Ultrasensitive Superconducting Cold-Electron Bolometer Coupled to Multi-frequency Phased Antenna Array for Polarization Detection of the Cosmic Microwave Background Thesis for the Degree of Erasmus Mundus Master of Nanoscience and Nanotechnology MOHAMED SALEH Quantum Device Physics Laboratory Department of Microtechnology and Nanoscience-MC2 CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden, 2014 Ultrasensitive Superconducting Cold-Electron Bolometer Coupled to Multi-frequency Phased Antenna Array for Polarization Detection of the Cosmic Microwave Background MOHAMED SALEH Promoter: Prof. Leonid Kuzmin Chalmers University of technology, Sweden Co-promoter: Prof. Jean-Pierre Locquet KU Leuven, Belgium External Examiner: Dr. Serguei Cherednichenko Chalmers University of technology, Sweden Quantum Device Physics Laboratory Department of Microtechnology and Nanoscience – MC2 CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2014 Ultrasensitive Superconducting Cold-Electron Bolometer Coupled to Multi-frequency Phased Antenna Array for Polarization Detection of the Cosmic Microwave Background MOHAMED SALEH © MOHAMED SALEH, 2014. Quantum Device Physics Laboratory Department of Microtechnology and Nanoscience – MC2 Chalmers University of Technology SE-412 96 Gothenburg Sweden Telephone + 46 (0)73-758 0788 Cover: Current distribution and beam pattern of the rectangular dual polarized phased array of slot antennas working at 75 GHz and 105 GHz simulated using CST MWS. Printed by Reproservice Gothenburg, Sweden 2014 Contents List of Figures v List of Tables vii Abstract ix Acknowledgements xi 1 Introduction 1 1.1 The Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Early Universe . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Anisotropies in CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Temperature Anisotropy . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Polarization Anisotropy . . . . . . . . . . . . . . . . . . . . . 4 1.3 The B-mode Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Disadvantages of Current Technologies . . . . . . . . . . . . . 5 1.3.2 Future Technologies . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Motivation for This Work . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Superconductivity 9 2.1 Superconductivity Basics . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Superconducting Gap . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Temperature Dependence of The Gap . . . . . . . . . . . . . . 10 2.1.4 Quasiparticles and The Semiconductor Picture . . . . . . . . . 11 2.1.5 BCS Density of States . . . . . . . . . . . . . . . . . . . . . . 11 2.1.6 Length Scales of Superconductivity . . . . . . . . . . . . . . . 12 2.2 The NIS Tunnel Junction . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 NIS Junction as an Electron Cooler . . . . . . . . . . . . . . . 13 2.2.2 NIS Tunneling Current . . . . . . . . . . . . . . . . . . . . . . 14 i ii Contents 2.3 Superconductors at High Frequencies . . . . . . . . . . . . . . . . . . 15 2.3.1 The Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 The Concept of Surface Impedance . . . . . . . . . . . . . . . 17 3 Cold-Electron Bolometer 19 3.1 Bolometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 What Is a Bolometer . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 Bolometer Technologies . . . . . . . . . . . . . . . . . . . . . 20 3.2 CEB Device Structure and Principles of Operation . . . . . . . . . . . 21 3.2.1 Filtering Capability . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 Advantages of CEB . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1 Responsivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.2 Noise and Noise Equivalent Power (NEP) . . . . . . . . . . . 25 3.3.3 Time Constant . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Modeling 27 4.1 Power Flow in CEB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.1 NIS Cooling Power . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.2 Electron-Phonon Coupling . . . . . . . . . . . . . . . . . . . . 28 4.1.3 Power Dissipated in the Superconductor . . . . . . . . . . . . . 29 4.1.4 Power Back-flow . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1.5 Subgap Leakage . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1.6 Joule Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1.7 Bringing It All Together: Heat Balance Equations . . . . . . . . 31 4.2 Parameters Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 Responsivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.4.1 Photon Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.4.2 Noise in the Absorber . . . . . . . . . . . . . . . . . . . . . . 37 4.4.3 NIS Junction Noise . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4.4 Amplifier Noise . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4.5 Total Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.5 Time Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.6 Microwave Absorption and Optical Conductivity of a Superconductor . 40 4.6.1 Nonlocality and The Anomalous Skin Effect . . . . . . . . . . 40 4.6.2 Simulation Setup in EM Simulators . . . . . . . . . . . . . . . 43 Contents iii 5 Antenna Design and Simulation 45 5.1 Antenna Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1.1 Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1.2 Directivity and Beam Width . . . . . . . . . . . . . . . . . . . 47 5.1.3 Co-polarization and Cross-polarization . . . . . . . . . . . . . 48 5.1.4 Input Impedance . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Simulation Methods and Software Packages . . . . . . . . . . . . . . . 49 5.3 Slot Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3.1 Rectangular Slot Antenna . . . . . . . . . . . . . . . . . . . . 50 5.3.2 Offset Feeding . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.3.3 Microstrip Feeding of Slot Antenna . . . . . . . . . . . . . . . 52 5.3.4 The Effect of the Substrate . . . . . . . . . . . . . . . . . . . . 54 5.4 Slot Antenna Design and Simulation . . . . . . . . . . . . . . . . . . . 55 5.4.1 Choosing Slot Dimensions . . . . . . . . . . . . . . . . . . . . 56 5.4.2 Choosing Feed Offset . . . . . . . . . . . . . . . . . . . . . . 58 5.4.3 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4.4 Choosing the Substrate Thickness . . . . . . . . . . . . . . . . 64 5.5 Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.6 Antenna Array Design and Simulation . . . . . . . . . . . . . . . . . . 68 5.6.1 Required Beam Width and Array Size . . . . . . . . . . . . . . 68 5.6.2 Rectangular Lattice Array . . . . . . . . . . . . . . . . . . . . 69 5.6.3 Triangular Lattice Array . . . . . . . . . . . . . . . . . . . . . 72 5.6.4 Performance Comparison . . . . . . . . . . . . . . . . . . . . . 72 5.7 Feed Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.7.1 Microstrip Bends . . . . . . . . . . . . . . . . . . . . . . . . . 76 6 Conclusions and Future Work 79 6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.1.1 Feeding Network . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.1.2 Increasing the Number of Frequency Bands . . . . . . . . . . . 80 6.1.3 Side lobe Reduction . . . . . . . . . . . . . . . . . . . . . . . 80 Appendix A Mesurement Setup 81 Appendix B MATLAB Routines 83 B.1 Solving Heat balance Equations . . . . . . . . . . . . . . . . . . . . . 83 B.2 Fitting the Experimental IV Curve . . . . . . . . . . . . . . . . . . . . 85 B.3 Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B.4 Mattis-Bardeen Nb Surface Impedance Calculator . . . . . . . . . . . . 89 iv Contents Bibliography 93 List of Figures 1.1 The inflation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Cartoon illustration of the origin of CMB. . . . . . . . . . . . . . . . . 3 1.3 Measured CMB Temperature Anisotropies across the sky. . . . . . . . . 3 1.4 Corrugated horns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Superconducting gap temperature dependence. . . . . . . . . . . . . . 10 2.2 The normalized BCS density of states and Quasiparticle occupation . . 12 2.3 The unbiased NIS junction. . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 The biased NIS junction. . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 NIS IV curves for different temperatures. . . . . . . . . . . . . . . . . 15 2.6 The two-fluid model of a superconductor. . . . . . . . . . . . . . . . . 17 3.1 A simple thermal model of a bolometer. . . . . . . . . . . . . . . . . . 19 3.2 Schematic of the Resonant Cold-Electron Bolometer. . . . . . . . . . . 22 3.3 Energy level diagram of the CEB. . . . . . . . . . . . . . . . . . . . . 23 4.1 NIS cooling power Pcool vs normalized biasing voltage. . . . . . . . . . 28 4.2 NIS PS vs normalized biasing voltage. . . . . . . . . . . . . . . . . . . 30 4.3 Modified BCS density of states. . . . . . . . . . . . . . . . . . . . . . 31 4.4 AutoCAD layout of the measured device. . . . . . . . . . . . . . . . . 33 4.5 Investigating the effect of different fitting parameters on IV curve. . . . 34 4.6 Temperature of electrons in the normal metal as a finction of bias voltage. 35 4.7 Fitted IV curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.8 Responsivity for different absorber resistance values. . . . . . . . . . . 36 4.9 Noise contributions for R=1KΩ. . . . . . . . . . . . . . . . . . . . . . 39 4.10 Time constant of the bolometer. . . . . . . . . . . . . . . . . . . . . . 40 4.11 Calculated complex conductivity of Nb film. . . . . . . . . . . . . . . . 42 4.12 The complex surface impedance of Nb film. . . . . . . . . . . . . . . . 43 5.1 Spherical coordinate system for representing an antenna radiation pattern. 46 5.2 Two-dimensional normalized power pattern to illustrate the beamwidth. 47 v vi List of Figures 5.3 The co-polarization and cross-polarization components according to Lud- wig second definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 Slot antenna on an infinite ground plane. . . . . . . . . . . . . . . . . . 50 5.5 Voltage and current distribution of half wavelength slot. . . . . . . . . . 51 5.6 Electric field distribution across the slot and the equivalent magnetic current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.7 Radiation Pattern of slot antenna in free space simulated using CST. . . 52 5.8 The microstrip line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.9 Substrate modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.10 A 3D view of the slot antenna and microstrips. . . . . . . . . . . . . . . 55 5.11 The coupling capacitor used for impedance matching. . . . . . . . . . . 55 5.12 The real part of the input impedance of the 75 GHz slot. . . . . . . . . . 57 5.13 The real part of the input impedance of the 105 GHz slot. . . . . . . . . 57 5.14 Offset feeding the slot antenna. . . . . . . . . . . . . . . . . . . . . . . 58 5.15 HFSS simulation setup for the auxiliary capacitor. . . . . . . . . . . . . 59 5.16 HFSS simulation setup for calculating the input impedance of the slot. . 59 5.17 Input impedance of the 75 GHz slot for different offset feeding distances. 60 5.18 Input impedance of the 105 GHz slot for different offset feeding distances. 61 5.19 Simulated reactances of the coupling capacitors to achieve matching. . . 63 5.20 Characteristic impedance of superconducting Nb microstrip line. . . . . 63 5.21 Reflection coefficient for matched slots. . . . . . . . . . . . . . . . . . 64 5.22 Moving the element from the origin to a point r0. . . . . . . . . . . . . 66 5.23 Planar rectangular antenna array. . . . . . . . . . . . . . . . . . . . . . 66 5.24 Normalized array factor of linear broadside array for different element spacing d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.25 Matching the beam to the telescope optics. . . . . . . . . . . . . . . . . 68 5.26 Current distribution of the 75 GHz perpendicular slots. . . . . . . . . . 69 5.27 Dual polarized dual frequency rectangular antenna array of slot antennas. 70 5.28 Simulation results for the rectangular array. . . . . . . . . . . . . . . . 71 5.29 Dual polarized dual frequency triangular antenna array of slot antennas. 