- Email: [email protected]
Ni multilayers
Journal of Physics and Chemistry of Solids 67 (2006) 610–612 www.elsevier.com/locate/jpcs
Matching effects in the field and temperature dependences of the magnetization of superconducting/ferromagnetic Nb/Ni multilayers S.A. Kryukov a, A. Bosomtwi a, L.E. De Long a,*, E.M. Gonzalez b, E. Navarro b, J.L. Vicent b, J.E. Villegas b, Wentao Xu a a
Department of Physics and Astronomy, University of Kentucky, CP 177, Lexington, KY 40506-0055, USA b Departamento de Fisica de Materiales, C.C. Fisicas, Universidad Complutense, 28040 Madrid, Spain
Abstract The temperature and field dependences of the magnetic moment of [Nb(x)/Ni(y)]5 multilayers are reported. Small kinks in the phase boundary defined in field-cooled AC experiments exhibit an approximate period of 733 Oe, and persist to a DC field HZ6.5 kOe applied parallel to the multilayer. A second group of cusp anomalies in DC magnetization are observed for H!100 Oe with approximate period 15–20 Oe. The absence of anomalies in perpendicular field and normal state data suggest they are due to finite-size effects expected when the superconducting layer is thinner than the magnetic penetration depth. q 2005 Elsevier Ltd. All rights reserved. Keywords: A: Multilayers; A: Superconductors; A: Magnetic materials; D: Magnetic properties; D: Phase equilibria
1. Introduction The fundamental properties of magnetic thin films and ‘multilayers’ (ML) have been extensively studied and exhibit many interesting properties [1]. In particular, a ML stack of ferromagnetic (FM) thin films interleaved with nonmagnetic layers can be configured as quasi-two-dimensional systems confined along the normal direction to the film plane. Finitetemperature phase transitions cannot be realized in twodimensional systems [2], and it is known that long-range FM order breaks down in favor of ‘superparamagnetism’ if a film is made thin enough [3]. Superconductivity is an established, sensitive probe of magnetic interactions and magnetic moment stability in dilute alloys and compounds [4] and more recently, ML [1]. Recent studies of bilayers, trilayers and ML have addressed a number of interesting effects of magnetic order on superconducting (SC) properties [5,6]. For example, both monotonic [7] and nonmonotonic [8,9] decreases of the SC transition temperature TC with FM layer thickness have been observed; and the
* Corresponding author. E-mail address: [email protected] (L.E. De Long).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.081
potential roles of pair breaking phase shifts and ‘p-junctions’ have been discussed [10]. An intriguing competition between FM and SC has been reported for [Nb(x)/Ni(y)]Z ML [11,12]. TC was found to vanish near a critical Ni thickness yz2.5 nm, above which FM develops in ML (for xZ10 nm Nb layers). In contrast, it was found that increasing the Nb layer thickness to 23 nm resulted in SC with a near-constant (with possible small-amplitude oscillations) TC over a wide range of Ni thickness 1%y%6 nm. The above results have led us to revisit magnetic ML as interesting nanoscale systems that might exhibit finite-size effects when the FM layer thickness is y!10 nm. In particular, the dependence of TC(y) for the xZ10 nm ML series is clear evidence for the destruction of SC by magnetic fluctuations near a quantum critical point (QCP) yz2.5 nm. Moreover, we expect their SC properties to sensitively reflect finite-size, shape and boundary effects on FM. FM/SC ML systems may display other interesting effects, including a number of flux line lattice phase transitions in strongly anisotropic superconductors [13,14]. We recently discovered that the onset of Nb superconductivity dramatically alters the switching dynamics of the FM Ni layers; and we found preliminary evidence that screening supercurrents sensitively probe the stability of FM layers [12]. These results suggest that the SC magnetization interacts with, and sensitively reflects, FM fluctuations or domain wall dynamics generated as Ni layers switch and force Nb supercurrents to redistribute.
