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Chapter 1 Macroscopic Quantum Phenomena R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) I. Foundations of the Josephson Effect 1. Macroscopic Quantum Phenomena 1.1 The Macroscopic Quantum Model of Superconductivity quantum mechanics: - physical quantities (momentum, energy) are quantized - usually thermal motion masks quantum properties no quantization effects on a macroscopic scale - superconductivity: macroscopic quantum effects are observable e.g. quantization of flux through a loop why: electrons form highly correlated electron system Chap. 1 - 2 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.1 Coherent Phenomena in Superconductivity milestones: 1911 discovery of superconductivity 1933 Meißner-Ochsenfeld effect 1935 London-Laue-theory: phenomenological model describing observations 1948 London treats superelectron fluid as a quantum mechanical entity superconductivity is an inherent quantum phenomenon manifested on a macroscopic scale macroscopic wave function Y(r,t) = Y 0 eiq(r,t) London equations 1952 Ginzburg-Landau theory description by complex order parameter Y(r) treatment of spatially inhomogeneous situations 1957 microscopic BCS theory (J. Bardeen, L.N. Cooper, J.R. Schrieffer) BCS ground state Y BCS (coherent many body state) 1962 prediction of the Josephson effect Chap. 1 - 3 1.1.1 Coherent Phenomena in Superconductivity R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Macroscopic quantum model of superconductivity: macroscopic wave function describes the behavior of the whole ensemble of superconducting electrons justified by microscopic BCS theory: - small portion of electrons close to Fermi level are bound to Cooper pairs - center of mass motion of pairs is strongly correlated - example: wave function every pair has momentum or velocity Chap. 1 - 4 1.1.1 Coherent Phenomena in Superconductivity R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Basic quantum mechanics: quantization of electromagnetic radiation (Planck, Einstein): photons represent smallest amount of energy: Luis de Broglie: describes classical particles as waves wave particle duality particle – wave interrelations: Erwin Schrödinger: development of wave mechanics for particles complex wave function describes quantum particle: free particle without spin (Epot = 0 E = Ekin): start with wave equation vph: phase velocity Chap. 1 - 5 1.1.1 Coherent Phenomena in Superconductivity R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Basic quantum mechanics: quantization of electromagnetic radiation (Planck, Einstein): photons represent smallest amount of energy: Luis de Broglie: describes classical particles as waves wave particle duality particle – wave interrelations: Erwin Schrödinger: development of wave mechanics complex wave function (wave packet) describes quantum particle: Chap. 1 - 6 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.1 Coherent Phenomena in Superconductivity with multiply by use since Chap. 1 - 7 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.1 Coherent Phenomena in Superconductivity with similar considerations Schrödinger postulated a general time dependent Schrödinger equation (differential equation) for massive quantum objects : Hamilton operator • we restrict ourselves to systems with constant total energy (conservative systems) due to E = ħw also the frequency is constant prefactor of 2nd term on lhs of is constant solutions can be split in a two parts depending only on space and time Chap. 1 - 8 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.1 Coherent Phenomena in Superconductivity stationary Schrödinger equation: Chap. 1 - 9 1.1.1 Coherent Phenomena in Superconductivity R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Probability currents: interpretation of complex wave function (note: em fields represented as the real or imaginary part of a complex expression) Schrödinger equation suggests that phase has physical significance Max Born: interpretation of absolute square as probability of quantum object conservation of probability density requires (continuity equation) describes evolution of probability in space and time statement of conservation of probability describes probabilistic flow of quantum object, not the motion of a charged particle in an electromagnetic field (forces depending on the motion of the particle itself) Chap. 1 - 10 1.1.1 Coherent Phenomena in Superconductivity R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • some math: derivation of probability current density (Schrödiger equation) (multiply from left by Y*) subtracting the complex conjugate yields vector identity: Chap. 1 - 11 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.1 Coherent Phenomena in Superconductivity time evolution of Task: find defines probability current: for a charged particle in em field start with classical equation of motion: canonical momemtum (kinetic and field momentum) Chap. 1 - 12 1.1.1 Coherent Phenomena in Superconductivity Lorentz’s law: R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Lorentz’s law with E and B expressed in terms of potential Φ and A with: generalized potential: Schrödinger equation: insert expression for V(r,t) Chap. 1 - 13 1.1.1 Coherent Phenomena in Superconductivity R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Schrödinger equation of a charged particle in an electromagnetic field: probability current of a charged particle in an electromagnetic field: central expression in quantum description of superconductivity wave function of single charged particle will be replaced by macroscopic wave function describing the