Journal of Low Temperature Physics, Vol. 137, Nos. 1/2, October 2004 (© 2004)

Repulsive Flux Pinning Force in NbTi/Nb
Superconductor/Superconductor Multilayers
Y. Obi,1 M. Ikebe,2 and H. Fujishiro2
1 Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aobaku,

Sendai 980-8577, Japan
E-mail: obi@imr.tohoku.ac.jp
2 Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan(Received December 8, 2003; revised May 14, 2004)

Flux pinning characteristics have been investigated for the N bx Ti100−x /N b
(x = 65 , 50 and 28) and N b28 T i72 /N b65 T i35 superconductor(S)/superconductor(S ) multilayers. The maximum of the pinning force Fp⊥max perpendicular to the layer plane as a function of the structure modulation length λhas a peak in the quasi-two-dimensional region (λ ∼20 nm). The maximum.
values of the Fp⊥max versus λ curve are proportional to the difference of the
GL coherence length (ξGL ) between the superconductive sublayers S and S .
The results suggest that the large Fp⊥max in the S/S multilayer is caused by
the repulsive pinning force due to Nb layers with larger ξGL .
KEY WORDS: superconductivity; critical current; pinning force; multiplayer.

1. INTRODUCTION
The study of the mechanism of ﬂux pinning is one of the most attractive subjects for superconducting multilayers because the superconducting
multilayer is considered to have intrinsic large pinning centers due to theintervening normal or superconducting layers with different superconducting characteristics. There have been several theoretical and experimental
investigations concerning the subject of the pinning mechanism for superconducting multilayers.1–5 Much of recent theories have been concentrated
on the high Tc oxide superconductors. Some of these theories can be applicable to the multilayered materials with a certain modiﬁcation becauseoxide superconductors have a layered structure composed of the strongly
superconductive Cu–O layer and weakly superconductive other layers. One
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of typical theories applicable to the transverse pinning force for multilayers
is the conception of the intrinsic pinning force introduced by Tachiki and
Takahashi.6 This idea is based on the fact that the layered structure itself
should play an important role for ﬂux pinning due to the oscillative spatial variation of the order parameter perpendicular to the periodic layered
structure; the weak superconducting layer should stabilize the ﬂux locatedin these layers.
Experimentally there have been several studies of the critical current
density Jc and the pinning force Fp for the multilayers. For example, Raffyand Renard7 have observed the peak effect in Jc versus H curves for
Pb/Bi superconducting multilayers. They interpreted the peak effect as aresult of the geometrical matching between the distance of the ﬂuxoids
and the distance of the pinning centers. Brossard and Geballe8 investigatedthe longitudinal Fp and transverse Fp⊥ pinning force, which act parallel and perpendicular to the layer plane, respectively, for the Nb/Ta multilayers. Estimating the reduced pinning force Fp /Fpmax as a function of
reduced magnetic ﬁeld H /Hc2 , they concluded that for large bilayer period
the pinning force behaves as a dislocation pinning and for small bilayer
period it originates from the collective pinning mechanism. We have investigated the pinning mechanism for Nb/Al2 O3 multilayers,9,10 and shownthat for the longitudinal pinning force Fp the surface pin plays an important role. We also have investigated the transverse pinning force Fp⊥ for
Nb/Ti multilayers.11 In this case the large enhancement of Fp⊥ as a function λ was observed for the quasi 2D-region. But until now the exact
mechanism of the transverse pinning for multilayers has not been clariﬁed
yet. Recently, we have investigated the pinning mechanism for both thesuperconductor(S)/normal metal(N) and S/S multilayers12,13 and foundthat the behavior of Fp⊥max as a function of λ and the size of the pinning are very similar for both systems despite that one has S intervening
layer and another has N intervening layer.
In S/N multilayers the transverse pinning force Fp⊥ is usually muchenhanced compared to the longitudinal pinning force Fp . In S/N multilayers the N layer usually works as an attractive pin center for vortices. So
the ﬂuxoids are stabilized in N sublayers. In this case the superconducting condensation energy is the main origin of the vortex pinning energy.
In contrast, in the S/S multilayers such as NbTi/Nb, which has almost thesame condensation energy between S and S layers, the main origin of thetransverse pinning force may come from the kinetic term of the GL freeenergy.14 This energy term stabilizes the ﬂuxoids in NbTi sublayers as suggested by Matsushita et al.15,16 In this case, Nb-layer works as the repulsive pin center. They demonstrated that in S/S multilayer (ξs < ξs  , whereRepulsive Flux Pinning Force in Superconducting Multilayers