73 5.30 Simulation results for the triangular array. . . . . . . . . . . . . . . . . 74 5.31 Current distribution of the arrays. . . . . . . . . . . . . . . . . . . . . . 75 5.32 The summing network. . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.33 Microstrip bends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.1 The T-junction used in signal distribution. . . . . . . . . . . . . . . . . 80 A.1 The measurement setup for the bolometer IV curve. . . . . . . . . . . . 81 A.2 The pulse tube cryostat used for measuring bolometer IV curve. . . . . 82 List of Tables 4.1 Fitted parameters values. . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 Performance comparison of the 8x8 rectangular and triangular arrays. . 72 vii This page intentionally contains only this sentence. Abstract We describe the design and simulation of a novel dual-polarized multi-frequency single- pixel for next-generation Cosmic Microwave Background (CMB) polarization detec- tion. The pixel consists of a phased array of planar antennas known as slot antennas coupled to millimeter-wave cryogenic detector by superconducting transmission lines. The pixel separates the incoming radiation into two linear polarizations with high po- larization purity and operates at two frequencies (75 GHz and 105 GHz) as well. This completely lithographed planar technology will enable us to squeeze the size of the focal plane to avoid aberration problems. This will also enable us to realize bolometer arrays with thousands of pixels needed to meet the sensitivity requirements of future CMB po- larization experiments and to achieve better resolutions and higher scan speeds. Having multiple frequencies is important to differentiate the CMB from the galactic foreground contamination. Those goals are difficult to achieve with the current technology which uses feed horn for each pixel. The detector is a superconducting Cold Electron Bolometer (CEB), which have sev- eral advantages over current technologies. Themain component of the CEB is aNormal- Insulator-Superconductor (NIS) tunnel junction, which acts as a thermometer. The CEB could implement an additional spectral filtering feature, which obviates the need for ex- ternal filters. This technology is a candidate for use in the European Space Agency (ESA) Cosmic Vision Program (2015-2025) to detect the B-mode of the CMB,whichwill be considered as an indirect detection of the gravitational waves that followed the Big Bang. ix This page intentionally contains only this sentence. Acknowledgements I would like thank my examiner and promoter, Prof. Leonid Kuzmin, for giving me the opportunity to be a member of his group. He was always supportive and had time to discuss the results and guide me no matter how busy he was. He was very friendly; he used to bring cakes during meetings, and invite the group members to lunch from time to time. I would also like to thank Anna Gordeeva for co-supervision and experimental help. Many thanks to Alexander Sobolev for his insights and help in CST simulations. I would like to express my gratitude to Sumedh Mahashabde for his continuous en- couragement and motivation. I appreciate your smartness. I considermyself lucky towork in an environment where thewelfare of the researcher is of a high priority. ThanksMC2 for the cake parties, the fruits, and the unlimited coffee. My thanks toMC2 IT support, especially Henric Fjellstedt, whowas always available to install the software packages needed to complete this project. xi This page intentionally contains only this sentence. Chapter 1 Introduction 1.1 The Cosmic Microwave Background Since the speed of light is finite, the images of far objects are late. For example, the image of our sun is eight minutes old, whereas the image of the nearest galaxy is 2 million years old! Therefore, a natural question arises: what is the oldest thing that we can observe? Fortunately, this thing turned out to be there. It is a dim radiation surrounding us. This light is the so-called Cosmic Microwave Background (CMB). The CMB is thought to have emerged about 13.7 billion years ago during the formation of our universe. Therefore, to understand the origin of CMB, we must understand how our early universe was. The next subsection elucidates briefly about the early universe. 1.1.1 The Early Universe After 10−36 seconds of the Big Bang “13.7 billion years ago”, and for a short period of about 10−33 seconds, the universe is thought to have expanded exponentially to reach 1078 of its initial volume, as shown if Figure 1.1. This theory is known as Inflation theory, and it is a key theory in the Big Bang cosmological model of the universe. The rapid expansion of universe during inflation have created ripples in the space known as primordial gravitational waves. Just as moving charges can generate electro- magnetic waves, moving masses can generate gravitational waves according to General Relativity as well. The Big Bang released hot and dense photons, electrons, and protons. At this point, photons scattered strongly by the charged electrons and protons and were not able to escape 1, so the universe was opaque. The universe cooled gradually, and after 378,000 years of the Big Bang, it was cool enough that electrons could bind to protons to form 1Charged particles are accelerated by the electric field component of the incident radiation (photons) and emit radiation at the same frequency as the incident wave, therefore the photons get scattered. 1 2 Chapter 1. Introduction Figure 1.1 – The inflation model. The universe has expanded exponentially within a very tiny fraction of a second. neutral hydrogen atoms. After this recombination, the scattering of photons decreased extremely since the vast majority of particles became neutral atoms rather than charged particles. The universe then became transparent and the photons could escape. Those photons still travel freely in the universe till now and they constitute the CMB radiation that we can observe today. Figure 1.2 shows a depiction of the origin of the CMB. The CMB has the spectrum of a black body that has a temperature of 2.725 K and peaks at about 160.2 GHz. Of course it has cooled during the past 14 billion year till it reached this temperature. 1.2 Anisotropies in CMB The CMB varies from a region to another in the sky. This spatial variation provides a snapshot of the early universe. There are two kinds of variations: intensity and polar- ization. 1.2.1 Temperature Anisotropy As was mentioned before, the average temperature of CMB is 2.725 K. However, the temperature of the CMB is anisotropic. It is not the same when looking in different directions in the sky. It has small variations at the level of about ±0.00335 K. CMB temperature anisotropies dictate that there were density fluctuations in the early universe 1.2. Anisotropies in CMB 3 Electron Proton PhotonLast scattering TransparentOpaque Big Bang Time 0 Recombination 378,000 years Today 13.7 billion years Figure 1.2 – Cartoon illustration of the origin of CMB. Before recombination, photons scattered strongly on the charged particles. After the charged particles were bound to form neutral atoms, the universe became transparent and CMB phototns could escape. It is those photon that we see today as CMB radiation. The neutral atoms joined each other due to gravity to form stars and galaxies. known as “primordial fluctuations”. WMAP 2003 C obe 1 992 Planck 2013 Figure 1.3 – Measured CMB Temperature Anisotropies across the sky. The figure shows how the resolution is being enhanced with time due to advancements in detectors technologies. This pattern gives information about initial conditions of the expansion. The Big Bangmodel predicts in detail these anisotropies. Therefore, those measured fluctuations lend credence to the Big Bang model in general. Figure 1.3 shows the measured CMB temperature anisotropy and how the accuracy and resolution are improving. These anisotropies were first observed in 1992 by The NASA COBE . The team received the 2006 Nobel Prize in physics for this achievement. 4 Chapter 1. Introduction 1.2.2 Polarization Anisotropy It has been recently detected that the CMB radiation is also polarized besides being anisotropic in temperature. Within a short period before the end of recombination, phonons were scattered by the still free electrons left. This scattering mechanism is known as Thomson Scattering and it has caused the CMB radiation to be polarized. Thomson scattering is the scattering of electromagnetic waves by a free charged parti- cles, where the electric field component of the incident radiation accelerates the charged particles in the direction of the electric field. This movement of the charged particles in turn causes emission of radiation at the same frequency as the incident wave, and thus the wave gets scattered. Consider incident unpolarized light on a charged particle, the two polarization com- ponents accelerate the particle in two perpendicular directions. However, since light cannot have a component polarized along its direction of propagation, only one linear polarization is scattered. Particle acceleration in a direction parallel to the direction of propagation of the scattered light produces no radiation. In other words, an unpolarized incident wave becomes polarized after being scattered by a charged particle. The CMB polarization is not constant through the sky. Temperature anisotropies gave rise to vari- ation in polarization magnitudes and orientations causing CMB polarization anisotropy. The polarization of CMB can be mathematically decomposed into what is called the E-mode and the B-mode. The E-mode describes magnitude changes “or gradient” of polarization, while the B-mode describes the rotation ”or curl” of CMB polarization. Those two modes originated from different physical origins and carry complementary information about the early universe. The B-mode is the subject of next section. Despite the fact that CMB polarization has been long predicted, it has remained un- detected until recently when the DASI team first detected the E-mode polarization in 2002 [12]. The B-mode polarization is predicted to have been produced by the gravi- tational waves generated during the inflation. Therefore, the detection of the B-mode signal would represent the final and definitive confirmation of the inflation model and evidence of primordial gravitational waves. By studying the CMB polarization pattern, we can know about the universe in its first tiny fraction of a second, among other things. The precise measurement of the B- mode will constrain possible inflation scenarios and fix its energy scale, which would be useful for developing theories of quantum gravity. The signal can also be used to determine the masses of the different types of neutrinos. 1.3. The B-mode Detection 5 1.3 The B-mode Detection Detecting B-modes is an extremely challenging task. It is extremely faint signal; whilst the temperature fluctuations of the CMB are around 160 µK, the E-mode signal is at an amplitude of order of 8 µK, and the fluctuations of B-mode signal is of the order 0.1 µK. B-modes have been measured by South Pole Telescope in 2013 [18], and in 2014 by BICEP2 experiment [1]. However, more sensitivity and multi-frequency measurements are still needed. 1.3.1 Disadvantages of Current Technologies Systems with millimetre-wave detectors require beam-collimation to control their illu- mination from the sky. The corrugated horn [9], see Figure 1.4, has been the standard antenna for this purpose for a long time due to the fact that its radiation patterns are rota- tionally symmetric , and it exhibits low cross-polarization (-30dB) 2, and high coupling efficiency to the beam. The corrugated horn is shown in Figure 1.4. (a) (b) Figure 1.4 – (a) Cross-section of a corrugated horn antenna. (b) The Focal Plane of Planck experiment [30]. This technology is very well established in CMB missions and has been the main beam forming technology in CMB experiments to date due to its unsurpassed RF per- formance. All the experiments that detected CMB E-mode polarization have used cor- rugated feed horns. These include Planck [31], WMAP [4], CAPMAP [3], DASI [12], CBI [44], Boomerang [23], and QUAD [19]. Future CMB polarization experiments will require more detectors “thousands” and measuring at larger range of frequencies. Unfortunately, corrugated feed horns have 2Cross polarization is simply the ratio between the power radiated in the unwanted polarization to the power radiated in the desired polarization, see Chapter 5 for more details. 