Spin Meissner effect in superconductors and the origin of the Meissner effect

1. Spin Meissner effect in superconductors and the origin of the Meissner effect Hvar, 2008 J.E. Hirsch, UCSD • Why the Meissner effect is not understood, and how it can be understood • Spin Meissner effect: spontaneous spin current in the ground state of superconductors • Charge expulsion, charge inhomogeneity in superconducting state • Electrodynamic (London-like) equations for charge and spin • Experiments

2. 3 key pieces of the physics that BCS theory got right: * Cooper pairs * Energy gap * Macroscopic quantum coherence * Electron-phonon-induced attraction between electrons

3. 6 key pieces of physics that BCS theory missed: (1) Key role of electron-hole asymmetry (2) Key role of kinetic energy lowering as driving force (3) Macroscopiccharge inhomogeneity and internal E-field http://physics.ucsd.edu/~jorge/hole.html (4) Key role of spin-orbit interaction Theory of hole superconductivity (5) Key role of mesoscopic orbits (6) Spontaneous currents in the absence of applied fields 3 key pieces of the physics that BCS theory got right: * Cooper pairs * Energy gap 1988-2008 * Macroscopic quantum coherence * Electron-phonon-induced attraction between electrons

4. cool 1933 lL=London penetration depth Meissner effect: expulsion of magnetic field from interior of superconductor The expulsion of magnetic flux from the interior of a superconducting metal when it is cooled in a magnetic field to below the critical temperature, near absolute zero, at which the transition to superconductivity takes place. It was discovered by Walther Meissner in 1933, when he measured the magnetic field surrounding two adjacent long cylindrical single crystals of tin and observed that at ?452.97°F (3.72 K) the Earth's magnetic field was expelled from their interior. This indicated that at the onset of superconductivity they became perfect diamagnets. This discovery showed that the transition to superconductivity is reversible, and that the laws of thermodynamics apply to it. The Meissner effect forms one of the cornerstones in the understanding of superconductivity, and its led F. London and H. London to develop their phenomenological electrodynamics of superconductivity. The magnetic field is actually not completely expelled, but penetrates a very thin surface layer where currents flow, screening the interior from the magnetic field.

5. apply B EFaraday cool I I same final state two pathways to Meissner current expel B cool B super normal super normal cool apply B I Faraday electric field points in opposite directions Meissner current I points in the same direction

6. Why the Meissner effect is a puzzle B EFarad I Lower the temperature... or lower slightly the applied H... Meissner state B=0 Current develops 'spontaneously' upon cooling or lowering H opposing EFarad What is the 'force'pushing the electrons near the surface to start moving all in the same direction, opposite to eEFarad ? How is angular momentum conserved???

7. r=2lL orbits! =angular momentum of 1 el 2lL bulk =# of electrons in surface layer The key to the Meissner effect cylinder B I lL R vs Angular momentum in Meissner current: (h=cylinder height, ns=superfluid density) . somevery complicated math. . . .

8. Some simple relations: Normal state: Superconducting state: London penetration depth: Density of states at the Fermi energy: Magnetic susceptibility: Magnetic susceptibility:

9. Larmor diamagnetism B or Dv v Orbital magnetic moment: r Apply magnetic field B: magnetic susceptibility per unit volume: n=electrons/unit vol

10. How the transition occurs B r kF-1 r=kF-1 orbits r=2lL orbits 2lL Superconducting state: Normal state: orbit expansion:

11. two pathways to Meissner current expel B apply B super normal EFaraday cool I I same final state

12. The two pathways to the Meissner current Dv F E BCS: Apply magnetic field B: Expand electron orbit in B: Faraday's law pushes e- Lorentz force pushes e- B B v r v r vf r=2lL

13. r=kF-1 orbits r=2lL orbits Why is there macroscopic phase coherence in superconductors? Superconducting state Highly overlapping orbits Phase coherence necessary to avoid collisions Normal state Non-overlapping orbits Relative phase doesn't matter

14. A little help from a friend... ==> , 137

15. A little help from a friend... ==> , The speed of light must enter into the superconducting wave function! 137

16. So we learn from the Meissner effect that: transition to superconductivity = expansion of electronic orbit from r=kF-1 to r=2lL What happens when there is no magnetic field? Spin-orbit force deflects electron in expanding orbit!

17. Spin orbit scattering spin-orbit spin-orbit m p v m m v m a moving magnetic moment is equivalent to an electric dipole (Goldberger&Watson) scattering center scattering center

18. m p v m So we learn from the Meissner effect that: transition to superconductivity = expansion of electronic orbit from r=kF-1 to r=2lL What happens when there is no magnetic field? Spin-orbit force deflects electron in expanding orbit! m E

19. m E m r=2lL For What's E? r . . . .

21. The two pathways to the Spin Meissner current t Dv 'Apply' electric field E: Expand electron orbit in E: Maxwell's law pushes m Lorentz torque pushes m E(t) B m m v r v r vf r=2lL

22. r=2lL orbits r=2lL orbits Ground state of a superconductor spin down electrons spin up electrons Currents in the interior cancel out, near the surface survive ==> there is a spontaneous spin current in the ground state of superconductors!

