C 2006)
Journal of Superconductivity and Novel Magnetism, Vol. 19, Nos. 1/2, February 2006 (DOI: 10.1007/s10948-006-0135-1

Analyses of the Pinning Mechanism for theSuperconductor/Normal-Metal andSuperconductor/Superconductor Multilayers
Y. Obi,1 M. Ikebe,2 and H. Fujishiro2
Published online: 3 August 2006

The vortex pinning force of superconductor(S)/normal-metal(N) multilayers, Nb/Ti and
Nb100−x Tix /Ti, and superconductor(S)/superconductor(S ) multilayers, Nb100−x Tix /Nb andNb28 Ti72 /Nb65 Ti35 , has been studied as a function of the layer thickness d for various compositions of x. The transverse pinning force density Fp⊥ acting perpendicular to the layer
planes makes a peak around d ∼ 2ξGL (ξGL : the GL coherence length) for both the S/N and
S/S multilayers, where they exhibit the quasi-two-dimensional superconductivity. In contrast,
the longitudinal pinning force density Fp acting parallel to the layer plane hardly depends on
d. The pinning mechanism inherent in the layered structure has been analyzed based on the
GL free energy, which clarified that the effective pinning force due to the multilayer structure originates from the difference in the potential energy for the S/N multilayer and from thedifference in the kinetic energy for the S/S multilayer between the constituent layers. The ddependence of the Fp⊥ , however, has been found to be qualitatively similar for the S/N and
S/S multilayers.
KEY WORDS: multilayer; superconductor; critical current; pinning force.

1. INTRODUCTION

the vortex is large and extends over the several sublayers. However, the nature of the 2D superconductivity below TD is different between the S/N and S/S
multilayers. In the S/N multilayer under H , the fluxoids are stabilized in the N layer, while in the S/Slayer, the fluxoids are stabilized in the S layer withshorter ξGL .
The flux pinning characteristics for the multilayered materials are very attractive subjects because
the multilayer structure itself is considered to play anessential role as a pinning center [2]. In the multilayers, the transverse pinning force density Fp⊥ acting in
the perpendicular direction to the layer is much enhanced compared to the longitudinal pinning force
density Fp [3] acting parallel to the layer. In the S/Nmultilayers, the N layer works as an attractive pin
center. The fluxoids are stabilized in the N layer andthe difference in the potential energy between the S
and N sublayers, i.e., the loss of the superconducting condensation energy due to the vortex formationin the S layer, is considered to be the main origin

The superconducting properties of the superconductor(S)/normal-metal(N) and superconductor(S)/superconductor(S ) multilayers are quite unique andinteresting, both type multilayers showing characteristic dimensional crossover in case of d ∼ 2ξGL
(d: sublayer thickness, ξGL : the GL coherence length)[1] as is known as quasi-two-dimensional superconductivity. Divided by the dimensional crossover temperature TD , the dimensionality of superconductivity
under the applied field parallel to the layer plane H
changes from three-dimensional(3D) for TD < T <Tc (Tc : superconducting transition temperature) to
2D for T < TD with decreasing temperature T. In the
3D-superconductivity region above TD , the size of
1 Institute for Materials Research, Tohoku University, Sendai 980-

8577, Japan; e-mail: obi@imr.tohoku.ac.jp.
of Engineering, Iwate University, Morioka 020-8551,
Japan.

2 Faculty

117
C 2006 Springer Science+Business Media, Inc.
1557-1939/06/0200-0117/0 

118

and longitudinal pinning force density Fp were
estimated from the critical current density J c by the
relations, Fp⊥ = (1/c)J c (H ) × H and Fp = (1/c)
J c (H⊥ ) × H⊥ , H and H⊥ being applied fields paralleland perpendicular to the layer and c being the light
velocity.
3. RESULTS AND DISCUSSION
Figure 1 shows the critical current density J c (H )and J c (H⊥ ) at 1.5 K for Nb65 Ti35 /Ti as the typicalcase of S/N (Fig. 1a) and for Nb65 Ti35 /Nb as the typical case of S/S (Fig. 1b) with various modulation
wavelengths. In this figure, it is clear that for both
samples J c (H⊥ ) hardly depends on λ, while J c (H )heavily depends on λ and has higher values compared to J c (H⊥ ). Although J c (H = 0) is rather highfor S/S relative to S/N, J c (H ) is more enhanced for

