20. The value of temperature coefficient of resistivity (-) for silver, is 3.8×10-11 m/K 33. The microwave induced voltage response ( V) is the difference between the peak voltage at the end of the microwave pulse and the sample voltage without microwave. This voltage response is linearly related to the local resistivity change as given by Ohm’s Law. The V response for different microwave incident position is calculated by solving the Equation 3.26 of Chapter 3. 4.2.1 Time dependence of V: The microwave pulse has duration of 0.7 sec and the measurement of V is made just when the pulse ends. Hence it is very important to study the variation 51 of V with time. Fig. 4.2a shows the time dependence of V at the edge and at the centre of the film for a bias current (Ib) of 25mA. The microwave pulse is also shown. The microwave probe tip diameter is ~ 200µm. From this figure it is observed that the V response at the edge of the film is larger than that of the centre. This is because the temperature change due to microwave absorption is larger at the edge as shown in Fig. 4.1. Pin(W) Pin(W) 1 a) 0 120 b) 0 80 60 5mA 15mA 25mA 35mA 45mA 200 V(µV) Ib=25mA Edge Centre 100 V(µV) 1 150 100 40 50 20 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time(sec) Time(sec) Fig: 4.2: Time dependence of V response a) at the edge ( ) and at the centre ( ) of the silver film and b) as a function of bias current (5mA-45mA) at the edge of the film as obtained from simulation. Tip diameter ~200µm. Thickness of the film ~ 0.1µm. 52 Fig. 4.2b shows the time dependence of V for different bias currents in the range 5mA-45mAwhen the probe is placed at one edge of the film. figure it is observed that From this V rises sharply during the initial 0.15 sec and then slows down until it reaches the maximum value. The maximum value is reached in 0.35sec. The initial sharp increase of V is attributed to the local excitation of charge carriers by electric field giving them additional kinetic energy. The growth of V then decreases because of the collision between the excited carriers and lattice which resulted in heat diffusion. Once the microwave pulse is switched off, V decreases rapidly. The relaxation time of V is observed to be ~ 0.3 sec. Fig. 4.3 shows the experimentally obtained time dependence of V for a silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~ 0.1µm 26. Fig: 4.3: Experimentally measured time dependence of V response of the silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~ 0.1µm at bias current 5mA for different width of the microwave input pulse26. 53 The time constant for both rise and decay of V is much larger in the case of the experimental curve compared to the simulated curve. This is because the heat content of the substrate is not taken into account in the simulation. The glass substrate has a heat capacity depending on its specific heat which will allow it to hold some heat. As the thermal conductivity of glass substrate (~1.0W/mK 1) is approximately two orders of magnitude less than that of silver film and the thickness of the substrate (~1.0mm) is ~105 times more than that of the film, the fall in substrate temperature will be less compared to the fall in film temperature when the microwave power source is switched off. Hence, when the temperature of the film reduces to a value close to Tair, the substrate temperature will be higher leading to heat flow into the film from substrate through the film-substrate interface. This will result in tailing in the time dependent response. This tailing becomes more pronounced only when the temperature of the film drops significantly and is close to Tair. 4.2.2 Input microwave power dependence of V: The microwave induced voltage response V will depend on the input microwave power. This is because the temperature rise depends on the amount of power absorbed