23. There is a spontaneous spin current in the ground state of superconductors, flowing within lL of the surface m For lL=400A, vs0=72,395cm/s vs0 m n # of carriers in the spin current: ns B vs The slowed-down spin component stops when (JEH, EPL81, 67003 (2008)) no external fields applied When a magnetic field is applied:

24. Summary of argument: (Ampere, Faraday, Newton, London) 1) 2) Orbits have radius (to explain origin of Meissner current) 3) Magnetic moment of electron is 4) Background positive charge density is = - superfluid dens. 1)+2)+3)+4) ==> magnetic field that stops the spin current is Hc1 1)+2)+3)+4) ==> Therefore: • Superconductivity is an intrinsically relativistic effect • Electron spin and associated magnetic moment plays a key role • The wavefunction of a superconductor contains c=speed of light

25. Back to: cooling a superconductor in the presence of a B-field: A clue from plasma physics B Vortex state B Meissner state I I B Intermediate state What makes electrons move in the direction needed to create all these currents when T is lowered from above to below Tc? I

26. A clue from plasma physics www.mpia-hd.mpg.de/homes/fendt/Lehre/Lecture_OUT/lect_jets4.pdf

27. But if there is charge flow, it will result in charge inhomogeneity and an electric field in the interior of superconductors. v v v Vortex state B FB FB FB B I B Le Intermediate state ve Electrons have to flow away from the interior of the superconductor, towards the surface and towards the normal regions! Le Meissner state I

28. But if there is charge flow, it will result in charge inhomogeneity and an electric field in the interior of superconductors. London says NO. First London equation (1934): free acceleration of electrons (n=density, v=speed, J=current) If E = 0, J increases to infinity, unless Newton’s law is violated? can be zero even if E is non-zero! Can there be an electric field inside superconductors? !

29. 2) , continuity equation: ==> ==> New electrodynamic equations for superconductors (JEH, PRB69, 214515 (2004)) 1) ==> Note: , NOT integrate in time, 1 integration constant r0 , ...

30. Relativistic form: 2 2 or equivalently Electrodynamics

31. lL Solution for sphere of radius R: Electrostatics: ; outside supercond. +assume f(r) and its normal derivative are continuous at surface No electric field outside sphere

32. Sample size dependence of expelled charge (Q) and E-field r- r0 lL E Em independent of R r R1 R2 sphere of radius R r-< 0 = charge density near surface r0> 0 = charge density in interior Q ~ r0 R3 ~ -r- R2 lL Electrostatic energy cost: UE ~ Q2/R ~ (r- R2 lL)2/R ~ (r-)2R3 ~ (r0)2R5 ~ Volume~R3 r- independent of R, r0 ~ 1/R ==> how big is r- , Em? Electric field vs. r:

33. Emax r R R How much charge is expelled?

34. E carries a spin current necessarily in the presence of internal E-field Flows within a London penetration depth of the surface Speed of spin current carriers: ~ 100,000 cm/s no charge current ==> no B-field Number of spin current carriers: =superfluid density spin current without charge current! Spin currents in superconductors (JEH, Phys. Rev. B 71, 184521 (2005)) Internal electric field (in the absence of applied B) pointing out

35. r- lL r0 r=2lL orbits We now have 2 new pieces of physics of superconductors: charge expulsion spin current How are they related? How much charge is expelled?

36. ==> (JEH) ~ Hc1

37. Emax (charge neutra- lity) R m vs0 m n ) (Recall

38. Spin current electrodynamics (4d formulation)

39. ~ condensation energy of sc Energetics Apply a magnetic field: ==> condensation energy per particle: energy lowering per particle in entering sc state: 2ec =ec + ec Coulomb energy cost + condensation energy

40. Type I vs type II materials x=distance between orbit centers Type I: x > 2lL Type II: x < 2lL Phase difference:

41. kF too many electrons! What drives superconductivity? 1) Excess negative charge (CuO2)=, (MgB2)-, (FeAs)- 2) Almost full bands (hole conduction in normal state) 3) Kinetic energy lowering (Kinetic energy is highest when band is almost full) kF-1 is small

42. Lions B B How is angular momentum conserved in the Meissner effect?? Electromagnetic field carries angular momentum! Le =-Le But r- is way too small to give enough Lfield Lfield Spin-orbit interaction transfers counter-L to ions! JEH, J. Phys.: Condens. Matter 20 (2008) 235233

43. Experimental tests? m vs0 m n 1) Detect spin current * polarized light scattering (PRL100, 086603 (08) * inelastic polarized neutron scattering * photoemission * Detect electric fields produced by spin current * Insert a 'spin current rectifier' 2) Detect internal electric field * positrons, muons, neutrons 3) Response of superconductor to applied electric field (Tao effect) 4) Detect change in plasmon dispersion relation in sc state .....

44. IT IS A FALSIFIABLE THEORY! * A superconductor that has no outward-pointing electric field in its interior * A superconductor that expels magnetic fields without expelling negative charge * A superconductor that has no spontaneous spin current near the surface, with carrier density ns/2 and speed * A superconductor that has no hole carriers in normal state * A high Tc superconductor with no excess negative charge anywhere * A superconductor with tunneling asymmetry of intrinsic origin that has opposite sign to the one usually observed * A superconductor with gap function that has noek dependence * A superconductor not driven by kinetic energy lowering To prove this theory wrong, find clear experimental evidence for any of the following: