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Lecture Notes: BCS Theory of Superconductivity
Lecture Notes: BCS theory of superconductivity Prof. Rafael M. Fernandes Here we will discuss a new ground state of the interacting electron gas: the superconducting state. In this macroscopic quantum state, the electrons form coherent bound states called Cooper pairs, which dramatically change the macroscopic properties of the system, giving rise to perfect conductivity and perfect diamagnetism. We will mostly focus on conventional superconductors, where the Cooper pairs originate from a small attractive electron-electron interaction mediated by phonons. However, in the so- called unconventional superconductors - a topic of intense research in current solid state physics - the pairing can originate even from purely repulsive interactions. 1 Phenomenology Superconductivity was discovered by Kamerlingh-Onnes in 1911, when he was studying the transport properties of Hg (mercury) at low temperatures. He found that below the liquifying temperature of helium, at around 4:2 K, the resistivity of Hg would suddenly drop to zero. Although at the time there was not a well established model for the low-temperature behavior of transport in metals, the result was quite surprising, as the expectations were that the resistivity would either go to zero or diverge at T = 0, but not vanish at a finite temperature. In a metal the resistivity at low temperatures has a constant contribution from impurity scattering, a T 2 contribution from electron-electron scattering, and a T 5 contribution from phonon scattering. Thus, the vanishing of the resistivity at low temperatures is a clear indication of a new ground state. Another key property of the superconductor was discovered in 1933 by Meissner. -
Progress in Wire Fabrication of Iron-Based Superconductors
Progress in wire fabrication of iron-based superconductors Yanwei Ma* Key Laboratory of Applied Superconductivity, Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China Abstract: Iron-based superconductors, with Tc values up to 55 K, are of great interest for applications, due to their lower anisotropies and ultrahigh upper critical fields. In the past 4 years, great progress has been made in the fabrication of iron-based superconducting wires and tapes using the powder-in-tube (PIT) processing method, including main three types of 122, 11, and 1111 iron-based parent compounds. In this article, an overview of the current state of development of iron-based superconducting wires and tapes is presented. We focus on the fabrication techniques used for 122 pnictide wires and tapes, with an emphasis on their meeting the critical current requirements for making high-performance conductors, such as a combination of using Ag sheath, addition element and optimized heat treatment to realize high Jc, ex situ process employed to reduce non-superconducting phases and to obtain a high relative density, and a texture control to improve grain connectivity. Of particular 4 2 interest is that so far transport Jc values above 10 A/cm at 4.2 K and 10 T are obtained in 122 type tapes, suggesting that they are prospective candidates for high-field applications. Finally, a perspective and future development of PIT pnictide wires are also given. * E-mail: [email protected] 1 Contents 1. Introduction ........................................................................................................... -
Selected High-Temperature Superconducting Electric Power Products
Selected High-Temperature Superconducting Electric Power Products Prototypes are outperforming design goals full-scale utility applications are moving onto the power system and ... Modernizing the Existing Electricity Infrastructure Superconductivity Program for Electric Systems•• Office of Power Delivery U.S. Department of Energy January 2000 HIGH TEMPERATURE SUPERCONDUCTING POWER PRODUCTS CAPTURE THE ATTENTION OF UTILITY ENGINEERS AND PLANNERS Soon superconductors could be so common that we will drop the “super” and refer to them simply as conductors Critical Role of Wire The application of superconductors to electric power systems has been pursued for more than 30 years. This persistence is starting to pay off and utilities that once politely acknowledged the long-term potential of superconducting applications are now paying close attention to several prototype devices that are being tested or are nearing testing on utility systems. What has prompted this interest is