Gross R., Sidorenko A., Tagirov L. Nanoscale Devices

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Superconductor-Ferromagnet Heterostructures 219

In presence of ‘perpendicular’ magnetic scatterings, up and down spins

states are mixed and equation (10) has to be used. When the temperature is

close to the critical temperature, this expression may be linearized. In that

case, the critical current becomes

2* *

cosh( ) sinh( )

Re 1

sinh(2 )(1 ) cosh( ) sinh( )

fff

qd qd

II qq

qd q d q d

ββ

∞

−∞



 

∆



=++

 



Ω

 



∑

%%%

where

(0)

eN D TS

= ,

222

Ω

=+∆ and

is defined by expression

(31).

Note that when

⊥

→ , the previous expression gives the critical

current when there is only spin flip in the direction of the exchange field.

The oscillations of the critical current (and the 0-

transitions) completely

disappear if 1

⊥

> . Then, the experimental observation of these

oscillations [14 - 16, 26] may be considered as an indication of the

weakness of spin-orbit and ‘perpendicular’ spin-flip scatterings effects.

Besides, this expression may be written in the same form as expression

(39) which is used to fit experimental data. However, the expressions for

the corresponding parameters are rather lengthy and therefore are not

presented here.

Electronic magnetization variation in S/F systems

The mutual influence of ferromagnetism and superconductivity has until

now been considered through its consequences on the evolution of the

superconducting critical temperature and critical current. Nevertheless, the

proximity effect can also manifest itself by a modification of the electronic

magnetization. Indeed, the presence of the ferromagnetic magnetization

leads to a magnetization onset in the S layer. Similarly, the ferromagnetic

layer may present a variation of its magnetization.

It should be underlined that this topic has already been investigated in the

dirty limit by Krivoruchko et al. [29] and Bergeret et al. [30] and in clean

multilayered S/F structures by Halterman and Valls [31].

220 A. I. Buzdin, M. Fauré, M. Houzet

Ferromagnet at the contact with a superconductor

We consider a S/F system, with a thickness

d of the S layer and an infinite

thickness of the F layer. The

x axis is chosen to be perpendicular to the

layer, with the origin at the vacuum-S layer interface.

The magnetization of the F layer is

MM=+,

(40)

where

is the magnetization due to the Pauli paramagnetism while

stems from the superconductivity contribution.

may be expressed as

(0) ( )

MiN T G G

↑

↑↓↓

=−

∑

(41)

where

G is the normal Green function in the F layer and may be deduced

from

F thanks to the normalization condition

−

= .

For

TT, the anomalous Green’s function is small, and therefore,

()~1 /2

− . Also taking into account that

FFF

↓↑ ↑↓

= , the

magnetization may be presented as

*2 2

(0) ( ( 0) ( 0))

sff

MiN T F F

πωω

=>−>

∑

(42)

Note that (42) gives in fact the part of the magnetization related to the

presence of superconducting correlations. Since

TT→ , the linearized

Usadel equation may be solved to find the anomalous Green’s function. If

, calculations give

the following final expression of the magnetization

2 (0) exp(2/ ) ()sin(2/ ) ()cos(2/ ),

sfff

MNT xA x B x

πξωξωξ





=− ∆ − +





(43)

with

12 122

() ( 2 )

(( ) ( ) )

ωω τ

ωτ τ

−

−−

∑

and

12 122

() 2 ( )

(( ) ( ) )

τωτ

ωτ τ

−−

∑

where

Ssf

−

= is the pair breaking parameter.

In the case of very low S/F interface transparency, the magnetization

becomes

the interface is supposed to be transparent, i.e.

Superconductor-Ferromagnet Heterostructures 221

122

(0) exp( 2 / )cos(2 / )

(/)

sff

MNT x

πξξ

τγγ

−

=− ∆ −

∑

(44)

A qualitative evolution of the magnetization variation is given in Fig. 10.

Fig. 10. Qualitative oscillating evolution of the magnetization variation in the F

layer

Magnetization variation in the S layer in a thin S/F bilayers

Let us now consider a thin S/F bilayer (see Fig. 11) with

<< and

<< .

Fig. 11. Geometry of the studied system.

SVacuum

Vacuum

SVacuum

Vacuum

222 A. I. Buzdin, M. Fauré, M. Houzet

The magnetization in the S layer may be easily determined in this case.

