Introduction to Unconventional Superconductivity Manfred Sigrist
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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. -
Supercritical Fluid Extraction of Positron-Emitting Radioisotopes from Solid Target Matrices
XA0101188 11. United States of America Supercritical Fluid Extraction of Positron-Emitting Radioisotopes From Solid Target Matrices D. Schlyer, Brookhaven National Laboratory, Chemistry Department, Upton, Bldg. 901, New York 11973-5000, USA Project Description Supercritical fluids are attractive as media for both chemical reactions, as well as process extraction since their physical properties can be manipulated by small changes in pressure and temperature near the critical point of the fluid. What is a supercritical fluid? Above a certain temperature, a vapor can no longer be liquefied regardless of pressure critical temperature - Tc supercritical fluid r«gi on solid a u & temperature Fig. 1. Phase diagram depicting regions of solid, liquid, gas and supercritical fluid behavior. The critical point is defined by a critical pressure (Pc) and critical temperature (Tc) for a particular substance. Such changes can result in drastic effects on density-dependent properties such as solubility, refractive index, dielectric constant, viscosity and diffusivity of the fluid[l,2,3]. This suggests that pressure tuning of a pure supercritical fluid may be a useful means to manipulate chemical reactions on the basis of a thermodynamic solvent effect. It also means that the solvation properties of the fluid can be precisely controlled to enable selective component extraction from a matrix. In recent years there has been a growing interest in applying supercritical fluid extraction to the selective removal of trace metals from solid samples [4-10]. Much of the work has been done on simple systems comprised of inert matrices such as silica or cellulose. Recently, this process as been expanded to environmental samples as well [11,12]. -
Singlet/Triplet State Anti/Aromaticity of Cyclopentadienylcation: Sensitivity to Substituent Effect
Article Singlet/Triplet State Anti/Aromaticity of CyclopentadienylCation: Sensitivity to Substituent Effect Milovan Stojanovi´c 1, Jovana Aleksi´c 1 and Marija Baranac-Stojanovi´c 2,* 1 Institute of Chemistry, Technology and Metallurgy, Center for Chemistry, University of Belgrade, Njegoševa 12, P.O. Box 173, 11000 Belgrade, Serbia; [email protected] (M.S.); [email protected] (J.A.) 2 Faculty of Chemistry, University of Belgrade, Studentski trg 12-16, P.O. Box 158, 11000 Belgrade, Serbia * Correspondence: [email protected]; Tel.: +381-11-3336741 Abstract: It is well known that singlet state aromaticity is quite insensitive to substituent effects, in the case of monosubstitution. In this work, we use density functional theory (DFT) calculations to examine the sensitivity of triplet state aromaticity to substituent effects. For this purpose, we chose the singlet state antiaromatic cyclopentadienyl cation, antiaromaticity of which reverses to triplet state aromaticity, conforming to Baird’s rule. The extent of (anti)aromaticity was evaluated by using structural (HOMA), magnetic (NICS), energetic (ISE), and electronic (EDDBp) criteria. We find that the extent of triplet state aromaticity of monosubstituted cyclopentadienyl cations is weaker than the singlet state aromaticity of benzene and is, thus, slightly more sensitive to substituent effects. As an addition to the existing literature data, we also discuss substituent effects on singlet state antiaromaticity of cyclopentadienyl cation. Citation: Stojanovi´c,M.; Aleksi´c,J.; Baranac-Stojanovi´c,M. Keywords: antiaromaticity; aromaticity; singlet state; triplet state; cyclopentadienyl cation; substituent Singlet/Triplet State effect Anti/Aromaticity of CyclopentadienylCation: Sensitivity to Substituent Effect. -
Colour Structures in Supersymmetric QCD