and the local resistivity change depends on the temperature change. Fig. 4.4 shows the experimentally observed V as a function of input microwave power for different bias currents in the range 15mA - 45mAfor the 54 silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~0.1µm 26 . The microwave probe tip of diameter ~20µm is placed at one edge of the film. Fig: 4.4: Experimentally obtained input microwave power dependence of V response at the edge of the silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~0.1µm for different bias currents. Tip diameter is ~20µm26. From this figure, it is observed that the slope of V increases with the increase in bias current. This is expected from Ohm’s Law. V is observed to have linear dependence on the input microwave power (Pin). This suggests that the temperature increase of the local sample spot due to microwave absorption is linearly proportional to the input microwave energy. To compare with the experimental results, V response as a function of input microwave power for different bias currents for the silver film of dimension 55 0.5cm×0.5cm and thickness ~ 0.1 µm is simulated and is shown in Fig. 4.5. The microwave probe tip of diameter ~200µm is placed at one edge of the film. From this figure it is observed that V does not have exactly linear dependence on Pin and shows a very small non-linearity. 450 2000 Ib=25mA 1800 1600 V(µV) 1400 0 1200 0.0 1000 0.5 1.0 1.5 5mA 15mA 25mA 35mA 45mA 800 600 2.0 2.5 3.0 3.5 Pin(W) 400 200 0 0 2 4 6 8 10 Pin (W) Fig: 4.5 Input microwave power dependence of V response at the edge of the silver film of dimensions 0.5cm×0.5cm and thickness ~ 0.1 µm for the bias current in the range 5mA-45mAobtained from simulation. Tip diameter is ~200µm. The inset shows the V response for Ib = 25mA 56 For a given bias current Ib, V is proportional to R (Ohm’s law). R is proportional to the local resistivity change. The local resistivity change is a linear function of the temperature change ( T) as given in Equation 3.20 of Chapter 3. Hence, for V to be a linear function of Pin, T has to be a linear function of Pin. However, the mechanism of microwave absorption into the film takes into account the heat loss through different mechanisms like conductive, convective and radiative losses. Out of them, the radiative loss is not linear in T as given in Equation 3.12. This leads to T not being a linear function of Pin and hence, V is not a linear function of Pin. The non-linearity in V response is not observed in the experimental case. This is because the Pin for the experiment is not very high and the radiative heat loss plays the important role in the high temperature range in determining the temperature profile. The inset of the Fig. 4.5 shows the input power dependence of V for the bias current 25mA. The power range is same as the experimental range. In this range it is observed that the simulated V response is also linear with Pin similar to that obtained experimentally. This also supports the fact that the radiative loss plays an important role in the high temperature regime to determine the voltage profile. Hence it can be concluded that even though the V response is not linear with the input microwave power, it can be considered linear in the power range used in the experiment. 