Ferromagnet superconductor hybrids

Transcription

1 Advances in Physics, Vol. 54, No. 1, January/February 2005, 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 the interaction between spatially separated, but closely located ferromagnets and superconductors, the so-called ferromagnet superconductor hybrids (FSH). Typical FSH are: coupled uniform and textured ferromagnetic and superconducting films, magnetic dots over a superconducting film, magnetic nanowires in a superconducting matrix, etc. The interaction is provided by the magnetic field generated by magnetic textures and supercurrents. The magnetic flux from magnetic structures or topological defects can pin vortices or create them, changing the transport properties and transition temperature of the superconductor. On the other hand, the magnetic field from supercurrents (vortices) strongly interacts with the magnetic subsystem, leading to formation of 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 suppresses the Cooper-pair wavefunction, but also leads to its oscillations, which in turn leads to oscillations of observable values: the transition temperature and Josephson current. In particular, in the ground state of the Josephson junction the relative phase of two superconductors separated by a layer of ferromagnetic metal is equal to p instead of the usual zero (the so-called p-junction). Such a junction carries a spontaneous supercurrent and possesses other unusual properties. Theory predicts that rotation of magnetization transforms s-pairing into p-pairing. The latter is not suppressed by the exchange field and serves as a carrier of long-range interaction between superconductors. Contents page 1. Introduction Basic equations Three-dimensional systems Two-dimensional systems The Eilenberger and Usadel equations Hybrids without proximity effect Magnetic dots Magnetic dot: perpendicular magnetization Magnetic dot: parallel magnetization Array of magnetic dots and superconducting film Vortex pinning by magnetic dots Vortex lattice symmetry versus magnetic dot array symmetry 86 *Corresponding author. ilx@physics.tamu.edu Advances in Physics ISSN print/issn online # 2005 Taylor & Francis Group Ltd DOI: /

2 68 I. F. Lyuksyutov and V. L. Pokrovsky Magnetic field induced superconductivity Magnetization controlled superconductivity Ferromagnet superconductor bilayer Topological instability in the FSB Superconducting transition temperature of the FSB Transport properties of the FSB Experimental studies of the FSB Thick films Domain wall superconductivity Proximity effects in layered ferromagnet superconductor systems Oscillations of the order parameter Resistance of the SF contact in wires and the spin-filtering effect Non-monotonic behaviour of the transition temperature Josephson effect in S/F/S-junctions Simplified approach and experiment Josephson effect in a clean system Half-integer Shapiro steps at the 0 p transition Spontaneous current and flux in a closed loop F/S/F-junctions Triplet pairing Conclusions 131 Acknowledgements 132 References Introduction We review a new avenue in solid state physics: studies of physical phenomena which appear when two mutually exclusive states of matter, superconductivity and ferromagnetism, are combined in a unified ferromagnet superconductor hybrid (FSH) system. In the hybrid systems fabricated from materials with different and even mutually exclusive properties, a strong mutual interaction between subsystems can dramatically change properties of the constituent materials. This approach offers vast opportunities for science and technology. The interplay of superconductivity and ferromagnetism has been thoroughly studied experimentally and theoretically [1, 2] for homogeneous systems. In such systems, both order parameters are homogeneous in space and suppress each other. As a result, one or both of the orderings are weak. A natural way to avoid the mutual suppression of the order parameter of the superconducting (S) and ferromagnetic (F) subsystems is to separate them by a thin but impenetrable insulator film. In such systems the S- and F-subsystems interact via a magnetic field induced by the non-uniform magnetization of the F-textures penetrating into the superconductor. If this field is strong enough, it can generate vortices in the superconductor. The textures can be either artificial (dots, wires) or topological like domain walls (DW). The inverse effect is also important: the S-currents generate a magnetic field interacting with the magnetization in the F-subsystem. Experimental work on FSH has been focused dominantly on pinning properties of magnetic dot arrays covered by a thin superconducting film [3 7]. The effect of commensurability on the transport properties was reported in [3 6]. This effect is not specific for magnets interacting with superconductors and was first observed in superconducting films in the 1970s. In these experiments the periodicity of the vortex lattice fixed by an external magnetic field competed with the periodicity of

3 Ferromagnet superconductor hybrids 69 an artificial array created by experimenters. Martinoli et al. [8 10] used grooves and Hebard et al. [11, 12] used arrays of holes. They observed well-pronounced minima of resistance or peaks of critical current at values of magnetic field corresponding to commensurate lattices. This approach was further developed by experimentalists in the 1990s [13 19]. Theoretical analysis was also performed in the last century [20 22]. Morgan and Ketterson [7] were the first to observe the change of resistivity induced by vortices pinned by magnetic dots during flipping of the external magnetic field orientation. This was the first direct indication of timereversal symmetry violation in FSH. New insight into the physics of FSH has been provided by the magnetic force microscope (MFM) and scanning Hall probe microscope (SHPM). By using such imaging technique the group at the University of Leuven has elucidated several pinning mechanisms in FSH [23 25]. Different mesoscopic magneto-superconducting systems have been proposed and studied theoretically: arrays of magnetic dots on the top of an S-film [26 29], ferromagnet superconductor bilayers (FSB) [28, 30 34], embedded magnetic nanowires combined with bulk superconductor [35, 36] or superconductor films [37, 38], a layer of magnetic dipoles between two bulk superconductors [39], an array of magnetic dipoles mimicking the F-dots on S-films [40], a giant magnetic dot which generates several vortices in bulk superconductor [41], a single magnetic dot on 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 these systems significantly exceeds the coherence length,. This means that they can be considered in the London approximation with good precision. In the next section we derive basic equations describing the FSH. Starting from the London Maxwell equations, we derive a variational principle (energy) containing only the values inside either S- or F-components. These equations allow us to study single magnetic dots 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- and S-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 a thick magnetic film on the bulk superconductor. An alternative approach to heterogeneous S/F-systems is just to employ the proximity effects instead of avoiding them. The exchange field existing in the ferromagnet splits the Fermi spheres for up and down spins. Thus, the Cooper pair acquires a non-zero total momentum and its wavefunction oscillates in space. This effect, 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 bound states of quasi-particles in the F-layer due to Andreev reflection [55] from the S/F-interface. One of their manifestations is the change of sign of the Cooper-pair tunnelling amplitude in space. Under certain conditions the Josephson current through a superconductor ferromagnet superconductor (S/F/S) junction has sign opposite to sin, where is the phase difference between the right and left superconducting layers. This type of junction was first proposed theoretically a long time ago by Bulaevsky et al. [56, 57] and was called a p-junction in contrast to the standard or 0-junction. It was first reliably observed in the experiment by Ryazanov 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

4 70 I. F. Lyuksyutov and V. L. Pokrovsky large review on this topic was published at the beginning of 2002 [61]. A more specialized survey was published at the same time by Garifullin [62]. We are not going to repeat what was already done in these reviews and will focus predominantly on work which has appeared after their publication. Only basic notions and ideas necessary for understanding the present discussion will be extracted from previous work. Quite recently a review by Golubov et al. [63] on the current phase relation in Josephson junctions was published. It has some overlap with our review. Most of the proximity phenomena predicted theoretically in SF-systems and found experimentally are based on the oscillatory behaviour of the Cooper-pair wavefunction penetrating in the ferromagnet. The oscillation of the wavefunction leads to oscillations of several important physical values. They are: oscillations of the transition temperature first predicted in [64, 65], and the critical current versus the thickness of the ferromagnetic layer, which are seen as oscillatory transitions from 0- to p-junctions [57]. Other proximity effects besides the usual suppression of the order parameters include the preferential antiparallel orientation of the F-layers in 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 by Bergeret et al. [71]: they have predicted that when the direction of magnetization varies in space, singlet Cooper pairs are transformed into triplet pairs. The triplet pairing is not suppressed by the exchange field and can propagate in the ferromagnet over large distances thus providing the long-range proximity between superconductors in S/F/S-junctions. The proximity effects may have technological applications as elements of high-speed magnetic electronics based on the spin-valve action [68] and also as elements of quantum computers [72]. Purely magnetic interaction between ferromagnetic and superconducting subsystems can also be used to design magnetic field controlled superconducting devices. A magnetic field controlled Josephson interferometer employing a thin magnetic F/S-bilayer has been demonstrated by Eom and Johnson [73]. In the next section we derive basic equations. The third section is focused on phenomena 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 the last section. 2. Basic equations In the proposed and experimentally realized FSH a magnetic texture interacts with the supercurrent. First we assume that F- and S-subsystems are separated by a thin insulating layer which prevents the proximity effect, focusing on the magnetic interaction only. Inhomogeneous magnetization generates a magnetic field outside ferromagnets. This magnetic field induces screening currents in superconductors which, in turn, change the magnetic field. The problem must be solved self-consistently. The calculation of the vortex and magnetization arrangement for interacting, spatially separated superconductors and ferromagnets is based on the static London Maxwell equations and corresponding energy. This description includes possible superconducting vortices. London s approximation works satisfactorily since the sizes of all structures in the problem exceed significantly the coherence length,. We recall that in the London approximation the modulus of the order parameter is constant

5 Ferromagnet superconductor hybrids 71 and the phase varies in space. Starting from the London Maxwell equation in the entire space, we eliminate the magnetic field outside F- and S-subsystems and obtain equations for the currents, magnetization and fields inside them. This is done in Section 2.1. In Section 2.2 we apply this method to very thin coupled ferromagnetic and superconducting films. When proximity effects dominate, the London approximation is invalid. The basic equations for this case will be described in Section Three-dimensional systems The total energy of a stationary F/S-system reads: ð" # B 2 H ¼ 8p þ m sn s v 2 s B M dv 2 ð1þ where B is the magnetic induction, M is the magnetization, n s is the density of S-electrons, m s is their effective mass and v s is their velocity. We assume that the S density n s and the magnetization M are separated in space. We assume also that the magnetic field B and its vector potential A asymptotically tend to zero at infinity. Employing the static Maxwell equations rb ¼ð4p=cÞ j and B ¼rA, the magnetic field energy can be transformed as follows: ð B 2 ð j A 8p dv ¼ 2c dv: ð2þ Though the vector potential enters explicitly in the last equation, it is gauge invariant due to current conservation, div j ¼ 0. When integrating by parts, we neglected the surface term. This is correct if the field, vector potential and the current decrease sufficiently fast at infinity. This condition is satisfied for the simple examples considered in this section. The current j can be represented as a sum: j ¼ j s þ j m of the SC and magnetic currents, respectively: j s ¼ n she 2m s r 2p 0 A j m ¼ crm: ð4þ We consider contributions from the magnetic and S-currents to the integral (2) separately. We start with the integral: ð 1 j 2c m A dv ¼ 1 ð ðrmþadv: ð5þ 2 Integrating by parts and neglecting the surface term again, we arrive at the following result: ð 1 j 2c m A dv ¼ 1 ð M B dv: ð6þ 2 We have omitted the integral over a remote surface H ðn MÞA ds. Such an omission is valid if the magnetization is confined to a limited volume. But for infinite magnetic systems it may be wrong even in the simplest problems. We will discuss this situation in the next section. Next we consider the contribution of the superconducting current j s to the integral (2). In the gauge-invariant equation (3), is the phase of the S-carrier ð3þ

6 72 I. F. Lyuksyutov and V. L. Pokrovsky (Cooper pair) wavefunction and 0 ¼ hc=2e is the flux quantum. Note that the phase gradient r can be included in A as a gauge transformation with the exception of vortex lines, where is singular. We employ equation (3) to express the vector potential A in terms of the supercurrent and the phase gradient: A ¼ 0 2p r m sc n s e 2 j s: ð7þ Plugging equation (7) into equation (2), we find: ð 1 j 2c s A dv ¼ h 4e ð r j s dv m s 2n s e 2 ð j 2 s dv: Since j s ¼ en s v s, the last term in this equation is equal to the kinetic energy taken with the sign minus. It exactly compensates the kinetic energy in the initial expression for the energy (1). Collecting all remaining terms, we obtain the following expression for the total energy: ð" # ns h 2 H ¼ ðr Þ 2 n she B M r A dv: ð9þ 8m s 4m s c 2 We remind the reader again about a possible surface term for infinite systems. Note that integration in the expression for energy (9) proceeds over the volumes occupied either by superconductors or by magnets. Equation (9) allows us to separate the energy of vortices from the energy of magnetization induced currents and fields and their interaction energy. Indeed, as we noted earlier, the phase gradient can be ascribed to the contribution of vortex lines only. It is representable as a sum of independent integrals over different vortex lines. The vector potential and the magnetic field can be represented as a sum of magnetization induced and vortex induced parts: A ¼ A m þ A v, B ¼ B m þ B v, where A k, B k (the index k is either m or v) are determined as solutions of the London Maxwell equations: ð8þ rðra k Þ ¼ 4p c j k: ð10þ The effect of the screening of the magnetization induced magnetic field by the superconductor is included in the vector fields A m and B m. Applying such a separation, we present the total energy (9) as a sum of terms containing only vortex contributions, only magnetic contributions and the interaction terms. The purely magnetic part can be represented as a non-local quadratic form of the magnetization. The purely superconducting part is representable as a non-local double integral over the vortex lines. Finally, the interaction term is representable as a double integral over the vortex lines and the volume occupied by the magnetization and is bilinear in the magnetization and vorticity. To avoid cumbersome formulas, we will not write these expressions explicitly 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 thickness, we assume that both films are located approximately at z ¼ 0. In some cases we need a more accurate treatment. Then we introduce a small distance d between the films, which in the end will be set to zero. Though the thickness of each film is

7 Ferromagnet superconductor hybrids 73 assumed to be small, the two-dimensional densities of S-carriers n ð2þ s ¼ n s d s and magnetization m ¼ Md m remain finite. Here we have introduced the thickness of the S-film d s and the F-film d m. The three-dimensional supercarrier density n s ðrþ can be represented as n s ðrþ ¼ðzÞn ð2þ s ðrþ and the three-dimensional magnetization MðRÞ can be represented as MðRÞ ¼ðz d ÞmðrÞ, where r is the two-dimensional radius vector and the z-direction is perpendicular to the films. In what follows n ð2þ s is assumed to be a constant and the index (2) is omitted. The energy (9) can be rewritten for this special case: ð" # ns h 2 H ¼ ðr Þ 2 n she b m r a d 2 r ð11þ 8m s 4m s c 2 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 AðzÞþ2phn s e m s c r ðzþ þ 4pr ðmðzþþ: ð12þ Here ¼ 2 L=d s is the effective screening length for the S-film, L is the London penetration depth and d s is the S-film thickness [74]. According to our general arguments, the term proportional to r in equation (13) describes vortices. A plane vortex characterized by its vorticity, q, an integer, and by the position of its centre on the plane, r 0, contributes a singular term to r : r 0 ðr, r 0 Þ¼q ^z ðr r 0 Þ jr r 0 j 2 ð13þ and generates a standard vortex vector potential: A v0 ðr r 0, zþ ¼ q 0 2p ^z ðr r 0 Þ jr r 0 j ð 1 0 J 1 ðkjr r 0 jþe kjzj dk: 1 þ 2k ð14þ Different vortices contribute independently to the vector potential and magnetic field. A peculiarity of this problem is that the usually applied gauge diva ¼ 0 becomes singular in the limit d s, d m! 0. Therefore, it is reasonable to apply another gauge, A z ¼ 0. The calculations are much simpler in the Fourier representation. Following the general procedure, we present the Fourier transform of the vector potential A k as a sum A k ¼ A mk þ A vk. The equation for the magnetic part of the vector potential reads: kðqa mk Þ k 2 A mk ¼ a mq 4pik m qe ik zd ð15þ where q is the projection of the wavevector k onto the plane of the films: k ¼ k z ^z þ q. An arbitrary vector field V k in the wavevector space can be represented by its local coordinates: V k ¼ V z k ^z þ V k k ^q þ V? k ð^z ^qþ: ð16þ