Colour structures in Supersymmetric QCD Jorrit van den Boogaard August 23, 2010 In this thesis we will explain a novel method for calculating the colour struc- tures of Quantum ChromoDynamical processes, or QCD prosesses for short, with the eventual goal to apply such knowledge to Supersymmetric theory. QCD is the theory of the strong interactions, among quarks and gluons. From a Supersymmetric viewpoint, something we will explain in the first chapter, it is interesting to know as much as possible about QCD theory, for it will also tell us something about this Supersymmetric theory. Colour structures in particular, which give the strength of the strong force, can tell us something about the interactions with the Supersymmetric sector. There are a multitude of different ways to calculate these colour structures. The one described here will make use of diagrammatic methods and is therefore somewhat more user friendly. We therefore hope it will see more use in future calculations regarding these processes. We do warn the reader for the level of knowledge required to fully grasp the subject. Although this thesis is written with students with a bachelor degree in mind, one cannot escape touching somewhat dificult matterial. We have tried to explain things as much as possible in a way which should be clear to the reader. Sometimes however we saw it neccesary to skip some of the finer details of the subject. If one wants to learn more about these hiatuses we suggest looking at the material in the bibliography. For convenience we will use some common conventions in this thesis. -
Equation of State and Phase Transitions in the Nuclear
National Academy of Sciences of Ukraine Bogolyubov Institute for Theoretical Physics Has the rights of a manuscript Bugaev Kyrill Alekseevich UDC: 532.51; 533.77; 539.125/126; 544.586.6 Equation of State and Phase Transitions in the Nuclear and Hadronic Systems Speciality 01.04.02 - theoretical physics DISSERTATION to receive a scientific degree of the Doctor of Science in physics and mathematics arXiv:1012.3400v1 [nucl-th] 15 Dec 2010 Kiev - 2009 2 Abstract An investigation of strongly interacting matter equation of state remains one of the major tasks of modern high energy nuclear physics for almost a quarter of century. The present work is my doctor of science thesis which contains my contribution (42 works) to this field made between 1993 and 2008. Inhere I mainly discuss the common physical and mathematical features of several exactly solvable statistical models which describe the nuclear liquid-gas phase transition and the deconfinement phase transition. Luckily, in some cases it was possible to rigorously extend the solutions found in thermodynamic limit to finite volumes and to formulate the finite volume analogs of phases directly from the grand canonical partition. It turns out that finite volume (surface) of a system generates also the temporal constraints, i.e. the finite formation/decay time of possible states in this finite system. Among other results I would like to mention the calculation of upper and lower bounds for the surface entropy of physical clusters within the Hills and Dales model; evaluation of the second virial coefficient which accounts for the Lorentz contraction of the hard core repulsing potential between hadrons; inclusion of large width of heavy quark-gluon bags into statistical description. -
Electron-Phonon Interaction in Conventional and Unconventional Superconductors
Electron-Phonon Interaction in Conventional and Unconventional Superconductors Pegor Aynajian Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 Electron-Phonon Interaction in Conventional and Unconventional Superconductors Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der W¨urde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung vorgelegt von Pegor Aynajian aus Beirut (Libanon) Hauptberichter: Prof. Dr. Bernhard Keimer Mitberichter: Prof. Dr. Harald Giessen Tag der m¨undlichen Pr¨ufung: 12. M¨arz 2009 Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 2 Deutsche Zusammenfassung Die Frage, ob ein genaueres Studium der Phononen-Spektren klassischer