ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2017, Vol. 125, No. 3, pp. 480–494. © Pleiades Publishing, Inc., 2017. Original Russian Text © V.D. Zhaketov, Yu.V. Nikitenko, A.V. Petrenko, A. Csik, V.L. Aksenov, F. Radu, 2017, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2017, Vol. 152, No. 3, pp. 565–580. ELECTRONIC PROPERTIES OF SOLID Relaxation of the Magnetic State of a Ferromagnetic–Superconducting Layered Structure V. D. Zhaketova, Yu. V. Nikitenkoa*, A. V. Petrenkoa, A. Csikb, V. L. Aksenova,c, and F. Radud a Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia MTA Atomki, Institute for Nuclear Research, Debrecen, H-4001 Hungary c Konstantinov St. Petersburg Institute for Nuclear Physics, National Research Center, “Kurchatov Institute,” Gatchina, Leningradskaya oblast, 188300 Russia d Helmoltz-Zentrum Berlin für Materialen und Energie, Berlin, 12489 Germany *e-mail: nikiten@nf.jinr.ru b Received February 1, 2017 Abstract—We have proposed a real-time method of neutron reflectometry. The magnetic state of the Ta/V/FM/Nb/Si ferromagnetic–superconducting system has been analyzed. Relaxation of the inhomogeneous magnetic state with a characteristic time of several hours, which depends on the magnetic field magnitude and temperature, has been observed. The relaxation of the domain structure has changed upon a transition of the V and Nb layers to the superconducting state. It has been concluded that real-time reflectometry data for polarized neutrons are important for determining the origin of magnetism in ferromagnetic–superconducting layered structures. DOI: 10.1134/S1063776117080210 1. INTRODUCTION Complex systems are characterized by relaxation behavior of physical quantities. This is typical, for example, of superconductors [1–3] and systems of magnetic nanoparticles [4, 5]. It was noted in [6] that close analogy exists in the behaviors of ferromagnetic ensembles of particles and superconductors. The logarithmic dependence of the temporal relaxation typical of such systems was also observed in thin magnetic films [7]. Magnetism in superconductor/ferromagnet/superconductor (S/FM/S) layered structures was analyzed in [8, 9]. It was found that the mean value of the magnetic induction, as well as the mean value of the local magnetic induction in domains, clusters, and superconducting vortices, varies with time. In this study, we propose a method of real-time reflectometry of polarized neutrons and report on the data concerning the effect of superconductivity on the magnetic state of Ta/V/FM/Nb/Si ferromagnetic– superconducting layered structure, which were obtained using this method. 2. Ta/V/FM/Nb/Si LAYERED STRUCTURE The structure under investigation was prepared by sequential magnetron sputtering of elements on the Si(0.5 mm) substrate in the MAGSSY chamber at the Helmholtz center (Berlin, Germany). The residual pressure in the chamber was below 5 × 10–9 mbar. The deposition was carried out in ultrapure gaseous argon with a partial pressure of 1.5 × 10–3 mbar. The deposition rate was calibrated using a microbalance of a quartz crystal. The substrate was the Si(100) crystal with the surface cleaned in a ultrasonic bath. During the deposition, the substrate was not heated, but was maintained at room temperature to exclude additional thermal diffusion and to ensure a smooth profile of the surface. The deposition of layers onto the substrate was carried out in the following sequence: Nb, Fe0.7V0.3, V, Fe0.7V0.3, V, and Ta. We assumed that the magnetic moments of two Fe0.7V0.3 layers were antiferromagnetically ordered, due to indirect exchange interaction via the intermediate V layer. We also assumed that due to suppression of the exchange interaction between iron atoms due to dilution with vanadium, as well as due to antiferromagnetic ordering of the magnetic moments of iron atoms relative to the magnetic moments of vanadium atoms induced by them [10, 11], the magnetization of Fe0.7V0.3 iron atoms is much lower than for pure Fe. These measurements were used to facilitate the realization of the superconducting state in Fe0.7V0.3. 480 RELAXATION OF THE MAGNETIC STATE 481 Fig. 1. Spatial profile of the elements of the Ta/V/FM/Nb/Si layered structure. which is close to the value of the magnetic moment of the saturated state. The negative value of the magnetic moment can apparently be explained by the interaction of clusters with magnetic domains. Indeed, the system of clusters has two energy minima [14] corresponding to the directions of the moments along the magnetic field (positive moment) and against it (negative moment). Since the minimum for the magnetic moments directed along the field is deeper, the probability of detecting the system of clusters in this state is higher. However, the energy minimum for the cluster and domain systems corresponds to a negative value of the total moment. Since the magnetic moment of clusters is larger than the magnetic moment of domains, the negative value of the total moment is apparently associated with the interaction between the two magnetic systems. It should be noted that the existence of such interaction will be confirmed below in dynamic experiments (see Section 3.4). Figure 1 shows the profile of the Ta/V/FM/Nb/Si structure, which was measured by neutral atomic mass spectrometry with a spatial resolution of 1 nm [12]. It can be seen from the curves that two magnetic Fe0.7V0.3 layers are not resolved and form in fact a single layer. The boundaries of the niobium and vanadium layers are extended towards the depth and have a thickness close to the width of the magnetic layer. The depth at which the boundaries are located increases with increasing distance from the silicon substrate. For example, for the Nb boundary closer to the substrate, its extension defined as the distance at which the concentration of the element varies in the range 50–100% amounts to 10 nm, while the extension of the distant boundary is 20 nm. The magnetic layer together with the vanadium interlayer has a thickness of 20 nm, which is 6.3 times larger than the thickness determined proceeding from the amount of deposited elements. As a result of mutual penetration of elements, the layer of iron atoms is additionally diluted by 14 times with Nb and V atoms so that the concentration of iron atoms is about 7% (over the width of spatial distribution). Figures 2a and 2b show the temperature dependences of the magnetic moment of the