127ξs and ξs  are the GL-coherence length of S and S layer, respectively), thelarge enhancement occurs in Fp , where S layer works as a repulsive pinning center.
We have investigated the Fp⊥ in three series of Nbx Ti100−x /Nb (withx = 65, 50 and 28) and Nb28 Ti72 /Nb65 Ti35 (S/S ) multilayers. We discuss
about the maximum value of Fp⊥max versus λ curve as a function of thedifference of the coherence length of both the sublayer.
2. EXPERIMENTAL
Superconductor(S)/ superconductor(S ) multilayers have been fabricated by an RF dual sputtering onto quartz substrates. Samples arecomposed of three series of NbTi/Nb (Nb65 Ti35 /Nb, Nb50 Ti50 /Nb andNb28 Ti72 /Nb) and one series of NbTi/NbTi (Nb28 Ti72 /Nb65 Ti35 ). Samples
thus made are designed to have equal sublayer thickness, dNbTi = dNb (ordNbTi ) = (1/2)λ, where dNbTi and dNb mean the thickness of the NbTilayer and Nb-layer, respectively, and λ is the modulation wavelength. The
sample thickness is about 500 nm. The critical current density was measured resistively at 4.2 and 1.5 K under the magnetic ﬁeld parallel to thelayer plane (Jc (H )) and perpendicular to the layer plane (Jc (H⊥ )), whereH and H⊥ are applied ﬁelds parallel and perpendicular to the layer,
respectively. The macroscopic transverse and longitudinal pinning forces
Fp⊥ and Fp were calculated from the critical current density Jc as Fp⊥ =(1/c)Jc (H ) × H and Fp = (1/c)Jc (H⊥ ) × H⊥ . The schematic conﬁgurations for these quantities J, H and F are illustrated in Fig. 1.
3. RESULTS AND DISCUSSION
The ﬁeld dependence of critical current densities Jc (H ) and Jc (H⊥ )at 1.5 K are shown in Fig. 2 a and b for Nb50 Ti50 /Nb-samples as a typicalFig. 1. Schematic conﬁguration expressing the relation among J, H and F for both the casesof (a) H being parallel to the layer plane, and (b) H being perpendicular to the layer plane.
J is always in the layer plane.

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Fig. 2. Critical current density Jc under (a) the parallel magnetic ﬁeld H and (b) the perpendicular magnetic ﬁeld H⊥ at 1.5 K for Nb50 Ti50 /Nb samples with changing of λ from 8.6
to 100 nm. An arrow in Fig. 2a indicates the jumping point of the vortex for the sample with
λ = 20 nm (see the text).

example. The macroscopic transverse pinning force Fp⊥ as a function ofH and longitudinal pinning force Fp as a function of H⊥ for the same
samples are shown in Fig. 3a and b, respectively. From these ﬁgures wecan see several characteristic features: (i) For small λ(λ = 8.6 nm) andlarge λ(λ = 100 nm) the pinning force Fp⊥ is relatively small because thelayer itself or layer boundary does not work effectively as a pinning center and also because of the relatively small upper critical ﬁeld Hc2 .17
Consequently, the peak of the pinning force Fp⊥max is also small. (ii) In
the quasi-2D region (λ ∼ 20 nm) the pinning force Fp⊥ is much enhanced
which may be due to the Nb layer acting as the repulsive pinning center as
discussed later. (iii) Fp scarcely depends on λ because in the layer planeRepulsive Flux Pinning Force in Superconducting Multilayers

129

Fig. 3. (a) Pinning force Fp⊥ of the same series as Fig. 2 under the parallel magnetic ﬁeld
H and (b) Fp under the perpendicular magnetic ﬁeld H⊥ at 1.5 K, which are calculated
from Jc using the relations, Fp⊥ = (1/c)Jc (H ) × H and Fp = (1/c)Jc (H⊥ ) × H⊥ . An arrow
in Fig. 3a indicates the vortex jumping point for the sample with λ = 20 nm.

direction there is no special pinning center which depends on the intervening layer thickness.
From the curve of Fp versus H , we can estimate the maximum valueof the pinning force Fpmax for each sample with different λ. Both of thus
obtained Fp⊥max and Fp max as a function of λ are illustrated in Fig. 4
for the present multilayers Nb65 Ti35 /Nb, Nb50 Ti50 /Nb, Nb28 Ti72 /Nb andNb28 Ti72 /Nb65 Ti35 . The characteristic results are as follows: (i) The values of Fp⊥max is about 1 ∼ 7 × 108 dyn/cm3 for all the present multilayers.
This value is nearly the same order of magnitude as that of commercial
NbTi alloy. (ii) For all series, Fp⊥max is always larger than Fp max except