Supraleiter wie Niob und Blei mittels inelastischer Neutronenstreuung der M¨uhe wert w¨are, w¨urde sicher von den meisten Wissenschaftlern verneint werden. Erstens erk¨art die ber¨uhmte mikroskopische Theorie von Bardeen, Cooper und Schrieffer (1957), bekannt als BCS Theorie, nahezu alle Aspekte der klassischen Supraleitung. Zweitens ist das aktuelle Interesse sehr stark auf die Hochtemperatur-Supraleitung in Kupraten und Schwere- Fermionen Systemen fokussiert. Daher waren die ersten Experimente dieser Arbeit, die sich mit der Bestimmung der Phononen-Lebensdauern in supraleitendem Niob und Blei befaßten, nur als ein kurzer Test der Aufl¨osung eines neuen hochaufl¨osenden Neutronen- spektrometers am Forschungsreaktor FRM II geplant. Dieses neuartige Spektrometer TRISP (triple axis spin echo) erm¨oglicht die Bestimmung von Phononen-Linienbreiten uber¨ große Bereiche des Impulsraumes mit einer Energieaufl¨osung im μeV Bereich, d.h. zwei Gr¨oßenordnungen besser als an klassische Dreiachsen-Spektrometern. Philip Allen hat erstmals dargelegt, daß die Linienbreite eines Phonons proportional zum Elektron-Phonon Kopplungsparameter λ ist. -
BCS Thermal Vacuum of Fermionic Superfluids and Its Perturbation Theory
www.nature.com/scientificreports OPEN BCS thermal vacuum of fermionic superfuids and its perturbation theory Received: 14 June 2018 Xu-Yang Hou1, Ziwen Huang1,4, Hao Guo1, Yan He2 & Chih-Chun Chien 3 Accepted: 30 July 2018 The thermal feld theory is applied to fermionic superfuids by doubling the degrees of freedom of the Published: xx xx xxxx BCS theory. We construct the two-mode states and the corresponding Bogoliubov transformation to obtain the BCS thermal vacuum. The expectation values with respect to the BCS thermal vacuum produce the statistical average of the thermodynamic quantities. The BCS thermal vacuum allows a quantum-mechanical perturbation theory with the BCS theory serving as the unperturbed state. We evaluate the leading-order corrections to the order parameter and other physical quantities from the perturbation theory. A direct evaluation of the pairing correlation as a function of temperature shows the pseudogap phenomenon, where the pairing persists when the order parameter vanishes, emerges from the perturbation theory. The correspondence between the thermal vacuum and purifcation of the density matrix allows a unitary transformation, and we found the geometric phase associated with the transformation in the parameter space. Quantum many-body systems can be described by quantum feld theories1–4. Some available frameworks for sys- tems at fnite temperatures include the Matsubara formalism using the imaginary time for equilibrium systems1,5 and the Keldysh formalism of time-contour path integrals3,6 for non-equilibrium systems. Tere are also alterna- tive formalisms. For instance, the thermal feld theory7–9 is built on the concept of thermal vacuum. -
Topological Superconductors, Majorana Fermions and Topological Quantum Computation
Topological Superconductors, Majorana Fermions and Topological Quantum Computation 0. … from last time: The surface of a topological insulator 1. Bogoliubov de Gennes Theory 2. Majorana bound states, Kitaev model 3. Topological superconductor 4. Periodic Table of topological insulators and superconductors 5. Topological quantum computation 6. Proximity effect devices Unique Properties of Topological Insulator Surface States “Half” an ordinary 2DEG ; ¼ Graphene EF Spin polarized Fermi surface • Charge Current ~ Spin Density • Spin Current ~ Charge Density Berry’s phase • Robust to disorder • Weak Antilocalization • Impossible to localize Exotic States when broken symmetry leads to surface energy gap: • Quantum Hall state, topological magnetoelectric effect • Superconducting state Even more exotic states if surface is gapped without breaking symmetry • Requires intrinsic topological order like non-Abelian FQHE Surface Quantum Hall Effect Orbital QHE : E=0 Landau Level for Dirac fermions. “Fractional” IQHE 2 2 e xy 1 2h B 2 0 e 1 xy n -1 h 2 2 -2 e n=1 chiral edge state xy 2h Anomalous QHE : Induce a surface gap by depositing magnetic material † 2 2 Hi0 ( - v - DM z ) e e - Mass due to Exchange field 2h 2h M↑ M↓ e2 xyDsgn( M ) EF 2h TI Egap = 2|DM| Chiral Edge State at Domain Wall : DM ↔ -DM Topological Magnetoelectric Effect Qi, Hughes, Zhang ’08; Essin, Moore, Vanderbilt ‘09 Consider a solid cylinder of TI with a magnetically gapped surface M 2 e 1 J xy E n E M h 2 J Magnetoelectric Polarizability topological “q term” 2 DL EB E e 1 ME n e2 h 2 q 2 h TR sym. -
