Coexistence of and magnetism at the nanoscale

A. Buzdin Institut Universitaire de France, Paris and Condensed Theory Group, University of Bordeaux

in collaboration with L. Bulaevskii, J. P. Brison, J. Cayssol, D. Denisov, M. Houzet, A. Melnikov, N. Pugach, V. Ryazanov, A. Samokhvalov…

Kourovka , 2012 1

Outline I • Recall on magnetism and superconductivity coexistence. • Ferromagnetic superconductors.Non-uniform magnetic structures. Re-entrant superconductivity.

• Origin and the main peculiarities of the proximity effect in superconductor-ferromagnet systems.

• Josephson π-junction.

• Spin-valve effet. Long ranged triplet correlations. Domain wall superconductivity.

• φ-junction. Direct coupling between ferromagnetism and superconductivity.

• Perspectives and possible applications. 2

Outline II 1. Larkin-Ovchinnikov-Fulde-Ferrell state. LOFF or FFLO states. The origin of the non-uniform modulated superconducting 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?

3 Magnetism and Superconductivity Coexistence

Ginzburg, 1956 – in the framework of the model of electromagnetic interaction.

dT c x dx m

(Abrikosov and Gorkov, 1960)

The critical temperature variation versus the concentration n of the Gd atoms in

La1-xGdxAl2 alloys (Maple, 1968). Tc0=3.24 K and ncr=0.590 atomic percent Gd.

The earlier experiments (Matthias et al., 1958) demonstrated that the presence of the magnetic atoms is very harmful for superconductivity. 4 5 Antagonism of magnetism (ferromagnetism) and superconductivity

• Orbital effect (Lorentz force)

Electromagnetic mechanism p B (breakdown of Cooper pairs by magnetic field FL F -p L induced by magnetic moment) • Paramagnetic effect (singlet pair)   μBH~Δ~Tc I S s Tc S =+1/2 S =-1/2 z z Exchange interaction 6 No antagonism between antiferromagnetism and superconductivity

Usually Tc>TN Tc (K) TN (K)

NdRh4B4 5.3 1.31

SmRh4B4 2.7 0.87

TmRh4B4 9.8 0.4

GdMo6S8 1.4 0.84

TbMo6S8 2.05 1.05

DyMo6S8 2.05 0.4

ErMo6S8 2.2 0.2

GdMo6Se8 5.6 0.75 Characteristic energy for magnetism (per one atom/electron) ErMo Se 6.0 1.1 6 8 DyNi B C 6.2 11 2 2 2 Emag ~ Θ ~ I /EF

ErNi2B2C 10.5 6.8 Characteristic energy for superconductivity (per one TmNi B C 11 1.5 2 2 electron)

HoNi2B2C 8 5 Es ~ Tc(Tc/EF) << Tc 7 8 FERROMAGNETIC CONVENTIONAL (SINGLET) SUPERCONDUCTORS

A. C. susceptibility and RE-ENTRANT resistance versus temperature SUPERCONDUCTIVITY in ErRh B ! in ErRh4B4 (Fertig et al.,1977) 4 4

9 Electromagnetic interaction

A ~ (T-θ)

d=2 /q a<

(Q) 2 Em hQ h=I Q 2

At T=0 and Q 0>>1 following (Anderson and Suhl, 1959)

s (Q) (Q)

(0) 2Q 0

d ~ (ξa2)1/3

Intensity of the neutron Bragg scattering and resistance as a function of temperature in an

ErRh4B4 (Sinha et al.,1982). The satellite position corresponds to the wavelength of the modulated magnetic d ~ (ξa)1/2 structure around 92 Å. a<

Tc=1.8 K Tm=0.74 K Tc=0.7 K

12 Auto-waves in reentrant superconductors?

current I

T

13 14

FERROMAGNETIC UNCONVENTIONAL (TRIPLET) SUPERCONDUCTORS

UGe2 (Saxena et al., 2000) and URhGe (Aoki et al., 2001) Triplet pairing

UGe2

URhGe (a) The total magnetic moment M total and

the component Mb measured for H// to the b axis . In (b), variation of the resistance at 40 mK and 500 mK with the field re-entrance of SC between 8-12 T 16 Very recently (2007): UCoGe θ=3K; Tc=0.8K (Levy et al 2005). 17 18 19 20 Superconducting order parameter behavior in ferromagnet

Standard Ginzburg-Landau functional:

2 1 2 b 4 F a 4m 2 The minimum energy corresponds to Ψ=const The coefficients of GL functional are functions of internal exchange field h !