6 Chapter 1. Introduction disadvantages with increasing the number of detectors, and with increasing or lowering the frequency. The following are the disadvantages of feed horn technology • As the number of detectors increase, more and more horns are needed to populate the focal plane 3. Those heavy metal horns will create a significant mass and volume penalty. They have to withstand accelerations during launch. They are too heavy to be cooled to very low temperatures that detectors work at. • Manufacturing detailed corrugations is limited by cost and time for the production of large detector arrays. • At high frequencies “short wavelengths”, the horns are relatively small and fabri- cating the corrugated structures is even harder. 1.3.2 Future Technologies Clearly, new technologies are required to realize focal planes with lower mass, lower cost, and less fabrication time. An alternative is to replace the horns with planar anten- nas. This will allow us to have entirely lithographed focal planes, and thus large and dense detector arrays could be realized. Planar antennas have several advantages over conventional feed horns such as • Densely packed focal planes are possible. • Low cost and low mass. • Parallel fabrication. • Fabrication simplicity. • Mechanical robustness. • Immunity to misalignment to the telescope optics. • Better immunity of the detectors to cosmic rays. • Can be dual-polarized operating in two polarizations simultaneously. • Multiple frequency bands. • Can include some level of spectral filtering. 3The focal plane is the plane of the detectors, it lies in the focus of the telescope optics, and hence the name. 1.4. Motivation for This Work 7 1.4 Motivation for This Work The detection of the B-modes component of the CMB is now the objective of the next generation CMB experiments. Those experiments will be required to reach an exquisite and unprecedented levels of sensitivity required to detect B-modes. To achieve these se- rious experimental challenging goals, extremely sensitive and ultra-low noise detectors are required. The pixel size also needs to be squeezed. This will help to avoid aberration problems and will enable us to populate the instrument focal plane with thousands of those detectors for increased resolution and scan speeds. Furthermore, multi-frequency coverage will be required for removal of foreground contamination, which is also po- larized. The goal of this thesis is double fold: firstly, to model the Cold-Electron Bolometer (a kind of sensitive superconducting detector) that have been fabricated in Chalmers Uni- versity of Technology. Secondly, investigation of an equivalent planar antenna which would replace conventional horn antennas to achieve increased packing density in the focal plane. This antenna should also allow for multi-frequency operations and simul- taneous measurement of two orthogonal polarizations. In the framework of the European Space Agency (ESA) Cosmic Vision Program (2015-2025), Chalmers University of Technology will lead the design and manufacture of a planar demonstrator (both the detector and the planar antenna) for next generation B-mode experiments. 1.5 Structure of the Thesis The thesis is structured as follows. Chapter 2 briefly touches upon the relevant basics of superconductivity that are necessary to understand how our bolometer works. Chapter 3 explains the idea behind bolometers generally and the Cold-Electron Bolometer espe- cially. In Chapter 4, the modeling of the device is shown and the theoretically generated curves are fitted to the experimentally measured ones. Chapter 5 deals with the design and simulation of the antenna and antenna array that is coupled to the bolometer. This page intentionally contains only this sentence. Chapter 2 Superconductivity 2.1 Superconductivity Basics In this chapter, we will cover some theoretical basics that will help to understand how a Cold-Electron Bolometer works. First, basics of superconductivity that are relevant are explained. Then the working principle of the Superconductor-Insulator-Normal tunnel junction, the heart of the Bolometer, will be touched upon. The concept of surface impedance of superconductors is finally introduced. Superconductivity is a quantum phenomenon characterized by the complete loss of resistivity in certain kinds of metals when the temperature of the metal is lower than a threshold temperature, known as the critical temperature Tc. In 1957 Bardeen, Cooper and Schrieffer developed a microscopic theory of superconductivity that can explain mostly every property of conventional low-Tc superconductors 1 . In the next section we give a brief introduction to the BCS theory, with only the results needed reviewed. 2.1.1 BCS Theory BCS Theory postulates that electrons in a superconductor weakly attract each other due to electron-phonon interaction at low temperatures. When a fast-moving electron passes through a lattice of heavy and stationary positive ion cores, it will attract the nearby ions. This causes lattice distortion in the vicinity of this electron and ions move slightly to- wards each other, increasing the positive charge density in this region. A second electron will then feel this increase in the positive charge and will move to lower its potential en- ergy, thus becoming effectively attracted to the first electron. When the temperature is low enough, the attraction force overcome the Coulumb repulsive force. The main consequence of this weak electron-electron attraction is the formation of bound electron pairs known as Cooper Pairs. 1High-Tc superconductors cannot be explained by BCS Theory. 9 10 Chapter 2. Superconductivity The two electrons forming Cooper Pairs have equal but opposite momentum and spin, therefore Cooper Pairs are bosons (zero spin) and will occupy the same lower energy state known as BCS ground state at zero temperature. They are able to carry non-dissipative super current. 2.1.2 Superconducting Gap The binding energy per electron of a Cooper pair is called the energy gap ∆. Therefore, breaking a pair requires energy of 2∆. Stable pair states cannot exist above the BCS ground state, so the minimum energy needed to create an excited state is that required to break a pair and remove it from the ground state, i.e. 2∆. BCS theory relates the gap ∆ at zero temperature to the superconductor’s critical temperature as follows: ∆(0) = 1.76KBTC (2.1.2.1) WhereKB is the Boltzmann constant. 2.1.3 Temperature Dependence of The Gap One of the BCS predictions is that the gap energy ∆ is temperature dependent. The temperature dependence of the gap can be obtained to a good approximation from the simple equation ∆(T ) ∆(0) = √ cos ( π 2 T TC ) (2.1.3.1) which agrees fairly well with experimental measurements. This gap temperature depen- dence is shown in Figure 2.1. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 T/T c ∆( T ) / ∆ (0 ) Figure 2.1 – Superconducting gap temperature dependence. 2.1. Superconductivity Basics 11 2.1.4 Quasiparticles and The Semiconductor Picture The two electrons comprising the Cooper pair occupy the state (k⃗ ↑,−k⃗ ↓), where k⃗ is the momentum and ↑ is the spin. At zero temperature all electrons are paired up in Cooper pairs, but at a finite tem- perature some electrons exist as excitations. The excited k state implies that (k⃗ ↑ is occupied, and (−k⃗ ↓) is empty. This picture has an analogy to semiconductors if we see this as creation of electron-like and hole-like excitations. However, in case of supercon- ductors, both the excitations have charge of –e. These particles are called quasiparticles. Breaking a pair corresponds to exciting a quasiparticle above the energy gap. It follows that the superconductor quasiparticle excitation spectrum is analogous to the semiconductor density of states. This is illustrated in Figure 2.2b of the next subsec- tion, where the BCS ground state of the superconductor corresponds to the ground state of the superconductor. The filled band is the valence-like band formed by the bound pairs at energies ∆ below the Fermi level, whereas the empty band is the conduction- like band formed by the broken pairs at energies ∆ above the Fermi level. An energy gap of 2∆ separates the two bands. At excitation, pairs of quasiparticles are created, the hole-like particles in the valence band and the electron-like particles in the conduction band. The Fermi energy EF lies exactly in the middle of the gap since there are always equal number of hole-like and electron-like excitations. This semiconductor picture is very helpful in understanding the tunneling properties of the NIS tunnel junction, and its utility will come to full expression in the next section. 2.1.5 BCS Density of States The BCS theory gives a density of states of DS(E) = DS(0) |E|√ E2 −∆2 (2.1.5.1) where DS(0) is the density of states “two-spin“ at the Fermi level when the supercon- ductor is in its normal state (above Tc). Figure 2.2a shows the normalized BCS density of states. The number of occupied states can be found by multiplying the Fermi distribution by the density of states. Figure 2.2b shows the quasiparticle occupation of a superconductor at a finite temperature below Tc. 12 Chapter 2. Superconductivity −2 −1 0 1 2 0 2 4 6 8 10 E/∆ D oS / D S (0 ) ∆ ∆ (a) −2 −1 0 1 2 0 0.5 1 1.5 2 2.5 E/∆ su pe rc on du ct or o cc up at io n / D S (0 ) ∆ ∆ E F (b) Figure 2.2 – (a) The normalized BCS density of states. (b) Quasiparticle occupation. There are no quasiparticles allowed in the gap. The lack of states in the regionEF < E < EF +∆ is compensated by a singularity at the density of states at the gap edges at energy E = EF ±∆ and the DOS diverges. This leads to the non-linear NIS IV curve, as we will see later. 2.1.6 Length Scales of Superconductivity The following are the characteristic lengths of a superconductor • London penetration depth λL: A superconductor excludes the magnetic field. The magnetic field falls off exponentially inside the superconductor. The penetration depth characterizes the distance over which the magnetic field decays to 1/e of its value at the surface of the superconductor. For typical superconductors, λL ranges from 50 to 500 nm. • Coherence length ζ0: It is the distance over which the correlation between the two electrons forming a cooper pair persists. It could be thought of as the size of the cooper pair. ζ0 typically ranges from 5 to 100 nm. We will see in Chapter 4 that these characteristic lengths, along with others, deter- mine the method by which we shall calculate the losses in the superconductor. 2.2 The NIS Tunnel Junction A Normal-Insulator-Superconductor tunnel junction (NIS) consists of a very thin layer of insulator (few nm) sandwiched between two electrodes. One electrode is a normal metal, while the other electrode is a superconductor. In this case, tunneling of electrons 2.2. The NIS Tunnel Junction 13 between the two electrodes is possible. Tunneling is a quantum mechanical process in which the classically impossible penetration of potential barrier is allowed. Tunneling in superconductors was first demonstrated by in 1960 Giaever [15], who was awarded the Nobel Prize in 1973. In this section, we will briefly outline the basic theoretical concepts of NIS tunneling and how it works as a microrefregirator. 