Non-uniform (FFLO) states and quantum oscillations in superconductors and superfluidultracold Fermi gases A. Buzdin University of Bordeaux I and Institut Universitaire de France in collaboration with L. Bulaevskii, J. P. Brison, M. Houzet, Y. Matsuda, T. Shibaushi, H. Shimahara, D. Denisov, A. Melnikov, A. Samokhvalov ECRYS-2011, August 15-27, 2011 Cargèse , France
Outline Singlet superconductivity destruction by the magneticfield: - The main mechanisms - Origin of FFLO state.2. Experimentalevidences of FFLO state.3. Exactly solvable models of FFLO state.4. Vortices in FFLO state. Role of the crystal structure.5. Supefluidultracold Fermi gases with imbalanced state populations: one more candidate for FFLO state?
1. Singlet superconductivity destruction by the magneticfield.
Orbital effect (Lorentz force) Electromagnetic mechanism (breakdown of Cooper pairs by magnetic field induced by magnetic moment) p B FL FL -p • Paramagnetic effect (singlet pair) μBH~Δ~Tc Sz=+1/2 Sz=-1/2 Exchange interaction
Flux quantum Hc2 p B FL FL Normal x(coherence length) -p D B Abrikosov Lattice l(penetration length） Meissner Vortex lattice in NbSe2 (STM) Tc T F0=hc/2e=2.07x10-7Oe・cm2z Orbital effect Vortex
p B FL FL -p Superconductivity is destroyed by magnetic field Orbital effect (Vortices) Zeeman effect of spin (Pauli paramagnetism) Maki parameter Usually the influence of Pauli paramagnetic effect is negligibly small
Superconducting order parameter behavior under paramagnetic effect Standard Ginzburg-Landau functional: The minimum energy corresponds to Ψ=const The coefficients of GL functional are functions of the Zeeman field h= μBH ! Modified Ginzburg-Landau functional ! : The non-uniform state Ψ~exp(iqr) will correspond to minimum energy and higher transition temperature
F q q0 Ψ~exp(iqr) - Fulde-Ferrell-Larkin-Ovchinnikovstate (1964). Only in pure superconductors and in the rather narrow region.
FFLO inventors Fulde and Ferrell Larkin and Ovchinnikov
E kF -dkF k kF +dkF The total momentum of the Cooper pair is -(kF -dkF)+ (kF -dkF)=2 dkF
Conventional pairing E k -k k -k kx ky E k k q -k+q q q~gmBH/vF kx ky -k+q ( k ,-k ) FFLO pairing ( k ,-k+q ) pairing between Zeeman split parts of the Fermi surface Cooper pairs have a single non-vanishing center of mass momentum
Pairing of electrons with opposite spins and momenta unfavourable : But : if 1d SC 2d SC 3d SC 1 At T = 0, Zeeman energy compensation is exact in 1d, partial in 2d and 3d. 0.8 0.6 • the upper critical field is increased • Sensivity to the disorder and to the orbital effect: • (clean limit) 0.4 0.2 0.56 0.2 0.4 0.6 0.8 1 0.56 0.2 0.4 0.6 0.8 1
FF state ( k ,-k ) ( k ,-k+q ) + + + uniform available for pairing depaired LO state spatially nonuniform The SC order parameter performs one-dimensional spatial modulations along H, forming planar nodes
ModifiedGinzburg-Landau functional: May be 1st order transitionat 1 0.8 0.6 0.4 0.2 0.56 0.2 0.4 0.6 0.8 1
2. Experimentalevidences of FFLO state. • Unusualform of Hc2(T) dependence • Change of the form of the NMR spectrum • Anomalies in altrasoundabsorbtion • Unusualbehaviour of magnetization • Change of anisotropy ….
Organic superconductor k-(BEDT-TTF)2Cu(NCS)2 (Tc=10.4K) Cu[N(CN)2]Br layer H or D H or D S S BEDT-TTF layer 15Å C C Suppression of orbital effects in H parallel to the planes Layered structure BEDT-TTF (donor molecule)
Talk of Stuart Brown about FFLO in this compound!