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 21 F

2 4 2 F (a q q ) q

q q0 Ψ~exp(iqr) - Fulde-Ferrell-Larkin-Ovchinnikov state (1964). Only in pure superconductors and in the very narrow region.

Proximity effect in ferromagnet ? In the usual case (normal metal):

1 2 qx a 0, and solution for T Tc is e , where q 4ma 4m 22 E

kF - kF

k

kF + kF

The total momentum of the is

-(kF - kF)+ (kF - kF)=2 kF

23 In ferromagnet ( in presence of exchange field) the equation for superconducting order parameter is different a 2 4 0

Its solution corresponds to the order parameter which decays with oscillations! Ψ~exp[-(q1 + iq2 )x] Ψ

Order parameter changes its sign!

x

24 Proximity effect as Andreev reflection

pF

Classical Andreev reflection Quantum Andreev reflection

F S

h>>Tc

pF pF

25 Eilenberger equations

26 Theory of S-F systems in dirty limit

Analysis on the basis of the Usadel equations

D 2 f F x, ,h ih F x, ,h 0 2 f f

2 * G f x, ,h F f x, ,h F f x, , h 1 .

leads to the prediction of the oscillatory - like dependence of the critical current on the exchange field h and/or thickness of ferromagnetic layer.

27 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)

S F S S F S

« phase » « 0 phase »

S/F bilayer S F f D f / h

h-exchange field,

Df - diffusion constant28 The oscillations of the critical temperature as a function of the thickness of the ferromagnetic layer in S/F multilayers has been predicted in 1990 and later observed on experiment by Jiang et al. PRL, 1995, in Nb/Gd multilayers

F

S

F S

F

29 SF-bilayer Tc-oscillations

Ryazanov et al. JETP Lett. 77, 39 V. Zdravkov, A. Sidorenko et al cond-mat/0602448 (2006) (2003) Nb-Cu0.43Ni0.57 Nb-Cu0.41Ni0.59

dFmin =(1/4) ex largest Tc-suppression 30 31 S-F-S Josephson junction in the clean/dirty limit

Damping oscillating dependence of the

critical current Ic as the function of the parameter =hd /v has been predicted. S F S F F (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

32 The oscillations of the critical current as a function of temperature (for different thickness of the ferromagnet) in S/F/S trilayers have been observed on experiment by Ryazanov et al. 2000, PRL

S S

F and as a function of a ferromagnetic layer thickness by Kontos et al. 2002, PRL

33 Phase-sensitive experiments -junction in one-contact interferometer

0-junction -junction I=Icsin( + )=-Icsin minimum energy at 0 minimum energy at

I I E= EJ[1-cos( + )]=EJ[1+cos ]

2 LIc > 0/2 L = = I (2 / 0) Adl = 2 / E E 0

Spontaneous circulating current in a closed superconducting loop when >1 with NO applied flux L

L = 0/(4 LIc) Bulaevsky, Kuzii, Sobyanin, JETP Lett. 1977 = 0/2 34 30 m Cluster Designs (Ryazanov et al.)

2 x 2

unfrustrated fully-frustrated checkerboard-frustrated

6 x 6

fully-frustrated checkerboard-frustrated 35 2 x 2 arrays: spontaneous vortices

Fully frustrated

Checkerboard frustrated 36 Scanning SQUID Microscope images (Ryazanov et al., Nature Physics, 2008))

Ic

T T T = 1.7K T = 2.75K T = 4.2K

37 Critical current density vs. F-layer thickness (V.A.Oboznov et al., PRL, 2006)

Ic=Ic0exp(-dF/ F1) |cos (dF / F2) + sin (dF / F2)|

dF>> F1

Spin-flip scattering decreases the decaying length and increases the “0”-state oscillation period. -state F2 > F1 0

Nb-Cu0.47Ni0.53-Nb “0”-state -state I=I sin c I=I sin( + )= - I sin( ) c c

38 Critical current vs. temperature

Nb-Cu0.47Ni0.53-Nb dF=9-24 nm “Temperature dependent” h=Eex 850 K (TCurie= 70 K) spin-flip scattering

