Chapter 1. Macroscopic Quantum Phenomena

Transcription

1 Chapter 1 Macroscopic Quantum Phenomena

2 Chap. 1-2 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

3 Chap 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 e iq(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

4 Chap Coherent Phenomena in Superconductivity 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

5 Chap Coherent Phenomena in Superconductivity 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 (E pot = 0 E = E kin ): start with wave equation v ph : phase velocity

6 Chap Coherent Phenomena in Superconductivity 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:

7 Chap Coherent Phenomena in Superconductivity with multiply by use since

8 Chap 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 2 nd term on lhs of is constant solutions can be split in a two parts depending only on space and time

9 Chap Coherent Phenomena in Superconductivity stationary Schrödinger equation:

10 Chap Coherent Phenomena in Superconductivity 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)

11 Chap Coherent Phenomena in Superconductivity some math: derivation of probability current density (Schrödiger equation) (multiply from left by Y*) subtracting the complex conjugate yields vector identity:

12 Chap Coherent Phenomena in Superconductivity time evolution of defines probability current: Task: find for a charged particle in em field start with classical equation of motion: canonical momemtum (kinetic and field momentum)

13 Chap Coherent Phenomena in Superconductivity Lorentz s law with E and B expressed in terms of potential Φ and A Lorentz s law: with: generalized potential: Schrödinger equation: insert expression for V(r,t)

14 Chap Coherent Phenomena in Superconductivity 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

15 Chap Macroscopic Quantum Currents in Superconductors 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

16 Chap Macroscopic Quantum Currents in Superconductors 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

17 Chap Macroscopic Quantum Currents in Superconductors 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: N s and n s (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

18 Chap Macroscopic Quantum Currents in Superconductors 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:

19 1.1.2 Macroscopic Quantum Currents in Superconductors Schrödinger equation for charged particle: electro-chemical potential insert macroscopic wave-function Y y, q q s, m m s split up into real and imaginary part and assume real part: energy-phase relation kinetic energy potential energy imaginary part: current-phase relation superelectron velocity v s J s = n s q s v s Chap. 1-19

20 Chap Macroscopic Quantum Currents in Superconductors we start from Schrödinger equation: electro-chemical potential we use the definition and obtain with

21 Chap Macroscopic Quantum Currents in Superconductors

22 1.1.2 Macroscopic Quantum Currents in Superconductors equation for real part: energy-phase relation (term of order ²n s is usually neglected) Chap. 1-22

23 Chap Macroscopic Quantum Currents in Superconductors interpretation of energy-phase relation corresponds to action in the quasi-classical limes the Hamilton-Jacobi equation the energy-phase-relation becomes

24 Chap Macroscopic Quantum Currents in Superconductors equation for imaginary part: conservation law for probability density continuity equation for probability density and probability current density

25 1.1.2 Macroscopic Quantum Currents in Superconductors 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 q s = -2e (ii) (iii) neutral Bose superfluid, e.g. 4 He neutral Fermi superfluid, e.g. 3 He we use equations (1) and (2) to derive London equations note: k drops out!! Chap. 1-25

26 Chap Macroscopic Quantum Currents in Superconductors 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

27 Chap Macroscopic Quantum Currents in Superconductors 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

28 Chap Macroscopic Quantum Currents in Superconductors using current-phase and energy-phase relation we can derive: 1. and 2. London equation Fluxoid quantization Josephson equations

29 1.1.3 The London Equations Fritz London ( ) Chap. 1-29

30 1.1.3 The London Equations Chap London equations are purely phenomenological - describe behavior of superconductors - starting point: supercurrent-phase relation (London parameter) 2 nd 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)

31 1.1.3 The London Equations Chap example: plane surface extending in yz-plane, magnetic field B z parallel to z-axis: exponential decay with

32 1.1.3 The London Equations Chap st London equation and perfect conductivity: take the time derivative use energy-phase relation: substitution and using first London equation: neglecting 2 nd term: (linearized 1 st London equation) for a time-independent supercurrent the electric field inside the superconductor vanishes dissipationless supercurrent

33 1.1.3 The London Equations Chap Processes that could cause a decay of J s : example: Fermi sphere in two dimensions in the k x k y plane T = 0: all states inside the Fermi circle are occupied current in x-direction shift of Fermi circle along k x by ±dk x normal state: superconducting state: normal state relaxation into states with lower energy (obeying Pauli principle) centered Fermi sphere, current relaxes all Cooper pairs must have same center of mass moment only scattering around the sphere no decay of supercurrent superconducting state

34 1.1.3 The London Equations additional topic: Linearized 1. London Equation usually, 1. London equation is linearized: allowed if E >> v s 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 J s in J n : Chap. 1-34