57 4.2.3 Thickness dependence of V: It is important to study the thickness dependence of V to determine an optimal thickness for experiment. Fig. 4.6 shows the experimental results of the thickness dependence of V for a silver micro-bridge. From this figure it is observed that V decreases almost linearly with the increase in thickness. Fig: 4.6: Experimentally measured thickness dependence of V response at the edge of the silver micro-bridge (by Dr. R. Aga). To compare with the experimental results, V response as a function of thickness for different bias currents in the range 5mA-45mAfor the silver film of dimension 0.5cm×0.5cm is simulated and is shown in Fig. 4.7. Input microwave 58 power is 1W. The microwave probe of diameter ~ 200µm is placed at the edge of the film. The thickness of the film ranges from 0.1µm-1.3µm which is less than the penetration depth of microwave in silver which is ~1.42 µm. 175 12 10 V(µvolts) 150 V(µV) 125 Ib=25mA 8 6 4 2 100 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Thickness(µm) 75 5mA 15mA 25mA 35mA 45mA 50 25 0 0.2 0.4 0.6 0.8 1.0 1.2 Thickness(µm) Fig: 4.7: Thickness dependence of V response at the edge of the silver film of dimensions 0.5cm×0.5cm for different bias currents in the range 5mA-45mA obtained from simulation. The inset of the figure shows the thickness dependence in the experimental range. Input microwave power is 1W. Tip diameter is ~ 200µm. 59 From this figure it is clearly observed that V is much higher when the thickness is less and decreases rapidly with the increase in thickness. This trend is not similar to the experimental results. The inset of this figure shows the thickness dependence in the experimental range. Comparing this with Fig. 4.6 it is observed that the decrease in V with thickness is much more rapid in the case of simulation compared to the experimental data. However, the experimental data has very few data point and hence is difficult to make any conclusion from it. The rapid decrease of V with thickness in simulation is because the net power absorbed depends on the thickness of the film as given in Equation 3.3 in Chapter 3. For the film of thickness 0.1µm, absorption ratio as given by Equation 3.3 is ~ 1.67×10-3 whereas for the film of thickness 1.3µm, it is ~1.29×10-4. This shows that the amount of power absorbed by the film decreases by almost 10 times for a thickness of 0.1µm to a thickness of 1.3µm. This will lead to rapid decrease in V with thickness. 4.2.4 V profile along the width of the film: The V profile along the width of the film is simulated by calculating the induced voltage response at different points along the width of the film. Fig. 4.8a gives the V profile across the width of the film of dimension 0.5cm×0.5cm and 60 thickness ~ 0.1 µm for different bias currents in the range 5mA-45mA. The tip diameter ~ 200µm and the microwave input power is 1W. a) 4.85 b) 5mA 15mA 25mA 35mA 45mA 4.80 R(m ) V(arb units) 5mA 15mA 25mA 35mA 45mA 4.75 4.70 0.0 0.1 0.2 0.3 0.4 4.65 0.0 0.5 Width(cm) 0.1 0.2 0.3 0.4 0.5 Width(cm) Fig: 4.8: The a) V profile and b) change in resistance ( R) across the width of the silver film of dimension 0.5cm×0.5cm and thickness ~ 0.1 µm for different bias currents obtained from simulation. The tip diameter ~ 200µm and the microwave input power is 1W. From this figure it is observed that the V is higher (~1.04 times) at the edge of the film compared to its centre and is fairly uniform between the two edges. The large V at the edge compared to that of the centre is because the temperature rise is more at the edge than at the centre. V has linear dependence on Ib from Ohm’s law. This linear dependence is clearly observed if the same plot normalized with respect to Ib ( R= V/Ib) as shown in Fig. 4.8b. From this figure 61 it is clearly observed that on normalizing with respect to Ib, curves with different Ib coincide. The ratio of V at edge to centre ( Vedge/ Vcentre) is dependent on various sample and environmental parameters like the thermal conductivity of the film and substrate, heat capacity of the film, heat transfer coefficient of air etc. In order to determine how these different parameters affect the ratio, it is important to compare the experimental curves with the simulated results. Fig. 4.9 shows the V profile obtained experimentally for a silver microbridge of thickness ~ 0.1µm and dimension 0.5cm×0.05cm with tip diameter ~ 20µm 26. From this figure, the Vedge/ Vcentre ratio is found to be ~3. 