8 74 I. F. Lyuksyutov and V. L. Pokrovsky In terms of these coordinates the solution of equation (15) reads: A k mk ¼ 4pim? q k z e ik zd ð17þ A? mk ¼ 1 k 2 a? q þ 4pi k zm k q qm qz k 2 e ikzd : ð18þ Integration of the latter equation over k z allows us to find the perpendicular component of a ðmþ q : a? mq ¼ 4pqðmk q þ im qz Þ e qd, 1 þ 2q ð19þ whereas it follows from equation (15) that a k mq ¼ 0. Note that the parallel component of the vector potential A k mk does not know anything about the S-film. It corresponds to the magnetic field being equal to zero outside the plane of the F-film. Therefore, it is inessential for our problem. The vortex part of the vector potential also does not contain a z-component since the supercurrents flow in the plane. The vortex solution in a thin film was first found by Pearl [75]. An explicit expression for the vortex-induced potential is: A vk ¼ 2i 0ð^z ^qþfðqþ k 2, ð20þ ð1 þ 2qÞ where FðqÞ ¼ P j eiqr j is the vortex form-factor; the index j labels the vortices and r j are coordinates of the vortex centres. The Fourier transform for the vortex-induced vector potential at the surface of the SC film a vq reads: a vq ¼ i 0ð^z ^qþfðqþ : ð21þ qð1 þ 2qÞ The z-component of magnetic field induced by the Pearl vortex in real space is: B vz ¼ 0 2p ð 1 0 J 0 ðqrþe qjzj qdq: ð22þ 1 þ 2q Its asymptotic at z ¼ 0 and r is B vz 0 =ðpr 3 Þ;atr it is B vz 0 =ðprþ. Each Pearl vortex carries the flux quantum 0 ¼ phc=e. The energy (11) can be expressed in terms of Fourier transforms: H ¼ H v þ H m þ H vm, where the vortex energy H v is the same as it would be in the absence of the F film: H v ¼ n sh 2 ð r 8m q r q 2p d 2 q a s vq 0 ð2pþ 2 : ð24þ ð23þ The magnetic self-energy H m is: H m ¼ 1 ð d 2 q m 2 q b mq ð2pþ 2 ð25þ

9 It contains the screened magnetic field and therefore differs from its value in the absence of the SC film. Finally the interaction energy reads: ð ð ðr Þ q a mq m q b vq ð26þ H mv ¼ n she 4m s c d 2 q ð2pþ d 2 q ð2pþ 2 Note that the information on the vortex arrangement is contained in the form-factor F(q) only. To illustrate how important the surface term can be, let us consider a homogeneous perpendicularly magnetized magnetic film and one vortex in a superconducting film. The authors [30] have shown that the energy of this system is " v ¼ " 0 v m 0, where " 0 v 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 flux quantum. Let us consider how this result appears from the microscopic calculations. The vortex energy (24) is just equal to " 0 v. The purely magnetic term (25) does not change in the presence of the vortex and is inessential. The first term in the interaction energy (26) is equal to zero since the infinite magnetic film does not generate a magnetic field outside. The second term of this energy is equal to m 0 =2. The second half of the interaction energy comes from the surface term. Indeed, it is equal to 1 2 lim r!1 ð 2p 0 Ferromagnet superconductor hybrids 75 mð^r ^zþard ¼ 1 2 I A dr ¼ m 0 = The Eilenberger and Usadel equations The essence of proximity phenomena is the change of the order parameter (Cooperpair wavefunction). Therefore, the London approximation is not valid in this case and equations for the order parameter must be solved. They are either the Bogolyubov de Gennes equations [76, 77] for the coefficients u and v or more conveniently the Gor kov equations [78] for the Green functions. Unfortunately the solution of these equations is not an easy problem in the spatially inhomogeneous case 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 layers are thin, the diffusion delivers atoms of one layer into another and the control of the structure 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 than atomic, the semiclassical approximation can be applied. Equations for the superconducting order parameter in the semiclassical approximation were derived a long time ago by Eilenberger [80] and by Larkin and Ovchinnikov [81]. They were further simplified in the case of strong elastic scattering (the diffusion approximation) by Usadel [82]. For the reader s convenience and for the unification of notation we demonstrate 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 functions integrated in momentum space over the momentum component perpendicular to the Fermi surface. Thus, they depend on a point of the Fermi surface characterized by two momentum components, on the coordinates in real space and time. It is more convenient in thermodynamics to use their Fourier components over

10 76 I. F. Lyuksyutov and V. L. Pokrovsky imaginary time, the so-called Matsubara representation [85]. The frequencies in this representation accept discrete real values! n ¼ð2n þ 1ÞpT, where T is the temperature. Wave functions of singlet Cooper pairs are represented by two Eilenberger anomalous Green functions Fð!, k, rþ and F y ð!, k, rþ (integrated along the 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 in real space (the coordinate of the Cooper-pair centre-of-mass). The function F is generally 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, rþš 2 þjfð!, k, rþj 2 ¼ 1: ð27þ Furthermore, the Eilenberger Green functions obey the following symmetry relations: Fð!, k, rþ ¼F ð!, k, rþ ¼F ð!, k, rþ ð28þ Gð!, k, rþ ¼ G ð!, k, rþ ¼ Gð!, k, rþ: ð29þ The Eilenberger equation reads: 2! þ v o 2e i or c AðrÞ Fð!, k, rþ ð ¼ 2Gð!, k, rþþ d 2 q ðqþwðk, qþ½gðkþfðqþ FðkÞGðqÞŠ; where ðrþ is the space (and time) dependent order parameter (local energy gap), v is the velocity on the Fermi surface, Wðk, qþ is the probability of a transition per unit time from the state with momentum q to the state with momentum k, andðqþ is the angular dependence of the density of states normalized by Ð d 2 q ðqþ ¼Nð0Þ. Here N(0) is the total density of states (DOS) in the normal state at the Fermi level. The Eilenberger equation has the structure of the Boltzmann kinetic equation, but it also incorporates quantum coherence effects. It must be complemented by the selfconsistency equation expressing the local value of ðrþ in terms of the anomalous Green function F: ðrþ ln T ð þ 2pT X1 ðrþ d 2 k ðkþfð! T c! n, k, rþ ¼ 0: ð31þ n¼0 n In the case of isotropic scattering frequently considered by theorists the collision integral in equation (30) is remarkably simplified: ð d 2 q ðqþwðk, qþ½gðkþfðqþ FðkÞGðqÞŠ ¼ 1 ½GðkÞhF i FðkÞhGiŠ, ð32þ where the relaxation time is equal to the inverse value of the angular independent transition probability W,andh...i means the angular average over the Fermi sphere. The Eilenberger equation is simpler than the complete Gor kov equations since it contains only one function depending on one fewer arguments. It could be expected that in the limit of very short relaxation time T c0 1 (T c0 is the transition temperature in the clean superconductor) the Eilenberger kinetic-like equation will become similar to the diffusion equation. Such a diffusion-like equation was indeed derived by Usadel [82]. In the case of strong elastic scattering and an isotropic Fermi ð30þ

11 Ferromagnet superconductor hybrids 77 surface (sphere), the Green function does not depend on the direction on the Fermi sphere and depends only on the frequency and the spatial coordinate r. The Usadel equation reads (we omit both arguments): 2!F D^o G^oF F ^og ¼ 2G: ð33þ In this equation D ¼ v 2 F=3 is the diffusion coefficient for electrons in the normal state and ^o stands for the gauge-invariant gradient: ^o ¼r 2ieA=hc. The Usadel equations must be complemented by the same self-consistency equation (31). It is also useful to keep in mind the expression for the current density in terms of the function F: j ¼ ie2ptnð0þd X! n >0ðF ^of F ^of Þ: ð34þ One can consider the set of Green functions G, F, F y as elements of the 2 2 matrix Green function ^g where the matrix indices can be identified with the particle and hole or Nambu channels. This formal trick becomes rather essential when the singlet and triplet pairing coexist and it is necessary to take into account the Nambu 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 proposed by Bergeret et al. [86]. Let us introduce a matrix gðr, t; r 0, t 0 Þ with matrix elements n, n0 gs, s 0, where n, n0 are the Nambu indices and s, s 0 are the spin indices, defined as follows: n, n0 gs, s ðr, t; 0 r0, t 0 Þ¼ 1 X ð ð ^ h 3 Þ n, n 00 d h n 00 sðr, tþ y n 0 s 0ðr0, t 0 Þi: ð35þ n 00 The matrix ^ 3 in the definition (35) is the Pauli matrix in the Nambu space. To clarify the Nambu indices we write explicitly what they mean in terms of the electronic -operators: 1s s ; 2s y s and s means s. The most general matrix g can be expanded in the Nambu space into a linear combination of four independent matrices ^ k, k ¼ 0, 1, 2, 3, where ^ 0 is the unit matrix and the three others are the standard Pauli matrices. Following [86], we introduce the following notations for the components of this expansion, which are matrices in the spin space: 1 g ¼ ^g 0 ^ 0 þ ^g 3 ^ 3 þ f ; f ¼ f ^ 1 i ^ 1 þ f ^ 2 i ^ 2 : ð36þ The matrix f describes Cooper pairing since it contains only antidiagonal matrices in the Nambu space. In turn, the spin matrices f ^ 1 and f ^ 2 can be expanded in the basis of spin Pauli matrices j, j ¼ 0, 1, 2, 3. Without loss of generality we can accept the following agreement about the scalar components of the spin-space expansion: ^ f 1 ¼ f 1 ^ 1 þ f 2 ^ 2 ; ^ f 2 ¼ f 0 ^ 0 þ f 3 ^ 3 : ð37þ 1 Each time the Nambu and spin matrices are written together we mean the direct product.

12 78 I. F. Lyuksyutov and V. L. Pokrovsky It is easy to check that the amplitudes f i, i ¼ 0,..., 3 are associated with the following combinations of wavefunction operators: f 0!h " # þ # " i f 1!h " " y # y # i f 2!h " " þ y # y # i f 3!h " # # " i: Thus, the amplitude f 3 corresponds to the singlet pairing, whereas the three others are responsible for the triplet pairing. In particular, in the absence of triplet pairing only the component f 3 survives and the matrix f is equal to 0 F F y : 0 Let us consider what modification must be introduced into the Eilenberger and Usadel equations to take in account the exchange interaction of Cooper pairs with the magnetization in the ferromagnet. Neglecting the reciprocal effect of the Cooper pairs on the d- or f-shell electrons responsible for the magnetization, we introduce the effective exchange field h(r) acting inside the ferromagnet. This produces a pseudo-zeeman splitting of the spin energy. 2 In the case of the singlet pairing the Matsubara frequency! must be replaced by! þ ihðrþ. When the direction of magnetization varies in space generating triplet pairing, the Usadel equation must be formulated in terms of the matrix g [86]: D 2 oð go gþ j!j½ ^ 3 ^ 3, gšþsign!½h, gš ¼ i½, gš, ð38þ where the operators of the magnetic field h and the energy gap are defined as follows: h ¼ 3 h ¼ i 2 2 : ð39þ ð40þ To find a specific solution of the Eilenberger and Usadel equations proper boundary conditions should be formulated. For the Eilenberger equations the boundary conditions at the interface of two metals were derived by Zaitsev [87]. They are most naturally formulated in terms of the antisymmetric ( g a ) and symmetric ( g s ) parts of the matrix g with respect to reflection of momentum p z! p z assuming that z is normal to the interface. One of them states that the antisymmetric part is continuous at the interface (z ¼ 0): g a ðz ¼ 0Þ ¼ g a ðz ¼þ0Þ: The second equation connects the discontinuity of the symmetric part at the interface g s ¼ g s ðz ¼þ0Þ g s ðz ¼ 0Þ with the reflection coefficient R and transmission ð41þ 2 In reality the exchange enegry has a quite different origin than the Zeeman interaction, but at a fixed magnetization there is a formal similarity in the Hamiltonians.

13 Ferromagnet superconductor hybrids 79 coefficient D of the interface and antisymmetric part g a at the boundary: D g s ð g s þ g a g s Þ¼R g a ½1 ðg a Þ 2 Š, ð42þ where g s ¼ g s ðz ¼þ0Þ g c ðz ¼ 0Þ. If the boundary is transparent (R ¼ 0, D ¼ 1), the symmetric part of the Green tensor g is also continuous. The boundary conditions for the Usadel equations, i.e. under the assumption that 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 the continuity of the current flowing through the interface: d g < g < < dz ¼ d g > g > > 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 condition connects the current with the discontinuity of the order parameter through the boundary and its transmission and reflection coefficients D() and R(): l > g > d g > dz ¼ 3 4 cos DðÞ RðÞ ½ g <, g > Š, where is the incidence angle of the electron at the interface and D(), R() are the corresponding transmission and reflection coefficients. This boundary condition can be rewritten in terms of measurable characteristics: ð44þ > g > d g > dz ¼ 1 R b ½ g <, g > Š, ð45þ where R b 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]: f < ¼ f > ; d f < dz ¼ d f > dz, where is the ratio of normal state resistivities. ð46þ 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 circular F-dot grown upon it, following the work [32]. The magnetization is assumed to be fixed, homogeneous inside the dot and directed either perpendicular or parallel to the S-film (see figure 1). This problem is a basic one for a class of more complicated problems incorporating arrays of magnetic dots. We will analyse what the conditions are for the appearance of vortices in the ground state, where they appear, and what the magnetic fields and currents are in these states. The S-film is assumed to be very thin, plane and infinite in the lateral directions. Since the magnetization is confined inside the finite dot no difficulties with the surface integrals over infinitely remote surfaces or contours arise.

14 80 I. F. Lyuksyutov and V. L. Pokrovsky Magnetic Dot Dot's Flux Lines MAGNETIC DOT ANTIVORTEX VORTEX Supercurrent Superconductor SUPERCONDUCTOR Figure 1. Magnetic dots with out-of-plane and in-plane magnetization and vortices Magnetic dot: perpendicular magnetization. The magnetic field generated by an infinitely thin circular magnetic dot of radius R with two-dimensional magnetization perpendicular to the plane mðrþ ¼m^zYðR rþðz dþ on the top of the S-film can be calculated using equations (18) and (19). The Fourier component of magnetization necessary for this calculation is: m k ¼ ^z 2pmR J q 1 ðqrþe ikzd, ð47þ where J 1 (x) is the Bessel function. The Fourier transforms of the vector potential reads: A? mk ¼ i8p2 mrj 1 ðqrþ k 2 e qd 2q 1 þ 2q þðeik zd e qd Þ : ð48þ Though the difference in the inner round brackets in equation (48) looks to be always small (we recall that d must be set to zero in the final answer), we cannot neglect it since it occurs to give a finite, non-negligible contribution to the parallel component of the magnetic field between the two films. From equation (48) we immediately find the Fourier transforms of the magnetic field components: B z mq ¼ iqa? mq; B? mq ¼ ik z A? mq: ð49þ For the reader s convenience we also present the Fourier transform of the vector potential at the superconductor surface: a? mq ¼ i8p2 mr 1 þ 2q J 1ðqRÞ: ð50þ In the last equation we have put e qd equal to 1. Performing the inverse Fourier transformation, we find the magnetic field in real space: ð 1 B z J mðr, zþ ¼4pmR 1 ðqrþj 0 ðqrþe qjzj q 2 dq ð51þ 1 þ 2q B r mðr, zþ ¼2pmR 0 ð 1 0 J 1 ðqrþj 1 ðqrþe qjzj 2q 1 þ 2q YðzÞþYðz dþ YðzÞ qdq, where Y(z) is the step function equal to þ1 at positive z and 1 at negative z. Note that B r m has discontinuities at z ¼ 0 and z ¼ d due to surface currents in the S- and F-films, respectively, whereas the normal component B z m is continuous. ð52þ

15 Ferromagnet superconductor hybrids no vortex with vortex 0.01 B z Figure 2. Magnetic field of dot with and without vortex for R= ¼ 5 and 0 =ð8p 2 mrþ ¼4. r/λ A vortex, if appears, must be located at the centre of the dot due to symmetry. If R, the direct calculation shows that the central position of the vortex provides minimum to energy. For small radius of the dot the deviation of the vortex from the central position seems even less probable. We have checked numerically that the central position is always energetically favourable for one vortex. Note that this fact is 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 by the 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 >shows that B z at the S-film ðz ¼ 0Þ changes sign at some r ¼ R 0 (see figure 2) in the presence of the vortex centred at r ¼ 0, but it is negative everywhere at r > R in the absence of the vortex. The physical explanation of this fact is as follows. The dot itself is an ensemble of parallel magnetic dipoles. Each dipole generates a magnetic field at the plane passing through the dot, which has sign opposite to its dipole moment. The fields from different dipoles compete at r<r, but they have the same sign at r > R. The S-current resists this tendency. The field generated by the vortex decays slower than the dipolar field (1/r 3 versus 1/r 4 ). Thus, the sign of B z is opposite to the magnetization 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 to detect an S-vortex confined by the dot. To our knowledge, so far there were no experimental measurements of this effect. In the presence of a vortex, the energy of the system can be calculated using equations (23) (26). The appearance of the vortex changes the energy by the amount: ¼ " v þ " mv ð53þ