S.A. Kryukov et al. / Journal of Physics and Chemistry of Solids 67 (2006) 610–612
2. Experimental results
611
7
[Nb(23nm)/Ni(5nm)]5
[Nb(23nm)/Ni(5nm)]5 30
25
f = 10 Hz, ho = 0.5 Oe H || ML
20
H (kOe)
1.0 0.8
15
0.6
10
0.4 0.2
5 0.0 5.90
5.95
6.00
6.05
6.10
0 0
1
2
3
4
5
6
T (K) Fig. 1. Superconducting field–temperature phase boundary for a yZ5 nm Nb/Ni ML as measured by the onset temperature T of an abrupt increase in the imaginary part m 00 of the AC magnetic moment in field-cooling experiments at AC frequency fZ10 Hz and amplitude h0Z0.5 Oe with applied field H parallel to the ML plane. Inset shows an enlarged view of the low-field regime.
6
f = 10 Hz, ho = 0.5 Oe H || ML
5
H (kOe)
We have used a Quantum Design MPMS5 SQUID Magnetometer to estimate the field-temperature phase boundary for SC of [Nb(23 nm)/Ni(y)]5 ML, which is signaled by an increase in imaginary part (m 00 ) of the AC magnetic moment in field cooling experiments [15,16]. The m 00 (H,T) data exhibit weak kinks below TC (see inset to Fig. 1), and these anomalies persist to fields of at least 6.5 kOe with an average period dB»733 Oe, as shown in Fig. 2. Similar ‘matching anomalies’ have been observed (albeit at very low fields) in the phase diagrams of SC films patterned with periodic arrays of submicron flux pinning centers that induce periodic extrema of the critical current when the average flux density is equal to an integral multiple of the pinning center density [15,16]. In the present case, the periodic pinning structure can only be the ML repeat unit. Since the thickness of a single Nb layer of the ML is wZ23 nm!l0z43 nm [17], the zero-temperature penetration depth, we assume that magnetic flux is confined by the Nb layer thickness. A simple approach is to assume NL SC FL enter a Nb layer in successive chains [18,19] along the length Lz3 mm for the yZ 5 nm sample) of the ML oriented perpendicular to the applied field H. The confinement of the vortices in the perpendicular direction (assuming the SC order parameterZ0 in the Ni layers!) implies they will be approximately elliptical with major and
4
3
2
1
0 5.6
5.7
5.8
5.9
6.0
6.1
6.2
T (K) Fig. 2. Superconducting field–temperature phase boundary data of Fig. 1, emphasizing the region just below the zero-field TC. Arrows denote kinks proposed as roughly periodic matching anomalies in the phase boundary.
minor axes of length ls and w, respectivelywhere ls represents an effective penetration depth parallel to the ML. The condition for vortex quantization and ML parameters yield dBwLzNLF0 and NLlszL, and imply NLz2.4!103 and lsz1.3!103 nm (F0 is the flux quantum). We use these results and a crude flux conservation relation pwlszpl2 to extract an ‘effective’ twodimensional penetration depth lz170 nmz4l0, where the factor of 4 might result from finite mean-free-path and twodimensional effects [20]. The geometry of a SC/FM ML of rectangular cross-section subjected to a parallel magnetic field (as in Figs. 1 and 2) meets the general conditions for the ‘terraced critical state’ proposed by Cooley and Grishin [21], and should lead to periodic structure in the DC magnetic moment as a function of parallel magnetic field. Indeed, small cusps in DC m(H,T) data are observed near HZ0 for temperatures below TC, as shown in Fig. 3. However, the rough spacing of these cusps is 15–20 Oe, and they extend to approximately G80 Oe, which is similar to the matching field scale in patterned films [15,16], but quite different from the kinks in the AC phase boundary in Fig. 2. We have not observed periodic structures in our field-cooled ‘critical field’ data for applied field oriented perpendicular to the ML plane, reinforcing the supposition that such effects are a consequence of matching flux to a confining cross-sectional area of the Nb layers. We have also argued that the SC layers act as a ‘current amplifier’ of the Ni moments and their fluctuations [12]. This line of reasoning suggests that the matching anomalies evident in Figs. 2 and 3 may reflect subtle FLL rearrangements [22,23] that could be affected, or even driven, by Ni domain dynamics.
612
S.A. Kryukov et al. / Journal of Physics and Chemistry of Solids 67 (2006) 610–612
5
m (10–4emu)
4
Ni(5nm)[Nb(23nm)/Ni(5nm)]5 3
H || ML T = 3.8 K –4
–5 –100
–50
0
50
100
H (Oe) Fig. 3. DC magnetic moment m versus magnetic field applied parallel to the yZ 5 nm Nb/Ni ML plane held at a temperature TZ3.8 K. Arrows denote rough positions of cusps that may signal either a flux matching effect, or ‘quasiperiodic’ FLL instabilities within a ‘terraced critical state’. Note the cut in the vertical axis.
Acknowledgements Research at University of Kentucky supported by US DoE Grant no. DE-FG02-97ER45653. Research at Universidad Complutense supported by Spanish CICY Grant no. MAT0204543 and ESF VORTEX Program. References [1] B.Y. Jin, J.B. Ketterson, Artificial metal superlattices, Adv. Phys. 38 (1989) 189–366. [2] N.D. Mermin, H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- and two-dimensional isotropic heisenberg models, Phys. Rev. Lett. 17 (1966) 1133–1136. [3] A.P. Kuprin, L. Cheng, D.W. Lee, Z. Altounian, D.H. Ryan, Magnetism and structure of Fe/Cu multilayers studied by low-temperature conversion electron mo¨ssbauer spectroscopy, J. Appl. Phys. 85 (1999) 5738–5740. [4] M.B. Maple, Interplay between superconductivity and magnetism, Physica B 215 (1995) 110. [5] F.Y. Ogrin, S.L. Lee, A.D. Hillier, A. Mitchell, T.H. Shen, Interplay between magnetism and superconductivity in Nb/Co multilayers, Phys. Rev. B 62 (2000) 6021–6026.
[6] L. Lazar, K. Westerholt, H. Zabel, L.R. Tagirov, Yu.V. Goryunov, N.N. Garif’yanov, I.A. Garifullin, Superconductor/ferromagnet proximity effect in Fe/Pb/Fe trilayers, Phys. Rev. B 61 (2000) 3711–3722. [7] J. Aarts, J.M.E. Geers, E. Bru¨ck, A.A. Golubov, R. Coehoorn, Interface transparency of superconductor/ferromagnetic multilayers, Phys. Rev. B 56 (1997) 2779–2787. [8] J.S. Jiang, D. Davidovic, D.H. Reich, C.L. Chien, Oscillatory superconducting transition temperatures in Nb/Gd multilayers, Phys. Rev. Lett. 74 (1995) 314–317. [9] M. Velez, M.C. Cyrille, S. Kim, J.L. Vicent, Ivan K. Schuller, Enhancement of superconductivity by decreased magnetic spin-flip scattering: nonmonotonic Tc dependence with enhanced magnetic ordering, Phys. Rev. B 59 (1999) 14659–14662. [10] Z. Radovic, M. Ledvij, L. Dobrosavljevic-Grujic, A.I. Buzdin, J.R. Clem, Transition temperatures of superconducting-ferromagnetic superlattices, Phys. Rev. B 44 (1991) 759–764. [11] J.E. Villegas, E. Navarro, D. Jaque, E.M. Gonzalez, J.I. Martı´n, J.L. Vicent, Mixed-state properties of superconducting Nb/Ni superlattices, Physica C 369 (2002) 213–216. [12] S.A. Kryukov, L.E. De Long, E. Navarro, J.E. Villegas, E.M. Gonzalez, J.L. Vicent, Magnetic switching of Nb/Ni multilayers near the superconducting critical temperature, IEEE Trans. Magn. 39 (2003) 2693– 2695. [13] A. Buzdin, A. Yu, On the possibility of a first-order phase transition to the vortex state in layered superconductors, Physica C 167 (1990) 388–394. [14] A.I. Buzdin, S.S. Krotov, D.A. Kuptsov, Attraction of inclined vortices in magnetic superconductors, Physica C 175 (1991) 42–46. [15] L.E. De Long, S. Lokhre, T. Yun, V.V. Metlushko, V.V. Moshchalkov, Y. Bruynseraede, Whatever happened to mott? Physica C 369 (2002) 118–124. [16] V.V. Metlushko, L.E. De Long, M. Baert, E. Rosseel, M.J. Van Bael, K. Temst, V.V. Moshchalkov, Y. Bruynseraede, Supermatching vortex phases in superconducting thin films with antidot lattices, Europhys. Lett. 41 (1998) 333–338. [17] D.K. Finnemore, T.F. Stromberg, C.A. Swenson, Superconducting properties of high-purity Nb, Phys. Rev. 149 (1966) 231–243. [18] A.I. Buzdin, A.Yu Simonov, Magnetic flux penetration into layered superconductors, Physica C 175 (1991) 143–155. [19] C.A. Bolle, P.L. Gammel, D.G. Grier, C.A. Murray, D.J. Bishop, D.B. Mitzi, A. Kapitulnik, Observation of a commensurate array of flux chains in tilted flux lattices in Bi–Sr–Ca–Cu–O single crystals, Phys. Rev. Lett. 66 (1991) 112–115. [20] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York, 1975 (Chapter 4). [21] L.D. Cooley, A.M. Grishin, Pinch effect in commensurate vortex-pin lattices, Phys. Rev. Lett. 74 (1995) 2788–2791. [22] L. De Long, J. Childers, A. Olinger Jr., Q. Chen, J.-C. Hou, U. Lahaise, J. Zhang, D.G. Hinks, P. Canfield, R. Schweinfurth, D. Van Harlingen, M. Norton, High-sensitivity vibrating reed studies of superconductors, Physica B 223&224 (1996). [23] L.S. Levitov, Phyllotaxis of flux lattices in layered superconductors, Phys. Rev. Lett. 66 (1991) 224–227.
Matching effects in the field and temperature dependences of the magnetization of superconducting/ferromagnetic Nb/Ni multilayers S.A. Kryukov a, A. Bosomtwi a, L.E. De Long a,*, E.M. Gonzalez b, E. Navarro b, J.L. Vicent b, J.E. Villegas b, Wentao Xu a a
Department of Physics and Astronomy, University of Kentucky, CP 177, Lexington, KY 40506-0055, USA b Departamento de Fisica de Materiales, C.C. Fisicas, Universidad Complutense, 28040 Madrid, Spain
Abstract The temperature and field dependences of the magnetic moment of [Nb(x)/Ni(y)]5 multilayers are reported. Small kinks in the phase boundary defined in field-cooled AC experiments exhibit an approximate period of 733 Oe, and persist to a DC field HZ6.5 kOe applied parallel to the multilayer. A second group of cusp anomalies in DC magnetization are observed for H!100 Oe with approximate period 15–20 Oe. The absence of anomalies in perpendicular field and normal state data suggest they are due to finite-size effects expected when the superconducting layer is thinner than the magnetic penetration depth. q 2005 Elsevier Ltd. All rights reserved. Keywords: A: Multilayers; A: Superconductors; A: Magnetic materials; D: Magnetic properties; D: Phase equilibria
1. Introduction The fundamental properties of magnetic thin films and ‘multilayers’ (ML) have been extensively studied and exhibit many interesting properties [1]. In particular, a ML stack of ferromagnetic (FM) thin films interleaved with nonmagnetic layers can be configured as quasi-two-dimensional systems confined along the normal direction to the film plane. Finitetemperature phase transitions cannot be realized in twodimensional systems [2], and it is known that long-range FM order breaks down in favor of ‘superparamagnetism’ if a film is made thin enough [3]. Superconductivity is an established, sensitive probe of magnetic interactions and magnetic moment stability in dilute alloys and compounds [4] and more recently, ML [1]. Recent studies of bilayers, trilayers and ML have addressed a number of interesting effects of magnetic order on superconducting (SC) properties [5,6]. For example, both monotonic [7] and nonmonotonic [8,9] decreases of the SC transition temperature TC with FM layer thickness have been observed; and the
* Corresponding author. E-mail address: [email protected] (L.E. De Long).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.081
potential roles of pair breaking phase shifts and ‘p-junctions’ have been discussed [10]. An intriguing competition between FM and SC has been reported for [Nb(x)/Ni(y)]Z ML [11,12]. TC was found to vanish near a critical Ni thickness yz2.5 nm, above which FM develops in ML (for xZ10 nm Nb layers). In contrast, it was found that increasing the Nb layer thickness to 23 nm resulted in SC with a near-constant (with possible small-amplitude oscillations) TC over a wide range of Ni thickness 1%y%6 nm. The above results have led us to revisit magnetic ML as interesting nanoscale systems that might exhibit finite-size effects when the FM layer thickness is y!10 nm. In particular, the dependence of TC(y) for the xZ10 nm ML series is clear evidence for the destruction of SC by magnetic fluctuations near a quantum critical point (QCP) yz2.5 nm. Moreover, we expect their SC properties to sensitively reflect finite-size, shape and boundary effects on FM. FM/SC ML systems may display other interesting effects, including a number of flux line lattice phase transitions in strongly anisotropic superconductors [13,14]. We recently discovered that the onset of Nb superconductivity dramatically alters the switching dynamics of the FM Ni layers; and we found preliminary evidence that screening supercurrents sensitively probe the stability of FM layers [12]. These results suggest that the SC magnetization interacts with, and sensitively reflects, FM fluctuations or domain wall dynamics generated as Ni layers switch and force Nb supercurrents to redistribute.
S.A. Kryukov et al. / Journal of Physics and Chemistry of Solids 67 (2006) 610–612
2. Experimental results
611
7
[Nb(23nm)/Ni(5nm)]5
[Nb(23nm)/Ni(5nm)]5 30
25
f = 10 Hz, ho = 0.5 Oe H || ML
20
H (kOe)
1.0 0.8
15
0.6
10
0.4 0.2
5 0.0 5.90
5.95
6.00
6.05
6.10
0 0
1
2
3
4
5
6
T (K) Fig. 1. Superconducting field–temperature phase boundary for a yZ5 nm Nb/Ni ML as measured by the onset temperature T of an abrupt increase in the imaginary part m 00 of the AC magnetic moment in field-cooling experiments at AC frequency fZ10 Hz and amplitude h0Z0.5 Oe with applied field H parallel to the ML plane. Inset shows an enlarged view of the low-field regime.
6
f = 10 Hz, ho = 0.5 Oe H || ML
5
H (kOe)
We have used a Quantum Design MPMS5 SQUID Magnetometer to estimate the field-temperature phase boundary for SC of [Nb(23 nm)/Ni(y)]5 ML, which is signaled by an increase in imaginary part (m 00 ) of the AC magnetic moment in field cooling experiments [15,16]. The m 00 (H,T) data exhibit weak kinks below TC (see inset to Fig. 1), and these anomalies persist to fields of at least 6.5 kOe with an average period dB»733 Oe, as shown in Fig. 2. Similar ‘matching anomalies’ have been observed (albeit at very low fields) in the phase diagrams of SC films patterned with periodic arrays of submicron flux pinning centers that induce periodic extrema of the critical current when the average flux density is equal to an integral multiple of the pinning center density [15,16]. In the present case, the periodic pinning structure can only be the ML repeat unit. Since the thickness of a single Nb layer of the ML is wZ23 nm!l0z43 nm [17], the zero-temperature penetration depth, we assume that magnetic flux is confined by the Nb layer thickness. A simple approach is to assume NL SC FL enter a Nb layer in successive chains [18,19] along the length Lz3 mm for the yZ 5 nm sample) of the ML oriented perpendicular to the applied field H. The confinement of the vortices in the perpendicular direction (assuming the SC order parameterZ0 in the Ni layers!) implies they will be approximately elliptical with major and
4
3
2
1
0 5.6
5.7
5.8
5.9
6.0
6.1
6.2
T (K) Fig. 2. Superconducting field–temperature phase boundary data of Fig. 1, emphasizing the region just below the zero-field TC. Arrows denote kinks proposed as roughly periodic matching anomalies in the phase boundary.