For a transparent interface, it reads

()

8(0)

MNTd

πτ

ωτ

−

=− ∆

∑

(45)

Therefore, the inverse proximity effect leads to the appearance of a negative

magnetization in the superconducting layer. Ref. [31] explains this fact

quite simply. Namely, although the Cooper pairs inside the F layer do not

contribute to a magnetization onset, a Cooper pair with one electron in each

layer does. Indeed, the electron in the F layer has its spin parallel to the

exchange field (up), while the second electron of the pair has a spin down.

Therefore, each of theses Cooper pairs gets a favorite orientation, with the

up spin electron in the F layer and the down spin electron in the S layer.

That is how a magnetization appears in the S layer, whose direction is the

opposite of the one in the F layer.

Conclusion

We have investigated the particularities of the proximity effect in S/F

multilayered systems in the dirty limit. We demonstrated that the spin-flip

scattering may strongly influence the behavior of the critical temperature

and critical current. Indeed, it decreases the decay length and increases the

oscillations period. Moreover, perpendicular magnetic scatterings can even

lead to the complete destruction of the oscillations. It should therefore be

taken into account for theoretical fits of experimental data for systems with

weak magnetic anisotropy.

Besides, another manifestation of the proximity effect in S/F bilayer is

the appearance of a negative magnetization in the S layer, while the

magnetization of the F layer is being oscillating.

Acknowledgments

The authors are grateful to V. Ryazanov for useful discussions and

J. Cayssol for critical reading of the manuscript. This work was supported

in part by ESF PI Shift Program.

Superconductor-Ferromagnet Heterostructures 223

References

1. Ginzburg VL (1956). Zh Eksp Teor Fiz 31:202 [(1957) Sov Phys JETP

4:153]

2. Anderson PW, Suhl H Phys (1959). Rev 116:898

3. Bulaevskii LN, Buzdin AI, Kulic ML, Panjukov SV (1985). Advances in

Physics 34:175

4. Larkin AI, Ovchinnikov YN (1964). Zh Eksp Teor Fiz. 47:1136 [(1965) Sov

Phys JETP 20:762]

5. Fulde P, Ferrell RA (1964). Phys Rev B 135:A550

6. Buzdin AI (2005). Rev Mod Phys 77:935

7. Golubov AA, Kuprianov MYu., Il’ichev E (2004). Rev Mod Phys 76:411

8. Tagirov LR (1998). Physica C 307:145

9. Demler EA, Arnold GB, Beasley MR (1997). Phys Rev B 55: 15174

10. De Gennes PG (1966) Superconductivity of Metals and Alloys. Benjamin,

New York

11. Deutscher G, De Gennes PG (1969). In: Parks RD (ed) Superconductivity:

1005-1034. Marcel Dekker, New York

12. Wong HK, Jin BY, Yang HQ, Ketterson JB, Hilliard JE (1986). J Low

Temp Phys 63:307

13. Jiang JS, Davidovic D, Reich DH, Chien CL (1995). Phys Rev Lett 74:314

14. Ryazanov VV, Oboznov VA, Veretennikov AV, Rusanov AYu, Golubov

AA, Aarts J (2001). Phys Rev Lett 86:2427

15. Sellier H, Baraduc C, Lefloch F, Calemczuk R (2003). Phys Rev B

68:054531

16. Kontos T, Aprili M, Lesuer J, Genet F, Stephanidis B, Boursier R (2002).

Phys Rev Lett 89:137007

17. Eilenberger G (1968). Z Phys 214:195

18. Usadel L (1970). Phys Rev Lett 25:507

19. Fulde P, Maki K (1966). Phys Rev 141:275

20. Abrikosov AA, Gor’kov LP (1960). Zh Eksp Theor Fiz 39:1781 [(1961) Sov

Phys JETP 12:1243]

21. Kuprianov MY, Lukichev VF (1982). Zh Eksp Teor Fiz 94:147 [(1988) Sov

Phys JETP 67:1163]

22. Abrikosov AA, Gorkov LP (1963). Methods of Quantum Field Theory in

Statistical Physics (English version). Prentice-Hall, New Jersy

23. Buzdin AI, Bulaevskii LN, Panyyukov SV (1982). Pis’ma Zh Eksp Teor

Phys 35:147 [(1982) JETP Lett 35:178]

24. Buzdin AI, Kuprianov MY (1991). Pis’ma Zh Eksp Teor Phys 53:308

[(1991) JETP Lett 53:321]

25. Ryazanov VV, Oboznov VA, Prokov’ev AS, Bolginov VV, Feofanov AK

(2004). J Low Temp Phys 136:385

26. Buzdin AI (2003). Pis'ma Zh Eksp Teor Fiz 78:1073 [(2003) JETP Lett

78:583]

224 A. I. Buzdin, M. Fauré, M. Houzet

27. Cretinon L, Kupta AK, Sellier H, Lefloch F, Fauré M, Buzdin AI, Courtois

H (2005). Cond-matt 0502050.

