Report

Non-uniform (FFLO) states and quantum oscillations in superconductors and superfluid ultracold 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 1 Outline 1. Singlet superconductivity destruction by the magnetic field: - The main mechanisms - Origin of FFLO state. 2. Experimental evidences of FFLO state. 3. Exactly solvable models of FFLO state. 4. Vortices in FFLO state. Role of the crystal structure. 5. Supefluid ultracold Fermi gases with imbalanced state populations: one more candidate for FFLO state? 1. Singlet superconductivity destruction by the magnetic field. • Orbital effect (Lorentz force) p FL B -p FL Electromagnetic mechanism (breakdown of Cooper pairs by magnetic field induced by magnetic moment) • Paramagnetic effect (singlet pair) μBH~Δ~Tc Sz=+1/2 Sz=-1/2 I S s Tc Exchange interaction Orbital effect p B FL FL -p Hc2 Hc2 Vortex Flux quantum 0 2 2 Normal coherence B length) D Abrikosov Lattice Meissner T Vortex lattice in NbSe2 (STM) Tc lpenetration length） 0hc/2e=2.07x10-7Oe・cm2z Superconductivity is destroyed by magnetic field Orbital effect (Vortices) p FL B H orb c2 FL -p 0 2 2 Zeeman effect of spin (Pauli paramagnetism) 1 1 N H 2 N(0)D2 2 2 1 N (g B ) 2 N (0) 2 H P c2 2D g B Maki parameter H corb 2 P2 H c2 ~ D F 1 Usually the influence of Pauli paramagnetic effect is negligibly small Superconducting order parameter behavior under paramagnetic effect Standard Ginzburg-Landau functional: 1 b 4 2 2 F a 4m 2 The minimum energy corresponds to Ψ=const The coefficients of GL functional are functions of the Zeeman field h= μBH ! Modified Ginzburg-Landau functional ! : 2 F a ... 2 2 2 The non-uniform state Ψ~exp(iqr) will correspond to minimum energy and higher transition temperature F F (a q q ) q 2 q0 4 q Ψ~exp(iqr) - Fulde-Ferrell-Larkin-Ovchinnikov state (1964). Only in pure superconductors and in the rather narrow region. 2 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 kx ky FFLO pairing k ( k ,-k ) -k E k q -k+q q q~gBH/vF ky k ( k ,-k+q ) kx -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 At T = 0, Zeeman energy compensation is exact in 1d, partial in 2d and 3d. 1d SC 2d SC 3d SC B H / D1 0.8 0.6 • the upper critical field is increased • Sensivity to the disorder and to the orbital effect: 0.4 0.2 0.2 0.56 0.4 0.6 0.8 1 T / Tc (clean limit) 0.56 0.2 0.4 0.6 0.8 1 T / Tc FF state D(r) D(q)exp( iq r) uniform ( k ,-k ) available for pairing depaired ( k ,-k+q ) LO state D(r) D 0 cos(q r) spatially nonuniform + + ~ 1 q + The SC order parameter performs one-dimensional spatial modulations along H, forming planar nodes Modified Ginzburg-Landau functional : 2 F a 2 2 2 *2 2 2 4 d * 2 6 2 2 ... May be 1st order transition at 1 B B / D 0.8 0.6 0.4 0.2 0.56 0.2 0.4 0.6 0.8 1 T / Tc 2. Experimental evidences of FFLO state. •Unusual form of Hc2(T) dependence •Change of the form of the NMR spectrum •Anomalies in altrasound absorbtion •Unusual behaviour of magnetization •Change of anisotropy …. Organic superconductor Layered structure k-(BEDT-TTF)2Cu(NCS)2 (Tc=10.4K) Suppression of orbital effects in H parallel to the planes Cu[N(CN)2]Br layer 15 Å BEDT-TTF layer H or D S C H or D S C 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) 18 Field induced superconductivity (FISC) in an organic compound l-(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 Eu-Sn Molybdenum chalcogenide (Eu0.75Sn0.25Mo6S7.2Se0.8) H. Meul et al, 1984 Critical field Hc2 [Tesla] Other example: S S Temperature [K] Strong evidence of inhomogeneous FFLO phase in CeCoIn5 H-T phase diagram of CeCoIn5 1st H// ab 2nd order 1st H// c 2nd 