2.2.1 NIS Junction as an Electron Cooler In this subsection we will show how an NIS tunnel junction can be used as an electron microrefregirator by removing power from the normal metal electrode. This technique was first demonstrated in 1994 [34, 40]. Figure 2.3 shows the structure of an unbiased NIS junction along with its energy band diagram. N I S (a) EF ∆ ∆ no states available states available states occupied states occupied states N I S (b) Figure 2.3 – The unbiased NIS junction. (a) Structure. (b) Energy band diagram. For zero and low biases, the occupied states of the normal metal lie within the energy gap of the superconductor. Electrons in these states cannot tunnel because there are no available states in the superconductor. However, for biases near Vb = ∆/e, as shown if Figure 2.4, the occupied states in the Fermi distribution tail in the normal metal are above the gap edge (have energy greater than∆) and can contribute to a tunneling current. A further small increase in the bias voltage will cause dramatic increase in the tunneling current since there is a large number of available states directly above the gap. 14 Chapter 2. Superconductivity N I S -V I e- (a) available states EF ∆ ∆ no states available states occupied states occupied states N I S -- --EF eVb (b) Figure 2.4 – The biased NIS junction. (a) Structure. (b) Energy band diagram. Therefore, a NIS junction extracts electrons having energy higher than the Fermi en- ergy (hot electrons) from the normal metal. The removal of those high-energy electrons is corresponds to cooling the electrons in the normal metal electrode. Temperatures below 100 mK are achievable. 2.2.2 NIS Tunneling Current As mentioned before, NIS junctions have a dependence of current on the temperature of the normal electrode. The temperature rise of the normal electrode is measured from this current temperature dependence. To find an expression of the NIS tunneling current, we first start by evaluating the tunneling rates of charge carriers. The tunneling rate from the normal electrode to the superconducting electrode ΓN→S is ΓN→S(Vb, TN , TS) = TDN(E − eVb)fN(E − eVb)DS(E)(1− fS(E)) (2.2.2.1) whereDN(E) is the density of states of the normal metal,DS(E) is the superconductor density of states and is given by Equation 2.1.5.1, and T is the tunneling rate probability. Similarly, the tunneling rate from the superconductor to the normal metal ΓS→N is ΓS→N(Vb, TN , TS) = TDS(E)fS(E)DN(E − eVb)(1− fN(E − eVb)) (2.2.2.2) 2.3. Superconductors at High Frequencies 15 The current associated with one tunneling event is the difference between the two tunneling rates multiplied by the electron charge e. To get the total current we then integrate over all energies. This gives a current of I(Vb, TN , TS) = e ∫ ∞ −∞ (ΓN→S − ΓS→N) dE = e e2Rn ∫ ∞ −∞ [fN(E − eVb)− fS(E)] ( |E|√ E2 −∆2 ) dE (2.2.2.3) where we have taken the density of states of both the normal metal and the superconduc- tor to be constants and equal the density of states around the Fermi Level (DN(E) ≈ DN(0), DN(E) ≈ DN(0)). This is a valid assumption because the density of states in metals varies slowly with energy, and for small temperatures, we can take it as a constant. We have also identified TDN(0)DS(0) = 1/(e2Rn), which comes from the requirement that Ohm’s law, Vb = IRn, is satisfied for large bias voltage where the gap is insignificant and the current is linear with bias voltage. Rn is the normal state resistance of the junction (see Figure 2.5). Figure 2.5 shown IV curves calculated using Equation 2.2.2.3 for different temperatures . −2 −1 0 1 2 −4 −3 −2 −1 0 1 2 3 4 x 10 −7 eV b /∆ I ( A ) T=0.01T c T=0.2T c T=0.5T c T>T c (a) 0 0.5 1 1.5 2 10 −20 10 −15 10 −10 10 −5 eV b /∆ I ( A ) T=0.01T c T=0.2T c T=0.5T c T>T c (b) Figure 2.5 – NIS IV curves for different temperatures on both linear and logarithmic scales. ∆ = 174µeV is used to generate these curves with no gap smearing. For T > Tc, we get a linear IV curve with slope of Rn. At much larger biases (eVb >> ∆), the energy gap is nearly invisible and we return back to the linear IV curve. 2.3 Superconductors at High Frequencies At zero frequency (DC current), the resistance of the superconductor is zero, and the current does not dissipate power. However, at finite frequencies, the superconductor 16 Chapter 2. Superconductivity resistance is not zero and there is power dissipation. It is important to account for this power loss in device modeling to get the required performance. The easiest way to understand why a superconductor dissipates power at higher that zero frequencies is to consider the two-fluid model of the superconductors, which is briefly explained in the subsection to follow. 2.3.1 The Two-Fluid Model The two-fluid model of superconductors is a pre-BCS simple model that models the free electrons in a superconductor as two fluids in parallel, “Normal” and ”Super” flu- ids. “Normal” electrons have a number density of nn and they act exactly as normal electrons in metals. They are scattered by lattice impurities, and they carry a dissipative current. Under applied electric field E, those normal electrons acquire a mean drift ve- locity ⟨Vn⟩ = −eτ m E due to scattering, where e is the electron charge, τ is the scattering mean free time, andm is the electron mass. The resulting current density Jn is Jn = −nne ⟨Vn⟩ = nne 2τ m E = σ1E (2.3.1.1) where σ is the conductivity. The conductivity is real, therefore, there is a resistance R associated with this fluid. The second fluid consists of “Super” electrons with number density ns. They are not scattered by the impurities, and carry non-dissipative super- current. Since there is no scattering, those electrons are freely accelerated under an external electric filed E with velocity Vs. The equation of motion is m dVs dt = −eE (2.3.1.2) assuming a sinusoidal electric field E = E0e iωt and solving we get Vs = −eE iωm (2.3.1.3) and the current density Js is Js = −nseVs = nse 2τ imω E = σ2E (2.3.1.4) The conductivity σ2 is imaginary, therefore, there is an inductance L associated with this “Super” fluid, known as kinetic inductance. Figure 2.6 depicts the two fluid model representation of a superconductor based on the aforementioned arguments. 2.3. Superconductors at High Frequencies 17 σ 1 , R σ 2 , L Figure 2.6 – The two-fluid model of a superconductor. According to Equation 2.3.1.4, σ2 is inversely proportional to the frequency. There- fore, at DC currents, the resistor R is shorted by the inductance L since it has zero impedance and all the current goes through the super fluid, dissipating no power. On the other hand, at higher frequencies, the impedance of the super fluid begins to increase and part of the current goes through the resistor, dissipating some power. 2.3.2 The Concept of Surface Impedance For a conductor of finite conductivity σ in an electromagnetic fields, solving Maxwell’s equations tells us that the intensity of the field inside the conductor decrease exponen- tially with the distance from the metal surface. The current flows through a thin sheet on the surface, and at a distance δ known as the skip depth, the intensity of the field decays to 1/e of its value at the surface. The skin depth is given by δ = √ 2 ωµσ (2.3.2.1) where ω is the angular frequency of the current, and µ is the magnetic permeability of the conductor. Therefore, for an ideal conductor of infinite conductivity, the electric field is com- pletely excluded from the interior of the conductor. On the other hand, the electromagnetic boundary conditions applied to the interface between two media (see for example [50]) tells us that the tangential component of the electric field in the two media should be equal at the interface. Hence, the tangential component of the electric field on the surface of an ideal conductor is always zero since the electric field inside a perfect conductor is zero. Whereas the tangential component of the electric field on the surface of a non-ideal conductor of finite conductivity is not zero since some field exists inside it. It follows that the simulation of real conductors would be a time consuming process since it requires solving Maxwell’s equations inside the conductor. The concept of surface impedance helps to avoid the complexity of solvingMaxwell’s equations inside the conductors. The surface impedance Zs is defined as the ratio be- 18 Chapter 2. Superconductivity tween the tangential components of the electric field E and the tangential component of the magnetic field H on the surface of the conductor Zs = Et Ht (2.3.2.2) The surface impedance therefore acts as a boundary condition for fields on the sur- face of the metal. By using surface impedance, the dissipation and energy stored in the superconductor are accounted for in the simulation. The two-fluid model could be used as a first-order approximation for the surface impedance because it is simple and intuitive. However, it does not consider the en- ergy gap, does not account for the temperature dependence of the penetration depth correctly. Hence, it is inaccurate. In Chapter 4, it will be presented how to get this sur- face impedance using the more accurate BCS Theory, while in Chapter 5, this surface impedance will be used in simulations. One may argues that this model might not be useful because for devices carrying AC current, the electromagnetic wave is not normally incident on the superconductor, instead it travels along it. However, Matick [36] showed that the analysis of wave propa- gation parallel to the conductor yields the same surface impedance as the normal incident wave. Chapter 3 Cold-Electron Bolometer 3.1 Bolometers 3.1.1 What Is a Bolometer Bolometers are direct detectors of infrared and millimeter waves. They are known to be the most sensitive and broadband detectors in this range of electromagnetic spec- trum. They are widely used in space-based astrophysical observations, since the early universe radiates strongly in this region of spectrum. To illustrate how a bolometer oper- ate, consider the simple thermal model in Figure 3.1. A bolometer consists of a radiation absorptive element of heat capacity C, like a thin layer of metal, connected to a body of constant temperature (thermal reservoir) through a thermal link of a small thermal conductance G. Thermal Conductance G Reservoir Tbath Absorber Incoming Radiation C Thermometer Figure 3.1 – A simple thermal model of a bolometer. The absorber absorbs the photons of the impinging radiation and converts their en- 19 20 Chapter 3. Cold-Electron Bolometer ergy to heat, leading to raising the temperature of the absorber above that of the reservoir. The absorber temperature change is proportional to the absorbed power; therefore mea- suring this temperature difference one can find the incident power. The temperature change can be measured by the attached thermometer. The thermometer is a device that has a temperature-dependent property that can be measured. Examples of those prop- erties are the resistance of a simple resistor, the resistance of a superconductor near its critical temperature, and the tunneling current in a tunnel junction. For the highest possible sensitivity, such detectors need to be cooled to cryogenic temperatures (below 1 K). Thermal noise decreases with decreasing the operating tem- perature of the bolometer. To be able to achieve the desired scientific goals, arrays of 103–105 pixels are required. 