structure for zero-field cooling (ZFC regime, curve 1) and cooling in a magnetic field (FC regime, curve 2). The curves in Fig. 2a diverge at blocking temperature T = 140 K. The blocking temperature is the temperature above which the orientations of the magnetic moments of clusters change due to thermal fluctuations of the medium. Using the value of T = 140 K and anisotropy constant Keff = 103–107 erg/cm3 [13], we find that the diameter of magnetic clusters lies in the range of d ≈ 1–20 nm. At T = 8 K, the magnetic moment is negative and is equal to –6.6 × 10–6 CGSM units in the ZFC regime, The insets to Fig. 2 show the temperature dependences of the magnetic moment in the temperature interval near temperatures of the superconducting transitions. The ZFC (1) and FC (2) curves for a magnetic field of 20 Oe (see Fig. 2a) demonstrate a complex temperature behavior. For example, in the ZFC regime, which is realized by cooling in zero field followed by the application of the field and the increase in temperature from 3 K, a diamagnetic moment of ‒2 × 10–6 CGSM units appears in the temperature interval from 4 to 5 K, which corresponds to a change in the magnetic induction in the Nb layer (it was found from magnetic measurements that TC = 4 K for the V layer and TC = 6.8 K for the Nb layer) equal to –7 G (35% of the magnetic field value). The existence of an extended interval in which the diamagnetic moment appears (T = 4–5 K) apparently indicates the existence of a surface potential barrier. The diamagnetic moment vanishes at T = 7 K. In the FC regime in which cooling from T = 300 K to T = 2.5 K occurs after the application of the field, a paramagnetic moment of 4 × 10–6 CGSM units appears in the temperature interval from 7 to 4.7 K, while in the interval from 4.7 to 3.8 K, a paramagnetic moment of 1.6 × 10‒6 CGSM units also appears. It is well known [15] that the emergence of a paramagnetic moment is explained by the nonuniformity of the superconducting properties of the layers. Both curves (ZFC and FC) have an inflection associated with a change in the magnetic moment in the mixed state of the V layer at T = 3.5 K. In the ZFC regime for a field of 1 kOe (see inset to Fig. 2b), the diamagnetic moment existing at T = 2 K decreases upon a change in temperature from 2 to 7 K. Consequently, the superconducting layers are in the mixed state for these values of the magnetic field Concentration, % 100 Ta 10 V O C H Nb Fe Si 1 0.1 0 50 100 150 200 250 300 350 400 z, nm JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 482 ZHAKETOV et al. y M × 106, CGSM units 0 Lx M × 106, CGSM units −2.0 2 −4.0 −1.0 −2.0 −6.0 −3.0 −8.0 2 6 (a) Ly θi 10 n −4.0 −5.0 ki Q z θf αf θi km xz 1 −6.0 kf x Qy Qx z 2 −7.0 20 60 100 140 M × 106, CGSM units 9.0 2 (b) 1 8.0 180 220 0 300 T, K 2 −8.0 7.0 260 M × 104, CGSM units 4.0 −4.0 −12.0 2 1 4 6 8 6.0 5.0 4.0 θi dy 8 xz αf αi 1 4 xy dx 20 60 100 140 180 220 260 300 T, K Fig. 2. Temperature dependence of the magnetic moment for the Ta/V/FM/Nb/Si structure upon zero field cooling (1) and cooling a magnetic field (2) at H = 20 Oe (a) and H = 1 kOe (b). The insets show the temperature dependences of the magnetic moment in the vicinity of the superconducting transition temperature TC = 6.9 K in the niobium layer. in this temperature range. In this case, almost the entire diamagnetic moment is due to superconductivity of the Nb layer. Further, it follows from Fig. 2b that at H = 1 kOe, the magnetic moment increases by 2.2 times to a value of 9 × 10–6 CGSM units upon the decrease in temperature from 300 to 11 K, which gives a value of 4πJ = 235 G for the magnetization of a magnetic layer of thickness 20 nm, while the magnetic moment per iron atom is μFeV = 0.33μΒ, which constitutes 0.15 of the magnetic moment of an atom in an iron crystal. Such a small value of μFeV is apparently associated with the anticollinear mutual directions of the mag- Fig. 3. (Color online) Geometry of reflection and scattering at grazing angles of radiation incidence. netic moments of iron and vanadium atoms [10, 11]. Extrapolating the curve in Fig. 2b to zero value of the magnetization, we obtain TC = 450 K for the Curie temperature. Therefore, we can state that the magnetic layers of the structure contain two phases of the substance. The first phase in the form of clusters contains 70% of iron atoms and 30% of vanadium atoms and has a Curie temperature of 450 K, while the second phase (medium surrounding the clusters) contains 7% of iron atoms and has a Curie temperature of about 150 K. 3. NEUTRON STUDIES We used reflectometry of polarized neutrons [16], which makes it possible to measure the profiles of magnetization and nuclear scattering length for neutrons in the bulk of the structure and to determine the structure of the inhomogeneous state in the plane of the layers. 3.1. Reflectometry of Polarized Neutrons A polarized neutron beam (denoted by n in Fig. 3) is incident at grazing angle θi on the sample. Neutrons reflected and scattered from the sample are detected by a 2D position-sensitive detector. Neutrons reflected specularly at angle θf = θi, having wavevector km, and transmitted through the structure provide information on the nuclear (Un(z)) and magnetic (Um(z)) complex potentials of interaction of a neutron with the structure, which are averaged over the plane of the sample [17]: U n(z) = V n(z ) − iW n(z ), (1) U m(z) = μ(B(z) − i Δ B(z)), where Vn(z) and Wn(z) are the real and imaginary parts of the nuclear interaction potential, μ is the magnetic moment of neutron, B(z) and ΔB(z) are the vectors of JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 RELAXATION OF THE MAGNETIC STATE UR, neV 125 1 100 3 75 50 25 2 0 −25 0 50 100 150 200 250 300 350 400 z, nm Fig. 4. (Color online) Coordinate dependence UR(z) from the data in Fig. 1 for the Ta/V/FM/Nb/Si structure (1) and for iron atoms (2) and from the fitting of the results of calculations to experimental data on neutron reflection from the Ta/V/FM/Nb/Si structure (3). 