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Fig. 4. Maximum pinning force Fp⊥max and Fpmax as a function of λ at 1.5 K for the present multilayer series, Nb65 Ti35 /Nb, Nb50 Ti50 /Nb, Nb28 Ti72 /Nb and Nb28 Ti72 /Nb65 Ti35 . Lines
are guide to the eyes.

for the smallest λ region. (iii) Fp max is almost invariable over the entire λrange for all the multilayers investigated, suggesting that the pinning force
acting parallel to the layer plane does not depend on the sublayer thickness, that is, in the direction of layer plane there exists no special pin center which depends on the layer thickness. It is quite reasonable because thepinning force acting parallel to the layer plane may be in the same situation to the case of Fp in a single layer. (iv) Fp⊥max as a function of λhas broad maximum around λ ∼ 20 nm for all the present multilayer series.
Therefore Fp⊥max is strongly enhanced in this region.
The scattering of the Fp values between sample to sample in thesame series and/or between series seen in this ﬁgure may come from thedifference in local pin centers formed by for example void, precipitation or dislocation. These local pins may inﬂuence equivalently both thetransverse and longitudinal directions. Therefore, in order to evaluate more
precisely the pinning force due to the pure effect of multilayering, the better method is to take the difference between. Fp⊥max and Fp max for eachsample. We introduce here the effective pinning force FpL max . The valueof FpLmax ≡ Fp⊥max − Fpmax . FpLmax as a function of λ for all the series
is shown in Fig. 5. In this ﬁgure the negative values of FpLmax seen in thesmall λ region of Nb28 Ti72 /Nb and Nb50 Ti50 /Nb are due to the larger longitudinal pinning compared to the transverse pinning. As the multilayering is not effective in this region, the negative values have no importantmeaning. Here we can see the effect of layering on the pinning force forall the series. The large enhancement is also seen around λ ∼ 20 nm for all
the multilayer series which is in the quasi-2D region for the superconductivity. The peak value (M(F )) of the FpLmax versus λ curve is the largest

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Fig. 5. Effective pinning force FpLmax (≡ Fp⊥max − Fpmax ) due to multilayering as a functionof λ at 1.5 K. Lines are guide to the eyes.

for Nb28 Ti72 /Nb. The difference of M(F ) among the series is expected tooriginate from the difference in GL-coherence length ξGL of S layer andS layer.
According to the measurement of parallel and perpendicular upper
critical ﬁeld,17,18 the GL-coherence lengthξGL (0) of sputtered Nb, Nb65 Ti35 ,Nb50 Ti50 and Nb28 Ti72 at 0 K is estimated as 10.00, 6.34, 5.89 and4.24 nm, respectively. The result of ξGL (0) is shown in Fig. 6. The ξGL (0)values have nearly a linear relation with respect to the Ti-concentration,suggesting that alloy-layer becomes dirtier by the substitution of Nb by Ti.

Fig. 6. GL-coherence length at 0K ξGL (0) versus Ti-concentration for the Nb and NbTi
alloy sublayers.

