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

GDR - MICO – Autrans 2008 1

• Recall on magnetism and superconductivity coexistence • Origin and the main peculiarities of the proximity effect in superconductor-ferromagnet systems. • Josephson π-junction, the role of the magnetic scattering. • Domain wall superconductivity. Spin-valve effet. • Inversion of the proximity effect in atomic F/S/F structures.

• φ-junctions. • Possible applications

2

Magnetism and Superconductivity Coexistence

dTc ≈ −Θ m x dx (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.

3

Antagonism of magnetism (ferromagnetism) and superconductivity

• 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

( )

r r I S ⋅ s ≈ Tc

Exchange interaction

4

No antagonism between antiferromagnetism and superconductivity 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

ErMo6Se8

6.0

1.1

DyNi2B2C

6.2

11

ErNi2B2C

10.5

6.8

TmNi2B2C

11

1.5

HoNi2B2C

8

5

Usually Tc>TN

5

FERROMAGNETIC CONVENTIONAL (SINGLET) SUPERCONDUCTORS

Tc2 A. C. susceptibility and resistance versus temperature in ErRh4B4 (Fertig et al.,1977).

Tc RE-ENTRANT SUPERCONDUCTIVITY in ErRh4B4 , HoMo6S8 6

Auto-waves in reentrant superconductors? current I

T>1 following (Anderson and Suhl, 1959)

χ s (Q) − χ (Q) π = χ (0) 2Qξ 0 Energy per atom/electron : magnetic, superconducting

≈θ

≈θ

≈ Tc2 / E F d ∝ aξ

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 structure around 92 Å. 8

FERROMAGNETIC UNCONVENTIONAL (TRIPLET) SUPERCONDUCTORS

UGe2 (Saxena et al., 2000) and URhGe (Aoki et al., 2001) Triplet pairing Moment [μB/URhGe]

0.6

M total

0.5 0.4 Mb

0.3 0.2

T=2 K

0.1

R [arb.units]

0.0 500 mK

8 6

40 mK

4 2 0

0

2

4

6

8

10

12

14

16

μ0H b [Tesla]

URhGe (a) The total magnetic moment M total and UGe2

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 9 (Levy et al 2005).

The coexistence of singlet superconductivity and ferromagnetism is basically impossible in the same compound but may be easily achieved in artificially fabricated superconductor/ferromagnet heterostructures.

S

S

F

S

F

S

h>>Tc

Due to the proximity effect, the Cooper pairs penetrate into the F layer and we have the unique possibility to study T the properties of superconducting electrons under the influence of the huge exchange field.

Varying in the controllable manner the thicknesses of the ferromagnetic and superconducting layers it is possible to change the relative strength of two competing ordering. Interesting effects at the nanoscopic scale. The Josephson junctions with ferromagnetic layers reveal many unusual properties quite interesting for applications, in particular the so-called πJosephson junction (with the π-phase difference in the ground state). 10

Superconducting order parameter behavior in ferromagnet Standard Ginzburg-Landau functional: 1 b 4 2 2 F =aΨ + ∇Ψ + Ψ 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

2

2

F = a Ψ − γ ∇Ψ +η ∇ Ψ + ... 2

The non-uniform state Ψ~exp(iqr) will correspond to minimum energy and higher transition temperature

11

F

F = (a − γq + ηq ) Ψq 2

q0

4

2

q

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

12

E

kF -δkF

k kF +δkF

The total momentum of the Cooper pair is -(kF -δkF)+ (kF -δkF)=2 δkF 13

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

1 2 ∇ Ψ = 0, and solution for T > Tc is Ψ ∝ e − qx , where q = 4m

4ma

In ferromagnet ( in presence of exchange field) the equation for superconducting order parameter is different

aΨ + γ∇ Ψ − η∇ Ψ = 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! x

14

Proximity effect as Andreev reflection

pF

Classical Andreev reflection

Quantum Andreev reflection

(A.F. Andreev, September, 2008)

pF↑ ≠ pF↓

F

S

h>>Tc

15

Theory of S-F systems in dirty limit Analysis on the basis of the Usadel equations

−

Df

∇ F f ( x,ω, h) + (ω + ih)F f ( x,ω, h) = 0 2

2 2 * G f ( x,ω, h) + F f ( x,ω, h)F f ( x,ω,−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. 16