2 1 1 1 1 1 , F1 F h s h s

2 F 2 F1 1 1 1 1 1 F 2 F h s h s

Effective spin-flip rate

(T)= cos (T)/ S; G = cos (T); F = sin (T) 39 Critical current vs. temperature (0- - and -0- transitions) Nb-Cu0.47Ni0.53-Nb dF1=10-11 nm dF2=22 nm

(V.A.Oboznov et al., PRL, 2006) 40 Triplet correlations Bergeret, Volkov Efetov - as a review see Bergeret et al., Rev. Mod. Phys. (2005).

41 Triplet proximity effect may substantially increase the decaying length in the dirty limit.

The same, but larger amplitude

No oscillations

42 f D f / h Some source of triplet correlations ?

Why difficult to observe ? Magnetic scattering and spin-orbit scattering are harmful for long ranged triplet component. Magnetic disorder, spin-waves… 43 Supercurrent measured in NbTiN/CrO2/NbTiN junctions

Klapwijk’s group in Delft

Long junctions with « large » Ic

CrO2 is half-metallic !

44 angle R S L S

(+ small term)45 Rather sharp maximum of the critical current at dL=dR=ξf 46 More details - M. Houzet and A. Buzdin « Long range triplet Josephson effect through a ferromagnetic trilayer », PRB, 2007. SFS junction with of domains

Different types of domains

47 Clean SFS junctions with sharp domain walls

Domain walls with collinear and non-collinear magnetic moments

S S S S Clean SFS junctions, BdG equations

g= (u,v), u and v are the electronlike and holelike parts of the quasiparticle wave function

2D : I~

3D: I~

I( ) sin2 I( ) 2 Long – range triplet proximity effect in dirty SF systems Bergeret – Volkov – Efetov (2001)

S F ξN Long – range triplet component x ξf

T. Champel et al. PRL (2008) Dirty SFS junctions with sharp domain walls. Usadel equations. Dirty SFS junctions with sharp domain walls. Collinear magnetic moments

No change in the length of decay of superconducting correlations

No long – range triplet component

In accordance with the result of Crouzy et al. PRB 76 (2007)

Dirty SFS junctions with sharp domain walls. Non- collinear magnetic moments

d N Clean SFS junctions with domain walls Dirty SFS junctions with domain walls

Maximum contribution to the critical Maximum contribution to the critical current: current: Domain walls with Domain walls with antiparallel magnetic moments non-collinear magnetic moments

Noncollinear domains in SFS junctions are sources of the long range triplet correlation.

The domain walls connecting superconducting leads can be considered as channels for triplet current. F/S/F trilayers, spin-valve effect

If ds is of the order of magnitude of s, the critical temperature is controlled by the proximity effect.

- F S F

R R

2d df s df Firstly the FI/S/FI trilayers has been studied experimentally in 1968 by Deutscher et Meunier. In this special case, we see that the critical temperature of the superconducting layers is reduced when the ferromagnets are polarized in the same direction

54 * In the dirty limit, we used the quasiclassical Usadel equations to find the new critical temperature T c. We solved it self-consistently supposing that the order parameter can be taken as :

2 x with L>>dS 0 1 L2

Buzdin, Vedyaev, Ryazhanova, Europhys Lett. 2000, Tagirov, Phys. Rev. Let. 2000. 1.0 In the case of a perfect 0.8 transparency of both

interfaces 0.6

* 0.4 Phase Tc /Tc

0.2 * h Ds d Phase Dn 4 Tc 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

* d / d s * T * 1 1 d *T T 1 1 d *T c c ln c c ln Re * 1 i * T 2 2 T 2 2 d T 55 c dsTc c s c Recent experimental verifications