whole entiry of all superelectrons Chap. 1 - 14 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) normal metals: electrons as weakly/non-interacting particles ordinary Schrödinger equation: where is the complex wave function of a particle in the stationary case: quantum behavior reduced to that of wave function phase Fermi statistics different energies: different time evolution of phase phases are uniformly distributed, phase drops out for macroscopic quantities Chap. 1 - 15 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Superconductors spin singlet Cooper pairs, S = 0 (in the simplest case) large size of Cooper pairs, wave functions are strongly overlapping pairs form phase-locked coherent state, description by single wave function with phase µ identical µ/t macroscopic variables (e.g. current) can depend on phase µ µ changes in quantum manner under the action of an electromagnetic field zero resistance, Meißner-Ochsenfeld effect, coherence effects (flux quantization, Josephson effect) similarity to electron orbiting around nucleus in an atom Chap. 1 - 16 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Central hypothesis of macroscopic quantum model: there exists a macroscopic wave function describing the behavior of all superelectrons in a superconductor (motivation: superconductivity is a coherent phenomenon of all sc electrons) normalization condition: Ns and ns (r,t) are the total and local density of superconducting electrons charged superfluid (analogy to fluid mechanics) similarities in description of superconductivity and superfluids no explanation of microscopic origin of superconductivity relevant issue: describe superelectron fluid as quantum mechanical entity Chap. 1 - 17 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Some basic relations: • general relations in electrodynamics: electric field: flux density: A = vector potential f = scalar potential • electrical current is driven by gradient of electrochemical potential: • canonical momentum: • kinematic momentum: Chap. 1 - 18 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • Schrödinger equation for charged particle: electro-chemical potential • insert macroscopic wave-function Y y, q qs , m ms • split up into real and imaginary part and assume real part: energy-phase relation kinetic energy imaginary part: potential energy current-phase relation superelectron velocity vs Js = ns qs vs Chap. 1 - 19 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • we start from Schrödinger equation: electro-chemical potential • we use the definition and obtain with Chap. 1 - 20 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.2 Macroscopic Quantum Currents in Superconductors Chap. 1 - 21 • equation for real part: ∆ R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.2 Macroscopic Quantum Currents in Superconductors energy-phase relation (term of order ²ns is usually neglected) Chap. 1 - 22 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • interpretation of energy-phase relation corresponds to action in the quasi-classical limes the Hamilton-Jacobi equation the energy-phase-relation becomes Chap. 1 - 23 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.2 Macroscopic Quantum Currents in Superconductors • equation for imaginary part: continuity equation for probability density and probability current density conservation law for probability density Chap. 1 - 24 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • energy-phase relation 1 • supercurrent density-phase relation (London parameter) 2 equations (1) and (2) have general validity for charged and uncharged superfluids (i) superconductor with Cooper pairs of charge qs = -2e (ii) neutral Bose superfluid, e.g. 4He (iii) neutral Fermi superfluid, e.g. 3He we use equations (1) and (2) to derive London equations • note: k drops out !! Chap. 1 - 25 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) additional topic: gauge invariance expression for the supercurrent density must be gauge invariant with the gauge invariant phase gradient: the supercurrent is London coefficient London penetration depth Chap. 1 - 26 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Summary: macroscopic wave function describes whole ensemble of superelectrons with the current-phase relation (supercurrent equation) is gauge invariant phase gradient the energy-phase relation is Chap. 1 - 27 1.1.2 Macroscopic Quantum Currents in Superconductors R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) using current-phase and energy-phase relation we can derive: • 1. and 2. London equation • Fluxoid quantization • Josephson equations Chap. 1 - 28 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.3 The London Equations Fritz London (1900 – 1954) Chap. 1 - 29 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.1.3 The London Equations • London equations are purely phenomenological - describe behavior of superconductors - starting point: supercurrent-phase relation (London parameter) 2nd London equation and the Meißner-Ochsenfeld effect: • take the curl second London equation: Maxwell: which leads to: • describes Meißner-Ochsenfeld effect: applied field decays exponentially inside superconductor, decay length: (London penetration depth) Chap. 1 - 30 1.1.3 The London Equations R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • example: plane surface extending in yz-plane, magnetic field Bz parallel to z-axis: exponential decay with Chap. 1 - 31 1.1.3 The London Equations R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1st London equation and perfect conductivity: • take the time derivative • use energy-phase relation: • substitution and