3.1.2 Bolometer Technologies The most sensitive detectors for cosmic applications in far-infrared are the cryogenic bolometers operating typically at a temperature of about 100 mK. The noise properties of those devices are considerably improved by decreasing the operating temperature. In this section, different types of cryogenic bolometers are briefly explained. Semiconductor Bolometers Semiconductor bolometers constitute the classical bolometer technology. They have been around for 20 years. They exploit the fact that the resistance of a semiconductor has an exponential dependence on temperature. The disadvantages of this technology include a small amplifier noise margin and a very difficult interface between the bolometers at 1000mK and the amplifiers at 100K [51]. They are rather difficult to fabricate and at low temperature they possess a very large time constant because of low temperature coefficient. Kinetic Inductance Detector (KID) This device is based on the modification of the inductance of a superconductor (called kinetic inductance) when it absorbs photons [38]. KID is very sensitive when cooled down to 100 mK, however, it is strongly influenced also by stray cosmic rays. Hot-Electron Bolometers (HEB) At low temperatures (less than 1K), the coupling between electrons and phonons is weak. This means that electrons are thermally isolated from phonons and can have a tempera- ture different from that of the surrounding phonons bath. This thermal decoupling results 3.2. CEB Device Structure and Principles of Operation 21 in large temperature rise of the electrons for a small absorbed power (increased sensitiv- ity). Those heated electrons are known as “hot electrons” [41]. Therefore for HEB, the electronic heat capacity is the relevant heat capacity, while the electron-phonon thermal conductance is the relevant thermal conductance. Transition Edge Sensor (TES) Transition Edge Sensor (TES) [32] exploits the fact that for a superconductor near its Tc, the change of the resistance is very sharp. TES bolometers have enhanced linear- ity, speed, and stability due to strong negative electrothermal feedback. The amplifiers are also superconducting, which mean they dissipate very small amount of power and produce very little noise. Its fabrication is simple, just thin-film deposition and optical lithography. One drawback of TES bolometers is that they operate at the temperature of the superconductor-normal transition. This fixed operation temperature makes the bolometers less flexible since it restricts the dynamic range. Cold-Electron Bolometers Like HEB bolometers, Cold-Electron Bolometer (CEB) [27, 28] make use of the weak electron-phonon at low temperatures, but they have no restrictions on the operating tem- perature. CEB also has the additional advantage of electron cooling thanks to NIS tunnel junction. This cooling feature results in high sensitivity, decreased noise, and high dy- namic range. The Cold-Electron Bolometer is the topic of the next section. 3.2 CEB Device Structure and Principles of Operation TheResonant CEB device considered in this work [29] is shown in Figure 3.2. It consists of two small normal metal strips acting as an absorber and forming the normal electrodes of two NIS tunnel junctions. The superconducting electrodes of the NIS junctions could be connected to the antenna or could acts themselves as an antenna. The antenna is needed to couple the radiation to the bolometer since the device is much smaller than the operating wavelength. The NIS junctions (see section 2.2) capacitively couple the RF power from the an- tenna to the absorber. Upon absorping this power, the electrons system of the absorber gets hotter. When the NIS junction is biased just below the gap voltage of the superconductor, the hottest and most energetic electrons at the tail of the Fermi distribution in the absorber rapidly tunnel through the NIS junctions, giving an increase in the bias voltage if the junctions are current biased, or an increase in the bias current if the junctions are voltage 22 Chapter 3. Cold-Electron Bolometer S Antenna Antenna S Absorber Superconducting NbN Strip -- Rabs CNIS Lkin NIS JunctionIb Ib Figure 3.2 – Schematic of the Resonant Cold-Electron Bolometer. The NIS junctions capcitively couples the absorber to the antenna. The NbN strip is a part of the absorber. The resistance of the absorber, the capacitance of the NIS junction, and the kinetic inductance of the NbN strip forms an RLC filter to select the frequency band of operation. biased. Thus, the increase in temperature is detected by the NIS junction. Due to the tunneling of those electrons, the absorber gets cooled. The role of the NIS junction is four-fold. It is used for capacitive RF coupling to the antenna, for thermal isolation of the absorber, for electron cooling, and as a thermometer as it experiences a current change if it is voltage biased, or voltage change if it is current biased. As will be shown in subsection 4.1.1, The NIS cooling power equation is similar for positive and negative bias voltages. It follows that the heat flows out of the normal metal regardless of the sign of the bias voltage. This feature can be exploited by connecting two NIS junctions in series such that the normal metal is shared. This combined junc- tion is known as SINIS junction. It was demonstrated experimentally [34] that SINIS structure is an efficient electron cooler. In SINIS junction, the current flows into the structure through one NIS junction, and out through the other one, whereas the heat is removed out of the normal metal absorber through both junctions. Therefore, refriger- ator performance is doubled. On top of that, when compared to single asymmetric NIS junction, SINIS structure provides more efficient thermal isolation of the central metal absorber and is easier to fabricate. The energy level diagram of the device is shown in Figure 3.3 at a voltage bias nearly equal double the tunneling threshold,V ≈ 2×∆/e. 3.2. CEB Device Structure and Principles of Operation 23 available states EF ∆ ∆no states available states occupied states occupied states N IS -- -- eVb / 2 occupied states -- -- EF ∆ ∆ no states eVb / 2 available states available states occupied states -- SI available states occupied states Metal TrapMetal Trap Figure 3.3 – Energy level diagram of the CEB at a voltage bias nearly equal double the tunneling threshold,V ≈ 2×∆/e. Shown also in Figure 3.3 is the metal traps. The power removed from the normal metal is deposited into the superconductor electrode in form of quasiparticles. This power heats up the superconductor and as a result the performance of the refrigerator is degraded. To cool down the superconductor, quasiparticles traps are used. Those quaiparticles traps acst as heatsink for the superconductor electrodes. They are basi- cally normal metals such that quasiparticles in the superconductor can thermalize easily in those normal metal traps. The idea of quasiparticles traps for NIS junctions was in- troduced by Fisher et al. [14]. 3.2.1 Filtering Capability The impedance of the CEB device shown in Figure 3.2 has three components; the resis- tance of the absorberRabs, the capacitance of the twoNIS tunnel junctionsCNIS , and the kinetic inductance of the NbN nanostrip Lkin. Those three components form together an integrated band-defining nano-filter, which obviates the need for the much larger external filters usually used in radio astronomy applications. Because of this resonant feature, the device is called Resonant Cold Electron Bolometer (RCEB) [29]. 24 Chapter 3. Cold-Electron Bolometer 3.2.2 Advantages of CEB In this subsection we summarize the advantages of CEB. The CEB has several advan- tages compared to current cryogenic detectors like TES: • The effective electron cooling by the tunneling current makes the operating tem- perature and noise lower. • High dynamic range. This is true because the absorbed power is removed from the absorber, delaying its saturation. • Easy to fabricate in arrays on planar substrate. • Temperature stability of a cryostat is not so important since it works at tempera- tures far below Tc. • Sensitivity to cosmic rays is considerably less. • Small footprints due to the embedded filtering feature. 3.3 Figures of Merit Figures of merit are those parameters based on which one can characterize and judge the performane of a bolometer. 3.3.1 Responsivity The thermal isolation of the bolometer’s absorber ultimately limits the sensitivity of the device. If the absorber is well isolated from the environment, a small amount of incident power will cause dramatic increase to its temperature. The volume of the absorber also plays an important role. A small absorber volumemeans that it heats upwith the smallest amount of power. The responsivity of the CEB is very high due to the fact that the absorber is isolated by weak electron-phonon coupling and by the NIS junctions, and its small volume. Responsivity is defined as the change in the output (voltage or current) signal per watt of absorbed power of the detected signal. In voltage biased mode, it is given by SV−bias(V ) = ∂I/∂P , and in current biased mode is given by SI−bias(I) = ∂V /∂P . Explicit expressions for CEB responsivity in both voltage and current biased mode are given in next chapter. 3.3. Figures of Merit 25 3.3.2 Noise and Noise Equivalent Power (NEP) Noise properties are characterized by the so-called Noise Equivalent Power (NEP). NEP is defined as the input power to a detector that will produce an output signal having the same power as the intrinsic noise of the device in one hertz output bandwidth [52]. The units of NEP is W/ √ Hz. Two kinds of noise are prevalent in bolometer. The first one is the so-called shot noise which arises from the inherent discrete nature of both electrons and photons. The second kind is the thermal noise, associated with the quantization of energy transfer between different subsystems in the bolometer The noise of the bolometer is the sum of the following four contributions: • The photon noise. It is a shot noise and arises from the intrinsic discrete nature of photons which causes fluctuation of the incident photon flux. • Noise in the absorber. It is a thermal fluctuation noise and associated with the quantization of energy transferred between electrons and phonons. • Noise in the NIS junction due to the discrete nature of the electrons that tunnel “shot noise”. NIS junction has also a thermal noise component that we will speak about later. • Noise in the amplifiers that amplify the detected signal. In the next chapter, each of these terms will be evaluated. The brightness of the cosmic microwave background sets a photon noise level of 1 × 10−17W/ √ Hz at the detector. Since current bolometers technology allows us to have a NEP level lower than that, we have a photon-noise limited detector, which means that the detector is limited by the photon noise of the CMB itself. 3.3.3 Time Constant The time constant τ is a measure of how fast the bolometer respond to the incoming radiation. Referring to the thermal model in Figure 3.1, τ can be written as τ = C/G [52]. Low time constant allows the scanning telescope to have high scan speed “high RPM”. This page intentionally contains only this sentence. Chapter 4 Modeling In this chapter, the modeling of the Cold-Electron Bolometer is presented. The purpose of the device modeling is to extract out the device’s important parameters like cooling power, sensitivity, responsivity, and noise characteristics. The methodology is to try to fit a mathematical model to the experimentally measured IV curves of the device. This is done by solving the so-called heat balance equations. In this chapter also, the modeling of the superconducting niobium microstrip lines at low temperatures and high frequencies will be presented. This is done by solvingMattis- Bardeen Theory in the local limit to get the frequency-dependent surface impedance, which is crucial for designing and simulating the superconducting antenna that is cou- pled to the bolometer. The design and simulation of the antenna itself is treated in the next chapter. Modelling of both the bolometer and the superconducting niobium was carried out using MATLAB (See Appendix B). 4.1 Power Flow in CEB This section introduces the different terms that contribute to the power flow in the device. Then those terms along with the experimentally measured IV curves of the device will be put together in one equation, the heat balance equation, whose solution gives the device parameters. 