5

10

Jc(H//)
Jc(H) (A/cm2)of enhanced Fp⊥ . In contrast, for the S/S multilayersuch as NbTi/Nb with almost the same condensationenergy of the S and S layer, the main origin of Fp⊥ is
to come from the kinetic energy term of the GL freeenergy between S and S , which stabilizes the vortices
in the superconducting layer with shorter ξGL [4,5].
The transverse pinning force density Fp⊥ is usually strongly dependent on the structural modulation
length λ. In case of the multilayer structure of thisstudy with equal sublayer thickness (λ = 2d), the Fp⊥
value has been found to take the maximum aroundd ∼ 2ξS , for both the S/N and S/S multilayers. This
result may originate from a kind of the matching effect between the vortex size and the layer thickness.
We have investigated [6] the difference of the pinning force between the S/N and S/S multilayers andfound that the behavior of the pinning force as a function of λ is qualitatively much similar in both systems despite the different type (superconducting S
or normal-metal N) intervening layers.
In this paper, we investigate Fp⊥ of S/N and S/S
multilayers with different ξGL and λ values. We examine the pinning mechanism for the S/N and S/S
multilayers, taking into account the difference in
the GL potential energy and/or the GL kinetic energy between the constituent layers. We discuss thestrength of the Fp⊥ max , the maximum value of Fp⊥ as
a function of applied field H , with respect to the coherence length ξGL . We also argue the λ dependenceof Fp⊥ max .

Obi, Ikebe, and Fujishiro

Nb65Ti35/Ti

4

10

(a)

Jc(H⊥)
λ (Å)
100
200
300
500
1000

3

10

2

10

0

2. EXPERIMENTAL

40

60

80

100

H (kOe)Nb65Ti35/Nb

5

10

Jc(H) (A/cm2)

Superconductor(S)/normal-metal(N) and superconductor(S)/superconductor(S ) multilayers havebeen fabricated by an RF dual sputtering onto quartz
substrates. Samples of each series are composed of
Nb/Ti and Nb100−x Tix /Ti (Nb65 Ti35 /Ti, Nb50 Ti50 /Ti
and Nb28 Ti72 /Ti) for S/N and Nb100−x Tix /Nb(Nb65 Ti35 /Nb, Nb50 Ti50 /Nb and Nb28 Ti72 /Nb) andNb28 Ti72 /Nb65 Ti35 for S/S multilayers. Samples were
designed so as to have an equal sublayer thickness,
dS = dN = d = (1/2)λ and dS = dS = d = (1/2)λ,where dS , dN and dS are the thickness of the S, N
and S sublayer and λ is the structural modulation
wavelength. The total sample thickness was designed
to be about 5000 Å. The critical current density (J c )was measured resistively with the current parallel tothe layer at 1.5 K under the magnetic field parallel(J c (H )) and perpendicular to the layer plane(J c (H⊥ )). The transverse pinning force density Fp⊥

20

(b)

4

10

Jc(H//)

Jc(H⊥)

λ (Å)
120
160
240
400
1000

3

10

2

10

0

10

20

30

40

50

60

70

H (kOe)Fig. 1. J c (H ) and J c (H⊥ ) at 1.5 K for the typical example of thepresent S/N and S/S multilayers as a function of the modulation
wavelength λ. (a) Nb65 Ti35 /Ti and (b) Nb65 Ti35 /Nb. In both thefigures open symbols denote the J c (H⊥ ) and closed symbols denote
the J c (H ).

