Superconductivity as a probe of anomalous magnetic

switching and ferromagnetic stability in Nb/Ni

multilayers

Lance De Long, Sergiy Kryukov, Asamoah Bosomtwi, Wentao Xu, E M

Gonzalez, E Navarro, J E Villegas, Jose Luis Vicent, Chengtao Yu, Michael J.

Pechan

To cite this version:

Lance De Long, Sergiy Kryukov, Asamoah Bosomtwi, Wentao Xu, E M Gonzalez, et al.. Su-
perconductivity as a probe of anomalous magnetic switching and ferromagnetic stability in
Nb/Ni multilayers. Philosophical Magazine, Taylor & Francis: STM, Behavioural Science and
Public Health Titles, 2006, 86 (17-18), pp.2735-2760. <10.1080/14786430500422308>. <hal-
00513634>

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Superconductivity as a probe of anomalous magnetic 
switching and ferromagnetic stability in Nb/Ni multilayers 

 
 

Journal: Philosophical Magazine & Philosophical Magazine Letters 

Manuscript ID: TPHM-05-Apr-0106.R1 

Journal Selection: Philosophical Magazine 

Date Submitted by the 
Author: 

24-Aug-2005 

Complete List of Authors: De Long, Lance; Univ of Kentucky, Physics and Astronomy 
Kryukov, Sergiy; University of Kentucky, Physics and Astronomy 

Bosomtwi, Asamoah; University of Kentucky, Physics and 
Astronomy 
Xu, Wentao; University of Kentucky, Physics and Astronomy 
Gonzalez, E M; Universidad Complutense de Madrid, Departamento 
de Fisica de Materiales, C C Fisicas 
Navarro, E; Universidad Complutense de Madrid, Departamento de 
Fisica de Materiales, C C Fisicas 
Villegas, J E; Universidad Complutense de Madrid, Departamento de 
Fisica de Materiales, C C Fisicas 
Vicent, Jose; Universidad Complutense de Madrid, Departamento de 
Fisica de Materiales 
Yu, Chengtao; Miami University, Physics 

Pechan, Michael J.; Miami University, Physics 

Keywords: 
magnetic multilayers, magnetometry, superconducting materials, 
low-dimensional nanomaterials 

Keywords (user supplied): Superconducting multilayers, Magnetic switching 

  
 
 

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Superconductivity as a probe of magnetic switching and ferromagnetic 

stability in Nb/Ni multilayers 

 

L. E. DE LONG*, S. A. KRYUKOV*, A. BOSOMTWI*, WENTAO XU* 

E. M. GONZALEZ**, E. NAVARRO**, J. E. VILLEGAS**, J. L. VICENT** 

CHENGTAO YU
†
 and M. J. PECHAN

†
 

 

*Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-

0055 U.S.A. 

lander@uky.edu, kryukov@uky.edu, bosomtwi@hotmail.com, wxu2@uky.edu     

**Departamento de Fisica de Materiales, C. C. Fisicas, Universidad Complutense, 28040 

Madrid, Spain 

jlvicent@fis.ucm.es 

  

†
Department of Physics, Miami University, Oxford, OH 45056 U.S.A. 

yuc@muohio.edu, pechanmj@muohio.edu  

 

Corresponding author:  Professor Lance E. De Long, Department of Physics and 

Astronomy, University of Kentucky, Lexington, KY 40506-0055, U.S.A. 

Tel: 1-859-257-4775 FAX:  1-859-323-2846 Email:  lander@uky.edu. 

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mailto:lander@uky.edu
mailto:sergey@pa.uky.edu
mailto:bosomtwi@hotmail.com
mailto:wxu2@uky.edu
mailto:jlvicent@fis.ucm.es
mailto:yuc@muohio.edu
mailto:pechanmj@muohio.edu


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The temperature and field dependences of the AC and DC magnetic 

moment of superconducting and ferromagnetic Nb/Ni multilayers were 

measured using a SQUID magnetometer with magnetic field applied 

parallel to the multilayer plane.  Periodic kinks in the superconducting 

upper critical field are evidence for nucleation of a hierarchy of Abrikosov 

vortex lattices aligned parallel to the multilayer.  Small cusps in the low-

field, isothermal DC magnetization are evidence that supercurrents are 

sensitive to extremely small changes in the Ni layer magnetization.  

Smooth ferromagnetic hysteresis is observed in the normal state, but is 

supplanted below the superconducting transition by two reproducible 

discontinuities that indicate magnetic switching of the Ni layers is tightly 

coupled to the supercurrents.  The discontinuities are attributed to the non-

dipole character of the moment near switching fields, and therefore cannot 

be analyzed by standard magnetometer software.  Ferromagnetic 

resonance spectra were measured in parallel and perpendicular DC 

magnetic fields at room temperature and 4.2 K; and resulting data suggest 

that Ni layers interact magnetically in the superconducting state.   

  

Keywords:  Multilayers, Superconducting thin films, Magnetic switching, 

Ferromagnetic resonance, Low-dimensional magnetism, SQUID 

magnetometer 

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1.  Introduction 

 

Ferromagnetic (FM) thin films and multilayers (ML) have been extensively studied and 

exhibit many interesting properties [1].    Finite-temperature phase transitions cannot be 

realized in ideal two-dimensional systems [2], and it is experimentally known that long-

range FM order breaks down in favor of “superparamagnetism” if a film is made thin 

enough [3].  In particular, a ML stack of FM thin films interleaved with nonmagnetic 

layers can be configured as a quasi-two-dimensional system confined along the normal 

direction to the film plane.   

 Of course, as the thickness of a magnetic film is decreased, it is increasingly difficult 

to maintain adequate signal strength from common experimental probes in order to 

accurately characterize film properties; and this problem is particularly difficult in the 

case of magnetic moment measurements performed in applied fields parallel to the film 

plane.  Fortunately, the scope of experimental tools is enhanced by superconductivity, 

which is an established, sensitive probe of magnetic interactions and magnetic moment 

stability in dilute alloys and compounds [4] and more recently, ML [1].  Previous 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.  These effects have been discussed in terms of the 

potential roles of pair breaking phase shifts and “π-junctions” [10].  FM/SC ML systems 

may display a variety of other interesting effects, including a number of vortex lattice 

phase transitions predicted for strongly anisotropic superconductors [11,12]. 

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 An intriguing competition between FM and SC has been reported for [Nb(x)/Ni(y)]Z 

ML [13,14].  The sensitivity of superconductivity to Ni layer thickness y appears to 

depend markedly on the Nb layer thickness x, as shown in Fig. 1.  The TC of the ML 

series with x = 10 nm Nb layers was found to vanish for Ni thickness yCR ≈ 2.5 nm, 

above which FM (as detected via Kerr effect) develops for temperatures T ≥ 10 K. The 

rapid variations of TC(y) and the Kerr reflectance for the x = 10 nm ML series constitute 

evidence for the destruction of SC by magnetic fluctuations near a quantum critical point 

(QCP) located near yCR.  On the other hand, TC(y) was found to be near-constant (with 

possible small-amplitude oscillations) over a range of Ni thickness 2.5 ≤ y ≤ 6 nm in a 

second ML series with thicker (x = 23 nm) Nb layers. 

 We recently discovered that the onset of Nb superconductivity dramatically alters the 

switching signature of the FM layers in SQUID magnetometer data; and we postulated 

that screening supercurrents sensitively probe the magnetic stability of the Ni layers [14].  

The above results have led us to revisit Nb/Ni ML as interesting nanoscale systems 

whose SC properties might exhibit finite-size, interface and critical fluctuation effects 

when the FM layer thickness y < 10 nm.  A closely related possibility is that the SC 

condensate interacts with local FM domain wall dynamics generated as Ni layers switch 

and force large (critical current density JC > 10
7
 A/cm

2
) Nb supercurrents to redistribute. 