483 The neutron beam scattered in the zx plane at angle θf (nonspecular scattering of neutrons) and associated with the transfer of wavevector Qx provides information on the roughness of the interfaces and inhomogeneities with a correlation length in the direction of the x axis ranging from 1 to 100 μm. The neutron beam scattered in the xy plane with grazing angle αf (grazing-incidence small-angle scattering (GISAS) or diffraction scattering (GIND)) associated with the transfer of wavevector Qy provides information on inhomogeneities with a correlation length in the direction of the y axis in the range 1–100 nm. Magnetic elastic scattering of neutrons in the case with energy transfer considerably exceeding the interaction potential is determined by vector M = Δm – e(eΔm) [18], where Δm is the difference in the amplitudes of magnetic scattering of neutrons for the scattering region of the medium and e is the unit vector in the direction of the transferred neutron wavevector. The value of M is maximal for Δm perpendicular to e and is zero when these vectors are parallel. The neutron scattering cross section can be written in terms of the components of M: σ(±) ~ [(M || ± Δ bnuc )2 + M ⊥2 ], magnetic induction and induction increment in the xy plane. The transmission of neutrons through the structure is described by the coefficients of neutron reflection (R+, R–), transmission (Tr+, Tr–), scattering (S+, S–), and absorption (M+, M–), which depend on the spin states of the neutron (“+” for neutrons with spin projection along the magnetic field and “—” with the spin projection against the field). The coherent propagation of neutrons is described by the reflection and transmission coefficients, while the incoherent process is described by scattering coefficients. The coefficients of neutron scattering and absorption are responsible to the leakage of neutrons from the coherent propagation channel and are therefore determined by the imaginary parts of the potential of interaction of neutrons with the structure. Neutron scattering coefficients S+(–)(Qz) can be determined from the data characterizing the neutron coherent propagation channel using the relations S +(−) = 1 − (R + Tr + M ) +(−). (2) The coefficient M of neutron absorption by the elements of the structure does not exceed 10–3; the coefficient of scattering by hydrogen impurity atoms is also smaller than 10–3; therefore, relation (2) was used for determining the value of S(Qz) from experimentally determined neutron reflection and transmission coefficients R(Qz) and Tr(Qz) to within 10–3. (3) where M|| and M⊥ are the components of M along and across the quantization axis (the axis on which the polarization of neutrons is fixed) and Δbnuc is the difference of the nuclear scattering lengths for the scattering region and the medium. For a nuclear-homogeneous medium, Δbnuc = 0 as well as the polarization of scattered neutrons (e.g., in the case of magnetic domains). The polarization of scattered neutrons in a heterogeneous medium is equal to zero in the case of the demagnetized magnetic state, for which the distribution of magnetic moments is isotropic. Nevertheless, neutron scattering exists due to the gradient of the local magnetic induction. The data represented in Fig. 1 were used for constructing the spatial dependences of the potential of nuclear interaction of neutrons (Fig. 4) with the structure under investigation (curve 1) and with a layer of iron atoms (curve 2). Figure 4 also shows the neutron interaction potential profile obtained by fitting to neutron experimental data on the intensity of neutrons reflected from the structure (curve 3). It can be seen that curves 1 and 3 do not coincide in the interval 125– 300 nm. The interval 175–300 nm corresponds to the niobium layer. Here, the potential calculated from the neutron data is lower that calculated from the concentration of individual elements. This can be due to the fact that niobium is packed with a nearly 90% density relative to its density in a crystal. Conversely, in the interval 125–175 nm, the potential obtained from neutron reflection data is higher than that calculated from the concentrations of the elements. This can be associated, for example, with the presence of vanadium in the form of compounds with a density higher than the JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 484 ZHAKETOV et al. density of the mixture of vanadium, hydrogen, and oxygen. The resultant potential of interaction of neutrons with the structure is such that the vanadium layer forms a potential well for neutrons, while the tantalum and niobium layers form potential barriers. For the given spatial dependence of the potential (see Fig. 4), two regimes of the neutron wave field are realized. For the energy of neutrons in the direction perpendicular to the interfaces and exceeding the potential of interaction with the Nb layer, the regime of a propagating neutron wave is realized. For the energy of neutrons lower than the potential of interaction with the Nb layer, the regime of a standing neutron wave takes place. As a result, we used in our measurements the potentialities of measuring the real and imaginary parts of the neutron–medium interaction potential, which are typical of these regimes. It is known that the imaginary part of the potential describes the absorption of neutrons by nuclei of the medium and the scattering of neutrons by nuclei, atoms, and the medium [19]. Scattering of neutrons by the magnetic layer located at the interface between the V and Nb layers is maximal for the transverse neutron wavevector component equal to the critical value of the neutron wavevector (0.007 Å–1). 3.1.1. Results of static measurements. Figure 5 shows the dependences of neutron scattering coefficient S integrated over the neutron wavelength (we assume that scattering at H = 0 is zero) out of solid angle Ω = 1.6 × 10–4 srad, which were obtained at T = 300 K. It can be seen that with increasing magnetic field, the scattering intensity increases, while the difference of the intensities for the “plus” and “minus” polarizations decreases. The intensification of scattering can be explained as follows. First, this means that the medium is magnetically heterogeneous and consists of atomic (nuclear) magnetic clusters (Fig. 5b) for which the density distribution of nuclear amplitude of neutron scattering coincides with the distribution of the magnetic scattering amplitude density. If the medium contained magnetic domains and were nuclear-homogeneous, an increase in the magnetic field would result in the disappearance of magnetic domains and, as a consequence, to a decrease in the neutron scattering intensity. With increasing magnetic field, the moments of clusters are aligned in the direction of magnetic field H. Therefore, we must consider two factors. First, magnetic channels are formed (see Fig. 5b), which intensifies scattering; however, scattering in this case becomes less dependent on the polarization of neutrons, which is precisely observed in experiments. Second, with increasing magnetic field, the “degree of orthogonality” of the magnetic moments of clusters to the transferred magnetic moment of scattered neutrons increases (e ⋅ Δm → 0), which also intensifies neutron scattering. Low-temperature measurements were carried out over 3.5 years. The first measurements were taken S 0.10 0.09 (a) 0.08 0.07 0.06 0.05 0.04 0.03 0.02 1 0.01 2 2000 1000 0 3000 4000 5000 H, Oe m m (b) Q H m m m 1 2 Fig. 5. (Color online) (a) S(H) dependence at T = 300 K for P0 = +1 (1) and –1 (2). (b) Magnetized (1) and demagnetized (2) systems of clusters with magnetic moments m and scattered field fluxes connecting them in magnetic field H; Q is the transferred wavevector of neutrons. 