1.6 5mA 15mA 25mA 35mA 45mA 1.4 1.2 V(µV) 1.0 0.8 0.6 0.4 0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Width(mm) Fig 4.9: Experimentally determined V line scans across the width of the silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~ 0.1µm for different bias currents. The tip-size ~ 20µm and the microwave input power is 1W 26. 62 Fig 4.10 gives the simulated V profile along the width of the bridge having the dimension similar to the experiment, for different bias currents in the range 5mA-45mA. The tip diameter is ~ 20µm. The input microwave power is 1W. Vedge/ Vcentre ratio was also found to be ~ 1.04 which is similar to the thin film case which had larger width. This suggests that the Vedge/ Vcentre ratio does not depend on the width or the lateral dimension of the film. However, the ratio is not close to the experimental ratio ~3. 0.775 a) 5mA 15mA 25mA 35mA 45mA b) 0.770 5mA 15mA 25mA 35mA 45mA R( ) V(arb units) 0.765 0.760 0.755 0.750 0.745 0.0 0.1 0.2 0.3 0.4 0.5 0.740 0.0 0.1 0.2 0.3 0.4 0.5 Width(mm) Width(mm) Fig: 4.10: The a) V and b) R profile along the width of the micro-bridge of dimension 0.5cm×0.05cm and thickness ~ 0.1µm for different bias currents in the range 5mA-45mA obtained from simulation. The tip diameter ~ 20µm and the microwave input power is 1W. 63 In all the simulated curves, the contribution of the substrate is not taken into account. The experimental measurements were recorded after 0.7 sec, just before the microwave was turned off. By that time, the temperature profile was substantially established or it was in the steady state. At this stage, there could be steady state heat loss due to conduction through the substrate. The amount of heat loss will depend on the conductivity of the substrate material. The thermal conductivity of glass substrate is ~102 times less than that of silver. However, the thickness of the film is much less than that of the substrate (~1.0mm). Hence, there will be some heat loss through the substrate due to conduction. Taking into account this steady state heat loss, the temperature profile is calculated by solving Equation 3.16 in Chapter 3. The V profile obtained for different values of f (the fraction of heat loss through the substrate) is given in Fig. 4.11. 1 .5 H eat Loss % 0% 5% 25% 50% 100% V/ V centre 1 .4 1 .3 1 .2 1 .1 1 .0 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 W id t h (m m ) Fig: 4.11: The V profile obtained from simulation for the silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~ 0.1µm for the different values of the fraction of heat loss through the substrate. The tip diameter is ~ 20µm. 64 From this figure it is observed that the Vedge/ Vcentre ratio improves to ~ 1.55 when taking into account 100% heat loss through the substrate, however, the profile becomes less flat in the centre. For the experimental data for silver microbridge given in Fig. 4.9, it is observed that the V stays almost constant in the centre, in between the two edges of the film. This suggests that in addition to the heat loss through substrate there can be other factors which will affect the ratio of Vedge/ Vcentre. The thermal conductivity (K) of silver can also affect this ratio because it determines the heat diffusion through the film. K value of ~ 417.1W/mK is for single crystal silver29,31. A thin film of silver can have a lower value of K depending on its crystalline quality. Fig. 4.12 gives the V profile with the decrease in K. V/ V centre 1 .1 1 K = 1 0 0 W /m K K = 2 0 0 W /m K K = 4 1 7 W /m K 1 .0 8 1 .0 5 1 .0 2 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 W id t h ( m m ) Fig: 4.12: The V profile obtained from simulation for silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~ 0.1µm for different values of thermal conductivity (K) of silver. The tip diameter is ~ 20µm. 65 From this figure it is observed that the Vedge/ Vcentre ratio increases slightly (to ~ 1.12) if K is reduced by ~1/4, however the profile is considerably flat in the centre. Fig. 4.11 and Fig. 4.12 indicate that the reduction in thermal conductivity of the film together with the heat loss through the substrate could increase the ratio of Vedge/ Vcentre to a value that matches the experimental data. Fig. 4.13 shows the V profile for the silver micro-bridge. Thermal conductivity (K) is ~100W/mK. The heat loss through substrate is varying from 0% to 100%. 5 Heat Loss % 0% 25% 50% 100% V/ Vcentre 4 3 2 1 0.0 0.1 0.2 0.3 0.4 0.5 W idth(mm) Fig: 4.13: The V profile obtained from simulation for silver micro-bridge of dimension 0.5cm×0.05cm and thickness ~ 0.1µm different values of the fraction of heat loss through the substrate. Thermal conductivity (K) = 100W/mK in this case. The tip diameter is ~ 20µm. 66 From this figure it is observed that the ratio of Vedge/ Vcentre has a value ~3 for about 50% heat loss through the substrate. This value is similar to that obtained experimentally. Also comparing the Fig. 4.10 and Fig. 4.12, it is observed that the V profile obtained when K = 100W/mK is much flat in the centre compared to those obtained by taking the K value for single crystal silver. This once again validates the use of reduced thermal conductivity for the simulation. 4.2.5 Two dimensional V profile: V profile along the width of the silver thin film or micro-bridge indicated that the V at the edge of the sample is much greater than the V at the centre of the sample. This can be clearly seen in the two-dimensional V map of the film where the microwave source scans the surface of the film. The two dimensional map of V can provide information about the surface morphology and the non-uniformity of the electromagnetic properties of the film. Fig. 4.14 shows the two dimensional V map of a silver film of dimension 0.5cm×0.5cm and thickness ~ 0.1µm. The heat loss through substrate is ~ 10%. From this figure it can be clearly seen that the V response is much higher at the edges of the film compared to its centre. 67 V (µV) Fig: 4.14: The two dimensional map of V obtained from simulation for a silver film of dimension 0.5cm×0.5cm and thickness ~ 0.1µm. The fraction of heat loss through the substrate is 10%. The tip diameter is ~ 200µm 4.3 COMPARISON OF LASER AND MICROWAVE HEATING: Scanning laser microscopy can also be used to map the surface morphology and identification of defects of a thin film. This is also based on the ‘hot-spot’ technique like microwave microscopy. The difference between laser heating and the microwave heating is due to the difference in their penetration depth. The penetration depth ( ) can be calculated from the Equation 3.1 of Chapter 3. For a microwave at frequency 2 GHz, whereas for a laser of wavelength 514nm, 68 is calculated to be ~1.42µm is calculated to be ~ 2.62 nm or ~ 2.62x10-3 µm. The film has a thickness ~ 0.1µm, which is much less than the penetration depth of microwave but much greater than that of laser. Hence, microwave penetrates through the film and only a small percent of it is absorbed whereas, for the laser the whole energy is absorbed within a small thickness near the top of the film. In the case of silver, it is generally used in coating mirrors and hence has very high reflection coefficient (R). Hence there is certain amount of reflection loss for laser microscopy on silver. Fig. 4.15 gives the temperature profile at the edge of the film for both microwave and laser for an input power of 30mW. The tip diameter is ~ 200µm. a) b) Fig: 4.15: The temperature profile at the edge of the silver film of thickness ~ 0.1µm for a) microwave and b) laser for an input power of 30mW. Tip diameter is ~200µm. 69 From this