4.1.1 NIS Cooling Power The power taken out of the normal metal due to tunneling can be evaluated by replacing a factor of e with a factorE−eVb in the NIS tunneling current equation Equation 2.2.2.3, which is the energy removed from the normal metal associated with a single tunneling event. This removed power results in cooling of the electrons system and is the basis of NIS cooling; hence it is called the cooling power. It is given by 27 28 Chapter 4. Modeling Pcool = 1 e2Rn ∫ ∞ −∞ (E − eVb)[fN(E − eVb)− fS(E)] |E|√ E2 −∆2 dE (4.1.1.1) Figure 4.1 shows Pcool calculated numerically for Al for different absorber electrons and superconductor temperatures. −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 eV b /∆ P co ol (p W ) T N =T S =0.1T c T N =T S =0.25T c T N =T S =0.5T c T N =T S =0.6T c Figure 4.1 – NIS cooling power PCool vs normalized biasing voltage for different temperatures for TN = TS . For biases near eVb ≈ ±∆, the cooling power is maximum, meaning that efficient cooling of the absorber electron system is being achieved. The fact that Pcool is even in Vb allows connecting two NIS junctions in series to double the cooling power. The two junctions have to be biased such that they have opposite sign on their voltage biases [42]. This structure is known as SINIS junction. 4.1.2 Electron-Phonon Coupling At low temperatures (less than 1K), the coupling between electrons and phonons is weak. This is true because at low temperatures, the electron-phonon collision rate is low, and the relaxation gets very slow. Phonons in the metal are well-coupled to phonons in the substrate and typically act as a thermal reservoir. This means that electrons are thermally isolated from phonons and can have a temperature different from that of the surrounding 4.1. Power Flow in CEB 29 phonons bath. In CEB, this thermal decoupling result in large temperature rise of the electrons system for a small absorbed power. This also makes significant cooling of the electrons system below the lattice temperature possible. The heat flow from the electrons to the phonons subsystem in the normal metal is given by [43, 53, 57, 58] Pe−p = ΣΩ(T 5 n − T 5 p ) (4.1.2.1) where Σ is a material constant that depends on electron-phonon coupling strength, Ω the volume of the absorber, Tn is the electron temperature in the normal metal, and Tp is the phonons temperature in the substrate. Σ ≈ 1nW/K5/µm3 for most metals. As a result of lattice mismatch between the absorber and the substrate, the phonons will be scattered at the interface resulting in thermal resistance, known as Kapitza acous- tic mismatch. Therefore, phonon systems in the absorber and in the substrate are two separate systems and can have different temperatures. However, at the low temper- atures, the wavelength of the thermal phonons is much larger than the thickness of a typical thin film of the absorber. Hence the interface should be quite transparent to these phonons and those two subsystems are considered as one system. The Kapitza resistance will be neglected throughout [42]. 4.1.3 Power Dissipated in the Superconductor Power is deposited in the superconductor in the form of quasiparticle excitations. Both the power removed from the normal-metal absorberPcool and the IV power are deposited in the superconductor. Hence the power deposited in the superconductor, PS , can be calculated from the relation PS = Pcool + IVb (4.1.3.1) Using Equation 2.2.2.3 and Equation 4.1.1.1, we arrive to the following expression for PS PS = 1 e2Rn ∫ ∞ −∞ E[fN(E − eVb)− fS(E)] |E|√ E2 −∆2 dE (4.1.3.2) Figure 4.2 shows PS calculated for Al for different absorber electrons temperatures and superconductor temperatures. Heating of the superconductor will decrease the cooling power. Heating of the su- perconductor can be reduced by using quasiparticles traps, which are normal metals. 30 Chapter 4. Modeling −1.5 −1 −0.5 0 0.5 1 1.5 0 5 10 15 20 25 30 35 eV b /∆ P S (p W ) T N =T S =0.1T c T N =T S =0.25T c T N =T S =0.5T c T N =T S =0.6T c Figure 4.2 – NIS PS vs normalized biasing voltage for different temperatures for TN = TS . 4.1.4 Power Back-flow Electrons that tunneled into the superconductor can recombine into Cooper pairs to pro- duce phonons that will heat the normal metal. This can be modeled as a small fraction β of the power deposited in the superconductor PS being returned back to the absorber [14, 55, 56]. This degrades the cooling capability of the NIS junction. In the thermal model presented in this chapter, β is one of the parameters to fit. 4.1.5 Subgap Leakage The DOS does not actually diverge but is smeared out due to the finite lifetime of the thermal electrons. The current-voltage (IV) relationship of actual tunnel junctions devi- ates from BCS theory, showing an extra leakage current when the bias voltage is below the gap (subgap conductance). This can be modeled by adding the so-called Dynes Pa- rameter Γ to the density of states [11]. The Dynes Parameter smears the gap by creating finite states within the gap and decreasing the singularity at the gap-edge. The modified normalized density of states with this parameter is ns(E) = ∣∣∣∣∣ℜ ( E − iΓ√ (E − iΓ)2 −∆2 )∣∣∣∣∣ 4.1. Power Flow in CEB 31 Figure 4.3 shows the modified BCS density of states with Dynes smearing. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 10 2 E/∆ D oS Γ/∆=10−10 Γ/∆=10−8 Γ/∆=10−6 Γ/∆=10−4 Figure 4.3 – Modified BCS density of states. 4.1.6 Joule Heating The current passing through the normal metal absorber causes resistive heating and the associated dissipated power in the absorberPjoule is simply given byOhm’s lawPjoule = I2Rabs, where Rabs is the resistance of the absorber. 4.1.7 Bringing It All Together: Heat Balance Equations The equation that summarizes all the pervious power terms together is what is called the heat balance equation. However, we have two systems: the normal electrode “the ab- sorber” and the superconducting electrode. Therefore, we have two heat balance equa- tions. Two complications arise, firstly those equations are transcendental equations so they have to be solved numerically, and secondly they are coupled equations so they have to be solved simultaneously. The next subsections deals with those two equations and their solution. Heat Balance Equation in Normal Electrode The incident power P0 to be measured heats the absorber along with other terms, while NIS tunneling and electron-phonon coupling cools the absorber. The heat balance equa- 32 Chapter 4. Modeling tion for the normal metal absorber is 2Pcool(Vb, Tn, Ts) + Pe−p,n(Tn, Tp) = Pjoule(Vb, Tn, Ts) + 2βPS(Vb, Tn, Ts) + P0 (4.1.7.1) The right-hand side describes the terms that causes heating, while the left-hand side describes the terms of cooling. Most of the terms are functions of the bias voltage Vb, the normal electrode temperature Tn, the quasiparticles temperature in the superconducting electrode Ts. Given Vb , Tp, and Ts one can solve for Tn, however, Ts itself is variable because the superconductor is often heated. Tn is related to Tn through another heat balance equation, which is the topic of the next subsection. Heat Balance Equation in Superconductor The heat balance equation in the superconductor is Pe−p,s(Ts, Tp) = (1− β)PS(Vb, Tn, Ts) (4.1.7.2) It simply says that part of the power that enters the superconductor, (1− β)PS , gets transferred to the phonon system of the superconductor through electron-phonon cou- pling. The factor of two signifies the fact that we have two NIS junctions. 4.2 Parameters Fitting The two aforementioned heat balance equations are both functions of the electron tem- perature in the normal metal Tn and the quasiparticles temperature in the superconductor Ts. Solving these two equations simultaneously for many values of Vb tells us how the junction performs at different bias voltages. One therefore can choose the value of Vb that minimized Tn and hence maximizes cooling. The layout of the measured device is shown in Figure 4.4. It consists of twelve bolometers connected in series. The measure- ment setup is described in Appendix A. In order to accurately fit the measured data, we require knowledge of all material parameters of the Aluminum used to fabricate the superconducting electrodes (transition temperature, gap ratio). Such measurements are not currently available. The transition temperature is different for thin film Al than the bulk Al. For a film of thickness of 70µm is 1.62 K [8] vs 1.2 for bulk. The gap ratio 2∆/KBTc is 3.37 according to [39] and 3.25 according to[5]. We use an average value between those two values. The 4.2. Parameters Fitting 33 (a) NIS junction Absorber (b) Figure 4.4 – AutoCAD layout of the measured device. (a) Twelve bolometers are connected in series. Every arm of the cross-slot antenna contains three bolometers. (b) Zoom in to one arm of the cross slot showing three bolometers. The absorber consists of two metal strips in parallel to lower the resistance. electron-phonon coupling strength parameter Σ in the normal metal is 1.1nW/µm3K5 [34, 40]. To solve those two equations, I assume initially that Ts = 0.3K then solve the first heat balance equation for Tn, then use this value of Tn to solve the second heat balance equation for Ts. This process is repeated four times to reach a self-consistent solution. I use the MATLAB function fzero to slove the equations numerically. See Appendix B.1 for the corresponding MATLAB code. After solving for Tn and Ts for a range of Vb, one can use those values to theoretically construct the IV curve of the junction using Equation 2.2.2.3. Those curves can then be compared to the experimentally measured IV curves of a given junction. The purpose of comparing the theoretically calculated IV curves to the experimen- tally measured ones is to deduce the parameters that cannot be measured experimentally. Those parameters are • Γ, the Dynes smearing parameter, see subsection 4.1.5. • β, the fraction of the power returned to the absorber, see subsection 4.1.4. • ΩS , the volume of the superconductor 1. 1Metal quasiparticles trap changes the effective volume of the superconductor electrode. 34 Chapter 4. Modeling • P0, the incident power 2. The strategy is to vary those parameters till the best fit is obtained. To make this process less time consuming, less error prone, and more efficient, one has to be able to make educated guesses while changing the parameters. Therefore, understanding the effect of each parameter individually on the IV curve is crucial. Figure 4.5 shows the effect of each of the four parameters on the IV curve, while the other three parameters are held constant. 0 0.2 0.4 0.6 0.8 1 10 −11 10 −10 10 −9 10 −8 eV b /∆ I( A ) Γ = 2x10−3 Γ = 4x10−3 Γ = 6x10−3 Γ = 8x10−3 Γ = 1x10−2 measured (a) 0 0.2 0.4 0.6 0.8 1 10 −11 10 −10 10 −9 10 −8 eV b /∆ I( A ) β = 0 β = 0.1 β = 0.2 β = 0.3 β = 0.4 β = 0.5 measured (b) 0 0.2 0.4 0.6 0.8 1 10 −11 10 −10 10 −9 10 −8 eV b /∆ I( A ) Ω S = 0.1 µm3 Ω S = 0.5 µm3 Ω S = 1 µm3 Ω S = 2 µm3 measured (c) 0 0.2 0.4 0.6 0.8 1 10 -11 10 -10 10 -9 10 -8 eV b /∆ I( A ) P0 = 0 pW P0 = 0.2 pW P0 = 0.4 pW P0 = 0.6 pW P0 = 0.8 pW P0 = 1 pW measured (d) Figure 4.5 – Investigating the effect of different fitting parameters on IV curve. (a) Effect of varying γ. (b) Effect of varying β. (c) Effect of varying the volume of the superconductor. (d) Effect of varying the incident power. After choosing the initial set of fitting parameters. I vary those parameters slightly around their initial values to obtain more fine-tuned fit. See Appendix A.2 for the cor- responding MATLAB code. Table Table 4.1 shows the final values of the fitting param- eters. 2In our measurement setup, the source of the power is a black body with known temperature and power, however, the power that the bolometer absorbs is unknown due to reflections. 4.2. Parameters Fitting 35 Table 4.1 – Fitted parameters values. Parameter Value Γ 6× 10−3 β 0.15 V olSC(µm 3) 2 Pbg(pW ) 0.2 Figure 4.6 shows the electron temperature Tn as a function of bias voltage. One finds that Tn is significantly lower than the substrate phonon temperature Tp, which is the effect of NIS electron cooling. The cooling is maximized near Vb ≈ ∆/e. 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 eV b /∆ E le ct ro n T em pe ra tu re ( K ) Figure 4.6 – Temperature of electrons in the normal metal as a finction of bias voltage. The temperature is lower than the phonon temperature (Tp = 0.3K), indicating cooling of electrons. Cooling is maximum near Vb ≈ ∆/e. Figure 4.7 shows the result of the fitted IV curve. 