16 82 I. F. Lyuksyutov and V. L. Pokrovsky where " v ¼ " 0 lnð=þ is the energy of the vortex without the magnetic dot, " 0 ¼ 2 0=ð16p 2 Þ, and " mv is the energy of interaction between the vortex and the magnetic dot given by equation (26). For this specific problem the direct substitution of the vector potential, magnetic field and the phase gradient (see equations 50 and 51) leads to the following result: ð 1 J " mv ¼ m 0 R 1 ðqrþ dq 0 1 þ 2q : ð54þ The vortex appears when tends to zero. This criterion determines a curve in the plane of two dimensionless variables R= and m 0 =" v. This curve separating regimes with and without vortices is depicted in figure 3. The asymptotic of " mv for large and small values of R= can be found analytically: R " mv m 0 1 " mv m 0 R 2 R 1 Thus, asymptotically the curve ¼ 0 turns into a horizontal straight line m 0 =" v ¼1 at large R= and a logarithmically distorted hyperbola ðm 0 =" v ÞðR=Þ ¼2 at small ratio R=. With further increasing of either m 0 =" v or R= the second vortex becomes energetically favourable. Due to symmetry the centres of the two vortices are located on the straight line including the centre of the dot at equal distances from it. The energy 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 there exists an infinite series of such transitions. However, here we limit ourselves to the first three since it is not quite clear what is the most energetically favourable vortex 2 vortices 3 vortices mφ 0 /ε v Figure 3. Phase diagram of vortices induced by a magnetic dot. The lines correspond to the appearance of 1, 2 and 3 vortices, respectively. R/λ

17 Ferromagnet superconductor hybrids 83 configuration for four vortices (for three it is a regular triangle). The role of configurations with several vortices confined inside the dot area and antivortices outside has not yet been studied Magnetic dot: parallel magnetization. Next we consider an infinitely thin circular magnetic dot whose magnetization M is directed in the plane and is homogeneous inside the dot. An explicit analytical expression for M reads as follows: M ¼ m 0 ðr ÞðzÞ ^x ð55þ where R is the radius of the dot, m 0 is the magnetization per unit area and ^x is the unit vector along the x-axis. The Fourier transform of the magnetization is: M k ¼ 2pm 0 R J 1ðqRÞ ^x: ð56þ q The Fourier representation for the vector potential generated by the dot in the presence of the magnetic film takes the form: " A? mk ¼ e ikd 8p 2 m 0 R k 2 z þ q 2 J {kz e {k # zd 1ðqRÞ cosð q Þ e qd : ð57þ q 1 þ 2q Let us introduce a vortex antivortex pair with the centres of the vortex and antivortex located at x ¼þ 0, x ¼ 0, respectively. Employing equations (23) (26) to calculate the energy, we find: E ¼ 2 0 ln ð 1 ð J ð2q 0 Þ þ 2q dq 2m J 0 0 R 1 ðqrþj 1 ðq 0 Þ dq þ E 0 1 þ 2q 0 ð58þ where E 0 is the dot self-energy. Numerical calculations [32] indicate that the equilibrium value of 0 is equal to R. The vortex antivortex creation changes the energy of the system by: ¼ 2 0 ln ð 1 ð J ð2qrþ þ 2q dq 2m J 0 0 R 1 ðqrþj 1 ðqrþ dq: ð59þ 0 1 þ 2q 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: m ln = ¼ ð Þ 4Ð 1 0 J 0ð2qRÞ=ð1 þ 2qÞ dq 2R Ð 1 0 J : ð60þ 1ðqRÞJ 1 ðqrþ=ð1 þ 2qÞ dq The critical curve in the plane of two dimensionless ratios m 0 0 = 0 and R= is plotted numerically in figure 4. The creation of vortex antivortex pairs is energetically unfavourable in the region below this curve and favourable above it. The phase diagram suggests that the smaller the radius R of the dot, the larger the value m 0 0 = 0 necessary to create the vortex antivortex pair. For large values of R and m 0 0 0, the vortex is separated by a large distance from the antivortex. Therefore, their energy is approximately equal to that of two free vortices. This positive energy is compensated by the attraction of the vortex and antivortex to the magnetic dot. The critical values of m 0 0 = 0 seem to be numerically large even for R= 1. This is probably a consequence of comparably ineffective interaction of the in-plane magnetization with the vortex.

18 84 I. F. Lyuksyutov and V. L. Pokrovsky mφ 0 /ε R/λ Figure 4. Phase diagram for vortices antivortices induced by the magnetic dot with in-plane magnetization. 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 Array of magnetic dots and superconducting film Vortex pinning by magnetic dots. Vortex pinning in superconductors is of the great practical importance. Artificial periodic vortex pinning was first produced with 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 additional opportunities to pin vortices. The first experiments with regular magnetic dot arrays were performed in the Louis Ne el Laboratory in Grenoble [3, 4]. The dots were several microns wide and their magnetization was parallel to the superconducting film. The authors reported oscillations of the magnetization versus magnetic field. These oscillations were attributed to a matching effect: pinning becomes stronger when the vortex lattice is commensurate with the lattice of pinning centres. Flux pinning by a triangular array of submicron-size dots with typical spacing nm and diameters close to 200 nm magnetized in-plane was first reported by Martin et al. [6]. They observed oscillations of the resistivity versus magnetic flux with 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. The pinning 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 by Morgan and Ketterson [7]. They measured the critical current as a function of the external magnetic field and found strong asymmetry of the pinning properties under

19 Ferromagnet superconductor hybrids 85 Figure 5. Left: field dependence of the resistivity of a Nb thin film with a triangular array of Ni dots. (From Martin et al. [6].) Right: critical current as a function of field for the highdensity triangular array at T ¼ 8:52 K, T c ¼ 8.56 K. (From Morgan and Ketterson [7].) magnetic field reversal (see figure 5, right). This was the first direct experimental evidence that vortex pinning by magnetic dots is different from that of non-magnetic pinning 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 of the dot size and the dot lattice constant to the effective penetration depth, the dot array magnetization, the strength and direction of the external field, etc. The use of magnetic 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 insight into vortex physics. This work was done mainly by the group at the University of Leuven. Below we briefly review several experimental studies of this group. Dots with parallel magnetization. Van Bael et al. [90] studied the magnetization and 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 probe microscope (SHPM). SHPM images revealed a magnetic field redistribution below the superconducting transition temperature in the 50 nm thin lead superconducting film. These data were interpreted by Van Bael et al. [90] as the formation of vortices of opposite sign on both sides of the dot. Applying a proper magnetic field, Van Bael et al. [90] generated the commensurate lattice of vortices residing at the ends of the magnetized dot diameter. This location is in agreement with theoretical predictions [32]. Remarkably, they were able to observe annihilation of the vortices created by the 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 and Ketterson [7] observed asymmetry of the vortex pinning with respect to magnetic field reversal in an S-film supplied with a periodic array of F-dots magnetized perpendicularly to the film. Employing SHPM images, Van Bael et al. [91] elucidated the nature of this phenomenon. They prepared a square F-dot lattice with lattice

20 86 I. F. Lyuksyutov and V. L. Pokrovsky Figure 6. Schematic presentation of the polarity-dependent flux pinning, presenting the 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 negative flux quantum is induced by the stray field (black arrows), and (b) a negative vortex is pinned at the pole where a positive flux quantum is induced by the stray field. (From Van Bael et al. [90].) constant equal to 1 mm. Each dot had the shape of a square with 400 nm side length and 14 nm thickness. They were made of Co/Pt multilayers and covered with 50 nm thick lead film. Zero field SHPM images showed the checkerboard-like distribution of magnetic field (see Section 3.3.4). The stray fields from the dots were not sufficient to create vortices. In a very weak (1.6 Oe) external field the average distance between vortices 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 the antiparallel field the SHPM shows vortices located at interstitial positions in the magnetic dot lattice (see figure 7b). It is plausible that the pinning barriers are lower in the second case. Figure 8 shows the dependence of S-film magnetization on the applied magnetic field normal to the film. Moshchalkov et al. [92] have shown that the magnetic field dependence of the S-film magnetization is similar to the critical current dependence on magnetic field. Figure 8 displays strong asymmetry of the pinning properties with respect to the external magnetic field orientation. The pinning is much stronger in the magnetic field parallel to the dot magnetization than in the antiparallel field Vortex lattice symmetry versus magnetic dot array symmetry. The vortex lattice induced by an external magnetic field in a homogeneous superconducting film has hexagonal symmetry. Its lattice constant is determined by the magnetic field strength so that each unit cell contains one flux quantum as originally predicted by Abrikosov [74]. An array of pinning centres generates a periodic potential for the vortices. Even if the dot lattice has hexagonal symmetry, the vortex lattice is commensurate with the pinning lattice only at special values of the external magnetic field. Generally symmetry and the phase state of the vortex lattice in the presence of a lattice of pinning centres depend in a complicated way on the external magnetic field, the relative strength of the pinning potential and its symmetry. As we mentioned in the Introduction, the first experiments with S-films, supplied with an artificial periodic pinning structure, were performed in the 1970s. In these experiments the periodicity of the vortex lattice induced by an external magnetic field competed with the periodicity of an artificial array created by experimenters [8 19]. The presence of magnetic dots introduces an additional dimension into the old problem. The most interesting is the situation when the dots generate vortices (and antivortices). The net magnetic flux from the magnetic dot perpendicular to

21 Ferromagnet superconductor hybrids 87 Figure 7. SHPM images of a (10.5 mm) 2 area of the sample in H ¼ 1:6 Oe (left panel) and H ¼ 1.6 Oe (right panel), at T ¼ 6.8 K (field-cooled). The tiny black/white dots indicate the positions of the Co/Pt dots, which are all aligned in the negative sense (m < 0). The flux lines emerge as diffuse dark (H < 0) or bright (H > 0) spots in the SHPM images. (From Van Bael et al. [91].) Figure 8. M(H/H 1 ) magnetization curves at different temperatures near T c (7.00 K open symbols, 7.10 K filled symbols) showing the superconducting response of the Pb layer on top of the Co/Pt dot array with all dots magnetized up (upper panel) or down (lower panel). H 1 ¼ 20:68 Oe is the first matching field, at which exactly one flux quantum is generated per unit cell of the dot array. (From Van Bael et al. [91].)

22 88 I. F. Lyuksyutov and V. L. Pokrovsky 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 if the lattice period is in the range of several coherence lengths, [97]. Below we review a few theoretical and experimental works about vortex lattice symmetry in the presence of an array of magnetic dots. Experimental studies are more numerous, but also cover only a small number of possible configurations. Spontaneous symmetry violation. Erdin [46] studied theoretically the simplest case of a square array of magnetic dots with perpendicular magnetization without an external magnetic field. Even in this case, Erdin found a range of parameters in which the tetragonal symmetry of the dot lattice is spontaneously broken by the vortex 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 spontaneous creation of vortex antivortex pairs for different values of two dimensionless ratios: Lm 0 =" 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 the vortex antivortex lattice is C 4, i.e. the same as that of the dot lattice. However, for small values of the ratio R/L (e.g. 0.10), the rotational symmetry of the vortex antivortex lattice is C 2, with 10 maximum deviation of the unit cell diagonal (see table 1 in Ref. [46]). Magnetic dot lattice near the superconducting transition temperature. Priour and Fertig [97] considered theoretically the square lattice of the F-dots upon an S-film at a temperature slightly below the S-transition temperature T c. In this situation, local magnetic fields generated by the dots can easily reach and exceed the second y antivortex magnetic dot x vortex Figure 9. Top view of the square magnetic dot array. The vortices are confined within the region of the dot, while the antivortices appear outside.

23 Ferromagnet superconductor hybrids 89 critical field of the S-film H c2, though the average magnetic field is zero. The strong inhomogeneous magnetic field induces vortices and antivortices. The authors assumed that the F-dots are cubes with side length equal to 2 ( is the coherence length) 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 magnetic flux through the cube face varying from 4.1 to 5.6 flux quanta. The authors solved numerically the GL equation in the presence of an inhomogeneous magnetic field generated by the F-dot array. The period of the magnetic dot lattice was taken to be 6:25. In the numerical procedure they solved the GL equation in a 4 4 lattice with periodic boundary conditions. The vortices and antivortices were identified as nodes of the GL wavefunction ðxþ. The resulting density diagram for j ðxþj 2 is shown in figure 10. Vortices and antivortices are clearly seen on it as light circular spots. The initial square symmetry of the dot lattice is violated by the vortex arrangement in all these configurations. Rectangular magnetic dot lattice. Schuller et al. [98] studied experimentally vortex lattices 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 ratio f ¼ c=d in the range Each dot was a circular cylinder with diameter 200 nm ρ pair ρ pair (a) = 2 (b) = (c) ρ pair = 2.5 (d) ρ pair = 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 fundamental flux units, respectively. Antivortices appear as small dark spots, while the large dark spots 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/ )

24 90 I. F. Lyuksyutov and V. L. Pokrovsky 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 was commensurate with the dot lattice; in higher fields the transition to the square vortex lattice has occurred so that the side of the cube was commensurate with the smaller side of the dot rectangle. Schuller et al. [98] have interpreted their data in terms of the balance between the periodic pinning and elastic energy of the vortex lattice Magnetic field induced superconductivity. Consider a regular array of magnetic dots placed upon a superconducting film with magnetization normal to the film. For simplicity we consider very thin magnetic dots. This situation is realized in magnetic films with normal magnetization used in the experiment [93]). The net flux from each magnetic dot through any plane including the surface of the superconducting film (see figure 11) is exactly zero. Suppose that on the top of the magnetic dot the z-component of the magnetic field is positive as shown in the mentioned figure. Due to the requirement of zero net flux the z-component of the magnetic field between the dots must be negative. Thus, a connected part of the S-film occurs in a negative magnetic field normal to the film. It can be partly or fully compensated 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 the S-transition temperature in an external magnetic field, the result looking counterintuitive if one forgets about the field generated by the dots. In this experiment a thin superconducting film was covered with a square array of CoPd magnetic dots perpendicular to the film magnetization. The dots had a square shape with side 0.8 mm, thickness 22 nm and dot array period 1.5 mm. The H T phase diagrams presented in [93] for zero and finite dot magnetization demonstrate the appearance of superconductivity in an applied magnetic field parallel to the dot magnetization. At T ¼ 7.20 K the system with magnetized dots is in the normal state. It transits (a) H= 0 B m z Co/Pd Ge Pb Ge Si/SiO 2 (b) H Figure 11. Schematic distribution of the magnetic field generated by an array of dots with normal to the superconducting film magnetization: (a) zero external field, (b) external field parallel to magnetization partly or completely compensates the magnetic field of dots between them. (From Lange et al. cond-mat/ )

25 Ferromagnet superconductor hybrids 91 to the superconducting state in the field 0.6 mt and back to the normal state at 3.3 mt. From the data in figure 3 of [93] one can conclude that the compensating field is about 2 mt Magnetization controlled superconductivity. Above (Section 3.2.3) we demonstrated an example when the application of a magnetic field transforms the FSH system from the normal to superconducting state due to compensation of the stray magnetic field of the dots with external magnetic field. Earlier Lyuksyutov and Pokrovsky [26] considered theoretically a randomly magnetized array of magnetic dots with normal magnetization. They argued that such an array induces the resistive state in the S-film. The superconducting state can be restored by regular magnetization of the dot array. This counterintuitive phenomena can be explained on a qualitative level. If a single dot induces one vortex, the magnetized array of dots induces a periodic vortex antivortex lattice with antivortices localized at the centres of the unit cells of the square dot lattice as shown in figure 12, left. This ordered vortex dot system provides a strong pinning. More interesting is a paramagnetic, i.e. randomly magnetized, dot array. Vortices and antivortices induced by a paramagnetic dot array generate a random field for a probe vortex. If the lattice constant of the array, a, is less than the effective penetration depth, the random fields from vortices are logarithmic. The effective number of random logarithmic potentials acting on a probe vortex is N p¼ð=aþ ffiffiffiffi 2 and the effective depth of the potential well for a vortex (antivortex) is N v. Under proper conditions, for example near the S-transition point, the potential wells can be very deep, enabling spontaneous generation of the vortex antivortex pairs at the edges between potential valleys and hills. The vortices and antivortices screen these deep potential wells and hills similarly 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 INDUCED VORTEX INDUCED VORTEX MAGNETIC DOT SUPERCURRENT MAGNETIC DOT SUPERCURRENT Figure 12. Left: magnetized magnetic dot array. Vortices of different signs are shown schematically by the supercurrent direction (dashed lines). The dot magnetic moment direction is indicated by. Vortices bound by dots and vortices in interstitial positions are shown. The magnetized array of dots induces a regular lattice of vortices and antivortices and provide strong pinning. Right: a demagnetized magnetic dot array results in a strongly fluctuating random potential, which generates unbound antivortices/vortices, transforming the S-film into a resistive film.