figure, it is observed that the temperature rise ( T) due to laser absorption ~ 4.50C when R 99% whereas it is ~ 0.350C for microwave absorption for the same input power. As V is proportional to the temperature rise, V induced due to laser absorption will be much greater than the microwave absorption. However, microwave absorption has other advantages. It has uniform penetration throughout the thickness of the film and hence can be used for thick films having thickness less than that of microwave penetration depth. Also, as only a very small percentage of input power is absorbed, there will be less damage to the film compared to laser. In the case of laser however, as the light does not penetrate through the whole thickness, it will give information only about the top layer. This is a disadvantage, especially for thick films. Summarizing, the model developed for heat diffusion due to microwave absorption can be used to calculate the effective temperature rise due to microwave absorption. The temperature rise will depend on various sample and environmental parameters such as the thermal conductivity and specific heat of the material of the film, heat transfer coefficient of air and the thermal conductivity of substrate. The induced voltage rise due to microwave absorption calculated using this model changes with input microwave power in a similar way as observed experimentally. However, the V variation with thickness is not similar to the experimental results. The one-dimensional voltage profile arising from microwave absorption is similar to that obtained experimentally however; it is not so flat in the centre compared to that obtained experimentally. This model 70 can also be used to obtain the two dimensional map of V which clearly shows that the V at the edge of the film is higher than the V at the centre of the film. Using this model it has been observed that the temperature rise due to microwave absorption is much less than that of laser absorption. However, microwave microscopy leads to more effective heating due to the large penetration depth of microwave. 71 CONCLUSIONS A theoretical model has been developed to study heat diffusion due to microwave absorption in conducting thin films. This model can be effectively used to calculate the effective temperature rise due to microwave absorption in the conducting thin films. The temperature rise at a given location will depend on various sample and environmental parameters like the thermal conductivity and specific heat of the material of the film, heat transfer coefficient of air, thermal conductivity of substrate etc. When the microwave probe is at the centre of the film, there is uniform heat diffusion due to thermal conduction through the neighboring atoms, but when the probe is at the edge, there is heat loss to the air by convection on one side and diffusion through the film on the other side. From this model it has been observed that the heat loss to air is much less compared to heat diffusion in a silver film leading to higher temperature rise at the edge of the film compared to the centre. The induced voltage rise ( V) due to microwave absorption calculated using this model changes with input microwave power in a similar way as observed experimentally. However, the V variation with thickness is not similar to the experimental results. Also, the time dependence of V obtained from this model is not similar to the experimental results as the effect of substrate is not taken into account in the simulation. The one-dimensional voltage profile arising from microwave absorption is qualitatively similar to that obtained 72 experimentally. As superconductors behave like normal metal in room temperatures, this model can be used effectively to determine the quantitative current distribution in high-TC superconductors in room temperature. 