 Herein, we address these possibilities as part of a review of the temperature and field 

dependences of the AC and DC magnetic moment of Nb/Ni ML in magnetic fields 

applied parallel to the film plane, as well as ferromagnetic resonance (FMR) spectra 

obtained for both parallel and perpendicular field geometries.  We also identify and 

discuss the limitations of using a point-dipole approximation (standard model for the 

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Quantum Design SQUID Magnetometer) to extract the DC magnetic moment of a 

SC/FM ML in parallel applied magnetic fields. 

 

2.  Experimental details 

 

Ni(y)/[Nb(x=10nm)/Ni(y)]8 and [Nb(x=23nm)/Ni(y)]5 ML with Nb layer thickness x and 

Ni layer thickness 1.5 ≤ y ≤ 5 nm, have been fabricated by DC magnetron sputtering on 

Si (100) substrates. The samples exhibit textured growth of Nb (110) and Ni (111) layers 

with negligible interdiffusion and interface roughness [13,14].  Samples used in FMR and 

magnetometer experiments were approximately square with areas in the range 4 x 10
-6

 m
2
 

to 9 x 10
-6

 m
2
.  Preliminary electron microscopy (EDAX) investigations reveal that 

oxygen contamination was not significant during deposition.  A typical resistance ratio 

(room temperature/TC
+
) for a ML sample was 8.  Structural data for the sample ML will 

be discussed in more detail in a separate publication. 

 We have used a Quantum Design MPMS5 SQUID Magnetometer to measure SC and 

FM properties of Nb/Ni ML with the applied magnetic field H parallel to the ML plane.  

Field-temperature phase boundaries between SC and normal states were measured using 

a quasi-static AC magnetic field modulation technique at frequencies 0.1 ≤ f ≤ 10 Hz and 

drive amplitudes 0.01 ≤ µοho ≤ 0.3 mT.  Isothermal magnetization hysteresis curves were 

obtained using the MPMS5 “Reciprocating Sample Option” (RSO), which optimizes 

magnetometer sensitivity by executing the low-frequency oscillatory movement of the 

sample through a superconducting flux-locked loop [15].   

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 Most of the RSO data reported herein were acquired using a short (3-cm) scan length 

and an “Iterative Regression” analysis that permits the sample position to vary in 

numerical fits of the raw output of the SQUID voltmeter.  These methodologies have 

been generally accepted as useful for improving the reliability of the MPMS5 data 

analysis software (which assumes the sample is an ideal point dipole [16]) in 

measurements of thin SC films having large demagnetization effects when oriented 

perpendicular to the applied magnetic field [17].  However, our results show that these 

data processing procedures fail in situations where macroscopic supercurrents strongly 

couple to FM domain patterns in SC/FM ML measured in the parallel geometry.   

 FMR measurements were carried out at 9.7 GHz in a rectangular cavity in the 

transverse electric mode (TE102, loaded Q ≈ 5000). Samples were fixed to a rotatable 

quartz rod by a small amount of vacuum grease so that DC magnetic field could be 

applied either parallel or perpendicular to the ML plane during measurements. 

Temperatures down to 4.2 K were attained using a helium gas-flow cryostat.  Additional 

TC data for FMR samples were acquired via standard AC magnetic susceptibility 

measurements using a Quantum Design PPMS System operating at f = 10 kHz and 

amplitude µοho =  0.4 mT, with applied DC magnetic fields oriented parallel to the ML 

plane.   

 

3.  Experimental results 

 

3.1.  Superconducting phase boundary measurements 

   

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Both AC and DC measurements of samples from the [Nb(23nm)/Ni(y)]5 ML series (with 

y = 2.5, 3.5 or 5 nm) yielded TC’s that were in the range 6.0 to 7.0 K, consistent with 

previous resistive data [13,14] shown in Fig. 1.  These values are to be compared with TC 

≈ 8.6 K, 7.0 K and 5.7 K, as observed for single-layer control films of Nb with thickness t 

≈ 100 nm, 20 nm and 10 nm, respectively.  These reference values reflect the depression 

of the transition well below the bulk value of 9.3 K by the effects of finite Nb layer 

thickness x << ξo ≈ 43 nm [18], the coherence length of bulk Nb [19]. 

 AC susceptibility data for the [Nb(23nm)/Ni(y)]5 ML set exhibited a nonmonotonic 

y-dependence of  TC = 6.88, 6.98, and 6.00 K, for y = 2.5, 3.5 and 5.0 nm, respectively.  

A small oscillation of TC(y) also has been observed in previous resistive measurements of 

the x = 23 nm ML series for 2.5 ≤ y ≤ 6.0 nm [13,14], as shown in Fig. 1.  A 

nonmonotonic dependence of TC on the thickness of the FM layer has also been reported 

in Nb/Gd multilayers [8], Fe/V/Fe trilayers [20], and Fe/Pb/Fe trilayers [6], and has been 

attributed to pairbreaking associated with an exchange coupling between magnetic layers 

that oscillates in magnitude and sign as a function of the superconducting layer thickness. 

 Clear SC transitions could not be observed by AC and DC magnetic measurements 

above 2 K for the Ni(y)/[Nb(10nm)/Ni(y)]8 set (y = 1.5, 2.5 and 3.5 nm).  Results of 

earlier resistive measurements [13] of [Nb(x)/Ni(y)]8 ML without an extra Ni capping 

layer are shown in Fig. 1.  The latter data indicate a strong depression of TC ≤ 5.3 K with 

increasing Ni layer thickness for y > 1.5 nm; and the apparent disagreement with the 

magnetic data for the capped ML series is not presently understood.  

 Measurements of the field-temperature phase boundary (upper critical magnetic field 

HC2(T)) yield information concerning the fundamental length scales governing the SC 

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state and the character of any coupling between the Nb layers in the presence of the FM 

Ni layers [1].  In particular, estimates of the coherence length ξ(T) and evidence for two-

dimensionality can be obtained from HC2(T) data, where we have assumed that the onset 

of superconductivity is marked by an abrupt increase in the imaginary part (m”) of the 

AC magnetic moment in field-cooling experiments.  The initial diamagnetic change of 

the real part (m’) of the AC moment can also be used to define an alternative phase 

boundary, but the m’ curve may lie slightly lower (of order 10
-2 

TC) in temperature than 

the m” data in cases of strong pinning of magnetic flux [21,22]. 

 The low-field portion of the m”(H,T) boundary does not exhibit deviations from 

three-dimensional (linear) behavior [1], except possibly very close to TC, as shown in the 

inset to Fig. 2a.  We have considered a simplified picture of a two-dimensional SC layer 

of thickness x between two FM layers with zero SC order parameter at the interfaces, as 

discussed by Jin and Ketterson [1], who give the following relation (gaussian units) for 

the zero-temperature value of HC2: 

 

      HC2(0) = (5.53)Φo/2πξox     (1) 

 

Estimating HC2(0) ≈ 3.5 x 10
4
 Oe from Fig. 2 and the Nb layer thickness x = 23 nm, we 

obtain a zero-temperature coherence length ξo ≈ 23 nm = x, which is at the borderline 

between two- and three-dimensional behavior for a single Nb layer.  Alternatively, a 

simple, three-dimensional Ginzberg-Landau model analogous to Eq. 1 yields 

 

     HC2(0) = Φo/2πξo
2
      (2) 

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The latter relation gives ξo ≈ 9.8 nm << x, which is consistent with three-dimensional 

behavior.   

 Electron scattering can have significant effects on the behavior of HC2(T), and can be 

roughly estimated from Eq. 2 by replacing ξo
2
 by ξoltr , which is valid in the “dirty limit” 

(transport mean-free path ltr < ξo [23]) with DC field applied perpendicular to the ML [1].  

For example, resistivity data [24] for a [Nb(10nm)/Ni(1.5nm)]8 ML yield a transport 

mean-free path ltr ≈ 4.3 nm parallel to the Nb planes (assuming a free-electron model with 

one conduction electron per Nb [25], and that the conductivity of the thinner Ni layers is 

negligible).  Perpendicular upper critical field data [13] then yield ξo ≈ 40 nm, close to 

the value for bulk Nb [19], and consistent with the dirty limit approach.   