Anomalous in-plane anisotropy of the onset of SC in (TMTSF)2ClO4 S.Yonezawa, S.Kusaba, Y.Maeno, P.Auban-Senzier, C.Pasquier, K.Bechgaard, and D. Jerome, Phys. Rev. Lett. 100, 117002 (2008)
Field induced superconductivity (FISC) in an organic compound -(BETS)2FeCl4 Metal FISC Insulator AF c-axis (in-plane) resistivity S. Uji et al., Nature 410 908 (2001) L. Balicas et al., PRL 87 067002 (2001)
Jaccarino-Peter effect Zeeman energy Exchange energy between conduction electrons in the BETS layers and magnetic ions Fe3+ (S=5/2) For some reason J > 0 : the paramagnetic effect is suppressed at Other example: Eu-Sn Molybdenum chalcogenide H. Meul et al, 1984 Critical field Hc2 [Tesla] (Eu0.75Sn0.25Mo6S7.2Se0.8) S S Temperature [K]
1st H// ab 2nd order 1st H// c 2nd H-T phase diagram of CeCoIn5 Pauli paramagnetically limited superconducting state
New high field phase of the flux line lattice in CeCoIn5 This 2nd order phase transition is characterized by a structural transition of the flux line lattice Ultrasound and NMR results are consistent with the FFLO state which predicts a segmentation of the flux line lattice
Proximity effect in a ferromagnet ? In ferromagnet ( in presence of exchange field) the equation for superconducting order parameter is different In the usual case (normal metal): Its solution corresponds to the orderparameterwhichdecayswithoscillations!Ψ~exp[-(q1± iq2 )x] Ψ Wave-vectors are complex! They are complex conjugate and we can have a real Ψ. Orderparameter changes itssign! Many new effects in S/F heterostructures! x
Remarkable effects come from the possible shift of sign of the wave function in the ferromagnet, allowing the possibility of a « π-coupling » between the two superconductors (π-phase difference instead of the usual zero-phase difference) F S S S F S « 0 phase » « phase » S/F bilayer F S h-exchange field, Df-diffusion constant
S-F-S Josephson junction in the clean/dirty limit • Damping oscillating dependence of the critical current Ic as the function of the parameter =hdF /vFhas been predicted. • (Buzdin, Bulaevskii and Panjukov, JETP Lett. 81) • h- exchange field in the ferromagnet, • dF - its thickness S F S E(φ)=- Ic (Φ0/2πc) cosφ Ic J(φ)=Icsinφ
Critical current density vs. F-layer thickness(V.A.Oboznov et al., PRL, 2006) Collaboration with V. Ryazanov group from ISSP, Chernogolovka Ic=Ic0exp(-dF/F1) |cos (dF /F2) + sin (dF /F2)| dF>> F1 Spin-flip scattering decreases the decaying length and increases the oscillation period. “0”-state -state F2 >F1 0 Nb-Cu0.47Ni0.53-Nbjunctions “0”-state -state I=Icsin I=Icsin(+ )= - Icsin()
Cluster Designs (Ryazanov et al.) 30m 2 x 2 fully-frustrated checkerboard-frustrated unfrustrated 6 x 6 fully-frustrated checkerboard-frustrated
Scanning SQUID Microscope images(Ryazanov et al., Nature Physics, 2008)) Ic T T T = 1.7K T = 2.75K T = 4.2K
Glitches FFLO State in Neutron Star Bose-Einstein-Condensate Color superconductivity Vortices R.Casalbuoni and G.Nardulli Rev. Mod. Phys. (2004) Supefluidultracold Fermi gases with imbalanced state populations: one more candidate for FFLO state? Massachusetts Institute of Technology: M.W. Zwierlein, A. Schirotzek, C. H. Schunck, W.Ketterle (2006) Rice University, Houston: Guthrie B. Partridge, Wenhui Li, Ramsey I. Kamar, Yean-an Liao, Randall G. Hulet (2006)
FFLO phase in the case of pure paramagnetic interaction and BCS limit Exact solution for the 1D and quasi-1D superconductors ! (Buzdin , Tugushev 1983) • The FFLO phase is the soliton lattice, • first proposed by Brazovskii, Gordyunin and Kirova in 1980 for polyacetylene. 1d SC 1 0.8 0.6 0.4 0.2 0.56 Magnetic moment x
Cu O Ge Chains direction Spin-Peierls transitions - e.g. CuGeO3 TSP=14.2 K
( k ,-k ) In 2D superconductors Y.Matsuda and H.Shimahara J.Phys. Soc. Jpn (2007)
In 3D superconductors The transition to the FFLO state is 1st order. The sequence of phases is similar to 2D case. Houzet et al. 1999; Mora et al. 2002
FFLO phase in the case of paramagnetic and orbital effect (3D BCS limit) – uppercriticalfield Note : The system withelliptic Fermi surface canbetranformed by scaling transformation to iheisotropic one. Sure the direction of the magneticfieldwillbechanged. Lowest m=0 Landau level solution, Gruenberg and Gunter, 1966 FFLO exists for Maki parameterα>1.8. For Maki parameter α>9 the highest Landau level solutions are realized – Buzdin and Brison, 1996.