73 REFERENCES 1. W. Prusseit, G. Sigl, R. Nemetschek, C. Hoffmann, J. Handke, A. Lümkemann, and H. Kinder, IEEE Transactions on Applied Superconductivity, 15(2), 2608 (2005). 2. T C Shields, K Kawano, T WButton and J S Abell, Supercond. Sci. Technol. 15, 99 (2002). 3. D. M. Feldmann, J. L. Reeves, A. A. Polyanskii, G. Kozlowski, R. R. Biggers, R. M. Nekkanti, I. Marteense, M. Tomsic, P. Barnes, C. E. Oberly, T. L. Paterson, S. E. Babcock, and D. C. Larbalestier, Appl. Phys. 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C., Physics, 4th Ed, Prentice Hall, (1995). 34. http://www.electro-optical.com/bb_rad/emissivity/matlemisivty.htm 77 APPENDIX: Simulation Code for 1-D profile c c external f, jac double precision atol, rtol, rwork, t, tout, y double precision si, store, epsilon, sigasi, sigasj, sigasij parameter (nx=26, ny=26, neq=nx*ny, nx2=2*nx) PARAMETER (LRW = 22+9*NEQ+NEQ**2, LIW = 20+NEQ) dimension y(nx,ny),atol(neq),rwork(lrw),iwork(liw),r(nx2,ny) dimension ynew(nx2,ny),si(nx2,ny),store(nx2,ny),epsilon(nx2,ny) common ro,cp,h,con,rc0,tair,power,eps,sconst,alpha common xl,yl,zl,gcon,zg,imy,fc open(unit=7,file='slsode.out',status='unknown') WRITE(6,*) ' TEMPERATURE PROFILE DUE TO MICROWAVE' c################################################################ C THE CONSTANTS: c c c c c c c c c c c C c c C density of silver in kg/m3 ro=10.5e3 specific heat of silver in J/kgK cp=235.0 heat transfer coefficient of air in W/Ksqm h=5.6 thermal conductivity of silver in W/mK con=417.1 thermal conductivity of glass in W/mK gcon=1.0 Coefficient of resistivity in ohm-m at 20 degC for silver rc0=1.59e-8 air temperature in Kelvin tair=293.0 microwave power in W power=1.0 surface emmisivity in silver eps=0.03 Stephan's constant in W/(m^2)/(K^4) sconst= 5.67e-8 Temperature coeff of resistivity in ohm-m per degK for silver alpha=3.8e-11 Base current in ampere basecur=25.0e-3 thickness of glass zg=1.0e-3 fraction of heat lost through substrate fc=0.0 DIMENSION OF THE FILM xl=0.25e-2 78 yl=0.5e-2 zl=0.1e-6 do 100 imy = 2, ny-1 define time at which output is required in second tout=0.7e0 c 1 t = 0.e0 itol = 2 rtol = 1.e-4 do 1 i=1,neq atol(i) = 1.e-6 continue itask = 1 istate = 1 iopt = 0 mf = 22 c########################################################## c 3 2 c Initialize all the grid points at the beginning of the microwave do 2 i=1,nx do 3 j=1,ny y(i,j)=1.0 continue continue Solve the ODE do 40 iout = 1,1 call slsode(f,neq,y,t,tout,itol,rtol,atol,itask,istate, 1 iopt,rwork,lrw,iwork,liw,jac,mf) do 4 j = 1, ny do 5 i = 1, nx ynew(i,j) = y(nx-i+1,j) continue do 6 i= nx+1, nx2 ynew(i,j)=y(i-nx,j) continue continue 5 6 4 c 8 7 do 7 i=1,nx2 do 8 j=1,ny r(i,j)=rc0+alpha*(ynew(i,j)-1.0)*tair write(*,*) r(i,j) continue continue dx=2.0*xl/(nx2-1.0) dy=yl/(ny-1.0) c Define dimensionless voltage vdim=basecur*rc0*2.0*xl/yl/zl 79 c c 10 9 C 70 11 Define the relaxation factor w=1.0 Define the initial dimensionless voltage V (Si here) in volts do 9 i=1,nx2 do 10 j=1,ny si(i,j)=10.0 epsilon(i,j)=0.0 store(i,j)=10.0 continue continue Define the boundary condition istep=0 istep=istep+1 do 11 j=1,ny store(nx2,j)=0.0 si(nx2,j)=0.0 store(nx2-1,j)=basecur*(r(nx21,j)+r(nx2,j))/2*dx/yl/zl/vdim si(nx2-1,j)=basecur*(r(nx2-1,j)+r(nx2,j))/2*dx/yl/zl/vdim continue do 12 ist=2,nx2-2 i=nx2-ist c Incorporate r(i,j) si(i,2)=0.25/(1/r(i,2)-0.25/r(i,1))*(si(i+1,2)/r(i+1,2)+ @ si(i,3)/r(i,3)+si(i-1,2)/r(i-1,2)) si(i,ny-1)=0.25/(1/r(i,ny-1)-0.25/r(i,ny))*(si(i+1,ny-1)/ @ r(i+1,ny-1)+si(i,ny-2)/r(i,ny-2)+si(i-1,ny-1)/r(i-1,ny- 1)) si(i,1)=si(i,2) si(i,ny)=si(i,ny-1) c 13 12 14 16 15 do 13 j=2,ny-1 sigasi=si(i+1,j)/r(i+1,j)+si(i-1,j)/r(i-1,j) sigasj=si(i,j+1)/r(i,j+1)+si(i,j-1)/r(i,j-1) write(6,*) sigasi,sigasj sigasij=0.25*(sigasi+sigasj) si(i,j)=store(i,j)+w*(sigasij*r(i,j)-store(i,j)) epsilon(i,j)=abs(100.0*(si(i,j)-store(i,j))/si(i,j)) store(i,j)=si(i,j) continue