3 months after the preparation of the sample. The results of first measurements are shown in Fig. 6a illustrating the temperature dependence of the scattering coefficient S (T ) = 1 − (R(T ) + Tr (T ))/(R(150 K) + Tr (150 K)) of polarized neutrons with wavelength λ = 1.28 ± 0.015 Å (scattering at T = 150 K is zero). At T = 8 K, neutron scattering intensity is maximal and amounts to 55–59%; the polarization of scattered neutrons is negative and equal to –0.035. It should be noted in this connection that a nonzero value of the polarization of scattered neutrons is determined by the system of clusters, while the neutron scattering intensity in a weak magnetic field is mainly determined by domains. On the other hand, the magnetization in a weak field is determined predominantly by clusters. Therefore, the negative value of polarization at T = 8 K indicates that the magnetic moments of clusters are oriented against the magnetic field, which is in conformity with the negative value of the magnetization (see Fig. 2a). Therefore, the neutron scattering intensity at H = 17 Oe at T = 8 K (see Fig. 6a) is 7 times JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 RELAXATION OF THE MAGNETIC STATE S 0.7 (a) 0.6 S 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 P0 = −1 +1 0.5 H = 17 Oe 0.4 0.3 H = 17 Oe, P0 = +1 2 4 6 8 10 12 T, K 9.5 kOe 0.2 0.1 +1 0 1 S 0.24 10 3 1000 T, K 100 1000 T, K (b) 0.20 0.16 100 2 0.12 0.08 1 0.04 0 1 αΔS 0.05 10 (c) 0.04 5 6 0.03 0.02 3 4 0.01 0 1 2 −0.01 1 2 3 4 5 6 7 8 9 10 11 12 T, K Fig. 6. S(T) dependence for the structure Ta/V/FM/Nb/Si: (a) 3 months after the preparation of the structure; (b) 1.5 years later for H = 17 Oe (1), H = 17 Oe after preliminary magnetization in a field of 2 kOe (2) and H = 1 kOe (3); (c) 3.5 years later for H = 28 Oe: λ = 2 ± 0.2 Å, P0 = +1 (1); λ = 2 ± 0.2 Å, P0 = –1 (2); λ = 4.6 ± 0.46 Å, P0 = +1 (3); λ = 4.6 ± 0.46 Å, P0 = –1 (4); λ = 7.2 ± 0.72 Å, P0 = +1 (5); λ = 7.2 ± 0.72 Å, P0 = –1 (6). 485 higher than the intensity at H = 5 kOe and T = 300 K (Fig. 5a), while the magnetization has increased by only 1.5 times (see Figs. 2a and 2b). This means that at H = 17 Oe and T = 8 K, about 70% of the neutron scattering intensity are determined by the domain structure and 30% are due to clusters. Figure 7 shows the pattern of scattering of neutrons with λ = 3.8 ± 0.2 Å on the cluster lattice at T = 10 K and H = 1 kOe. The first (Ny = 135 and Nz = 135), third (Ny = 90 and Nz = 90), and fifth (Ny = 25 and Nz = 40) scattering (reflection) orders corresponding to an interplanar spacing of the cluster array of 6.5 nm can be seen. The reflection planes of the lattice are inclined at an angle of about 45° to the y axis in the yz plane (Fig. 8). Near the spots of the first and third reflection orders, the satellite spots corresponding to spatial periods of 20 and 60 nm along the y axis are observed [9]. These periods can be attributed to longperiodic magnetic structures formed by clusters. However, in all probability, these values correspond to the domain wall thickness and the magnetic period of the domain structure (the domain size is 40 nm) in the medium containing clusters. It follows from Fig. 6a that at T = 3 K, the polarization has increased to 0.11 and has become positive. The increase in the neutron polarization is associated with an increase in the fraction of neutrons scattered by clusters. At T = 1.6 K, the scattering intensity is even lower and amounts to 0.1. The scattering intensity has decreased by approximately 5.7 times as compared to the intensity at T = 8 K. The decrease in the neutron scattering intensity by 70% due to the domain structure can be explained by an increase in the domain size and disordering of the directions of magnetic moments of clusters (demagnetization of the cluster system). The increase in the domain size is favorable for the superconducting state if the magnetic moments of domains are oriented perpendicularly to the planes of the structure [19]. In this case, upon an increase in the domain size, the magnetic induction in a domain decreases due to the demagnetization factor, which facilitates the passage of superconducting pairs through domains. Figures 9a and 9b show the magnetic state of the structure in a weak magnetic field of 17 Oe at 8 K and 1.6 K. The circles and rectangles indicate the clusters and domains, respectively; arrows show the directions of the magnetic moments of clusters and domains. In the nonsuperconducting state (at T = 8 K) in a weak magnetic field (H = 17 Oe), the magnetic moments of domains are disordered and directed at right angles to the magnetic layer, while the moments of clusters are ordered and directed against the field. In the superconducting state (T = 1.6 K, H = 17 Oe), the domains grow, and the magnetic moments of clusters become disordered in direction. Figure 6a also shows the data for a strong magnetic field of 9.5 kOe, which demonstrate scattering by the third magnetic system (namely, superconducting vor- JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 486 ZHAKETOV et al. Ny 180 160 y ×10−6 5.0 (a) 2d 4.5 n 4.0 140 3.5 120 3.0 100 2.5 80 2.0 2d 1.5 60 1.0 40 0.5 40 60 80 100 120 140 160 180 Nz Ny 180 160 z Detector 1 2 x ×10−6 4.5 3 (b) 4.0 140 3.5 120 3.0 2.5 100 2.0 80 Fig. 8. Schematic diagram of the experiment on detection of neutron scattering from the cluster lattice in the magnetic layer (reflection plane is shown by the gray rectangle) in the case of grazing incidence of neutrons on the Ta/V/FM/Nb/Si layered structure: (1) transmitted