the development of electric wires that become superconducting when cooled to the affordable operating-temperature realm of liquid nitrogen as well as the development of coils, magnets, conductors, and machines and power components made with these wires. Superconducting wires have as much as 100 times the current carrying capacity as (Courtesy of American Superconductor Corp.) ordinary conductors. Flexible HTS conductors promise reduced A Real Need operating costs and many other benefits when incorporated into electric power devices. They can change the way power is managed and The timing is right for superconducting solutions consumed. to emerging business problems. Power generation and transmission equipment is aging and must be replaced. Environmental considerations are increasing. Utilities are changing the way they evaluate capital investments. -
Sn and Ti DIFFUSION, PHASE FORMATION, STOICHIOMETRY, and SUPERCONDUCTING PROPERTIES of INTERNAL-Sn-TYPE Nb3sn CONDUCTORS
Sn AND Ti DIFFUSION, PHASE FORMATION, STOICHIOMETRY, AND SUPERCONDUCTING PROPERTIES OF INTERNAL-Sn-TYPE Nb3Sn CONDUCTORS A Thesis Presented in Partial Fulfillment of the Requirements for The Degree of Master of Science in the Graduate School of the Ohio State University By Rakesh Kumar Dhaka, B. Tech. ***** The Ohio State University 2007 Masters Examination Committee Professor Mike Sumption, Adviser Professor John Morral, Adviser Professor Katharine Flores ABSTRACT In the present work the diffusion of Sn through the interfilamentary matrix within a subelement and the formation of the associated Cu-Sn intermetallics were observed experimentally for several different Nb3Sn internal-Sn type strands during the pre- reaction part of the heat treatment. An analytical-based model was then developed to determine the time and temperature dependence of Sn-diffusion through the Cu matrix of Nb3Sn subelements. The output of the model is in the form of radial positions of , and phases as a function of time during pre-reaction heat treatment process. These predicted radial positions can be used to determine the optimum heat treatment parameters. The model was then compared to the experimental results. Experimental results for Ti-bearing superconductor strands were also discussed. Following the pre-reaction heat treatment studies, the effects of Titanium doping in presence of Tantalum on the kinetics of Nb3Sn formation and superconducting properties of internal-tin type strands were examined. A series of internal-tin type Nb3Sn sublements which had Nb-7.5wt%Ta filaments and various levels of Ti doping were investigated. Titanium was introduced into the Sn core of the subelements such that the core contained 0 at% to 2.8 at% Ti before reaction. -
Introduction to Unconventional Superconductivity Manfred Sigrist
Introduction to Unconventional Superconductivity Manfred Sigrist Theoretische Physik, ETH-Hönggerberg, 8093 Zürich, Switzerland Abstract. This lecture gives a basic introduction into some aspects of the unconventionalsupercon- ductivity. First we analyze the conditions to realized unconventional superconductivity in strongly correlated electron systems. Then an introduction of the generalized BCS theory is given and sev- eral key properties of unconventional pairing states are discussed. The phenomenological treatment based on the Ginzburg-Landau formulations provides a view on unconventional superconductivity based on the conceptof symmetry breaking.Finally some aspects of two examples will be discussed: high-temperature superconductivity and spin-triplet superconductivity in Sr2RuO4. Keywords: Unconventional superconductivity, high-temperature superconductivity, Sr2RuO4 INTRODUCTION Superconductivity remains to be one of the most fascinating and intriguing phases of matter even nearly hundred years after its first observation. Owing to the breakthrough in 1957 by Bardeen, Cooper and Schrieffer we understand superconductivity as a conden- sate of electron pairs, so-called Cooper pairs, which form due to an attractive interaction among electrons. In the superconducting materials known until the mid-seventies this interaction is mediated by electron-phonon coupling which gises rise to Cooper pairs in the most symmetric form, i.e. vanishing relative orbital angular momentum and spin sin- glet configuration (nowadays called s-wave pairing). After the introduction of the BCS concept, also studies of alternative pairing forms started. Early on Anderson and Morel [1] as well as Balian and Werthamer [2] investigated superconducting phases which later would be identified as the A- and the B-phase of superfluid 3He [3]. In contrast to the s-wave superconductors the A- and B-phase are characterized by Cooper pairs with an- gular momentum 1 and spin-triplet configuration. -
