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Coexistence of Ferromagnetism and Superconductivity at the Nano-Scale

A. BuzdinInstitut Universitaire de France, Paris

and Condensed Matter Theory Group, University of Bordeaux

MISIS, Moscow 2009

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• Recall on magnetism and superconductivity coexistence

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

• Josephson π-junction.

• Domain wall superconductivity.

• Spin-valve effet.

• Possible applications

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Magnetism and Superconductivity

Coexistence

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.

xdx

dTm

c

(Abrikosov and Gorkov, 1960)

4

5

Antagonism of magnetism (ferromagnetism) and superconductivity

• Orbital effect (Lorentz force)

B

-pFL

p

FL

• Paramagnetic effect (singlet pair)

Sz=+1/2 Sz=-1/2

μBH~Δ~Tc cTsSI

Electromagneticmechanism

(breakdown of Cooper pairsby magnetic field

induced by magnetic moment)

Exchange interaction

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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

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FERROMAGNETIC CONVENTIONAL (SINGLET)

SUPERCONDUCTORS

A. C. susceptibility and resistance versus temperature

in ErRh4B4 (Fertig et al.,1977)

RE-ENTRANT SUPERCONDUCTIVITY in ErRh4B4 !

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2

2

)(Q

Qm h

QE

At T=0 and Q0>>1 following (Anderson and Suhl, 1959)

02)0(

)()(

Q

QQs

Coexistence phase

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.

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HoMo6S8

Tc=1.8 KTm=0.74 KTc=0.7 K

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Auto-waves in reentrant superconductors?

current I

T<Tc2

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FERROMAGNETIC UNCONVENTIONAL (TRIPLET) SUPERCONDUCTORS

Triplet pairing

UGe2 (Saxena et al., 2000)

and URhGe (Aoki et al., 2001)

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 (Levy et al 2005).

UGe2

Very recently (2007): UCoGe θ=3K; Tc=0.8K

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16

17

18

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Superconducting order parameter behavior in ferromagnet

Standard Ginzburg-Landau functional:

...2222 aF

The minimum energy corresponds to Ψ=const

The coefficients of GL functional are functions of internal exchange field h !

Modified Ginzburg-Landau functional ! :

422

24

1

b

maF

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

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Proximity effect in ferromagnet ?

q

F

q0

242 )( qqqaF

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

In the usual case (normal metal):

4maq where, is TTfor solution and ,04

1c

2 qxem

a

21

E

kkF +kF

kF -kF

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

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

042 a

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

x

Ψ

Order parameter changes its sign!

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Theory of S-F systems in dirty limit

Analysis on the basis of the Usadel equations

.

1,,,,,,

0,,,,2

*2

2

hxFhxFhxG

hxFihhxFD

fff

fff

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

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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

« phase  » « 0 phase »

S F S

S/F bilayer hD ff /

h-exchange field,Df-diffusion constant

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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

F

F

S

S

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SF-bilayer Tc-oscillations

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

V. Zdravkov, A. Sidorenko et alcond-mat/0602448 (2006)Nb-Cu0.41Ni0.59

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

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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 Ic as the function of the parameter =hdF /vF has been predicted.

h- exchange field in the ferromagnet,dF - its thickness

Ic

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S S

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

Fand as a function of a ferromagnetic layerthickness by Kontos et al. 2002, PRL

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

interferometer0-junction

minimum energy at 0

I

-junctionminimum energy at

I

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

I=Icsin(+)=-Icsin

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

L = 0/(4 LIc)

= 0/2

E

E

Bulaevsky, Kuzii, Sobyanin, JETP Lett. 1977

2LIc > 0/2

= =(2 / 0)Adl

= 2 /0

L

I

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Current-phase experiment. Two-cell interferometer

Ic

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Cluster Designs (Ryazanov et al.)

2 x 2

6 x 6

fully-frustrated checkerboard-frustrated

fully-frustrated unfrustrated checkerboard-frustrated

30m

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Checkerboardfrustrated

Fullyfrustrated

2 x 2 arrays: spontaneous vortices

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T

T = 1.7K T = 4.2KT = 2.75K

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

T

Ic

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

-stateI=Icsin

I=Icsin(+ )= - Icsin()

Nb-Cu0.47Ni0.53-Nb

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

dF>> F1

“0”-state

F2 >F1

“0”-state-state

0

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

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

“Temperature dependent” spin-flip scattering

G = cos (T); F = sin (T)

Effective spin-flip rate (T)= cos (T)/S;

ssFF

ssFF

hh

hh

111

11

,11

111

2

2

2

1

12 FF

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

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

Nb-Cu0.47Ni0.53-NbdF1=10-11 nmdF2=22 nm

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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.