How the Electron-Phonon Coupling Mechanism Work in Metal Superconductor
How the electron-phonon coupling mechanism work in metal superconductor Qiankai Yao1,2 1College of Science, Henan University of Technology, Zhengzhou450001, China 2School of physics and Engineering, Zhengzhou University, Zhengzhou450001, China Abstract Superconductivity in some metals at low temperature is known to arise from an electron-phonon coupling mechanism. Such the mechanism enables an effective attraction to bind two mobile electrons together, and even form a kind of pairing system(called Cooper pair) to be physically responsible for superconductivity. But, is it possible by an analogy with the electrodynamics to describe the electron-phonon coupling as a resistivity-dependent attraction? Actually so, it will help us to explore a more operational quantum model for the formation of Cooper pair. In particularly, by the calculation of quantum state of Cooper pair, the explored model can provide a more explicit explanation for the fundamental properties of metal superconductor, and answer: 1) How the transition temperature of metal superconductor is determined? 2) Which metals can realize the superconducting transition at low temperature? PACS numbers: 74.20.Fg; 74.20.-z; 74.25.-q; 74.20.De ne is the mobile electron density, η the damping coefficient 1. Introduction that is determined by the collision time τ . In the BCS theory[1], superconductivity is attributed to a In metal environment, mobile electrons are usually phonon-mediated attraction between mobile electrons near modeled to be a kind of classical particles like gas molecules, Fermi surface(called Fermi electrons). The attraction is each of which performs a Brown-like motion and satisfies the sometimes referred to as a residual Coulomb interaction[2] that Langevin equation can glue Cooper pair together to cause superconductivity. -
Theory of Superconductivity
Theory of Superconductivity Kwon Park KIAS-SNU Physics Winter Camp Camp Winter KIAS-SNU Physics Phoenix Park Jan 20 – 27, 2013 Outline • Why care about superconductivity? • BCS theory as a trial wave function method • BCS theory as a mean-field theory • High-temperature superconductivity and strong correlation • Effective field theory for superconductivity: Ginzburg- Landau theory Family tree of strongly correlated electron systems Quantum magnetism Topological Mott insulator FQHE insulator HTSC Quantum Hall effect Superconductivity Wigner crystal Breakdown of the Landau-Fermi liquid Collective behavior of a staring crowd Disordered State Ordered State Superconductivity as an emergent phenomenon Superconducting phase coherence: Josephson effect • Cooper-pair box: An artificial two-level system composed of many superconducting electron pairs in a “box.” reservoir box - - - - + + + + gate Josephson effect in the Cooper-pair box Nakamura, Pashkin, Tsai, Nature 398, 786 (99) • Phase vs. number uncertainty relationship: When the phase gets coherent, the Cooper-pair number becomes uncertain, which is nothing but the Josephson effect. Devoret and Schoelkopf, Nature 406, 1039 (00) Meissner effect • mv-momentum = p-momentum − e/c × vector potential • current density operator = 2e × Cooper pair density × velocity operator • θ=0 for a coherent Cooper-pair condensate in a singly connected region.region London equation • The magnetic field is expelled from the inside of a superconductor: Meissner effect. London equation where Electromagnetic field, or wave are attenuated inside a superconductor, which means that, in quantum limit, photons become massive while Maxwell’s equations remain gauge-invariant! Anderson-Higgs mechanism • Quantum field theories should be renormalizable in order to produce physically meaningful predictions via systematic elimination of inherent divergences. -