26 92 I. F. Lyuksyutov and V. L. Pokrovsky self-consistent potential relief the vortices move along percolated infinite trajectories passing through the saddle points [29]. The drift motion of the delocalized vortices and antivortices in the external field generates dissipation and transfers the S-film into the resistive state (see figure 12, right). Replacing the slowly varying logarithmic potential by a constant at distances less than and zero at larger distances, Feldman et 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: V 0 ¼2 0 at the distance r <and zero at r >, where 0 ¼ 2 0=ð16p 2 Þ. 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 U X i n i þ X X n 2 i þ 2 0 n i n j, ð61þ i i i>j where n i is an integer characterizing an individual vortex located either at a dot or at a site of the dual lattice (between the dots) which we conventionally associate with the location of unbound vortices; i ¼1 is the random sign of the dot magnetic moments; and i ¼ 0 relates to the sites of the dual lattice. The first term of the Hamiltonian (61) describes the binding energy of the vortex at the magnetic dot and U 0 d = 0, with d being the magnetic flux through a single dot. The second term in the Hamiltonian is the sum of single vortex energies, ¼ 0 lnð=aþ, where a is the period of the dot array. The third term mimics the vortex interaction. Redefining the constant, one can replace the last term of equation (61) by 0 ð P n i Þ 2. The sign of the vorticity n i at a dot follows two possible ( up or down ) orientations of its magnetization. The vortices located between the dots (n i on the dual lattice) are correlated at distances of the order of and form the above-mentioned irregular checkerboard potential relief. To find the ground state, we consider a cell with a large number of dots of each sign ð=aþ 2 1. The energy (61) is minimal when the neutrality condition Q P n i ¼ 0 is satisfied. Indeed, if Q 6¼ 0 the interaction energy grows as Q 2, whereas the first term of the Hamiltonian behaves as jqj and cannot compensate the last one unless Q 1. The neutrality constraint means that the unbound vortices screen almost p completely the charge of those bound by dots, that is K ðn þ N Þ ffiffiffiffiffiffiffi N =a where K is the difference between the numbers of the positive and negative dots and N are the numbers of the positive and negative vortices, respectively. Neglecting the total charge jqj as compared with =a, we minimize the energy (61) accounting for the neutrality constraint. At Q ¼ 0 the Hamiltonian (61) can be written as the sum of single-vortex energies: H ¼ X H i ; H i ¼ U i n i þ n 2 i : ð62þ The minima for any H i are achieved by choosing n i ¼ n 0 i, an integer closest to the magnitude i ¼ i U=ð2Þ. The global minimum consistent with the neutrality is realized by values of n i which differ from the local minima values n 0 i by not more than 1. Indeed, in the configuration with n i ¼ n 0 i, the total charge is j P n 0 i jj P i j¼jkj. Hence, if =a, then the change of the vorticity at a small number of sites by 1 restores neutrality. To be more specific let us consider K > 0. Let n be the integer closest to, and consider the case < n. Then the minimal energy corresponds to a configuration with vorticity n i ¼ n at each negative dot and with vorticity n or n 1

27 Ferromagnet superconductor hybrids 93 at positive dots. The neutrality constraint implies that the number of positive dots with vorticity n 1isM ¼ K n. In the opposite case > n the occupancies of all the positive dots are n, whereas the occupancies of the negative dots are either n or nþ1. Note that in 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 the energy by E ¼ 2ð n þ 2Þ. Hence, the energy transfer is zero if and only if is an integer, otherwise the energy change upon vortex transfer is positive. For integer, the number of unbound vortices can vary from 0 to K n without change of energy. The ground state is degenerate at any non-integer since, while the total number of dots with different vorticities is fixed, the vortex exchange between two dots with vorticities n and n 1 does not change the total energy. Thus, the model predicts a step-like dependence of dot occupancies on at zero temperature and peaks in the concentration of unbound vortices as shown in figure 13. The data for a finite temperature were calculated in [29]. The dependencies of the unbound vortex concentration on for several values of x ¼ =T are shown in figure 13. Oscillations are well pronounced for x 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 and the heights of peaks are n, where ¼ K/N. Vortex transport. For moderate external currents j the vortex transport and dissipation are controlled by unbound vortices. The typical energy barrier associated with the vortex motion is 0. The unbound vortex density is m a 2 ðaþ 1 and oscillates p with as shown above. The average distance between the unbound vortices is l ffiffiffiffiffi a. The transport current exerts a Magnus (Lorentz) force FM ¼ j 0 =c acting on a vortex. Since the condition T 0 is satisfied in the vortex state everywhere except in the close vicinity of the transition temperature, T c, the vortex motion occurs via thermally activated jumps with rate: ¼ 0 expð 0 =T Þ¼ðj 0 =clþ expð 0 =T Þ, ð63þ where ¼ð 2 n Þ=ð4pe 2 Þ is the Bardeen Stephen vortex mobility [95]. The induced 6 4 q/γ κ Figure 13. Left: the checkerboard average structure of the vortex plasma. Right: the average number of unbound vortices in the cell of size a versus a parameter proportional to the dot magnetic moment. The dot-dashed line corresponds to T= 0 ¼ 0:15, the solid line corresponds to T= 0 ¼ 0:4, and the dashed line corresponds to T= 0 ¼ 2.

28 94 I. F. Lyuksyutov and V. L. Pokrovsky electric field is accordingly E c ¼ l _B=c ¼ m 0 l=c: ð64þ The ohmic losses per unbound vortex are W c ¼ je c a ¼ j 0 l=c giving rise to the dc resistivity as dc ¼ W c j 2 2 ¼ 2 0 c 2 2 exp½ 0ðT Þ=T Š: ð65þ Note the non-monotonic dependence of dc on temperature T (figure 14). The density of the unbound vortices is an oscillating function of the flux through the dot. The resistivity of such a system is determined by thermally activated jumps of vortices through the corners of the irregular checkerboard formed by the positive or negative unbound vortices and oscillates with d. These oscillations can be observed by additional deposition (or removal) of the magnetic material to the dots Ferromagnet superconductor bilayer Topological instability in the FSB. Let us consider a F/S-bilayer with both layers infinite and homogeneous. An infinite magnetic film with ideal parallel surfaces and homogeneous magnetization generates no magnetic field outside. Indeed, it can be considered as a magnetic capacitor, the magnetic analogue of an electric capacitor, and therefore its magnetic field is confined inside. Thus, there is no direct interaction between the homogeneously magnetized F-layer and a homogeneous S-layer in the absence of currents in it. However, Lyuksyutov and Pokrovsky argued [30] that such a system is unstable with respect to spontaneous formation of vortices in the S-layer. Below we reproduce these arguments. Assume the magnetic anisotropy to be sufficiently strong to keep the magnetization 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 Pearl vortex somehow appears in the superconducting film, it generates a magnetic field 4 3 ρ mω t Figure 14. The static resistance of the film versus dimensionless temperature t ¼ T=T c at typical values of the parameters.

29 Ferromagnet superconductor hybrids 95 interacting with the magnetization m per unit area of the F-film. For a proper circulation direction in the vortex and rigid magnetization m this field decreases the total energy by the amount m Ð B z ðrþ d 2 x ¼ m, where is the total flux. We recall that each Pearl vortex carries flux equal to the famous flux quantum 0 ¼ phc=e. The energy necessary to create the Pearl vortex in the isolated S-film is v ð0þ ¼ 0 lnð=þ [75], where 0 ¼ 2 0=16p 2, ¼ 2 L=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 ð0þ v m 0, ð66þ and the FSB becomes unstable with respect to spontaneous vortex formation as soon as v becomes negative. Note that close enough to the S-transition temperature T s, v is definitely negative since the S-electron density n s and, therefore, ð0þ v is zero at T s. If m is so small that v > 0atT¼0, the instability exists in a temperature interval T v < T < T s, where T v is defined by the equation v ðt v Þ¼0. Otherwise instability persists until T ¼ 0. A newly appearing vortex phase cannot consist of the vortices of one sign. A more accurate statement is that any finite density of vortices independent on the size of the film, L f, is energetically unfavourable in the thermodynamic limit L f!1. Indeed, any system with non-zero average vortex density n v generates a constant magnetic field B z ¼ n v 0 along the z-direction. The energy of this field for a large but finite film of linear size L f grows as L 3 f, exceeding the gain in energy due to creation of vortices proportional to L 2 f in the thermodynamic limit. Thus, paradoxically the vortices appear, but cannot proliferate to a finite density. This is a manifestation of the long-range character of magnetic forces. The way out of this controversy is similar to that in ferromagnets: the film should split into domains with alternating magnetization and vortex circulation directions. Note that these are combined topological defects: vortices in the S-layer and domain walls in the F-layer. They attract with each other. The vortex density is higher near the domain walls. The described texture represents a new class of topological defects which does not appear in isolated S- and F-layers. We show below that if the linear size, L, of the domain is much larger than the effective penetration length,, the most favourable arrangement is the stripe domain structure (see figure 15). The quantitative theory of this structure was given by Erdin et al. [31]. VORTEX ANTIVORTEX VORTEX VORTEX SUPERCURRENT ANTIVORTEX ANTIVORTEX FERROMAGNET SUPERCONDUCTOR DOMAIN WALL Figure 15. Magnetic domain wall and coupled arrays of superconducting vortices with opposite vorticity. Arrows show the direction of the supercurrent.

30 96 I. F. Lyuksyutov and V. L. Pokrovsky The total energy of the bilayer can be represented by a sum: U ¼ U sv þ U vv þ U vm þ U mm þ U dw where U sv is the sum of energies of single vortices, U vv is the vortex vortex interaction energy, U vm is the energy of interaction between the vortices and magnetic field generated by the domain walls, U mm is the self-interaction energy of the magnetic layer, and U dw is the linear tension energy of the domain walls. We assume the twodimensional periodic domain structure consisting of two equivalent sublattices. The magnetization m z ðrþ and density of vortices n(r) alternate when passing from one sublattice to another. Magnetization is supposed to have a constant absolute value: m z ðrþ ¼msðrÞ, where s(r) is the periodic step function equal to þ1 at one sublattice and 1 at the other one. We consider a dilute vortex system in which the vortex spacing is much larger than. Then the single-vortex energy is: U sv ¼ v ð nðrþsðrþ d 2 x; ð67þ v ¼ ð0þ v m 0 ð68þ The vortex vortex interaction energy is: U vv ¼ 1 ð nðrþvðr r 0 Þnðr 0 Þ d 2 xd 2 x 0, 2 ð69þ where Vðr r 0 Þ is the pair interaction energy between vortices located at points r and r 0. Its asymptotics at large distances j r r 0 j is Vðr r 0 Þ¼ 2 0=ð4p 2 j r r 0 jþ [96]. This long-range interaction is induced by the magnetic field generated by the Pearl vortices and their slowly decaying currents. 3 The energy of vortex interaction with the magnetic field generated by the magnetic film looks as follows [32]: U vm ¼ 0 8p 2 ð r ðr r 0 Þnðr 0 Þa ðmþ ðrþ d 2 xd 2 x 0 : Here ðr r 0 Þ¼arctan½ðy y 0 Þ=ðx x 0 ÞŠ is a phase shift created at a point r by a vortex centred at a point r 0 and a ðmþ ðrþ is the value of the vector potential induced by the F-film upon the S-film. Similarly to what we did for one vortex, this part of the energy can be reduced to the renormalization of the single vortex energy with the final result already shown in equation (68). The magnetic self-interaction reads: U mm ¼ m ð B ðmþ z ðrþsðrþ d 2 x: ð71þ 2 Finally, the linear energy of the domain wall is U dw ¼ dw L dw where dw is the linear tension of the domain wall and L dw is the total length of the domain walls. Erdin et al. [31] have compared energies of stripe, square and triangular domain wall lattices, and found that the stripe structure has the lowest energy. For details of the calculation see [31] (and the correction in [33]). The equilibrium domain width and the equilibrium energy for the stripe structure are: L s ¼ 4 exp " dw 4 ~m 2 C þ 1 ð70þ ð72þ 3 From this long-range interaction of the Pearl vortices it is easy to derive that the energy of a system of vortices with the same circulation, located with the permanent density n v on a film having lateral size L, is proportional to n 2 vl 3.

31 U s ¼ 16em2 exp " dw 4em 2 þ C 1 where ~m ¼ m 0 v= 0 and C ¼ is the Euler constant. The vortex density for the stripe domain case is: nðxþ ¼ 4p~ v L s sinðpx=l s Þ : ð74þ Note the strong singularity of the vortex density near the domain walls. The continuous approximation is invalid at distances of the order of, and the singularities must be smeared out in a band of width around the domain wall. The domains become infinitely wide at T ¼ T s and at T ¼ T v.if dw 4 ~m 2, the continuous approximation becomes invalid (see Section 3.2.3) and instead a discrete lattice of vortices must be considered. It is possible that the long nucleation time can interfere with the observation of the described textures. We expect, however, that the vortices 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 experimentally. The most popular material with the magnetization perpendicular to film is a multilayer made from ultrathin Co and Pt films (see Section 3.3.4). This material has a very large coercive field and rather chaotic morphology. Therefore, the domain walls 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 be overcome and spontaneous vortex structures will be discovered before long Superconducting transition temperature of the FSB. The superconducting phase transition in a ferromagnet superconductor bilayer was studied by Pokrovsky and Wei [33]. They have demonstrated that in the FSB the transition proceeds discontinuously as a result of competition between the stripe domain structure in an F-layer at suppressed superconductivity and the combined vortex-domain structure in the FSB. Spontaneous vortex-domain structures in the FSB tend to increase the transition temperature, whereas the effect of the F-self-interaction decreases it. The final shift of transition temperature T c depends on several parameters characterizing the S- and F-films and varies typically between 0:03T c and 0.03T c. As we demonstrated earlier, the homogeneous state of the FSB with the magnetization perpendicular to the layer is unstable with respect to formation of a stripe domain structure, in which both the direction of the magnetization in the F-film and the circulation of the vortices in the S-film alternate together. The energy of the stripe structure per unit area U and the stripe equilibrium width L s is given in equations (72) and (73). To find the transition temperature, we combine the energy given by equation (73) with the Ginzburg Landau free energy. The total free energy per unit area reads: F ¼ U þ F GL ¼ 16 ~m2 e Ferromagnet superconductor hybrids 97 exp dw 4 ~m 2 þ C 1 ð73þ þ n s d s ðt T c Þþ 2 n s : ð75þ Here and are the Ginzburg Landau parameters. We omit the gradient term in the Ginzburg Landau equation since the gradients of the phase are included in the energy (73), whereas the gradients of the superconducting electron density can be

32 98 I. F. Lyuksyutov and V. L. Pokrovsky neglected everywhere beyond the vortex cores. Minimizing the total free energy, Pokrovsky and Wei [33] have found that near T c the FSB free energy can be represented as F s ¼ 2 ðt T r Þ 2 d 2 s ð76þ where T r is given by the equation: T r ¼ T c þ 64pm2 e 2 m s c 2 exp dw 4m 2 þ C 1 : ð77þ The S-phase is stable if its free energy equation (76) is less than the free energy of a single F-film with the stripe domain structure, which has the following form [99, 100]: F m ¼ 4m2 ð78þ L f where L f is the stripe width of the single F-film. Near the S-transition point the temperature dependence of the variation of this magnetic energy is negligible. Hence, when T increases, the S-film transforms into a normal state at some temperature Tc below T r. This is the first-order phase transition. At the transition point, both energies F s and F m are equal to each other. The shift of the transition temperature is determined by the following equation: T c T c T c ¼ 64pm2 e 2 m s c 2 exp sffiffiffiffiffiffiffiffiffiffiffiffiffiffi dw 4m 2 þ C 1 8m 2 2 : ð79þ d s L f Two terms in equation (79) play opposite roles. The first one is due to the appearance of spontaneous vortices which lowers the free energy of the system and it enhances the transition temperature. The second term is the contribution of the purely magnetic energy, which tends to decrease the transition temperature. The values of the parameters entering equation (79) can be estimated as follows. The dimensionless Ginzburg Landau parameter is ¼ 7:04T c = F, where F is the Fermi energy. A typical value of is about 10 3 for low-temperature superconductors. The second Ginzburg Landau parameter is ¼ T c =n e, where n e is the electron density. For estimates, Pokrovsky and Wei [33] took T c 3K,n e cm 3. The magnetization per unit area m is the product of the magnetization per unit volume M and the thickness of the FM film d m. For typical values of M 10 2 Oe and d m 10 nm, m 10 4 Gs/cm 2. In an ultrathin magnetic film the observed values of L f vary in the range mm [101, 102]. If L f 1 mm, d s ¼ d m ¼ 10 nm, and expð ~ dw =4m 2 þ C 1Þ 10 3, T c =T c 0:03. For L f ¼ 100 mm, d s ¼ 50 nm, and expð ~ dw =4m 2 þ C 1Þ 10 2, T c =T c 0: Transport properties of the FSB. The spontaneous domain structure violates the initial rotational symmetry of the FSB. Therefore, it makes the transport properties of the FSB anisotropic. Kayali and Pokrovsky [34] have calculated the periodic pinning force in the stripe vortex structure resulting from a highly inhomogeneous distribution of the vortices and antivortices in the FSB. The transport properties of the FSB are associated with the driving force acting on the vortex lattice from an external electric current. In the FSB the pinning force is due to the interaction of the domain walls with the vortices and antivortices and the vortex vortex