 Close examination of the m”(H,T) boundary reveals weak “kinks” below TC, and 

these anomalies persist to fields of at least 0.65 T with an average period µoδH ≈ 73.3 

mT, as shown in Fig. 2.  Periodic extrema (“matching anomalies”) in the critical current, 

magnetization and HC2(T) of SC materials can be observed at applied fields where the 

average density of flux lines (FL) is equal to an integral multiple of the pinning center 

density.  Matching anomalies are relatively well defined in the case of thin films and ML 

if the FL pinning centers are located on a periodic lattice and have a characteristic size 

ξ(T) < D < λ(T) [21,22].  The only apparent periodic structure capable of pinning FL in 

the present case is the ML repeat unit (see Fig. 3a), which leads to the supposition that 

such effects are a consequence of the confinement of quantized FL to the cross-sectional 

area of individual Nb layers in the parallel field geometry.   

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 A simple approach to explain the kinks is to assume that a number NL FL enter a 

single Nb layer in successive chains [26-28] that form along the perpendicular edge of 

length L (≈ 3 mm for the y = 5 nm sample) when the ML is oriented parallel to the 

applied field H.  A schematic arrangement of the FL in an isolated Nb plane is shown in 

Fig. 3b.  It is important to note that the thickness of a single Nb layer is w = x = 23 nm <  

λo ≈ 43 nm [19], the zero-temperature penetration depth of clean, bulk Nb.  If the SC 

order parameter = 0 in the Ni layers (total confinement of supercurrents within a Nb 

plane) we must assume that the FL cross-sections are approximately elliptical with major 

and minor axes of length λ|| and w, respectively, where λ|| represents the penetration 

depth parallel to the ML.    The condition for flux quantization and the known ML 

parameters then yield (δH)wL ≈ NLΦo and 2NLλ|| ≈ L, and imply NL ≈ 2.4 x 10
3
 and λ|| ≈ 

614 nm (Φo is the flux quantum).  We use these results and a relation πwλ|| ≡ πλΙ
2
 to 

extract an “effective” isotropic penetration depth λΙ ≈ 84 nm ≈ 2λo, where the rough 

factor of 2 could easily be accounted for by finite mean-free-path and two-dimensional 

confinement effects [29].  Since NL >> 1, and the ML were not precisely aligned with the 

applied DC field, neither do we expect the FL chains to be perfectly ordered, nor the 

matching anomalies to be perfectly periodic in applied field.  This crude model therefore 

yields a plausible explanation of the phase boundary kinks. 

 However, noting that x < ξo ≈ 30 - 40 nm for [Nb(10nm)/Ni(y)]8 ML [13] and clean, 

bulk Nb [19], one must consider the possibility that the Nb layers are proximity-coupled 

via the Ni layers (of thickness y << ξo).  Specifically, if the superconducting order 

parameter is nonzero in the Ni layers, the presumed “chain” of FL should re-distribute 

among the coupled Nb and Ni layers available (i.e. the assumed SC layer spans the entire 

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ML cross-section, and Ni layers present periodic FL pinning sites).  A corresponding 

estimate that assumes that one row of NL cylindrical FL enters the ML cross section of 

total width w = 140 nm (five repeat units of a Nb(x = 23 nm)/Ni(y = 5 nm) bilayer) yields 

λ = 101 nm ≈ (2.4)λo.   

 Note that the values of λ >> x generated by the last set of estimates support a scenario 

in which supercurrents flow between Nb layers via a finite SC order parameter within the 

Ni layers.  This scenario is also consistent with the relatively small value x = 23 nm of 

the Nb layer thickness compared to the bulk Nb FL core size ≈ 2ξo ≈ 80 nm, and the 

existence of oscillations of TC(y) generated by pairbreaking within the Ni layers, as 

shown in Fig. 1.  Moreover, these characteristic lengths are all well below the total 

thickness (140 nm) of the [Nb(23nm)/Ni(5nm)]5  ML, and are therefore consistent with 

the linear, three-dimensional behavior observed in the m” phase boundary shown in Fig. 

2. 

 However, magnetic pairbreaking, proximity effects and interface scattering also must 

be taken into account in a quantitative model of SC/FM ML data [1].  In the absence of a 

rigorous microscopic theory for the Nb/Ni ML case, we must extract estimates of 

characteristic length scales that govern the interlayer couplings by investigation of 

additional physical properties. 

 

3.2  Magnetic moment measurements 

 

The “standard” manifestation of flux matching in patterned SC thin films in 

perpendicular magnetic fields is observation of sharp cusp or peak anomalies in the 

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isothermal magnetization curves [21,22].  Alternatively, in the more relevant situation 

where field was applied parallel to SC planes, peaks have been observed in the m(H) 

curves of SC/normal-metal ML [26], SC/FM ML [30,31] and high-TC [31-33] 

superconductors.  Although these anomalies were attributed to flux matching effects, the 

observed peaks were separated by relatively large field intervals of 10
-2

 to 1 T, were not 

precisely periodic in field, and were often a mix of narrow and broad features.   

 In the present case of [Nb(23nm)/Ni(y)]5 ML, we observed small cusp anomalies in 

the field dependence of the SC state magnetic moment m(H) (see Figs. 4a and 4b) near H 

= 0; whereas we have observed only smooth FM hysteresis in the normal state for fields 

applied parallel to the ML plane and fixed temperatures just above TC  ≈ 5.8 K, as shown 

in Fig. 5.   It is apparent that the rough spacing between the cusps in Figs. 4a and 4b is 

not strongly affected by changes in temperature just below TC, as expected for a flux 

matching effect that is mainly dependent on the arrangement and density of pinning 

centers.  On the other hand, the strong temperature dependences (very close to TC ≈ 6 K) 

of the coherence length ξ(T) and penetration depth λ(T) will affect the FL pinning 

strength and electromagnetic coupling between Ni layers, respectively.  For example, 

smearing (without changing the spacing) of matching anomalies by random pinning 

forces generally strengthens with decreasing temperature and higher FL density [21,22].  

Such an effect may be evident in Fig. 5a, where the small cusp and abrupt switching 

anomalies are absent in an initial magnetization loop measured well below TC at T = 3.5 

K.  However, increasing T to just below TC (Fig. 5c), or changing the magnetic history 

(e.g., “training” on successive field cycling) of the sample (Fig. 5b) results in the 

appearance of the cusps.  Furthermore, the smooth FM reversal observed in the normal 

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state hysteresis curves is replaced by a remarkable effect below TC, where two 

symmetric, highly reproducible (on repeated field cycling) jumps of the diamagnetic 

signal occur near or above the FM coercive fields, which suggests that they mark a 

sudden switching of the Ni magnetization.  Another interesting feature is the “latching” 

of m(H) at exactly zero at the midpoints of the switching anomalies, as shown in Fig. 5b 

(also see Fig. 6, below).   

 The proposed QCP (defined by a critical Ni layer thickness yCR ≈ 2.5 nm, at which 

superconductivity and FM order simultaneously decrease toward T = 0, as shown in Fig. 

1) is presumably caused by the instability of long-range FM order in [Nb(10nm)/Ni(y)]8 

ML with Ni layer thickness y ≤ 2.5 nm [14].  Therefore, strong effects of temperature and 

finite Ni layer thickness on the magnetization data also might be expected for the other 

series of [Nb(23nm)/Ni(y)]5 ML with y ≈ 2.5 nm, as shown in Fig. 6.   Indeed, the initial 

magnetization curve (Fig. 6a) of the [Nb(23nm)/Ni(2.5nm)]5 ML exhibits strong 

discontinuities for µο|H| > 50 mT at T = 5.0 K.  These apparently “random instabilities” 

are absent (even in the initial magnetization) at a lower temperature T = 3.5 K (as shown 

in Fig. 6b), consistent with a rapidly increasing stability of the FM state with decreasing 

temperature in this temperature and composition regime. 