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) Δ

Δ

S

Δ

S

F

« 0 phase »

« π phase » S

F

S

S −Δ

Δ

F

S/F bilayer

−Δ

ξ f = Df / h h-exchange field, Df-diffusion constant

17

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

18

SF-bilayer Tc-oscillations Ryazanov et al. JETP Lett. 77, 39 (2003) Nb-Cu0.43Ni0.57

V. Zdravkov, A. Sidorenko et al

PRL (2007)

Nb-Cu0.41Ni0.59

dFmin =(1/4) λ ex largest Tc-suppression

19

S-F-S Josephson junction in the clean limit (Buzdin, Bulaevskii and Panjukov, JETP Lett. 81)

S

F

S

Damping oscillating dependence of the critical current Ic as the function of the parameter α=hdF /vF has been predicted. h- exchange field in the ferromagnet, dF - its thickness Ic

E(φ)=- Ic (Φ0/2πc) cosφ

J(φ)=Icsinφ α

20

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 21

Phase-sensitive experiments π-junction in one-contact interferometer 0-junction minimum energy at 0

π-junction minimum energy at π

I

I=Icsin(π+φ)=-Icsinφ E= EJ[1-cos(π+φ)]=EJ[1+cosφ]

I

L

φ

φ

I

2πLIc > Φ0/2 φ =π = (2π / Φ0)∫Adl = 2π Φ/Φ0

E

E

φ

φ

Spontaneous circulating current in a closed superconducting loop when βL>1 with NO applied flux βL = Φ0/(4 π

Bulaevsky, Kuzii, Sobyanin, JETP Lett. 1977

LIc)

Φ = Φ0/2

22

Current-phase experiment. Two-cell interferometer

Ic

23

Cluster Designs (Ryazanov et al.)

30μm

2x2

unfrustrated

fully-frustrated

checkerboard-frustrated

6x6

fully-frustrated

checkerboard-frustrated

24

2 x 2 arrays: spontaneous vortices

Fully frustrated

Checkerboard frustrated 25

Scanning SQUID Microscope images (Ryazanov et al.)

Ic Tπ

T = 1.7K

T = 2.75K

T T = 4.2K

26

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

Ic=Ic0exp(-dF/ξF1) |cos (dF /ξF2) + sin (dF /ξF2)| dF>> ξF1

“0”-state

Spin-flip scattering decreases the decaying length and increases the oscillation period.

π-state

ξF2 >ξF1 0 Nb-Cu0.47Ni0.53-Nb

“0”-state I=Icsinϕ

π-state I=Icsin(ϕ+ π)= - Icsin(ϕ) 27

Critical current vs. temperature Nb-Cu0.47Ni0.53-Nb dF=9-24 nm h=Eex ∼ 850 K (TCurie= 70 K)

1

ξ F1 1

ξF2

=

=

1

ξF 1

ξF

⎛ 1 1 + ⎜⎜ ⎝ hτ s ⎛ 1 1 + ⎜⎜ ⎝ hτ s

“Temperature dependent” spin-flip scattering 2

⎞ ⎛ 1 ⎟⎟ + ⎜⎜ ⎠ ⎝ hτ s 2

⎞ ⎛ 1 ⎟⎟ − ⎜⎜ ⎠ ⎝ hτ s

G = cos Θ(T); F = sin Θ(T)

⎞ ⎟⎟, ⎠ ⎞ ⎟⎟ ⎠

ξ F 2 > ξ F1

Effective spin-flip rate Γ(T)= cos Θ(T)/τS; 28

Critical current vs. temperature (0-π- and π-0- transitions)

Nb-Cu0.47Ni0.53-Nb dF1=10-11 nm dF2=22 nm

(V.A.Oboznov et al., PRL, 2006)

29

Density of states in the ferromagnet in contact with superconductor (Buzdin, PRB, 2000; Baladie and Buzdin, PRB, 2001)

F

S df

In the case of a weak proximity effect (weak influence of the ferromagnet on the superconducting order parameter) we can derive the superconducting density of states induced in the ferromagnet by the proximity effect. In the clean limit and in the dirty limit, far away from Tc , close to Tc we see that the oscillatory behavior of the density of states close to the S/F interface is really robust to the variation of parameters characterizing the system. 30

DOS structures at different distances from S layer

31

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

32

Triplet correlations Bergeret, Volkov Efetov -as a review see Bergeret et al., Rev. Mod. Phys. (2005).

