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Nonlinear Dynamics of Josephson Junction Chains

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Nonlinear Dynamics of Josephson Junction Chains Nonlinear dynamics of Josephson Junction Chains and Superconducting Resonators ADEM ERGUL¨ Doctoral Thesis in Physics Stockholm, Sweden 2013 KTH-Royal Institute of Technology TRITA FYS 2013:52 School of Engineering Sciences ISSN 0280-316X Department of Applied Physics ISRN KTH/FYS/- -13:52- -SE SE-100 44 Stockholm ISBN 978-91-7501-869-0 SWEDEN Akademisk avhandling som med tillst˚and av Kungliga Tekniska H¨ogskolan framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen i fysik torsda- gen den 28 november 2013 klockan 13:00 i sal FA32, AlbaNova Universitetscentrum, Kungliga Tekniska H¨ogskolan, Roslagstullsbacken 21, Stockholm. Opponent: Prof. Per Delsing Huvudhandledare: Prof. David B. Haviland © ADEM ERGUL,¨ 2013 Tryck: Universitetsservice US AB iii Abstract This thesis presents the results of the experimental studies on two kind of Superconducting circuits: one-dimensional Josephson junction chains and superconducting coplanar waveguide (CPW) resonators. One-dimensional Josephson junction chains are constructed by connecting many Supercon- ducting quantum interference devices (SQUIDs) in series. We have stud- ied DC transport properties of the SQUID chains and model their nonlinear dynamics with Thermally Activated Phase-Slips (TAPS). Experimental and simulated results showed qualitative agreement revealing the existence of a uniform phase-slipping and phase-sticking process which results in a voltage- independent current on the dissipative branch of the current-voltage char- acteristics (IVC). By modulating the effective Josephson coupling energy of the SQUIDs (EJ ) with an external magnetic field, we found that the ra- tio EJ /EC is a decisive factor in determining the qualitative shape of the IVC. A quantum phase transition between incoherent Quantum Phase Slip, QPS (supercurrent branch with a finite slope) to coherent QPS (IVC with well-developed Coulomb blockade) via an intermediate state (supercurrent branch with a remnant of Coulomb blockade) is observed as the EJ /EC ra- tio is tuned. This transition from incoherent QPS to the intermediate-state ∼ 2 happens around R0 = RQ (RQ = h/4e = 6.45kΩ). We also fabricated struc- tured chains where a SQUID at the middle of the chain (central SQUID) has different junction size and loop area compared to other SQUIDs in the chain. Results showed that with these structured chains it is possible to localize and tune the amplitude of both TAPS and QPS at the central SQUID. The second part of the thesis describes the fabrication process and the measurement results of superconducting CPW resonators. Resonators with different design parameters were fabricated and measured. The transmission spectra showed quality factors up to, Q ∼ 5 × 105. We have observed bending of the resonance curves to the lower frequencies due to existence of a non- linear kinetic inductance. The origin of the nonlinear kinetic inductance is the nonlinear relation between supercurrent density, Js and superfluid veloc- ity, vs, of the charge carriers on the center line of the resonators. A simple model based on the Ginzburg-Landau theory is used in order to explain ob- served nonlinear kinetic inductance and estimates using this model showed good agreement with the experimental results. iv Sammanfattning Denna avhandling presenterar resultaten av de experimentella studierna p˚atv˚atyper av supraledande kretsar: endimensionella Josephson¨overg˚angskedjor och supraledande plana v˚agledar- (CPW) resonatorer. Endimensionella Joseph- son¨overg˚angskedjor konstrueras genom att ansluta flera supraledande kvantin- terferensenheter (SQUID) i serie. Vi har studerat likstr¨omsegenskaper av SQUID kedjor och modellerat deras ickelinj¨ara dynamik som termiskt aktiver- ade fas-hopp (TAPS). Experimentella och simulerade resultat visade kvalita- tiv ¨overenskommelse vilket tyder p˚af¨orekomsten av en enhetlig fas-hopp/fas- fastnar mekanism som resulterar i en sp¨annings-oberoende str¨om p˚a den dissipativa grenen av str¨om-sp¨annings karakteristiken (IVC). Genom att mod- ulera den effektiva Josephsonkopplingsenergin (EJ ) av SQUID:er med ett yt- tre magnetf¨alt, fann vi att f¨orh˚allandet EJ /EC ¨ar en avg¨orande faktor f¨or den kvalitativa formen av IVC. En kvantfas¨overg˚ang mellan icke-koherenta kvant-fas-hopp, QPS (superstr¨om-gren med en ¨andlig lutning) till koher- ent QPS (IVC med v¨alutvecklad Coulombblockad) via ett mellanliggande tillst˚and (superstr¨om-gren med en tecken av Coulombblockad) observeras n¨ar f¨orh˚allandet EJ /EC justeras. Denna ¨overg˚ang fr˚an icke-koherent QPS till det ∼ 2 mellanliggande tillst˚andet sker runt R0 = RQ (RQ = h/4e = 6.45kΩ). Vi tillverkade ocks˚astrukturerade kedjor d¨ar en SQUID vid mitten av