Key symmetries of superconductivity

Key symmetries of superconductivity Inversion and time reversal symmetry Sendai, March 2009 1 st GCOE International Symposium 3 He A 1 phase Manfred Sigrist, ETH Zürich UGe 2 paramagnetic CePt 3 Si ferromagnetic SC

Key symmetries of superconductivity Inversion and time reversal symmetry Sendai, March 2009 Manfred Sigrist, ETH Zürich superconductivity: general introduction Cooper pairing: symmetry aspects role of inversion and time reversal symmetry superconductivity in the absence of inversion and time reversal sym. lack of inversion symmetry: non-centrosymmetric superconductors and some of their physical properties

Superconductivity Electrical resistance (1911) Field expulsion (1933) Meissner-Ochsenfeld effect resistivity B B=0 B T c temperature T>T c T<T c Superconductivity as a thermodynamic phase B London theory (1935) " # $ 2r j = % B r " # r B = 4$ c r j density of superconducting electrons " 2 r B = # $2 r B " #2 = 4$e2 n s mc 2 London penetration depth λ x

Superconductivity Electrical resistance (1911) Field expulsion (1933) Meissner-Ochsenfeld effect resistivity B B=0 B T c temperature T>T c T<T c Superconductivity as a thermodynamic phase Ginzburg-Landau theory (1950) Superconductivity described by a complex macroscopic wave function order parameter Phenomenology of superconductivity

Superconductivity Electrical resistance (1911) Field expulsion (1933) Meissner-Ochsenfeld effect resistivity B B=0 B T c temperature Anderson-Higgs T>T c mechanism Superconductivity as a thermodynamic phase Ginzburg-Landau violates theory U(1)-gauge (1950) symmetry Superconductivity described by a complex macroscopic wave function superconductivity Phenomenology of superconductivity order parameter T<T c

Phenomenological point of view Spontaneous symmetry breaking

Ginzburg-Landau theory supplement 2 nd order phase transition from normal to superconducting state spontaneous symmetry breaking order parameter: macroscopic wave function normal phase T > T c superconducting phase T < T c Free energy expansion at T c F T>T c scalar under U(1)-gauge operation T<T c

Ginzburg-Landau theory 2 nd order phase transition from normal to superconducting state spontaneous symmetry breaking supplement order parameter: macroscopic wave function phase Free energy expansion at T c normal phase T > T c determined superconducting phase T < T c free spontaneous breaking of U(1) symmetry F T>T c scalar under U(1)-gauge operation T<T c

Ginzburg-Landau theory local U(1)-gauge invariance supplement Free energy functional gradient vector potential variational equations inhomogeneous oder parameter structures domain walls, vortices etc supercurrent London equation massive photon Anderson-Higgs mechanism

Ginzburg-Landau theory local U(1)-gauge invariance supplement Free energy functional single-valued macroscopic wave function gradient variational equations vector potential flux quantization inhomogeneous oder parameter structures domain walls, vortices etc persistent current dissipation free supercurrent London equation massive photon Anderson-Higgs mechanism

Microscopic point of view Cooper pairing of electrons

Microscopic theory of superconductivity Bardeen-Cooper-Schrieffer (BCS) (1957) Superconducting state as a coherent state of electron Cooper pairs with " r k # Cooper pairs r k " Fermi sea 0 electron 1 electron total momentum in each single-electron state superconductor coherent state free adding / removing of Cooper pairs number of pairs not fixed

Microscopic theory of superconductivity Bardeen-Cooper-Schrieffer (BCS) (1957) Superconducting state as a coherent state of electron Cooper pairs with pair wave function U(1) gauge operation phase 2α conjugate to pair number free adding / removing of Cooper pairs order parameter macroscopic wavefunction superconductor coherent state number of pairs not fixed

Pairing interaction - electron phonon (BCS) Cooper pair formation (bound state of 2 electrons) needs attractive interaction k k k -k k -k -k -k electron phonon interaction: attractive interaction scattering between electron states with degenerate energy

Alternative mechanism for Cooper pairing Pairing by magnetic fluctuations: Berk & Schrieffer (1966) T Quantum Critical Point AF SC easily spin polarizable medium longer ranged interaction pairing for higher angular momentum CeIn 3