33 Ferromagnet superconductor hybrids 99 Y b - a L X Figure 16. Schematic vortex distribution in the FSB. The sign refers to the vorticity of the trapped flux. From Kayali and Pokrovsky [34]. interaction, U vv. Periodic pinning forces in the direction parallel to the stripes do not appear in continuously distributed vortices. Therefore, one needs to modify the theory [31] to incorporate the discreteness effects. Kayali and Pokrovsky [34] have shown that, in the absence of a driving force, the vortices and antivortices line themselves up in straight chains (see figure 16) and the force between two chains of vortices falls off exponentially as a function of the distance separating the chains. They also argued that the pinning force in the direction parallel to the domains drops faster in the vicinity of the superconducting transition temperature T s and vortex disappearance temperature T v. In the presence of a permanent current there are three kinds of forces acting on a vortex. They are: 1. the Magnus force proportional to the vector product of the current density and 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 domain walls. In the continuous limit the parallel component obviously vanishes. This means that it is exponentially small if the distances between vortices are much less than the domain width. The sum of all three forces must be zero. This equation determines the dynamics of the vortices. It was solved under the simplifying assumption that vortices inside one domain move with the same velocity. The critical currents have been calculated for the parallel and perpendicular orientations. Theory predicts a strong anisotropy of the critical current. The ratio of the parallel to perpendicular critical current is expected to be in the range at temperatures close to the superconducting transition temperature T s or to the vortex disappearance temperature T v. The anisotropy decreases rapidly when the temperature moves away 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

34 100 I. F. Lyuksyutov and V. L. Pokrovsky in the current perpendicular and parallel to the domain walls. When the direction of the current is perpendicular to the domains, the friction force makes all the vortices drift in the direction of the current, whereas the Magnus force induces the motion of vortices (antivortices) in neighbouring domains in opposite directions, both perpendicular to the current. The motion of all vortices perpendicular to the domains forces domain walls to move in the same direction. This is a Goldstone mode; no perpendicular pinning force appears in this case. The periodic pinning in the parallel direction and together with that in the perpendicular critical current is exponentially small. In the case of parallel current the viscous force draws all vortices into the parallel motion along the domain walls and in alternating motion perpendicularly to them. The domain walls remain unmoving and provide very strong periodic pinning force in the perpendicular direction. This anisotropic transport behaviour could serve as a diagnostic tool to discover spontaneous topological structures in magnetic superconducting systems Experimental studies of the FSB. In the preceding theoretical section we assumed that the magnetic film changes its magnetization direction in a weak external field and achieves the equilibrium state. In all experimental works, magnets with magnetization perpendicular to the film were Co/Pt or Co/Pd multilayers. They have a large coercive field and are quenched at low temperature, necessary for the S-state. Lange et al. [ ] have studied the phase diagram and pinning properties of such magnetically quenched FSB. In these works the average magnetization was characterized by a parameter s, the fraction of spins directed up. Magnetic domains in Co/Pd(Pt) multilayers look like meandering irregular bands at s ¼ 0.5 (zero magnetization) (see figure 17b) and like irregular localized domains (see figure 17d) with typical size mm near fully magnetized states (s ¼ 0ors ¼ 1). The stray field from domains is maximal at s ¼ 0:5. It suppresses the superconducting transition temperature T c of the Pb film by 0.2 K (see figure 18). The effective penetration depth is about 0.76 mm at 6.9 K. When s is close to 0 or 1, Lange et al. [ ] observed asymmetry of the physical properties in the applied magnetic field similar to the asymmetry of magnetization observed in the hybrid of an S-film and array of magnetic dots with magnetization normal to the film (see Section 3.2.1). The asymmetry occurred in the dependence of T c (H) and in transport properties. The localized domains have a perpendicular magnetic moment. If the thickness and magnetization are sufficiently large, these domains can pin vortices induced by the external magnetic field. In this respect 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 this situation, when s 0:5, the randomly bent band domains destroy the long-range order 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 a resistive state. This qualitative difference between the magnetized and demagnetized state was observed in the experiments by Lange et al. [ ]. The above qualitative picture of vortex pinning is close to that developed by Lyuksyutov and Pokrovsky [26] and by Feldman et al. [29] for the transport properties of a regular array of magnetic dots with random normal magnetization (see Section 3.2.4). In this model the demagnetized state of the dot array is associated with vortex creep through the percolating network. The regularly magnetized state, on the other hand, provides more regular vortex structure and enhances pinning.

35 Ferromagnet superconductor hybrids (a) (d) s = 1 M fm / M sat (c) µh 0 H (mt) (mt) s = 0 (b) (c) (d) Figure 17. Magnetic properties of the Co/Pt multilayer: (a) hysteresis loop measured by the magneto-optical Kerr effect with H perpendicular to the sample surface. MFM images (5 5 mm 2 ) show that at s ¼ 0.5 the domains look like bands (b); localized domains at s ¼ 0.3 (c); and s ¼ 0.93 (d). (From Lange et al. cond-mat/ ) 7.25 T c (H=0) (K) Figure 18. Dependence of the critical temperature at zero field T c ðh ¼ 0Þ on the parameter s. The minimum value of T c is observed for s ¼ 0.5. (From Lange et al. cond-mat/ ). s

36 102 I. F. Lyuksyutov and V. L. Pokrovsky (a) l y δ FM d M x (b) y FM SC x Figure 19. Magnetic charges (þ and ) and magnetic flux (thin lines with arrows) in a ferromagnetic film (FM) without (a) and with (b) a superconducting substrate (SC). The magnetization vectors in domains are shown by thick arrows. (From Sonin cond-mat/ ) Thick films. Above we reviewed properties of S/F-bilayers when both the F- and S-films are thin, namely, d s L and d m L f. In this subsection we consider, following work by Sonin [48], the situation when both films are thick: d s L and d m L f. We neglect the domain wall width, considering it as a line. Let us study the properties of an isolated F-film without a superconductor. This problem was solved exactly by Sonin [106]. Figure 19(a) shows schematically the magnetic field distribution around a thick ferromagnetic film. The analytic expression for the magnetic field at the boundary of the ferromagnetic film y ¼ 0 þ reads: H x ðxþ ¼ 4Mln tan px 2L f : ð80þ H y ðxþ ¼2pMsign tan px at y!0: 2L f ð81þ The field pattern is periodic with period 2L f along the x-axis. Sonin argued [48] that in the bulk superconductor, under the additional constraint L =L f! 0, the magnetic flux from the F-film does not penetrate into the superconductor even to the London penetration depth. Instead the magnetic force lines turn near the interface back to the ferromagnet, rearranging the domains. Therefore the energy changes in the presence of the superconducting substrate by a 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

37 Ferromagnet superconductor hybrids 103 proportional to the domain width L f with the coefficient (the domain wall energy per unit length in the perpendicular direction, dw ) changing in the presence of the S-film since the effective length of the domain changes. The stray field energy is proportional to L f, but the coefficient of this linear function does not change. The minimum p ffiffiffiffiffiffi of the energy is proportional to dw, whereas the equilibrium domain width Lf is inversely proportional to the same value. Therefore, the domain width L f decreases discontinuously at the S-phase transition, reducing by the factor 1.5. The correction to the energy has relative order of magnitude L =L f [48] Domain wall superconductivity. Buzdin and Melnikov [107] and Aladyshkin et al. [52] considered the nucleation of superconductivity in a thin S-film (d s L )in the presence of a magnetic field varying in space and generated by a thick F-film (d m L f ). We review their results in this subsection. The distribution of magnetic field around a thick magnetic film shown in figure 19 is only slightly modified by a thin S-film. For suppression of the order parameter in a thin S-film only the component of the magnetic field normal to the film matters. It changes sign at the domain wall and vanishes at the domain wall centre, facilitating nucleation of superconductivity. 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 variable B z ðxþ ¼H þ bðxþ, where H is the uniform external magnetic field far from the S/F-interface and b(x) is the z-component of the field induced by the magnetization M ¼ MðxÞz 0. The linearized GL equation for the order parameter reads: rþ 2pi 2 A ¼ 1 0 2, ð82þ ðtþ where AðrÞ p is the vector potential, BðrÞ ¼rAðrÞ, 0 is the flux quantum, ðt Þ¼ 0 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 T=T c0 is the coherence length, and T c0 is the critical temperature of the bulk superconductor at B ¼ 0. For a single domain wall of negligible thickness the magnetization M can be represented in the form M ¼ MsignðxÞz 0. Then the dependence of the magnetic field on the coordinate at the S/F-interface reads B z ¼ 4pMsignðxÞþH. The vector potential corresponding to this field has only a y-component, A y ¼ 4pMjxjþHx. The nucleation of superconductivity in such a field is similar to the surface nucleation of superconductivity near H c3 [96, 108]. Aladyshkin et al. [52] found the nucleation temperature for an isolated domain wall in an external magnetic field: TðhÞ T c0 2p2 0B 0 a þ bh 2 þ ch 4 ð83þ T c0 0 where h ¼ H=B 0, and B 0 is the maximum absolute value of the field b(x) (B 0 ¼ 4pM). a, b and c are constants obeying two relations: a þ c ¼ 1=2 and 2a b ¼ 1=2, so that only one of these constants is independent. For the specific model they calculated a ¼ The negative coefficient a in equation (83) is due to the stray fields of the domain walls, which suppress superconductivity. The coefficient b in this equation is positive. This means that the transition temperature increases in a weak external field. This enhancement of the transition temperature is due to the decrease of the total field (stray field 4pM plus external field) in half of the domains. The local decrease of field enhances the superconductivity nucleation. The shift of transition temperature at nucleation near the domain wall according to

38 104 I. F. Lyuksyutov and V. L. Pokrovsky calculations in [52] is rather large: for niobium film with T c 9 K and magnetization in the range 4pM 1 10 koe it may be 1 3 K. In a recent experiment, Yang et al. [109] prepared a 50 nm thick Nb film upon a single-crystal ferromagnet BaFe 12 O 19 separated by a 10 nm thick Si insulating layer. Due to the strong magnetic anisotropy of BaFe 12 O 19 its magnetization is perpendicular to the superconducting Nb film. The domain width is much larger than the domain wall width and the S-film thickness. Therefore, the demagnetizing effect of the S-film is very weak. The external magnetic field normal to the surface of the superconducting film weakens the total field above the domains with the magnetization antiparallel to the field resulting in an increase of T c in a field up to 0.4 T. The field dependence of T c observed by Yang et al. [109] was similar to that predicted by Aladyshkin 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-layered systems are based on the Larkin Ovchinnikov Fulde Ferrel (LOFF) effect, first proposed for homogeneous systems with coexisting superconductivity and ferromagnetism [53, 54]. They predicted that the superconducting order parameter in the presence of an exchange field should oscillate in space. The physical picture of this oscillation is as follows. In a singlet Cooper pair the electron with spin projection parallel to the exchange field acquires the energy h, whereas the electron with antiparallel spin acquires the energy þh. Their Fermi momenta therefore split by the value q ¼ 2h=v F. The Cooper pair acquires such a momentum and therefore its wavefunction oscillates. The direction of the modulation vector in the bulk superconductor is arbitrary, but in the S/F-bilayer the preferential direction of the modulation is determined by the normal to the interface (z-axis). There exist two kinds of Cooper pairs differing in the direction of the momentum of the electron whose spin is parallel to the exchange field. The interference of the wavefunctions for these two kinds of pairs leads to the standing wave FðzÞ ¼F 0 cos qz: ð84þ A modification of this consideration for the case when the Cooper pair penetrates into a ferromagnet from a superconductor was proposed by Demler et al. [112]. They argued that the energy of a singlet pair is higher than the energy of two electrons in the bulk ferromagnet by the value 2h (the difference of exchange energy between spin-up and spin-down electrons). This can be compensated if the electrons slightly change their momentum so that the pair will acquire the same total momentum q ¼ 2h=v F. The value l m ¼ v F =h, called the magnetic length, is a natural lengthscale for the LOFF oscillations in a clean ferromagnet. Anyway, equation (84) shows that the sign of the order parameter changes in the ferromagnet. This oscillation leads to a series of interesting phenomena that will be listed here and considered in some detail in next subsections.

39 Ferromagnet superconductor hybrids periodic transitions from the 0- to p-phase in the S/F/S Josephson junction when varying the thickness d f of the ferromagnetic layer and temperature T; 2. oscillations of the critical current versus d f and T; 3. oscillations of the critical temperature versus the thickness of magnetic layer. The penetration of the magnetized electrons into superconductors strongly suppresses the superconductivity. This obvious effect is accompanied with the appearance of magnetization in the superconductor. It penetrates to the depth of the 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 be considered later is the preferential antiparallel orientation of the two F-layers in the S/F/S-trilayer. This simple physical picture can also be treated in terms of the Andreev reflection at the boundaries [55], long known to form the in-gap bound states [96, 114]. Due to the exchange field the phases of Andreev reflection in the S/F/S-junction are different from 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 the interfaces [122]. The pair of electrons emitted from this point at angle, ðp Þ to the z-axis will be reflected as a hole along the same lines and returns to the same point (figure 20). The interference of the Feynman amplitudes for these four trajectories creates an oscillating wavefunction of the Cooper pair. The main contribution to the total wavefunction arises from the vicinity of ¼ 0. Taking only this direction, we find for the phases: S 1 ¼ S 2 ¼ qz; S 3 ¼ S 4 ¼ qð2d f zþ. Summing up all Feynman s amplitudes, e is k ; k ¼ 1,..., 4, we find the spatial dependence of the order parameter: F / cos qd f cos qðd f 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. The solid lines correspond to electrons, the dashed lines to holes; the arrows indicate the direction of the velocity. (From Fominov et al. cond-mat/ )

40 106 I. F. Lyuksyutov and V. L. Pokrovsky At the interface F /ðcos qd f Þ 2. It oscillates as a function of magnetic layer thickness with period l m =2 ¼ p=q ¼ pv F =2h and decays due to the interference of trajectories with different. In a real experimental set-up, the LOFF oscillations are strongly suppressed by elastic impurity scattering. The trajectories are diffusive random paths and a simple geometrical picture is no longer valid. However, as long as the exchange field h exceeds or is of the same order of magnitude as the scattering rate in the ferromagnet, 1= f, the oscillations do not disappear completely. Unfortunately, experiments with strong magnets possessing large exchange fields are not reliable since the period of oscillations goes to the atomic scale. Two layers with different thickness when they are so thin can have different structural and electronic properties. In this situation it is very difficult to ascribe unambiguously the oscillations of properties to quantum interference. The Andreev reflection at the interface of non-magnetic S- and metallic F-wires changes the contact resistance in comparison to the contact of the same wires when non-magnetic wire is in the normal state (NF contact) [115]. In the next subsection we consider this phenomenon in some detail Resistance of the SF contact in wires and the spin-filtering effect De Jongh and Beenakker [115] have noted that the superconducting transition changes drastically the reflection and transmission of the carriers at the interface of magnetic and non-magnetic metals. If the contact was completely penetrable for normal electrons, it becomes impenetrable for electrons in the ferromagnet if their energy is less than the gap in superconducting wire. Nevertheless, a current through the contact is possible due to the Andreev process: an electron approaching the boundary is 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 the reflected hole has spin opposite to that of the incident electron, the Andreev reflection becomes impossible if the F-wire is completely polarized and there are no electrons with opposite spin in it. This phenomenon is called the spin-filtering effect. For a simple qualitative treatment of the contact conductance, de Jongh and Beenakker applied the Landauer formalism (see, e.g. [116]). Let the numbers of channels with spins up and down near the interface be N " and N #, respectively, and all modes pass through 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: G FN ¼ e2 h N " þ N # : ð86þ When one of the wires becomes superconducting, all minority carriers are reflected by the Andreev process and the holes with opposite spin can find a place inside the big 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 Fermi momentum of minority carriers p F#. In other words only the fraction N # =N " of the majority carriers can be reflected as holes to the ferromagnet. At each Andreev process a new Cooper pair with charge 2 enters the superconductor. Therefore: ¼ 4 e2 h N #: G FS ¼ e2 h 2 N # þ N # N " N " ð87þ