 The effects of training and interactions between supercurrents and the FM layers are 

also exibited by the [Nb(23nm)/Ni(2.5nm)]5 ML at a slightly higher temperature T = 5.5 

K, as shown in Fig. 7a.  These data exhibit anomalies for µο|H| > 30 mT as emphasized in 

Fig. 7b, where we have subtracted the normal state FM hysteresis curve (which is 

essentially temperature-independent near TC) from the SC state data.  The difference 

curve emphasizes the contribution of supercurrents to the sample magnetization.  The 

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initial linear magnetization curve (blue line) is consistent with a Meissner diamagnetic 

response that is followed by entry into the mixed state, accompanied by a number of 

anomalies that could be generated by random flux avalanche events [34].  Finally, there is 

an abrupt collapse of the inferred SC response near µοH = 35 mT.  However, further field 

cycling generates two highly symmetric groups of magnetization anomalies near ±35 mT, 

and sharp collapses of the supercurrent magnetization shown in Fig. 7b near ±50 mT.  

Note that the unstable field interval at T = 5.5 K shown in Fig. 7 is consistent with that 

observed at T = 5 K, as shown in Fig. 6a.  

 Additional experiments were conducted to determine if the stability of the Ni 

magnetization is strongly dependent on Ni layer thickness near y ≥ 2.5 nm, and 

temperatures in the range 4 – 10 K.  Magnetic moment data at T = 5.0 K for a sample 

with a slightly thicker Ni layers (y = 3.5 nm) are shown in Fig. 8.  Moment 

discontinuities are absent in these data, which is consistent with a comparatively high 

stability of the FM state for y = 3.5 nm under these measuring conditions.   

 The apparent change in character of the magnetization curves upon cooling into the 

SC state can be interpreted as evidence for a close coupling between the supercurrent 

response of the SC condensate and the FM switching of the Ni layers.  We have 

speculated that the highly reproducible, discontinuous jumps in m(H) might be due to 

“supercurrent-induced” switching of the Ni moments [14].  Crude estimates of the current 

density JC necessary to account for the SC state magnetization near zero field in Fig. 5 

yield values as high as 10
8
 to 10

9
 A/cm

2
, depending upon the value of penetration depth 

chosen.   

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 We also have argued that the SC condensate within Nb/Ni ML acts as a “current 

amplifier” of the Ni moments and their fluctuations [14].  This line of reasoning assumes 

that the cusp anomalies evident in Figs. 4a and 4b are due to abrupt rearrangements of 

the FL lattice driven by subtle (i.e., hard to detect in normal state data) FM domain wall 

motion or local magnetization dynamics, as have been observed in sensitive vibrating 

reed measurements on bulk FM superconductors [35].  Such magnetic anomalies are 

difficult to observe in the normal state of FM materials, where they are responsible for 

“Barkhausen noise” [36]. 

 However, it is important to keep in mind that interpretations of the SC state moment 

data depend upon the quantitative reliability of the software fitting routines applied to the 

raw MPMS5 SQUID voltmeter output.  Surprisingly, we have found that the compelling 

reproducibility and symmetry of the sharp anomalies shown in Figs. 5-8 does not even 

guarantee their qualitative validity, as discussed below.   

 

3.3  Caveats Concerning SQUID Magnetometer Data 

 

It is widely known that SQUID magnetometer data processing software can yield 

erroneous results when SC samples having large demagnetization coefficients (thin plates 

and films are oriented perpendicular to the applied magnetic field) [17].  In these cases, 

small magnetic field gradients caused by imperfections and flux trapping in the 

magnetometer SC solenoid can alter the magnetic response of the sample, which no 

longer behaves as a stable point-dipole.  The effects of non-uniform field can be largely 

avoided by minimizing the distance the sample travels (“scan length”) during each 

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measurement, and executing procedures that eliminate trapped flux in the SC solenoid 

[15-17]. 

 Nevertheless, we have encountered a novel effect that complicates the evaluation of 

MPMS SQUID Magnetometer data for FM thin film and ML samples oriented parallel to 

the applied field.  This is normally a situation in which large demagnetization effects and 

nonuniform applied fields are not expected to be serious problems.  On the other hand, 

the magnetization of a FM layer of finite length aligned parallel to the applied magnetic 

field will generally be composed of domains of variable size and orientation that form a 

mosaic of magnetic dipoles, as opposed to the single point-dipole assumed by the MPMS 

Magnetometer software.  This situation generally results in the movement of the apparent 

sample position (“center”) as the domain structure reverses and rearranges in variable 

temperature and field environments.  The Quantum Design MPMS Magnetometer 

software offers the “Iterative Regression” option, which treats the sample position as a fit 

variable in the evaluation of the SQUID voltmeter output.  It is therefore tempting to use 

the Iterative Regression option to improve fit quality in cases where the sample does not 

move, but the magnetization does distort during FM reversal.  As long as the FM layer 

can be approximated by a dominant single domain of a size that is much smaller than the 

separation (1.5 cm) between SQUID sense coils, satisfactory fits of RSO data can be 

obtained, which is apparently the case for the “fully trained”, normal-FM-state response 

of Nb/Ni ML, as reflected in Figs. 5 and 9.   

 We have regularly monitored the “regression fit” parameter R that assesses the 

quality of fit of the SQUID voltmeter output that is converted into a calibrated magnetic 

moment datum. We have determined that values of R > 0.90 are typical for the normal-

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FM-state hysteresis data (see Fig. 9), indicating that the Ni layers behave very consistent 

with the MPMS5 software requirement that the sample be a single, point-magnetic-

dipole.  Excursions (greater than the sample size!) in the fitted sample position are, 

however, apparent in the initial (virgin) magnetization of the [Nb(23nm)/Ni(5nm)]5 ML 

at T = 6.0 K (see Figs. 5c and 9); and this behavior cannot be simply dismissed as due to 

a FM domain distribution that is not yet “well trained”.   

 The indicators of fit quality change more dramatically in the SC state (see Fig. 10), 

where unphysically large movement of the sample position apparently occurred over a 

wide range of applied field.  Brief summaries of the behavior of the R parameter are 

given in the figure captions to help the reader assess the reliability of the fitted data.  We 

emphasize that the SC state moment data were extremely reproducible once the two 

switching anomalies were established in the initial magnetization cycle, and the MPMS 

software yielded satisfactory fits (which we arbitrarily define by R ≥ 0.80) except for 

narrow field intervals surrounding the abrupt switching anomalies (see Fig. 10). 

 Alternatively, use of the standard “Linear Regression” option of the MPMS software, 

which assumes a fixed geometrical sample position, will yield inferior fit quality (R) in 

measurements of finite-sized FM ML in the parallel field geometry.  One manifestation 

of such a failure is the “latching” of the Nb/Ni ML moment at zero in the middle of an 

abrupt switch in the SC state, as shown in Figs. 5 and 6.  When the best-fit sample 

position varies by more than 1 cm, the MPMS software automatically replaces the 

Iterative Regression with the Linear Regression procedure [17] which, in the present 

case, returns a default, zero moment and quality of fit R = 0, as shown in Fig. 10.  

Moreover, none of the standard MPMS Magnetometer software options are quantitatively 

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reliable at DC fields near the apparent switching anomalies, which is also evident in Fig. 

10.   