beam; (2) reflected beam; (3) neutron beam diffracted on the cluster lattice. The interplanar distance for the reflection being detected is 21/2d = 6.5 nm. 1.5 60 1.0 40 0.5 40 60 80 100 120 140 160 180 Nz 0 Fig. 7. (Color online) Neutron scattering intensity on the [Ny, Nz] plane for (a) P0 = +1 and (b) –1 for λ = 3.8 Å, where Ny and Nz are the numbers of channels of the neutron detector along the y and z axes, respectively. tices). Measurements in a magnetic field of 9.5 kOe were taken using the following procedure. A magnetic field of 9.5 kOe was set at a temperature of 3 K. Then the temperature was reduced to the minimal value (1.6 K), and measurements were taken again. After this, the temperature was raised to the next closest higher value, and measurements were carried out, and so on up to the maximal value of T = 10 K. It can be seen that in the range of 3–7 K, scattering of neutrons is observed, which is attributed to scattering by vortices in the mixed state of the superconducting vanadium and niobium layers. At T < 3 K, the Meissner vortexfree state is realized, while at T > 7 K, superconductivity disappears. In both cases, neutron scattering inten- sity decreases to the level determined by scattering from magnetic clusters. The second series of measurements was carried out 1.5 years after the sample preparation (Fig. 6b). This figure shows the S(T) dependence for a magnetic field of 17 Oe (curve 1), in the same field after preliminary magnetization of the structure in a magnetic field of 2 kOe (2) and in a magnetic field of 1 kOe (3). Curve 1 is generally analogous to the dependence in Fig. 6a. It can be seen, however, that scattering intensity increases in a narrow interval of 5–6 K, which probably indicates the formation of a domain structure of the cryptoferromagnetic phase [20–25]. The intensification of scattering on curves 2 and 3 is more pronounced. Since these dependences correspond to a higher magnetization, we can propose that the magnetizations of domains now lie in the plane. Therefore, a decrease in the domain size upon cooling now becomes favorable for the superconducting state [26– 28]. Figures 9c and 9d show the magnetic structure in a moderate magnetic field of 1 kOe at T = 2 K (see curve 3 in Fig. 6b) and 10 K, respectively. With increasing magnetic field in the superconducting state (2 K, 1 kOe), the magnetic moments of domains become oriented along the field, and their size decreases. In the nonsuperconducting state in a magnetic field of a moderate magnitude (10 K, 1 kOe), the JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 RELAXATION OF THE MAGNETIC STATE 487 8 K, 17 Oe 1.6 K, 17 Oe 2 K, 1 kOe 10 K, 1 kOe (a) (b) (c) (d) H Fig. 9. (Color online) Magnetic state of the structure in the ranges of T = 1.6–10 K and H = 17 Oe–1 kOe according to the neutron scattering data: (a) magnetic moments of clusters are oriented against the field; magnetic moments of domains are oriented perpendicularly to the magnetic field; (b) magnetic moments of clusters are disordered, the domain size has increased as compared to state (1); (c) clusters are magnetized in the direction of the field, the magnetic moments of domains are oriented along the magnetic field, but their density has increased due to a decrease in the domain size; (d) magnetic moments of clusters are oriented along the magnetic field; the density of domains has decreases. density of domains decreases, and the magnetic moments of clusters are oriented along the field. The third neutron measurement was taken 3.5 years after the preparation of the structure. The results of these measurements are represented in Fig. 6c. The results of these measurements at H = 28 Oe are shown in Fig. 6c. It can be seen that the neutron scattering intensity in the range from 10 to 1.5 K changes by just 1.5–3%. The small value of scattering intensity indicates that the nuclear and magnetic contrasts have decreased. This can be primarily due to diffusion of iron atoms from the magnetic layer. This follows from the results of analogous repeated measurements (see Fig. 1) of the spatial profiles, indicating the decrease in the spatial distribution width for iron atoms by 11.4% from 22.8 to 20.2 nm over 3.5 years. 3.2. Real-Time Reflectometry of Neutrons The real-time experiments with neutrons, in which specularly reflected and scattered neutrons are detected during time t in the n = t/Δt time interval of duration Δt, make it possible to measure nonstationary processes with period of variation T exceeding Δt (T > Δt). In this case, Δt obviously determines the time resolution of measurements. The minimal value of Δt is determined by neutron intensity J and change ΔJ of the neutron intensity due to the temporal process under investigation. Let us suppose that the change in the intensity of neutron count by the detector over the entire measurement time t is ΔJ = βJ, where β < 1. Let the minimal change during time interval Δt be determined by parameter μ and amount to δJ = μΔJ, where μ < 1. Obviously, the statistical error of neutron count measurement Nst = (JΔt)1/2 over time interval Δt must be smaller than count increment ΔN = βμJΔt. We can write the relation connecting these quantities in the form Nst = αΔN, where α < 1. For time measurement interval Δt, we can write from this relation (4) Δ t = 1/(α 2β 2μ 2 J ). It follows from this relation that to reduce Δt, we must increase neutron count intensity J, which corresponds to an increase in the neutron spectrometer luminosity. High-transmission methods of measurements include, for example, diffraction and depolarization measurements. In [29–31], neutron-diffraction and spin-precession (variant of depolarization) real-time measurements were reported. In [29, 32], various possibilities of diffraction studies were also indicated. It was noted that for different experimental setups, irreversible transition processes can be measured with a resolution from 1 min to 1 ms, while the resolution for reversible processes can be down to 1 μs. In [31], the magnetic field relaxation averaged over the volume exposed to the neutron beam was measured in a superconducting ceramic with the minimal value of Δt = 1000 s, which made it possible to observe magnetic field inductions with frequencies lower than 0.1 mHz to a high statistical degree of accuracy. Because of the small viewing solid angle of the neutron source (~10–5 srad) and the small neutron beam cross section on the