AF F layer with F1 fixed magnetization

S « free » F layer F2

CuNi/Nb/CuNi Gu, You, Jiang, Pearson, Bazaliy, Bader, 2002

Ni/Nb/Ni Moraru, Pratt Jr, Birge, 2006

56 Evolution of the difference between the critical temperatures as a function of interfaces’ transparency

B 0 Infinite transparency

B 5 Finite transparency

Tc Tc ~ Tc Tc d / ds

* ~ Tc 1 1 dTc ln Re 1 i h T 2 2 * c dsTc ~ Ds Dn * ~ d T 1 1 dT 4 Tc h c c 1 1 i ln * B T 2 2 Dn 57 c dsTc Ni0.80Fe0.20/Nb (20nm) Similar physics in F/S bilayers Thin films : Néel domains Rusanov et al., PRL, 2004 In practice, magnetic domains appear quite easily in ferromagnets

w: width of the domain wall

H=0 mT H=4.2 mT

Localized (domain wall) superconducting phase. Theory - Houzet and Buzdin, PRB (2006). 58 z F

Inverse effect: appearence of the dense domain structure under the influence of superconductivity. S Not observed yet.

d dss D<<ξ

D

x

-ds df 59 Domain wall superconductivity in purely electromagnetic model

2 Ferromagnetic 2 i   1 A(r) layer w 2 (T ) 0 D M Superconducting film w>>D w<

Hc3 E.B.Sonin (1988) ?

Pb-Co/Pt 60 Superconductivity nucleation at a single domain wall

Thin domains Step-like magnetic field profile M H w<

x

wavefunction η+ x 61 S/F bilayer with domain structure

S

Domain N Wall S

Nb/F dH c2 ~ 0.5kOe/ K B0 4 M ~ 1 10kOe dT

Tc ~ 9K Tc ~ 1 3K 62 w>>D Thick domains

B0-maximum field induced by the domain wall

Local approximation: Particle in a linear B profile

63 Superconducting nucleus in a periodic domain structure in an external field H 0 B w2 0 5 0

B w2 0 1 0 Domain wall superconductivity

64 65 Nb/BaFe12O19

Z. YANG et al, Nature Materials, 2004

66 Atomic layered S-F systems (Andreev et al, PRB 1991, Houzet et al, PRB 2001, Europhys. Lett. 2002)

Magnetic layered superconductors like RuSr2GdCu2O8

F exchange field h S « 0 » BCS coupling F S « π » t F S « 0 »

Also even for the quite small exchange field (h>Tc) the π-phase must appear.

67 Crystal structure of layered HTS

• Bi2Sr2CaCu2O8+x ! (Bi2212)

• Bi2Sr2CuO6+x (Bi2201)

1.5 nm 1.5 Also Tl2212, Tl2201 etc.

68 Hamiltonian of the system

BCS coupling

Exchange field

It is possible to obtain the exact solution of this model and to find all Green functions. 69 T/Tco

1

0-phase -phase

h/Tco 2

The limit t <

Ic

-phase 0-phase

70 h/Tco Atomic thickness F/S/F heterostructure. Inversion of the proximity effect at low température.

F S t F

A Also the exact solution is possible. (Tollis et al., PRB, 2005)

TcP < TcAP

71 A P

Half-metal

T* T P AP c Tc T

The inversion of the AP F S F proximity effect occurs at the temperature * T =0.47Tc0 for the weak field h

72 F S F Superconducting multilayered systems

(Buzdin, Cayssol and Tollis, PRL 2005,)

layered superconductors with a structure like high-Tc

Zeeman effect, i.e. the exchange field μBH, or ferromagnetic superconductor S BCS coupling and exchange t 1 S field h=μBH t2<

exceed μBH~t1. π-phase with FFLO modulation in plane.

73 Superconducting bilayer

S « 0 » BCS coupling t1 S « π »

Energy spectrum

π-phase Compensation of the Zeeman splitting

at μBH=h~t

74 At low temperature the paramagnetic limit may be strongly exceed μBH~t1. π-phase with FFLO modulation in plane. 75 Field-induced superconductivity!

More complicated phase diagramm 76 Mechanism for the φ0 - Josephson junction realization. Recently the broken inversion symmetry (BIS) superconductors (like

CePt3Si) have attracted a lot of interest.

Very special situation is possible when the weak link in Josephson junction is a non-centrosymmetric magnetic metal with broken inversion symmetry ! Suitable candidates : MnSi, FeGe.