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We now consider about the large enhancement of FpLmax aroundλ ∼20 nm as seen in Fig. 5. In this region, as mentioned above, thesuperconducting state is quasi-2D as indicated by the dimensional crossover in Hc2 .17,18 Therefore the quasi-2D state is related to the large
enhancement of FpLmax . One reason of the large enhancement is that thesize of vortex just matches to the sublayer thickness in this region andconsequently the vortex is hard to move even under the applied highmagnetic ﬁeld and/or the large current. Another reason of large enhancement is as follows. The phenomenon of the large enhancement of Fp⊥
at the quasi-2D state in the S/S multilayers is similar to that of S/Nmultilayers.13 Moreover the size of Fp⊥max is nearly the same for both
multilayer systems. S/N multilayer such as NbTi/Ti has pin center in thenormal Ti-layer, which acts as attractive pin. The same situation is anticipated in the S/S multilayer such as NbTi/Nb. In this case Nb-layer, which
has larger coherence length than NbTi layer, is considered to work as arepulsive pin,15,16 signifying that the vortices preferably occupy the NbTi
layer. This situation is realized as follows. In the quasi-2D region of S/S
multilayer with different electron diffusion constant between S and S , thetwo stage crossover occurs in the Hc2 versus T curve as predicted theoretically by Tachiki and Takahashi.3,4 Near Tc , Hc2 − T curve behaves
as anisotropic three dimension (3D), where the superconducting orderparameter spreads over both the S and S layers. With decreasing temperature, Hc2 − T curve upturns accompanying a kink at a certain temperature T = T + and the system behaves as two dimensional (2D+ state),
where the superconducting order parameter locates in Nb layer (having alonger coherence length) and vortex is mainly located in NbTi layer. Withfurther decreasing T , Hc2 − T curve upturns again at another temperature
T = T ∗ and system enters into another two dimensional state (2D* state).
The order parameter then is located in NbTi layer (having a shorter coherence length) near Hc2 (T ) and vortices moves to the Nb layer from NbTi
layer with decreasing temperature. A typical example is shown in Fig. 7afor the case of Nb65 Ti35 /Nb(λ = 20 nm). In this sample, as T + is very close
to Tc the upturn of Hc2 at T + is not apparently visible in the ﬁgure.
T ∗ can be identiﬁed at about 6 K as a kink of the Hc2 (T ) curve. If weassume that the order parameter persists in the Nb layer over the entire
temperature range below Tc , the hypothetical upper critical ﬁeld H ∗ (T )is expected to follow the dotted line in Fig. 7a.18 Figure 7b presents theJc versus H curves at 1.5 and 4.2 K for Nb65 Ti35 /Nb (λ = 20 nm). The
Jc (H ) has a kink around 50 kOe at 1.5 K and around 30 kOe at 4.2 K.
It is quite interesting to note that 50 kOe corresponds to H ∗ (1.5 K) and30 kOe corresponds to H ∗ (4.2 K) in Fig. 7a. These kinks indicate the jumpRepulsive Flux Pinning Force in Superconducting Multilayers

133

Fig. 7. (a) H–T diagram of Nb65 Ti35 /Nb for λ = 20 nm sample: circle: Hc2 data, triangle:
transition ﬁeld at 4.2 and 1.5 K from the measurement of Jc (H ). Solid line and dottedline show Hc2 curves when order parameter is located in S -sublayer (NbTi) and S-sublayer (Nb). T + , T ∗ and, H ∗ are the transition temperature and transition ﬁeld. 3D, 2D+ and2D* are the corresponding superconducting states. (b) Jc versus H curve at 1.5 and 4.2 K
of Nb65 Ti35 /Nb for λ = 20 nm sample. Kink appearing for each curve at H ∗ (1.5) and H ∗ (4.2)(arrow indicated) corresponds to the transitions from 2D+ to 2D* state.

of the order parameter from the Nb-sublayer (low ﬁeld side) to the NbTisublayer (higher ﬁeld side). This phenomenon clearly occurs when the sizeof vortex (∼ ξ ) becomes comparable to the sublayer thickness. The vortices arrangement at the 2D+ -state and 2D*-state is schematically drawn in
Fig. 8. As shown in this ﬁgure, in the H < H ∗ region (2D+ -state) vortices
are located in the NbTi-layer while in the H > H ∗ region (2D*-state) they
are located in the Nb-layer. Therefore if the Nb-layer (having larger coherence length) works as the repulsive pin center, 2D+ state has larger pinning force than 2D* state. This situation is substantiated by the behavior

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Fig. 8. Vortices arrangement at (a) the 2D+ -state and (b) the 2D*-state.

of the Jc (H )-curve as shown in Fig. 7b, where the large enhancement ofJc (H ) is observable at ﬁeld below H ∗ . The Jc (H ) kink and the enhancement of Fp⊥ at the boundary between 2D+ and 2D* can be also seen forother series, for example in Nb50 Ti50 /Nb-sample of λ = 20 nm in Figs. 2aand 3a (marked by an arrow). Thus the kink of the Jc (H ) at H ∗ provides
a direct evidence of the repulsive pinning due to the Nb-layer.
Mechanism of repulsive pinning has been ﬁrst treated for the longitudinal pin by Matsushita et al.16 for NbTi/Nb lamella structure. They havededuced the elementary pinning force fp (see Eq. 11 of Ref. 16) acting
on the longitudinal direction to the layer plane based on the Ginzburg–
Landau theory. The repulsive pinning force comes from the kinetic energy
difference between the both layers due to the spatial variation of the orderparameter. Based on their theory we have calculated the transverse pinning
force for the present S/S multilayers. The free energy in the superconducting state Fs is written down as follows.
Fs = Fn (0) + α||2 +β1
1|(−i –h∇ + 2eA)|2 + µ0 (rotA)2 , (1)||4 +
22m∗
2