minor axes of length ls and w, respectivelywhere ls represents an effective penetration depth parallel to the ML. The condition for vortex quantization and ML parameters yield dBwLzNLF0 and NLlszL, and imply NLz2.4!103 and lsz1.3!103 nm (F0 is the flux quantum). We use these results and a crude flux conservation relation pwlszpl2 to extract an ‘effective’ twodimensional penetration depth lz170 nmz4l0, where the factor of 4 might result from finite mean-free-path and twodimensional effects [20]. The geometry of a SC/FM ML of rectangular cross-section subjected to a parallel magnetic field (as in Figs. 1 and 2) meets the general conditions for the ‘terraced critical state’ proposed by Cooley and Grishin [21], and should lead to periodic structure in the DC magnetic moment as a function of parallel magnetic field. Indeed, small cusps in DC m(H,T) data are observed near HZ0 for temperatures below TC, as shown in Fig. 3. However, the rough spacing of these cusps is 15–20 Oe, and they extend to approximately G80 Oe, which is similar to the matching field scale in patterned films [15,16], but quite different from the kinks in the AC phase boundary in Fig. 2. We have not observed periodic structures in our field-cooled ‘critical field’ data for applied field oriented perpendicular to the ML plane, reinforcing the supposition that such effects are a consequence of matching flux to a confining cross-sectional area of the Nb layers. We have also argued that the SC layers act as a ‘current amplifier’ of the Ni moments and their fluctuations [12]. This line of reasoning suggests that the matching anomalies evident in Figs. 2 and 3 may reflect subtle FLL rearrangements [22,23] that could be affected, or even driven, by Ni domain dynamics.
612
S.A. Kryukov et al. / Journal of Physics and Chemistry of Solids 67 (2006) 610–612
5
m (10–4emu)
4
Ni(5nm)[Nb(23nm)/Ni(5nm)]5 3
H || ML T = 3.8 K –4
–5 –100
–50
0
50
100
H (Oe) Fig. 3. DC magnetic moment m versus magnetic field applied parallel to the yZ 5 nm Nb/Ni ML plane held at a temperature TZ3.8 K. Arrows denote rough positions of cusps that may signal either a flux matching effect, or ‘quasiperiodic’ FLL instabilities within a ‘terraced critical state’. Note the cut in the vertical axis.
Acknowledgements Research at University of Kentucky supported by US DoE Grant no. DE-FG02-97ER45653. Research at Universidad Complutense supported by Spanish CICY Grant no. MAT0204543 and ESF VORTEX Program. References [1] B.Y. Jin, J.B. Ketterson, Artificial metal superlattices, Adv. Phys. 38 (1989) 189–366. [2] N.D. Mermin, H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- and two-dimensional isotropic heisenberg models, Phys. Rev. Lett. 17 (1966) 1133–1136. [3] A.P. Kuprin, L. Cheng, D.W. Lee, Z. Altounian, D.H. Ryan, Magnetism and structure of Fe/Cu multilayers studied by low-temperature conversion electron mo¨ssbauer spectroscopy, J. Appl. Phys. 85 (1999) 5738–5740. [4] M.B. Maple, Interplay between superconductivity and magnetism, Physica B 215 (1995) 110. [5] F.Y. Ogrin, S.L. Lee, A.D. Hillier, A. Mitchell, T.H. Shen, Interplay between magnetism and superconductivity in Nb/Co multilayers, Phys. Rev. B 62 (2000) 6021–6026.