kedjan (centrala SQUID:en) hade olika ¨overg˚angsstorlek och lop area j¨amf¨ort med andra SQUID:ar i kedjan. Resultaten visade att med dessa strukturerade ked- jor var det m¨ojligt att lokalisera och justera amplituden f¨or b˚ade TAPS och QPS vid den centrala SQUID:en. Den andra delen av avhandlingen beskriver tillverkningsprocessen och m¨atning av supraledande CPW resonatorer. Resonatorer med olika tillv¨arknings- parametrar fabrikerades och m¨attes. Transmissionsspektra visade kvalitets- faktorer upp till, Q ∼ 5 × 105. Vi observerade b¨ojning av resonanskurvor till l¨agre frekvenser p˚agrund av f¨orekomsten av en icke-linj¨ar kinetisk induktans. Ursprunget f¨or den icke-linj¨ara kinetiska induktansen ¨ar den icke-linj¨ara rela- tionen mellan superstr¨omt¨atheten, Js och den superflytande hastigheten, vs, av laddningsb¨ararna p˚amittlinjen av resonatorerna. En enkel modell baserad p˚aGinzburg-Landau teori anv¨andes f¨or att f¨orklara den observerade icke- linj¨ara kinetiska induktansen och ber¨akningar med denna modell visade god ¨overensst¨ammelse med de experimentella resultaten. Contents Contents v 1 Introduction 3 1.1 JosephsonJunctionChains . 4 1.2 SuperconductingCPWResonators . 6 1.3 OutlineoftheThesis........................... 7 2 Josephson Junction Chains 9 2.1 Theory................................... 9 2.1.1 Characteristic Energies of a Josephson Junction . .... 10 2.1.2 DynamicsofaJosephsonJunction . 11 2.1.3 ClassicalPhaseDynamics . 13 2.1.4 TiltedWashboardModel . 15 2.1.5 PhasePortraits.......................... 16 2.1.6 Superconducting Quantum Interference Device . .. 18 2.1.7 Semi-classicalChargeDynamics . 19 2.1.8 Coulomb Blockade of Cooper pair Tunneling . 22 2.1.9 PhaseDynamicsoftheSQUIDChain . 24 2.2 ExperimentalTechniques . 27 2.2.1 SampleFabrication. 27 2.2.2 SampleGeometries. 32 2.2.3 MeasurementTechniques . 33 2.3 OverviewofMeasuredResults. 35 2.3.1 Chaincurrent........................... 43 2.3.2 ZenerCurrent........................... 45 2.3.3 ThresholdVoltage . 46 2.4 Discussion................................. 47 2.4.1 ThermallyActivatedPhaseSlips . 47 2.4.2 QuantumPhaseSlips . 50 3 Superconducting CPW Resonators 51 3.1 Theory................................... 51 3.1.1 Current and Magnetic Field Distribution . 51 v vi CONTENTS 3.1.2 Kinetic and Electromagnetic Inductance . 52 3.1.3 NonlinearKineticInductance . 54 3.2 ExperimentalTechniques . 57 3.2.1 SampleFabrication. 57 3.3 ResultsandDiscussion. 61 3.3.1 ParametricAmplification . 64 3.3.2 Determining the Nonlinear Kinetic Inductance . .. 65 4 Conclusions 69 Bibliography 73 A Fabrication of Josephson Junction Chains 83 A.1 FabricationRecipes. .. .. .. .. .. .. .. .. .. .. .. 83 A.2 SpinCoating ............................... 85 A.3 Photolithography.. .. .. .. .. .. .. .. .. .. .. .. 87 A.4 MetalDeposition ............................. 88 A.5 Electronbeamlithography. 89 A.6 AngledEvaporation ........................... 93 A.7 BarrierOxidation............................. 96 B Fabrication of CPW Resonators 101 B.1 FabricationRecipes. 101 B.2 Most Frequently Encountered Fabrication Problems . .. 102 B.3 SputnikDepositionSystem . 107 B.4 ThinFilmFabrication . 109 Acknowledgements 117 Appended Papers 119 To my family... 1 Chapter 1 Introduction Since its initial introduction in the 1920’s, quantum physics has evoked a tremen- dous amount of interest in the scientific community and it has been considered not only an emerging branch of physics but also one of the most important scientific achievements of the 20th century. The prominence that quantum physics gained through the last century can be attributed to the fact that, beyond its contribu- tions to a new specific field of physics, it introduces a radically new point of view and interpretation of physics. Unlike classical physics’ perspective, which usually conforms to common sense and ordinary experiences of daily life, quantum physics draws a novel picture of the physical world that is beyond simple human concep- tions of the ”real” world. Furthermore, quantum physics and its new point of view provides us with new opportunities to explore and understand the behavior of mat- ter. Quantum physics has also spurred many new applications that were impossible to build and operate with mere knowledge of classical physics. The most prominent examples of such applications are Lasers (Light Amplification by Stimulated Emis- sion of Radiation), STM (Scanning Tunneling Microscope), and Quantum Comput- ers which process qbits (quantum bits). The development of the quantum computer, which could offer the next major leap forward in information technology, will be pos- sible only by fabricating and manipulating qbits. There are several systems which can be utilized as a qbit and prominent systems are based on Josephson Junctions. Furthermore, the reading and writing process
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