where Fn (0) is the free energy in the normal state in the absence of magnetic ﬁeld,  the order parameter, m* the mass of superconducting electron, µ0 the permeability of vacuum, A the vector potential, and α andβ are the numerical coefﬁcients. In the NbTi/Nb, Tc of both the layers
is nearly the same and the density of state is also comparable.17,19 The
vortex pinning is mainly correlated to the change of kinetic and potential
energy caused by the spatial variation of the order parameter. Here we canput  = ||/|∞ | = 3r/2a0 − r 3 /2a03 (r  a0 ) and  = 1(r > a0 ) arounda ﬂuxoid,20 where ∞ is the equilibrium value of order parameter and r
the position within the ﬂuxoid, r = 0 is the position of the center of theﬂuxoid core and a0 = (8/3)1/2 ξ . The increase of the energy E1 by having
the ﬂuxoid in the layer is;21 a0 –h  ∂|| 2β24α|| + || +2π r dr
E1 =
2
2m∗
∂r
0

Repulsive Flux Pinning Force in Superconducting Multilayersβ−πa0 α|∞ |2 + |∞ |42∼ πµ0 Hc2 ξ 2 .

135

(2)

At the boundary region between NbTi-layer and Nb-layer, a0 must
be taken as the average value of both the coherence lengths as a0 =(8/3)1/2 (ξNT + ξN )/2, where ξN and ξNT are the GL-coherence length of
Nb and NbTi layer, respectively. The elementary pin fp⊥ working at theboundary of both layers NbTi and Nb acting on the perpendicular direction to the layer plane is thus,
2 ξ2 − πµ H 2 ξ2
∂E1 E1N − E1NT πµ0 HcN
0 cNT NT
N
=
=
∂r
2a0
2(8/3)1/2 (ξNT + ξN )/2∼ (3/8)1/2 πµ0 Hc2 (ξN − ξNT ),

fp⊥ =

(3)

where sufﬁx N and NT mean Nb and NbTi layer, respectively. Here weassume that the thermodynamical critical ﬁeld Hc for both NbTi and Nb
layer are almost same as Hc ∼ HcNT ∼ HcN . Usually the macroscopic pinning force is written as Fp ∼ εNfp ,22 where ε is the efﬁciency factor ofpin and N the effective pin density. Therefore FpLmax is regarded to be
proportional to fp⊥max as FpLmax ∝ fp⊥ . So, from the Eq. (3) we can conclude that if the pinning force originates from the repulsive pin, the relation FpLmax ∝ ξN − ξNT must hold for the multilayers systems.
We plot the values of M(F ) as a function of ξN − ξNT at 1.5 and
4.2 K shown in Fig. 9, where the values listed in Fig. 6 are taken for ξNand ξNT . In this ﬁgure we can see that M(F ) and ξN − ξNT have a roughlyFig. 9. Peak of maximum, pinning force M(F) versus ξN − −ξNT for the present multilayer
series at 1.5 and 4.2 K.

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linear relation. Thus in the present S/S multilayers, the repulsive pinning
force actually acts on the systems as the main pin, the strength of which
is of the same order as the usual attractive pinning force.
4. SUMMARY
The superconductor/superconductor multilayer NbTi/Nb has the large
enhanced transverse pinning force Fp⊥max compared to the longitudinal
pinning force Fpmax . This large Fp⊥max is comparable to that of superconductor/normal metal multilayer NbTi/Ti. Present series of Nbx Ti100−x /Nbwith x = 65, 50 and 28 and Nb28 Ti72 /Nb65 Ti35 show the broad peak in the
FpLmax (the effective pinning force due to multilayering) versus λ at thequasi-2D region. The peak value of FpLmax (M(F )) as a function of ξN −
ξNT satisﬁes nearly a linear relation. This should originate from the superconducting Nb layers acting as repulsive pin centers. With increasing H ,the vortices jump from the NbTi sublayer to the Nb sublayer at H ∗ (T )because of the repulsive pinning force, H ∗ (T ) being the extrapolated hypothetical Hc2 (T ) of the 2D3+ state of the superconductor/superconductor
multilayer where the order parameter is ﬁxed in the Nb sublayer.
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