33

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

The same, but larger amplitude

No oscillations

34

Some source of triplet correlations ?

ξ f = Df / h

Why difficult to observe ?

Magnetic disorder, spin-waves…

Magnetic scattering and spin-orbit scattering are harmful for long ranged triplet component.

35

Supercurrent measured in NbTiN/CrO2/NbTiN junctions Klapwijk’s group in Delft

Long junctions with « large » Ic CrO2 is half-metallic ! 36

S

angle θL

θR

S

37 (+ small term)

IcΠG2 4eTc

0.2

ΦLΠ2

ΦR Π2 Π3 Π4 Π6 Π12 0

0.1 0

ΦR Π12 Π6 Π4 Π3 Π2

0.1 0.2 0.5

1

1.5 dLΞf dR Ξf

2

2.5

3

Rather sharp maximum of the critical current at dL=dR=ξf 38 More details - M. Houzet and A. Buzdin « Long range triplet Josephson effect through a ferromagnetic trilayer », PRB, 2007.

F/S/F trilayers, spin-valve effect If ds is of the order of magnitude of ξs, the critical temperature is controlled by the proximity effect.

φ F

-φ

S

F R↑↑

df

2ds

R↑↓

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 39

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 :

⎛ x2 ⎞ Δ = Δ 0 ⎜1 − 2 ⎟ ⎜ L ⎟ ⎝ ⎠

with L>>dS

Buzdin, Vedyaev, Ryazhanova, Europhys Lett. 2000, Tagirov, Phys. Rev. Let. 2000. 1.0

In the case of a perfect transparency of both interfaces *

Tc / Tc

h Ds d* =γ Dn 4πTc

0.8

0.6

Phase ↑↓

0.4

0.2

Phase ↑↑ 0.0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

d* /ds

⎛ 1 d *Tc ⎞ ⎛ Tc↑↑* ⎞ 1 ⎛ ⎞ ⎜ ⎜ ⎟ ( ln 1 + i )⎟ = Ψ⎜ ⎟ − Re Ψ + * ⎜2 d T ⎟ ⎜ Tc ⎟ 2 ⎝ ⎠ s c↑↑ ⎝ ⎠ ⎝ ⎠

⎛ 1 d *Tc ⎞ ⎛ Tc↑↓* ⎞ 1 ⎛ ⎞ ⎟ ⎟ = Ψ⎜ ⎟ − Ψ⎜ + ln⎜ * ⎜ 2 d T 40 ⎟ ⎜ Tc ⎟ ⎝ 2⎠ s c↑↓ ⎠ ⎝ ⎝ ⎠

Recent experimental verifications

AF F1

F layer with 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 41

Evolution of the difference between the critical temperatures as a function of interfaces’ transparency γB =0 Infinite transparency

γB ≈5

Finite transparency

Tc↑↑

Tc↑↓

Tc↑↑

~ ⎛1 ⎞ ⎛ Tc ↑↑* ⎞ d Tc 1 ⎛ ⎞ ⎜ ⎜ ⎟ ( 1 + i )⎟ ln = Ψ⎜ ⎟ − Re Ψ + * ⎜ T ⎟ ⎟ ⎜2 d T ⎝ 2⎠ s c↑↑ ⎝ c ⎠ ⎝ ⎠ ⎛ 1 ⎛ d~T ⎞ ⎞ ⎛ Tc ↑↓* ⎞ 1 ⎛ ⎞ c ⎟⎟ ⎟ = Ψ⎜ ⎟ − Ψ⎜ + ⎜ ln⎜ ⎜ T ⎟ ⎜ 2 ⎜ d sT * ⎟ ⎟ 2 ⎝ ⎠ c ⎝ ⎠ ⎝ c↑↓ ⎠ ⎠ ⎝

Tc↑↓

~ d / ds

D ~ d= s 4πTc

γ

h Dn

1 + (1 + i )γ B γ

h Dn

42

z

F

Inverse effect: appearence of the dense domain structure under the influence of superconductivity.

S

Not observed yet.

d ds

s

DD

?

E.B.Sonin (1988)

Pb-Co/Pt

w

D w