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 thesuperconducting layers is reduced when the ferromagnets are polarized in the same direction

F FS

-

df df2ds

R R

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sdd /*

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.0

0.2

0.4

0.6

0.8

1.0

cc TT /* Phase

Phase

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 :

with L>>dS

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

iTd

Td

T

T

cs

c

c

c 12

1Re

2

1ln *

**

c

s

n T

D

D

hd

4*

2

2

0 1L

x

In the case of a perfect transparency of both interfaces

*

**

2

1

2

1ln

cs

c

c

c

Td

Td

T

T

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Recent experimental verifications

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

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

AF

F1

S

F2

F layer with fixed

magnetization

« free » F layer

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

n

B

n

c

s

D

hi

D

h

T

Dd

114

~

Tc TcTc Tc

0BInfinite transparency

5BFinite transparency

sdd /~

*

*

*

*

~

2

1

2

1ln

1~

2

1Re

2

1ln

cs

c

c

c

cs

c

c

c

Td

Td

T

T

iTd

Td

T

T

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Similar physics in F/S bilayersRusanov et al., PRL, 2004

Ni0.80Fe0.20/Nb (20nm)

Thin films : Néel domains

w: width of the domain wall

In practice, magnetic domains appear quite easily in ferromagnets

Localized (domain wall) superconducting phase.

Theory - Houzet and Buzdin, PRB (2006).

H=4.2 mTH=0 mT

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x

S

df

ds

-ds

D

z

x

S

F

df

ds

-ds

D

z

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

Not observed yet.

D<<ξ

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?Pb-Co/Pt

M

Ferromagneticlayer

Superconducting film

E.B.Sonin (1988)

D

w>>D w<<D

Hc3

w

Domain wall superconductivity in purelyelectromagnetic model

)(

1)(

22

2

0 TrA

i

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Mx

η+

η-H

x

U

x

η+

Superconductivity nucleation at a single domain wall

Step-like magnetic field profile

wavefunction

Thin domains

D

w<<D

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kOeMB 101~40 KTc 31~

KkOedT

dH c /5.0~2Nb/F

KTc 9~

S

NDomain Wall S

S/F bilayer withdomain structure

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w>>D Thick domains

B0-maximum field induced by the domain wall

Local approximation:Particle in a linear B profile

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Superconducting nucleus in a periodic domain structure in an external field

10

20

wB

50

20

wB0H

Domain wallsuperconductivity

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Nb/BaFe12O19

Z. YANG et al, Nature Materials, 2004

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Atomic layered 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 RuSr2GdCu2O8

« π »

« 0 »

« 0 »

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

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• Bi2Sr2CaCu2O8+x (Bi2212)• Bi2Sr2CuO6+x (Bi2201)

Also Tl2212, Tl2201 etc.

Crystal structure of layered HTS1.

5 nm

!

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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.

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h/Tco

T/Tco

1

2

0-phase-phase

The limit t <<Tco

h/Tco

Ic

-phase0-phase

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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.

np

Recently the broken inversion symmetry (BIS) superconductors (like CePt3Si) have attracted a lot of interest.

Rashba-type spin-orbit coupling

n

is the unit vector along the asymmetric potential gradient.

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).

Mechanism for the φ0 - Josephson junction realization.

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Geometry of the junction with BIS magnetic metal

M

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iii eAiD

DDhnb

DaF

2

,2

**422

,022

2

x

hix

a

h

whereaa

xxi c

~ ),exp()~exp(

φ0 - Josephson junction (A. Buzdin, PRL, 2008).

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h

whereaa

xxi c

~ ),exp()~exp(

In contrast with a π junctionit is not possible to choose a

real ψ function !

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ocjj sin

hL

o

2where

Energy EJ(φ)~-jccos(φ +φ0)

φ0 Josephson junction

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.

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φ

E

EJ(φ)~-jccos(φ +φ0)

E(φ =φ0)≠ E(φ =-φ0)

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

hL

o

2

2)cos(-

2)E(

2

0

k

e

jc

cLj

ck

20

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

o

Very important : The φ0 junction provides a mechanism of a direct coupling

between supercurrent (superconducting phase) and magnetic moment (z component).

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Let us consider the following geometry :

voltage-biased Josephson junction

θ

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Magnetic anisotropy (easy z-axis) energy :

Coupling parameter :

Let us consider first the φ0 - junction when a constant current I<Ic is applied :

I

Minimum energy condition:

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

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I

M

θ

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!

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I

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

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

0 π

θ

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.

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Mz

My(t)

Magnetic moment precession – voltage-biased φ0- junction

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Landau-Lifshitz equation :

Magnetic moment precession :

M

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With damping

Weak coupling regime : Γ<<1.

The current acquires ad..c. component !

Without damping

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Srong coupling regime : Γ>>1.

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

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

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Complicated regime of the magnetic dynamics :

For more details – see ( F. KonschelIe and A. Buzdin, PRL, 2009 ).

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A very rich physics emerges if the φ0 – junction is exposed to the microwave radiation ω1.

In addition to the Shapiro steps at ωJ=n ω1 it will appear the half-interger-steps.

The microwave field may also generate the additional precession with ω1 frequency.

Dramatic increase of the amplitude of the Shapiro steps near the ferromagnetic resonancewould be expected.

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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

LIc >0 RSFQ -cell

-RSFQ –ToggleFlip-Flop

L 0Fluxon memorizing cell

LJ= 0 /(2 Ic)

~ 1/(Ic R)

To operate at 20 GHz clock rate Ic R~50 V has to beWe have Ic R > 0.1 V for the present

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Superconducting phase qubit

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Conclusions• 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, and it exists even for thin F-layers (d<ξf), in the case of low interface transparency.

• Transition to φ- junction state can be observed by decreasing the temperature from Tc. New types of solitons, unusual current - phase relations.

• Domain wall superconductivity. Spin - valve effects.

• Superconductor-ferromagnet heterostructures are very promising for superconducting spintronics.

Some 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).