Josephson junctions with time reversal symmetry: j(-φ)= - j(φ); i.e. higher harmonics can be observed ~jnsin(nφ) –the case of the π junctions. Without this restriction a more general dependence is possible

j(φ)= j0sin(φ+ φ0).    Rashba-type spin-orbit coupling p n  n is the unit vector along the asymmetric potential gradient. 77 Geometry of the junction with BIS magnetic metal

 M

78 2  2 b 4    *  F a D n h D * D , 2

Di i i 2eAi

2 a 2i h 0, x 2 x

a a h exp(i~x)exp( x c ), where ~

φ0 - Josephson junction (A. Buzdin, PRL, 2008). 79 a a h exp(i~x)exp( x c ), where ~

In contrast with a π junction it is not possible to choose a real ψ function !

80 φ0 Josephson junction

j jc sin o

2 hL where o

The phase shift φ0 is proportional to the length and the strength of the BIS magnetic interaction.

The φ0 Junction is a natural phase shifter.

Energy EJ(φ)~-jccos(φ +φ0)

81 EJ(φ)~-jccos(φ +φ0)

E

φ

E(φ =φ0)≠ E(φ =- φ0)

82 Spontaneous flux (current) in the superconducting ring with φ0 - junction.

2 L j k E( ) c - cos( ) 2e 0 2 o c k 0 2 Lj c

In the k<<1 limit the junction generates the flux Φ=Φ0(φ0/2π)

2 hL o

Very important : The φ0 junction provides a mechanism of a direct coupling between supercurrent (superconducting phase) and magnetic moment (z component). 83 Let us consider the following geometry :

θ

voltage-biased Josephson junction

84 Magnetic anisotropy (easy z-axis) energy :

Coupling parameter : Weak coupling regime : Γ<1. Srong coupling regime : Γ>1.

Let us consider first the φ0 - junction when a constant current I

I

Minimum energy condition: 85 M θ

I

The current provokes rotation of the magnetic moment :

For the case Γ>1 when I>Ic / Γ the moment will be oriented along the y-axis.

Applying to the φ0 - junction a current (phase difference) we can generate the magnetic moment rotation. a.c. current -> moment’s precession!

86 What happens if the SO gradient is along y-axis?

I

The total energy has two minima θ=(0, π).

θ 0 π I>Ic / Γ

The current pulses would provoke the switches of M between θ=0 and θ= π orientations.

This corresponds to the transition of the junction from +φ0 to -φ0 state. 87 Magnetic moment precession – voltage-biased φ0- junction

Mz

My(t)

88 Landau-Lifshitz equation :

M

Magnetic moment precession :

89 Weak coupling regime : Γ<<1.

Without damping

With damping

The current acquires a d..c. component !

90 Srong coupling regime : Γ>>1.

if r <<1 then and without damping we have :

Comparison between analytic results (dashed line) and numerical computation.

91 Complicated regime of the magnetic dynamics :

For more details – see ( F. KonschelIe and A. Buzdin, PRL, 2009 ). 92 Complementary Josephson logic RSFQ-logic using -shifters A.V.Ustinov, V.K.Kaplunenko. Journ. Appl. Phys. 94, 5405 (2003)

RSFQ- logic: Rapid Single Quantum logic

Conventional RSFQ-cell LJ= 0 /(2 Ic) ~ 1/(Ic R) LIc > 0 RSFQ - cell

L 0 Fluxon memorizing cell

To operate at 20 GHz clock rate -RSFQ –Toggle Ic R~50 V has to be Flip-Flop We have I R > 0.1 V for the present c 93 July 2010

94 Superconducting phase qubit

95 July 2010

96 I. conclusions • Unusual non-uniform magnetic and superconducting phases.

• Ferromagnetic superconductors –unconventional types of superconductivity. Fundamental properties of this superconductivity?

• Superconductor-ferromagnet heterostructures permit to study superconductivity under huge exchange field (h>>Tc).

• The -junction realization in S/F/S structures is quite a general phenomenon.

• The BIS magnets provide a mechanism of the realization of the

novel φ0 - junctions . In these φ0 - junctions a direct (linear) coupling between superconductivity and magnetism is realized. Seems to be ideal for superconducting spintronics.

Some general Refs.: Magnetic superconductors- M. Kulic and A. Buzdin in Superconductivity, Springer, 2008 (eds. Benneman and Ketterson). S/F proximity effect - A. Buzdin, Rev. Mod. Phys. (2005). S. Bergeret et al. Rev. Mod. Phys. (2005). Coexistence of Superconductivity and Magnetism –Bulaevskii et al., ADV. IN PHYSICS, 1985, VOL. 34,97 175.