41 Ferromagnet superconductor hybrids 107 This can be larger or smaller than G FN depending on what is larger 3N # and N ". Note that G FS ¼ 0ifN # ¼ 0. A more accurate treatment is based on the Bogolyubov de Gennes formalism. In the framework of this formalism the quasi-particle wavefunctions are described by the Bogolyubov two-dimensional vector u ", v # depending on the coordinates and satisfying the following equation: H 0 h u" ¼ " u " ð88þ H 0 þ h v # v # with H 0 ¼ðp 2 =2mÞ E F 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 and that the F-wire occupies the negative x half-axis, whereas the S-wire occupies the positive half-axis. Therefore they accepted hðxþ ¼h 0 Yð xþ for the exchange energy and ðxþ¼yðxþ for the superconducting gap, where Y(x) is the step function. In the standard approximation of tunnelling amplitudes the FS-junction conductance was calculated by Takane and Ebisawa [117]. It reads: G FS ¼ 2 e2 h X ¼", # Tr r y h, e r h, e, ð89þ where the Andreev reflection matrix r is given by its matrix elements r h, e ¼ 2iq p ffiffiffiffiffiffiffiffiffiffi k " k # k " k # þ q 2 : ð90þ The wavevectors for electrons and holes propagating in the S- and F-wires are: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mðE q ¼ F E n Þ h 2 ð91þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mðE k ", # ¼ F E n h 0 Þ h 2, ð92þ where E n is the energy in the n-th channel. The trace in equation (89) can be replaced by integration with the following result: G FS ¼ 4 e2 h N 4 hpffiffiffiffiffiffiffiffiffiffiffiffiffi þ 4 i 6 þ ð93þ where ¼ h 0 =E F and N 0 ¼ k 2 F=4p is the number of channels, and is the cross-sectional area. Since N " þ N # ¼ 2N 0, one finds that G FS < G FN if >0:47 or N # =N " <0:36 instead of 1/3 following from qualitative theory. This result is impressive, however the assumption of very clean wires and ballistic transport in 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 the S- and F-wires. The F-wire is characterized by the current polarization s defined as follows: s ¼ j " j # j " þ j # ¼ D " " D # # D " " þ D # # ð94þ where D ¼ð1=3Þv l are diffusion coefficients, are densities of states (DOS) and l are the mean free paths for different directions of spin. The authors assumed that

42 108 I. F. Lyuksyutov and V. L. Pokrovsky the scattering in the F-wire is strong, but completely elastic. To calculate the resistance (conductance) of a junction with a diffusive F-layer it is necessary to solve modified diffusion equations for the density N ðz, "Þ, ¼"; # at fixed energy: D o 2 zn ðz, "Þ ¼w "# ½N ðz, "Þ N ðz, "ÞŠ, ð95þ where w "# is the rate of spin-reversal processes. They are equivalent to the following two equations with separated variables: X o 2 z D N ¼ 0; o 2 z L 2 s N" N # ¼ 0 L 2 s ¼ w "# ð " # Þ: ð96þ ¼", # These equations must be solved with certain boundary conditions. At the left end the condition is N ð L, "Þ ¼1=2 ½n T ð" evþþn T ð " evþš, where n T ð"þ is the equilibrium occupation number for quasi-particles. Here the electron hole symmetry is used; " is the energy of the quasi-particle minus the chemical potential. The geometry is shown in figure 21. As already indicated earlier, the Andreev reflection is forbidden for those majority electrons whose momentum is larger than the Fermi momentum of the minority carriers. Since the projection of momentum along the interface is conserved (interfaces are ideal), this constraint can be formulated in terms of projections instead of full momenta (see figure 21). It is clear that such processes depend crucially on the ratio of Fermi momenta ¼ p F# =p F". In the case of a long F-wire L l #, the Andreev reflection is effective at a short distance L s from the interface and can be accounted for by effective boundary conditions, which match the isoenergetic diffusive currents and densities from inside to that near the interface with normal ballistic layer for "<: D " " o z N " ¼ D # # o z N # ð97þ N " þ N # þ 2 3=2l# 1 2 o 3 z N # ¼ n T ð"þþn T ð "Þ: ð98þ F - Reservoir L Diffusive F - Wire L s N S-Reservoir II I e h h I F S II e F e N S e e l α e h Figure 21. Pictorial representation of the FS-junction, double Andreev reflection processes in it, and of possible relations between the Fermi surfaces of spin-up and spin-down electrons in the F-wire (left) and in the N/S metal (right). (From Falko et al. cond-mat/ )

43 Ferromagnet superconductor hybrids 109 The interface resistance is determined by the following equation: r i ¼ R L L s 2 s 1 s 2 þ l 1 2 3=2! " 2 ð1 þ sþ ð99þ where R is the total resistance of the F-wire in the absence of the SF-junction. Both ratios L s =L and l " =L are small, but this smallness can be compensated either by almost complete spin polarization of the currents ðs 1Þ or by the large ratio of numbers of majority and minority carriers 1 1. The most important and experimentally verifiable result of this work is the change of the total resistance of the F-wire in transition from the normal to superconducting state in the S-wire: R s ðt Þ R N R N ¼ s2 L s 1 s 2 L tanh ðtþ 2T : ð100þ It is always positive since the superconductivity enhances the reflection. It becomes infinite at s ¼ 1. The influence of spin-filtering on the subgap I(V ) characteristics of FS-junctions has been studied by McCann and Fal ko [120] and by Tkachov, McCann, and Fal ko [121] Non-monotonic behaviour of the transition temperature This effect was first predicted by Radovic et al. [65]. Its reason is the LOFF oscillations described in Section 4.1. If the transparency of the S/F-interface is low, one can expect that the order parameter in the superconductor is not strongly influenced by the ferromagnet. On the other hand, the condensate wavefunction at the interface in the F-layer F /ðcos 2d f =l m Þ 2 becomes zero at d f ¼ pl m ðn þ 1=2Þ=2 (n is an integer). At these values of thickness the discontinuity of the order parameter at the boundary and, together with it, the current of Cooper pairs into the ferromagnet, has a maximum. Therefore one can expect that the transition temperature is minimal [123]. Experimental attempts to observe this effect were made many times with S/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 description can be found in the cited reviews [61, 62]. Unfortunately, in these experiments the magnetic component was a strong ferromagnet and, therefore, they faced all the difficulties mentioned in Section 3.3.1: the F-layer must be a few angstroms thin to be comparable with the magnetic length and its variation produces uncontrollable changes in the sample, and the influence of the growth defects is too strong. Furthermore, in multilayers the reason of the non-monotonic dependence of T c on d f may be the 0 p transition. Therefore, a reliable experiment should be performed with a bilayer possessing a sufficiently thick F-layer. Such experiments have been performed recently [129, 130]. The idea was to use a weak ferromagnet (the dilute ferromagnetic alloy p Cu Ni) with rather small exchange field h to increase the magnetic length l m ¼ ffiffiffiffiffiffiffiffiffiffi D f =h. 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 the F-layer was of the same order of magnitude as the magnetic length. Therefore, for the quantitative description of the experiment, theory should not be restricted by

44 110 I. F. Lyuksyutov and V. L. Pokrovsky limiting cases only. Such a theory was developed by Fominov et al. [122]. In the pioneering work by Radovic et al. [65] the exchange field was assumed to be very strong. At critical temperature, the energy gap and anomalous Green function F are infinitely small. Therefore one needs to solve the linearized equations of superconductivity. The approach by Fominov et al. is based on solution of the linearized Usadel equation and is valid in the diffusion limit s T c 1; f T c 1; f h 1. In fact, this situation was realized in the cited experiments [129, 130]. The work by Fominov et al. [122] covers numerous works by their predecessors [68, 112, 123, 131, 132] clarifying and improving their methods. Therefore in the presentation of this subsection we follow mainly the cited work [122] and briefly describe specific results of other works. The starting point is the linearized Usadel equations for anomalous Green functions F s in the superconductor and F f in the ferromagnet: D s o 2 F s oz 2 j! njf s þ ¼ 0; 0 < z < d s ð101þ o 2 F f D f oz 2 ðj! njþih sgn! n ÞF f ¼ 0; d f < z < 0: ð102þ Thus, we accept a simplified model in which ¼ 0 in the ferromagnetic layer and h ¼ 0 in the superconducting layer. The geometry is schematically shown in figure 22. Equations (102) and (101) must be complemented with the self-consistency equation: ðrþ ln T cs T ¼ pt X n ðrþ j! n j F sð! n, rþ, ð103þ where T cs is the bulk SC transition temperature, and with linearized boundary conditions at the interface: df s s dz ¼ df f f ð104þ dz df f A f dz ¼ G b F s ð0þ F f ð0þ, ð105þ F S d f 0 d s Figure 22. FS-bilayer. The F- and S-layers occupy the regions d f < z < 0 and 0 < z < d s, respectively.

45 Ferromagnet superconductor hybrids 111 where s, f is the conductivity of the superconducting (ferromagnetic) layer in the normal state, G b 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 to zero at the interface with the vacuum: df f dz j z ¼ d f ¼ df s dz j z ¼ d s ¼ 0: ð106þ The condition of solvability of the linear equations (101) (103) with the boundary conditions (104) (106) determines the value of transition temperature T c for the F/S-bilayer. The solution F f ð! n, zþ in the F-layer satisfying the boundary condition (106) reads: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j! F f ð! n, zþ ¼Cð! n Þ cosh½k fn ðz þ d f ÞŠ; k fn ¼ n jþih sgn! n, ð107þ where Cð! n Þ is the integration constant to be determined from the matching condition at the F/S-interface z ¼ 0. From the two boundary conditions at z ¼ 0 (104), (105) it is possible to eliminate F f and df f /dz and reduce the problem to finding the function F s from equation (101) and the effective boundary condition at z ¼ 0: df s s dz ¼ b þ B f ð! n Þ F s, ð108þ p where s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D s =ð2pt cs Þ, ¼ f = s, b ¼ðA f Þ=ð s G b Þ and B f ð! n Þ¼ ðk fn s tanhðk fn d f ÞÞ 1. Since k fn is complex the parameter B f ð! n Þ and consequently the function F s 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 F s traditionally denoted as Fs þ ð! n, zþ 1=2ðF s ð! n, zþþf s ð! n, zþþ. The boundary condition for this function reads: dfs þ s dz ¼ Wð! nþfs þ A sn ð b þ<b f Þþ j z¼0 ; Wð! n Þ¼ A sn j b þ B f j 2 þ ð b þ<b f Þ, ð109þ p where A sn ¼ k sn d s tanhðk sn d s Þ and k sn ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D s =j! n j. To derive this boundary condition we accept the function (z) to be real (this will be justified later). Then the imaginary part of the anomalous Green function Fs ð! n, zþ obeys the homogeneous linear differential equation d 2 F s dz 2 ¼ k 2 snf s and the boundary condition dfs =dz ¼ 0 at z ¼ d s. Its solution is Fs ð! n, zþ ¼ Eð! n Þ cosh½k sn ðz d s ÞŠ. At the interface z ¼ 0 its derivative dfs =dzj z¼0 is equal to k sn tanhðk sn d s ÞFs ð! n, z ¼ 0Þ. Eliminating Fs and its derivative from the real and imaginary parts of the boundary condition (108), we arrive at the boundary condition (109). Note that only Fs þ participates in the self-consistence equation (103). This fact serves as justification of our assumption on the reality of the order parameter (z). Simple analytic solutions of the problem are available for different limiting cases. 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 D f

46 112 I. F. Lyuksyutov and V. L. Pokrovsky parameters vary. Let us consider the case of a very thin S-layer d s s. In this case the order parameter is almost a constant. The solution of equation (101) in such a situation is Fs þ ð! n, zþ ¼ j! n j þ n cosh k sn ðz d s Þ, where n is an integration constant. From the boundary condition (109) we find: 2Wð! n ¼ n Þ j! n jða sn þ Wð! n ÞÞ, ð110þ where the coefficients A sn are the same as in equation (109). We assume that k sn d s 1. Then A sn k 2 sn s d s ¼ðd s = s Þðn þ 1=2Þ. The function Fs þ ð! n, zþ almost does not depend on z. The self-consistence equation reads: ln T cs ¼ 2 X Wð! n Þ T c ðn þ 1=2Þðd n0 s = s ðn þ 1=2ÞþWð! n ÞÞ : ð111þ The summation can be performed explicitly in terms of digamma functions (F): ln T cs ¼ s T c 2ð b þ<b f Þd s < 1 ið b þ<b f Þ F 1 =B f 2 þ s 1 i=b f F 1 : ð112þ d s b þ<b f 2 Possible p oscillations are associated with the coefficients B f. If the magnetic length l m ¼ ffiffiffiffiffiffiffiffiffiffi D f =h is much less than d f, then sffiffiffiffiffiffiffiffiffiffiffi h B f exp 4pT cs 2id f l m ip 4 In the opposite limiting case l m d f there are no oscillations of the transition temperature. Note that lnðt cs =T c Þ can be rather large, s =d s, i.e the transition temperature in the F/S-bilayer with a very thin S-layer can be exponentially suppressed. 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 parameters d s = s, d f =l m,, b are very small or very large and an exact method of solution should be elaborated. The authors propose to separate explicitly the oscillating part of the functions F þ s ð! n, zþ and (z) and the reminders: : F þ s cos qðz d ð! n, zþ ¼f s Þ n cos qd s þ X1 m¼1 f nm cosh q m ðz d s Þ cosh q m d s ð113þ ðzþ ¼ cos qðz d sþ þ X1 cosh q m ðz d s Þ cos qd m s cosh q m¼1 m d s ð114þ where the wavevectors q and q m, as well as the coefficients of the expansion, must be found from the boundary conditions and self-consistency equation. Equation (101)

47 Ferromagnet superconductor hybrids 113 results in relations between the coefficients of the expansion: f n ¼ j! n jþd s q 2 ; f nm ¼ m j! n j D s q 2 : ð115þ m Substituting the values of the coefficients f n, f nm from equation (115) into the boundary condition (109), we find an infinite system of homogeneous linear equations for the coefficients and m : q tan qd s Wð! n Þ j! n jþd s q 2 ¼ X1 m¼1 q m tanh q m d s þ Wð! n Þ m j! n j D s q 2 : ð116þ m Equating the determinant of this system D to zero, we find a relation between q and q m. 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 that all coefficients m, m ¼ 1, 2,..., are zero and only the coefficient survives. The system (116) implies that this is only possible when the coefficients Wð! n Þ do not depend on their argument! n. This happens indeed in the limit d s = s 1 and h T. For a more realistic regime the equation D¼0 must be solved numerically together with the self-consistency condition, which becomes the system of equations:! ln T cs ¼ F 1 T c 2 þ D sq 2 F 1 2 ln T cs T c ¼ F T c 1 2 D sq 2 m T c! F 1 : ð117þ 2 This systems was truncated and solved with all data extracted from the experimental set-up used by Ryazanov et al. [130]. The only two fitting parameters were h ¼ 130 K and b ¼ 0:3. Figure 23 demonstrates rather good agreement between theory and experiment. Various types of curves T c (d f ) are shown in figure 24. Note that the minimum on these curves eventually turns into a plateau at T c ¼ 0, the re-entrant phase transition into the superconducting state. Some of the curves have a well-pronounced discontinuity, which can be treated as a first-order phase transition. The possibility of a first-order transition to the superconducting state in the F/S-bilayer was first indicated by Radovic et al. [65] Josephson effect in S/F/S-junctions As we already mentioned, the exchange field produces oscillations of the order parameter inside the F-layer. This effect in turn can change the sign of the Josephson current in the S/F/S-junction compared to the standard S/I/S- or S/N/S-junctions. As a result the 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 magnetic flux and spontaneous current appear in the ground state. These phenomena were first predicted by Bulaevsky et al. [56] for p-junctions independently of how it was realized. 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 Cu x Ni 1 x as a ferromagnetic layer. A similar approach was developed a little later by Kontos et al. [60], who used a diluted PdNi

48 114 I. F. Lyuksyutov and V. L. Pokrovsky Figure 23. Theoretical fit to the experimental data [130]. In the experiment, Nb was the superconductor (with d s ¼ 11 nm, T cs ¼ 7 K) and Cu 0.43 Ni 0.57 was the weak ferromagnet. The best fit corresponds to h 130 K and b 0:3. (From Fominov et al. cond-mat/ ) p Figure 24. Characteristic types of T c (d f ) dependence. The length-scale is ex ¼ ffiffiffiffiffiffiffiffiffiffi h=d f ; the exchange energy is h ¼ 150 K; other parameters are the same as in figure 23. There are three types of curves: (1) non-monotonic decay to a finite T c with a minimum ( b ¼ 0:07 2); (2) re-entrant behaviour with the first-order transitions at small d f ( b ¼ 0:02, 0:05); (3) monotonic decay to T c ¼ 0 at finite d f ( b ¼ 0). (From Fominov et al. cond-mat/ ) alloy. The weakness of the exchange field allowed them to drive the oscillations and in particular the 0 p transition by temperature at a fixed magnetic field. The success of this experiment has generated an extended literature. Theoretical and experimental studies of this and related phenomena are still active. In what follows we present a brief description of relevant theoretical ideas and experiments Simplified approach and experiment. Here we present a simplified picture of the S/F/S-junction based on the following assumptions: 1. The transparency of the S/F-interfaces is small. Therefore the anomalous Green function in the F-layer is small and it is possible to use the linearized Usadel equation.