continue do 14 j=1,ny si(1,j)=si(2,j)+basecur*(r(1,j)+r(2,j))/2*dx/yl/zl/vdim continue do 15 i=1,nx2 do 16 j=1,ny if (epsilon(i,j).gt.5.0e-15) go to 70 continue continue 80 17 40 c 60 =,i4) 500 100 80 90 sum=0.0 do 17 j=1,ny sum=sum+si(1,j) continue avg=sum/ny if (istate .lt. 0) go to 80 tout = tout+0.05 continue write(6,60)iwork(11),iwork(12),iwork(13) format(/12h no. steps =,i4,11h no. f-s =,i4,11h no. j-s write(7,*) (imy-1)*dy*100, (avg-1.0)*vdim*1.0e6 format(2f6.2) write(*,*) imy continue stop write(6,90)istate format(///22h error halt.. istate =,i3) STOP END C*************************************************************** subroutine f (neq, t, y, ydot) c double precision t, y, ydot parameter (nx=26, ny=26) dimension y(nx,ny), ydot(nx,ny), y2(nx+2,ny+2), @ dmx(nx+1), dmyold(ny+1), dmy(ny+1) common ro,cp,h,con,rc0,tair,power,eps,sconst,alpha common xl,yl,zl,gcon,zg,imy,fc If (t.LE.0.70005) then pow=power else pow=0.0 endif C c Ditermining the grid size Uniform spacing dx=1.0/(nx-1) dy=1.0/(ny-1) c Non uniform grid size kparami=20 kparamj=20 c X grid spacing do 1 i=2,nx c dmx(i)=log(kparami+1.0)/nx/kparami*exp((i1)*log(kparami+1.0)/nx) c********************************************************** c To be used for uniform spacing dmx(i)=dx 81 c********************************************************** 1 continue dmx(1)=dmx(2) dmx(nx+1)=dmx(nx) c########################################################## c Y grid spacing ystep=0.0 detay=(log(ystep*kparamj+1.0)+log((1.0ystep)*kparamj+1.0))/ny rnyminus=(log(kparamj*ystep+1.0))/detay If ((rnyminus-int(rnyminus)).lt. 0.5) then nyminus=int(rnyminus) else nyminus=int(rnyminus)+1 endif do 2 j=1,ny c********************************************************** c To be used for uniform spacing dmy(j)=dy c********************************************************** c dmyold(j)=detay*exp((j)*detay)/kparamj 2 continue c sumdmy=0.0 c do 3 jnew =2,ny c If (jnew.le.nyminus)then c dmy(jnew)=dmyold(nyminus-jnew+1) c else c dmy(jnew)=dmyold(jnew-nyminus) c endif c sumdmy=sumdmy+dmy(jnew) c 3 continue c dmy(ny)=dmy(ny)+1.0-sumdmy dmy(1)=dmy(2) dmy(ny+1)=dmy(ny) C Taking into account the boundary condition 5 4 do 4 i=2,nx+1 do 5 j=2,ny+1 y2(i,j)=y(i-1,j-1) continue continue 6 do 6 j=2,ny+1 y2(nx+2,j)=-2.0*dmx(nx+1)*xl*h/con*(y(nx,j-1)-1)+y(nx-1,j1) y2(1,j)=y(2,j-1) continue do 7 i=2,nx+1 y2(i,1)=-2.0*dmy(1)*yl*h/con*(y(i-1,1)-1)+y(i-1,2) y2(i,ny+2)=-2.0*dmy(ny+1)*yl*h/con*(y(i-1,ny)-1)+y(i-1,ny- 1) 82 7 continue C########################################################## c Solving the differential equations gama=con/(ro*cp*xl*xl) beta=con/(ro*cp*yl*yl) do 8 i=2,nx+1 do 9 j=2,ny+1 xterm=((y2(i+1,j)-y2(i,j))/dmx(i))-((y2(i,j)-y2(i1,j))/dmx(i-1)) yterm=((y2(i,j+1)-y2(i,j))/dmy(j))-((y2(i,j)-y2(i,j1))/dmy(j-1)) ydot(i-1,j-1)=2*gama/(dmx(i)+dmx(i-1))*xterm+ yterm* @ 2*beta/(dmy(j)+dmy(j-1))-(h+fc*gcon/zg)/ro/cp/zl*(y2(i,j)1.0) c c c********************************************************** c The following equation is for uniform grid size ydot(i-1,j-1)= -(h+fc*gcon/zg)/ro/cp/zl*(y2(i,j)-1.0) + @ gama/dx/dx*(y2(i+1,j)-2.0*y2(i,j)+y2(i-1,j)) * +beta/dy/dy*(y2(i,j+1)-2.0*y2(i,j)+y2(i,j-1)) # - eps*sconst*(tair**3.0)*((y2(i,j)**4.0)-1.0)/ro/cp/zl c********************************************************** 9 8 c c continue continue Inserting power in the grid Scale length s=2.0*rc0/(377.0) absorbtion ratio in silver aratio=(2.0*zl/s)/((1.0+zl/s)**2.0) pownet=pow*aratio c ydot(2,imy)=ydot(2,imy)+pownet/2/tair/ro/cp/(xl*yl*zl) & /dmx(2)/dmy(imy) return end C**************************************************************** c 2 1 subroutine jac (neq, t, y, ml, mu, pd, nrpd) double precision pd, t, y parameter (nx=13, ny=13) dimension y(nx,ny), pd(nrpd,neq) do 1 i=1,nrpd do 2 j=1,neq pd(i,j)=0.0 continue continue return end 83

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