 Even though the standard MPMS5 software may not be appropriate for quantitative 

processing of the raw input data from the SQUID voltmeter under experimental 

conditions where the point-dipole model fails, the SQUID voltmeter signal remains a 

reliable measure of the sample’s magnetic response, and is not dependent upon which 

fitting procedure is adopted to process the raw voltmeter output [16,17].  Indeed, the 

remarkable reproducibility of the fitted MPMS5 RSO data strongly suggests that once a 

well defined pattern of FM domains is established within the Ni layers, a corresponding, 

unique supercurrent response that is tightly coupled to the Ni magnetization is also 

established.  The reproducibility of the magnitude of abrupt changes in the fitted output 

and the fields at which they occur also rules out simple, random FL avalanche events as a 

probable mechanism for such anomalies.  Specialized techniques (valid for a particular 

sample geometry and magnetic behavior) can be developed for a quantitative evaluation 

of the SQUID voltmeter output observed for Nb/Ni ML; and our recent success in 

developing a novel fitting routine for the data of Figs. 5-8 will be described in a separate 

publication. 

 

3.4  Ferromagnetic Resonance Results 

 

Ferromagnetic resonance (FMR) is a powerful probe of the FM properties of ML and thin 

films, including magnetic moment, magneto-elastic coupling coefficients, and magnetic 

anisotropy [37,38].   In spite of the extensive research on the physical properties of 

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FM/SC ML, FMR has not been widely applied to these systems.  We are interested in 

FMR characterization of Nb/Ni ML since the magnetic coupling between FM Ni layers 

and the boundary conditions appropriate to FMR in Nb/Ni ML may be affected by a 

transition to a SC state (that may, or may not be localized within the Nb spacer layers); 

and this situation could be further modified by a nonzero SC order parameter and 

screening in thin FM Ni layers, or a magnetic polarization induced within the SC layers.  

 Fig. 11 shows FMR absorption derivative of three Ni(y)/[Nb(10 nm)/Ni(y)]8 samples  

at room temperature and 4 K for DC magnetic field applied in the ML plane.  The FMR 

spectra change markedly with the Ni layer thickness y; in particular, the resonance 

amplitude of the uniform mode increases much more rapidly than linearly with y for 

samples that all had comparable planar area.  The trends in resonance amplitude are not, 

therefore, due to the relative amount of Ni present in the samples.  The strongest peak 

(presumed to be the uniform mode [39]) was down-shifted by 39.0, 32.4, and 20.7 mT 

(field at the absorption peak for y = 1.5, 2.5, and 3.5 nm, respectively) when measured at 

4 K, compared to the room-temperature value.  Since the x = 10 nm ML series samples 

were not superconducting, these shifts cannot be attributed to field inhomogeneities or 

screening by supercurrents, and probably reflect a temperature dependence in the FM 

state of the Ni layers. 

 The FMR spectra of the same Ni(y)/[Nb(10 nm)/Ni(y)]8 samples with DC magnetic 

field applied perpendicular to the ML plane are more complex, as shown in Fig. 12; and 

the field scale of the observed modes is roughly twice that of the parallel field case shown 

in Fig. 11.  A single (which we assume is either a broadened uniform mode or two 

crossing modes) resonance signature appears for Ni layer thickness y = 2.5 nm.  At least 

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two modes are observed for samples with y = 1.5 and 3.5 nm.  The sample with y = 3.5 

nm exhibits at least one resonance (at 452.5 mT) below the uniform mode (518.7 mT) at 

RT, whereas two resonances (at 395.3 and 505.4 mT) appear below the uniform mode 

(611.2 mT) at 4 K.  The broadening and changes in amplitude of the resonances again 

suggest that the FM state is temperature dependent between room temperature and 4 K.   

    The FMR modes of the second [Nb(23nm)/Ni(y)]5 ML series of samples have 

relatively weak amplitude and therefore details of complex spectra, if present, could not 

be detected.  Note that the resonance amplitude decreases with decreasing Ni layer 

thickness y; and no FMR signal could be detected for either applied field orientation for 

the smallest Ni layer thickness y = 2.5 nm.  Only the uniform mode is observed for y = 

3.5 and 5 nm with DC magnetic field applied either parallel or perpendicular to the 

sample plane, as shown in Figs. 13 and 14. This suggests that the interlayer coupling is 

very weak and each Ni layer resonates independently in the same mode. Although the 

samples are superconducting below 6 K at zero magnetic field, they may not remain 

superconducting at the fields applied during the FMR measurements.  In particular, there 

is no big difference between the RT and 4 K spectra in Fig. 14, which were taken at 

perpendicular applied fields above 0.1 T, where the ML are expected to be in the normal 

FM state at 4 K, based upon perpendicular HC2(T) data not shown here.  On the other 

hand, the 4 K data for the parallel field orientation (Fig. 13) are slightly shifted and 

broadened compared to the room temperature data, as expected if magnetic induction 

gradients due to the shape anisotropy of the superconducting magnetization are present. 

 

4.  Discussion 

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4.1  Superconducting and  ferromagnetic phase boundaries 

 

First, it is important to note that our magnetometer data do not yield any evidence for 

superparamagnetic behavior in the normal state for any Ni layer thickness studied, which 

implies that at least some type of short-range FM order is present, as opposed to an ideal 

“quantum critical” case in which no long-range FM order should be present at finite 

temperatures near a QCP.  The Kerr effect data of Fig. 1, which suggest there is no FM 

order below 10 K for Ni layer thicknesses y < 2.5 nm, might be better explained by a 

systematic reduction in the size of well-ordered Ni domains with decreasing y to a value 

comparable to, or below, the wavelength of the probe radiation (of order 100 nm).  Of 

course, this scenario also implies that the Ni domains are both small and not well aligned 

for y ≈ 2.5 nm.  Using data for the spontaneous moment (0.604 Bohr magnetons) of bulk 

Ni [40] and the approximate dimensions of the [Nb(23nm)/Ni(5nm)]5 ML (square sample 

of side L = 2.5 x 10
-3

 m), we calculate an in-plane Ni moment = 8.05 x 10
-5

 emu for the 

normal FM state, which is in very good agreement with the data in Fig. 6c.  Precise data 

for the y-dependence of the volume-scaled saturation magnetization in the normal state 

would be useful in assessing the stability of the FM state of the Ni layers.    

 Second, the Ni(y)/[Nb(x)/Ni(y)]8 ML set studied herein exhibited no clear evidence of 

superconductivity at temperatures above 2.2 K, whereas previous resistive data [13] for a 

different set of [Nb(10nm)/Ni(y)]8 ML yielded a dramatic decrease of TC from 5.3 K for 

y ≈ 0 to well below 2 K for y ≥ 2.2 nm, as shown in Fig. 1.  A possibly crucial difference 

is that the SC samples possessed an even number (8) of both Ni and Nb layers, but 

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stacked in an asymmetric manner (one Nb layer on the Si substrate was not covered by a 

Ni layer); whereas the Ni(y)/[Nb(10nm)/Ni(y)]8 ML set newly investigated in the present 

study had an odd number (9) of Ni layers with a even number (8) of Nb layers that were 

all bounded by two Ni layers. 

 In contrast, the [Nb(23nm)/Ni(y)]5 samples, which were taken from the same batch of 

ML for both the present and previous studies [13,14], have an odd number of  Ni layers 

and an odd number of Nb layers.  The latter ML set exhibits robust superconductivity 

over a range of temperatures below 7 K, and the data from AC and DC magnetization and 

electrical resistance measurements of TC are in good agreement.  The behavior of the ML 

series with x = 23 nm make it unlikely (but not impossible) that an unintentional mishap 

in sample preparation is responsible for the different behaviors of the two ML series with 

x = 10 nm, but having different configurations of Ni capping layers. 

 Further research should address the possibility that the combined actions of the SC 

proximity effect and magnetic pairbreaking interactions within the Ni layers lead to phase 

shifts or changes in boundary conditions governing the SC condensate wavefunction that 

are sensitive to the presence of an additional Ni capping layer between the Si substrate 

and the first Nb layer.  Such a possibility could provide interesting explanations for a 

number of aspects of our results. 