using first London equation: • neglecting 2nd term: (linearized 1st London equation) • for a time-independent supercurrent the electric field inside the superconductor vanishes dissipationless supercurrent Chap. 1 - 32 1.1.3 The London Equations R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • Processes that could cause a decay of Js: example: Fermi sphere in two dimensions in the kxky – plane • T = 0: all states inside the Fermi circle are occupied • current in x-direction shift of Fermi circle along kx by ±dkx • normal state: relaxation into states with lower energy (obeying Pauli principle) centered Fermi sphere, current relaxes • superconducting state: all Cooper pairs must have same center of mass moment only scattering around the sphere no decay of supercurrent normal state superconducting state Chap. 1 - 33 1.1.3 The London Equations R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) additional topic: Linearized 1. London Equation • usually, 1. London equation is linearized: allowed if |E| >> |vs | |B| condition is satisfied in most cases kinetic energy of superelectrons Note: the nonlinear first London equation results from the Lorentz's law and the second London equation exact form of the expression describing the phenomenon of zero dc resistance in superconductors the first London equation is derived using the second London equation Meißner-Ochsenfeld effect is the more fundamental property of superconductors than the vanishing dc resistance additional topic: The London Gauge (see lecture notes) • rigid phase: • no conversion of Js in Jn: Chap. 1 - 34 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.2. Fluxoid Quantization • direct consequence of macroscopic quantum model Gedanken-experiment: generate supercurrent in a ring zero dc-resistance stationary state • quantum treatment: stationary state determined by quantum conditions • cf. Bohr‘s model for atoms: quantization condition for angular momentum no destructive interference of electron wave stationary state stationary supercurrent: macroscopic wave function is not allowed to interfere destructively quantization condition superconducting cylinder • derivation of quantization condition (based on macroscopic quantum model of superconductivity) start with supercurrent density: Chap. 1 - 35 1.2. Fluxoid Quantization R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • integration of expression for supercurrent density around a closed contour • Stoke‘s theorem (path C in simply or multiply connected region): • applied to supercurrent: • integral of phase gradient: • if r1 r2 (closed path), then integral → zero but: phase is only specified within modulo 2p of its principal value [-p, p]: qn = q0 + 2p n Chap. 1 - 36 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.2. Fluxoid Quantization • then: fluxoid is quantized • flux quantum: • quantization condition holds for all contour lines including contour that has shrunk to single point r1 = r2 in limit r1 → r2 n = 0 • contour line can no longer shrink to single point inclusion of nonsuperconducting region in contour r1 ≠ r2 in limit r1 → r2 n ≠ 0 possible Chap. 1 - 37 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.2.1 Fluxoid and Flux Quantization • Fluxoid Quantization: total flux = external applied flux + flux generated by induced supercurrent must have discrete values • Flux Quantization: - superconducting cylinder, wall much thicker than ¸L - application of small magnetic field at T < Tc screening currents, no flux inside application of Hext during cool down: screening current on outer and inner wall amount of flux trapped in cylinder: satisfies fluxoid quantization condition wall thickness >> l J: closed contour deep inside with Js = 0 then: flux quantization remove field after cooling down: trapped flux: integer of the flux quantum Chap. 1 - 38 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.2.1 Fluxoid and Flux Quantization magnetic flux Hext Js Js inner surface current r r outer surface current Chap. 1 - 39 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.2.1 Fluxoid and Flux Quantization Bext > 0 Bext > 0 Bext = 0 T > Tc T < Tc T < Tc Chap. 1 - 40 1.2.1 Fluxoid and Flux Quantization R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • Flux Trapping: why is flux not expelled after switching of external field Js /t = 0 according to 1st London equation: E = 0 deep inside (supercurrent only on surface within l J ) • with and we get: F: magnetic flux enclosed in loop contour deep inside the superconductor: E = 0 and therefore flux enclosed in cylinder stays constant Chap. 1 - 41 1.2.2 Experimental Proof of Flux Quantization R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • 1961 by Doll/Näbauer at Munich, Deaver/Fairbanks at Stanford quantization of magnetic flux in superconducting cylinder Cooper pairs with qs = - 2e cylinder with wall thickness À l J different amounts of flux are frozen in during cooling down in Bcool measure amount of trapped flux required: large relative changes of magnetic flux small fields and small diameter d for d = 10 µm we need 2x10-5 T for one flux quantum measurement of (very small) torque D = ¹ x Bp due to probe field Bp resonance method Bp amplitude of rotary oscillation ≈ exciting torque quartz thread quartz cylinder Pb d ≈ 10 µm Chap. 1 - 42 1.2.2 Experimental Proof of Flux Quantization 3 4 R. Doll, M. Näbauer Phys. Rev. Lett. 7, 51 (1961) trapped magnetic flux (h/2e) resonance amplitude (mm/Gauss) R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 4 2 1 F0 0 -1 -0.1 0.0 0.1 0.2 0.3 0.4 B.S. Deaver, W.M. Fairbank Phys. Rev. Lett. 7, 43 (1961) 3 2 1 0 0.0 Bcool (Gauss) 0.1 0.2 0.3 0.4 Bcool (Gauss) prediction by Fritz London: h/e experimental proof for existence of Cooper pairs qs = - 2e Paarweise im Fluss D. Einzel, R. Gross, Physik Journal 10, No. 6, 45-48 (2011) Chap. 1 - 43 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.3 Josephson Effect Brian David Josephson (born 04. 