Non-uniform (FFLO) states and quantum oscillations in superconductors and superfluid ultracold Fermi gases 1. Larkin-Ovchinnikov-Fulde-Ferrell state. LOFF or FFLO states. The origin of the non-uniform modulated superconducting 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?

98 FFLO inventors

Fulde and Ferrell

Larkin and Ovchinnikov E

kF - kF

k

kF + kF

The total momentum of the Cooper pair is

-(kF - kF)+ (kF - kF)=2 kF

Pairing of electrons with opposite spins and momenta unfavourable :

But :

if

1d SC H / 1 At T = 0, Zeeman energy compensation B 2d SC 0.8 is exact in 1d, partial in 2d and 3d. 3d SC 0.6

• the upper critical field is increased 0.4 • Sensivity to the disorder and to the orbital 0.2 0.56 effect: 0.2 0.4 0.6 0.8 1 (clean limit)

0.56 T /Tc 0.2 0.4 0.6 0.8 1 FF state uniform (r) (q)exp(iq r) ( k ,-k ) available for pairing

depaired

( k ,-k+q )

LO state spatially nonuniform (r) 0 cos(q r)

+

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 4 2 2

2 *2 2 2 * 6 ...

May be 1st order transition at

1 B B /

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 -(BEDT-TTF)2Cu(NCS)2 (Tc=10.4K)

Suppression of orbital effects in H parallel to the Layered structure 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)

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) 107 Field induced superconductivity (FISC) in an organic compound -(BETS)2FeCl4

Metal

FISC Insulator AF

c-axis (in-plane) S. Uji et al., Nature 410 908 (2001) resistivity L. Balicas et al., PRL 87 067002 (2001) 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 FFLO State in Bose-Einstein-Condensate Color superconductivity Vortices R.Casalbuoni and G.Nardulli Rev. Mod. Phys. (2004)

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, 1d SC H / 1 first proposed by Brazovskii, Gordyunin B 0.8 and Kirova in 1980 for polyacetylene. 0.6

0.4 (x) 0sn(x / ,k) 0.2

0.56 T /Tc 2 atT 0 B H Magnetic moment

(x) x

Spin-Peierls transitions - e.g. CuGeO3

Cu O Ge

n un ( 1) (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 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

orb H c2 2 P FFLO exists for Maki parameter α>1.8. H c2

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 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 Instability toward 1st FFLO state 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 : 3

Analysis of phase diagram : • cascade of 2nd and 1st order transitions n=2 between S and N phases 2 n=1 • 1st order transitions within the S phase n=0

• exotic vortex lattice structures field Magnetic 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. 124 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

cos(φ - φ ) Fn,n+1=[-I0cos(αn_- αn+1)+I2cos(qr) cos(αn_+ αn+1)] n n+1

φ - φ =2πxHs/Φ + πn n n+1 0 s-interlayer distance, x-coordinate 125 along q CeCoIn5

Quasi-2D heavy fermion

CeIn3 (Tc=2.3K) Strong antiferromagnetic CoIn2 fluctuation z d-wave symmetry CeIn3 Tetragonal symmetry Modified Ginzburg-Landau functional: isotropic part x y

2e iA j j c j No orbital effect

~

= = /2 z-axis modulation xy-plane modulation z

~ =0.5 arccos( z/( - z)) ~ =2 z

~ 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 q =0, n=max z ~ = x q =max, n=0 z z

q >0, n>0 z

~ =2 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

~ = z x q =0, n=max z xy ~ = x q =max, n=0 z z

z

intermediateq >0, n>0 z ~ xy ~ =2 = = /2 z z-axis modulation xy-plane modulation z z

intermediate~ =0.5 arccos( z/( - z)) ~ =2 z Magnetic field along x axis

~ ~ =-3 z

qx=max, n=0 z

~ =- x q 0, n=max x H||x

qx>0, n>0

~ =3 z-2 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

~ ~ =-3 z xy qx=max, n=0 z z ~ =- x qx 0, n=max

q >0, n>0 ~ intermediatex xy ~ =3 -2 = = /2 z x z-axis modulation xy-plane modulation z z

intermediate~ =0.5 arccos( z/( - z)) ~ =2 z Small angle neutron scattering from the vortex lattice for H //c

FFLO? Neutron form factor

Neutron form factor seems to be consisitent with FFLO state

A.D.Bianch et al.Science (2007)  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 hyperfine 76 MHz 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 Images of vortex lattices

MIT: Ketterle et al (2005) 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

vs -1 1 0 A.Bezryadin, A.I. Buzdin, B. Pannetier (1994) Ф/Ф

o L=-1 L=0 L=1 c -ΔTc/Tc0 0 e Ф/Ф

o Examples of quantum oscillation effects.