Cooper pair symmetry Symmetry of pairs of identical electrons: Pauli principle: wave function totally antisymmetric under particle exchange orbital spin " r k s' r k s r k s Fermi sea " r k s' P tot =0 k r # " k r s! s' even parity: odd parity: angular momentum L =0,2,4,, S=0 spin singlet even odd L = 1,3,5,, S=1 spin triplet odd spin even

Cooper pair symmetry Symmetry of pairs of identical electrons: Pauli principle: wave function totally antisymmetric under particle exchange Classification " k r s' r k s r k s Fermi sea " r k s' P tot =0 k r # " k r s! L = 0, S = 0: most symmetric conventional pairing L > 0 : lower symmetry unconventional pairing s' even parity: odd parity: angular momentum L =0,2,4,, S=0 spin singlet even odd L = 1,3,5,, S=1 spin triplet odd spin even

Cooper pair symmetry Symmetry of pairs of identical electrons: Pauli principle: wave function totally antisymmetric under particle exchange " k r s' r k s r k s Fermi sea " r k s' P tot =0 key symmetries for this classification k r # " k r s! time reversal & inversion s' even parity: odd parity: angular momentum L =0,2,4,, S=0 spin singlet even odd L = 1,3,5,, S=1 spin triplet odd spin even

Anderson s Theorems (1959,1984) Cooper pairs with total momentum P tot =0 form from degenerate quasiparticle states. with How to guarantee existence of degenerate partners? Spin singlet pairing: time reversal symmetry harmful: magnetic impurities, ferromagnetism, Zeemann fields (paramagnetic limiting) Spin triplet pairing: inversion symmetry harmful: crystal structure without inversion center

Basic Model of systems without inversion center

Key symmetries and band structure Electron state: time reversal: inversion: Electron spectrum: orbital and spin part distinctly treated charge density spin density conserved time reversal inversion

Lack of time reversal symmetry UGe 2 paramagnetic superconductivity in a ferromagnet M, H ferromagnetic C 4v SC Electrons in a ferromagnet / magnetic field: and Zeeman field

Band structure - band splitting Spin split energy spectrum: E k spin polarization E k

Band structure - band splitting Spin split energy spectrum: k y minority k x Fermi surface splitting E k majority

Lack of inversion symmetry - non-centrosymmetric CePt 3 Si A B C A B C C B A C B A E C 4v motion of electron in electric field: and special relativity spin-orbit coupling: Rashba-like spin-orbit coupling

Band structure - band splitting Spin split energy spectrum: E k spin-orbit couping E k

Band structure - band splitting Spin split energy spectrum: k y k x Fermi surface splitting E k

Superconducting phase

Superconducting phases pair wave function spin singlet, even parity spin triplet, odd parity 1 configuration 3 configurations

Anderson theorem for small perturbation supplement superconducting phase: bare T c = T c0 and singlet pairing no inversion no time reversal

Anderson theorem for small perturbation supplement superconducting phase: bare T c = T c0 and triplet pairing no inversion no time reversal otherwise spin structure adapted to perturbation

Anderson theorem for small perturbation supplement superconducting phase: bare T c = T c0 and triplet pairing H equal spin pairing parallel to magnetic field no inversion no time reversal otherwise spin structure adapted to perturbation

Anderson theorem for small perturbation supplement superconducting phase: bare T c = T c0 and triplet pairing no inversion no time reversal otherwise spin structure adapted to perturbation

Anderson theorem for perturbations time reversal symmetry breaking inversion symmetry breaking Zeeman coupling Rashba spin-orbit coupling spin-singlet even-parity pairing T c T c time reversal breaking inversion breaking

Anderson theorem for perturbations time reversal symmetry breaking inversion symmetry breaking Zeeman coupling Rashba spin-orbit coupling spin-triplet odd-parity pairing T c T c time reversal breaking inversion breaking

Structure of pairing state

Anderson theorem for perturbations - summary spin singlet even parity spin triplet odd parity T c T c T c time reversal breaking time reversal breaking time reversal breaking T c T c T c inversion breaking inversion breaking inversion breaking

Anderson theorem for perturbations - summary spin singlet even parity broken time reversal symmetry spin triplet odd parity T c T c T c time reversal breaking time reversal breaking time reversal breaking T c T c T c inversion breaking inversion breaking inversion breaking