01. 1940) Nobel Prize in Physics 1973 "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" (together with Leo Esaki and Ivar Giaever) Chap. 1 - 45 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.3.1 Josephson Equations • what happens if we weakly couple two superconductors? - coupling by tunneling barriers, point contacts, normal conducting layers, etc. - do they form a bound state such as a molecule? - if yes, what is the binding energy? • B.D. Josephson in 1962 (nobel prize with Esaki and Giaever in 1973) Cooper pairs can tunnel through thin insulating barrier expectation: - tunneling probability for pairs ≈ (|T|²)2 extremely small ≈ (10-4)2 Josephson: - tunneling probability for pairs ≈ |T|² - coherent tunneling of pairs („tunneling of macroscopic wave function“) finite supercurrent at zero applied voltage Josephson effects oscillation of supercurrent at constant applied voltage finite binding energy of coupled SCs = Josephson coupling energy Chap. 1 - 46 • coupling is weak supercurrent density is small |ª|2 = ns is not changed • supercurrent density depends on gauge invariant phase gradient: • then: - replace gauge invariant phase gradient ° by gauge invariant phase difference: g(x) dx,g(x) (arb. units) • simplifying assumptions: - current density is homogeneous - ° varies negligibly in electrodes - Js same in electrodes and junction area ° in superconducting electrodes much smaller than in insulator I 1.0 ns(x) 0.5 ns /n R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.3.1 Josephson Equations g(x) 0.0 S1 I S2 -0.5 -1.0 -6 g(x) dx -4 -2 0 2 4 6 x (arb. units) Chap. 1 - 47 1.3.1 Josephson Equations R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) first Josephson equation: • we expect: • for Js = 0: phase difference must be zero: therefore: Jc: critical/maximum Josephson current density general formulation of 1st Josephson equation • in most cases: keep only 1st term (especially for weak coupling): 1. Josephson equation: current – phase relation • generalization to spatially inhomogeneous supercurrent density: derived by Josephson for SIS junctions supercurrent density varies sinusoidally with = q2- q1 w/o external potentials Chap. 1 - 48 1.3.1 Josephson Equations R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • other argument why there are only sin contributions to Josephson current time reversal symmetry • if we reverse time, the Josephson current should flow in opposite direction - t → -t, Js → - Js • the time evolution of the macroscopic wave functions is exp [iq(t)] = exp [iwt] - if we reverse time, we have t → -t • if the Josephson effect stays unchanged under time reversal, we have to demand satisfied only by sin-terms Chap. 1 - 49 1.3.1 Josephson Equations R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) second Josephson equation: • time derivative of the gauge invariant phase difference: • substitution of the energy-phase relation gives: • supercurrent density across the junction is continuous (Js (1) = Js(2)): (term in parentheses = electric field) voltage – phase relation 2. Josephson equation: voltage drop across barrier Chap. 1 - 50 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.3.1 Josephson Equations • for a constant voltage across the junction: • Is is oscillating at the Josephson frequency n = V/F0: voltage controlled oscillator • applications: - Josephson voltage standard - microwave sources Chap. 1 - 51 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.3.2 Josephson Tunneling • what determines the maximum Josephson current density Jc ? - insulating tunneling barrier, thickness d • calculation by wave matching method: • energy-phase relation: E0 = kinetic energy time depedent macroscopic wave function: • wave function within barrier with height V0 > E0: only elastic processes time evolution is the same outside and inside barrier consider only time independent part time independent Schrödinger(-like) equation for region of constant potential Chap. 1 - 52 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) 1.3.2 Josephson Tunneling • assumption: homogeneous barrier and supercurrent flow 1d problem • solutions: - in superconductors - in insulator: sum of decaying and growing exponentials - characteristic decay constant: barrier properties • coefficients A and B are determined by the boundary conditions at x = ± d/2: n1,2 , q1,2: Cooper pair density and wave function phase at the boundaries x = ± d/2 Chap. 1 - 53 1.3.2 Josephson Tunneling R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) • solving vor A and B: • supercurrent density: • substituting the coefficients A and B: current-phase relation maximum Josephson current density: • real junctions: V0 ≈ few meV 1/κ < 1 nm, d few nm κd À 1, then: • maximum Josephson current decays exponentially with increasing thickness d: Chap. 1 - 54 R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Summary Macroscopic wave function ª: describes ensemble of macroscopic number of superconducting pairs |ª|2 describes density of superconducting pairs Current density in a superconductor: Gauge invariant phase gradient: Phenomenological London equations: flux/fluxoid quantization: Chap. 1 - 55 Summary R. Gross, A. Marx and F. Deppe, © Walther-Meißner-Institut (2001 - 2013) Josephson equations: maximum Josephson current density Jc: wave matching method Chap. 1 - 56