49 Ferromagnet superconductor hybrids The energy gap inside each of the S-layers is constant and equal to 0 e i =2 (the sign relates to the left S-layer, þ to the right one). 3. ¼ 0 in the F-layer and h ¼ 0 in the S-layers. The geometry of the system is shown in figure (22). From the second assumption it follows that the qanomalous ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Green function F is also constant within each of the S-layers: F ¼ = j! n j 2 þ 2 0. The linearized Usadel equation in the F-layer (102) has the following general solution: where Fð! n, zþ ¼ n e k fnz þ n e k fnz, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j! k fn ¼ n jþih sgn! n D f ð118þ ð119þ (compare equation 107). The boundary condition at the two interfaces follows from the second boundary condition of the previous section (105) in which F f is neglected: df f f dz ¼ bf s : ð120þ The coefficients n and n are completely determined by the boundary conditions (120): cosðð ik fn d f Þ=2Þ n ¼ Q n ð121þ sinhðk fn d f Þ n ¼ Q n cosðð þ ik fn d f Þ=2Þ sinhðk fn d f Þ ð122þ where 0 Q n ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : b f k fn j! n j 2 þ 2 0 Equation (34) for the electric current must be slightly modified to incorporate the exchange field h: j ¼ ieptnð0þd X n ð F~ ^of F ^o FÞ, ~ ð123þ where Fð! ~ n, zþ ¼F ð! n, zþ. Note that under this transformation the wavevectors k fn remain invariant. After substitution of the solution (118) we find that j ¼ j c sin with the following expression for the critical current [58, 132, 133]: " j c ¼ 4pT2 0 < X # ð! 2 n þ 2 1 er N 0Þk fn d f sinhðk fn d f Þ, ð124þ b! n >0 where R N is the normal resistance of the ferromagnetic layer and b is a dimensionless parameter characterizing the ratio of the interface resistance to that of the F-layer. Kupriyanov and Lukichev [88] have found a relationship between b and

50 116 I. F. Lyuksyutov and V. L. Pokrovsky the barrier transmission coefficient D b () ( is the angle between the electron velocity and the normal to the interface): b ¼ 2l f cos D b ðþ : ð125þ 3d f 1 D b ðþ As we explained earlier, the oscillations pffiffiffiffiffiffiffiffiffiffiffiffiffi appear since k fn are complex values. If h 2pT! n, then k fn ð1þiþ h=2d f and oscillations are driven only by the thickness. It was very important to use a weak ferromagnet with exchange field h comparable to pt. Then the temperature also drives the oscillations. In the Cu Ni alloys used in the experiment [58] the Curie point T m was between 20 and 50 K. Nevertheless, the ratio h=pt was about 10 even for the lowest T m. In this situation k fn does not depend on n for the large number of terms in the sum (124). This p is the reason why the sum in total is a periodic function of d f with period m ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi 2D f =h. The dependence on temperature is generally weak. However, if the thickness is close to the value at which j c tends to zero at T ¼ 0, the variation of temperature can change the sign of j c. In figure (25, bottom left) theoretical curves j c (T ) from the cited work [122] are compared with the experimental data by Ryazanov et al. [58, 79]. The curves are plots of the modulus of j c versus T. Therefore, the change of sign of j c is seen as a cusp on such a curve. The position of the cusp determines the temperature of transition from the 0- to p-state of the junction. The change of sign is clearly seen on the curve corresponding to d f ¼ 27 nm. The experimental S/F/S-junction is schematically shown in figure (25, top). The details of the experiment are described in the original paper [58] and in the reviews [79, 61]. No less impressive agreement between theory and 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 in recent theoretical work by Buzdin and Baladie [134] who solved the Eilenberger equation. Zyuzin et al. [135] found that in a dirty sample the amplitude of the Josephson current j c is a random value with an indefinite sign. They estimated the average square fluctuations p of this amplitude for the interval of the F-layer thickness s < d f < ffiffiffiffiffiffiffiffiffiffi D=T as:! h jc 2 i¼as 2 g 4 2 D f 8pNð0ÞD f 2p 2 Tdf 2 ð126þ where A is the area of the interface, g is the conductance of a square and N(0) is the DOS in the F-layer. The fluctuations pffiffiffiffiffiffiffiffiffiffiffiffi are significant when d f becomes smaller than the diffusive thermal length D f =T Josephson effect in a clean system. Recently, Radovic et al. [136] considered the 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 this consideration is the simplicity of the model and very clear representation of the solution. Though in the existing experimental systems the oscillations are not disguised by impurity scattering, it is useful to have an idea of what maximal effect

51 Ferromagnet superconductor hybrids 117 Figure 25. (Top) schematic cross-section of the sample. (Bottom) left: critical current I c as a function of temperature for Cu 0.48 Ni 0.52 junctions with different F-layer thicknesses between 23 and 27 nm as indicated; right: model calculations of the temperature dependence of the critical current in an SFS-junction. (From Ryazanov et al. cond-mat/ ) 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 Gennes equations: ^H u ¼ E v u v, ð127þ

52 118 I. F. Lyuksyutov and V. L. Pokrovsky 80 T=1.5 K SIS ξ10 0 I c R n (µv) 40 I (ma) -10 SIFS V (mv) 20 0 π d F (Å) Figure 26. Josephson coupling as a function of thickness of the PdNi layer (full circles). The critical current vanishes at d F 65 A indicating the transition from 0- to p-coupling. The full line is the best fit obtained from the theory. The insert shows typical I V characteristics of two junctions with (full circles), and without (empty circles) the PdNi layer. (From Kontos et al. cond-mat/ ) where means and the effective Hamiltonian reads: ^H ¼ H 0ðrÞ hðrþ ðrþ ðrþ H 0 þ hðrþ, ð128þ H 0 ðrþ ¼ h2 2m r2 þ WðrÞ: ð129þ In the last equation is the chemical potential and W(r) is the barrier potential: WðrÞ ¼W½ðz þ d=2þþðz d=2þš: ð130þ The assumptions about the exchange field h(r) and the order parameter ðrþ are the same as in the previous subsection. We additionally assume that the left and right S-layers are identical and semi-infinite, extending from z ¼ 1 to z ¼ d=2 and from z ¼ d=2 to z ¼1. Due to translational invariance in the (x, y)-plane the dependence of the solution on the lateral coordinates is a plane wave: u ¼ e ik kr ðzþ: ð131þ v There are eight fundamental solutions of these equations corresponding to the injection of the quasi-particle or quasi-hole from the left or from the right with spin up or spin down. We will write one of them explicitly as 1(z), corresponding to the injection of the quasi-particle from the right. In the superconducting area z < d=2

53 we will see the incident quasi-particle wave with coefficient 1 and normal wavevector k þ, the reflected quasi-particle with reflection coefficient b þ and normal wavevector k þ ; the reflected quasi-hole (Andreev reflection) with reflection coefficient a 1 and wavevector k, where ðk Þ 2 ¼ 2m h 2 ðe F Þ 2 k 2 k, p E F is the Fermi energy and ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2 jj 2. Thus the solution 1(z) atz < d=2 reads: 1ðzÞ ¼ðe ikþz þ b 1 e ikþz ue Þ i =2 ve i =2 þ a 1 e ik z ve i =2 ue i =2 ð132þ p where u and v are the bulk Bogolyubov Valatin coefficients: u ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1 þ =EÞ=2; v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 =EÞ=2. In the F-layer d=2 < z < d=2 there appear transmitted and reflected electrons and transmitted and reflected holes. Since according to our assumption ¼ 0 in the F-layer, there is no mixing of the electron and hole. With this explanation we can write directly the solution 1(z) in the F-layer: 1ðzÞ ¼ðC 1 e iqþ z þ C 2 e iqþ z 1 Þ þðc 0 3 e iq z þ C 4 e iq z 0 Þ, ð133þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where q ¼ 2m=h 2 ðe f F þ h E Þ k2 k. Finally in the right S-layer z > d=2 only the transmitted quasi-particle and quasi-hole propagate: 1ðzÞ ¼c 1 e ikþ z ue i =2 ve i =2 þ d 1 e ik z ve i =2 ue i =2 : ð134þ The value of all coefficients can be established by matching of solutions at the interfaces: d 2 0 Ferromagnet superconductor hybrids 119 ¼ d 2 þ 0 ; d dz j d=2þ0 d dz j d=2 0 ¼ 2mW h 1 : ð135þ Other fundamental solutions can be found by symmetry relations: a 2 ð Þ ¼a 1 ð Þ; a 3 ¼ a 2 ; a 4 ¼ a 1 ; b 3 ¼ b 1 ; b 4 ¼ b 2, ð136þ where the index 2 relates to the hole incident from the left, and indices 3, 4 relate to the electron and hole incident from the right. Each mode generates the current independently of the others. The critical current reads: j c ¼ i et X þ kn þ k n a 1n h 2,! n, k k n k þ a 2n n k : ð137þ n Here all the values with the index n mean functions of energyp Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi denoted by the same symbols in which E is replaced by i! n, for example n ¼ i! 2 n þ 2. We will not demonstrate here the straightforward, but somewhat cumbersome calculations and transit to the conclusions. The critical current displays oscillations originating from two different types of bound states. One of them appears if the barrier transmission coefficient is small. This is geometrical resonance. The superconductivity is irrelevant for it. Another one appears even in the case of ideal transmission: this is the resonance due to the Andreev reflection. When the transmission coefficient is not small and not close to 1, it is not easy to separate these two type of resonances and the

54 120 I. F. Lyuksyutov and V. L. Pokrovsky oscillations picture becomes rather chaotic. The LOFF oscillations are better seen when the transmission coefficient is close to 1 since geometrical resonances do not interfere. Varying the thickness, one observes periodic transitions from the 0- to p-state with the period equal to f =2 ¼ 2pv F =h. The lowest value of d at which the 0 p transition takes place is approximately f =4. The temperature changes this picture only slightly, but near the thickness corresponding to the 0 p transition the non-monotonic behaviour of j c versus temperature, including the temperature driven 0 p transition, can be found. An intermediate case between the diffusion and clean limits was considered by Bergeret et al. [139]. They assumed that the F-layer is so clean that h f 1, whereas T c s 1. Therefore the Usadel equation is not valid for the F-layer and they solved the Eilenberger equation. They have found that the superconducting condensate oscillates as a function of the thickness with period f and penetrates into the F-layer over the depth equal to the electron mean free path, l f. The period of oscillation of the critical current is f =2. No qualitative differences with the considered cases appear unless the magnetization is inhomogeneous. Even a very small inhomogeneity can completely suppress the 0 p transitions. This is a consequence of the generation of triplet pairing, which will be considered later. An interesting opportunity for the Andreev bound states was indicated by Gyo rffy and coworkers [ ]. They considered an I/S/F-trilayer, where I stays for insulator. Their numerical studies as well as semiclassical solutions demonstrated that in such a system there appear many Andreev bound states close to the middle of the energy gap. They argued that such a situation is unstable and any spontaneous current will shift the levels from the centre of the gap decreasing the energy. While the idea seems interesting, it is not clear how the current flowing partly in the F-layer can avoid losses 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 the voltage V n ¼ nh!=2e (! is the frequency of the applied ac current). Let us recall that the standard (integer) Shapiro steps appear as a consequence of the resonance between the external ac field and the time-dependent Josephson energy E J ¼ ðhj c =ed f Þ cos ðtþ where the phase is proportional to time due to the external permanent voltage through the contact: ðtþ ¼2eVt=h. Just at the 0 p transition point j c tends to zero. Then the next term in the Fourier expansion of the Josephson energy proportional to cosð2 Þ dominates. That means that the Josephson current is proportional to sinð2 Þ. Such a term leads to the Shapiro steps not only at integer, but also at half-integer values, since the resonance now happens at ð4ev=hþ ¼!. Normally the term with sinð2 Þ is so small that it was always assumed to vanish completely. The resonance hf method used by the authors had sufficient sensitivity to discover this term. The authors prepared the Nb=Cu 52 Ni 48 =Nb junction by the photolithography method. The Curie temperature of the F-layer is 20 K. The two samples they used had thicknesses 17 and 19 nm. The 0 p transition was driven by temperature. The transition temperatures in the first and second sample were 1.12 and 5.36 K, respectively. The external ac current had the frequency! ¼ 800 khz and amplitude about 18 ma. The voltage current curves for d f ¼ 17 nm and temperatures close to 1.12 K are shown in figure 27. The fact that the half-integer steps disappear at very small

55 Ferromagnet superconductor hybrids Figure 27. Shapiro steps in the voltage current curve of a 17 nm thick junction with an excitation 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 1.07 K are shifted by 10 and 20 ma for clarity. (From Sellier et al. cond-mat/ ) deviation from the transition temperature proves convincingly that it is associated with the 0 p transition Spontaneous current and flux in a closed loop. Bulaevsky et al. [56] argued that a closed loop containing the p-junction may carry a spontaneous current and flux in the ground state. We reproduce their arguments below. The energy of the closed superconducting loop depends on the total flux through the loop: EðÞ ¼ h 2e J c cos þ p 2 Lc 2, ð138þ where ¼ 2p= 0, J c is the critical current and L is the inductance of the loop. The first term in equation (138) is the Josephson energy; the second is the energy of the magnetic field. The location of the energy minimum depends on the parameter k ¼ 0 =4pLJ c c. At k > 1 the absolute minimum of E c ( ) 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 the equation sin = ¼ k. Thus, the spontaneous flux appears at sufficiently large inductance of the loop. Ryazanov et al. [59] proposed a way to avoid this limitation by measuring the dependence of the current inside the loop on the external flux through it. They used a triangular bridge array with p-junctions in each shoulder (see figure 28). Due to the central p-junction the phases of the current in two subloops of the bridge differ by p. Therefore the critical current between the two contacts of the bridge is equal to zero in the absence of a magnetic field. If the flux inside the loop reaches half of the flux quantum, it compensates the indicated phase difference and the currents from both subloops are in the same phase. Thus, the shift of the current maximum from ¼ 0to¼ 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 nonzero magnetic field. Figure 30 shows the shift of the flux through the loop 0 to 1/2 of the flux quantum at the temperature driven 0 p transition.

56 122 I. F. Lyuksyutov and V. L. Pokrovsky Figure 28. Real (upper) and schematic (lower) picture of the network of five SFS-junctions Nb Cu 0.46 Ni 0.54 Nb (d F ¼ 19 nm), which was used in the phase-sensitive experiment. (From Ryazanov et al. cond-mat/ ) Quite recently Bauer et al. [145] have discovered experimentally spontaneous currents and flux in p-junctions in agreement with theoretical predictions made by Bulaevsky et al. many years ago [56] and presented at the beginning of this subsection. The experimenters deposited a Nb square loop with average linewidth 700 nm and the side of the loop 7.4 mm over a GaAs AlGaAs heterostructure. The latter served as a micro-hall sensor with a high resolution of the magnetic flux due to high mobility of the two-dimensional electron gas in it ( cm/vs at electron density 2: cm 2 ). The Josephson junction inside the Nb loop was made from a weak ferromagnet Pd 0:82 Ni 0:18. To avoid flux trapping they cooled the system in a weak magnetic field 1:7 mt down from 10 K. Such a loop had sufficiently large inductance to ensure the appearance of spontaneous current. One flux quantum in such a loop corresponded to the field 37.5 mt. For control they compared all

57 Ferromagnet superconductor hybrids 123 Figure 29. Magnetic field dependences of the critical transport current for the structure depicted in figure 28 at temperature above (a) and below (b) T cr. (From Ryazanov et al. cond-mat/ ) Figure 30. (a) Temperature dependence of the critical transport current for the structure depicted in figure 28 in the absence of a magnetic field; (b) temperature dependence (jump) of the position of the maximal peak on the curves I m ðhþ, corresponding to the two limiting temperatures depicted in figure 29. (From Ryazanov et al. cond-mat/ )

58 124 I. F. Lyuksyutov and V. L. Pokrovsky a down sweep up sweep Φ 0 c 0 Φ 0 δv (nv) H B cool =-1,72 µ T "0" Φ Φ 0 0 " π" "0" " π" -Φ 0 Φ 0 0 Φ 0 B applied ( µ T) Φ applied δv (nv) H b down sweep B cool =1,72 µ T up sweep "0" δv (nv) H d down sweep B cool =19,2 µ T up sweep "0" " π" " π" B applied ( µ T) B applied ( µ T) Figure 31. Signature of the p-state in the switching behaviour during magnetic field sweeps. (a), (b) Magnetization traces V H I S for a p- and a 0-loop. All data are taken at 2 K using a measurement current of 20 ma. The critical currents are 65 ma for the 0-junction and 27 ma for the p-junction. By choosing cooling fields slightly below and above zero applied flux and performing sweeps of the magnetic fields in both field directions, an asymmetric switching behaviour of the p-loop is observed, while the 0-loop shows symmetric jumps with respect to the cooling field. (c) Relation between the total magnetic flux, applied flux applied (linear background), and the flux LI S generated by the supercurrent in the loop. The ð applied ) 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 leads to a hysteretic switching behaviour for LI C > 0 =2. The three vertical lines in each curve indicate the applied flux during field cooling below the critical temperature for the three sets of measurements discussed. (d) Reversal of the symmetry properties of the p- and 0-loop when choosing a cooling field equal to half a flux quantum. (From Bauer et al. cond-mat/ ) 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) excited by 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 the voltage, i.e. the flux through the p-junction, is not zero, whereas it remains zero for 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 hysteresis loops in the graph (figure 31c). A non-zero flux in the p-junction is explicitly seen in this 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 case the signal from the p-loop is close to zero, whereas the signal from the 0-loop is about half of its maximal value. They also measured the temperature dependence of the same Hall voltage or the magnetic flux in the loop at zero external field and at the field corresponding to the flux quantum through the loop (figures 32 a, b).