experimental sample (~0.1 cm2), neutron reflectometry is not a high-transmission method of measurements. Let us estimate the value of Δt for the realization of the method on the REMUR spectrometer of the IBR-2 pulsed reactor [33]. The neutron flux recorded by a position-sensitive detector at a reactor power of 2 MW in the operation conditions (e.g., fan polarization analyzer) amounts to 1.6 × 105 (s cm2)–1, which gives the count intensity J0 = JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 488 ZHAKETOV et al. 2.4 × 10 4 s–1 on the neutron detector for a neutron beam cross section of 0.15 cm2. Let us consider the channel of specularly reflected neutrons with intensity JR = J0 of reflection from the interface between the media (total reflection of neutrons). For realistic values of parameters α = 1/3 (statistical error is 30% of the effect), β = 0.1 (total change in the count intensity amounts to 10% of the count intensity), and μ = 0.03 (change in the count intensity on interval Δt is 3% of the total change), we obtain Δt = 9 × 106/(2.4 × 105) ≈ 40 s. This gives 5 min for the minimal value of the period of temporal variation Tmin ≈ 8Δt (maximal variation frequency is fmax = 3.3 mHz). Let us now consider the neutron scattering channel. Let the intensity of scattered neutrons amount to 10% of the intensity of specularly reflected neutrons. The change in the intensity of scattering of neutrons with time is assumed to be 100% (β = 1). Then for parameters α = 1/3 and μ = 0.03 (same as for specular reflection), we obtain the same value of 5 min for the minimal measured period Tmin. 3.3. Results of Dynamic Measurements Let us now consider the results of measurements of the dynamic behavior (namely, relaxation of the magnetic state of the layered structure). The main dynamic measurements were taken in the second series of neutron measurements. Figure 10 shows the S(t) dependences for λ = 1.8 Å after the step variation of the field and temperature. The structure was magnetized in a magnetic field of 2 kOe at a temperature of 10 K. Since the magnetic field lies in the plane of the layer, the domains oriented perpendicularly to the layer were oriented in the plane of the layer. Then the field was reduced to 17 Oe; a temperature of 10, 6, 3, or 2 K was set, and measurements were taken. It can be seen that neutron scattering increases with time, and the polarization changes its sign at T = 6, 3, and 2 K at tch1 = 17000, 9000, and 7500 s, respectively. The intensification of scattering was associated with the increase in the density of domain walls, while the change in polarization was due to rotation of the magnetic moments of clusters. The part of the intensity of neutrons independent of their polarization (averaged over polarizations +1 and –1) for T = 6, 3, and 2 K consists of two segments separated by instant tch2 = 17000, 15000, and 10000 s, respectively. The inflection on the curve indicates the emergence of a new domain structure (namely, still smaller domains are formed). As a result, the density of domain walls increases still further and, accordingly, the neutron scattering intensity increases. It can be seen that tch1 and tch2 decrease upon cooling, indicating that the processes of motion of cluster magnetic moments and the formation small domains are associated with superconductivity. Thus, we can state that relaxation with the superconducting layers of the structure (T = 2, 3, and 6 K) substantially differs from relaxation with nonsuperconducting layers (T = 10 K). It occurs in two stages in the former case and in one stage in the latter case. Superconductivity leads to the formation of domains with a size smaller than usual. It cannot be ruled out that this is connected with the cryptoferromagnetic phase predicted and described in [20–25]. In the case of superconducting layers, the directions of the magnetic moments of the clusters relative to the magnetic field direction change simultaneously with the changes in the domain structure. Both processes are interrelated and are controlled by superconductivity. The change in the scattering intensity during relaxation attains 100%. In the interval of measurements for 1 h (frequency range f < 3 × 10–4 Hz), the statistical error of measurements is 3% of the change in the neutron scattering intensity. Figures 11a–11c show the S(t) dependences obtained in a magnetic field of 1 kOe in the temperature interval 2–10 K for neutron polarization P0 = +1 and neutron wavelengths of 1.8, 3, and 6 Å. The sequence of measurements was as follows. The magnetic field of 1 kOe was stabilized at 10 K; then the temperature was reduced to the value at which the measurement was taken. It can be seen that the scattering intensity maximum corresponds to a certain wavelength of 3 Å, which indicates an admixture of neutron scattering from the magnetic cluster lattice. Figures 12a and 12b show the S(t) dependences for P0 = +1, –1, and 0 in a magnetic field of 1 kOe at temperature (a) 10 K and (b) 2 K. It can be seen that dependences for P0 = +1 and P0 = –1 oscillate and intersect. This corresponds to the situation when the magnetic moments of clusters are directed either along the field (the curve for P0 = +1 is above the curve for P0 = –1) or against it (curve with P0 = +1 is below the curve with P0 = –1). The points of intersection of the curves with P0 = +1 and P0 = –1 correspond to zero mean magnetization of clusters, when all magnetic moments are perpendicular to the magnetic field vector or when the directions of the moments are distributed symmetrically relative to the perpendicular to the magnetic field vector. The dependences obtained at T = 2 K indicate that the magnetization changes by more than +40%. Let us estimate the value of the magnetic induction at clusters. In the case where all magnetic moments are oriented against the magnetic field, the magnetic induction in their neighborhood is in the range 1 kOe–235 Oe (2.2/0.33)/1.4 = –120 Oe ≈ 0. For a nearly zero magnetic induction at a cluster, the formation of the domain structure apparently becomes favorable. In this case, the spatial distribution of the domains structure in the magnetic layer must probably repeat the distribution of clusters oriented against the field. Figure 12c shows the S(t) JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 RELAXATION OF THE MAGNETIC STATE S 0.20 S 0.20 0.18 T = 10 K (a) 0.16 −1 0.14 0.14 0.12 0.12 0.10 0.10 0.08 0.08 5000 