Anderson theorem for perturbations - summary spin singlet even parity spin triplet odd parity T c T c T c T c time reversal breaking time reversal breaking broken inversion symmetry T c T c time reversal breaking inversion breaking combination parity-mixing inversion breaking inversion breaking

Structure of pairing states time reversal symmetry broken (e.g. magnetic field) spin triplet state: Cooper pair spin expectation value with spin parallel to field inversion symmetry broken (e.g. non-centrosymmetric crystal) combination of spin singlet and spin triplet state: mixed parity state with

Mixed parity states are non-unitary unitary superconducting states: 2x2 unit matrix CePt 3 Si inversion symmetry violated time reversal symmetry violated UGe 2 3 He A 1 phase

Non-centrosymmetric superconductors

Ce-based heavy Fermion superconductors CePt 3 Si CeRhSi 3 CeIrSi 3 ambient pressure Bauer et al (2004) Kimura et al. (2005) Onuki et al. (2005) Rashba-type of spin-orbit coupling tetragonal crystal lattice temperature AF T N coexistence of AF and SC! T c SC pressure quantum critical point

Upper critical field How to destroy Cooper pairs by a magnetic field? Two types of depairing orbital depairing B Lorentz force Heavy Fermion superconductors: effective electron mass: very slow Fermions coherence length: paramagnetic depairing B orbital depairing mechanism weak paramagnetic limiting important!? magnetic energy condensation energy spin polarization strong magnetic correlations/order unconventional pairing likely

Spin polarization - spin susceptibility spin singlet pairing pair breaking by spin polarization Yosida behavior of spin susceptibility χ χ p T c T spin triplet pairing χ χ = const. for note: no pair breaking for equal-spin pairing with r d ( k r )" H r =0 χ p T c T

Paramagnetic limiting field destruction of superconductivity due to Zeeman splitting of electron spins Compare the two energies at T=0K superconducting condensation energy thermodynamic critical field paramagnetic energy paramagnetic limiting field Pauli susceptibility spin susceptibility at T=0K

Paramagnetic limiting field destruction of superconductivity due to Zeeman splitting of electron spin states Compare the two energies at T=0K superconducting condensation energy thermodynamic critical field paramagnetic energy paramagnetic limiting field χ Pauli susceptibility spin singlet pairing spin susceptibility at T=0K χ p T c T

Paramagnetic limiting field destruction of superconductivity due to Zeeman splitting of electron spin states Compare the two energies at T=0K superconducting condensation energy thermodynamic critical field paramagnetic energy paramagnetic limiting field Pauli susceptibility spin triplet pairing χ spin susceptibility at T=0K χ p T c T

Paramagnetic limiting field - mixed parity state spin susceptibility of non-centrosymmetric SC χ/χ p 1 0.5 r H z ˆ r H " z ˆ Yosida as in CePt 3 Si, CeRhSi 3, CeIrSi 3 paramagnetic limiting field 0 0 1 T/T c paramagnetic limiting no paramagn. limiting

Paramagnetic limiting field - mixed parity state spin susceptibility of non-centrosymmetric SC χ/χ p 1 0.5 r H z ˆ r H " z ˆ Yosida as in CePt 3 Si, CeRhSi 3, CeIrSi 3 CePt 3 Si 0 0 1 T/T c paramagnetic limiting T. Yasuda et al. no paramagn. limiting does not follow the expectations!

Upper critical field and paramagnetic limiting CeRhSi 3 CeIrSi 3 P c ~ 25-30 kbar no limiting no limiting limited limited Kimura et al. Onuki et al. fits very well to theoretical expectations of paramagnetic limiting

Upper critical field and paramagnetic limiting CeRhSi 3 Comparison with non-heavy Fermion SC CeIrSi 3 P c ~ 25-30 kbar LaIrSi 3 no limiting no limiting limited limited Kimura et al. Onuki et al. fits very well to theoretical expectations of paramagnetic limiting

Special features of mixed-parity states intrinsic multi-band aspect twin boundaries surface states

Structure of the pair wave function k y 2 Fermi surfaces k x 2 different pair wave functions:

States at twin boundaries non-centrosymmetric crystals can be twinned inversion Twin boundaries: Fermi surfaces exchange role

States at twin boundaries non-centrosymmetric crystals can be twinned inversion broken time reversal symmetry on twin boundary 0 < φ < π φ = π φ = 0 Twin boundaries: Fermi surfaces exchange role Iniotakis et al