B q FFLO phase in 2D superconductors in the tiltedmagneticfield - uppercriticalfield Highest Landau level solutions are realized – Bulaevskii, 1974; Buzdin and Brison, 1996; Houzet and Buzdin, 2000.
Exotic vortex lattice structures in tilted magnetic field Generalized Ginzburg-Landau functional Near the tricritical point, the characteristic length is Microscopic derivation of the Ginzburg-Landau functional : Instability toward FFLO state Instability toward 1st order transition Next orders are important : • Validity: • large scale for spatial variation of : vicinity of T * • small orbital effect, introduced with • we neglect diamagnetic screening currents (high- limit)
2nd order phase transition at higher Landau levels • Near the transition: minimization of the free energy with solutions in the form gauge Parametrizes all vortex lattice structures at a given Landau level N is the unit cell All of them are decribed in the subset :
__ 1st order transition __ 2nd order transition __ 2nd order transition in the paramagnetic limit • Analysis of phase diagram : • cascade of 2nd and 1st order transitions between S and N phases • 1st order transitions within the S phase • exotic vortex lattice structures 3 n=2 Landau levels Magnetic field n=1 2 Rectang. n=0 Triangular 1 square Tricritical point Rectangular 0 Triangular lattice -2 -1 0 Temperature At Landau levels n > 0, we find structures with several points of vanishment of the order parameter in the unit cell and with different winding numbers w = 1, 2 …
Order parameter distribution for the asymmetric and square lattices with Landau level n=1. The dark zones correspond to the maximum of the order parameter and the white zones to its minimum.
Intrinsic vortex pinning in LOFF phase for parallelfield orientation Δn= Δ0cos(qr+αn)exp(iφn(r)) t – transfer integral Josephson couplingbetweenlayersismodulated Fn,n+1=[-I0cos(αn_- αn+1)+I2cos(qr) cos(αn_+ αn+1)] cos(φn- φn+1) φn- φn+1=2πxHs/Φ0 + πn s-interlayer distance, x-coordinatealongq
CeIn3 CoIn2 CeIn3 CeCoIn5 • Quasi-2D heavy fermion (Tc=2.3K) • Strong antiferromagnetic fluctuation • d-wave symmetry • Tetragonal symmetry z Modified Ginzburg-Landau functional: isotropic part x y
No orbital effect Modulation diagram in the case of the absence of the orbital effect (pure paramagnetic limit). Areas with different patterns correspond to different orientation of the wave-vector modulation. The phase diagram does not depend on the εx value.
Magnetic field along z axis H||z Modulation diagram in the case when the magnetic field applied along z axis. There are 3 areas on the diagram corresponding to 3 types of the solution for modulation vector qz and Landau level n. Modulation direction is always parallel to the applied field and εx here is treated as a constant.
Magnetic field along z axis xy z intermediate xy z intermediate
Magnetic field along x axis H||x Modulation diagram (ἕ, εz) in the case when the magnetic field is applied along x axis. There are three areas on the diagram corresponding to different types of the solution for modulation vector qx and Landau level n. Modulation direction is always parallel to the applied field. The choice of the intersection point is determined by the coefficient εx.
Magnetic field along x axis xy z intermediate xy z intermediate