59 Ferromagnet superconductor hybrids 125 (a) 0, ,6 0 (b) Φ( Φ 0 ) Φ( Φ 0 ) -0,25-0,50 0,25 0-0,25-0,50 π π 0 Φ applied =0 Φ applied = Φ 0 / T (K) -12,6-25,1 9,5 0-9,5-18,9 V H (nv) V H (nv) Figure 32. Temperature dependence of the spontaneous current. The measurements show the 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. Below T 5.5 K the p-loop develops a spontaneous current when cooling down in zero field, while the magnetic flux through the 0-loop remains zero. When applying a field equal to half a flux quantum 0 the roles of the p- and 0-loop are exchanged. At low temperatures the spontaneous flux of both loops saturates close to 0 =2, as expected. To avoid heating of the sensor at the lowest temperatures, the measurement current has been reduced to 7 ma. (From Bauer et al. cond-mat/ ) Figure 32(a) demonstrates that the 0-loop at zero external field retains zero flux below 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, the 0-loop develops a screening current, whereas the p-loop retains zero current F/S/F-junctions The F/N/F-trilayers (N is normal non-magnetic metal) have attracted much attention starting from the discovery by Baibich et al. [146] of giant magnetoresistance (GMR). The direction of magnetization of ferromagnetic layers in these systems may be either parallel or antiparallel in the ground state oscillating with the thickness of the normal layer on the scale of a few nanometres. The mutual orientation can be changed from antiparallel to parallel by a rather weak magnetic field. Simultaneously the resistance changes by the relative value reaching 50%. This phenomenon has already obtained a technological application in the magnetic transistors and valves used in computers [147]. A natural question is what happens if the central layer is superconducting: will it produce the spin-valve effect (a preferential mutual orientation of F-layer magnetization) and how does it depend on the thicknesses of the S- and F-layers? This question was considered theoretically by

60 126 I. F. Lyuksyutov and V. L. Pokrovsky several authors [66 69, 89, 148]. The spin-valve effect in F/S/F-junctions was experimentally discovered by Tagirov et al. [151]. Even without calculations it is clear that, independently of the thicknesses of the S- and F-layers, the antiparallel orientation of magnetizations in the F-layers always has lower energy than the parallel orientation. This happens because the exchange field always suppresses superconductivity. When the fields from different layers are parallel, they enhance this effect and increase the energy, and vice versa. The effect strongly depends on the transparency of the interfaces. Usually transparency is very small, 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 with the 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 with the S-layer whose thickness does not exceed s. The choice of the material and thickness of F-layers is dictated by the requirement that they could be reoriented by 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 fixed mutual orientation of magnetic moments in the F/S/F-trilayer. A theoretical study was performed by Baladie and Buzdin [148] for the case of a very thin superconducting layer d s s. They considered F s almost as a constant, but incorporated small linear and quadratic deviations and solved the linearized Usadel equation as shown in Section 4.3 to find the critical temperature versus thickness of the ferromagnetic layers. They found that at large b (low interface transparency) the transition temperature monotonically decreases with d f increasing from its value in the absence of the F-layers to some saturation value. There occurred no substantial difference between parallel and antiparallel orientations. At smaller values of b the suppression of T c increases and at parallel orientation the re-entrant transition occurs at d f f, but still the transition temperature saturates at large d f. For b smaller than a critical value the transition temperature becomes zero at a finite thickness d f for both parallel and antiparallel orientations. The authors also found some evidence that at low b the S-transition becomes discontinuous for the parallel orientation. This conclusion was confirmed by a recent theoretical study by Tollis [152], who proved 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. For the case of thick ferromagnetic layers d f f they found that the energy gap is a monotonically decreasing function of the dimensionless collision frequency ð f 0 Þ 1, where 0 is the value of the energy gap in the absence of the ferromagnetic layers. It tends to zero at ð f 0 Þ 1 ¼ 0:25 for the parallel and for the antiparallel orientation Triplet pairing If the direction of the magnetization in the F-layer varies due to a domain wall or artificially, the singlet Cooper pairs penetrating into the F- from S-layers will be partly transformed into triplet pairs. This effect was first predicted by Kadigrobov et al. [70] and by Bergeret et al. [71]. The singlet p pairs cannot penetrate into the F-layer deeper than to the magnetic length l m ¼ ffiffiffiffiffiffiffiffiffiffi D f =h (or v F / h for a clean ferromagnet), but neither exchange interaction nor the elastic scattering suppresses the triplet pairs.

61 p Therefore, they can penetrate to a much longer distance T ¼ ffiffiffiffiffiffiffiffiffiffiffiffi D f =T. Even if the triplet pairing is weak, it provides the long-range coupling between two superconducting layers in an S/F/S-junction. Moreover, if the thickness d f exceeds l m significantly, only triplet pairs survive at distances much larger than l m, completely changing the symmetry properties of the superconducting condensate. Earlier the idea that the helical magnetic order in bulk can transform the singlet pairing into triplet pairing was formulated by Bulaevsky et al. [149, 150]. The exchange field rotating in the (y, z)-plane is naturally described by the operator ^h ¼ hð ^ 3 cos þ ^ 2 sin Þ in spin space, where h and are scalar functions of the coordinates, ^ 2 and ^ 3 are the Pauli matrices. It is clear that the non-diagonal part of ^h flips one of the spins of the pair, transforming the singlet into the triplet. It does not appear if the magnetization is collinear ( ¼ 0). To make things more explicit, let us consider the Usadel equation in the F-layer, i.e. equation (38) of Section 2. First we simplify it by linearization, which is valid if the transparency of the interface barrier is small [86]. Then the condensate Green tensor f in the F-layer is small. The linearized Usadel equation reads: D f 2 o 2 f oz 2 j!j Ferromagnet superconductor hybrids 127 h f þ ih ^ 0 f ^ 3, f cos þ ^ 3 ½ ^ 2, f i Š sin g ¼ 0, ð139þ where fa, Bg means the anticommutator of the operators A and B. If ¼ const, equations (139) have an exponential solution f ¼ e kz f 0. The secular equation for k is: ðk 2 k 2!Þ 2 ðk 2 k 2!Þ 2 þ 2h ¼ 0, ð140þ D f where k 2! ¼ 2j!j=D f. Note that the secular equation does not depend on. Itisa consequence of the rotational invariance of the exchange interaction. For ¼ 0 the two-fold eigenvalue k 2 ¼ k 2! corresponds to f 1, 2 (triplet pairing with projection 1 onto the magnetic field). Since! n is proportional to T, these modes have a longrange character. Two other modes have wavevectors k ¼ k h and k ¼ k h, where k 2 h ¼ 2ðj!jþih sgn!þ=d f. They penetrate no deeper than the magnetic length. These short-range modes are linear combinations of the singlet and triplet with spin projection zero, i.e. orthogonal to the magnetic field. Bergeret et al. considered two different geometries. In the first one [71] they considered an S/F-bilayer. The angle was a linear function of the coordinates starting from 0 at the S/F-interface, reaching a value w at the distance w from the interface and remaining constant at larger distances. They solved the linearized Usadel equation (139) with boundary condition f df f =dz ¼ b F s valid at small transparency of the interface by a clever unitary transformation f! ^UðzÞ f ½ ^UðzÞŠ 1 with ^UðzÞ ¼expðiQ ^ 1 z=2þ and Q ¼ d=dz ¼ w =w. This transformation turns the rotating magnetic field into a constant field, directed along the z-axis, but the differential term generates perturbations proportional to Q and Q 2. By this trick the initial equations with the coordinate-dependent hðzþ are transformed into an ordinary differential equation with constant (operator) coefficients. The generation of the triplet component is weak if b is large and it acquires an additional small factor if 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 a strong enhancement of the F-layer conductivity. Such an enhancement was observed in the experiment by Petrashov et al. [153] in 1999, two years afore the theoretical

62 128 I. F. Lyuksyutov and V. L. Pokrovsky work. They studied an F/S-bilayer made from 40 nm thick Ni and 55 nm thick Al films. The interface was about nm 2. The samples were prepared by e-beam lithography. They measured the resistivity and the barrier resistance directly. They also found the diffusion coefficients D s ¼ 100 cm 2 /s and D f ¼ 10 cm 2 /s, which we cite here to give an idea of the order of magnitudes. They claimed that the large drop in the resistance of the sample cannot be explained by the existing singlet pairing mechanism. We are not aware of detailed comparison of the theory [71] and experiment [153]. Dubonos et al. [155] argued that the long-range effects observed by Petrashov et al. [153] may be caused by the stray fields of the domain walls in the F-layer rather than the triplet pairing. They studied a thin Ni disk deposited upon Al disks. The two disks were surmounted upon two of five contacts in a Hall bridge made from a GaAs AlGaAs heterostructure with high mobility destined to measure local magnetic fields. On the third contact they deposited only an Al disk for control. They observed at zero field cooling sometimes (not always) spontaneous magnetization of the Ni disk at T < T c 1:25 K. Statistical analysis of these random fields shows that the average spontaneous field modulus behaves as the square 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 somehow the presence of the F-disk influences the motion of the vortices. They also have found a non-local effect, studying a long Ni wire, which had a small contact with an S-disk. They measured the field at a distance of about 1.5 mm from the contact and discovered spontaneous fields there. These could appear as a result of domain wall motions. They assumed that the vortices do not appear since the magnetization M in nickel is about 500 G, too small to create vortices in Al (Hc ¼ 0 =pd 2 ¼ 1 kg). One more piece of indirect evidence of long-range penetration of the superconducting order parameter through the ferromagnet was reported in [154]. The authors measured the resistance of a 0.5 mm Ni loop connected with superconducting Al wire. They extracted the decay length for the proximity effect in the ferromagnet from differential resistance and concluded that it is much larger than could be expected for singlet pairing. In their second publication on triplet pairing [86] the authors proposed an interesting six-layer structure presented in figure 33. They assumed that the magnetization in each layer is constant, but its direction is different in different layers. It is assumed to stay in the (y, z)-plane and thus it can be characterized by S 0 F 1 S A F 2 S B F 3 Figure 33. Six-layer structure.

63 Ferromagnet superconductor hybrids 129 one angle. Let this angle be in the layer F 1, 0 in the layer F 2, and in the layer F 3. They speak about the positive chirality if the sign is þ and negative chirality if the sign is. They prove that, if the thickness of the F-layers is larger than l m, the superconducting layers S A and S B are connected by an 0-junction if the chirality is positive and by a p-junction if the chirality is negative. This phenomenon is completely due to the triplet pairing since it dominates at this distance. Kulic and Kulic [158] considered two bulk magnetic superconductors with rotating magnetization separated by an insulating layer. They also found that the sign of the Josephson current can be negative depending on the relative chirality. In this system singlet and triplet pairs coexist in the bulk, whereas in the system proposed by Bergeret et al. the triplet 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 without linearization, since the coefficients of the differential equations are constant) and matched these solutions using the Kupriyanov Lukichev boundary conditions. The current density in the F 2 -layer can be calculated using the modified Eilenberger Usadel expression: j ¼ f Tr ^ 3 ^ 0 pt X! n f d f! : ð141þ dz The maximum effect is reached when the magnetic moment of the central layer is perpendicular 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 completely reflects electrons into the F-layer, but the reflection coefficient is spin-sensitive. It follows from arguments developed earlier (see Section 4.2) that the F/S-interface reflects electrons completely when the ferromagnet carriers are fully polarized. The full polarization is a property of the so-called half-metals, such as La 0.7 Sr 0.3 MnO 3 or CrO 2. At complete reflection the reflection amplitude is a phase factor. Without loss of generality one can assume that the reflection phases for opposite projections of spin have different sign. Thus, at any reflection the states are transformed as follows: jk z "i! e i=2 j k z #i; jk z p#i! e i=2 j k z "i. The singlet pair wavefunction in the superconductor jsi ¼ð1= ffiffi 2 Þ½jk ", k #i jk #, k "iš after reflection will be transformed p into cos jsiþsin jti where jti denotes the triplet pair state jti ¼ ð1= ffiffi 2 Þjk ½ ", k #i þ jk #, k "iš. The triplet pair can penetrate in the halfmetal if the spin-flip processes happen near the interface. As soon as they appear in the F-layer they propagate to long distances and become responsible for the Josephson coupling in the SFS-junction. Quantitative theory is considered for a clean ferromagnet and is based on the Eilenberger 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, ^S and ^S, and two Green functions, ^g 0 and ^g 0 for impenetrable, but active interfaces on either side. The connection between the in and out propagators reads: ^g 0 out ¼ ^S ^g 0 ^S y in, ^g 0 ¼ ^S ^g 0 ^S y out in. Transmission is described by hopping amplitudes. The scattering geometry is depicted in figure 34. The tunnelling process with spin-flips was described by the matrix ^. The full propagators ^g and ^g are expressed in terms of

64 130 I. F. Lyuksyutov and V. L. Pokrovsky Figure 34. Scattering geometry illustrating the scattering channels and the corresponding transfer amplitudes for the model discussed in the text. (From Cuevas et al. [157].) decoupled propagators as follows: ^g in ¼ ^g 0 in þð^g 0 in þ ii^ Þ ^ð ^g 0 in ii^ Þ ^g out ¼ ^g 0 out þð^g 0 out þ ii^ Þ ^ð ^g 0 out ii^ Þ ð142þ ð143þ with analogous equations for ^g. The Eilenberger equation without the collision term with these boundary conditions was solved for a heterostructure consisting of a halfmetal L H <x<l H between two superconductors L H < jxj < L. The modulus of the gap at both interfaces is assumed to be identical, but the phase difference is introduced as a boundary condition so that ðlþ ¼ð LÞe i. The spin rotation by the interface can be described by the scattering matrix ^S ¼ expði z =2Þ at the S-side of interface. Generally depends on the impact angle. For numerical calculations they assumed ¼ 0:75 cos. On the half-metal side of the interface ^S ¼ 1. In this case the tunnelling matrix is a 2 1 matrix ^ ¼ð1þ ^S y Þ 0 cos, where 0 ¼ð "" #" Þ determines tunnelling without and with spin-flop. Spin rotation at transmission is half of spin rotation at reflection. For numerics the authors accepted #" = "" ¼ 0:7 and 0.1 and 2L H ¼ 3 0, L L H. The triplet order parameter was defined by the following integral: F t ðxþ ¼ ð "c " c d" 2pi hðpþf ðp, x, "Þi p tanh " 2T, ð144þ where ðpþ is the projector onto the p-state and averaging proceeds over the Fermi surface. The s- and t-order parameters versus x are plotted in figure 35. The upper panel corresponds to the phase ¼ p, the lower one corresponds to ¼ 0. For a sufficiently long F-layer the triplet pairing dominates. Numerical calculations showed 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 and an 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 numerical results with the triplet pairing dominating show that the current is equal to zero at ¼ 0 and ¼ p, but is negative at any intermediate value of the phase. Thus, the S/half-metal/S junction is of p-type. It is sensitive to the value of the spin-flop transmission coefficient as seen in figure 36.

65 Ferromagnet superconductor hybrids 131 Figure 35. Self-consistent order parameter and triplet correlations in an S/HM/S heterostructure for a 0-junction and a p-junction. The relative signs of the pairing correlations in the s-wave singlet and three p-wave triplet channels are indicated. A 0-junction for the singlet order parameter leads to a relative phase difference of p for the triplet correlations, and vice versa. The calculations are for temperature T ¼ 0:05T c, and for " = # ¼ 0:7. (From Cuevas et al. [157].) Figure 36. Critical Josephson current density as a function of temperature for an S/HM/S heterostructure. 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=T c ¼ 0:05 (dashed line), 0.2, 0.3, 0.4, and 0.5 (full lines from bottom to top). The unit is the Landau critical current density J L ¼ ev f N f 0, with the zero temperature bulk superconducting gap 0 ¼ 1:76T c. (From Cuevas et al. [157].) 5. Conclusions This review shows that, though studies of ferromagnet superconductor hybrids are coming of age, we are still at the beginning of an interesting voyage into this emerging field. The most active development is undoubtedly taking place in the