35 1.2. Fluxoid Quantization Chap 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 superconducting cylinder stationary supercurrent: macroscopic wave function is not allowed to interfere destructively quantization condition derivation of quantization condition (based on macroscopic quantum model of superconductivity) start with supercurrent density:

36 1.2. Fluxoid Quantization Chap 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 r 1 r 2 (closed path), then integral zero but: phase is only specified within modulo 2p of its principal value [-p, p]: q n = q 0 + 2p n

37 1.2. Fluxoid Quantization Chap then: fluxoid is quantized flux quantum: quantization condition holds for all contour lines including contour that has shrunk to single point r 1 = r 2 in limit r 1 r 2 n = 0 contour line can no longer shrink to single point inclusion of nonsuperconducting region in contour r 1 r 2 in limit r 1 r 2 n 0 possible

38 1.2.1 Fluxoid and Flux Quantization Chap 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 < T c screening currents, no flux inside application of H ext 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 J s = 0 then: flux quantization remove field after cooling down: trapped flux: integer of the flux quantum

39 1.2.1 Fluxoid and Flux Quantization Chap H ext magnetic flux J s J s inner surface current r r outer surface current

40 1.2.1 Fluxoid and Flux Quantization Chap B ext > 0 B ext > 0 B ext = 0 T > T c T < T c T < T c

41 1.2.1 Fluxoid and Flux Quantization Chap Flux Trapping: why is flux not expelled after switching of external field J s / t = 0 according to 1 st 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

42 1.2.2 Experimental Proof of Flux Quantization 1961 by Doll/Näbauer at Munich, Deaver/Fairbanks at Stanford quantization of magnetic flux in superconducting cylinder Cooper pairs with q s = - 2e cylinder with wall thickness À l J different amounts of flux are frozen in during cooling down in B cool 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 B p due to probe field B p resonance method B amplitude of rotary oscillation exciting torque p quartz thread Pb quartz cylinder d 10 µm Chap. 1-42

43 resonance amplitude (mm/gauss) trapped magnetic flux (h/2e) Chap Experimental Proof of Flux Quantization 4 3 R. Doll, M. Näbauer Phys. Rev. Lett. 7, 51 (1961) 4 3 B.S. Deaver, W.M. Fairbank Phys. Rev. Lett. 7, 43 (1961) F B cool (Gauss) B cool (Gauss) prediction by Fritz London: h/e experimental proof for existence of Cooper pairs q s = - 2e Paarweise im Fluss D. Einzel, R. Gross, Physik Journal 10, No. 6, (2011)

44 1.3 Josephson Effect (together with Leo Esaki and Ivar Giaever) Chap Brian David Josephson (born ) 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"

45 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 oscillation of supercurrent at constant applied voltage Josephson effects finite binding energy of coupled SCs = Josephson coupling energy Chap. 1-46

46 n s /n Chap Josephson Equations coupling is weak supercurrent density is small ª 2 = n s is not changed supercurrent density depends on gauge invariant phase gradient: simplifying assumptions: - current density is homogeneous - varies negligibly in electrodes - J s same in electrodes and junction area in superconducting electrodes much smaller than in insulator I then: - replace gauge invariant phase gradient by gauge invariant phase difference: n s (x) S 1 I S 2 g(x) dx x (arb. units) g(x) g(x) dx, g(x) (arb. units)

47 1.3.1 Josephson Equations first Josephson equation: we expect: for J s = 0: phase difference must be zero: therefore: J c : critical/maximum Josephson current density general formulation of 1 st Josephson equation in most cases: keep only 1 st 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 = q 2 - q 1 w/o external potentials Chap. 1-48

48 Chap Josephson Equations 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, J s - J s 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

49 1.3.1 Josephson Equations 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 (J s (1) = J s (2)): (term in parentheses = electric field) 2. Josephson equation: voltage phase relation voltage drop across barrier Chap. 1-50

50 Chap Josephson Equations for a constant voltage across the junction: I s is oscillating at the Josephson frequency n = V/F 0 : voltage controlled oscillator applications: - Josephson voltage standard - microwave sources

51 1.3.2 Josephson Tunneling Chap what determines the maximum Josephson current density J c? - insulating tunneling barrier, thickness d calculation by wave matching method: energy-phase relation: E 0 = kinetic energy time depedent macroscopic wave function: wave function within barrier with height V 0 > E 0 : 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

52 1.3.2 Josephson Tunneling Chap 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: n 1,2, q 1,2 : Cooper pair density and wave function phase at the boundaries x = ± d/2

53 1.3.2 Josephson Tunneling Chap solving vor A and B: supercurrent density: substituting the coefficients A and B: maximum Josephson current density: current-phase relation real junctions: V 0 few mev 1/κ < 1 nm, d few nm κd À 1, then: maximum Josephson current decays exponentially with increasing thickness d:

54 Summary Chap 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:

55 Summary Chap Josephson equations: maximum Josephson current density J c : wave matching method