 

4.2  Coupling between superconducting and  ferromagnetic layers 

 

The finite thickness of both the Ni and Nb layers could influence the coupling between 

the SC and FM order parameters via quasiparticle or pair tunneling.  The supercurrent 

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density within Nb layers in the critical state is expected to be at least 10
6
 or 10

7
 A/cm

2
 

[41], which is large enough (consider a tunneling supercurrent bridging adjacent Nb 

layers through the intervening Ni layer) to initiate the abrupt switching of the Ni via a 

“spin torque” mechanism [42,43].  The abrupt switching anomalies in Figs. 5-8 are at 

least superficial evidence for such an effect.  However, we caution that the standard 

MPMS5 Magnetometer fits of the SC state data are not quantitatively valid near these 

switching anomalies.   

 Another important consideration is how differences in (even/odd) number of Ni 

layers could lead to significant changes in the electromagnetic coupling between different 

Ni layers and the occurrence of “flux closure” states among pairs of oppositely 

magnetized Ni layers, especially for applied magnetic fields near zero. The “latching” of 

the magnetization at precisely zero at the midpoint of the switching anomalies observed 

for the ML series with x = 23 nm is indicative of such a flux closure state, assuming that 

the magnetization of an odd Ni layer is exactly screened by the Nb layers.  However, the 

output of the MPMS5 SQUID voltmeter is severely distorted from that expected for a 

single point-dipole near the latching points; and these special points correspond to fields 

at which the MPMS Magnetometer fitting procedure fails completely (R = 0). 

 In spite of the ambiguities introduced by the occasional failure of the MPMS5 data 

analysis software, the data in Figs. 4-8 provide clear evidence for the sensitivity of the 

SC condensate to small changes in the Ni magnetization.  Furthermore, these data 

provide excellent illustrations of the strong synergy between the reversal of the Ni 

moments and the SC magnetic response (compare the large magnitude of the SC 

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magnetic moment with the normal state FM moment, as well as a strong shift of the SC 

state switching field to well above the normal state coercive field in Fig. 6, for example).  

 

4.3  Flux matching effects in the superconducting state 

 

The cusp and kink anomalies in the magnetization and SC-normal phase boundary data 

are evidence for “flux matching”, and there are several models that could be relevant to 

our results.  For example, the geometry of a SC/FM ML of rectangular cross-section 

subjected to a parallel magnetic field (as in Fig. 3a) meets the general conditions for the 

“terraced critical state” proposed by Cooley and Grishin [44].  Specifically, they assumed 

a rectangular superconducting slab whose plane was parallel to an external applied field; 

but the slab was also assumed to have a periodic array of strong pinning centers that were 

shown to lead to periodic, “saw tooth” anomalies in the DC magnetic moment as a 

function of parallel magnetic field. 

 Another type of matching effect has been predicted [27] and observed [28] in highly 

anisotropic bulk superconductors and ML [26] when FL enter a sample in successive 

chains that extend along the length of crystal planes oriented approximately parallel to 

the applied field H.  Indeed, the rough spacing of the cusps shown in Fig. 4 is 1.5-3.0 

mT, and they extend to approximately ± 8.0 mT, which is similar to the matching field 

scale in thin films patterned with antidot lattices [21,22].  The asymmetry of the 

magnetization curve near H = 0 could be due to the Bean-Livingston penetration barrier 

[45] that acts against vortex penetration, but not against vortex removal.  We note the 

existence of very subtle variations of the magnetic moment (on the field-decreasing sides 

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of H = 0) having a rough periodicity comparable to the cusps in Fig. 4; but we do not 

discuss these features since they are not sufficiently defined above the scatter in the data. 

 On the other hand, the spacing of the cusps in the isothermal m(H) data is not uniform 

and is very different from the kink spacing in the AC phase boundary shown in Fig. 2b.  

The irregular spacing of the cusps in field sweep data could be due to a strong and 

inhomogeneous vortex pinning landscape [21,22] that generates a nonequilibrium 

“critical state” [29], whereas the larger spacing of kinks in the phase boundary of Fig. 2b 

is determined by AC experiments with DC field-cooling, which would involve a different 

degree of equilibration.  Calculations [26,46,47] predict that a succession of FL lattices 

will form as vortices penetrate a single, thin film superconductor in an increasing parallel 

field; but these phases will have different packing topologies whose ranges of stability 

may not be strictly periodic in applied field.  This situation is at least qualitatively 

consistent with the anomalies shown in Fig. 4. 

 A novel, alternative explanation of the cusp anomalies in m(H) originates from a 

primary motivation of this investigation:  We hoped to discover evidence that the 

superconducting state can amplify very small changes in the Ni layer magnetization and 

sensitively reflect details of the interactions between the Ni and Nb layers at the ML 

interfaces.  The small jumps in the m(H) data of Fig. 4 could be due to an amplified SC 

response to small rearrangements of Ni domain walls or local magnetic reversals within 

Ni layers.  Ordinarily, the microscopic dynamics of the FM state are extremely small, and 

are relegated to Barkhausen noise or tiny “stick-slip” events that make up the fine 

structure of the normal FM hysteresis curves.  The nearly reproducible patterns of these 

jumps apparent in Fig. 4 would then point to a remarkably reproducible sequence of 

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domain wall motion and growth patterns during the magnetic reversal of the Ni layers.  It 

remains to unambiguously identify mechanisms governing the influence of the proximate 

SC Nb layers on the symmetry and near periodicity of the cusps in m(H) shown in Fig. 4. 

 

4.4  Ferromagnetic resonance 

 

The microscopic mechanism of a particular FMR resonance mode is not obvious in the 

absence of micromagnetic simulation results for ML.  In the case of FM permalloy dot 

lattices, “hybrid” FMR modes that arise from both exchange and dipolar interactions 

between spins have been reported [48].  Spin wave resonances with finite wavevector k > 

0 are hard to observe in very thin films (< 10 nm), and the field separation between the 

modes should be much larger than observed here [38].  Ajan et al. reported that three-

dimensional coupling between FM layers exists in Nb/Co ML with Nb layer thickness up 

to 13.5 nm [49]. Therefore, 2D and 3D layer couplings may be possible in the Nb/Ni ML 

having Nb layer thickness x ≈ 10 nm. 

 Nevertheless, information concerning the stability of the FM state is apparent from 

the FMR data.  The lack of FMR signal for the [Nb(23nm)/Ni(2.5nm)]5 ML is consistent 

with the expectation generated by Fig. 1 that this sample may have relatively unstable 

FM order or a fine-grain domain structure that reduces the FMR response at room 

temperature or 4 K.  The magnetization data of Figs. 6 and 7 are consistent with the 

presence of FM domain instabilities at temperatures close to the SC TC.  The stabilization 

of the magnetization signal at temperatures well below TC is also consistent with an 

enhancement of long-range FM order, or simply the growth of more ordered, larger 

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domains at lower temperature.  The weak amplitude and slight broadening of the FMR 

spectra at 4K for the [Nb(23nm)/Ni(y)]5 ML with y = 3.5 and 5.0 nm are consistent with 

inhomogeneities of magnetic induction in the SC state.   

 It is useful to compare the rather weak room temperature FMR data for the 

[Nb(23nm)/Ni(y)]5 ML series to the corresponding data for the Ni(y)/[Nb(10nm)/Ni(y)]8 

ML series.  The comparison suggests that interactions between Ni layers are stronger for 

the x = 10 nm ML, and these “three-dimensional” interactions help stabilize the FM state 

and yield the relatively strong FMR signals shown in Figs. 11 and 12.  The fact that the 

FMR resonances actually strengthen for the Ni(1.5nm)/[Nb(10nm)/Ni(1.5nm)]8 sample at 

4 K, and do not dramatically change with temperature for the other two samples with 

thicker Ni layers, is consistent with the absence of superconductivity and the increased 

FM stability of the x = 10 nm ML compared to the x = 23 nm series.   We also reiterate 

that the small differences in sample area and the number of Ni layers cannot explain the 

observed differences in FMR signal between the x = 10 and 23 nm ML series. 