High Performance Magnesium Diboride (Mgb ) Superconductors
High Performance Magnesium Diboride (MgB2) Superconductors: Towards the Prospect for Commercialization M.S.A. Hossain (ARC DECRA Fellow) The Australian National University– 16/04/2014 Research Programs at ISEM/AIIM Faculty . Applied Superconductivity Group • Bulk • Wire • Tape • Cable • Thin Film . Energy Storage Group . Spintronics and Electronic Materials Group . Thin Film Technology Group . Terahertz Science, Solid State Physics Group . Nanostructure Materials Group . Advanced Photovoltaic Materials Group ISEM Performance Profile highlight ISEM Team: 40 research staff (12 ARC fellows) and more than 80 PhD Seven research program centred on energy and electronics Electrification Program leader in Automotive CRC 2020 More than 50% citations in Li ion battery and superconductivity from ISEM in Australia ISEM is ranked at first place in magnesium diboride supercondcutors and eighth place in Li ion battery research in terms of outputs since 2001 105 PhD graduates widely spread across five continents since 1994 50 ARC fellowship awards to ISEM since 1995 80 ARC projects since 2000 12% publications, 15% ARC funding and 24% citations of UoW are from ISEM $80m for building & $20m for facilities for research infrastructure Member of CoE, ANFF and Flagship Bao Steel Joint Centre with other 3 Universities; Network with more than 50 institutions world-wide Strong links with more than 10 industry partners ERA assessment ranked at 5 for materials engineering, materials chemistry, physical chemistry and interdisciplinary engineering of UOW Superconductivity? Applications • Making a good superconducting product is a formidable interdisciplinary problem Wire cost Wire performance Engineering Cryogenics MgB2 Very simple crystal structure Very high current densities Polycristalline materials observed in films carry large currents Moderately high Tc Tc of 39K Factor of 10 larger than in bulks; room for large improvement in wires still available from R&D MgB2 presents very Good mechanical properties Potentially high critical field 60 1.2 promising features Sample No. -
High Temperature Superconductivity: Cooper Pairs in Trap
High temperature superconductivity: Cooper pairs in trap Wei Ruan Affiliations: Key Laboratory of Inorganic Functional Materials and Devices, Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, China Email: [email protected] or [email protected] Abstract The tremendous efforts to unveil high temperature superconductivity (HTSC) have been devoted to the search of the mechanism underlying Cooper pairs which, however, remains a mysterious subject of vigorous debate, let alone many other mysteries like the pseudogap state, the peculiar Homes’ law, the unnegligible electron-phonon interaction, the stripe, the universal nodal Fermi velocity, etc. Most of subsequent works either bring in more controversies or swell the list of mysteries. Here I tentatively propose a whole new perspective on the basis of low-dimensional Bose-Einstein Condensation (LDBEC), which possibly makes lots of those mysteries correlated and understood in single picture. 1 It is believed that we know as much about the cuprate HTSC as we don’t, due to so many controversies and mysteries. To find out what exactly underlies the cuprate HTSC, we first requires an in-depth understanding of the cuprate pairing mechanism. Anderson’s ‘resonating valence bond’ (RVB) theory expresses a magnetic pairing idea — the strong Coulomb repulsion in these two-dimensional (2D) quantum spin-1/2 systems yields an antiferromagnetic (AF) Mott insulator where the singlet electron pairs naturally form arising from the AF superexchange interaction [1,2]. Recent works further discussed and reviewed the magnetic pairing mechanism to elucidate the cuprate HTSC — superconductivity without phonon [3,4]. No matter in what specific form or model, the magnetically mediated pairing mechanism has provided so many significant insights and guidances in pursuit of the essence of the cuprate HTSC. -
Cooper Pairs, Superconductivity and Flash Mobs