28. Ryazanov VV, Oboznov VA, Bolginov VV, Feofanov AK, Buzdin AI

(2005). To be published

29. Krivoruchko VN, Koshina EA (2002). Phys Rev B 66:014521

30. Halterman K, Valls OT (2004). Phys Rev B 69:014517

31. Bergeret FS, Volkov AF, Efetov KB (2004). Phys Rev B 69:174504

Superconducting/Ferromagnetic Nanostructures:

Spin Fluctuations and Spontaneous

Supercurrents

M. Aprili

1,2

, M. L. Della Rocca

2,3

, T. Kontos

LPQ-ESPCI, 10 rue Vauquelin, 75005 Paris, France

CSNSM-CNRS, Bât.108 Université Paris-Sud, 91405 Orsay, France.

Dipartimento di Fisica,, Università di Salerno, via S. Allende,84081

Baronissi, Italy

Abstract: We present a quantitative analysis of the superconducting proximity

effect at the paramagnetic-ferromagnetic transition. In the

ferromagnetic state we have shown than the normal layer for some

thickness becomes a phase provider and can be used as a “quantum

battery”.

Keywords: proximity effect, superconductivity, hybrid nanostructures.

Introduction

Superconducting correlations change when Cooper pairs leave from a

superconductor and enter in a strong interacting normal metal. This simple

consideration has two main consequences. First, a vanishing

superconducting state with new specific properties can be induced in the

normal metal. Second, the modifications of the BCS ground state can be

used as a probe of the interactions. Thus, on an energy scale comparable or

even larger than the superconducting energy gap, interactions can be

investigated by spectroscopy of the induced superconducting state.

In this paper, we investigate these two points separately. First of all, we

shall

show that the strong exchange enhancement of spin fluctuations at

the

paramagnetic-ferromagnetic transition can be directly measured by

225

R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 225–237.

226 M. Aprili, M. L. Della Rocca, T. Kontos

proximity effect tunneling spectroscopy (PETS) [1]. Second, the

superconducting proximity effect in a ferromagnet results in a oscillating

superconducting wave function due to the finite momentum transferred

from the exchange energy to the Cooper pairs [2]. This vanishing

superconducting state can be used as a “quantum battery”. Specifically, by

connecting the ferromagnet on both sides to the same superconductor, the

phase difference resulting from the oscillations in the ferromagnetic layer,

can generate a spontaneous supercurrent [3].

The proximity effect at the paramagnetic-ferromagnetic

transition

The electron-spin fluctuation coupling constant and the exchange field at

the paramagnetic-ferromagnetic transition are obtained by fitting the

tunneling spectra within the theory of proximity effect in the dirty limit [4].

The DOS is measured by planar tunneling spectroscopy in Pd thin films

with different Ni concentrations either in the paramagnetic or in

ferromagnetic regime. As a comparison, the superconducting density of

states of pure Ag, Pt and Pd is also investigated.

Minigap in N/S bilayers

Spin fluctuations as well as the exchange field are expected to modify the

DOS of an SN structure over the whole energy range relevant to

superconductivity. This is better understood if one considers the basic

mechanism responsible for the proximity effect: the Andreev reflection [5].

When an electron reaches an SN interfaces, it cannot enter into the

superconducting region if its energy E is smaller than the energy gap

∆ of

the superconductor. It is then backscattered as a hole of energy -E. The

electron and the hole are coupled coherently. Their phase relationship

2/Et

∆= h , t being the relevant time scale in the normal region of the

structure, controls the quantum interferences in the normal metal. When

∆ is an integer multiple of 2

, a constructing interference occurs,

thereby defining the energy scale of the SN structure. If the N metal

consists of a thin layer whose thickness d

is smaller than the

superconducting coherence length

, the relevant time scale is the dwell

time

in the structure. In this case, the criterium 2

∆

= may be

fulfilled only for a finite energy E

, which can be of the same order of

magnitude than

∆ . Therefore, no quasiparticle can be found with energy

below E

. This energy scale is commonly referred to as a minigap and

Superconducting/Ferromagnetic Nanostructures 227

depends also on the amplitude of the Andreev process which can be altered

by imperfect S/N boundaries [6].