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 a Proximity effect in a ferromagnet ? In the usual case (normal 1metal): 2 qx 4m 0, and solutionfor T Tc is e , where q 4ma In ferromagnet ( in presence of exchange field) the equation for superconducting order parameter is different a 0 2 Ψ 4 Its solution corresponds to the order parameter which decays with oscillations! Ψ~exp[-(q1 ± iq2 )x] Wave-vectors are complex! They are complex conjugate and we can have a real Ψ. Order parameter changes its sign! Many new effects in S/F heterostructures! x 24 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) D D S F S S D D S D S « 0 phase » « phase » F F S/F bilayer D f D f / h (1 10)nm h-exchange field, 25 Df-diffusion constant S-F-S Josephson junction in the clean/dirty limit S F S Damping oscillating dependence of the critical current Ic as the function of the parameter =hdF /vF has been predicted. (Buzdin, Bulaevskii and Panjukov, JETP Lett. 81) h- exchange field in the ferromagnet, dF - its thickness Ic E(φ)=- Ic (Φ0/2πc) cosφ J(φ)=Icsinφ 26 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 “0”-state Spin-flip scattering decreases the decaying length and increases the oscillation period. -state F2 >F1 0 “0”-state I=Icsin -state Nb-Cu0.47Ni0.53-Nb junctions I=Icsin(+ )= - Icsin() 27 Cluster Designs (Ryazanov et al.) 30m 2x2 unfrustrated fully-frustrated checkerboard-frustrated 6x6 fully-frustrated checkerboard-frustrated 28 Scanning SQUID Microscope images (Ryazanov et al., Nature Physics, 2008)) Ic T T = 1.7K T = 2.75K T T = 4.2K 29 FFLO State in Neutron Star Color superconductivity R.Casalbuoni and G.Nardulli Rev. Mod. Phys. (2004) Bose-Einstein-Condensate Vortices Glitches Supefluid ultracold 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) 3. Exactly solvable models of FFLO state. 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 B H / D1 0.8 0.6 D( x) D0 sn( x / , k ) 0.4 0.2 0.56 at T 0 D(x) B H 2 T / Tc D Magnetic moment x Spin-Peierls transitions - e.g. CuGeO3 Cu O Ge un (1)n D( x na) Chains direction TSP=14.2 K In 2D superconductors ( k ,-k ) 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 4. Vortices in FFLO state. Role of the crystal structure. FFLO phase in the case of paramagnetic and orbital effect (3D BCS limit) – upper critical field Note : The system with elliptic Fermi surface can be tranformed by scaling transformation to ihe isotropic one. Sure the direction of the magnetic field will be changed. Lowest m=0 Landau level solution, Gruenberg and Gunter, 1966 H corb 2 2 P H c2 FFLO exists for Maki parameter α>1.8. For Maki parameter α>9 the highest Landau level solutions are realized – Buzdin and Brison, 1996. FFLO phase in 2D superconductors in the tilted magnetic field - upper critical field Highest Landau level solutions are realized – Bulaevskii, 1974; Buzdin and Brison, 1996; Houzet and Buzdin, 2000. B q 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 Next orders are important : Validity: • large scale for spatial variation of D : vicinity of T * small orbital effect, introduced with • we neglect diamagnetic screening currents (high-k limit) Instability toward 1st order transition • 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 : • cascade of 2nd and 1st order transitions between S and N phases • 1st order transitions within the S phase • exotic vortex lattice structures 3 Magnetic field Analysis of phase diagram : n=2 2 n=1 n=0 1 Tricritical point 0 -2 -1 0 Temperature __ 1st order transition __ 2nd order transition __ 2nd order transition in the paramagnetic limit 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. 