10000 15000 20000 25000 t, s T=3K +1 0 5000 10000 15000 20000 25000 t, s +1 0.12 0.10 0.10 0.08 0.08 5000 10000 −1 0.14 0.12 15000 20000 25000 t, s (d) P0 = +1 0.16 (c) 0.14 T=2K 0.18 P0 = −1 0.16 0 P0 = −1 S 0.20 S 0.20 0.18 (b) T=6K 0.18 P0 = +1 0.16 0 489 0 5000 10000 15000 20000 25000 t, s Fig. 10. S(t) dependence in a magnetic field of 17 Oe in the case of preliminarily magnetized sample in a magnetic field of 2 kOe at different temperatures for λ = 1.8 Å. dependences for P0 = 0 (mean magnetization of clusters is zero). It can be seen that at T = 2 K, scattering coefficient S(t) increases with time. This means that the magnetic domain structure that does not correlate with the nuclear structure changes or is generated. Obviously, the system of clusters oscillates near the equilibrium state right to t = 460 min, while the domain structure has not yet attained saturation. Consequently, the evolution of the domain structure with time at T = 2 K is accompanied with an increase in the amplitude of oscillations of the magnetic moments of clusters. Therefore, in a moderate magnetic field of strength 1 kOe, the behaviors of the two magnetic systems are rigidly coupled and depend on time, and the angular range of oscillations of the magnetic moments is determined by the domain structure. At T = 2 K, the angular range increases together with the density of domain walls due to the formation of the structure of small domains at tch2 = 10000 s, while at T = 10 K, both parameters become saturated with time. It is important to know whether small domains are formed inside large domains or large domains first disappear and then small domains are formed. There is no answer to this question as yet. For example, the curve in Fig. 12c for T = 2 K appears as monotonically increasing, which speaks in favor of the first version. However, the curve in the same figure for T = 3 K has a weakly pronounced minimum, which indicates the second version. Furthermore, since the magnetic field scattered from the domain structure decreases upon an increase in the domain wall density, the magnetic action of the domain structure on the system of clusters also becomes weaker. In this case, the system of clusters is in an unstable oscillatory state, passing from one state in the magnetic field to the other and back. Obviously, JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 490 ZHAKETOV et al. S 0.12 S 0.28 (a) (a) 0.24 0.08 0.20 0.04 T=2K 0.16 6 K 3K 0.12 10 K 0.08 0 0 2 λ = 1.8 Å 5000 S 0.32 10000 15000 20000 25000 30000 t, s T = 10 K −0.04 3 −0.08 1 0 S 0.24 5000 10000 15000 20000 25000 30000 t, s (b) 0.28 0.20 0.24 0.16 T=2K 6K 0.20 (b) 2 3 0.12 1 0.16 0.12 0.08 3K λ=3Å 5000 S 0.20 10000 15000 20000 25000 30000 t, s (c) 0.16 0.08 0.04 1 T=3K λ=6Å −0.04 0 5000 (c) 0.12 0.04 0 5000 10000 15000 20000 25000 30000 t, s 0.16 10 K 6K 0.08 0 S 0.24 0.20 2K 0.12 −0.08 T=2K 0.04 10 K 0 0.08 10000 15000 20000 25000 30000 t, s 0 2 3 −0.04 4 −0.08 5000 10000 15000 20000 25000 30000 t, s Fig. 11. S(t) dependence in a magnetic field of 1 kOe at different temperatures for P0 = +1. Fig. 12. S(t) dependence for H = 1 kOe: (a, b) P0 = –1 (2), 0 (3), +1 (1); (c) P0 = 0, T = 10 (4), 6 (3), 3 (2), and 2 K (1). the oscillatory motion of the magnetic moments of clusters with increasing amplitude in the presence of two magnetic systems is a manifestation of the domain phase consisting of small domains. Figure 13 shows the S(t) dependences at T = 3 K in the range H = 17 Oe–8 kOe of magnetic field variation. The value of the magnetic field was stabilized at T = 12 K, after which the temperature was decreased JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 RELAXATION OF THE MAGNETIC STATE S 0.26 T=3K λ = 1.8 Å 0.24 0.22 6 7 0.20 4 5 0.18 8 0.16 3 2 1 0.14 0.12 0 5000 10000 15000 20000 25000 t, s Fig. 13. S(t) dependence for P0 = +1 and H = 17 Oe (1), 1 (2), 2 (3), 4 (4), 6 (6), 7 (7), and 8 kOe (8); P0 = –1, H = 4 kOe (5). to 3 K, and measurements were taken. Curves 4–8 at H = 4–8 kOe are characterized by a decrease in the neutron scattering intensity on the initial time interval, which can be explained by a decrease in the density of the pinning centers in the superconducting layers. This is confirmed by the fact that the polarization of neutrons scattered by the superconducting current loops around the pinning centers is negative right to t = 1.5 × 10 4 (curves 4 and 5). The latter mechanism is also confirmed by the observed form of the neutron scattering law, which corresponds to the theory of magnetization relaxation in low-temperature superconductors [1, 2] and explains the relaxation by the separation of vortices from the pinning centers. The curve for H = 4 kOe in the interval 3000–10000 s is described by the logarithmic law S = 0.25 – 0.0725 × ln(t(min)/60). Beginning from instant t = 15000 s, curves 4 and 5 demonstrate the reciprocal nature with a rather sharp increase in the neutron scattering intensity. We can assume that the end of the straightening of the vortex lattice somehow increases the rate of formation of the domain structure. It is important to find out whether the value of magnetization averaged in the plane of the layer changes due to coherent reflection and transmission of neutrons along with a local change in the magnetization determined by neutron scattering. It would also be interesting to determine the position of the part of magnetization varying with time over the depth. This can be done using the time dependences of three coefficients R(t), Tr(t), and S(t) (Fig. 14). It can be seen that there exist time intervals t ≈ 1000 s (a) and t ≈ 12500 s (b) in which the behaviors of coefficients R(t) 491 R, Tr, S 1.0 T=2K 0.9 (a) H = 1 kOe 0.8 0.7 0.6 Tr 0.5 0.4 0.3 R 0.2 0.1 S 0 −0.1 0 5000 10000 15000 20000 25000 30000 t, s R, Tr, S 1.0 T=3K 0.9 (b) H = 4 kOe 0.8 0.7 Tr 0.6 0.5 0.4 0.3 0.2 R 0.1 0 S −0.1 0 5000 10000 15000 20000 t, s R, Tr, S 1.0 (c) 0.9 Tr 0.8 0.7 0.6 T = 10 K 0.5 H = 4.5 kOe 0.4 0.3 R 0.2 0.1 S 0 −0.1 0 5000 10000 15000 20000 25000 30000 t, s Fig. 14. S(t), R(t), and Tr(t) dependences for λ = 6 Å. Solid and dotted curves correspond P0 = + 1 and –1, respectively. and Tr(t) correlate. This means that apart from a change in the local magnetization of the scattering object, the value of magnetization averaged over the plane also changes with time. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 492 ZHAKETOV et al. αΔS 0.040 αΔS 0.030 0.025 8 7 H = 28 Oe λ = 2 ± 0.2 Å 0.020 0.015 5 6 0.035 T = 1.68 K λ = 2 ± 0.2 Å 0.030 5 6 3 4 0.025 0.010 0.005 1 2 0.020 0.015 0 3 4 −0.005 −0.010 7 8 0 10000 2 1 0.010 20000 30000 40000 50000 t, s Fig. 15. Dependence αΔS(t): (1) T = 6 K, P0 = +1; (2) T = 6 K, P0 = –1; (3) T = 4 K, P0 = +1; (4) T = 4 K, P0 = –1; (5) T = 3 K, P0 = +1; (6) T = 3 K, P0 = –1; (7) T = 1.68 K, P0 = +1; (8) T = 1.68 K, P0 = –1. It follows from the curves corresponding to H = 4.5 kOe and T = 10 K (Fig. 14c) that while S and Tr mainly anticorrelate for t < 250 min, S and R anticorrelate predominantly for t > 250 min. This means that the relaxation of the magnetic state shifts with time from the deeper regions of the magnetic layer contacting with the Nb layer to the region located closer to the surface and contacting with the V layer. For T = 3 K and H = 4 kOe (Fig. 14b), prior to processes described above (curves 4 and 5 from Fig. 13), an increase in the average magnetization of the magnetic layer is first observed. When T = 2 K and H = 1 kOe, the relaxation in the surface regions of the structure is observed only at the initial stage lasting to instant of 5000 s. At the same time, for T = 3 K and H = 4 kOe (see Fig. 12b), the transmission coefficient is determined by neutron scattering more strongly as compared to the reflection coefficient. This means that neutron scattering mainly occurs at vortices in the niobium and not the vanadium layer. Therefore, the curves in Fig. 14 demonstrate that real-time neutron reflectometry makes it possible to simultaneously trace the temporal and spatial variation of the magnetic state in the layered structure. Some dynamic measurements were also performed in the third series of experiments. Figure 15 shows the αΔS(t) dependences in a magnetic field of 28 Oe, where α = 1/(1 – S(T = 10 K)), which were obtained upon a gradual cooling in the temperature range from 10 to 1.68 K. It can be seen that after a short-term increase in the scattering intensity at T = 1.68 K and T = 3 K (reduction of the domain size), its decrease is observed (increase in the 0.005 0 20000 40000 60000 80000 t, s Fig. 16. Dependence αΔS(t): (1, 2) T = 1.68 K, H = 500 Oe, P0 = +1 (1) and –1 (2); (3, 4) T = 1.68 K, H = 1000 Oe, P0 = +1 (3) and –1 (4); (5, 6) T = 1.68 K, H = 28 Oe, P0 = +1 (5) and –1 (6); (7, 8) T = 10 K, H = 1000 Oe, P0 = +1 (7) and –1 (8). domain size). Upon a decrease in temperature, the range of neutron scattering intensity oscillations increases. At T = 1.68 K, the curve resembles a transient process when the amplitude of oscillations relative to the equilibrium state decreases with time. Along with the oscillations in the domain sizes, oscillations of the magnetic moments of clusters are observed. No such interrelation between the systems of domains and clusters was observed in the first and second series of experiments. Figure 16 shows the S(t) dependences at T = 1.68 K, which were obtained for different values of the magnetic field. The change in the field magnitude leads to a change in the domain structure; however, in contrast to temperature variations, the long-term relaxation occurs with a substantially smaller amplitude, although it demonstrates a quasi-periodic nature (change in the domain sizes). Ir can also be seen that the effect of superconductivity leads to a decrease in the domain size with time in a magnetic field of the moderate value (H = 1 kOe). 4. CONCLUSIONS Static and dynamic neutron studies have revealed that the characteristics of the Ta/V/FM/Nb/Si magnetic structure have changed significantly during 3.5 years. For example, the distribution of elements in the plane of the magnetic layer has become more uniform with time, which follows from an order-of-magnitude decrease in the neutron scattering intensity. We have studied the relaxation of the magnetic state of the hybrid layered structure consisting of ferromagnetic JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 125 No. 3 2017 RELAXATION OF THE MAGNETIC STATE and superconducting layers. The magnetic state relaxation was studied using polarized neutron reflectometry. Real-time dynamic neutron studies have confirmed their importance in determining the origin of magnetism in ferromagnetic–superconducting layered structures. It has been shown that the systems of magnetic clusters and ferromagnetic domains interact, and their behavior is determined by the superconducting layers of the structure. The manifestations of the interaction between the two systems change with time. For example, it has been established in experiments performed during the first 1.5 years that the formation of ferromagnetic domains of the same size changes with time under the effect of superconductivity to the formation of domains of a smaller size. The rate of formation of the new domain phase increases upon cooling to below the critical point for the niobium layer. It has been proposed that the structure with small domains is probably the cryptoferromagnetic phase of the magnetic layer. It should be noted that the manifestation of the cryptophase in the increase in the intensity of scattering of polarized neutrons is one of the first direct indications of its formation in the layered structure. Simultaneously with the formation of the smalldomain structure in a magnetic field of 1 kOe right to an instant of 6 h, the amplitude of oscillations of the magnetic moments in the cluster system from the direction along the magnetic field to the opposite direction increased. This was the result of suppression of the magnetic interaction of clusters with the smalldomain structure and, as a consequence, the emergence of an unstable state of oscillation of the system of clusters between two minima of its energy in the magnetic field [14]. It was found in analysis of the structure 3.5 years after its preparation that the interaction between the two magnetic systems was manifested differently as compared to that observed earlier, namely, in the simultaneous change in the direction of the magnetic moments of clusters and in the domain size. The amplitude of oscillations of the domain sizes increased upon cooling. The intensity of the polarized neutron beam in the REMUR spectrometer of the pulsed IBR-2 reactor used in our experiment can in principle be increased, which will make it possible to improve the time resolution in the investigation of irreversible processes to a few minutes. 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