In a strong paramagnet, spin fluctuations are known to renormalize the

quasiparticle energy [7], just like phonons do in strong coupling

superconductors, when their energy is much larger than

∆ . This

renormalization factor relates the energy of bare quasiparticles to dressed

quasiparticles by

(1 )

=+,

being the electron-spin fluctuation

coupling constant.

diverges at the ferromagnetic-paramagnetic

transition. As a consequence, the condition

∆

= becomes

2(1 )/ 2

τπ

+=h and the minigap is expected to provide a direct and

very sensitive measurement of

near the paramagnetic-ferromagnetic

transition.

In a ferromagnet the exchange field acts like a spin dependent potential,

as Andreev reflexions revert the spin. The condition

∆

= becomes

2( ) / 2

+=h . If

E >∆, which is usually the case, the condition

for constructive interferences becomes independent of E and E

vanishes.

The energy dependence of quantum interferences cannot be related

anymore to constructives interferences between electrons and holes. The

only energy dependence expected is that of the Andreev process itself and

the effect of the exchange field is to scale the amplitude of these

correlations by a cosine exponentially damped whose phase is just

h [8]. We will see that this provides a direct way to determine the

exchange field in the ferromagnetic layer.

The Usadel equations

The Usadel equations provide the formalism to implement the qualitative

arguments developed above. The effect of a high Stoner enhancement as

well as that of an exchange field are particularly simple to include. In the

paramagnetic regime, the S/N hybrid structure is described in the dirty limit

by the following equations:

(1 ) sin 2 sin cos cos 0

sf AG N

∂

λθ θθ θ

∂

++ +Γ +∆ =

(1)

2sin

λθ

∆=

∑

228 M. Aprili, M. L. Della Rocca, T. Kontos

∆ and

are respectively the diffusion constant and the effective

electron-electron coupling constant in the normal slab,

is the electron-

spin fluctuations renormalization constant and

is an Abrikosov-

Gor'kov (AG) pair-breaking term . The pairing angle

contains all the

information about equilibrium properties and the spatially resolved DOS,

N(E,x), is found by N(E,x)=Re (cos

In the ferromagnetic regime, the pairing angle depends on the spin

=± of the quasiparticles, because of the exchange field

E , eq. (1)

reads:

()

(1 ) sin 2 sin cos

sf ex AG

iEE

σσ

∂

σθ θθ

∂

++ + +Γ

cos 0

+∆ = .

(2)

The spatially resolved DOS N(E,x) is found by

{

}

(,) Recos cos /2NEx

θθ

+−

=+. As predicted above, one can see from

eqs. (1) and (2) that the exchange field and the electron-spin fluctuation

coupling constant both affect directly the energy E of quasiparticles. While

spin fluctuations renormalize it, the exchange field acts like a spin

dependent potential. Note that, as the Coulomb interaction is expected to be

non-negligible, one has to consider a priori a non zero negative gap,

∆ .

Tunnel junction fabrication

We measure the density of states by planar tunneling spectroscopy in

Counter-Electrode/Insulator/Normal-Metal/Superconductor (CE/I/N/S)

junctions. The structures are fabricated in a UHV system of a base pressure

of 10

-9

Torr. A thick Al layer of 1500 Å is deposited at room temperature

on a Si(100) substrate with a SiO buffer of 500 Å. We oxidize the Al layer

with an O

glow discharge. The typical pressure of the plasma is 8×10

-2

mbar. Depending on the time of oxidation (several minutes), we get

junction resistances ranging from 10

Ω

to 10 k

Ω

, for a typical area of

100

× 100 m

. The size of the junctions is defined by evaporating

two

500 Å SiO layers. A normal metal layer of 50 Å is then evaporated

and it is after covered by a Nb layer of 500 Å whose T

is typically 8.7 K.

The 4-probes cross-junction geometry is shown in the inset of fig. 2. The

Gross R., Sidorenko A., Tagirov L. Nanoscale Devices - Fundamentals and Applications

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