[6] L. Lazar, K. Westerholt, H. Zabel, L.R. Tagirov, Yu.V. Goryunov, N.N. Garif’yanov, I.A. Garifullin, Superconductor/ferromagnet proximity effect in Fe/Pb/Fe trilayers, Phys. Rev. B 61 (2000) 3711–3722. [7] J. Aarts, J.M.E. Geers, E. Bru¨ck, A.A. Golubov, R. Coehoorn, Interface transparency of superconductor/ferromagnetic multilayers, Phys. Rev. B 56 (1997) 2779–2787. [8] J.S. Jiang, D. Davidovic, D.H. Reich, C.L. Chien, Oscillatory superconducting transition temperatures in Nb/Gd multilayers, Phys. Rev. Lett. 74 (1995) 314–317. [9] M. Velez, M.C. Cyrille, S. Kim, J.L. Vicent, Ivan K. Schuller, Enhancement of superconductivity by decreased magnetic spin-flip scattering: nonmonotonic Tc dependence with enhanced magnetic ordering, Phys. Rev. B 59 (1999) 14659–14662. [10] Z. Radovic, M. Ledvij, L. Dobrosavljevic-Grujic, A.I. Buzdin, J.R. Clem, Transition temperatures of superconducting-ferromagnetic superlattices, Phys. Rev. B 44 (1991) 759–764. [11] J.E. Villegas, E. Navarro, D. Jaque, E.M. Gonzalez, J.I. Martı´n, J.L. Vicent, Mixed-state properties of superconducting Nb/Ni superlattices, Physica C 369 (2002) 213–216. [12] S.A. Kryukov, L.E. De Long, E. Navarro, J.E. Villegas, E.M. Gonzalez, J.L. Vicent, Magnetic switching of Nb/Ni multilayers near the superconducting critical temperature, IEEE Trans. Magn. 39 (2003) 2693– 2695. [13] A. Buzdin, A. Yu, On the possibility of a first-order phase transition to the vortex state in layered superconductors, Physica C 167 (1990) 388–394. [14] A.I. Buzdin, S.S. Krotov, D.A. Kuptsov, Attraction of inclined vortices in magnetic superconductors, Physica C 175 (1991) 42–46. [15] L.E. De Long, S. Lokhre, T. Yun, V.V. Metlushko, V.V. Moshchalkov, Y. Bruynseraede, Whatever happened to mott? Physica C 369 (2002) 118–124. [16] V.V. Metlushko, L.E. De Long, M. Baert, E. Rosseel, M.J. Van Bael, K. Temst, V.V. Moshchalkov, Y. Bruynseraede, Supermatching vortex phases in superconducting thin films with antidot lattices, Europhys. Lett. 41 (1998) 333–338. [17] D.K. Finnemore, T.F. Stromberg, C.A. Swenson, Superconducting properties of high-purity Nb, Phys. Rev. 149 (1966) 231–243. [18] A.I. Buzdin, A.Yu Simonov, Magnetic flux penetration into layered superconductors, Physica C 175 (1991) 143–155. [19] C.A. Bolle, P.L. Gammel, D.G. Grier, C.A. Murray, D.J. Bishop, D.B. Mitzi, A. Kapitulnik, Observation of a commensurate array of flux chains in tilted flux lattices in Bi–Sr–Ca–Cu–O single crystals, Phys. Rev. Lett. 66 (1991) 112–115. [20] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York, 1975 (Chapter 4). [21] L.D. Cooley, A.M. Grishin, Pinch effect in commensurate vortex-pin lattices, Phys. Rev. Lett. 74 (1995) 2788–2791. [22] L. De Long, J. Childers, A. Olinger Jr., Q. Chen, J.-C. Hou, U. Lahaise, J. Zhang, D.G. Hinks, P. Canfield, R. Schweinfurth, D. Van Harlingen, M. Norton, High-sensitivity vibrating reed studies of superconductors, Physica B 223&224 (1996). [23] L.S. Levitov, Phyllotaxis of flux lattices in layered superconductors, Phys. Rev. Lett. 66 (1991) 224–227.