36 Chapter 4. Modeling −2 −1 0 1 2 −200 −150 −100 −50 0 50 100 150 200 eV b /∆ I ( nA ) Experimental Data Theoretical Fit (a) 0 0.5 1 1.5 2 10 −4 10 −2 10 0 10 2 10 4 eV b /∆ I ( nA ) Experimental Data Theoretical Fit (b) Figure 4.7 – Fitted IV curve on both linear and logarithmic scales. 4.3 Responsivity Low frequency responsivity in both voltage-biased mode SV−bias and in current-biased mode SI−bias at low frequencies are given by the following two expressions respectively SV−bias(V ) = ∂I ∂P = ∂I ∂T 5ΣΩT 4 e + ∂P ∂T (4.3.0.3) SI−bias(I) = ∂V ∂P = − ∂I/∂T ∂I/∂V 5ΣΩT 4 e + ∂P ∂T − ∂I/∂T ∂I/∂V ∂P ∂V (4.3.0.4) Figure 4.8 shows how the responsivity changes with changing the resistance of the absorber for both voltage biased and current biased modes of operation. 0 0.5 1 1.5 −5 0 5 10 15 20 25 30 35 40 eV/∆ S V − bi as ( nA /p W ) R=1 KΩ R=5 KΩ R=10 KΩ (a) 0 0.5 1 1.5 −80 −70 −60 −50 −40 −30 −20 −10 0 10 eV/∆ S I− bi as ( µV /p W ) R=1 KΩ R=5 KΩ R=10 KΩ (b) Figure 4.8 – Responsivity for different absorber resistance values. (a) In voltage biased mode. (b) In current biased mode. 4.4. Noise Model 37 Having solved for temperatures Tn and Ts, we can evaluate the responsivity. Heat is relaxed through two mechanisms, through electron-phonon interaction, and through electron tunneling in the NIS junction. Therefore, in the expression for the voltage biased responsivity, the sum of the two heat conductancesGe−ph = 5ΣΩT 4 e andGNIS = ∂P ∂T appears in the denominator. To make the evaluation of the responsivities easier, analytical expressions for the dif- ferentials was obtained first using differentiation under integration principle. Appendix B.3 shows the responsivities functions. 4.4 Noise Model Aswas mentioned in subsection 3.3.2, the noise characteristics of the bolometer consists of four contributions. This section is devoted to the evaluation of these terms. Here we follow the noise model presented in [16]. The corresponding MATLAB code is in Appendix B.3. 4.4.1 Photon Noise As was mentioned in the Introduction, CMB spectrum follows a perfect black body distribution at a temperature of 2.725K. Therefore the spectral density of radiation can be obtained from Planck’s distribution and then can be used to calculate the noise in radiation received by the bolometer. An approximate result for the NEP due to photon fluctuation is [7] NEP 2 photon = 2KBTb(hf0 +KBTb)BW (4.4.1.1) where Tb is the black body temperature and equals 2.725K for CMB, f0 is the center frequency of the received band, and BW is the bandwidth of operation. As will be shown in the next chapter, the bolometer considered in this work operates at a frequency of nearly 100 GHz and a bandwidth of 15 GHz. Plugging the numbers givesNEPphoton = 1× 10−17W/ √ Hz. 4.4.2 Noise in the Absorber This contribution to noise is due to the quantization of the heat flow between electrons and phonons, and also known as thermal fluctuation noise or phonon noise. Phonon noise in systems where electrons have different temperature than phonons was studied in [17]. A simplified version was presented in [16] and the results is 38 Chapter 4. Modeling NEP 2 e−p = 10KBΣΩ(T 6 e + T 6 p ) (4.4.2.1) where Te and Tp are the temperature of the electrons and phonons respectively. 4.4.3 NIS Junction Noise Part of this noise is associated with electrons that tunnel through the junction. Also every electron that tunnel carries energy, which introduces a thermal noise. These two noises are correlated. For voltage-biased mode, the NIS junction noise is given by NEP 2 NISV −bias = ⟨δI2⟩ S2 I (V ) + ⟨δP 2⟩ − 2 ⟨δPδI⟩ SI(V ) (4.4.3.1) and in current-biased mode by NEP 2 NISI−bias = ⟨δI2⟩ S2 V (I) ( ∂V ∂I )2 + ⟨δP 2⟩ − 2 ⟨δPδI⟩ SV (I) ( ∂V ∂I ) (4.4.3.2) where ⟨δI2⟩ is the spectral density of the shot noise due to current fluctuations, ⟨δP 2⟩ is the spectral density of heat flow noise, and ⟨δP 2δI2⟩ is the spectral density of the correlator of those two noises. 4.4.4 Amplifier Noise The noise due to the readout amplifier in voltage biased mode is given by NEP 2 amp = ⟨δI2⟩amp S2 V−bias(V ) (4.4.4.1) where ⟨δI2⟩amp is the current sensitivity of the current amplifier, and in current biased mode is given by NEP 2 amp = ⟨δV 2⟩amp S2 I−bias(V ) (4.4.4.2) where ⟨δV 2⟩amp is the voltage sensitivity of the voltage amplifier. 4.4.5 Total Noise The total noise is now given by NEPtotal = NEP 2 e−p +NEP 2 NIS +NEP 2 amp (4.4.5.1) 4.5. Time Constant 39 Figure 4.9 shows the components of the NEP for both voltage biased and current biased modes for absorber resistance of Rabs = 1KΩ. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10 −17 eV b /∆ N E P V − bi as (W /H z1/ 2 ) NEP NIS NEP e−p NEP AMP Total NEP (a) 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10 −16 eV b /∆ N E P I− bi as (W /H z1/ 2 ) NEP NIS NEP e−p NEP AMP Total NEP (b) Figure 4.9 – Noise contributions for R=1KΩ. (a) In voltage biased mode, assuming an amplifier noise of 0.05pA/ √ Hz. (b) In current biased mode, assuming an amplifier noise of 3nV / √ Hz. 4.5 Time Constant As mentioned in subsection 3.3.3, the time constant is given by τ = C/G, where C is the heat capacity of the absorber, and G is the thermal conductance. Heat is relaxed through two paths; one path is the electron-phonon coupling and the other is the NIS junction through tunneling. Hence, G can be written as G = Ge−ph +GNIS , and τ can be written as τ−1 = τ−1 e−ph + τ−1 NIS . For the voltage biased operation, the time constant is therefore τ−1 V−bias = 1 cvΩ ( 5ΣΩT 4 e + ∂P ∂T ) (4.5.0.2) and for the current biased is τ−1 I−bias = 1 cvΩ ( 5ΣΩT 4 e + ∂P ∂T − ∂I/∂T ∂I/∂V ∂P ∂V ) (4.5.0.3) The heat capacity is related to the electron temperature by cv = γTe, where γ is a material constant. The time constant for both modes of operations is shown in Figure 4.10. 40 Chapter 4. Modeling 0 0.5 1 1.5 0 0.5 1 1.5 2 eV b /∆ T im e co ns ta nt τ ( µs ) τ e−ph τ NIS τ total (a) 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 eV b /∆ T im e co ns ta nt τ ( µs ) τ e−ph τ NIS τ total (b) Figure 4.10 – Time constant of the bolometer for an absorber volume of 0.05µm3, absorber resistance of 100 Ω and γ = 9.77 × 10−17J/µm3K2. (a) For the voltage biased mode of operation. (b) For the current biased mode of operation. 4.6 Microwave Absorption and Optical Conductivity of a Superconductor Understanding the characteristics of superconductors at high frequencies is crucial for designing devices out of them. In this section, we will introduce the anomalous skin effect, which occurs at high frequencies and very low temperatures, where Ohm’s law is no longer valid. We will also calculate the surface impedance of superconducting niobium by solving Mattis-Bardeen equation in the extreme anomalous limit. Niobium will be used for microstrip lines, as will be shown in next chapter. 4.6.1 Nonlocality and The Anomalous Skin Effect Consider a normal metal in an external field E. The celebrated Ohm’s law J = σE, where J is the current density and σ is the complex conductivity, is valid as long as the penetration depth is much larger than the electron mean free path, i.e. δ >> l. In this case the electric filed that the electrons see between scattering events does not vary significantly and this yields a linear relationship between J and E. This is known as local limit. However, δ decreases with increasing frequency (see Equation 2.3.2.1), and l in- creases with going to lower temperatures 3. In this case σ is of the same order of magni- tude of l, and the electric field varies rapidly between scattering events and the velocities of the electrons is now dependent on the previous states. Therefore, at high frequencies 3Low temperature means reduced lattice vibrations, which in turn causes a longer mean free path. 4.6. Microwave Absorption and Optical Conductivity of a Superconductor 41 and/or low temperatures Ohm’s law stops to be valid, and the current density at a given point no longer depends on the electric field at this point, but it depends on the electric fields in the surrounding volume as well. This non-local relationship between J and E gives rise to what is called anomalous skin effect. At even higher frequencies, the phenomenon is named extreme anomalous effect. Due to the anomalous skin effect phenomenon, the surface impedance is much higher than that obtained by applying the normal skin effect theory at higher frequencies and low temperatures. In the case of superconductors we replace the skin depth δ by the Lon- don penetration depthλL, and themean free path l by the coherence length ζ0. Therefore, when ζ0 > λL, the local J-E relation fails. This analogy was first realized by Pippared [45]. Just one year after the debut of the BCS Theory, Mattis and Bardeen [37] used the BCS theory to derive an expression that relates the current density J(r) at any point r to the magnetic vector potential A(r) of a superconductor. This expression is general. It takes into account the superconductivity gap and the non-local electrodynamics of the superconductor. It is valid for all temperatures < Tc and for all frequencies even those higher than the gap energy ~ω > 2∆. Mattis-Bardeen Theory allows calculating the complex surface impedance of a thin film as done by Pöpel [46], however, those equations are cumbersome. Fortunately, approximations are possible. Pöpel found that if the film thickness is larger than the penetration depth by at least three times, then bulk limit is a very good approximation. Those calculations can be simplified further when considering Mattis-Bardeen Theory in the extreme anomalous limit. For Nb at 300 mK, λL is roughly 50 µm, and for high quality films, l is in the range of 100-1000 µm, therefore the extreme anomalous limit is applicable here. For this approximation, it is convenient to introduce a complex conductivity σs = σ1 − iσ2, which is a function of temperature and temperature, and is given by [46] σ1 σn = 2 ~ω ∫ ∞ ∆ [f(E)− f(E + ~ω]g(E) dE + 1 ~ω ∫ −∆ ∆−~ω [1− 2f(E + ~ω]g(E) dE (4.6.1.1) σ2 σn = 1 ~ω ∫ ∆ ∆−~ω,−∆ [1− 2f(E + ~ω)][E2 +∆2 + ~ωE] (∆2 − E2)1/2[(E + ~ω)2 −∆2]1/2 dE (4.6.1.2) where σn is the normal state conductivity, f(E) is the Fermi distribution, and g(E) is given by g(E) = E2 +∆2 + ~ωE (E2 −∆2)1/2[(E + ~ω)2 −∆2]1/2 (4.6.1.3) 42 Chapter 4. Modeling The second integral in Equation 4.6.1.1 is set to zero for sub-gap frequencies ~ω < 2∆. The lower limit on the integral of Equation 4.6.1.2 is−∆ for ~ω > 2∆. Figure 4.11 shows the computed results for σ1 and σ2 for Nb at 500 mK. 50 400 700 1000 1300 10 −15 10 −10 10 −5 10 0 10 5 Frequency (GHz) C om pl ex c on du ct iv ity σ 1 /σ n σ 2 /σ n Figure 4.11 – Calculated complex conductivity of Nb film having a thickness of 400 nm at a temperature of 500 mK. For Nb, Tc=9.2 K and ∆0 = 1.8KBTc. To calculate the surface impedance, the following equation can be used [36] Zs = (iωµ0/σ) 1/2coth[(iωµ0σ) 1/2d] (4.6.1.4) where d is the film thickness, and µ0 is the free space permeability. Figure 4.12 shows the calculated of the real and the imaginary parts of the surface impedance. Nb has a gap energy that corresponds to 700 GHz. Above this frequency, the real part of the sur- face impedance becomes much higher and losses are introduced. Below this frequency losses are negligible; however, the imaginary part of the surface impedance is significant and gives rise to the kinetic inductance of the superconductor. These values of surface impedance will be used in simulating the superconductor in electromagnetic simulation programs as described in the next subsection. 4.6. Microwave Absorption and Optical Conductivity of a Superconductor 43 50 400 700 1000 1300 0 0.2 0.4 0.6 0.8 Frequency (GHz) S ur fa ce im pe da nc e Z s ( Ω / s qu ar e) Re(Z s ) Im(Z s ) (a) 50 400 700 1200 10 −4 10 −3 10 −2 10 −1 10 0 Frequency (GHz) S ur fa ce im pe da nc e Z s ( Ω / s qu ar e) Re(Z s ) Im(Z s ) (b) Figure 4.12 – The real and imaginary parts of the complex surface impedance of Nb film of thickness 400nm, and at a temperature of 500 mK. (a) Linear scale. (b) Logarithmic scale. 