43 Intrinsic vortex pinning in LOFF phase for parallel field orientation Δn= Δ0cos(qr+αn)exp(iφn(r)) t – transfer integral Josephson coupling between layers is modulated Fn,n+1=[-I0cos(αn_- αn+1)+I2cos(qr) cos(αn_+ αn+1)] cos(φ - φ n φn- φn+1=2πxHs/Φ0 + πn s-interlayer distance, x-coordinate along q n+1) 44 CeCoIn5 Quasi-2D heavy fermion CeIn3 (Tc=2.3K) Strong antiferromagnetic fluctuation CoIn2 z d-wave symmetry CeIn3 Tetragonal symmetry Modified Ginzburg-Landau functional: isotropic part x j j 2e iA j c y No orbital effect ~ q=0 z-axis modulation q=/2 xy-plane modulation z ~ q=0.5 arccos(z/(-z)) ~ =2z Modulation (~, z ) 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 ~ z=x H||z qz=0, n=max ~ =x qz=max, n=0 z qz>0, n>0 ~ =2z 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 ~ z=x xy qz=0, n=max ~ =x qz=max, n=0 z z intermediate q >0, n>0 z ~ =2z ~ xy q=0 z-axis modulation q=/2 xy-plane modulation z ~ intermediate q=0.5 arccos(z/(-z)) ~ =2z z Magnetic field along x axis ~ ~ =-3z qx=max, n=0 z ~ =-x qx 0, n=max H||x qx>0, n>0 ~ =3z-2x 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 ~ ~ =-3z xy qx=max, n=0 z z ~ =-x qx 0, n=max qx>0, n>0 intermediate ~ xy ~ =3z-2x q=0 z-axis modulation q=/2 xy-plane modulation z intermediate~ q=0.5 arccos(z/(-z)) ~ =2z z Small angle neutron scattering from the vortex lattice for H //c FFLO? Neutron form factor seems to be consisitent with FFLO state A.D.Bianch et al.Science (2007) Neutron form factor The crystal structure effects influence on the FFLO state is very important. The FFLO states with higher Landau level solutions could naturally exist in real 3D compounds (without any restrictions to the value of Maki parameter). Wave vector of FFLO modulation along the magnetic field could be zero. In the presence of the orbital effect the system tries in some way to reproduce optimal directions of the FFLO modulation by varying the Landau level index n and wave-vector of the modulation along the field. 5. Supefluid ultracold 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) Experimental system: Fermionic 6Li atoms cooled in magnetic and optical traps (mixture of the two lowest hyperfine states with different populations) Experimental result: phase separation rf transitions 76 MHz hyperfine states Supefluid core Normal gas Rotating supefluid ultracold Fermi gases in a trap Coils generating magnetic field Fermion condensate Vortex as a test for superfluidity Laser beams MIT: Ketterle et al (2005) Images of vortex lattices Questions: 1. What is the effect of confinement (finite system size) on FFLO states? 2. Effect of rotation on FFLO states in a trap (effect of magnetic field on FFLO state in a small superconducting sample). 3. Possible quantum oscillation effects. Examples of quantum oscillation effects. Little-Parks effect. Switching between the vortex states. Multiply-connected systems Superconducting thin-wall cylinder Superconductor with a columnar defect or hole Tc (H) oscillations Multiquantum vortices -1 vs 1 Ф/Ф 0 L=-1 L=0 -ΔTc/Tc0 L=1 o c 0 e Ф/Ф A.Bezryadin, A.I. Buzdin, B. Pannetier (1994) Examples of quantum oscillation effects. Little-Parks effect. Simply-connected systems Mesoscopic samples dimensions ~ several coherence lengths O.Buisson et al (1990) R.Benoist, W.Zwerger (1997) V.A.Schweigert, F.M.Peeters (1998) H.T.Jadallah, J.Rubinstein, Sternberg (1999) H R~ξ Tc (H) oscillations Origin of Tc oscillations: Transitions between the states with different vorticity L iLq D | D r | e L - Vorticity (orbital momentum) Multiquantum vortices Examples of quantum oscillation effects. FFLO states and Tc(H) oscillations in infinite 2D superconductors A.I. Buzdin, M.L. Kulic (1984) Hz H|| ~ Hp Hz a 0 k02 Model: Modified Ginzburg-Landau functional (2D) 2 2 2 2 F (( a V (r )) | | | D | | D | )dxdy a (T Tc0 ) T N Trapping potential FFLO instability Range of validity: vicinity of tricritical point S D 2iM [, r ] D 2ieA c Confinement mechanisms: 1. Zero trapping potential. Boundary condition at the system edge nD 0 FFLO H 2. nonzero trapping potential V (r ) M 2r 2 2 FFLO states in a 2D mesoscopic superconducting disk Hz z q H|| ~ Hp r R~ξ Interplay between the system size, magnetic length, and FFLO length scale Perpendicular magnetic field component Hz = 0 The critical temperature: Eigenvalue problem: Wave number of FFLO instability H - T Phase diagram: Hz = 0 T 2 k0 ( H , T ) H ↑ → L↓ L=1 L=2 FFLO state L=0 H Tilted magnetic field: Hz ≠ 0 Field induced superconductivity Eigenvalue problem: a R2 H z 0 Tilted magnetic field: Hz ≠ 0 Transitions with large jumps in vorticity Vortex solutions beyond the range of FFLO instability. Critical field of the vortex entry. Hz z q H|| ~ Hp r 2 4 2 2 F ( | D | 2 | D | )dxdy 2 1 We focus on the limit1 2 T eiLq N S (H*,T*) Condition of the first vortex entry: Beyond the range of FFLO instability F ( L 1) F ( L 0) 0 FFL O H|| Condition of the first vortex entry: R m2 2 R 3 ln 2 3 4 ln 0 m 2 R m m max(1, 2 ) H z R 2 0 24 2 2 m1 Limiting cases: 2 1 2 1 R 1 m H * H|| ln R 1 R 2 Hz 23 1 R ln R 2 1 1 Hz R 43 Change in the scaling law FFLO states in a trapping potential Interplay between the rotation effect, confinement, and FFLO instability FFLO states in a 2D system in a parabolic trapping potential (no rotation) 4 2 2 D 2D ( v0 ) 0 k0 2 FFLO length scale a k 4 0 6 (Trapping frequency)2 0 k0r e d q iq Temperature shift v0 M / 2k 2 Fourier transform: L0 Dimensionless coordinate U (q ) U (q) q 4 2q 2 1 4 x 2 q e 2 2 v0 2 q 4 2q 2 q q 1 x lx 2 q 1 l v0 v0 0 q FFLO states in a 2D system in a parabolic trapping potential (no rotation) Phase diagram Condensate wave function 1 k0r 1 k r cos(k0 r 4)e k0 r 1 2 v0 2 2 0 v0 v0 4 Suppression of wave function oscillations by the increase in the trapping frequency k0 r 2 1/ 4 0 v 1 =Number of observable oscillations FFLO states in a rotating 2D gas in a parabolic trapping potential. (r ) f L (r )eiLq D4 f L 2D2 f L ( v0 2 ) f L 0 1 L D a 2 2 Expansion in eigenfunctions of the problem without trap: f L ( ) cnunL ( ) n 0 2 D2unL qnL unL 2a (2n | L | L 1)unL 2 4 (2qnL qnL )cn vnmcm cn m vnm v0 umL unL 3d 0 k0 2 FFLO length scale a k04 Temperature shift v0 M 2 / 2k06 a 2M k02 k0r (Trapping frequency)2 rotation frequency Dimensionless coordinate FFLO states in a rotating 2D gas in a parabolic trapping potential. Suppression of quantum oscillations by the increase in the trapping frequency. First-order perturbation theory: max (4a v0 a )( 2 L 1) v0 L a 4a2 (2 L 1)2 L0 rotation induced superfluid phase Conclusions • There are strong experimental evidences of the existence of the the FFLO state in organic layered superconductors and in heavy fermion superconductor CeCoIn5 • FFLO –type modulation of the superconducting order parameter plays an important role in uperconductorferromagnet heterostructures. The -junction realization in S/F/S structures is quite a general phenomenon. • The interplay between FFLO modulation and orbital effect results in new type of the vortex structures, non-monotonic critical field behavior in layered superconductor in tilted field. • Special behavior of fluctuations near the FFLO transition – vanishing stiffness. • FFLO phase in ultracold Fermi gases with imbalanced state populations? 73