 

Summary and conclusions 

 

The stability of the FM state of two series of [Nb(x)/Ni(y)] ML was studied using FMR 

with applied DC magnetic field parallel and perpendicular to the sample plane at room 

temperature and 4 K.    We observed a strong dependence of the FMR spectra on the 

thickness of both Ni and Nb layers, and weak shifts in the resonance fields and modest 

line broadening were observed when the ML were superconducting.  Our results suggest 

that various degrees of three-dimensional magnetic interactions exist in Nb/Ni ML, 

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depending upon the thickness of Ni and Nb layers, and whether or not the Nb layers are 

superconducting.  Detailed analyses of these data using micromagnetic simulation 

techniques will be required to help sort out the contributions of “hybrid” dipole and 

exchange couplings [48] to the FMR response and their effect on the coupling between 

Ni layers. 

 AC measurements of the approximate field-temperature phase boundaries for the SC 

state revealed three-dimensional behavior and weak kink anomalies in HC2(T) for DC 

fields applied parallel to the ML plane; and FL pinning or transitions between FL lattice 

phases are mechanisms consistent with the near-periodic spacing (of order 0.1 T) of the 

kinks.  It is possible that the deposition of two (as opposed to one) Ni capping layers 

strongly suppresses the TC of the Ni(y)/[Nb(10nm)/Ni(y)]8 ML.  

 Measurements of the SC magnetization in parallel applied fields near zero revealed 

small cusp anomalies spaced by field intervals of 1 - 3 mT.  Although the cusps were not 

exactly periodic in field, they were approximately reproducible on field cycling, and were 

prominent when field was increasing in magnitude after crossing H = 0, suggesting they 

are due to very subtle changes in the domain structure of the FM Ni layers that are 

amplified by the supercurrent response.   

 The parallel magnetization of the SC state exhibited two symmetric, abrupt jumps at, 

or well beyond (depending upon the Ni layer thickness y), the coercive magnetic fields 

seen in the normal FM state.  These jumps were extremely reproducible after no more 

than one field cycle, and often exhibited “latching” of the sample moment at exactly zero 

over a very small field interval at the midpoint of a jump (making it appear to be a two-

step process).   Moreover, these data suggest that the magnetic reversal of the Ni layers is 

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strongly coupled to the supercurrent response of the Nb layers via long-range dipolar 

and/or shorter-range “spin torque” mechanisms that could provide a novel means of 

controlling the magnetic switching of a FM layer.  However, the failure of the MPMS5 

SQUID Magnetometer software to fit SC state moment data near the jumps prevents a 

clear interpretation of these anomalies.   

 Ongoing, extensive examinations of a large body of SC state data taken with the 

Quantum Design MPMS5 SQUID Magnetometer under different field-temperature 

histories and employing novel data fitting procedures have revealed a subtle, reproducible 

magnetic reversal process coupled to the supercurrent response that we will discuss in 

more detail in a separate publication. 

      

Acknowledgments 

Research at University of Kentucky supported by U.S. DoE Grant #DE-FG02-

97ER45653.  Research at Universidad Complutense supported by Spanish CICY Grant 

#MAT02-04543 and ESF VORTEX Program.  Research at Miami University was 

supported by U.S. DoE Grant #DE-FG02-86ER45281. 

 

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Figure Captions 

 

Fig. 1.  Superconducting transition temperature TC(y) (left vertical axis) for 

[Nb(10nm)/Ni(y)]8 (open symbols) and [Nb(23nm)/Ni(y)]5 (solid symbols) ML series as 

functions of Ni layer thickness y (horizontal axis).  Fractional change in reflectivity δkerr 

= ∆Rkerr/R (right vertical axis) of the magneto-optical Kerr effect signal (at saturation) at 

temperature T = 10 K for the series with 10 nm Nb layers.  After [14]. 

 

Fig. 2a.  Magnetic field - temperature phase boundary between the superconducting and 

normal states for a [Nb(23nm)/Ni(5nm)]5 ML, as measured by the onset temperature of 

an abrupt increase in the imaginary part m” of the AC magnetic moment.  Data were 

taken in field-cooling experiments at AC frequency f = 10 Hz and amplitude µoho = 0.05 

mT with applied DC field H parallel to the ML plane.  Inset shows an enlarged view of 

the low-field regime. 

 

Fig. 2b.  Superconducting field - temperature phase boundary data of Fig. 2a, 

emphasizing the region just below the zero-field TC.  Arrows denote “kinks” that are 

proposed as roughly periodic “matching anomalies” in the phase boundary (see text for 

discussion). 

 

Fig. 3a.  Schematic cross-section (not drawn to scale) of a [Nb(x)/Ni(y)]5 ML in an 

external magnetic field H oriented parallel to the ML plane.  The black field denotes the 

Si substrate, the gray fields the superconducting Nb layers, and the white fields the 

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ferromagnetic Ni layers.  Note the absence of a Ni capping layer between the Si substrate 

and the first Nb layer. 

 

Fig. 3b.  Schematic diagram of FL confinement in one Nb layer (length L and width w) 

of a Nb/Ni ML.  The applied field H is oriented parallel to the ML plane and 

perpendicular to the layer side of length L.  The diagram depicts an artificially small 

number NL = 3 FL forming a “chain” within the  Nb layer.  Assuming the SC order 

parameter is zero in neighboring FM Ni layers, confinement of the supercurrents within 

the Nb layer demands that FL have elliptical cross-section (white field with crosses) with 

major and minor axes of length λ|| and w, respectively. 

 

Fig. 4a.  DC magnetic moment m versus magnetic field H applied parallel to the 

[Nb(23nm)/Ni(5nm)]5 ML held at a temperature T = 3.8 K.  Arrows denote rough 

positions of cusps that may signal a flux matching effect.  These data are re-plotted from 

the low-field regions of the isothermal (T = 3.8 K) magnetization curve shown in Fig. 5b.  

Note the cut in the vertical axis. 

 

Fig. 4b.  DC magnetic moment m versus magnetic field H applied parallel to the 

[Nb(23nm)/Ni(5nm)]5 ML held at a temperature T = 5.0 K.  Arrows denote rough 

positions of cusps that may signal a flux matching effect.  These data are re-plotted from 

the low-field regions of the isothermal (T = 5.0 K) magnetization curve shown in Fig. 5c.  

Note the cut in the vertical axis. 

 

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Fig. 5a.  DC magnetic moment m versus magnetic field H applied parallel to a 

[Nb(23nm)/Ni(5nm)]5 ML for temperatures T = 6.0 K (normal FM state, black curve) and 

T = 3.5 K (SC state, red points).  Normal state data were taken after the sample was 

initially degaussed at T = 250 K, then zero-field cooled to T = 6 K for field-cycling.  The 

quality of fit for the nearly all of the normal FM state data exceeds 0.85 (see Fig. 9).  The 

SC state data were obtained for the sample initially at µoH = 0.1 T and T = 4.0 K after a 

previous field cycle; the sample was then field-cooled to 3.5 K and field-cycled from H = 

0.  The quality of fit R > 0.80 for SC state data for µo|H| < 20 mT (excepting the initial 

magnetization data for the Meissner state). 

 

Fig. 5b.  DC magnetic moment m versus magnetic field H applied parallel to a 

[Nb(23nm)/Ni(5nm)]5 ML for temperatures T = 6.0 K (normal FM state, black curve) and 

T = 3.8 K (below TC, red points).  The normal FM state data are the same for Figs. 5a, 5b 

and 5c.  The dashed lines denote two symmetric switching anomalies located near ± 20 

mT in the SC state.  The quality of fit R > 0.80 for SC state data everywhere except over 

narrow intervals of width ≈ 5 mT about the switching anomalies.  The SC state data were 

taken in a manner similar to those of Fig. 6c; and an expanded view of the data near H = 

0 is given in Fig. 4a. 