Little-Parks effect. O.Buisson et al (1990) Simply-connected systems R.Benoist, W.Zwerger (1997) Mesoscopic samples V.A.Schweigert, F.M.Peeters dimensions ~ several coherence lengths (1998) H.T.Jadallah, J.Rubinstein, Sternberg (1999) H R~ξ Origin of Tc oscillations: Transitions between the states with different vorticity L | r | eiL Tc (H) oscillations 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

H z a 2 0k0 Model: Modified Ginzburg-Landau functional (2D)

   F ((a V (r))| |2 | D |2 | D2 |2 )dxdy   a (T Tc0 )  Trapping D 2iM[ ,r]  potential FFLO instability   D 2ieA c T Range of validity: N Confinement mechanisms: vicinity of tricritical

point 1. Zero trapping potential. S Boundary condition at the system edge   nD 0 H FFLO 2. nonzero trapping potential  V (r) M 2r 2 2 FFLO states in a 2D mesoscopic superconducting disk

Hz z 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

2 k0 (H,T ) T H ↑ → L↓ L=1 L=2

FFLO state L=0

H Tilted magnetic field: Hz ≠ 0

Field induced superconductivity

Eigenvalue problem:

2 R H z a 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 H|| ~ Hp r   2 2 4 2 2 F ( 1 | D | 2 | D | )dxdy

We focus on the limit 1 2 T N eiL S (H*,T*) Condition of the first vortex entry: Beyond the range of FFLO F(L 1) F(L 0) 0 instability FFL H || O Condition of the first vortex entry:

2 R m 2 R 3 ln 2 3 4 ln 0 m 2 R m

max( , ) m 1 2 Limiting cases: H R2 R 1 R z 2 1 ln H z 2 ln 1 R 1 0 4 2 3 4 3 2 R 1 2 2 2 1 H z m 1 2 R R 1 Change in the scaling m law * H H|| 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 2 ( v0 ) 0

FFLO length scale Fourier transform: k0 2   iq 2  4 Temperature shift e d q a k0 q 2 2 6 (Trapping frequency)2 4 2 v0 M / 2 k0 v  q 2q 0 q 2   Dimensionless k0r coordinate L 0

U(q) q 1 x q 4 2 2 U (q) q 2q 1 4x v0 0

x2  1 q e v0 FFLO states in a 2D system in a parabolic trapping potential (no rotation) Phase diagram Condensate wave function

k0r 1 1 1 2 v 2 2 0 cos(k r 4)e k0 r v0 4 v 0 0 k0r

Suppression of wave function oscillations by the increase in the trapping frequency

k0r

2 1 =Number of observable oscillations 1/ 4 v0 FFLO states in a rotating 2D gas in a parabolic trapping potential.

 iL 4 2 2 (r) fL (r)e D fL 2D fL ( v0 ) fL 0

2 k 2 FFLO length scale 2 1 L 0 D a 4 Temperature shift a k0

Expansion in eigenfunctions of the problem without 2 6 (Trapping frequency)2 trap: v0 M / 2 k0 f ( ) c u ( ) L n nL 2 rotation frequency n 0 a 2M  k0 2 2 D unL qnLunL 2 a (2n | L | L 1)unL   Dimensionless 2 4 k0r coordinate (2qnL qnL )cn vnmcm cn m 3 vnm v0 umLunL d 0 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: 2 2 max (4 a v0 a )(2L 1) v0L a 4 a (2L 1) L 0

rotation induced superfluid phase II. Conclusions

• There are strong experimental evidences of the existence of the the FFLO state in organic layered superconductors and in heavy fermion superconductor CeCoIn5

• 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?

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