States at twin boundaries non-centrosymmetric crystals can be twinned inversion broken time reversal symmetry on twin boundary 0 < φ < π φ = π φ = 0 Twin boundaries: broken time reversal symmetry Fermi surfaces exchange role line defects of phase φ on twin boundary impediment for flux flow PrOs 4 Sb 12 UBe 13 fractional vortices strongly pinned on twin boundary Iniotakis, Fujimoto, Savary & MS high barrier CePt 3 Si Miclea, Mota et al.(2008)

Quasiparticle tunneling multigap features and zero-bias anomalies ratio s- vs p-wave even vs odd NS-tunneling spectroscopy conductance dominant odd-parity dominant even-parity non-centrosym SC y normal metal tunneling x + - max

Quasiparticle tunneling multigap features and zero-bias anomalies ratio s- vs p-wave even vs odd conductance dominant odd-parity + - dominant even-parity surface bound states for dominantly odd-parity pairing state max C. Iniotakis et al. (2007)

Quasiparticle tunneling multigap features and zero-bias anomalies different spin structure for two branches ratio s- vs p-wave even vs odd conductance spin currents carried by surface states Vorontsov et al. dominant odd-parity max + - dominant even-parity surface bound states for dominantly odd-parity pairing state C. Iniotakis et al. (2007)

Li Pd B & Li Pt B 2 3 2 3 Unequal twins

Li 2 Pd 3 B, Li 2 Pt 3 B Togano et al. (2004) Space group: P4 3 32 cubic alloy interpolation: Li 2 (Pd x Pt 1-x ) 3 B Nuclear magnetic resonance Li 2 Pd 3 B conventional Li 2 Pt 3 B unconventional London penetration depth Li 2 Pd 3 B T c = 6 K full gap Li 2 Pt 3 B T c = 2.5 K nodes Yuan et al. (2005) Zheng et al. (2007)

LaAlO / SrTiO 3 3 Heterostructure Environment for non-centrosymmetric superconductivity

LaAlO 3 / SrTiO 3 heterostructure + + + 0 0 0 0 0 0 polar band insulator non-polar band insulator

LaAlO 3 / SrTiO 3 heterostructure + + + 0 0 0 0 0 0 polar band insulator non-polar band insulator electronic reconstruction metallic layer

LaAlO 3 / SrTiO 3 heterostructure E interface superconductivity + + polar sheet band insulator resistance T c + 0 0 0 0 0 0 non-polar band insulator superconductivity electric field / gate voltage Reyren, Triscone, Mannhart et al. (2008)

LaAlO 3 / SrTiO 3 heterostructure E interface superconductivity + + + polar sheet band insulator resistance non-centrosymmetric superconductivity with variable 0electric field induced spin-orbit 0 couplig 0 0 0 0 non-polar band insulator T c superconductivity electric field / gate voltage Reyren, Triscone, Mannhart et al. (2008)

Conclusions Cooper Pairing involves two key symmetries time reversal inversion non-unitary states: spin polarized pairing mixed-parity pairing lack of time reversal lack of inversion non-centrosymmetric superconductors with unconventional pairing rich in phenomena with a complex phenomenology magnetoelectric phenomena connection to spintronics and multiferroics Edelstein, Mineev, Samokhin, Eschrig, Josephson effect phase sensitive probes Hayashi, Linder, Subdo, Borkje, Coexistence of magnetism and superconductivity at quantum critical points Yanase, Fujimoto,

Collaborators: Theory ETH Zurich: P.A. Frigeri, N. Hayashi, K. Wakabayashi, I. Milat, Y. Yanase C. Iniotakis, D. Perez, L. Savary, T. Neupert, Uni Wisconsin: D.F. Agterberg, R.P. Kaur Japan: A. Koga, Y. Tanaka, S. Fujimoto,. Experiment TU Wien: E. Bauer and team Japan: T. Shibauchi, Y. Matsuda, N. Kimura, Y. Onuki, Z. Hiroi,. IBM Watson Lab: L. Kruzin-Elbaum Uni Illinois UC: H. Yuan, M. Salamon and team Venecuela: I. Bonalde Geneva & Augsburg: J.M. Triscone, J. Mannhart and teams Funding: