Ferromagnet–superconductor hybridspeople.physics.tamu.edu/ilx/adv.pdf · Advances in Physics, Vol. 54, No. 1, January/February 2005, 67–136 ... We review a new avenue in solid

Advances in Physics,Vol. 54, No. 1, January/February 2005, 67–136

Ferromagnet–superconductor hybrids

I. F. LYUKSYUTOV* and V. L. POKROVSKY

Department of Physics, Texas A&M University

(Received 1 June 2004; Accepted in revised form 20 December 2004)

The new class of phenomena described in this review is based on theinteraction between spatially separated, but closely located ferromagnets andsuperconductors, the so-called ferromagnet–superconductor hybrids (FSH).Typical FSH are: coupled uniform and textured ferromagnetic andsuperconducting films, magnetic dots over a superconducting film, magneticnanowires in a superconducting matrix, etc. The interaction is provided by themagnetic field generated by magnetic textures and supercurrents. The magneticflux from magnetic structures or topological defects can pin vortices or createthem, changing the transport properties and transition temperature of thesuperconductor. On the other hand, the magnetic field from supercurrents(vortices) strongly interacts with the magnetic subsystem, leading to formationof coupled magnetic–superconducting topological defects.The proximity of ferromagnetic layer dramatically changes the properties of

the superconducting film. The exchange field in ferromagnets not only suppressesthe Cooper-pair wavefunction, but also leads to its oscillations, which in turnleads to oscillations of observable values: the transition temperature andJosephson current. In particular, in the ground state of the Josephson junctionthe relative phase of two superconductors separated by a layer of ferromagneticmetal is equal to p instead of the usual zero (the so-called p-junction). Such ajunction carries a spontaneous supercurrent and possesses other unusualproperties. Theory predicts that rotation of magnetization transforms s-pairinginto p-pairing. The latter is not suppressed by the exchange field and serves as acarrier of long-range interaction between superconductors.

Contents page

1. Introduction 68

2. Basic equations 702.1. Three-dimensional systems 712.2. Two-dimensional systems 722.3. The Eilenberger and Usadel equations 75

3. Hybrids without proximity effect 793.1. Magnetic dots 79

3.1.1. Magnetic dot: perpendicular magnetization 803.1.2. Magnetic dot: parallel magnetization 83

3.2. Array of magnetic dots and superconducting film 843.2.1. Vortex pinning by magnetic dots 843.2.2. Vortex lattice symmetry versus magnetic

dot array symmetry 86

*Corresponding author. Email: [email protected]

Advances in Physics

ISSN 0001–8732 print/ISSN 1460–6976 online # 2005 Taylor & Francis Group Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/00018730500057536

3.2.3. Magnetic field induced superconductivity 903.2.4. Magnetization controlled superconductivity 91

3.3. Ferromagnet–superconductor bilayer 943.3.1. Topological instability in the FSB 943.3.2. Superconducting transition temperature of the FSB 973.3.3. Transport properties of the FSB 983.3.4. Experimental studies of the FSB 1003.3.5. Thick films 1023.3.6. Domain wall superconductivity 103

4. Proximity effects in layered ferromagnet–superconductor systems 1044.1. Oscillations of the order parameter 1044.2. Resistance of the SF contact in wires and the spin-filtering effect 1064.3. Non-monotonic behaviour of the transition temperature 1094.4. Josephson effect in S/F/S-junctions 113

4.4.1. Simplified approach and experiment 1144.4.2. Josephson effect in a clean system 1164.4.3. Half-integer Shapiro steps at the 0–p transition 1204.4.4. Spontaneous current and flux in a closed loop 121

4.5. F/S/F-junctions 1254.6. Triplet pairing 126

5. Conclusions 131

Acknowledgements 132

References 133

1. Introduction

We review a new avenue in solid state physics: studies of physical phenomenawhich appear when two mutually exclusive states of matter, superconductivity andferromagnetism, are combined in a unified ferromagnet–superconductor hybrid(FSH) system. In the hybrid systems fabricated from materials with different andeven mutually exclusive properties, a strong mutual interaction between subsystemscan dramatically change properties of the constituent materials. This approach offersvast opportunities for science and technology. The interplay of superconductivityand ferromagnetism has been thoroughly studied experimentally and theoretically[1, 2] for homogeneous systems. In such systems, both order parameters are homog-eneous in space and suppress each other. As a result, one or both of the orderings areweak. A natural way to avoid the mutual suppression of the order parameter of thesuperconducting (S) and ferromagnetic (F) subsystems is to separate them by a thinbut impenetrable insulator film. In such systems the S- and F-subsystems interact viaa magnetic field induced by the non-uniform magnetization of the F-textures pene-trating into the superconductor. If this field is strong enough, it can generate vorticesin the superconductor. The textures can be either artificial (dots, wires) or topolo-gical like domain walls (DW). The inverse effect is also important: the S-currentsgenerate a magnetic field interacting with the magnetization in the F-subsystem.

Experimental work on FSH has been focused dominantly on pinning propertiesof magnetic dot arrays covered by a thin superconducting film [3–7]. The effectof commensurability on the transport properties was reported in [3–6]. This effectis not specific for magnets interacting with superconductors and was first observedin superconducting films in the 1970s. In these experiments the periodicity of thevortex lattice fixed by an external magnetic field competed with the periodicity of

68 I. F. Lyuksyutov and V. L. Pokrovsky

an artificial array created by experimenters. Martinoli et al. [8–10] used groovesand Hebard et al. [11, 12] used arrays of holes. They observed well-pronouncedminima of resistance or peaks of critical current at values of magnetic field corre-sponding to commensurate lattices. This approach was further developed byexperimentalists in the 1990s [13–19]. Theoretical analysis was also performed inthe last century [20–22]. Morgan and Ketterson [7] were the first to observe thechange of resistivity induced by vortices pinned by magnetic dots during flipping ofthe external magnetic field orientation. This was the first direct indication of time-reversal symmetry violation in FSH. New insight into the physics of FSH has beenprovided by the magnetic force microscope (MFM) and scanning Hall probemicroscope (SHPM). By using such imaging technique the group at theUniversity of Leuven has elucidated several pinning mechanisms in FSH [23–25].

Different mesoscopic magneto-superconducting systems have been proposedand studied theoretically: arrays of magnetic dots on the top of an S-film [26–29],ferromagnet–superconductor bilayers (FSB) [28, 30–34], embedded magnetic nano-wires combined with bulk superconductor [35, 36] or superconductor films [37, 38], alayer of magnetic dipoles between two bulk superconductors [39], an array ofmagnetic dipoles mimicking the F-dots on S-films [40], a ‘giant’ magnetic dotwhich generates several vortices in bulk superconductor [41], a single magnetic doton a thin superconducting film [42–46], a thick magnetic film combined with a thick[47–50] or thin superconducting film [51, 52].

The characteristic length of the magnetic field and current variation in all thesesystems significantly exceeds the coherence length, . This means that they can beconsidered in the London approximation with good precision. In the next section wederive basic equations describing the FSH. Starting from the London–Maxwellequations, we derive a variational principle (energy) containing only the valuesinside either S- or F-components. These equations allow us to study single magneticdots coupled with superconducting film (Section 3.1) as well as arrays of such dots(Section 3.2). The simplest possible FSH system, a sandwich formed by F- andS-layers, separated by ultrathin insulating film (FSB), displays unusual behaviour:spontaneous formation of coupled system of vortices and magnetic domains.These phenomena are reviewed in Section 3.3. We also discuss the influence of athick magnetic film on the bulk superconductor.

An alternative approach to heterogeneous S/F-systems is just to employ theproximity effects instead of avoiding them. The exchange field existing in the ferro-magnet splits the Fermi spheres for up and down spins. Thus, the Cooper pairacquires a non-zero total momentum and its wavefunction oscillates in space. Thiseffect, first predicted by Larkin and Ovchinnikov [53] and by Ferrel and Fulde [54],will be cited further as the LOFF effect. These oscillations can be treated as boundstates of quasi-particles in the F-layer due to Andreev reflection [55] from theS/F-interface. One of their manifestations is the change of sign of the Cooper-pairtunnelling amplitude in space. Under certain conditions the Josephson currentthrough a superconductor–ferromagnet–superconductor (S/F/S) junction has signopposite to sin ’, where ’ is the phase difference between the right and left super-conducting layers. This type of junction was first proposed theoretically a long timeago by Bulaevsky et al. [56, 57] and was called a p-junction in contrast to thestandard or 0-junction. It was first reliably observed in the experiment byRyazanov and coworkers in 2001 [58, 59] and a little later by Kontos et al. [60].The experimental findings of these groups have generated an extended literature. A

Ferromagnet–superconductor hybrids 69

large review on this topic was published at the beginning of 2002 [61]. A morespecialized survey was published at the same time by Garifullin [62]. We are notgoing to repeat what was already done in these reviews and will focus predominantlyon work which has appeared after their publication. Only basic notions and ideasnecessary for understanding the present discussion will be extracted from previouswork. Quite recently a review by Golubov et al. [63] on the current–phase relation inJosephson junctions was published. It has some overlap with our review.

Most of the proximity phenomena predicted theoretically in SF-systems andfound experimentally are based on the oscillatory behaviour of the Cooper-pairwavefunction penetrating in the ferromagnet. The oscillation of the wavefunctionleads to oscillations of several important physical values. They are: oscillations of thetransition temperature first predicted in [64, 65], and the critical current versusthe thickness of the ferromagnetic layer, which are seen as oscillatory transitionsfrom 0- to p-junctions [57]. Other proximity effects besides the usual suppression ofthe order parameters include the preferential antiparallel orientation of the F-layersin a F/S/F-trilayer, the so-called spin-valve effect [66–69].

More recently a new idea was proposed by Kadigrobov et al. [70] and byBergeret et al. [71]: they have predicted that when the direction of magnetizationvaries in space, singlet Cooper pairs are transformed into triplet pairs. The tripletpairing is not suppressed by the exchange field and can propagate in the ferromagnetover large distances thus providing the long-range proximity between superconduc-tors in S/F/S-junctions.

The proximity effects may have technological applications as elements ofhigh-speed magnetic electronics based on the spin-valve action [68] and also aselements of quantum computers [72]. Purely magnetic interaction between ferro-magnetic and superconducting subsystems can also be used to design magneticfield controlled superconducting devices. A magnetic field controlled Josephsoninterferometer employing a thin magnetic F/S-bilayer has been demonstrated byEom and Johnson [73].

In the next section we derive basic equations. The third section is focused onphenomena in FSH due only to magnetic interaction between F- and S-subsystems.Recent results on proximity phenomena in bi- and trilayer FSH are presented in thelast section.

2. Basic equations

In the proposed and experimentally realized FSH a magnetic texture interacts withthe supercurrent. First we assume that F- and S-subsystems are separated by a thininsulating layer which prevents the proximity effect, focusing on the magnetic inter-action only. Inhomogeneous magnetization generates a magnetic field outside ferro-magnets. This magnetic field induces screening currents in superconductors which, inturn, change the magnetic field. The problem must be solved self-consistently. Thecalculation of the vortex and magnetization arrangement for interacting, spatiallyseparated superconductors and ferromagnets is based on the static London–Maxwellequations and corresponding energy. This description includes possible super-conducting vortices. London’s approximation works satisfactorily since the sizes ofall structures in the problem exceed significantly the coherence length, . We recallthat in the London approximation the modulus of the order parameter is constant


and the phase varies in space. Starting from the London–Maxwell equation in theentire space, we eliminate the magnetic field outside F- and S-subsystems and obtainequations for the currents, magnetization and fields inside them. This is done inSection 2.1. In Section 2.2 we apply this method to very thin coupled ferromagneticand superconducting films. When proximity effects dominate, the London approx-imation is invalid. The basic equations for this case will be described in Section 2.3.

2.1. Three-dimensional systems

The total energy of a stationary F/S-system reads:

H ¼

ðB2

8pþmsnsv

2s

2 B M

" #dV ð1Þ

where B is the magnetic induction, M is the magnetization, ns is the density ofS-electrons, ms is their effective mass and vs is their velocity. We assume that theS density ns and the magnetizationM are separated in space. We assume also that themagnetic field B and its vector potential A asymptotically tend to zero at infinity.Employing the static Maxwell equations r B ¼ ð4p=cÞ j and B ¼ r A, themagnetic field energy can be transformed as follows:ð

B2

8pdV ¼

ðj A

2cdV : ð2Þ

Though the vector potential enters explicitly in the last equation, it is gauge invariantdue to current conservation, div j ¼ 0. When integrating by parts, we neglected thesurface term. This is correct if the field, vector potential and the current decreasesufficiently fast at infinity. This condition is satisfied for the simple examplesconsidered in this section. The current j can be represented as a sum: j ¼ js þ jm ofthe SC and magnetic currents, respectively:

js ¼nshhe

2ms

r’2p0

A

ð3Þ

jm ¼ cr M: ð4Þ

We consider contributions from the magnetic and S-currents to the integral (2)separately. We start with the integral:

1

2c

ðjmA dV ¼

1

2

ðr Mð Þ A dV : ð5Þ

Integrating by parts and neglecting the surface term again, we arrive at the followingresult:

1

2c

ðjmA dV ¼

1

2

ðM B dV : ð6Þ

We have omitted the integral over a remote surfaceHnMð Þ A dS. Such an

omission is valid if the magnetization is confined to a limited volume. But for infinitemagnetic systems it may be wrong even in the simplest problems. We will discuss thissituation in the next section.

Next we consider the contribution of the superconducting current js to theintegral (2). In the gauge-invariant equation (3), ’ is the phase of the S-carrier


(Cooper pair) wavefunction and 0 ¼ hc=2e is the flux quantum. Note that the phasegradient r’ can be included in A as a gauge transformation with the exception ofvortex lines, where ’ is singular. We employ equation (3) to express the vectorpotential A in terms of the supercurrent and the phase gradient:

A ¼0

2pr’

msc

nse2js: ð7Þ

Plugging equation (7) into equation (2), we find:

1

2c

ðjsA dV ¼

hh

4e

ðr’ js dV

ms

2nse2

ðj2s dV : ð8Þ

Since js ¼ ensvs, the last term in this equation is equal to the kinetic energy takenwith the sign minus. It exactly compensates the kinetic energy in the initialexpression for the energy (1). Collecting all remaining terms, we obtain the followingexpression for the total energy:

H ¼

ðnshh

2

8ms

ðr’Þ2 nshhe

4mscr’ A

B M

2

" #dV : ð9Þ

We remind the reader again about a possible surface term for infinite systems. Notethat integration in the expression for energy (9) proceeds over the volumes occupiedeither by superconductors or by magnets. Equation (9) allows us to separate theenergy of vortices from the energy of magnetization induced currents and fields andtheir interaction energy. Indeed, as we noted earlier, the phase gradient can beascribed to the contribution of vortex lines only. It is representable as a sum ofindependent integrals over different vortex lines. The vector potential and themagnetic field can be represented as a sum of magnetization induced and vortexinduced parts: A ¼ Am þ Av, B ¼ Bm þ Bv, where Ak, Bk (the index k is either m or v)are determined as solutions of the London–Maxwell equations:

r r Akð Þ ¼4pcjk: ð10Þ

The effect of the screening of the magnetization induced magnetic field by the super-conductor is included in the vector fields Am and Bm. Applying such a separation, wepresent the total energy (9) as a sum of terms containing only vortex contributions,only magnetic contributions and the interaction terms. The purely magnetic partcan be represented as a non-local quadratic form of the magnetization. The purelysuperconducting part is representable as a non-local double integral over the vortexlines. Finally, the interaction term is representable as a double integral over thevortex lines and the volume occupied by the magnetization and is bilinear in themagnetization and vorticity. To avoid cumbersome formulas, we will not write theseexpressions explicitly.

2.2. Two-dimensional systems

Below we perform a more explicit analysis for the case of two parallel films, one F,another S, both very thin and each very close to one other. Neglecting their thick-ness, we assume that both films are located approximately at z¼ 0. In some cases weneed a more accurate treatment. Then we introduce a small distance d between thefilms, which in the end will be set to zero. Though the thickness of each film is


assumed to be small, the two-dimensional densities of S-carriers nð2Þs ¼ nsds andmagnetization m ¼ Mdm remain finite. Here we have introduced the thickness ofthe S-film ds and the F-film dm. The three-dimensional supercarrier density nsðRÞcan be represented as nsðRÞ ¼ ðzÞnð2Þs ðrÞ and the three-dimensional magnetizationMðRÞ can be represented as MðRÞ ¼ ðz d ÞmðrÞ, where r is the two-dimensionalradius vector and the z-direction is perpendicular to the films. In what follows nð2Þs isassumed to be a constant and the index (2) is omitted. The energy (9) can berewritten for this special case:

H ¼

ðnshh

2

8ms

ðr’Þ2 nshhe

4mscr’ a

b m

2

" #d2r ð11Þ

where a ¼ Aðr, z ¼ 0Þ and b ¼ Bðr, z ¼ 0Þ. The vector potential satisfies the Maxwell–London equation:

r ðr AÞ ¼ 1

AðzÞ þ

2phhnsemsc

r’ðzÞ

þ 4pr ðmðzÞÞ: ð12Þ

Here ¼ 2L=ds is the effective screening length for the S-film, L is the Londonpenetration depth and ds is the S-film thickness [74].

According to our general arguments, the term proportional to r’ inequation (13) describes vortices. A plane vortex characterized by its vorticity, q,an integer, and by the position of its centre on the plane, r0, contributes a singularterm to r’:

r’0ðr, r0Þ ¼ qzz ðr r0Þ

jr r0j2

ð13Þ

and generates a standard vortex vector potential:

Av0ðr r0, zÞ ¼q0

2pzz ðr r0Þ

jr r0j

ð10

J1ðkjr r0jÞekjzj

1þ 2kdk: ð14Þ

Different vortices contribute independently to the vector potential and magneticfield. A peculiarity of this problem is that the usually applied gauge divA ¼ 0becomes singular in the limit ds, dm ! 0. Therefore, it is reasonable to apply anothergauge, Az ¼ 0. The calculations are much simpler in the Fourier representation.Following the general procedure, we present the Fourier transform of the vectorpotential Ak as a sum Ak ¼ Amk þ Avk. The equation for the magnetic part of thevector potential reads:

kðqAmkÞ k2Amk ¼amq

4pikmqe

ikzd ð15Þ

where q is the projection of the wavevector k onto the plane of the films: k ¼ kzzzþ q.An arbitrary vector field Vk in the wavevector space can be represented by its localcoordinates:

Vk ¼ Vzkzzþ Vk

k qqþ V?k ðzz qqÞ: ð16Þ


In terms of these coordinates the solution of equation (15) reads:

Ak

mk ¼ 4pim?

q

kzeikzd ð17Þ

A?mk ¼

1

k2a?q þ

4pi kzmkq qmqz

k2

eikzd : ð18Þ

Integration of the latter equation over kz allows us to find the perpendicularcomponent of aðmÞ

q :

a?mq ¼ 4pqðmk

q þ imqzÞ

1þ 2qeqd , ð19Þ

whereas it follows from equation (15) that akmq ¼ 0. Note that the parallel componentof the vector potential Ak

mk does not know anything about the S-film. It correspondsto the magnetic field being equal to zero outside the plane of the F-film. Therefore, itis inessential for our problem.

The vortex part of the vector potential also does not contain a z-component sincethe supercurrents flow in the plane. The vortex solution in a thin film was first foundby Pearl [75]. An explicit expression for the vortex-induced potential is:

Avk ¼2i0ðzz qqÞFðqÞ

k2ð1þ 2qÞ, ð20Þ

where FðqÞ ¼P

j eiqrj is the vortex form-factor; the index j labels the vortices and rj

are coordinates of the vortex centres. The Fourier transform for the vortex-inducedvector potential at the surface of the SC film avq reads:

avq ¼i0ðzz qqÞFðqÞ

qð1þ 2qÞ: ð21Þ

The z-component of magnetic field induced by the Pearl vortex in real space is:

Bvz ¼0

2p

ð10

J0ðqrÞeqjzj

1þ 2qq dq: ð22Þ

Its asymptotic at z¼ 0 and r is Bvz 0=ðpr3Þ; at r it is Bvz 0=ðprÞ.

Each Pearl vortex carries the flux quantum 0 ¼ phhc=e.The energy (11) can be expressed in terms of Fourier transforms:

H ¼ Hv þHm þHvm, ð23Þ

where the vortex energy Hv is the same as it would be in the absence of the F film:

Hv ¼nshh

2

8ms

ðr’q r’q

2p0

avq

d2q

ð2pÞ2: ð24Þ

The magnetic self-energy Hm is:

Hm ¼ 1

2

ðmqbmq

d2q

ð2pÞ2ð25Þ


It contains the screened magnetic field and therefore differs from its value in theabsence of the SC film. Finally the interaction energy reads:

Hmv ¼ nshhe

4msc

ððr’Þqamq

d2q

ð2pÞ21

2

ðmqbvq

d2q

ð2pÞ2ð26Þ

Note that the information on the vortex arrangement is contained in the form-factorF(q) only.

To illustrate how important the surface term can be, let us consider a homo-geneous perpendicularly magnetized magnetic film and one vortex in a super-conducting film. The authors [30] have shown that the energy of this system is"v ¼ "0v m0, where "

0v is the energy of the vortex in the absence of the magnetic

film, m is the magnetization per unit area and 0 ¼ hc=2e is the magnetic fluxquantum. Let us consider how this result appears from the microscopic calculations.The vortex energy (24) is just equal to "0v . The purely magnetic term (25) does notchange in the presence of the vortex and is inessential. The first term in the inter-action energy (26) is equal to zero since the infinite magnetic film does not generatea magnetic field outside. The second term of this energy is equal to m0=2.The second half of the interaction energy comes from the surface term. Indeed, itis equal to

1

2limr!1

ð2p0

mðrr zzÞ Ar d’ ¼ 1

2

IA dr ¼ m0=2

2.3. The Eilenberger and Usadel equations

The essence of proximity phenomena is the change of the order parameter (Cooper-pair wavefunction). Therefore, the London approximation is not valid in thiscase and equations for the order parameter must be solved. They are either theBogolyubov–de Gennes equations [76, 77] for the coefficients u and v or more con-veniently the Gor’kov equations [78] for the Green functions. Unfortunately thesolution of these equations is not an easy problem in the spatially inhomogeneouscase when combined with the scattering by impurities and/or irregular boundaries.This is a typical situation for experiments with F/S proximity effects, since the layersare thin, the diffusion delivers atoms of one layer into another and the control of thestructure and morphology is not so strict as for three-dimensional single crystals.Sometimes experimenters deliberately use amorphous alloys as magnetic layers [79].Fortunately, if the scale of variation of the order parameter is much larger thanatomic, the semiclassical approximation can be applied. Equations for the super-conducting order parameter in the semiclassical approximation were derived a longtime ago by Eilenberger [80] and by Larkin and Ovchinnikov [81]. They were furthersimplified in the case of strong elastic scattering (the diffusion approximation) byUsadel [82]. For the reader’s convenience and for the unification of notation wedemonstrate them here, referring the reader to the original works or to the textbooks[83, 84] for the derivation.

The Eilenberger equations are written for the electronic Green functionsintegrated in momentum space over the momentum component perpendicular tothe Fermi surface. Thus, they depend on a point of the Fermi surface characterizedby two momentum components, on the coordinates in real space and time.It is more convenient in thermodynamics to use their Fourier components over


imaginary time, the so-called Matsubara representation [85]. The frequencies inthis representation accept discrete real values !n ¼ ð2nþ 1ÞpT , where T isthe temperature. Wave functions of singlet Cooper pairs are represented by twoEilenberger anomalous Green functions Fð!, k, rÞ and Fyð!, k, rÞ (integrated alongthe normal to the Fermi-surface Gor’kov anomalous functions), where ! stands for!n, k is the wavevector on the Fermi sphere and r is the position vector of a point inreal space (the coordinate of the Cooper-pair centre-of-mass). The function F isgenerally complex, in contrast to the integrated normal Green function Gð!, k, rÞ,which is real. Eilenberger has proved that the functions G and F are not independent:they obey the normalization condition:

½Gð!, k, rÞ2 þ jFð!, k, rÞj2 ¼ 1: ð27Þ

Furthermore, the Eilenberger Green functions obey the following symmetryrelations:

Fð!,k, rÞ ¼ F ð!, k, rÞ ¼ F

ð!, k, rÞ ð28Þ

Gð!,k, rÞ ¼ G ð!, k, rÞ ¼ Gð!, k, rÞ: ð29Þ

The Eilenberger equation reads:

2!þ vo

or i

2e

cAðrÞ

Fð!, k, rÞ

¼ 2Gð!, k, rÞ þ

ðd 2q ðqÞWðk, qÞ½GðkÞFðqÞ FðkÞGðqÞ; ð30Þ

where ðrÞ is the space (and time) dependent order parameter (local energy gap), v isthe velocity on the Fermi surface, Wðk, qÞ is the probability of a transition per unittime from the state with momentum q to the state with momentum k, and ðqÞ is theangular dependence of the density of states normalized by

Ðd2q ðqÞ ¼ Nð0Þ. Here

N(0) is the total density of states (DOS) in the normal state at the Fermi level. TheEilenberger equation has the structure of the Boltzmann kinetic equation, but italso incorporates quantum coherence effects. It must be complemented by the self-consistency equation expressing the local value of ðrÞ in terms of the anomalousGreen function F:

ðrÞ lnT

Tc

þ 2pT

X1n¼0

ðrÞ

!n

ðd2k ðkÞFð!n, k, rÞ

¼ 0: ð31Þ

In the case of isotropic scattering frequently considered by theorists the collisionintegral in equation (30) is remarkably simplified:ð

d2q ðqÞWðk, qÞ½GðkÞFðqÞ FðkÞGðqÞ ¼1

GðkÞhF i FðkÞhGi½ , ð32Þ

where the relaxation time is equal to the inverse value of the angular independenttransition probability W, and h. . .i means the angular average over the Fermi sphere.

The Eilenberger equation is simpler than the complete Gor’kov equations since itcontains only one function depending on one fewer arguments. It could be expectedthat in the limit of very short relaxation time Tc0 1 (Tc0 is the transitiontemperature in the clean superconductor) the Eilenberger kinetic-like equation willbecome similar to the diffusion equation. Such a diffusion-like equation was indeedderived by Usadel [82]. In the case of strong elastic scattering and an isotropic Fermi


surface (sphere), the Green function does not depend on the direction on the Fermisphere and depends only on the frequency and the spatial coordinate r. The Usadelequation reads (we omit both arguments):

2!F Doo GooF F ooG

¼ 2G: ð33Þ

In this equation D ¼ v2F=3 is the diffusion coefficient for electrons in the normalstate and oo stands for the gauge-invariant gradient: oo ¼ r 2ieA=hhc. The Usadelequations must be complemented by the same self-consistency equation (31). It isalso useful to keep in mind the expression for the current density in terms of thefunction F:

j ¼ ie2pTNð0ÞDX!n>0

ðF ooF F ooF Þ: ð34Þ

One can consider the set of Green functions G, F, Fy as elements of the 2 2matrix Green function gg where the matrix indices can be identified with the particleand hole or Nambu channels. This formal trick becomes rather essential when thesinglet and triplet pairing coexist and it is necessary to take into account theNambu indices and spin indices simultaneously. Eilenberger, in his original article[80], indicated a way to implement the spin degrees of freedom in his scheme.Below we demonstrate a convenient modification of this representation proposedby Bergeret et al. [86]. Let us introduce a matrix ggðr, t; r0, t0Þ with matrix elementsgn, n

0

s, s0 , where n, n0 are the Nambu indices and s, s0 are the spin indices, defined asfollows:

gn, n0

s, s0 ðr, t; r0, t0Þ ¼

1

hh

Xn00

ð3Þn, n00

ðd h n00sðr, tÞ

yn0s0 ðr

0, t0Þi: ð35Þ

The matrix 3 in the definition (35) is the Pauli matrix in the Nambuspace. To clarify the Nambu indices we write explicitly what they mean in terms ofthe electronic -operators: 1s s; 2s

yss and ss means s. The most general

matrix gg can be expanded in the Nambu space into a linear combination of fourindependent matrices k, k ¼ 0, 1, 2, 3, where 0 is the unit matrix and the threeothers are the standard Pauli matrices. Following [86], we introduce the followingnotations for the components of this expansion, which are matrices in the spinspace:1

gg ¼ gg00 þ gg33 þ ff ; ff ¼ ff1i1 þ ff2i2: ð36Þ

The matrix ff describes Cooper pairing since it contains only antidiagonal matrices inthe Nambu space. In turn, the spin matrices ff1 and ff2 can be expanded in the basis ofspin Pauli matrices j, j ¼ 0, 1, 2, 3. Without loss of generality we can accept thefollowing agreement about the scalar components of the spin-space expansion:

ff1 ¼ f11 þ f22; ff2 ¼ f00 þ f33: ð37Þ

1Each time the Nambu and spin matrices are written together we mean the direct product.


It is easy to check that the amplitudes fi, i ¼ 0, . . . , 3 are associated with the follow-ing combinations of wavefunction operators:

f0 ! h " # þ # "i

f1 ! h " " y#

y#i

f2 ! h " " þ y#

y#i

f3 ! h " # # "i:

Thus, the amplitude f3 corresponds to the singlet pairing, whereas the three othersare responsible for the triplet pairing. In particular, in the absence of triplet pairingonly the component f3 survives and the matrix ff is equal to

0 FFy 0

:

Let us consider what modification must be introduced into the Eilenberger andUsadel equations to take in account the exchange interaction of Cooper pairs withthe magnetization in the ferromagnet. Neglecting the reciprocal effect of the Cooperpairs on the d- or f-shell electrons responsible for the magnetization, we introducethe effective exchange field h(r) acting inside the ferromagnet. This produces apseudo-Zeeman splitting of the spin energy.2 In the case of the singlet pairing theMatsubara frequency ! must be replaced by !þ ihðrÞ. When the direction ofmagnetization varies in space generating triplet pairing, the Usadel equation mustbe formulated in terms of the matrix gg [86]:

D

2oð ggo ggÞ j!j½33, gg þ sign!½ hh, gg ¼ i½ , gg, ð38Þ

where the operators of the magnetic field hh and the energy gap are defined asfollows:

hh ¼ 3 h ð39Þ

¼ i22: ð40Þ

To find a specific solution of the Eilenberger and Usadel equations proper boundaryconditions should be formulated. For the Eilenberger equations the boundaryconditions at the interface of two metals were derived by Zaitsev [87]. They aremost naturally formulated in terms of the antisymmetric ( gga) and symmetric ( ggs)parts of the matrix gg with respect to reflection of momentum pz ! pz assumingthat z is normal to the interface. One of them states that the antisymmetric part iscontinuous at the interface (z¼ 0):

ggaðz ¼ 0Þ ¼ ggaðz ¼ þ0Þ: ð41Þ

The second equation connects the discontinuity of the symmetric part at the interfaceggs ¼ ggsðz ¼ þ0Þ ggsðz ¼ 0Þ with the reflection coefficient R and transmission

2In reality the exchange enegry has a quite different origin than the Zeeman interaction, but ata fixed magnetization there is a formal similarity in the Hamiltonians.


coefficient D of the interface and antisymmetric part gga at the boundary:

D ggsð ggsþ gga ggsÞ ¼ R gga½1 ð ggaÞ2, ð42Þ

where ggs ¼ ggsðz ¼ þ0Þ ggcðz ¼ 0Þ. If the boundary is transparent (R ¼ 0,D ¼ 1),the symmetric part of the Green tensor gg is also continuous.

The boundary conditions for the Usadel equations, i.e. under the assumptionthat the mean free path of the electron, l, is much shorter than the coherence length,, were derived by Kupriyanov and Lukichev [88]. The first of them ensures thecontinuity of the current flowing through the interface:

< gg<d gg<dz

¼ > gg>d gg>dz

, ð43Þ

where the subscripts < and > relate to the left and right sides of the interface; and<,> denotes the conductivity of the proper metal. The second boundary conditionconnects the current with the discontinuity of the order parameter through theboundary and its transmission and reflection coefficients D() and R():

l> gg>d gg>dz

¼3

4

cos DðÞ

RðÞ

½ gg<, gg>, ð44Þ

where is the incidence angle of the electron at the interface and D(), R() are thecorresponding transmission and reflection coefficients. This boundary condition canbe rewritten in terms of measurable characteristics:

> gg>d gg>dz

¼1

Rb

½ gg<, gg>, ð45Þ

where Rb is the resistance of the interface. In the case of high transparency (R 1)the boundary conditions (43), (44) can be simplified as follows [89]:

ff< ¼ ff>;d ff<dz

¼ d ff>dz

, ð46Þ

where is the ratio of normal state resistivities.

3. Hybrids without proximity effect

3.1. Magnetic dots

In this subsection we consider the ground state of a S-film with a very thin circularF-dot grown upon it, following the work [32]. The magnetization is assumed to befixed, homogeneous inside the dot and directed either perpendicular or parallel to theS-film (see figure 1). This problem is a basic one for a class of more complicatedproblems incorporating arrays of magnetic dots.

We will analyse what the conditions are for the appearance of vortices in theground state, where they appear, and what the magnetic fields and currents are inthese states. The S-film is assumed to be very thin, plane and infinite in the lateraldirections. Since the magnetization is confined inside the finite dot no difficulties withthe surface integrals over infinitely remote surfaces or contours arise.


3.1.1. Magnetic dot: perpendicular magnetization. The magnetic field generated byan infinitely thin circular magnetic dot of radius R with two-dimensional magnetiza-tion perpendicular to the plane mðrÞ ¼ mzzYðR rÞðz dÞ on the top of the S-filmcan be calculated using equations (18) and (19). The Fourier component of magne-tization necessary for this calculation is:

mk ¼ zz2pmR

qJ1ðqRÞe

ikzd , ð47Þ

where J1(x) is the Bessel function. The Fourier transforms of the vector potentialreads:

A?mk ¼

i8p2mRJ1ðqRÞ

k2eqd 2q

1þ 2qþ ðeikzd eqd

Þ

: ð48Þ

Though the difference in the inner round brackets in equation (48) looks to be alwayssmall (we recall that d must be set to zero in the final answer), we cannot neglect itsince it occurs to give a finite, non-negligible contribution to the parallel componentof the magnetic field between the two films. From equation (48) we immediately findthe Fourier transforms of the magnetic field components:

Bzmq ¼ iqA?

mq; B?mq ¼ ikzA

?mq: ð49Þ

For the reader’s convenience we also present the Fourier transform of the vectorpotential at the superconductor surface:

a?mq ¼ i8p2mR

1þ 2qJ1ðqRÞ: ð50Þ

In the last equation we have put eqd equal to 1.Performing the inverse Fourier transformation, we find the magnetic field in real

space:

Bzmðr, zÞ ¼ 4pmR

ð10

J1ðqRÞJ0ðqrÞeqjzj

1þ 2qq2 dq ð51Þ

Brmðr, zÞ ¼ 2pmR

ð10

J1ðqRÞJ1ðqrÞeqjzj 2q

1þ 2qYðzÞ þYðz dÞ YðzÞ

q dq, ð52Þ

where Y(z) is the step function equal to þ1 at positive z and 1 at negative z. Notethat Br

m has discontinuities at z¼ 0 and z¼ d due to surface currents in the S- andF-films, respectively, whereas the normal component Bz

m is continuous.

Superconductor

Magnetic DotDot's Flux Lines

Supercurrent SUPERCONDUCTOR

ANTIVORTEX VORTEX

MAGNETIC DOT

Figure 1. Magnetic dots with out-of-plane and in-plane magnetization and vortices.


A vortex, if appears, must be located at the centre of the dot due to symmetry. IfR , the direct calculation shows that the central position of the vortex providesminimum to energy. For small radius of the dot the deviation of the vortex from thecentral position seems even less probable. We have checked numerically that thecentral position is always energetically favourable for one vortex. Note that this factis not trivial since the magnetic field of the dot is stronger near its boundary.However, the gain in energy due to interaction of the magnetic field generated bythe vortex with magnetization decreases when the vortex approaches the boundary.The normal magnetic field generated by the Pearl vortex is given by equation (22).Numerical calculations based on equations (51) and (22) for the case R > showsthat Bz at the S-film ðz ¼ 0Þ changes sign at some r ¼ R0 (see figure 2) in the presenceof the vortex centred at r¼ 0, but it is negative everywhere at r > R in the absence ofthe vortex.

The physical explanation of this fact is as follows. The dot itself is an ensemble ofparallel magnetic dipoles. Each dipole generates a magnetic field at the plane passingthrough the dot, which has sign opposite to its dipole moment. The fields fromdifferent dipoles compete at r<R, but they have the same sign at r>R. TheS-current resists this tendency. The field generated by the vortex decays slowerthan the dipolar field (1/r3 versus 1/r4). Thus, the sign of Bz is opposite to themagnetization at small values of r (but larger than R) and positive at large r.The measurement of magnetic field near the film may serve as a diagnostic tool todetect an S-vortex confined by the dot. To our knowledge, so far there were noexperimental measurements of this effect.

In the presence of a vortex, the energy of the system can be calculatedusing equations (23)–(26). The appearance of the vortex changes the energy by theamount:

¼ "v þ "mv ð53Þ

5 6 7 8 9 10

r/λ

-0.02

-0.01

0

0.01

0.02

Bz

no vortexwith vortex

Figure 2. Magnetic field of dot with and without vortex for R= ¼ 5 and 0=ð8p2 mRÞ ¼ 4.


where "v ¼ "0 lnð=Þ is the energy of the vortex without the magnetic dot,"0 ¼ 2

0=ð16p2Þ, and "mv is the energy of interaction between the vortex and the

magnetic dot given by equation (26). For this specific problem the direct substitutionof the vector potential, magnetic field and the phase gradient (see equations 50and 51) leads to the following result:

"mv ¼ m0R

ð10

J1ðqRÞ dq

1þ 2q: ð54Þ

The vortex appears when tends to zero. This criterion determines a curve in theplane of two dimensionless variables R= and m0="v. This curve separating regimeswith and without vortices is depicted in figure 3. The asymptotic of "mv for large andsmall values of R= can be found analytically:

"mv m0

R

1

"mv m0

R

2

R

1

Thus, asymptotically the curve ¼ 0 turns into a horizontal straight line m0="v ¼1at large R= and a logarithmically distorted hyperbola ðm0="vÞðR=Þ ¼ 2 at smallratio R=.

With further increasing of either m0="v or R= the second vortex becomesenergetically favourable. Due to symmetry the centres of the two vortices are locatedon the straight line including the centre of the dot at equal distances from it. Theenergy of the two-vortex configuration can be calculated by the same method.Curve 2 in figure 3 corresponds to this second phase transition. In principle thereexists an infinite series of such transitions. However, here we limit ourselves to thefirst three since it is not quite clear what is the most energetically favourable

0 1 2 3 4 5 6

R/λ

0

1

2

3

4

5

6

7

8

mΦ

0/εv

1 vortex2 vortices3 vortices

Figure 3. Phase diagram of vortices induced by a magnetic dot. The lines correspond to theappearance of 1, 2 and 3 vortices, respectively.


configuration for four vortices (for three it is a regular triangle). The role ofconfigurations with several vortices confined inside the dot area and antivorticesoutside has not yet been studied.

3.1.2. Magnetic dot: parallel magnetization. Next we consider an infinitelythin circular magnetic dot whose magnetization M is directed in the plane andis homogeneous inside the dot. An explicit analytical expression for M reads asfollows:

M ¼ m0ðR ÞðzÞxx ð55Þ

where R is the radius of the dot, m0 is the magnetization per unit area and xx is theunit vector along the x-axis. The Fourier transform of the magnetization is:

Mk ¼ 2pm0RJ1ðqRÞ

qxx: ð56Þ

The Fourier representation for the vector potential generated by the dot in thepresence of the magnetic film takes the form:

A?mk ¼ eikd

"8p2m0R

k2z þ q2J1ðqRÞ cosðqÞ

kze

kzd

q

eqd

1þ 2q

#: ð57Þ

Let us introduce a vortex–antivortex pair with the centres of the vortex and anti-vortex located at x ¼ þ0, x ¼ 0, respectively. Employing equations (23)–(26)to calculate the energy, we find:

E ¼ 20 ln

40

ð10

J0ð2q0Þ

1þ 2qdq 2m00R

ð10

J1ðqRÞJ1ðq0Þ

1þ 2qdqþ E0 ð58Þ

where E0 is the dot self-energy. Numerical calculations [32] indicate that theequilibrium value of 0 is equal to R. The vortex–antivortex creation changes theenergy of the system by:

¼ 20 ln

40

ð10

J0ð2qRÞ

1þ 2qdq 2m00R

ð10

J1ðqRÞJ1ðqRÞ

1þ 2qdq: ð59Þ

The instability to the appearance of the vortex–antivortex pair develops when changes sign. The curve that corresponds to ¼ 0 is given by the following equation:

m00

0¼

2 ln =ð Þ 4Ð10 J0ð2qRÞ=ð1þ 2qÞ dq

2RÐ10 J1ðqRÞJ1ðqRÞ=ð1þ 2qÞ dq

: ð60Þ

The critical curve in the plane of two dimensionless ratios m00=0 and R= isplotted numerically in figure 4. The creation of vortex–antivortex pairs is energeti-cally unfavourable in the region below this curve and favourable above it. The phasediagram suggests that the smaller the radius R of the dot, the larger the valuem00=0 necessary to create the vortex–antivortex pair. For large values of R andm00 0, the vortex is separated by a large distance from the antivortex.Therefore, their energy is approximately equal to that of two free vortices. Thispositive energy is compensated by the attraction of the vortex and antivortex tothe magnetic dot. The critical values of m00=0 seem to be numerically largeeven for R= 1. This is probably a consequence of comparably ineffective inter-action of the in-plane magnetization with the vortex.


Magnetic dots with a finite thickness were considered by Milosevic et al. [42–44].No qualitative changes of the phase diagram or magnetic fields were reported.

3.2. Array of magnetic dots and superconducting film

3.2.1. Vortex pinning by magnetic dots. Vortex pinning in superconductors is ofthe great practical importance. Artificial periodic vortex pinning was first producedwith S-film thickness modulation by Martinoli et al. [8]. A little later Hebard et al.[11, 12] used triangular arrays of holes. Magnetic structures provide additionalopportunities to pin vortices. The first experiments with regular magnetic dot arrayswere performed in the Louis Neel Laboratory in Grenoble [3, 4]. The dots wereseveral microns wide and their magnetization was parallel to the superconductingfilm. The authors reported oscillations of the magnetization versus magnetic field.These oscillations were attributed to a matching effect: pinning becomes strongerwhen the vortex lattice is commensurate with the lattice of pinning centres.

Flux pinning by a triangular array of submicron-size dots with typical spacing400–600 nm and diameters close to 200 nm magnetized in-plane was first reported byMartin et al. [6]. They observed oscillations of the resistivity versus magnetic fluxwith period corresponding to one flux quantum per unit cell of magnetic dot lattice(see figure 5, left). The oscillations can be explained by the matching effect. Thepinning by the dots with parallel to the film magnetization was rather strong [6].

A dot array with out-of-plane magnetization was first prepared and studied byMorgan and Ketterson [7]. They measured the critical current as a function of theexternal magnetic field and found strong asymmetry of the pinning properties under

0 1 2 3 4 5 6 7R/λ

0

5

10

15

20

25

30

mφ 0

/ε0

Figure 4. Phase diagram for vortices–antivortices induced by the magnetic dot with in-planemagnetization.


magnetic field reversal (see figure 5, right). This was the first direct experimentalevidence that vortex pinning by magnetic dots is different from that of non-magneticpinning centres.

Pinning properties of the magnetic dot array depend on several factors:orientation of the magnetic moment, the strength of the stray field, the ratios ofthe dot size and the dot lattice constant to the effective penetration depth, the dotarray magnetization, the strength and direction of the external field, etc. The use ofmagnetic imaging techniques, namely the scanning Hall probe microscope (SHPM)and magnetic force microscope (MFM), revealed exciting pictures of the vortex‘world’. Such studies in combination with traditional measurements give new insightinto vortex physics. This work was done mainly by the group at the University ofLeuven. Below we briefly review several experimental studies of this group.

Dots with parallel magnetization. Van Bael et al. [90] studied the magnetizationand vortex distribution in a square array (1.5 mm period) of rectangular (540 nm

360 nm) dots fabricated from cobalt–gold trilayer Au(7.5 nm)/Co(20 nm)/Au(7.5 nm)magnetized along one of the edges of the dot lattice with the scanning Hall probemicroscope (SHPM). SHPM images revealed a magnetic field redistribution belowthe superconducting transition temperature in the 50 nm thin lead superconductingfilm. These data were interpreted by Van Bael et al. [90] as the formation of vorticesof opposite sign on both sides of the dot. Applying a proper magnetic field, Van Baelet al. [90] generated the commensurate lattice of vortices residing at the ends of themagnetized dot diameter. This location is in agreement with theoretical predictions[32]. Remarkably, they were able to observe annihilation of the vortices created bythe stray field of the dots by the antivortices induced by the applied normal field(see figure 6).

Dots with normal magnetization. We have already mentioned that Morgan andKetterson [7] observed asymmetry of the vortex pinning with respect to magneticfield reversal in an S-film supplied with a periodic array of F-dots magnetized per-pendicularly to the film. Employing SHPM images, Van Bael et al. [91] elucidatedthe nature of this phenomenon. They prepared a square F-dot lattice with lattice

Figure 5. Left: field dependence of the resistivity of a Nb thin film with a triangular array ofNi dots. (From Martin et al. [6].) Right: critical current as a function of field for the high-density triangular array at T ¼ 8:52K, Tc¼ 8.56K. (From Morgan and Ketterson [7].)


constant equal to 1 mm. Each dot had the shape of a square with 400 nm side lengthand 14 nm thickness. They were made of Co/Pt multilayers and covered with 50 nmthick lead film. Zero field SHPM images showed the checkerboard-like distributionof magnetic field (see Section 3.3.4). The stray fields from the dots were not sufficientto create vortices. In a very weak (1.6 Oe) external field the average distance betweenvortices was about four lattice spacings. In the field parallel to the dot magnetization,vortices reside on the dots, as the SHPM image shows (see figure 7a). In theantiparallel field the SHPM shows vortices located at interstitial positions in themagnetic dot lattice (see figure 7b). It is plausible that the pinning barriersare lower in the second case.

Figure 8 shows the dependence of S-film magnetization on the applied magneticfield normal to the film. Moshchalkov et al. [92] have shown that the magnetic fielddependence of the S-film magnetization is similar to the critical current dependenceon magnetic field. Figure 8 displays strong asymmetry of the pinning properties withrespect to the external magnetic field orientation. The pinning is much stronger in themagnetic field parallel to the dot magnetization than in the antiparallel field.

3.2.2. Vortex lattice symmetry versus magnetic dot array symmetry. The vortexlattice induced by an external magnetic field in a homogeneous superconductingfilm has hexagonal symmetry. Its lattice constant is determined by the magneticfield strength so that each unit cell contains one flux quantum as originally predictedby Abrikosov [74]. An array of pinning centres generates a periodic potential forthe vortices. Even if the dot lattice has hexagonal symmetry, the vortex lattice iscommensurate with the pinning lattice only at special values of the external magneticfield. Generally symmetry and the phase state of the vortex lattice in the presence ofa lattice of pinning centres depend in a complicated way on the external magneticfield, the relative strength of the pinning potential and its symmetry.

As we mentioned in the Introduction, the first experiments with S-films, suppliedwith an artificial periodic pinning structure, were performed in the 1970s. In theseexperiments the periodicity of the vortex lattice induced by an external magnetic fieldcompeted with the periodicity of an artificial array created by experimenters [8–19].The presence of magnetic dots introduces an additional dimension into the oldproblem. The most interesting is the situation when the dots generate vortices(and antivortices). The net magnetic flux from the magnetic dot perpendicular to

Figure 6. Schematic presentation of the polarity-dependent flux pinning, presentingthe cross-section of a Pb film deposited over a magnetic dipole with in-plane magnetization:(a) a positive vortex (wide grey arrow) is attached to the dot at the pole where a negativeflux quantum is induced by the stray field (black arrows), and (b) a negative vortex is pinnedat the pole where a positive flux quantum is induced by the stray field. (From Van Baelet al. [90].)


Figure 7. SHPM images of a (10.5 mm)2 area of the sample in H ¼ 1:6 Oe (left panel) andH¼ 1.6 Oe (right panel), at T¼ 6.8K (field-cooled). The tiny black/white dots indicate thepositions of the Co/Pt dots, which are all aligned in the negative sense (m < 0). The flux linesemerge as diffuse dark (H < 0) or bright (H > 0) spots in the SHPM images. (From Van Baelet al. [91].)

Figure 8. M(H/H1) magnetization curves at different temperatures near Tc (7.00K – opensymbols, 7.10K – filled symbols) showing the superconducting response of the Pb layer on topof the Co/Pt dot array with all dots magnetized up (upper panel) or down (lower panel).H1 ¼ 20:68 Oe is the first matching field, at which exactly one flux quantum is generated perunit cell of the dot array. (From Van Bael et al. [91].)


the plane magnetization is zero. Although the average magnetic field is equal to zero,fields locally varying in space may be large enough to destroy superconductivity ifthe lattice period is in the range of several coherence lengths, [97]. Below we reviewa few theoretical and experimental works about vortex lattice symmetry in the pre-sence of an array of magnetic dots. Experimental studies are more numerous, butalso cover only a small number of possible configurations.

Spontaneous symmetry violation. Erdin [46] studied theoretically the simplest caseof a square array of magnetic dots with perpendicular magnetization without anexternal magnetic field. Even in this case, Erdin found a range of parameters inwhich the tetragonal symmetry of the dot lattice is spontaneously broken by thevortex–antivortex lattice. He assumed that each dot creates only one vortex in the S-film located just against the dot centre and one antivortex in an interstitial position(see figure 9). Employing the approach of Section 3.1, Erdin studied spontaneouscreation of vortex–antivortex pairs for different values of two dimensionless ratios:Lm0="v and R/L, where R is the dot radius and L is the dot lattice constant.For rather large values of the ratio R/L (e.g. 0.35), the rotational symmetry of thevortex–antivortex lattice is C4, i.e. the same as that of the dot lattice. However, forsmall values of the ratio R/L (e.g. 0.10), the rotational symmetry of thevortex–antivortex lattice is C2, with 10 maximum deviation of the unit cell diagonal(see table 1 in Ref. [46]).

Magnetic dot lattice near the superconducting transition temperature. Priour andFertig [97] considered theoretically the square lattice of the F-dots upon an S-filmat a temperature slightly below the S-transition temperature Tc. In this situation,local magnetic fields generated by the dots can easily reach and exceed the second

y

x

magnetic dot

vortex

antivortex

Figure 9. Top view of the square magnetic dot array. The vortices are confined within theregion of the dot, while the antivortices appear outside.


critical field of the S-film Hc2, though the average magnetic field is zero. The stronginhomogeneous magnetic field induces vortices and antivortices. The authorsassumed that the F-dots are cubes with side length equal to 2 ( is the coherencelength) and accepted the Ginzburg–Landau (GL) parameter ¼ = to be equal to 4.The magnetization is assumed to be perpendicular to the S-film and to create mag-netic flux through the cube face varying from 4.1 to 5.6 flux quanta. The authorssolved numerically the GL equation in the presence of an inhomogeneous magneticfield generated by the F-dot array. The period of the magnetic dot lattice was takento be 6:25. In the numerical procedure they solved the GL equation in a 4 4 latticewith periodic boundary conditions. The vortices and antivortices were identified asnodes of the GL wavefunction ðxÞ. The resulting density diagram for jðxÞj

2 isshown in figure 10. Vortices and antivortices are clearly seen on it as light circularspots. The initial square symmetry of the dot lattice is violated by the vortex arrange-ment in all these configurations.

Rectangular magnetic dot lattice. Schuller et al. [98] studied experimentally vortexlattices induced by a rectangular magnetic dot array in an external magnetic field.The unit cell of the dot array, a rectangle with sides c and d, had the asymmetry ratiof ¼ c=d in the range 1–2.25. Each dot was a circular cylinder with diameter 200 nm

0

ρpair (c) = 2.5 ρ

pair (d) = 3

ρpairρ

pair (a) = 2 (b) = 2

0

6

0

6

6 12 18 24 12 18 24600

12

18

24

12

18

24

0

6

12

18

24

0 6 12 18 24 0 6 12 18 24

6

12

18

24

0

Figure 10. Density plots of the GL order parameter in stable phases. The images in panels(a), (b), (c) and (d) correspond to dot magnetic flux equal to 4.10, 4.62, 4.91 and 5.62 funda-mental flux units, respectively. Antivortices appear as small dark spots, while the large darkspots indicate dot regions; the Cooper-pair density is depressed in regions with lighter shading.pair is the vortex–antivortex pair density per unit cell. (From Priour and Fertig cond-mat/0310221.)


and height 40 nm made from nickel. Performing magnetotransport measurements,Schuller et al. [98] found two regimes. In a low external field the vortex lattice wascommensurate with the dot lattice; in higher fields the transition to the square vortexlattice has occurred so that the side of the cube was commensurate with the smallerside of the dot rectangle. Schuller et al. [98] have interpreted their data in terms of thebalance between the periodic pinning and elastic energy of the vortex lattice.

3.2.3. Magnetic field induced superconductivity. Consider a regular array of mag-netic dots placed upon a superconducting film with magnetization normal to thefilm. For simplicity we consider very thin magnetic dots. This situation is realized inmagnetic films with normal magnetization used in the experiment [93]). The net fluxfrom each magnetic dot through any plane including the surface of the supercon-ducting film (see figure 11) is exactly zero. Suppose that on the top of the magneticdot the z-component of the magnetic field is positive as shown in the mentionedfigure. Due to the requirement of zero net flux the z-component of the magneticfield between the dots must be negative. Thus, a connected part of the S-film occursin a negative magnetic field normal to the film. It can be partly or fully compen-sated by an external magnetic field parallel to the dot magnetization (see figure 11).Lange et al. [93] proposed this experiment and found a positive shift of theS-transition temperature in an external magnetic field, the result looking counter-intuitive if one forgets about the field generated by the dots. In this experiment athin superconducting film was covered with a square array of CoPd magnetic dotsperpendicular to the film magnetization. The dots had a square shape with side0.8 mm, thickness 22 nm and dot array period 1.5 mm. The H–T phase diagramspresented in [93] for zero and finite dot magnetization demonstrate the appearanceof superconductivity in an applied magnetic field parallel to the dot magnetization.At T ¼ 7.20K the system with magnetized dots is in the normal state. It transits

Si/SiO2

GeCo/Pd

Ge

Pb

z(a) H= 0

(b)

mB

H

Figure 11. Schematic distribution of the magnetic field generated by an array of dots withnormal to the superconducting film magnetization: (a) zero external field, (b) external fieldparallel to magnetization partly or completely compensates the magnetic field of dots betweenthem. (From Lange et al. cond-mat/0209101.)


to the superconducting state in the field 0.6mT and back to the normal state at3.3mT. From the data in figure 3 of [93] one can conclude that the compensatingfield is about 2mT.

3.2.4. Magnetization controlled superconductivity. Above (Section 3.2.3) we demon-strated an example when the application of a magnetic field transforms the FSHsystem from the normal to superconducting state due to compensation of the straymagnetic field of the dots with external magnetic field. Earlier Lyuksyutov andPokrovsky [26] considered theoretically a randomly magnetized array of magneticdots with normal magnetization. They argued that such an array induces the resistivestate in the S-film. The superconducting state can be restored by regular magnetiza-tion of the dot array. This counterintuitive phenomena can be explained on aqualitative level. If a single dot induces one vortex, the magnetized array of dotsinduces a periodic vortex–antivortex lattice with antivortices localized at the centresof the unit cells of the square dot lattice as shown in figure 12, left. This orderedvortex–dot system provides a strong pinning. More interesting is a ‘paramagnetic’,i.e. randomly magnetized, dot array. Vortices and antivortices induced by a para-magnetic dot array generate a random field for a probe vortex. If the lattice constantof the array, a, is less than the effective penetration depth , the random fields fromvortices are logarithmic. The effective number of random logarithmic potentialsacting on a probe vortex is N ¼ ð=aÞ2 and the effective depth of the potentialwell for a vortex (antivortex) is

ffiffiffiffiN

pv. Under proper conditions, for example near

the S-transition point, the potential wells can be very deep, enabling spontaneousgeneration of the vortex–antivortex pairs at the edges between potential valleysand hills. The vortices and antivortices screen these deep potential wells and hillssimilarly to the screening in a plasma. The difference is that, in contrast to a plasma,the screening ‘charges’ do not exist without an external potential. In such a flattened

MAGNETIC DOT SUPERCURRENT

INDUCED VORTEX

MAGNETIC DOT SUPERCURRENT

INDUCED VORTEX

Figure 12. Left: magnetized magnetic dot array. Vortices of different signs are shown sche-matically by the supercurrent direction (dashed lines). The dot magnetic moment direction isindicated by . Vortices bound by dots and vortices in interstitial positions are shown. Themagnetized array of dots induces a regular lattice of vortices and antivortices and providestrong pinning. Right: a demagnetized magnetic dot array results in a strongly fluctuatingrandom potential, which generates unbound antivortices/vortices, transforming the S-film intoa resistive film.


self-consistent potential relief the vortices move along percolated infinite trajectoriespassing through the saddle points [29]. The drift motion of the delocalized vorticesand antivortices in the external field generates dissipation and transfers the S-filminto the resistive state (see figure 12, right). Replacing the slowly varying logarithmicpotential by a constant at distances less than and zero at larger distances, Feldmanet al. [29] found the thermodynamic and transport characteristics of this system.Below we briefly outline their main results.

For the sake of simplicity we replace the slowly varying interaction potential V(r)by a constant value within the single cell: V0 ¼ 20 at the distance r < and zero atr > , where 0 ¼ 2

0=ð16p2Þ. Considering the film as a set of almost unbound cells

of linear size we arrive at the following Hamiltonian for such a cell:

H ¼ UXi

ini þ Xi

n2i þ 20Xi>j

ninj, ð61Þ

where ni is an integer characterizing an individual vortex located either at a dot or ata site of the dual lattice (between the dots) which we conventionally associate withthe location of unbound vortices; i ¼ 1 is the random sign of the dot magneticmoments; and i ¼ 0 relates to the sites of the dual lattice. The first term of theHamiltonian (61) describes the binding energy of the vortex at the magnetic dot andU 0d=0, with d being the magnetic flux through a single dot. The second termin the Hamiltonian is the sum of single vortex energies, ¼ 0 lnð=aÞ, where a is theperiod of the dot array. The third term mimics the vortex interaction. Redefining theconstant , one can replace the last term of equation (61) by 0ð

PniÞ

2. The sign ofthe vorticity ni at a dot follows two possible (‘up’ or ‘down’) orientations of itsmagnetization. The vortices located between the dots (ni on the dual lattice) arecorrelated at distances of the order of and form the above-mentioned irregularcheckerboard potential relief.

To find the ground state, we consider a cell with a large number of dots of eachsign ð=aÞ2 1. The energy (61) is minimal when the ‘neutrality’ conditionQ

Pni ¼ 0 is satisfied. Indeed, if Q 6¼ 0 the interaction energy grows as Q2,

whereas the first term of the Hamiltonian behaves as jQj and cannot compensatethe last one unless Q 1. The neutrality constraint means that the unboundvortices screen almost completely the ‘charge’ of those bound by dots, that isK ðNþ NÞ

ffiffiffiffiffiffiffiN

p =a where K is the difference between the numbers of

the positive and negative dots and N are the numbers of the positive and negativevortices, respectively. Neglecting the total charge jQj as compared with =a, weminimize the energy (61) accounting for the neutrality constraint. At Q¼ 0 theHamiltonian (61) can be written as the sum of single-vortex energies:

H ¼X

Hi; Hi ¼ Uini þ n2i : ð62Þ

The minima for any Hi are achieved by choosing ni ¼ n0i , an integer closest to themagnitude i ¼ iU=ð2Þ. The global minimum consistent with the neutrality is rea-lized by values of niwhich differ from the local minima values n0i by not more than1.Indeed, in the configuration with ni ¼ n0i , the total charge is j

Pn0i j j

Pij ¼ jK j.

Hence, if =a, then the change of the vorticity at a small number of sites by 1restores neutrality. To be more specific let us consider K>0. Let nn be the integerclosest to , and consider the case < nn. Then the minimal energy corresponds to aconfiguration with vorticity ni ¼ nn at each negative dot and with vorticity nn or nn1


at positive dots. The neutrality constraint implies that the number of positive dots withvorticity nn1 isM ¼ K nn. In the opposite case > nn the occupancies of all the positivedots are nn, whereas the occupancies of the negative dots are either nn or nnþ1. Note thatin this model the unbound vortices are absent in the ground state unless is an integer.Indeed, the transfer of a vortex from a dot with occupancy n to a dual site changes theenergy by E ¼ 2ð nþ 2Þ. Hence, the energy transfer is zero if and only if is aninteger, otherwise the energy change upon vortex transfer is positive. For integer , thenumber of unbound vortices can vary from 0 to K nn without change of energy. Theground state is degenerate at any non-integer since, while the total number of dotswith different vorticities is fixed, the vortex exchange between two dots with vorticitiesn and n 1 does not change the total energy. Thus, the model predicts a step-likedependence of dot occupancies on at zero temperature and peaks in the concentra-tion of unbound vortices as shown in figure 13. The data for a finite temperature werecalculated in [29]. The dependencies of the unbound vortex concentration on forseveral values of x ¼ =T are shown in figure 13. Oscillations are well pronounced forx 1 and are suppressed at small x (large temperatures). At low temperatures, x 1,the half-widths of the peaks in the density of the unbound vortices are 1=x andthe heights of peaks are n, where ¼K/N.

Vortex transport. For moderate external currents j the vortex transport and dis-sipation are controlled by unbound vortices. The typical energy barrier associatedwith the vortex motion is 0. The unbound vortex density is m a2 ðaÞ1 andoscillates with as shown above. The average distance between the unbound vorticesis l

ffiffiffiffiffiffia

p. The transport current exerts a Magnus (Lorentz) force FM ¼ j0=c

acting on a vortex. Since the condition T 0 is satisfied in the vortex state every-where except in the close vicinity of the transition temperature, Tc, the vortex motionoccurs via thermally activated jumps with rate:

¼ 0 expð0=T Þ ¼ ð j0=clÞ expð0=T Þ, ð63Þ

where ¼ ð2nÞ=ð4pe2Þ is the Bardeen–Stephen vortex mobility [95]. The induced

0 1 2 3 4 5κ

0

2

4

6

q/γ

Figure 13. Left: the checkerboard average structure of the vortex plasma. Right: the averagenumber of unbound vortices in the cell of size a versus a parameter proportional to the dotmagnetic moment. The dot-dashed line corresponds to T=0 ¼ 0:15, the solid line correspondsto T=0 ¼ 0:4, and the dashed line corresponds to T=0 ¼ 2.


electric field is accordingly

Ec ¼ l _BB=c ¼ m0l=c: ð64Þ

The ohmic losses per unbound vortex are Wc ¼ jEca ¼ j0l=c giving rise to the dcresistivity as

dc ¼Wc

j22¼ 2

0

c22exp½0ðT Þ=T : ð65Þ

Note the non-monotonic dependence of dc on temperature T (figure 14). Thedensity of the unbound vortices is an oscillating function of the flux through thedot. The resistivity of such a system is determined by thermally activated jumps ofvortices through the corners of the irregular checkerboard formed by the positive ornegative unbound vortices and oscillates with d. These oscillations can be observedby additional deposition (or removal) of the magnetic material to the dots.

3.3. Ferromagnet–superconductor bilayer

3.3.1. Topological instability in the FSB. Let us consider a F/S-bilayer with bothlayers infinite and homogeneous. An infinite magnetic film with ideal parallelsurfaces and homogeneous magnetization generates no magnetic field outside.Indeed, it can be considered as a magnetic capacitor, the magnetic analogue of anelectric capacitor, and therefore its magnetic field is confined inside. Thus, there is nodirect interaction between the homogeneously magnetized F-layer and a homo-geneous S-layer in the absence of currents in it. However, Lyuksyutov andPokrovsky argued [30] that such a system is unstable with respect to spontaneousformation of vortices in the S-layer. Below we reproduce these arguments.

Assume the magnetic anisotropy to be sufficiently strong to keep the magnetiza-tion perpendicular to the film (in the z-direction). According to the above arguments,the homogeneous F-film creates no magnetic field outside itself. However, if a Pearlvortex somehow appears in the superconducting film, it generates a magnetic field

0 0.2 0.4 0.6 0.8 1t

0

1

2

3

4

ρ m

Ω

Figure 14. The static resistance of the film versus dimensionless temperature t ¼ T=Tc attypical values of the parameters.


interacting with the magnetization m per unit area of the F-film. For a propercirculation direction in the vortex and rigid magnetization m this field decreasesthe total energy by the amount m

ÐBzðrÞ d

2x ¼ m, where is the total flux.We recall that each Pearl vortex carries flux equal to the famous flux quantum0 ¼ phhc=e. The energy necessary to create the Pearl vortex in the isolated S-filmis ð0Þv ¼ 0 lnð=Þ [75], where 0 ¼ 2

0=16p2, ¼ 2L=d is the effective penetration

depth [74], L is the London penetration depth, and is the coherence length. Thus,the total energy of a single vortex in the FSB is:

v ¼ ð0Þv m0, ð66Þ

and the FSB becomes unstable with respect to spontaneous vortex formation as soonas v becomes negative. Note that close enough to the S-transition temperature Ts,v is definitely negative since the S-electron density ns and, therefore,

ð0Þv is zero at Ts.

If m is so small that v > 0 at T¼ 0, the instability exists in a temperature intervalTv < T < Ts, where Tv is defined by the equation vðTvÞ ¼ 0. Otherwise instabilitypersists until T¼ 0.

A newly appearing vortex phase cannot consist of the vortices of one sign. Amore accurate statement is that any finite density of vortices independent on the sizeof the film, Lf, is energetically unfavourable in the thermodynamic limit Lf ! 1.Indeed, any system with non-zero average vortex density nv generates a constantmagnetic field Bz ¼ nv0 along the z-direction. The energy of this field for a large butfinite film of linear size Lf grows as L

3f , exceeding the gain in energy due to creation

of vortices proportional to L2f in the thermodynamic limit. Thus, paradoxically the

vortices appear, but cannot proliferate to a finite density. This is a manifestation ofthe long-range character of magnetic forces. The way out of this controversy issimilar to that in ferromagnets: the film should split into domains with alternatingmagnetization and vortex circulation directions. Note that these are combinedtopological defects: vortices in the S-layer and domain walls in the F-layer. Theyattract with each other. The vortex density is higher near the domain walls. Thedescribed texture represents a new class of topological defects which does not appearin isolated S- and F-layers. We show below that if the linear size, L, of the domain ismuch larger than the effective penetration length, , the most favourable arrange-ment is the stripe domain structure (see figure 15). The quantitative theory of thisstructure was given by Erdin et al. [31].

SUPERCONDUCTOR

FERROMAGNET

SUPERCURRENT

VORTEX

VORTEX

ANTIVORTEX

ANTIVORTEX

ANTIVORTEX

DOMAIN WALL

VORTEX

Figure 15. Magnetic domain wall and coupled arrays of superconducting vortices withopposite vorticity. Arrows show the direction of the supercurrent.


The total energy of the bilayer can be represented by a sum:

U ¼ Usv þUvv þUvm þUmm þUdw ð67Þ

where Usv is the sum of energies of single vortices, Uvv is the vortex–vortex interac-tion energy, Uvm is the energy of interaction between the vortices and magnetic fieldgenerated by the domain walls, Umm is the self-interaction energy of the magneticlayer, and Udw is the linear tension energy of the domain walls. We assume the two-dimensional periodic domain structure consisting of two equivalent sublattices. Themagnetization mzðrÞ and density of vortices n(r) alternate when passing from onesublattice to another. Magnetization is supposed to have a constant absolute value:mzðrÞ ¼ msðrÞ, where s(r) is the periodic step function equal to þ1 at one sublatticeand 1 at the other one. We consider a dilute vortex system in which the vortexspacing is much larger than . Then the single-vortex energy is:

Usv ¼ v

ðnðrÞsðrÞ d2x; v ¼ ð0Þv m0 ð68Þ

The vortex–vortex interaction energy is:

Uvv ¼1

2

ðnðrÞVðr r

0Þnðr0Þ d2x d2x0, ð69Þ

where Vðr r0Þ is the pair interaction energy between vortices located at points r

and r0. Its asymptotics at large distances j r r

0j is Vðr r

0Þ ¼ 2

0=ð4p2j r r

0jÞ

[96]. This long-range interaction is induced by the magnetic field generated by thePearl vortices and their slowly decaying currents.3 The energy of vortex interactionwith the magnetic field generated by the magnetic film looks as follows [32]:

Uvm ¼ 0

8p2

ðr’ðr r

0Þnðr0Þ aðmÞ

ðrÞ d2x d2x0: ð70Þ

Here ’ðr r0Þ ¼ arctan½ðy y0Þ=ðx x0Þ is a phase shift created at a point r by a

vortex centred at a point r0 and aðmÞ

ðrÞ is the value of the vector potential induced bythe F-film upon the S-film. Similarly to what we did for one vortex, this part of theenergy can be reduced to the renormalization of the single vortex energy with thefinal result already shown in equation (68). The magnetic self-interaction reads:

Umm ¼ m

2

ðBðmÞz ðrÞsðrÞ d2x: ð71Þ

Finally, the linear energy of the domain wall is Udw ¼ dwLdw where dw is the lineartension of the domain wall and Ldw is the total length of the domain walls.

Erdin et al. [31] have compared energies of stripe, square and triangular domainwall lattices, and found that the stripe structure has the lowest energy. For details ofthe calculation see [31] (and the correction in [33]). The equilibrium domain widthand the equilibrium energy for the stripe structure are:

Ls ¼

4exp

"dw4 ~mm2

C þ 1

ð72Þ

3From this long-range interaction of the Pearl vortices it is easy to derive that the energy of asystem of vortices with the same circulation, located with the permanent density nv on a filmhaving lateral size L, is proportional to n2vL

3.


Us ¼ 16emm2

exp

"dw4emm2

þ C 1

ð73Þ

where ~mm ¼ m 0v=0 and C¼ 0.57721 is the Euler constant. The vortex density forthe stripe domain case is:

nðxÞ ¼ 4p ~v2

0Ls

1

sinðpx=LsÞ: ð74Þ

Note the strong singularity of the vortex density near the domain walls. The con-tinuous approximation is invalid at distances of the order of , and the singularitiesmust be smeared out in a band of width around the domain wall.

The domains become infinitely wide at T¼Ts and at T¼Tv. If dw 4 ~mm2, thecontinuous approximation becomes invalid (see Section 3.2.3) and instead a discretelattice of vortices must be considered. It is possible that the long nucleation time caninterfere with the observation of the described textures. We expect, however, that thevortices that appear first will reduce the barriers for domain walls and, subsequently,expedite domain nucleation.

Despite theoretical simplicity, the ideal bilayer is not easy to realize experimen-tally. The most popular material with the magnetization perpendicular to film is amultilayer made from ultrathin Co and Pt films (see Section 3.3.4). This material hasa very large coercive field and rather chaotic morphology. Therefore, the domainwalls in such a multilayer are chaotic and almost unmovable at low temperatures(see Section 3.3.4). We hope, however, that these experimental difficulties will beovercome and spontaneous vortex structures will be discovered before long.

3.3.2. Superconducting transition temperature of the FSB. The superconduct-ing phase transition in a ferromagnet–superconductor bilayer was studied byPokrovsky and Wei [33]. They have demonstrated that in the FSB the transitionproceeds discontinuously as a result of competition between the stripe domain struc-ture in an F-layer at suppressed superconductivity and the combined vortex-domainstructure in the FSB. Spontaneous vortex-domain structures in the FSB tend toincrease the transition temperature, whereas the effect of the F-self-interactiondecreases it. The final shift of transition temperature Tc depends on severalparameters characterizing the S- and F-films and varies typically between 0:03Tc

and 0.03Tc.As we demonstrated earlier, the homogeneous state of the FSB with the magne-

tization perpendicular to the layer is unstable with respect to formation of a stripedomain structure, in which both the direction of the magnetization in the F-film andthe circulation of the vortices in the S-film alternate together. The energy of the stripestructure per unit area U and the stripe equilibrium width Ls is given in equations(72) and (73). To find the transition temperature, we combine the energy given byequation (73) with the Ginzburg–Landau free energy. The total free energy per unitarea reads:

F ¼ U þ FGL ¼16 ~mm2

eexp

dw

4 ~mm2þ C 1

þ nsds ðT TcÞ þ

2ns

: ð75Þ

Here and are the Ginzburg–Landau parameters. We omit the gradient term inthe Ginzburg–Landau equation since the gradients of the phase are included in theenergy (73), whereas the gradients of the superconducting electron density can be


neglected everywhere beyond the vortex cores. Minimizing the total free energy,Pokrovsky and Wei [33] have found that near Tc the FSB free energy can berepresented as

Fs ¼ 2ðT TrÞ

2

2ds ð76Þ

where Tr is given by the equation:

Tr ¼ Tc þ64pm2e2

msc2

expdw4m2

þ C 1

: ð77Þ

The S-phase is stable if its free energy equation (76) is less than the free energyof a single F-film with the stripe domain structure, which has the following form[99, 100]:

Fm ¼ 4m2

Lf

ð78Þ

where Lf is the stripe width of the single F-film. Near the S-transition point thetemperature dependence of the variation of this magnetic energy is negligible.Hence, when T increases, the S-film transforms into a normal state at sometemperature T

c below Tr. This is the first-order phase transition. At the transitionpoint, both energies Fs and Fm are equal to each other. The shift of the transitiontemperature is determined by the following equation:

Tc T c Tc ¼

64pm2e2

msc2

expdw4m2

þ C 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi8m2

2dsLf

s: ð79Þ

Two terms in equation (79) play opposite roles. The first one is due to the appearanceof spontaneous vortices which lowers the free energy of the system and it enhancesthe transition temperature. The second term is the contribution of the purelymagnetic energy, which tends to decrease the transition temperature. The valuesof the parameters entering equation (79) can be estimated as follows. The dimension-less Ginzburg–Landau parameter is ¼ 7:04Tc=F , where F is the Fermi energy.A typical value of is about 103 for low-temperature superconductors. The secondGinzburg–Landau parameter is ¼ Tc=ne, where ne is the electron density. Forestimates, Pokrovsky and Wei [33] took Tc 3 K, ne 1023 cm3. The magnetizationper unit area m is the product of the magnetization per unit volume M and thethickness of the FM film dm. For typical values of M 102 Oe and dm 10 nm,m 104 Gs/cm2. In an ultrathin magnetic film the observed values of Lf vary inthe range 1–100 mm [101, 102]. If Lf 1 mm, ds ¼ dm ¼ 10 nm, and expð ~dw=4m

2þ

C 1Þ 103, Tc=Tc 0:03. For Lf¼ 100 mm, ds ¼ 50 nm, and expð ~dw=4m2þ

C 1Þ 102, Tc=Tc 0:02.

3.3.3. Transport properties of the FSB. The spontaneous domain structure violatesthe initial rotational symmetry of the FSB. Therefore, it makes the transport proper-ties of the FSB anisotropic. Kayali and Pokrovsky [34] have calculated the periodicpinning force in the stripe vortex structure resulting from a highly inhomogeneousdistribution of the vortices and antivortices in the FSB. The transport propertiesof the FSB are associated with the driving force acting on the vortex lattice froman external electric current. In the FSB the pinning force is due to the interactionof the domain walls with the vortices and antivortices and the vortex–vortex


interaction, Uvv. Periodic pinning forces in the direction parallel to the stripes do notappear in continuously distributed vortices. Therefore, one needs to modify thetheory [31] to incorporate the discreteness effects.

Kayali and Pokrovsky [34] have shown that, in the absence of a driving force, thevortices and antivortices line themselves up in straight chains (see figure 16) andthe force between two chains of vortices falls off exponentially as a function ofthe distance separating the chains. They also argued that the pinning force in thedirection parallel to the domains drops faster in the vicinity of the superconductingtransition temperature Ts and vortex disappearance temperature Tv.

In the presence of a permanent current there are three kinds of forces acting on avortex. They are:

1. the Magnus force proportional to the vector product of the current densityand the velocity of the vortex;

2. the viscous force directed oppositely to the vortex velocity;3. the periodic pinning force acting on a vortex from other vortices and domain

walls.

The pinning forces have components perpendicular and parallel to the domainwalls. In the continuous limit the parallel component obviously vanishes. This meansthat it is exponentially small if the distances between vortices are much less than thedomain width. The sum of all three forces must be zero. This equation determinesthe dynamics of the vortices. It was solved under the simplifying assumption thatvortices inside one domain move with the same velocity. The critical currents havebeen calculated for the parallel and perpendicular orientations. Theory predicts astrong anisotropy of the critical current. The ratio of the parallel to perpendicularcritical current is expected to be in the range 102–104 at temperatures close tothe superconducting transition temperature Ts or to the vortex disappearancetemperature Tv. The anisotropy decreases rapidly when the temperature movesaway from the ends of this interval, reaching its minimum somewhere inside it.The anisotropy is associated with the fact that the motion of vortices is very different

+

+

+

+

+

+

-

-

-

-

-

- +

+

+

+

+

+

-

-

-

-

-

-

X

Y

b

-

-

-

-

-

-

+

+

+

+

+

+

-

-

-

-

-

-

+

+

+

+

+

+

a

L

Figure 16. Schematic vortex distribution in the FSB. The sign refers to the vorticity of thetrapped flux. From Kayali and Pokrovsky [34].


in the current perpendicular and parallel to the domain walls. When the directionof the current is perpendicular to the domains, the friction force makes all thevortices drift in the direction of the current, whereas the Magnus force induces themotion of vortices (antivortices) in neighbouring domains in opposite directions,both perpendicular to the current. The motion of all vortices perpendicular to thedomains forces domain walls to move in the same direction. This is a Goldstonemode; no perpendicular pinning force appears in this case. The periodic pinning inthe parallel direction and together with that in the perpendicular critical current isexponentially small. In the case of parallel current the viscous force draws allvortices into the parallel motion along the domain walls and in alternating motionperpendicularly to them. The domain walls remain unmoving and provide verystrong periodic pinning force in the perpendicular direction. This anisotropic trans-port behaviour could serve as a diagnostic tool to discover spontaneous topologicalstructures in magnetic–superconducting systems.

3.3.4. Experimental studies of the FSB. In the preceding theoretical sectionwe assumed that the magnetic film changes its magnetization direction in a weakexternal field and achieves the equilibrium state. In all experimental works, magnetswith magnetization perpendicular to the film were Co/Pt or Co/Pd multilayers.They have a large coercive field and are quenched at low temperature, necessaryfor the S-state. Lange et al. [103–105] have studied the phase diagram and pinningproperties of such magnetically quenched FSB. In these works the averagemagnetization was characterized by a parameter s, the fraction of spins directedup. Magnetic domains in Co/Pd(Pt) multilayers look like meandering irregularbands at s¼ 0.5 (zero magnetization) (see figure 17b) and like irregular localizeddomains (see figure 17d) with typical size 0.25–0.35 mm near fully magnetized states(s¼ 0 or s¼ 1). The stray field from domains is maximal at s ¼ 0:5. It suppresses thesuperconducting transition temperature Tc of the Pb film by 0.2K (see figure 18).The effective penetration depth is about 0.76 mm at 6.9K.

When s is close to 0 or 1, Lange et al. [103–105] observed asymmetry ofthe physical properties in the applied magnetic field similar to the asymmetry ofmagnetization observed in the hybrid of an S-film and array of magnetic dots withmagnetization normal to the film (see Section 3.2.1). The asymmetry occurred in thedependence of Tc(H) and in transport properties. The localized domains have aperpendicular magnetic moment. If the thickness and magnetization are sufficientlylarge, these domains can pin vortices induced by the external magnetic field. In thisrespect they are similar to randomly distributed dots with normal magnetization.Thus, when s is close to 0 or 1, the critical current is maximal. Contrary to thissituation, when s 0:5, the randomly bent band domains destroy the long-rangeorder of the vortex lattice and provide percolation ‘routes’ for the vortex motion.The pinning becomes weaker, resulting either in a smaller critical current or in aresistive state. This qualitative difference between the magnetized and demagnetizedstate was observed in the experiments by Lange et al. [103–105]. The abovequalitative picture of vortex pinning is close to that developed by Lyuksyutov andPokrovsky [26] and by Feldman et al. [29] for the transport properties of a regulararray of magnetic dots with random normal magnetization (see Section 3.2.4). In thismodel the demagnetized state of the dot array is associated with vortex creepthrough the percolating network. The regularly magnetized state, on the otherhand, provides more regular vortex structure and enhances pinning.


(a)

(b)

(c) (d)

-150 -100 -50 0 50 100 150

1.0

-0.5

0.0

0.5

1.0

Mfm

/Msa

tH (mT)

(c)

(d)s = 1

s = 0

µ H0 (mT)

Figure 17. Magnetic properties of the Co/Pt multilayer: (a) hysteresis loop measured bythe magneto-optical Kerr effect with H perpendicular to the sample surface. MFM images(5 5 mm2) show that at s¼ 0.5 the domains look like bands (b); localized domains at s¼ 0.3(c); and s¼ 0.93 (d). (From Lange et al. cond-mat/0310132.)

0.0 0.2 0.4 0.6 0.8 1.0

7.00

7.05

7.10

7.15

7.20

7.25

T c(H=0

)(K

)

s

Figure 18. Dependence of the critical temperature at zero field Tc ðH ¼ 0Þ on the parameter s.The minimum value of Tc is observed for s¼ 0.5. (From Lange et al. cond-mat/0310132).


3.3.5. Thick films. Above we reviewed properties of S/F-bilayers when both theF- and S-films are thin, namely, ds L and dm Lf . In this subsection weconsider, following work by Sonin [48], the situation when both films are thick:ds L and dm Lf . We neglect the domain wall width, considering it as a line.Let us study the properties of an isolated F-film without a superconductor. Thisproblem was solved exactly by Sonin [106]. Figure 19(a) shows schematically themagnetic field distribution around a thick ferromagnetic film. The analyticexpression for the magnetic field at the boundary of the ferromagnetic film y ¼ 0þ

reads:

HxðxÞ ¼ 4M ln tanpx2Lf

: ð80Þ

HyðxÞ ¼ 2pMsign tanpx2Lf

at y ! 0: ð81Þ

The field pattern is periodic with period 2Lf along the x-axis.Sonin argued [48] that in the bulk superconductor, under the additional

constraint L=Lf ! 0, the magnetic flux from the F-film does not penetrate intothe superconductor even to the London penetration depth. Instead the magneticforce lines turn near the interface back to the ferromagnet, rearranging the domains.Therefore the energy changes in the presence of the superconducting substrate bya numerical factor of about 1.5 of purely geometrical origin and, therefore,independent of the parameters characterizing the superconductor and ferromagnet.The energy of the domain walls per unit length in the x-direction is inversely

++++++ ++++++ ++++++- - - - - - - - - - - - - - - - - -

++++++ ++++++ ++++++- - - - - - - - - - - - - - - - - -

- - - - - - ++++++ - - - - - - ++++++ - - - - - - ++++++

- - - - - -++++++ - - - - - - ++++++ - - - - - - ++++++

++++++ - - - - - - ++++++ - - - - - - ++++++- - - - - -

- - - - - - - - - - - - - - - - - -++++++ ++++++ ++++++

(a)

(b)

δ

l

dMFM

FM

SC

x

y

x

y

Figure 19. Magnetic charges (þ and ) and magnetic flux (thin lines with arrows) in aferromagnetic film (FM) without (a) and with (b) a superconducting substrate (SC). Themagnetization vectors in domains are shown by thick arrows. (From Sonin cond-mat/0102102.)


proportional to the domain width Lf with the coefficient (the domain wall energy perunit length in the perpendicular direction, dw) changing in the presence of the S-filmsince the effective length of the domain changes. The stray field energy is propor-tional to Lf, but the coefficient of this linear function does not change. The minimumof the energy is proportional to

ffiffiffiffiffiffiffidw

p, whereas the equilibrium domain width Lf is

inversely proportional to the same value. Therefore, the domain width Lf decreasesdiscontinuously at the S-phase transition, reducing by the factor 1.5. The correctionto the energy has relative order of magnitude L=Lf [48].

3.3.6. Domain wall superconductivity. Buzdin and Melnikov [107] and Aladyshkinet al. [52] considered the nucleation of superconductivity in a thin S-film (ds L) inthe presence of a magnetic field varying in space and generated by a thick F-film(dm Lf ). We review their results in this subsection. The distribution of magneticfield around a thick magnetic film shown in figure 19 is only slightly modified bya thin S-film. For suppression of the order parameter in a thin S-film only thecomponent of the magnetic field normal to the film matters. It changes sign at thedomain wall and vanishes at the domain wall centre, facilitating nucleation of super-conductivity. For a quantitative description of this nucleation, Aladyshkin et al. [52]modelled the real magnetic field of the domain wall by a function of one variableBzðxÞ ¼ H þ bðxÞ, where H is the uniform external magnetic field far from theS/F-interface and b(x) is the z-component of the field induced by the magnetizationM ¼ MðxÞz0.

The linearized GL equation for the order parameter reads:

r þ2pi0

A

2

¼1

2ðTÞ, ð82Þ

where AðrÞ is the vector potential, BðrÞ ¼ r AðrÞ, 0 is the flux quantum,ðT Þ ¼ 0=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 T=Tc0

pis the coherence length, and Tc0 is the critical temperature

of the bulk superconductor at B¼ 0. For a single domain wall of negligible thicknessthe magnetization M can be represented in the form M ¼ MsignðxÞz0. Then thedependence of the magnetic field on the coordinate at the S/F-interface readsBz ¼ 4pMsignðxÞ þH. The vector potential corresponding to this field has only ay-component, Ay ¼ 4pMjxj þHx. The nucleation of superconductivity in such afield is similar to the surface nucleation of superconductivity near Hc3 [96, 108].

Aladyshkin et al. [52] found the nucleation temperature for an isolated domainwall in an external magnetic field:

TðhÞ Tc0

Tc0

’2p20B0

0

aþ bh2 þ ch4

ð83Þ

where h ¼ H=B0, and B0 is the maximum absolute value of the field b(x)(B0 ¼ 4pM). a, b and c are constants obeying two relations: aþ c ¼ 1=2 and2a b ¼ 1=2, so that only one of these constants is independent. For the specificmodel they calculated a¼ 0.59. The negative coefficient a in equation (83) is dueto the stray fields of the domain walls, which suppress superconductivity. Thecoefficient b in this equation is positive. This means that the transition temperatureincreases in a weak external field. This enhancement of the transition temperature isdue to the decrease of the total field (stray field 4pM plus external field) in half ofthe domains. The local decrease of field enhances the superconductivity nucleation.The shift of transition temperature at nucleation near the domain wall according to


calculations in [52] is rather large: for niobium film with Tc 9K and magnetizationin the range 4pM 1–10 kOe it may be 1–3K.

In a recent experiment, Yang et al. [109] prepared a 50 nm thick Nb film upon asingle-crystal ferromagnet BaFe12O19 separated by a 10 nm thick Si insulating layer.Due to the strong magnetic anisotropy of BaFe12O19 its magnetization is perpen-dicular to the superconducting Nb film. The domain width is much larger than thedomain wall width and the S-film thickness. Therefore, the demagnetizing effect ofthe S-film is very weak. The external magnetic field normal to the surface of thesuperconducting film weakens the total field above the domains with the magnetiza-tion antiparallel to the field resulting in an increase of Tc in a field up to 0.4 T. Thefield dependence of Tc observed by Yang et al. [109] was similar to that predicted byAladyshkin et al. [52]. More details can be found in the original work by Yang et al.[109] and in the comment by Buzdin [110].

4. Proximity effects in layered ferromagnet–superconductor systems

4.1. Oscillations of the order parameter

All oscillatory phenomena theoretically predicted and partly observed in S/F-layeredsystems are based on the Larkin–Ovchinnikov–Fulde–Ferrel (LOFF) effect, firstproposed for homogeneous systems with coexisting superconductivity and ferro-magnetism [53, 54]. They predicted that the superconducting order parameter inthe presence of an exchange field should oscillate in space. The physical picture ofthis oscillation is as follows. In a singlet Cooper pair the electron with spin projectionparallel to the exchange field acquires the energy h, whereas the electron withantiparallel spin acquires the energy þh. Their Fermi momenta therefore split bythe value q ¼ 2h=vF . The Cooper pair acquires such a momentum and thereforeits wavefunction oscillates. The direction of the modulation vector in the bulksuperconductor is arbitrary, but in the S/F-bilayer the preferential direction of themodulation is determined by the normal to the interface (z-axis). There exist twokinds of Cooper pairs differing in the direction of the momentum of the electronwhose spin is parallel to the exchange field. The interference of the wavefunctionsfor these two kinds of pairs leads to the standing wave

FðzÞ ¼ F0 cos qz: ð84Þ

A modification of this consideration for the case when the Cooper pair penetratesinto a ferromagnet from a superconductor was proposed by Demler et al. [112]. Theyargued that the energy of a singlet pair is higher than the energy of two electrons inthe bulk ferromagnet by the value 2h (the difference of exchange energy betweenspin-up and spin-down electrons). This can be compensated if the electrons slightlychange their momentum so that the pair will acquire the same total momentumq ¼ 2h=vF . The value lm ¼ vF=h, called the magnetic length, is a natural length-scale for the LOFF oscillations in a clean ferromagnet. Anyway, equation (84)shows that the sign of the order parameter changes in the ferromagnet. This oscilla-tion leads to a series of interesting phenomena that will be listed here and consideredin some detail in next subsections.


1. periodic transitions from the 0- to p-phase in the S/F/S Josephson junc-tion when varying the thickness df of the ferromagnetic layer and tempera-ture T;

2. oscillations of the critical current versus df and T;3. oscillations of the critical temperature versus the thickness of magnetic layer.

The penetration of the magnetized electrons into superconductors stronglysuppresses the superconductivity. This obvious effect is accompanied with theappearance of magnetization in the superconductor. It penetrates to the depth ofthe coherence length and is directed opposite to the magnetization of the F-layer[113]. Another important effect which does not have oscillatory character and will beconsidered later is the preferential antiparallel orientation of the two F-layers in theS/F/S-trilayer.

This simple physical picture can also be treated in terms of the Andreev reflectionat the boundaries [55], long known to form the in-gap bound states [96, 114]. Due tothe exchange field the phases of Andreev reflection in the S/F/S-junction are differentfrom those in S/I/S- or S/N/S-junctions (with non-magnetic normal metal N).Indeed, let us consider a point P inside the F-layer at a distance z from one of theinterfaces [122]. The pair of electrons emitted from this point at angle , ðp Þto the z-axis will be reflected as a hole along the same lines and returns to the samepoint (figure 20). The interference of the Feynman amplitudes for these fourtrajectories creates an oscillating wavefunction of the Cooper pair. The main con-tribution to the total wavefunction arises from the vicinity of ¼ 0. Taking onlythis direction, we find for the phases: S1 ¼ S2 ¼ qz; S3 ¼ S4 ¼ qð2df zÞ.Summing up all Feynman’s amplitudes, eiSk ; k ¼ 1, . . . , 4, we find the spatial depen-dence of the order parameter:

F / cos qdf cos qðdf zÞ: ð85Þ

Figure 20. Four types of trajectories contributing (in the sense of Feynman’s path integral)to the anomalous wavefunction of correlated quasi-particles in the ferromagnetic region. Thesolid lines correspond to electrons, the dashed lines to holes; the arrows indicate the directionof the velocity. (From Fominov et al. cond-mat/0202280.)


At the interface F / ðcos qdf Þ2. It oscillates as a function of magnetic layer thickness

with period lm=2 ¼ p=q ¼ pvF=2h and decays due to the interference of trajectorieswith different .

In a real experimental set-up, the LOFF oscillations are strongly suppressedby elastic impurity scattering. The trajectories are diffusive random paths and asimple geometrical picture is no longer valid. However, as long as the exchangefield h exceeds or is of the same order of magnitude as the scattering rate in theferromagnet, 1=f , the oscillations do not disappear completely. Unfortunately,experiments with strong magnets possessing large exchange fields are not reliablesince the period of oscillations goes to the atomic scale. Two layers with differentthickness when they are so thin can have different structural and electronic proper-ties. In this situation it is very difficult to ascribe unambiguously the oscillations ofproperties to quantum interference.

The Andreev reflection at the interface of non-magnetic S- and metallic F-wireschanges the contact resistance in comparison to the contact of the same wires whennon-magnetic wire is in the normal state (NF contact) [115]. In the next subsectionwe consider this phenomenon in some detail.

4.2. Resistance of the SF contact in wires and the spin-filtering effect

De Jongh andBeenakker [115] have noted that the superconducting transition changesdrastically the reflection and transmission of the carriers at the interface of magneticand non-magnetic metals. If the contact was completely penetrable for normalelectrons, it becomes impenetrable for electrons in the ferromagnet if their energy isless than the gap in superconducting wire. Nevertheless, a current through thecontact is possible due to the Andreev process: an electron approaching the boundaryis reflected as a hole and one new Cooper pair enters the S-wire. It is clear, however,that the effective transmission coefficient is no longer equal to 1. Moreover, since thereflected hole has spin opposite to that of the incident electron, the Andreev reflectionbecomes impossible if the F-wire is completely polarized and there are no electronswith opposite spin in it. This phenomenon is called the spin-filtering effect. For asimple qualitative treatment of the contact conductance, de Jongh and Beenakkerapplied the Landauer formalism (see, e.g. [116]). Let the numbers of channels withspins up and down near the interface be N" and N#, respectively, and all modes passthrough the contact without reflection when both wires are in the normal state. Then,according to the Landauer formula, the conductance of the contact FN reads:

GFN ¼e2

hN" þN#

: ð86Þ

When one of the wires becomes superconducting, all minority carriers are reflectedby the Andreev process and the holes with opposite spin can find a place inside thebig majority Fermi sphere, but the majority carriers turn into minority holes.Therefore, they can find their place only if their momentum is less than the Fermimomentum of minority carriers pF#. In other words only the fraction N#=N" of themajority carriers can be reflected as holes to the ferromagnet. At each Andreevprocess a new Cooper pair with charge 2 enters the superconductor. Therefore:

GFS ¼e2

h2 N# þ

N#

N"

N"

¼ 4

e2

hN#: ð87Þ


This can be larger or smaller than GFN depending on what is larger 3N# and N".Note that GFS ¼ 0 if N# ¼ 0.

A more accurate treatment is based on the Bogolyubov–de Gennes formalism. Inthe framework of this formalism the quasi-particle wavefunctions are described bythe Bogolyubov two-dimensional vector u", v# depending on the coordinates andsatisfying the following equation:

H0 h

H0 þ h

u"v#

¼ "

u"v#

ð88Þ

with H0 ¼ ð p2=2mÞ EF and h denoting the exchange field. For simplification they

assumed that the kinetic energy of electrons is the same in the F- and S-wires andthat the F-wire occupies the negative x half-axis, whereas the S-wire occupies thepositive half-axis. Therefore they accepted hðxÞ ¼ h0YðxÞ for the exchange energyand ðxÞ ¼ YðxÞ for the superconducting gap, where Y(x) is the step function. Inthe standard approximation of tunnelling amplitudes the FS-junction conductancewas calculated by Takane and Ebisawa [117]. It reads:

GFS ¼ 2e2

h

X¼",#

Tr ryh , erh , e

, ð89Þ

where the Andreev reflection matrix r is given by its matrix elements

rh , e ¼ 2iq

ffiffiffiffiffiffiffiffiffiffik"k#

pk"k# þ q2

: ð90Þ

The wavevectors for electrons and holes propagating in the S- and F-wires are:

q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m EF Enð Þ

hh2

rð91Þ

k",# ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m EF En h0ð Þ

hh2

r, ð92Þ

where En is the energy in the n-th channel. The trace in equation (89) can be replacedby integration with the following result:

GFS ¼ 4e2

hN0

4

154

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

p6 72 þ 4

6þ 102 44h i

ð93Þ

where ¼ h0=EF and N0 ¼ k2F=4p is the number of channels, and is thecross-sectional area. Since N" þN# ¼ 2N0, one finds that GFS < GFN if > 0:47or N#=N"<0:36 instead of 1/3 following from qualitative theory. This resultis impressive, however the assumption of very clean wires and ballistic transportin the F-wire is not realistic. It was essentially lifted by Fal’ko et al. [118, 119].To simplify the problem these authors introduced a ballistic N-layer between theS- and F-wires. The F-wire is characterized by the current polarization s defined asfollows:

s ¼j" j#

j" þ j#¼

D"" D##D"" þD##

ð94Þ

where D ¼ ð1=3Þvl are diffusion coefficients, are densities of states (DOS) and lare the mean free paths for different directions of spin. The authors assumed that


the scattering in the F-wire is strong, but completely elastic. To calculate theresistance (conductance) of a junction with a diffusive F-layer it is necessary tosolve modified diffusion equations for the density Nðz, "Þ, ¼";# at fixed energy:

Do2zNðz, "Þ ¼ w"# Nðz, "Þ Nðz, "Þ½ , ð95Þ

where w"# is the rate of spin-reversal processes. They are equivalent to the followingtwo equations with separated variables:

o2z

X¼",#

DN ¼ 0; o2z L2s

N" N#

¼ 0 L2

s ¼ w"#ð" #Þ: ð96Þ

These equations must be solved with certain boundary conditions. At the left endthe condition is NðL, "Þ ¼ 1=2 nT " eVð Þ þ nT " eVð Þ½ , where nT ð"Þ is theequilibrium occupation number for quasi-particles. Here the electron–hole symmetryis used; " is the energy of the quasi-particle minus the chemical potential. Thegeometry is shown in figure 21. As already indicated earlier, the Andreev reflectionis forbidden for those majority electrons whose momentum is larger than the Fermimomentum of the minority carriers. Since the projection of momentum along theinterface is conserved (interfaces are ideal), this constraint can be formulated interms of projections instead of full momenta (see figure 21). It is clear that suchprocesses depend crucially on the ratio of Fermi momenta ¼ pF#=pF". In the caseof a long F-wire L l#, the Andreev reflection is effective at a short distance Ls fromthe interface and can be accounted for by effective boundary conditions, whichmatch the isoenergetic diffusive currents and densities from inside to that near theinterface with normal ballistic layer for " < :

D""ozN" ¼ D##ozN# ð97Þ

N" þN# þ2

31 2 3=2

l#ozN# ¼ nT ð"Þ þ nT ð"Þ: ð98Þ

F -

Re

se

rvo

ir

S-ReservoirDiffusive F - Wire N

LLs

↑

↑

F S

↑

↑

II

I

F N S

lα

e↑

h ↑

e↑

e↑

e↑

e↑

e↑

h ↑

h ↑I

II

Figure 21. Pictorial representation of the FS-junction, double Andreev reflection processes init, and of possible relations between the Fermi surfaces of spin-up and spin-down electrons inthe F-wire (left) and in the N/S metal (right). (From Falko et al. cond-mat/9901051.)


The interface resistance is determined by the following equation:

ri ¼R

LLs

s2

1 s2þ l"

1 2 3=22ð1þ sÞ

!ð99Þ

where R is the total resistance of the F-wire in the absence of the SF-junction.Both ratios Ls=L and l"=L are small, but this smallness can be compensatedeither by almost complete spin polarization of the currents s 1ð Þ or by thelarge ratio of numbers of majority and minority carriers 1

1. The most impor-tant and experimentally verifiable result of this work is the change of the totalresistance of the F-wire in transition from the normal to superconducting state inthe S-wire:

RsðT Þ RN

RN

¼s2

1 s2Ls

Ltanh

ðTÞ

2T: ð100Þ

It is always positive since the superconductivity enhances the reflection. It becomesinfinite at s¼ 1. The influence of spin-filtering on the subgap I(V ) characteristics ofFS-junctions has been studied by McCann and Fal’ko [120] and by Tkachov,McCann, and Fal’ko [121].

4.3. Non-monotonic behaviour of the transition temperature

This effect was first predicted by Radovic et al. [65]. Its reason is the LOFF oscilla-tions described in Section 4.1. If the transparency of the S/F-interface is low, one canexpect that the order parameter in the superconductor is not strongly influenced bythe ferromagnet. On the other hand, the condensate wavefunction at the interface inthe F-layer F / ðcos 2df =lmÞ

2 becomes zero at df ¼ plmðnþ 1=2Þ=2 (n is an integer).At these values of thickness the discontinuity of the order parameter at the boundaryand, together with it, the current of Cooper pairs into the ferromagnet, has amaximum. Therefore one can expect that the transition temperature is minimal[123]. Experimental attempts to observe this effect were made many times withS/F-multilayers Nb/Gd [124, 125], Nb/Fe [126], V/V-Fe [127], and V/Fe [128].More references and details about these experiments and their theoretical descriptioncan be found in the cited reviews [61, 62]. Unfortunately, in these experiments themagnetic component was a strong ferromagnet and, therefore, they faced all thedifficulties mentioned in Section 3.3.1: the F-layer must be a few angstroms thinto be comparable with the magnetic length and its variation produces uncontrollablechanges in the sample, and the influence of the growth defects is too strong.Furthermore, in multilayers the reason of the non-monotonic dependence of Tc ondf may be the 0–p transition. Therefore, a reliable experiment should be performedwith a bilayer possessing a sufficiently thick F-layer. Such experiments have beenperformed recently [129, 130]. The idea was to use a weak ferromagnet (the diluteferromagnetic alloy Cu–Ni) with rather small exchange field h to increase the mag-netic length lm ¼

ffiffiffiffiffiffiffiffiffiffiffiDf =h

p. The experiments were performed with S/F-bilayers to be

sure that the non-monotonic behaviour does not originate from the 0–p transition.In these experiments the transparency of the interface was not too small or too large,the exchange field was of the same order as the temperature and the thickness of theF-layer was of the same order of magnitude as the magnetic length. Therefore, forthe quantitative description of the experiment, theory should not be restricted by


limiting cases only. Such a theory was developed by Fominov et al. [122]. In thepioneering work by Radovic et al. [65] the exchange field was assumed to be verystrong.

At critical temperature, the energy gap and anomalous Green function F areinfinitely small. Therefore one needs to solve the linearized equations of supercon-ductivity. The approach by Fominov et al. is based on solution of the linearizedUsadel equation and is valid in the diffusion limit sTc 1; f Tc 1; f h 1. Infact, this situation was realized in the cited experiments [129, 130]. The work byFominov et al. [122] covers numerous works by their predecessors [68, 112, 123, 131,132] clarifying and improving their methods. Therefore in the presentation of thissubsection we follow mainly the cited work [122] and briefly describe specific resultsof other works.

The starting point is the linearized Usadel equations for anomalous Greenfunctions Fs in the superconductor and Ff in the ferromagnet:

Ds

o2Fs

oz2 j!njFs þ ¼ 0; 0 < z < ds ð101Þ

Df

o2Ff

oz2 ðj!nj þ ih sgn!nÞFf ¼ 0; df < z < 0: ð102Þ

Thus, we accept a simplified model in which ¼ 0 in the ferromagnetic layerand h¼ 0 in the superconducting layer. The geometry is schematically shown infigure 22. Equations (102) and (101) must be complemented with the self-consistencyequation:

ðrÞ lnTcs

T¼ pT

Xn

ðrÞ

j!nj Fsð!n, rÞ

, ð103Þ

where Tcs is the bulk SC transition temperature, and with linearized boundaryconditions at the interface:

sdFs

dz¼ f

dFf

dzð104Þ

AfdFf

dz¼ Gb Fsð0Þ Ff ð0Þ

, ð105Þ

F S

df 0 ds

Figure 22. FS-bilayer. The F- and S-layers occupy the regions df < z < 0 and 0 < z < ds,respectively.


where s, f is the conductivity of the superconducting (ferromagnetic) layer in thenormal state, Gb is the conductance of the interface and A is the area of the interface.We assume that the normal derivative of the anomalous Green function is equal tozero at the interface with the vacuum:

dFf

dzjz¼df

¼dFs

dzjz¼ ds

¼ 0: ð106Þ

The condition of solvability of the linear equations (101)–(103) with the boundaryconditions (104)–(106) determines the value of transition temperature Tc for theF/S-bilayer.

The solution Ff ð!n, zÞ in the F-layer satisfying the boundary condition (106)reads:

Ff ð!n, zÞ ¼ Cð!nÞ cosh½kfnðzþ df Þ; kfn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij!nj þ ih sgn!n

Df

s, ð107Þ

where Cð!nÞ is the integration constant to be determined from the matching con-dition at the F/S-interface z¼ 0. From the two boundary conditions at z¼ 0 (104),(105) it is possible to eliminate Ff and dFf /dz and reduce the problem to finding thefunction Fs from equation (101) and the effective boundary condition at z¼ 0:

sdFs

dz¼

b þ Bf ð!nÞFs, ð108Þ

where s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDs=ð2pTcsÞ

p, ¼ f =s, b ¼ ðAf Þ=ðsGbÞ and Bf ð!nÞ ¼

ðkfns tanhðkfndf ÞÞ1. Since kfn is complex the parameter Bf ð!nÞ and consequently

the function Fs is complex. The coefficients of the Usadel equation (101) are real.Therefore, it is possible to solve it for the real part of the function Fs traditionallydenoted as F þ

s ð!n, zÞ 1=2ðFsð!n, zÞ þ Fsð!n, zÞÞ. The boundary condition for thisfunction reads:

sdF þ

s

dz¼ Wð!nÞF

þs jz¼0; Wð!nÞ ¼

Asnðb þ<Bf Þ þ

Asnjb þ Bf j2 þ ðb þ <Bf Þ

, ð109Þ

where Asn ¼ ksnds tanhðksndsÞ and ksn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDs=j!nj

p. To derive this boundary condition

we accept the function (z) to be real (this will be justified later). Then the imaginarypart of the anomalous Green function F

s ð!n, zÞ obeys the homogeneous lineardifferential equation

d2Fs

dz2¼ k2snF

s

and the boundary condition dF s =dz ¼ 0 at z¼ ds. Its solution is F

s ð!n, zÞ ¼Eð!nÞ cosh½ksnðz dsÞ. At the interface z¼ 0 its derivative dF

s =dzjz¼0 is equalto ksn tanhðksndsÞF

s ð!n, z ¼ 0Þ. Eliminating F

s and its derivative from the realand imaginary parts of the boundary condition (108), we arrive at the boundarycondition (109). Note that only F þ

s participates in the self-consistence equation (103).This fact serves as justification of our assumption on the reality of the orderparameter (z).

Simple analytic solutions of the problem are available for different limitingcases. Though these cases are unrealistic at the current state of experimental art,they help to understand the properties of the solutions and how they change when


parameters vary. Let us consider the case of a very thin S-layer ds s. In this casethe order parameter is almost a constant. The solution of equation (101) in such asituation is

F þs ð!n, zÞ ¼

j!njþ n cosh ksnðz dsÞ,

where n is an integration constant. From the boundary condition (109) we find:

n ¼ 2Wð!nÞ

j!nj Asn þWð!nÞð Þ, ð110Þ

where the coefficients Asn are the same as in equation (109). We assume thatksnds 1. Then Asn k2snsds ¼ ðds=sÞðnþ 1=2Þ. The function F þ

s ð!n, zÞ almostdoes not depend on z. The self-consistence equation reads:

lnTcs

Tc

¼ 2Xn0

Wð!nÞ

ðnþ 1=2Þ ds=sðnþ 1=2Þ þWð!nÞð Þ: ð111Þ

The summation can be performed explicitly in terms of digamma functions (F):

lnTcs

Tc

¼s

2ðb þ <Bf Þds

< 1iðb þ <Bf Þ

=Bf

F

1

2þsds

1i=Bf

b þ<Bf

F

1

2

: ð112Þ

Possible oscillations are associated with the coefficients Bf. If the magnetic lengthlm ¼


pis much less than df, then

Bf

ffiffiffiffiffiffiffiffiffiffiffiffih

4pTcs

sexp

2idf

lmip4

:

In the opposite limiting case lm df there are no oscillations of the transitiontemperature. Note that lnðTcs=TcÞ can be rather large, s=ds, i.e the transitiontemperature in the F/S-bilayer with a very thin S-layer can be exponentiallysuppressed. This tendency is reduced if the resistance of the interface is large(b 1).

In a more realistic situation considered in the work [122] none of the parametersds=s, df =lm, , b are very small or very large and an exact method of solution shouldbe elaborated. The authors propose to separate explicitly the oscillating part of thefunctions F þ

s ð!n, zÞ and (z) and the reminders:

F þs ð!n, zÞ ¼ fn

cos qðz dsÞ

cos qdsþX1m¼1

fnmcosh qmðz dsÞ

cosh qmdsð113Þ

ðzÞ ¼ cos qðz dsÞ

cos qdsþX1m¼1

mcosh qmðz dsÞ

cosh qmdsð114Þ

where the wavevectors q and qm, as well as the coefficients of the expansion, must befound from the boundary conditions and self-consistency equation. Equation (101)


results in relations between the coefficients of the expansion:

fn ¼

j!nj þDsq2; fnm ¼

mj!nj Dsq

2m

: ð115Þ

Substituting the values of the coefficients fn, fnm from equation (115) into theboundary condition (109), we find an infinite system of homogeneous linearequations for the coefficients and m:

q tan qds Wð!nÞ

j!nj þDsq2

¼X1m¼1

mqm tanh qmds þWð!nÞ

j!nj Dsq2m

: ð116Þ

Equating the determinant of this system D to zero, we find a relation between qand qm. It is worth mentioning a popular approximation adopted by several theorists[68, 112, 123], the so-called single-mode approximation. In our terms it means thatall coefficients m,m ¼ 1, 2, . . ., are zero and only the coefficient survives. Thesystem (116) implies that this is only possible when the coefficients Wð!nÞ do notdepend on their argument !n. This happens indeed in the limit ds=s 1 and h T .For a more realistic regime the equation D ¼ 0 must be solved numerically togetherwith the self-consistency condition, which becomes the system of equations:

lnTcs

Tc

¼ F1

2þDsq

2

Tc

! F

1

2

lnTcs

Tc

¼ F1

2Dsq

2m

Tc

! F

1

2

: ð117Þ

This systems was truncated and solved with all data extracted from the experimentalset-up used by Ryazanov et al. [130]. The only two fitting parameters were h¼ 130Kand b ¼ 0:3.

Figure 23 demonstrates rather good agreement between theory and experiment.Various types of curves Tc(df) are shown in figure 24. Note that the minimum onthese curves eventually turns into a plateau at Tc¼ 0, the re-entrant phase transitioninto the superconducting state. Some of the curves have a well-pronounced dis-continuity, which can be treated as a first-order phase transition. The possibilityof a first-order transition to the superconducting state in the F/S-bilayer was firstindicated by Radovic et al. [65].

4.4. Josephson effect in S/F/S-junctions

As we already mentioned, the exchange field produces oscillations of the order param-eter inside the F-layer. This effect in turn can change the sign of the Josephson currentin the S/F/S-junction compared to the standard S/I/S- or S/N/S-junctions. As a resultthe relative phase of the S-layers in the ground state is equal to p (the so-called p-junction). In a closed superconducting loop with such a junction spontaneous mag-netic flux and spontaneous current appear in the ground state. These phenomena werefirst predicted by Bulaevsky et al. [56] for p-junctions independently of how it wasrealized. Buzdin et al. [57] first argued that such a situation can be realized in the S/F/S-junction for a proper choice of its length. Ryazanov et al. [58, 59] have realized a p-junction employing a weak ferromagnet CuxNi1x as a ferromagnetic layer. A similarapproach was developed a little later by Kontos et al. [60], who used a diluted PdNi


alloy. The weakness of the exchange field allowed them to drive the oscillations and inparticular the 0–p transition by temperature at a fixed magnetic field. The success ofthis experiment has generated an extended literature. Theoretical and experimentalstudies of this and related phenomena are still active. In what follows we present abrief description of relevant theoretical ideas and experiments.

4.4.1. Simplified approach and experiment. Here we present a simplified picture ofthe S/F/S-junction based on the following assumptions:

1. The transparency of the S/F-interfaces is small. Therefore the anomalousGreen function in the F-layer is small and it is possible to use the linearizedUsadel equation.

03

4

5

6

7

5 10 15 20

Figure 23. Theoretical fit to the experimental data [130]. In the experiment, Nb was thesuperconductor (with ds¼ 11 nm, Tcs ¼ 7K) and Cu0.43Ni0.57 was the weak ferromagnet.The best fit corresponds to h 130K and b 0:3. (From Fominov et al. cond-mat/0202280.)

0.0 0.5

0.0

0.2

0.4

0.6

0.8

1.0

1.0 1.5 2.0 2.5

Figure 24. Characteristic types of Tc(df) dependence. The length-scale is ex ¼ffiffiffiffiffiffiffiffiffiffiffih=Df

p;

the exchange energy is h¼ 150K; other parameters are the same as in figure 23. There arethree types of curves: (1) non-monotonic decay to a finite Tc with a minimum (b ¼ 0:07–2);(2) re-entrant behaviour with the first-order transitions at small df (b ¼ 0:02, 0:05); (3) mono-tonic decay to Tc¼ 0 at finite df (b ¼ 0). (From Fominov et al. cond-mat/0202280.)


2. The energy gap inside each of the S-layers is constant and equal to 0ei’=2

(the sign relates to the left S-layer, þ to the right one).3. ¼ 0 in the F-layer and h¼ 0 in the S-layers.

The geometry of the system is shown in figure (22). From the second assumptionit follows that the anomalous Green function F is also constant within each of the

S-layers: F ¼ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij!nj

2 þ20

q. The linearized Usadel equation in the F-layer (102)

has the following general solution:

Fð!n, zÞ ¼ nekfnz þ ne

kfnz, ð118Þ

where

kfn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij!nj þ ih sgn!n

Df

sð119Þ

(compare equation 107). The boundary condition at the two interfaces followsfrom the second boundary condition of the previous section (105) in which Ff isneglected:

fdFf

dz¼ bFs: ð120Þ

The coefficients n and n are completely determined by the boundary conditions(120):

n ¼ Qn

cosðð’ ikfndf Þ=2Þ

sinhðkfndf Þð121Þ

n ¼ Qn

cosðð’þ ikfndf Þ=2Þ

sinhðkfndf Þð122Þ

where

Qn ¼0

bf kfn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij!nj

2 þ20

q :

Equation (34) for the electric current must be slightly modified to incorporate theexchange field h:

j ¼ iepTNð0ÞDXn

ð ~FF ooF F oo ~FFÞ, ð123Þ

where ~FFð!n, zÞ ¼ F ð!n, zÞ. Note that under this transformation the wavevectors

kfn remain invariant. After substitution of the solution (118) we find that j ¼ jc sin ’with the following expression for the critical current [58, 132, 133]:

jc ¼4pT2

0

eRNb

<X!n>0

ð!2n þ2

0Þkfndf sinhðkfndf Þ 1

" #, ð124Þ

where RN is the normal resistance of the ferromagnetic layer and b is a dimension-less parameter characterizing the ratio of the interface resistance to that of theF-layer. Kupriyanov and Lukichev [88] have found a relationship between b and


the barrier transmission coefficient Db() ( is the angle between the electron velocityand the normal to the interface):

b ¼2lf

3df

cos DbðÞ

1DbðÞ

: ð125Þ

As we explained earlier, the oscillations appear since kfn are complex values.If h 2pT !n, then kfn ð1þ iÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffih=2Df

pand oscillations are driven only by the

thickness. It was very important to use a weak ferromagnet with exchange field hcomparable to pT. Then the temperature also drives the oscillations. In the Cu–Nialloys used in the experiment [58] the Curie point Tm was between 20 and 50K.Nevertheless, the ratio h=pT was about 10 even for the lowest Tm. In this situationkfn does not depend on n for the large number of terms in the sum (124). This is thereason why the sum in total is a periodic function of df with period m ¼ p

ffiffiffiffiffiffiffiffiffiffiffiffiffi2Df =h

p.

The dependence on temperature is generally weak. However, if the thickness is closeto the value at which jc tends to zero at T¼ 0, the variation of temperature canchange the sign of jc.

In figure (25, bottom left) theoretical curves jc(T ) from the cited work [122] arecompared with the experimental data by Ryazanov et al. [58, 79]. The curves areplots of the modulus of jc versus T. Therefore, the change of sign of jc is seen as acusp on such a curve. The position of the cusp determines the temperature of transi-tion from the 0- to p-state of the junction. The change of sign is clearly seen on thecurve corresponding to df¼ 27 nm. The experimental S/F/S-junction is schematicallyshown in figure (25, top). The details of the experiment are described in the originalpaper [58] and in the reviews [79, 61]. No less impressive agreement between theoryand experiment is reached by Kontos et al. [60] (the theory was given by T. Kontos)(see figure 26).

Very good agreement with the same experiment was reached also inrecent theoretical work by Buzdin and Baladie [134] who solved the Eilenbergerequation.

Zyuzin et al. [135] found that in a dirty sample the amplitude of the Josephsoncurrent jc is a random value with an indefinite sign. They estimated the averagesquare fluctuations of this amplitude for the interval of the F-layer thicknesss < df <

ffiffiffiffiffiffiffiffiffiffiD=T

pas:

h j2c i ¼ A2sg

8pNð0ÞDf

4 Df

2p2Td2f

!2

ð126Þ

where A is the area of the interface, g is the conductance of a square and N(0) is theDOS in the F-layer. The fluctuations are significant when df becomes smaller thanthe diffusive thermal length

ffiffiffiffiffiffiffiffiffiffiffiffiDf =T

p.

4.4.2. Josephson effect in a clean system. Recently, Radovic et al. [136] consideredthe same effect in a clean S/F/S-trilayer. A similar, but somewhat different in detail,approach was developed by Halterman and Olives [137, 138]. The motivation for thisconsideration is the simplicity of the model and very clear representation of thesolution. Though in the existing experimental systems the oscillations are not dis-guised by impurity scattering, it is useful to have an idea of what maximal effect


could be reached and what role is played by the finite transparency of the interface.The authors employed the simplest version of the theory, the Bogolyubov–de Gennesequations:

HHuv

¼ E

uv

, ð127Þ

Figure 25. (Top) schematic cross-section of the sample. (Bottom) left: critical current Ic as afunction of temperature for Cu0.48Ni0.52 junctions with different F-layer thicknesses between23 and 27 nm as indicated; right: model calculations of the temperature dependence of thecritical current in an SFS-junction. (From Ryazanov et al. cond-mat/0008364.)


where means and the effective Hamiltonian reads:

HH ¼H0ðrÞ hðrÞ ðrÞ

ðrÞ H0 þ

hhðrÞ

, ð128Þ

H0ðrÞ ¼ hh2

2mr

2 þWðrÞ: ð129Þ

In the last equation is the chemical potential and W(r) is the barrier potential:

WðrÞ ¼ W ½ðzþ d=2Þ þ ðz d=2Þ: ð130Þ

The assumptions about the exchange field h(r) and the order parameter ðrÞ are thesame as in the previous subsection. We additionally assume that the left and rightS-layers are identical and semi-infinite, extending from z ¼ 1 to z ¼ d=2 andfrom z ¼ d=2 to z ¼ 1. Due to translational invariance in the (x , y)-plane thedependence of the solution on the lateral coordinates is a plane wave:

uv

¼ eikkrðzÞ: ð131Þ

There are eight fundamental solutions of these equations corresponding to the injec-tion of the quasi-particle or quasi-hole from the left or from the right with spin upor spin down. We will write one of them explicitly as 1(z), corresponding to theinjection of the quasi-particle from the right. In the superconducting area z < d=2

80

60

40

20

0

I cRn

(µV

)

140120100806040dF (Å)

T=1.5 K

-20

-10

0

10

20

I (m

A)

-3 -2 -1 0 1 2 3

V (mV)

ξ10

SIS

SIFS

0 π

Figure 26. Josephson coupling as a function of thickness of the PdNi layer (full circles). Thecritical current vanishes at dF ’ 65 A indicating the transition from 0- to p-coupling. The fullline is the best fit obtained from the theory. The insert shows typical I–V characteristics of twojunctions with (full circles), and without (empty circles) the PdNi layer. (From Kontos et al.cond-mat/0201104.)


we will see the incident quasi-particle wave with coefficient 1 and normal wavevectorkþ, the reflected quasi-particle with reflection coefficient bþ and normal wavevectorkþ; the reflected quasi-hole (Andreev reflection) with reflection coefficient a1 andwavevector k, where

ðkÞ2 ¼2m

hh2ðEF Þ2 k

2k,

EF is the Fermi energy and ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 jj2

p. Thus the solution 1(z) at z < d=2

reads:

1ðzÞ ¼ ðeikþz

þ b1eikþz

Þuei’=2

vei’=2

þ a1e

ikz vei’=2

uei’=2

ð132Þ

where u and v are the bulk Bogolyubov–Valatin coefficients: u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ =EÞ=2

p;

v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 =EÞ=2

p. In the F-layer d=2 < z < d=2 there appear transmitted and

reflected electrons and transmitted and reflected holes. Since according to ourassumption ¼ 0 in the F-layer, there is no mixing of the electron and hole. Withthis explanation we can write directly the solution 1(z) in the F-layer:

1ðzÞ ¼ ðC1eiqþ z þ C2e

iqþ zÞ10

þ ðC3e

iq z þ C4eiq zÞ

01

, ð133Þ

where q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m=hh2ðE f

F þ h E Þ k2k

q. Finally in the right S-layer z > d=2 only

the transmitted quasi-particle and quasi-hole propagate:

1ðzÞ ¼ c1eikþz uei’=2

vei’=2

þ d1e

ikz vei’=2

uei’=2

: ð134Þ

The value of all coefficients can be established by matching of solutions at theinterfaces:

d

2 0

¼

d

2þ 0

;

d

dzjd=2þ0

d

dzjd=20 ¼

2mW

hh1: ð135Þ

Other fundamental solutions can be found by symmetry relations:

a2ð’Þ ¼ a1ð’Þ; a3 ¼ a2; a4 ¼ a1; b3 ¼ b1; b4 ¼ b2, ð136Þ

where the index 2 relates to the hole incident from the left, and indices 3, 4 relate tothe electron and hole incident from the right. Each mode generates the currentindependently of the others. The critical current reads:

jc ¼ ieT

hh

X,!n, kk

kþn þ kn2n

a1nkþn

a2nkn

: ð137Þ

Here all the values with the index n mean functions of energy E denoted by the samesymbols in which E is replaced by i!n, for example n ¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2n þ2

p. We will not

demonstrate here the straightforward, but somewhat cumbersome calculations andtransit to the conclusions. The critical current displays oscillations originating fromtwo different types of bound states. One of them appears if the barrier transmissioncoefficient is small. This is geometrical resonance. The superconductivity is irrelevantfor it. Another one appears even in the case of ideal transmission: this is the reso-nance due to the Andreev reflection. When the transmission coefficient is not smalland not close to 1, it is not easy to separate these two type of resonances and the


oscillations picture becomes rather chaotic. The LOFF oscillations are better seenwhen the transmission coefficient is close to 1 since geometrical resonances do notinterfere. Varying the thickness, one observes periodic transitions from the 0- top-state with the period equal to f =2 ¼ 2pvF=h. The lowest value of d at which the0–p transition takes place is approximately f =4. The temperature changes thispicture only slightly, but near the thickness corresponding to the 0–p transitionthe non-monotonic behaviour of jc versus temperature, including the temperaturedriven 0–p transition, can be found.

An intermediate case between the diffusion and clean limits was considered byBergeret et al. [139]. They assumed that the F-layer is so clean that hf 1, whereasTcs 1. Therefore the Usadel equation is not valid for the F-layer and they solvedthe Eilenberger equation. They have found that the superconducting condensateoscillates as a function of the thickness with period f and penetrates into theF-layer over the depth equal to the electron mean free path, lf. The period of oscilla-tion of the critical current is f =2. No qualitative differences with the consideredcases appear unless the magnetization is inhomogeneous. Even a very smallinhomogeneity can completely suppress the 0–p transitions. This is a consequenceof the generation of triplet pairing, which will be considered later.

An interesting opportunity for the Andreev bound states was indicated byGyorffy and coworkers [140–143]. They considered an I/S/F-trilayer, where I staysfor insulator. Their numerical studies as well as semiclassical solutions demonstratedthat in such a system there appear many Andreev bound states close to the middle ofthe energy gap. They argued that such a situation is unstable and any spontaneouscurrent will shift the levels from the centre of the gap decreasing the energy. Whilethe idea seems interesting, it is not clear how the current flowing partly in the F-layercan avoid losses.

4.4.3. Half-integer Shapiro steps at the 0–p transition. Recently Sellier et al. [144]reported the observation of additional half-integer values of Shapiro steps at thevoltage Vn ¼ nhh!=2e (! is the frequency of the applied ac current). Let us recall thatthe standard (integer) Shapiro steps appear as a consequence of the resonancebetween the external ac field and the time-dependent Josephson energyEJ ¼ ðhhjc=edf Þ cos ’ðtÞ where the phase is proportional to time due to the externalpermanent voltage through the contact: ’ðtÞ ¼ 2eVt=hh. Just at the 0–p transitionpoint jc tends to zero. Then the next term in the Fourier expansion of theJosephson energy proportional to cosð2’Þ dominates. That means that theJosephson current is proportional to sinð2’Þ. Such a term leads to the Shapirosteps not only at integer, but also at half-integer values, since the resonance nowhappens at ð4eV=hhÞ ¼ !. Normally the term with sinð2’Þ is so small that it wasalways assumed to vanish completely. The resonance hf method used by the authorshad sufficient sensitivity to discover this term.

The authors prepared the Nb=Cu52Ni48=Nb junction by the photolithographymethod. The Curie temperature of the F-layer is 20K. The two samples they usedhad thicknesses 17 and 19 nm. The 0–p transition was driven by temperature. Thetransition temperatures in the first and second sample were 1.12 and 5.36K, respec-tively. The external ac current had the frequency !¼ 800 kHz and amplitude about18 mA. The voltage current curves for df¼ 17 nm and temperatures close to 1.12Kare shown in figure 27. The fact that the half-integer steps disappear at very small


deviation from the transition temperature proves convincingly that it is associatedwith the 0–p transition.

4.4.4. Spontaneous current and flux in a closed loop. Bulaevsky et al. [56] arguedthat a closed loop containing the p-junction may carry a spontaneous current andflux in the ground state. We reproduce their arguments below. The energy of theclosed superconducting loop depends on the total flux through the loop:

EðÞ ¼ hh

2eJc cos ’þ

20’

2

8p2Lc2, ð138Þ

where ’ ¼ 2p=0, Jc is the critical current and L is the inductance of the loop. Thefirst term in equation (138) is the Josephson energy; the second is the energy of themagnetic field.

The location of the energy minimum depends on the parameter k ¼ 0=4pLJcc.At k > 1 the absolute minimum of Ec(’) is located at ’¼ 0. At k < 1, the point’¼ 0 corresponds to a maximum; the minimum is located at the first root of theequation sin ’=’ ¼ k. Thus, the spontaneous flux appears at sufficiently largeinductance of the loop. Ryazanov et al. [59] proposed a way to avoid this limitationby measuring the dependence of the current inside the loop on the external fluxthrough it.

They used a triangular bridge array with p-junctions in each shoulder (seefigure 28). Due to the central p-junction the phases of the current in two subloopsof the bridge differ by p. Therefore the critical current between the two contacts ofthe bridge is equal to zero in the absence of a magnetic field. If the flux inside theloop reaches half of the flux quantum, it compensates the indicated phase differenceand the currents from both subloops are in the same phase. Thus, the shift of thecurrent maximum from ¼ 0 to ¼ 0=2 is direct evidence of the 0–p transition.Such experimental evidence was first obtained in the same work [59].

The graphs of the current versus magnetic field for two different temperatures(figure 29) clearly demonstrate the shift of the current maximum from zero to non-zero magnetic field. Figure 30 shows the shift of the flux through the loop 0 to 1/2 ofthe flux quantum at the temperature driven 0–p transition.

0 10 20 30 40 50 60 700

1

2

3

Figure 27. Shapiro steps in the voltage–current curve of a 17 nm thick junction with anexcitation at 800 kHz (amplitude about 18 mA). Half-integer steps (n ¼ 1=2 and n ¼ 3=2)appear at the 0–p crossover temperature T *. Curves at 1.10 and 1.07K are shifted by 10and 20 mA for clarity. (From Sellier et al. cond-mat/0406236.)


Quite recently Bauer et al. [145] have discovered experimentally spontaneouscurrents and flux in p-junctions in agreement with theoretical predictions madeby Bulaevsky et al. many years ago [56] and presented at the beginning of thissubsection. The experimenters deposited a Nb square loop with average linewidth700 nm and the side of the loop 7.4 mm over a GaAs–AlGaAs heterostructure. Thelatter served as a micro-Hall sensor with a high resolution of the magnetic flux due tohigh mobility of the two-dimensional electron gas in it (750 000 cm/Vs at electrondensity 2:66 1011 cm2). The Josephson junction inside the Nb loop was madefrom a weak ferromagnet Pd0:82Ni0:18. To avoid flux trapping they cooled the systemin a weak magnetic field 1:7 mT down from 10K. Such a loop had sufficiently largeinductance to ensure the appearance of spontaneous current. One flux quantum insuch a loop corresponded to the field 37.5 mT. For control they compared all

Figure 28. Real (upper) and schematic (lower) picture of the network of five SFS-junctionsNb–Cu0.46Ni0.54–Nb (dF ¼ 19 nm), which was used in the phase-sensitive experiment.(From Ryazanov et al. cond-mat/0103240.)


Figure 29. Magnetic field dependences of the critical transport current for the structuredepicted in figure 28 at temperature above (a) and below (b) Tcr. (From Ryazanov et al.cond-mat/0103240.)

Figure 30. (a) Temperature dependence of the critical transport current for the structuredepicted in figure 28 in the absence of a magnetic field; (b) temperature dependence (jump) ofthe position of the maximal peak on the curves ImðHÞ, corresponding to the two limitingtemperatures depicted in figure 29. (From Ryazanov et al. cond-mat/0103240.)


measurements made with this loop to the same measurements made with the normal,non-magnetic junction. Figures 31(a) and (b) display the signal (Hall voltage) excitedby the loops in the sensor versus the external magnetic field for the ferromagnetic (p)and non-magnetic (0) junctions. It is clearly seen that at zero applied field thevoltage, i.e. the flux through the p-junction, is not zero, whereas it remains zerofor the 0-junction (the small displacement is due to the cooling field). Figure 31(c)shows the same result recalculated in terms of the magnetic flux through the loops.Different curves for sweeping the field up and down (figures 31 a, b) imply hysteresisloops in the graph (figure 31c). A non-zero flux in the p-junction is explicitly seen inthis graph. Figure 31(d) demonstrates the Hall voltage in the sensor at the cooling(background) field 19.2 mT close to the half-quantum through the loop. In this casethe signal from the p-loop is close to zero, whereas the signal from the 0-loop isabout half of its maximal value. They also measured the temperature dependence ofthe same Hall voltage or the magnetic flux in the loop at zero external field andat the field corresponding to the flux quantum through the loop (figures 32 a, b).

b

δV(n

V)

H

" "π

"0"

B ( T)applied µ

down sweep up sweep

δV(n

V)

H

B ( T)applied µ

" "π

"0"

down sweep up sweepa

down sweep up sweep

" "π

"0"

B ( T)applied µ

d

δ V(n

V)

H

c

0

Φapplied

Φ

" "π0

Φ0

-Φ0

0

Φ0

Φ0

−Φ0

−Φ0

"0"

B =19,2 Tcool µB =1,72 Tcool µ

B =-1,72 Tcool µ

Figure 31. Signature of the p-state in the switching behaviour during magnetic field sweeps.(a), (b) Magnetization traces VH IS for a p- and a 0-loop. All data are taken at 2K usinga measurement current of 20 mA. The critical currents are 65mA for the 0-junction and 27 mAfor the p-junction. By choosing cooling fields slightly below and above zero applied flux andperforming sweeps of the magnetic fields in both field directions, an asymmetric switchingbehaviour of the p-loop is observed, while the 0-loop shows symmetric jumps with respect tothe cooling field. (c) Relation between the total magnetic flux , applied flux applied (linearbackground), and the flux LIS generated by the supercurrent in the loop. The ðapplied )relation of a p-loop is shifted by 0=2 with respect to that of a conventional 0-loop.During the field sweep, only the sections with positive slope can be traced, which leadsto a hysteretic switching behaviour for LIC > 0=2. The three vertical lines in each curveindicate the applied flux during field cooling below the critical temperature for the threesets of measurements discussed. (d) Reversal of the symmetry properties of the p- and0-loop when choosing a cooling field equal to half a flux quantum. (From Bauer et al.cond-mat/0312165.)


Figure 32(a) demonstrates that the 0-loop at zero external field retains zero fluxbelow the transition temperature, whereas the p-loop repels the magnetic flux, i.e.it develops a spontaneous current. At external flux equal to half flux quantum, the0-loop develops a screening current, whereas the p-loop retains zero current.

4.5. F/S/F-junctions

The F/N/F-trilayers (N is normal non-magnetic metal) have attracted muchattention starting from the discovery by Baibich et al. [146] of giant magnetoresis-tance (GMR). The direction of magnetization of ferromagnetic layers in these sys-tems may be either parallel or antiparallel in the ground state oscillating with thethickness of the normal layer on the scale of a few nanometres. The mutual orienta-tion can be changed from antiparallel to parallel by a rather weak magnetic field.Simultaneously the resistance changes by the relative value reaching 50%. Thisphenomenon has already obtained a technological application in the magnetictransistors and valves used in computers [147]. A natural question is what happensif the central layer is superconducting: will it produce the spin-valve effect (a pre-ferential mutual orientation of F-layer magnetization) and how does it depend on thethicknesses of the S- and F-layers? This question was considered theoretically by

π

0

T (K)

ΦΦ(

0)Φ

Φ(0)

V(n

V)

HV

(nV

)H

π

0

(a)

(b)

0,25

0

-0,25

-0,50

0,25

0

-0,25

-0,50

0 2 4 6 8 10

0 2 4 6 8 10

0

0

12,6

-12,6

-25,1

9,5

-9,5

-18,9

Φapplied=0

Φapplied 0= /2Φ

Figure 32. Temperature dependence of the spontaneous current. The measurements showthe temperature-dependent magnetic flux produced by a p- and a 0-loop when cooling in(a) zero field and in (b) a magnetic field equal to half a flux quantum in the loop. BelowT 5.5K the p-loop develops a spontaneous current when cooling down in zero field, whilethe magnetic flux through the 0-loop remains zero. When applying a field equal to half a fluxquantum 0 the roles of the p- and 0-loop are exchanged. At low temperatures the sponta-neous flux of both loops saturates close to 0=2, as expected. To avoid heating of the sensor atthe lowest temperatures, the measurement current has been reduced to 7 mA. (From Bauer et al.cond-mat/0312165.)


several authors [66–69, 89, 148]. The spin-valve effect in F/S/F-junctions was experi-mentally discovered by Tagirov et al. [151].

Even without calculations it is clear that, independently of the thicknesses of theS- and F-layers, the antiparallel orientation of magnetizations in the F-layers alwayshas lower energy than the parallel orientation. This happens because the exchangefield always suppresses superconductivity. When the fields from different layers areparallel, they enhance this effect and increase the energy, and vice versa. The effectstrongly depends on the transparency of the interfaces. Usually transparency is verysmall, and the effect is so weak that its experimental observation is rather difficult.Even in the case of almost ideally transparent interfaces, the majority electrons withthe preferential spin orientation cannot penetrate from the F-layer to the S-layer deeper than the coherence length, s. Therefore, it is reasonable to work withthe S-layer whose thickness does not exceed s. The choice of the material andthickness of F-layers is dictated by the requirement that they could be reorientedby a sufficiently weak magnetic field. Thus, the coercive force must be small enough.We refer the reader to the original works for quantitative details.

Another value worth studying is the critical temperature of the S-layer at a fixedmutual orientation of magnetic moments in the F/S/F-trilayer. A theoretical studywas performed by Baladie and Buzdin [148] for the case of a very thin superconduct-ing layer ds s. They considered Fs almost as a constant, but incorporated smalllinear and quadratic deviations and solved the linearized Usadel equation as shownin Section 4.3 to find the critical temperature versus thickness of the ferromagneticlayers. They found that at large b (low interface transparency) the transitiontemperature monotonically decreases with df increasing from its value in the absenceof the F-layers to some saturation value. There occurred no substantial differencebetween parallel and antiparallel orientations. At smaller values of b the sup-pression of Tc increases and at parallel orientation the re-entrant transition occursat df f , but still the transition temperature saturates at large df. For b smallerthan a critical value the transition temperature becomes zero at a finite thickness dffor both parallel and antiparallel orientations. The authors also found some evidencethat at low b the S-transition becomes discontinuous for the parallel orientation.This conclusion was confirmed by a recent theoretical study by Tollis [152], whoproved that the S-transition for the antiparallel orientation is always of second order,whereas for the parallel orientation it becomes of first order for small b [152].Baladie and Buzdin [148] considered also the energy gap at low temperature. Forthe case of thick ferromagnetic layers df f they found that the energy gap is amonotonically decreasing function of the dimensionless collision frequency ðf0Þ

1,where 0 is the value of the energy gap in the absence of the ferromagnetic layers.It tends to zero at ðf0Þ

1¼ 0:25 for the parallel and 0.175 for the antiparallel

orientation.

4.6. Triplet pairing

If the direction of the magnetization in the F-layer varies due to a domain wall orartificially, the singlet Cooper pairs penetrating into the F- from S-layers will be partlytransformed into triplet pairs. This effect was first predicted by Kadigrobov et al. [70]and by Bergeret et al. [71]. The singlet pairs cannot penetrate into the F-layer deeperthan to the magnetic length lm ¼


p(or vF / h for a clean ferromagnet), but

neither exchange interaction nor the elastic scattering suppresses the triplet pairs.


Therefore, they can penetrate to a much longer distance T ¼ffiffiffiffiffiffiffiffiffiffiffiffiDf =T

p. Even if the

triplet pairing is weak, it provides the long-range coupling between two supercon-ducting layers in an S/F/S-junction. Moreover, if the thickness df exceeds lm signifi-cantly, only triplet pairs survive at distances much larger than lm, completelychanging the symmetry properties of the superconducting condensate. Earlier theidea that the helical magnetic order in bulk can transform the singlet pairing intotriplet pairing was formulated by Bulaevsky et al. [149, 150].

The exchange field rotating in the (y, z)-plane is naturally described by theoperator hh ¼ hð3 cosþ 2 sin Þ in spin space, where h and are scalar functionsof the coordinates, 2 and 3 are the Pauli matrices. It is clear that the non-diagonalpart of hh flips one of the spins of the pair, transforming the singlet into the triplet. Itdoes not appear if the magnetization is collinear (¼ 0). To make things moreexplicit, let us consider the Usadel equation in the F-layer, i.e. equation (38) ofSection 2. First we simplify it by linearization, which is valid if the transparencyof the interface barrier is small [86]. Then the condensate Green tensor ff in theF-layer is small. The linearized Usadel equation reads:

Df

2

o2 ff

oz2 j!j ff þ ih 0f3, ff cosþ 3½2, ff sin g

h i¼ 0, ð139Þ

where fA,Bg means the anticommutator of the operators A and B. If ¼ const,equations (139) have an exponential solution ff ¼ ekz ff0. The secular equation for k is:

ðk2 k2!Þ2ðk2 k2!Þ

2þ

2h

Df

¼ 0, ð140Þ

where k2! ¼ 2j!j=Df . Note that the secular equation does not depend on . It is aconsequence of the rotational invariance of the exchange interaction. For ¼ 0 thetwo-fold eigenvalue k2 ¼ k2! corresponds to f1, 2 (triplet pairing with projection 1onto the magnetic field). Since !n is proportional to T, these modes have a long-range character. Two other modes have wavevectors k¼ kh and k ¼ kh, wherek2h ¼ 2ðj!j þ ih sgn!Þ=Df . They penetrate no deeper than the magnetic length.These short-range modes are linear combinations of the singlet and triplet withspin projection zero, i.e. orthogonal to the magnetic field.

Bergeret et al. considered two different geometries. In the first one [71] theyconsidered an S/F-bilayer. The angle was a linear function of the coordinatesstarting from 0 at the S/F-interface, reaching a value w at the distance w fromthe interface and remaining constant at larger distances. They solved the linearizedUsadel equation (139) with boundary condition f dFf =dz ¼ bFs valid at small

transparency of the interface by a clever unitary transformation ff ! UUðzÞ ff ½UUðzÞ1

with UUðzÞ ¼ expðiQ1z=2Þ and Q ¼ d=dz ¼ w=w. This transformation turns therotating magnetic field into a constant field, directed along the z-axis, but the differ-ential term generates perturbations proportional to Q and Q2. By this trick the initialequations with the coordinate-dependent hhðzÞ are transformed into an ordinarydifferential equation with constant (operator) coefficients. The generation of thetriplet component is weak if b is large and it acquires an additional small factorif the ratio f =w is small (w mimics the domain wall width), but, as we demonstrated,this component has a large penetration depth. Experimentally it could produce astrong enhancement of the F-layer conductivity. Such an enhancement was observedin the experiment by Petrashov et al. [153] in 1999, two years afore the theoretical


work. They studied an F/S-bilayer made from 40 nm thick Ni and 55 nm thick Alfilms. The interface was about 100 100 nm2. The samples were prepared by e-beamlithography. They measured the resistivity and the barrier resistance directly. Theyalso found the diffusion coefficients Ds¼ 100 cm2/s and Df¼ 10 cm2/s, which wecite here to give an idea of the order of magnitudes. They claimed that the largedrop in the resistance of the sample cannot be explained by the existing singletpairing mechanism. We are not aware of detailed comparison of the theory [71]and experiment [153]. Dubonos et al. [155] argued that the long-range effectsobserved by Petrashov et al. [153] may be caused by the stray fields of the domainwalls in the F-layer rather than the triplet pairing. They studied a thin Ni diskdeposited upon Al disks. The two disks were surmounted upon two of five contactsin a Hall bridge made from a GaAs–AlGaAs heterostructure with high mobilitydestined to measure local magnetic fields. On the third contact they deposited onlyan Al disk for control. They observed at zero field cooling sometimes (not always)spontaneous magnetization of the Ni disk at T < Tc 1:25K. Statistical analysis ofthese random fields shows that the average spontaneous field modulus behaves as thesquare of the order parameter. The hysteresis loop shows a remanent (diamagnetic)moment in the bilayer, but no hysteresis in the Al disk itself. This shows that some-how the presence of the F-disk influences the motion of the vortices. They also havefound a non-local effect, studying a long Ni wire, which had a small contact with anS-disk. They measured the field at a distance of about 1.5 mm from the contact anddiscovered spontaneous fields there. These could appear as a result of domain wallmotions. They assumed that the vortices do not appear since the magnetization M innickel is about 500G, too small to create vortices in Al (Hc ¼ 0=pd

2¼ 1 kG).

One more piece of indirect evidence of long-range penetration of the supercon-ducting order parameter through the ferromagnet was reported in [154]. The authorsmeasured the resistance of a 0.5 mmNi loop connected with superconducting Al wire.They extracted the decay length for the proximity effect in the ferromagnet fromdifferential resistance and concluded that it is much larger than could be expected forsinglet pairing.

In their second publication on triplet pairing [86] the authors proposedan interesting six-layer structure presented in figure 33. They assumed that themagnetization in each layer is constant, but its direction is different in differentlayers. It is assumed to stay in the (y, z)-plane and thus it can be characterized by

F2F1S0 SA SB F3

Figure 33. Six-layer structure.


one angle. Let this angle be in the layer F1, 0 in the layer F2, and in the layerF3. They speak about the positive chirality if the sign is þ and negative chirality if thesign is . They prove that, if the thickness of the F-layers is larger than lm, thesuperconducting layers SA and SB are connected by an 0-junction if the chirality ispositive and by a p-junction if the chirality is negative. This phenomenon is com-pletely due to the triplet pairing since it dominates at this distance. Kulic and Kulic[158] considered two bulk magnetic superconductors with rotating magnetizationseparated by an insulating layer. They also found that the sign of the Josephsoncurrent can be negative depending on the relative chirality. In this system singlet andtriplet pairs coexist in the bulk, whereas in the system proposed by Bergeret et al. thetriplet dominates. We will give a brief description of how they derived their results.They solved the Usadel equation in each layer separately (this can be done withoutlinearization, since the coefficients of the differential equations are constant) andmatched these solutions using the Kupriyanov–Lukichev boundary conditions.The current density in the F2-layer can be calculated using the modifiedEilenberger–Usadel expression:

j ¼ fTr 3300pTX!n

ffd ff

dz

!: ð141Þ

The maximum effect is reached when the magnetic moment of the central layer isperpendicular to the other two.

A modification of the triplet pairing idea was proposed by Eshrig et al. [156].Instead of rotating magnetization they consider the F/S-interface which completelyreflects electrons into the F-layer, but the reflection coefficient is spin-sensitive. Itfollows from arguments developed earlier (see Section 4.2) that the F/S-interfacereflects electrons completely when the ferromagnet carriers are fully polarized. Thefull polarization is a property of the so-called half-metals, such as La0.7Sr0.3MnO3 orCrO2. At complete reflection the reflection amplitude is a phase factor. Without lossof generality one can assume that the reflection phases for opposite projections ofspin have different sign. Thus, at any reflection the states are transformed as follows:jkz "i ! ei=2j kz #i; jkz #i ! ei=2

j kz "i. The singlet pair wavefunction in thesuperconductor jsi ¼ ð1=

ffiffiffi2

pÞ½jk " , k #i jk # , k "i after reflection will be

transformed into cos jsi þ sin jti where jti denotes the triplet pair state jti ¼ð1=

ffiffiffi2

pÞ jk " , k #i þ jk # , k "i½ . The triplet pair can penetrate in the half-

metal if the spin-flip processes happen near the interface. As soon as they appearin the F-layer they propagate to long distances and become responsible for theJosephson coupling in the SFS-junction.

Quantitative theory is considered for a clean ferromagnet and is based on theEilenberger equation. Since the number of states in the S- and F-layers are different(no spin-down channels in F-layers), the boundary conditions must be modified[157]. The authors introduce two surface scattering matrices, SS and SS, and twoGreen functions, gg0 and gg0 for impenetrable, but active interfaces on either side.The connection between the in and out propagators reads: gg0out ¼ SSgg0inSS

y,gg0out

¼ SS gg 0in SS

y. Transmission is described by hopping amplitudes. The scatteringgeometry is depicted in figure 34. The tunnelling process with spin-flips wasdescribed by the matrix . The full propagators gg and gg are expressed in terms of


decoupled propagators as follows:

ggin ¼ gg0in þ ðgg0in þ iII Þðgg0in iII Þ ð142Þ

ggout ¼ gg0out þ ðgg0out þ iII Þðgg0out iII Þ ð143Þ

with analogous equations for gg. The Eilenberger equation without the collision termwith these boundary conditions was solved for a heterostructure consisting of a half-metal LH<x<LH between two superconductors LH < jxj < L. The modulus ofthe gap at both interfaces is assumed to be identical, but the phase difference isintroduced as a boundary condition so that ðLÞ ¼ ðLÞei’. The spin rotationby the interface can be described by the scattering matrix SS ¼ expðiz=2Þ at theS-side of interface. Generally depends on the impact angle . For numericalcalculations they assumed ¼ 0:75 cos . On the half-metal side of the interfaceSS ¼ 1. In this case the tunnelling matrix is a 2 1 matrix ¼ ð1þ SSyÞ0 cos ,where 0 ¼ ð""#"

Þ determines tunnelling without and with spin-flop. Spin rotation attransmission is half of spin rotation at reflection. For numerics the authors accepted#"="" ¼ 0:7 and 0.1 and 2LH ¼ 30, L LH . The triplet order parameter wasdefined by the following integral:

FtðxÞ ¼

ð"c"c

d"

2pihðpÞf ðp, x, "Þip tanh

"

2T, ð144Þ

where ðpÞ is the projector onto the p-state and averaging proceeds over the Fermisurface. The s- and t-order parameters versus x are plotted in figure 35. The upperpanel corresponds to the phase ’ ¼ p, the lower one corresponds to ’¼ 0. For asufficiently long F-layer the triplet pairing dominates. Numerical calculationsshowed that the phase p is energetically favourable under the conditions assumed.Note that the triplet order parameter is an even function of coordinates at ’ ¼ p andan odd function at ’¼ 0. The singlet order parameter behaves in the opposite way.The current through the junction can be expressed in the standard way. The numer-ical results with the triplet pairing dominating show that the current is equal to zeroat ’¼ 0 and ’ ¼ p, but is negative at any intermediate value of the phase. Thus, theS/half-metal/S junction is of p-type. It is sensitive to the value of the spin-floptransmission coefficient as seen in figure 36.

Figure 34. Scattering geometry illustrating the scattering channels and the correspondingtransfer amplitudes for the model discussed in the text. (From Cuevas et al. [157].)


5. Conclusions

This review shows that, though studies of ferromagnet–superconductor hybrids arecoming of age, we are still at the beginning of an interesting voyage into thisemerging field. The most active development is undoubtedly taking place in the

Figure 35. Self-consistent order parameter and triplet correlations in an S/HM/S hetero-structure for a 0-junction and a p-junction. The relative signs of the pairing correlations inthe s-wave singlet and three p-wave triplet channels are indicated. A 0-junction for the singletorder parameter leads to a relative phase difference of p for the triplet correlations, and viceversa. The calculations are for temperature T ¼ 0:05Tc, and for "=# ¼ 0:7. (From Cuevaset al. [157].)

Figure 36. Critical Josephson current density as a function of temperature for an S/HM/Sheterostructure. The two curves are for "=# ¼ 0:1 (dashed line) and for "=# ¼ 0:7 (full line).The inset shows the current–phase relationships for "=# ¼ 0:7 for temperatures T=Tc ¼ 0:05(dashed line), 0.2, 0.3, 0.4, and 0.5 (full lines from bottom to top). The unit is the Landaucritical current density JL ¼ evf Nf0, with the zero temperature bulk superconducting gap0 ¼ 1:76Tc. (From Cuevas et al. [157].)


field of proximity based phenomena in layered ferromagnet–superconductorsystems. The strong point of this thrust is fruitful collaboration between experimentand theory. Significant progress has been achieved due to a new idea proposed byRyazanov and coworkers to use weak ferromagnets in experiments. This ideaallowed them to increase the thickness of ferromagnetic layers to a macroscopicscale and simultaneously to drive non-monotonic behaviour of the Josephsoncurrent by temperature. In this way experimenters reliably found several interestingphenomena predicted many years ago, such as the 0–p transition and oscillationsof critical temperature versus the thickness of the F-layer and also some newphenomena such as the valve effect in F/S/F-junctions and the Shapiro steps athalf-integer frequencies.

The experimental studies of ordering/transport in FSH have greatly benefitedfrom the introduction of imaging techniques (SHPM, MFM) in the field. We expectthat several experimental groups will have access to this technique in the near future,which will result in more exciting experiments. Theoretical and experimental studiesof ordering and transport phenomena in the FSH have surprisingly little overlap,especially in comparison with studies of proximity based phenomena. The materialsused in experiment are far from being regular, whereas theorists so far prefer simpleproblems with regular, homogeneous or periodic systems. Even the simplest ideaabout topological instability in the S/F-bilayer has not been checked experimentally.It would be very instructive to find experimentally the phase diagram of a singlemagnetic dot using a SQUID magnetometer or the MFM. Finally the transportproperties of the S/F-bilayer and the S-films supplied with regular or randomlymagnetized arrays of F-dots should be measured. On the other hand, experimentdictates new problems for theory: a description of a random set of strongly pinneddomain walls, their magnetic field and its effect on S-films. We think that both theexperimental and theoretical communities can find systems of common interest. Weexpect another possibility for an interesting development in the FSH field with theintroduction of new types of FSH, e.g. arrays of magnetic nanowires in aluminatemplates, covered with superconducting film. Such arrays generate non-uniformmagnetic fields of high strength with short scale variation.

Acknowledgements

The authors acknowledge the support by NSF under grants DMR-0103455and DMR-0321572, by DOE under grant DE-FG03-96ER45598, by Telecommuni-cations and Informatics Task Force at Texas A&M University and by DeutscheForschungsgemeinschaft. I.L. acknowledge the support by the George P. andCynthia W. Mitchell Institute for Fundamental Physics. V.P. acknowledges thesupport from the Humboldt Foundation, Germany. He is indebted to theUniversity of Cologne and to Prof. T. Nattermann for the hospitality extended tohim during his stay at Cologne University. I.L. is grateful to Prof. H. Pfnur, for thekind hospitality during his stay at Hannover University, where part of this work wasdone. Our thanks are due to A. Buzdin, A. Volkov, Ya. Fominov, M. Feigel’man,M. Kulic and A. Golubov for useful discussions. We also are indebted to our refereesfor attentive reading of the manuscript, constructive criticism and proposals.


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Ferromagnet–superconductor hybridspeople.physics.tamu.edu/ilx/adv.pdf · Advances in Physics, Vol. 54, No. 1, January/February 2005, 67–136 ... We review a new avenue in solid