 

Fig. 5c.  DC magnetic moment m versus magnetic field H applied parallel to a 

[Nb(23nm)/Ni(5nm)]5 ML for temperatures T = 6.0 K (normal FM state, black curve) and 

T = 5.0 K (below TC, red points).  Dashed lines denote two symmetric switching 

anomalies near µoH = ± 20 mT in the SC state data, which were taken after the sample 

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was degaussed at T = 250 K, then field-cooled in 50 mT to T = 5 K for field cycling from 

µoH = 0 T.  R > 0.80 for SC state data (see Fig. 10) everywhere except for intervals of 

width ≈ 4 mT about the switching anomalies.  An expanded view of the SC state data 

near H = 0 is given in Fig. 4b. 

 

Fig. 6a.  DC magnetic moment m versus magnetic field H applied parallel to a 

[Nb(23nm)/Ni(2.5nm)]5 ML for temperatures T = 6.5 K (normal FM state, black curve) 

and T = 5.0 K (below TC, red points).  The normal state fit quality R > 0.90 except for a 

narrow interval of around 2 mT about the coercive fields.  SC state data were obtained 

after the sample was degaussed at T = 250 K, then field-cooled in 50 mT to T = 5 K, 

whereupon H was set to 0, and the sample field-cycled.  Abrupt switching anomalies in 

the SC state are located near ± 0.11 T, where R < 0.80 (for -0.14 T ≤ µoH ≤ 0.09 T and 

0.1 T ≤ µoH ≤  0.13 T). 

 

Fig. 6b.  DC magnetic moment m versus magnetic field H applied parallel to a 

[Nb(23nm)/Ni(2.5nm)]5 ML for temperatures T = 6.5 K (normal FM state, black curve) 

and T = 3.5 K (below TC, red points).  The dashed lines denote two symmetric switching 

anomalies located near ± 0.16 T, which are clearly displaced from the normal FM 

coercive fields observed at 6.5 K.  The quality of fit R for the SC state data was in the 

range 0.75 < R < 0.85 for |H| < 0.15 T, and 0.64 < R < 0.80 for |H| > 0.15 T.  The SC 

state data were taken after previous field cycling (see Fig. 6a for comparison). 

 

Fig. 7a.  DC magnetic moment m versus magnetic field H applied parallel to a 

[Nb(23nm)/Ni(2.5nm)]5 ML for temperatures T = 8.5 K (normal FM state, black curve) 

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and T = 5.5 K (below TC, red points).  The quality of fit R > 0.90 for the normal state FM 

data, except for fields within intervals of approximate width = 2 mT about the coercive 

fields.  SC state data were obtained after the sample was field-cooled from 10 K in µoH = 

50 mT, and then field cycled from H = 0.  R > 0.80 for SC state data for |µoH| > 50 mT, 

but dropped rapidly below 0.60 within intervals of approximate width = 60 mT about the 

switching anomalies.  During an initial magnetization from 0 to 0.3 T, irreproducible 

anomalies set in near µoH = + 20 mT, and precede full reversal of the magnetization at 

higher applied fields. 

 

Fig. 7b.  Plot of the difference between the DC magnetic moment m of Fig. 7a in the SC 

state (T = 5.5 K) and the normal state (T = 8.5 K).  Blue points were taken in the initial 

magnetization sweep from 0 to 0.3 T, and red points were measured upon further field 

cycling.  Note the bands of reproducible anomalies located beyond ± 30 mT that precede 

reversal of the magnetization at higher applied fields.  The initial linear section of the 

blue trace behaves like a Meissner state, then exhibits an abrupt collapse near 35 mT, 

after which it is indistinguishable from the normal FM state data. 

 

Fig. 8.  DC magnetic moment m versus magnetic field H applied parallel to a 

[Nb(23nm)/Ni(3.5nm)]5 ML for temperatures T = 6.5 K (normal state above TC, black 

points) and T = 5.0 K (below TC, red points).  The blue trace denotes the initial (virgin) 

magnetization curve that begins in the SC Meissner state.  The quality of fit R > 0.99 for 

normal state data everywhere except at the coercive fields.  Sample was degaussed at T = 

250 K, then field-cooled in 50 mT to 5K, and field-cycled from H = 0.  The quality of fit 

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R > 0.78 for SC state data everywhere except for intervals of approximate width = 20 mT 

about the switching anomalies, and the initial magnetization branch.  Note the 

nonmonotonic behavior preceding the switching anomalies at  ± 60 mT. 

 

Fig. 9.  Fit quality R (left axis, red and black curves) and Center Offset (right axis, blue 

curve) from fits of the SQUID voltmeter data versus applied magnetic field H parallel to 

a [Nb(23nm)/Ni(5nm)]5 ML at temperature T = 6.0 K (normal FM state).  The red (black) 

curve is for field decreasing (increasing).  A plot of the corresponding magnetic moment 

data is shown in Fig. 5. 

 

Fig. 10.  Fit quality R (left axis, red and black curves) and Center Offset (right axis, blue 

curve) from fits of the SQUID voltmeter data versus applied magnetic field H parallel to 

a [Nb(23nm)/Ni(5nm)]5 ML at temperature T = 5.0 K (SC state).    The red (black) curve 

is for field decreasing (increasing).  A plot of the corresponding magnetic moment data is 

shown in Fig. 5c. 

 

Fig. 11.  FMR absorption derivative versus applied magnetic field applied parallel to 

Ni(y)/[Nb(10 nm)/Ni(y)]8 sample plane for different Ni layer thickness y = 1.5 (a), 2.5 

(b), and 3.5 nm (c).  Dashed and solid curves correspond to room temperature and 4 K, 

respectively. 

 

Fig. 12.  FMR absorption derivative for applied DC magnetic field perpendicular to 

Ni(y)/[Nb(10 nm)/Ni(y)]8 sample plane for different Ni layer thickness y = 1.5 (a), 2.5 

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(b), and 3.5 nm (c).  Dashed and solid curves correspond to room temperature and 4 K, 

respectively. 

 

Fig. 13.  FMR absorption derivative measured with DC magnetic field parallel to [Nb(23 

nm)/Ni(y)]5 sample plane for different Ni layer thickness y = 3.5 (a) and 5.0 nm (b).  

Dashed and solid curves correspond to room temperature and 4 K, respectively.  No 

resonance was observed for the sample with y = 2.5 nm. 

 

Fig. 14.  FMR absorption derivative measured with DC magnetic field perpendicular to 

[Nb(23 nm)/Ni(y)]5 sample plane for different Ni layer thickness y = 3.5 (a) and 5.0 nm 

(b).  Dashed and solid curves correspond to room temperature and 4 K, respectively.  No 

resonance was observed for the sample with y = 2.5 nm.  

 

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Fig. 1  De Long et al.  
 

10 15 20 25 30 35 40 45 50 55 60
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∆
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rr /R

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(K
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Ni 

 (Α)

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-5

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-4

4.2x10
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Fig. 2a  De Long et al. 

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Fig. 2b  De Long et al. 

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Fig. 3b   De Long et al. 

 

 

 

 
 

 

Fig. 3a  De Long et al. 

 

H 

             λλλλ||    

 w 

     L 

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Fig. 4a  De Long et al. 

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Fig. 4b  De Long et al. 

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Fig. 5a  De Long et al. 

 

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Fig. 5b  De Long et al.

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Fig. 5c  De Long et al. 

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Fig. 6a  De Long et al. 

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Fig. 6b  De Long et al. 

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Fig. 7a  De Long et al. 

 
 

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Fig. 7b  De Long et al. 

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Fig. 8  De Long et al. 

 

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Fig. 9  De Long et al.

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Fig. 10  De Long et al. 

 
 
 

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Fig. 11  De Long et al. 

0.0 0.1 0.2 0.3 0.4

-400

-200

0

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400

 

(c)
 

 

Field (T)

-100

-50

0

50

 

(a)

 

 

 

-100

0

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Fig. 14  De Long et al. 

 

 

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