Cooper Pairs, Superconductivity and Flash Mobs PHYS 498 ART Spring 2018 Karmela Padavić Superconductivity: History • Heike Kamerlingh Onnes, 1911, “Kwik nagenoeg nul” Nobel Prizes for superconductivity: • Heike Kamerlingh Onnes (1913), "for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium" • John Bardeen, Leon N. Cooper, and J. Robert Schrieffer (1972), "for their jointly developed theory of superconductivity, usually called the BCS-theory" • Leo Esaki, Ivar Giaever, and Brian D. Josephson (1973), "for their experimental discoveries regarding tunneling phenomena in semiconductors and superconductors, respectively," and "for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects" • Georg Bednorz and K. Alex Müller (1987), "for their important break-through in the discovery of Heike Kamerlingh Onnes (right), the discoverer of superconductivity. Paul Ehrenfest, Hendrik Lorentz, Niels Bohr stand to his left. (from Wikipedia) superconductivity in ceramic materials" • Alexei A. Abrikosov, Vitaly L. Ginzburg, and Anthony J. Leggett (2003), "for pioneering contributions to the theory of superconductors and superfluids" Superconductivity: Experimental Signatures No dissipation below critical temperature Meissner effect Superconductivity: Cooper Pairs (BCS theory) • Two electrons (fermions) can interact due to the presence of a crystal lattice and form a bosonic pair – these bosonic pairs condense into one common ground state • Wikipedia: “Although Cooper pairing is a quantum effect, the reason for the pairing can be seen from a simplified classical explanation. An electron in a metal normally behaves as a free particle. The electron is repelled from other electrons due to their negative charge, but it also attracts the positive ions that make up the rigid lattice of the metal. -
Topological Superconductivity
Topological Superconductivity Annica Black-Schaffer [email protected] Materials Theory 6th MaNEP winter school, Saas Fee, January 2015 Uppsala Open PD and PhD positions! Ångström laboratory Contents • Introduction to superconductivity – BCS theory – BdG formulation • Unconventional superconductivity • Topological matter • Topological superconductivity – Chiral superconductors – “Spinless” superconductors – Majorana fermions Introduction to Superconductivity What is it? How do we describe it? What is Superconductivity? • What is superconductivity? – Electric transport without resistance – Meissner effect Superconductor Magnet History of Superconductivity Superconductivity in Hg: Heike Kammerlingh Onnes Tc Superconductors Department of Energy Nobel Prizes How? • But how do the electrons move without resistance? – The electrons are all in a coherent quantum state with a fixed phase (condensate) i' = ∆0e – One of the few macroscopic manifestations of QM – Spontaneous gauge symmetry breaking – Higgs mechanism (discovered by P. W. Anderson before Higgs et al.) Higgs Mechanism • In the Standard Model: – Interactions with the Higgs field (Higgs bosons) give masses to the W+,- and Z bosons due to electroweak (gauge) symmetry breaking • In superconductors: – The superconducting pairs give mass to the electromagnetic field (photons) due to spontaneous symmetry breaking of the gauge symmetry (fixed phase on Ψ) – Explored by P.W. Anderson for superconductivity 2 years before Higgs et al. Meissner effect = The “Higgs mass” of the electromagnetic field expels it from the superconductor But Wait! • All electrons in one state: a superconducting condensate – Pauli exclusion principle?? How do electrons end up in the same state? Statistics • Fermions (half-integer spin): – Electrons, quarks, neutrinos – Only one electron per quantum state • Bosons (integer spin): – Photons, gauge, and Higgs boson – As many particles as you wish in one quantum state (Bose-Einstein condensate, BEC) Maybe two electrons can form an electron pair!? Sort of, but not the whole story…. -
Superfluidity on Neutron Matter
Superfluidity in neutron star matter Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Fisica Candidato Giulia De Rosi Matricola 1217074 Relatore Prof. Omar Benhar Noccioli Anno Accademico 2012/2013 Superfluidity in neutron star matter Tesi di Laurea Magistrale. Sapienza – Università di Roma © 2013 Giulia De Rosi. Tutti i diritti riservati Questa tesi è stata composta con LATEX e la classe Sapthesis. Email dell’autore: [email protected] A mio nipote Diego “Qualunque cosa tu possa fare, qualunque sogno tu possa sognare, comincia. L’audacia reca in se genialità, magia e forza. Comincia ora” Johann Wolfgang Göethe v Contents Introduction 1 1 Neutron stars 3 1.1 Neutron star formation . .4 1.2 Neutron star structure . .6 2 Superfluidity, superconductivity and Cooper pairs 9 2.1 The electron-phonon interaction . 