4.6.2 Simulation Setup in EM Simulators The characteristics of superconducting thin film microstrip lines depend on the surface impedance of the film. Surface impedance calculations usually encompass complicated numerical computations, such as solution of integral equations or numerical integration. Those calculations are not available in commercial electromagnetic software packages, and so those packages do not provide a means for simulating superconductors directly. Superconductors cannot be represented by just a perfect conductors with zero resistance. The kinetic inductance associated with the superconductors will cause a significant de- viation from the expected performance. To obtain accurate results, surface impedances should be computed first then imported into standard microwave programs as a look-up table over the required range of frequencies. AppendixA.4 presents aMATLAB script that is used to generate frequency-dependent surface impedance tables to be used in electromagnetic simulation. It calculates the sur- face impedance of Nb thin film using Mattis-Bardeen Theory in the extreme anomalous limit, as was described in the previous subsection, for given material parameters. In this thesis, two software packages (HFSS [20] and Sonnet em [22]) are used to simulate the superconducting Nb microstrip lines. This page intentionally contains only this sentence. Chapter 5 Antenna Design and Simulation This chapter deals with the design and simulation of the antenna that is coupled to the bolometer. The antenna is needed to couple the radiation to the bolometer since the device is much smaller than the operating wavelength. As we mentioned in the intro- duction chapter, planar antennas solve many of the problems of horn antenna, however, the field of view of a single antenna is wide and it is needed to restrict the field of view of the detectors to reduce pickup of stray radiation and to correctly couple the detector to the telescope lenses and mirrors. Antenna arrays solves this problem since they could produce narrow beams. We will begin this chapter by some antenna definitions that we will regularly use. Then, the requirements of the antenna are presented. After that, the design and simu- lation of the proposed antenna element will be treated. The next natural step, which is making an array out of the antenna element, is then shown. 5.1 Antenna Basic Definitions 5.1.1 Radiation Pattern The radiation pattern of an antenna is defined as the distribution of the radiated electric field intensity over a sphere having the antenna in its center. The field is considered only as a function of two coordinates (θ and ϕ), while the radial distance remains constant. Figure 5.1 shows the coordinate system for pattern analysis. Shown also is a typical 3D radiation pattern. The pattern consists of a main beam in the direction of maximum radiation (called main lobe). There is also some smaller lobes (called side lobes). Cuts through this pattern in various planes are usually used to represent patterns. Those cuts are often made for fixed ϕ values, making the pattern only a function of θ. Those cuts are usually made in both E andH-planes. The E-plane is the plane containing the electric field vector 45 46 Chapter 5. Antenna Design and Simulation y x z Main lobe φ-plane Sidelobes Azimuth plane θ-cone Φ r dΦ r dθ rθ θ Figure 5.1 – Spherical coordinate system for representing an antenna radiation pattern. in the direction of the main beam, while the H-plane is perpendicular to the E-plane and containing the magnetic field vector in the direction of the main beam. At a very large distance r from the antenna 1, the field dependence on distance r and direction (θ, ϕ) is separable and can be expressed as E⃗(r, θ, ϕ) = A e−jk0r r G⃗(θ, ϕ) (5.1.1.1) where A is a constant related to the excitation of the antenna, the factor (1 r ) indicates that field decays with distance from the antenna, k0 is the wave-number in free space, e−jk0r is a phase factor, and G⃗(θ, ϕ) is the element radiation pattern function. Therefore, to construct the field at any point far from the antenna for a given amplitude and phase, we need only to know its direction dependence G⃗(θ, ϕ). 1This region is known as the farfield region, where r >> λ0, where λ0 is the wavelength in free space. 5.1. Antenna Basic Definitions 47 5.1.2 Directivity and BeamWidth Directivity is the antenna characteristic that tells how much it concentrates energy in one direction compared to other directions. Directivity D is defined as the ratio of the power radiated per unit solid angle in a certain direction to the power radiated by an isotropic radiator that radiates an equal amount of power in all directions. Usually the directivity is expressed in units of dBi, where “i” stands for isotropic. For example, if the directivity of an antenna is 3 (4.77 dBi), it receives 3 times more power in its peak direction than an isotropic radiator would receive. For an antenna with the effective area “effective aperture” A, the maximum directiv- ity is given by Dmax = 4π λ2 A (5.1.2.1) One representation of the beam width is what is called Full Width at Half Maximum (FWHM), which is the angular width at which the power is equal to half of its maximum value, as shown in Figure 5.2. 0 -3 dB-3 dB FWHM x y z Back lobe Side lobe (a) θππ /2−π −π /2 0 Main lobe Side lobe Back lobe FWHM (b) Figure 5.2 – Two-dimensional normalized power pattern to illustrate the beamwidth. (a) Polar plot. (b) Linear plot. A measure for how much power is concentrated in the main beam compared to the side lobe is the “side lobe level”. It is defined as the ratio between the maximum pattern value of a side lobe to the maximum pattern value of the main beam, and is usually expressed in dB. 48 Chapter 5. Antenna Design and Simulation 5.1.3 Co-polarization and Cross-polarization As mentioned in subsection 5.1.1, the antenna field has two orthogonal components along θ and ϕ. The co-polarization vector is defined as the preferred electric field vec- tor. The cross-polarization vector is orthogonal to co-polarization vector and represents the undesired polarization. Both the co-polarization vector and the cross-polarization vectors are orthogonal to direction of propagation. The selection of the preferred polar- ization direction is arbitrary, however, we will follow Ludwig’s second definition [35] 2. According to this definition, the co-polarization and cross-polarization components of the field at a point (θ , ϕ ) are defined as Eco(θ, ϕ) = E(θ, ϕ) . ( sinϕ cosθ θ̂ + cosϕ ϕ̂√ 1− sin2θ sin2ϕ ) (5.1.3.1a) Ecross(θ, ϕ) = E(θ, ϕ) . ( cosϕ θ̂ − cosθ sinϕ ϕ̂√ 1− sin2θ sin2ϕ ) (5.1.3.1b) Figure 5.3 shows those two components for easier visualization. z z x x yy Eco Ecross Figure 5.3 – The co-polarization and cross-polarization components according to Ludwig second definition. According to the “IEEE Standard Definition of Terms for Antennas” , we define the cross-polarization level (CPL) as CPL ≡ |Ecross|max |Eco|max (5.1.3.2) The cross polarization level is usually specified in negative dB, indicating howmany decibels the undesired polarization is below the desired polarization. For our detector, 2On the other hand, Ludwig’s third definition has more practical significance since it is easier to measure practictice. However, since we are doing simulations here and no pattern measurements are carried out, we will follow the second definition which is more meaningful. 5.2. Simulation Methods and Software Packages 49 having a low cross-polarization is crucial. 5.1.4 Input Impedance The radiation resistance for an antenna is defined as the resistance that dissipates the same amount of power as the antenna radiated power, if the antenna was replace by this resistance. In this situation, the transmitter sees exactly the same thing either with the antenna or with the radiation resistor. However, while the energy dissipated in an ohmic resistance is converted to heat, the energy dissipated by radiation resistance is converted to electromagnetic radiation. Generally, the input impedance is complex, with the real part (resistance) related to the radiated power, and the imaginary part (reactance) related to the energy stored by fields in the vicinity of the antenna. Both influence the matching between the antenna and the feeding structure, and hence affecting the antenna efficiency. 5.2 Simulation Methods and Software Packages In this thesis, simulations were carried out using a combination of commercial packages (Ansys HFSS [20], CST Microwave Studio [21], Sonnet em [22]). Generally, electromagnetic simulators are divided into two types: (i) three-dimensional solvers, such as HFSS and CST, which divide the structure to three-dimensional mesh, and solve Maxwell’s equations in each mesh element, and (ii) two-dimensional solvers, like Sonnet, which divides the conducting surfaces to two-dimensional mesh and uses method of moment to solve for the currents. Furthermore, the three-dimensional solvers are divided to frequency domain solvers (e.g. HFSS), and time domain solvers (e.g. CST). The advantage of using different solvers is double fold. Firstly, depending on the structure, some solvers are more appropriate than the others. For instance, for planar circuits, 2-D simulators can produce accurate results much faster and with much lower memory requirements than 3-D solvers. Secondly, the use of different packages based on different theoretical methods (time and frequency domain for example) provides a means to judge the quality of simulation results. If the results are in good agreement, then the results are trustable, otherwise, further investigations are necessary. 50 Chapter 5. Antenna Design and Simulation 5.3 Slot Antenna 5.3.1 Rectangular Slot Antenna A rectangular slot is a narrow opening in a large conductive sheet known as the ground plane as shown in Figure 5.4. y z x W L .. P1 P2 Figure 5.4 – Slot antenna on an infinite ground plane. The key motivations for choosing a slot antenna are: • The ground plane isolates the incoming radiation and feeding circuit, which pre- vents the feeding structure from degrading the cross-polarization level. This also allows for independent design procedures for the antenna and the complicated feeding network required in arrays. • Narrow slots are intrinsically polarization sensitive. • cheap and accurate to manufacture. WhenW << λ, the semi-infinite ground planes along the slot length can be consid- ered as a section of two-wire transmission line [13]. Those two lines are being shorted at both ends of the slot. When the slot is excited by connecting a source to points P1 and P2, the excited voltage propagates towards the slot’s short-circuited ends. This wave gets reflected by the slot termination. When the slot length L is odd multiple of half wave length (λ/2), the two waves reflected by both ends create a standing wave on the slot. A current standing wave also exists. Since the slot end is shorted, the voltage at the end is zero, while the current is maximum. The current and voltage standing waves for lambda/2 slot are shown in Figure 5.5. 5.3. Slot Antenna 51 Voltage distribution Current distributionλ/4 (a) (b) Figure 5.5 – (a)Standing voltage and current waves in a slot. (b) Simulated current distribution using Sonnet package. The figure shows two current maximums at both ends of the slot and a current null at the center. The slot can also be conceived as a short waveguide, whith the thickness of the ground plane acts as the length of this wave guide. For narrow (W << λ) half wavelength slot, the E-field distribution over the slot is reasonably approximated by the TE10mode of the rectangular waveguide [24]. For an x-directed slot having the origin of the coordinate system in its center, the slot field becomes Eslot(x, y) = E0 cos (π L x ) ŷ (5.3.1.1) Figure 5.6a shows the E-field distribution on the slot. The E-field can be replaced by an equivalent magnetic currents M 3. M = Eslot × ẑ = E0 cos (π L x ) x̂ (5.3.1.2) The equivalent magnetic current is shown in Figure 5.6b. This fact can be exploited to speed up the simulation of a slot antenna. Instead of solving for the currents in the ground plane, we can j