Lecture 5 Non-Aqueous Phase Liquid (NAPL) Fate and Transport
Lecture 5 Non-Aqueous Phase Liquid (NAPL) Fate and Transport Factors affecting NAPL movement Fluid properties: Porous medium: 9 Density Permeability 9 Interfacial tension Pore size 9 Residual saturation Structure Partitioning properties Solubility Ground water: Volatility and vapor density Water content 9 Velocity Partitioning processes NAPL can partition between four phases: NAPL Gas Solid (vapor) Aqueous solution Water to gas partitioning (volatilization) Aqueous ↔ gaseous Henry’s Law (for dilute solutions) 3 Dimensionless (CG, CW in moles/m ) C G = H′ CW Dimensional (P = partial pressure in atm) P = H CW Henry’s Law Constant H has dimensions: atm m3 / mol H’ is dimensionless H’ = H/RT R = gas constant = 8.20575 x 10-5 atm m3/mol °K T = temperature in °K NAPL to gas partitioning (volatilization) NAPL ↔ gaseous Raoult’s Law: CG = Xt (P°/RT) Xt = mole fraction of compound in NAPL [-] P° = pure compound vapor pressure [atm] R = universal gas constant [m3-atm/mole/°K] T = temperature [°K] Volatility Vapor pressure P° is measure of volatility P° > 1.3 x 10-3 atm → compound is “volatile” 1.3 x 10-3 > P° > 1.3 x 10-13 atm → compound is “semi-volatile” Example: equilibrium with benzene P° = 76 mm Hg at 20°C = 0.1 atm R = 8.205 x 10-5 m3-atm/mol/°K T = 20°C (assumed) = 293°K Assume 100% benzene, mole fraction Xt = 1 3 CG = Xt P°/(RT) = 4.16 mol/m Molecular weight of benzene, C6H6 = 78 g/mol 3 3 CG = 4.16 mol/m × 78 g/mol = 324 g/m = 0.32 g/L 6 CG = 0.32 g/L x 24 L/mol / (78 g/mol) x 10 = 99,000 ppmv One mole of ideal gas = 22.4 L at STP (1 atm, 0 C), Corrected to 20 C: 293/273*22.4 = 24.0 L/mol Gas concentration in equilibrium with pure benzene NAPL Example: equilibrium with gasoline Gasoline is complex mixture – mole fraction is difficult to determine and varies Benzene = 1 to several percent (Cline et al., 1991) Based on analysis reported by Johnson et al. -
Deuterium Isotope Effect in the Radiative Triplet Decay of Heavy Atom Substituted Aromatic Molecules
Deuterium Isotope Effect in the Radiative Triplet Decay of Heavy Atom Substituted Aromatic Molecules J. Friedrich, J. Vogel, W. Windhager, and F. Dörr Institut für Physikalische und Theoretische Chemie der Technischen Universität München, D-8000 München 2, Germany (Z. Naturforsch. 31a, 61-70 [1976] ; received December 6, 1975) We studied the effect of deuteration on the radiative decay of the triplet sublevels of naph- thalene and some halogenated derivatives. We found that the influence of deuteration is much more pronounced in the heavy atom substituted than in the parent hydrocarbons. The strongest change upon deuteration is in the radiative decay of the out-of-plane polarized spin state Tx. These findings are consistently related to a second order Herzberg-Teller (HT) spin-orbit coupling. Though we found only a small influence of deuteration on the total radiative rate in naphthalene, a signi- ficantly larger effect is observed in the rate of the 00-transition of the phosphorescence. This result is discussed in terms of a change of the overlap integral of the vibrational groundstates of Tx and S0 upon deuteration. I. Introduction presence of a halogene tends to destroy the selective spin-orbit coupling of the individual triplet sub- Deuterium (d) substitution has proved to be a levels, which is originally present in the parent powerful tool in the investigation of the decay hydrocarbon. If the interpretation via higher order mechanisms of excited states The change in life- HT-coupling is correct, then a heavy atom must time on deuteration provides information on the have a great influence on the radiative d-isotope electronic relaxation processes in large molecules.