11 2.2 Origin of the gap and Cooper pairs . 14 3 Variational formulation of BCS theory 19 3.1 BCS theory without spin . 19 3.2 Variational Valatin-BCS theory . 24 3.3 BCS theory at finite temperature . 30 3.4 Predictions of the BCS theory . 32 4 Nuclear matter and nuclear forces 35 4.1 Empirical information on nuclei and nuclear matter . 35 4.2 Nuclear forces . 38 4.2.1 The two-nucleon system . 39 4.2.2 The three-nucleon system . 44 5 The CBF effective interaction 47 5.1 Non-relativistic many-body theory . 47 5.1.1 Correlated basis function (CBF) formalism . 48 5.1.2 Cluster expansion formalism . 50 5.2 Derivation of the effective interaction . 50 5.3 Binding energy per particle of nucleon matter . -
Effect of Plasmons on Superconducting Transition Temperature of Bao.6Ko.4Bi03: a Three Square Well Approach Dinesh Varshney', R P Kumhar' , Sanjay Shah' & R K Singht
Indian Journal of Pure & Applied Physics Vol. 40, December 2002, pp. 879-886 Effect of plasmons on superconducting transition temperature of Bao.6Ko.4Bi03: A three square well approach Dinesh Varshney', R P Kumhar' , Sanjay Shah' & R K Singht *School of Physics, Vi gyan Bhawan, Devi Ahilya University, Kh andwa Road Campus, Indore 452017 t Department of Physics, PMB Gujarati Science College, Devi Ahilya University, Indore 452 00 I tSchool of Pure and Applied Physics, Guru Ghasidas University, Bilaspur 495009 Received 12 June 200 I; revised December 200 I; accepted 29 Apri I 2002 By assuming thc existence of a low energy plasmon in attractive intcraction, the cooper pairing theory is cxtendcd for cubic pcrovskite Bao.{,Ko.4 BiO,. Three square well model for the three interactions namely, electron-phonon, electron plasmon and Coulomb in the calculation of superconducting transition temperat ure (T..) has been employed. The analyti cal solutions for the energy gap equation allow one to visualise the relative interplay of various intcractions. To correlatc th e T.. with various coupling strengths (A'ph , Api and ~ *) curvcs of T,.. are presented with them. The values of the coupling strength and of the Coulomb interaction parameter indicate that, the superconductor is in the intermediate coupling regime. The superconducting transiti on temperature of optimall y doped Ba-K-BiO is estimated as 25 K, for Aph = 0.6, Api = 0. 1 and ~ * = 0. 18 . The low energy plasmons play a key role in raising T .. with the increased Api values. The prcsent analysis points to the importance of both plasmons and optical phonons in determining thc effective clectron-elcctron interaction leading to superconductivity in doped cubic perovskites. -
(Nb-Ti) Superconducting Joints for Persistent-Mode Operation
www.nature.com/scientificreports OPEN Niobium-titanium (Nb-Ti) superconducting joints for persistent-mode operation Received: 21 June 2019 Dipak Patel1,2, Su-Hun Kim1,3, Wenbin Qiu1, Minoru Maeda4, Akiyoshi Matsumoto2, Accepted: 26 August 2019 Gen Nishijima2, Hiroaki Kumakura2, Seyong Choi4 & Jung Ho Kim1 Published: xx xx xxxx Superconducting joints are essential for persistent-mode operation in a superconducting magnet system to produce an ultra-stable magnetic feld. Herein, we report rationally designed niobium- titanium (Nb-Ti) superconducting joints and their evaluation results in detail. For practical applications, superconducting joints were fabricated by using a solder matrix replacement method with two types of lead-bismuth (Pb-Bi) solder, including Pb42Bi58 as a new composition. All the joints attained a critical current of >200 A below 1.43 T at 4.2 K. Our optimized superconducting joining method was tested in a closed-loop coil, obtaining a total circuit resistance of 3.25 × 10−14 Ω at 4.2 K in self-feld. Finally, persistent-mode operation was demonstrated in an Nb-Ti solenoid coil with a persistent-current switch. This work will pave the way to developing high-performance Nb-Ti superconducting joints for practical applications. Superconducting magnets are used to produce a strong magnetic feld in a compact system, which otherwise cannot be attained using conventional copper (Cu) based magnets. Niobium-titanium (Nb-Ti) with a critical temperature (Tc) of 9.2 K is usually used to fabricate superconducting magnets for various practical applications. One of the distinct features of Nb-Ti that makes it ideal for most commercial applications is the possibility to produce reliable “superconducting joints”.