1 Non-uniform superconductivity in superconductor/ferromagnet nanostructures A. Buzdin Institut Universitaire de France, Paris and Condensed Matter Theory Group, University of Bordeaux in collaboration with M. Daumens, J. Cayssol, S. Tollis University of Bordeaux A. Koshelev, Argonne National Laboratory

2 Antagonism of magnetism (ferromagnetism) and superconductivity Orbital effect (Lorentz force) B -p FLFL p FLFL Paramagnetic effect (singlet pair) S z =+1/2S z =-1/2 μ B H~Δ~T c

3 Superconducting order parameter behavior in ferromagnet Standard Ginzburg-Landau functional: The minimum energy corresponds to Ψ=const The coefficients of GL functional are functions of internal exchange field! Modified Ginzburg-Landau functional : The non-uniform state Ψ~exp(iqr) will correspond to minimum energy and higher transition temperature

4 Proximity effect in ferromagnet ? q F q0q0 Ψ~exp(iqr) - Fulde-Ferrell-Larkin-Ovchinnikov state (1964) In the usual case (normal metal):

5 In ferromagnet ( in presence of exchange field) the equation for superconducting order parameter is different Its solution corresponds to the order parameter which decays with oscillations! Ψ~exp[-(q 1 + iq 2 )x] x Ψ Order parameter changes its sign!

6 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) SFS S F «  phase » « 0 phase » SFS S/F bilayer

7 The oscillations of the critical temperature as a function of the thickness of the ferromagnetic layer in S/F multilayers has been predicted by Buzdin and Kuprianov, JETPL, 1990 and observed on experiment by Jiang et al. PRL, 1995, in Nb/Gd multilayers F F F S S

8 S-F-S Josephson junction in the clean limit (Buzdin, Bulaevskii and Panjukov, JETP Lett. 81) S SF Damping oscillating dependence of the critical current I c as the function of the parameter  =hd F /v F has been predicted. h- exchange field in the ferromagnet, d F - its thickness IcIc 

9 SS 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 F and as a function of a ferromagnetic layer thickness by Kontos et al 2002 PRL

10 Critical current density vs. F-layer thickness (Ryazanov et al. 2005)  -state I=I c sin  I=I c sin(  +  )= - I c sin(  ) Nb-Cu 0.47 Ni Nb I c =I c0 exp(-d F /  F1 ) | cos (d F /  F2 ) + sin (d F /  F2 ) | d F >>  F1 d F,  1 =(3/4)  F2 =(3/8) ex “0”-state  F2 >  F1 d F,  2 =(7/4)  F2 =(7/8) ex “0”-state  -state 0

11 In the clean limit (h  >>1), we find oscillations of period v f /h, oscillating like sin(x)/x Density of states at Fermi level In the dirty limit (h  <<1), we find oscillations of period oscillating like exp(-x)/x T/T c variation Density of states

12 Density of states measured by Kontos et al (PRL 2001) on Nb/PdNi bilayers

13 Atomic layer S-F systems F F F S S S exchange field h BCS coupling t (Andreev et al, PRB 1991, Houzet et al, PRB 2001, Europhys. Lett. 2002) Magnetic layered superconductors like RuSr 2 GdCu 2 O 8 « π » « 0 » Also even for the quite small exchange field (h>T c ) the π-phase must appear.

14 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.

15 h/T co T/Tco phase  -phase The limit t<<Tco h/T co IcIc  -phase 0-phase

16 Superconducting multilayered systems S S Zeeman effect, i.e. the exchange field μ B H BCS coupling t1t1 (Buzdin, Cayssol and Tollis, to be published, PRL 2005 ) layered superconductors with a structure like high-T c « π » « 0 » At low temperature the paramagnetic limit may be strongly exceed μ B H~t 1. π-phase with FFLO modulation in plane. S S BCS coupling t1t1 « π » « 0 » t 2 <<t 1

17 The mechanism of the  -junction realization due to the tunneling through thin ferromagnetic layer (Buzdin, 2003) SS d/2-d/2 The large and small

18 At T=0, and γ B >>h/T c SS F(x)

19 How the transition from 0- to  – state occurs? J(φ)=I c sinφ ; I c >0 in the 0- state and I c <0 in the  – state J(φ)=I 1 sinφ +I 2 sin2φ Energy E(φ)=(Φ 0 /2πc)[-I 1 cosφ –(I 2 /2)sin2φ] 0  φ E I 2 >0 I c =|I 2 |

20 J(φ)=I 2 sin2φ The realization of the equilibrium phase difference 0< φ 0 <π 0  φ E I 2 <0

21 (Manhnhart, van-Harlingen et al ) Grain boundaries in YBaCuO YBaCuO-Nb Josephson junctions of zig-zag geometry (Hilgenkamp, Smilde et al. 2002) YBaCuO Nb 0 π 0 0 π π Possibility to fabricate different alternating 0- and π- junctions Arbitrary equilibrium phase difference: φ- junction (Buzdin, Koshelev, 2003) S S F

22 We will study the properties of long Josephson junctions with lengths d 0 of 0-junctions and d π of π-junctions in the limit d 0, d π <<λ J, λ J is the Josephson length of individual junction (for simplicity we assume that it is the same for 0- and π-junctions) π 0 The energy per period of our system is: 00 π π dπdπ d0d0 x is the coordinate along the zig-zag boundary φ(x)

23 The current-phase relation for φ - junction : The current-phase relation is quite peculiar, the current has two maxima and two minima at

24 New kinds of solitons in φ - junctions Besides 2π degeneracy, there is ±φ 0 degeneracy! New solitons - φ 0 →+φ 0 or φ 0 → 2π -φ 0, The flux of the first type of solitons is Φ 0 (φ 0 /π). The flux of the second type of solitons is Φ 0 ((π-φ 0 ) /π).

25 Conclusions The  -junction realization in S/F/S structures is quite a general phenomenon, and it exists even for thin F- layers (d<ξ f ), in the case of low interface transparency. New non-uniform superconducting phases in superconducting layered structures with alternating electron transfer integrals Transition to the φ- junction state can be observed by decreasing the temperature from T c. For review see - A. Buzdin, Rev. Mod. Phys. (July 2005)