Analyses of the Pinning Mechanism for the Superconducting Multilayers
483

Fp (10 dyn/cm )Nb65Ti35/Ti

(a)

λ (Å)
3
100
200
300
500
2
1000

Fp⊥

1

Fp//
0

0

20

40

60

80

100

H (kOe)432

Fp (10 dyn/cm )3

8

Nb65Ti35/Nb

λ (Å)
120
160
240
400
1000

(b)

1

0

Fp⊥

Fp//
0

10

20

30

40

50

60

70

H (kOe)Fig. 2. Transverse pinning force density Fp⊥ as a function of parallel field H and longitudinal pinning force density Fp as a function of perpendicular field H⊥ at 1.5 K for (a) Nb65 Ti35 / Ti and(b) Nb65 Ti35 / Nb with various λ values. Open symbols denote Fp
and closed symbols denote Fp⊥ .

S/N than S/S in high magnetic field region because of
the higher parallel critical field Hc2 of the S/N multilayer [3].
Figure 2 presents the macroscopic transverse
pinning force density Fp⊥ and the macroscopic longitudinal pinning force density Fp as a function of theapplied magnetic field at 1.5 K for the same samples
as Fig. 1. The characteristic features are as follows:
Fp is always smaller compared to Fp⊥ for both the
S/N and S/S multilayer series and Fp⊥ is very sensitive to the modulation wavelength λ for the both
series, i.e., Fp⊥ is rather small at small and large λvalues but it is enhanced at the medium λ region. On
the other hand, the longitudinal pinning force Fp isnearly independent of λ. This implies that for the perpendicular direction, the layer structure is intrinsically involved in the pinning mechanism as expected,
while for the parallel direction, the size of layer thickness does not directly affect the pinning force.

119

Figure 3 shows the maximum transverse pinning force density Fp⊥ max and the maximum longitudinal pinning force density Fpmax at 1.5 K as afunction of λ for the S/N (Fig. 3a) and S/S multilayers (Fig. 3b), respectively. Fpmax is defined as
the maximum value of Fp versus H curve for eachsample. Fp⊥max takes a broad peak around λ = 200–
300 Å for all the samples of both the S/N and S/S
series studied. In contrast, Fpmax does not have adefinite λ-dependence but all the Fpmax values for
S/N series slightly increase with increasing λ. In thecase of Fp , vortices penetrate perpendicular to thelayer planes and move parallel to the layer plane. So
this slight increase of Fpmax suggests the growing upof the local pin inside of the superconducting NbTi
layer with the increase of the layer thickness. In thecase of S/S multilayers, the slight increase of Fpmax
is not discernible and the Fpmax data scatter may
come from the different sample quality which yields
the different size of local pinning center. We assume that these local pins may act isotropically in themultilayers.
In order to estimate correctly the net pinning
force working only in the perpendicular direction, it
is necessary to eliminate the influence of the sample
dependent isotropic local pins. We define the effective transverse pinning force density FpL by FpL =Fp⊥ − Fp , which is to describe the pinning force inherent to the layer structure. The results of FpL max(= Fp⊥max − Fpmax ) are shown in Fig. 4 as a functionof d/2ξS (ξS : ξGL of the S layer) for the S/N (Fig. 4a)and the S/S (Fig. 4b) multilayer series. Here we haveset ξS at T = 1.5 K as 100.0, 63.4, 58.9 and 42.4 Å
for Nb, Nb65 Ti35 , Nb50 Ti50 and Nb28 Ti72 , respectively[7]. In these figures, the layer thickness is normalizedwith respect to 2ξS because the core size of the vortex should approximately be 2ξS . As schematically
pictured in Fig. 5, the results of Fig. 4 suggest thatthe layer structure works most effectively as a pinning center at around d ∼ 2ξS possibly because of
the matching effect between the S thickness and thesize of the vortex core. It is also to be noticed thatd/2ξS ∼ 1 corresponds to the typical case of the quasi2D state for the superconductivity of the multilayer
system [3].
In Fig. 4a and b, the peak value of FpL max as
a function of d/2ξS , which we define as M(F ) =[FpL max ]max , depends on the alloy composition of the
S layer of each series and the difference of the sizeof the GL coherence length may be the origin of thecomposition dependence of M(F ). In order to estimate the size of M(F ), it is useful to start from the120

Obi, Ikebe, and Fujishiro
6

6

Fp⊥maxFp//max(a)83

Fpmax (10 dyn/cm )5

Nb65Ti35/Ti

Nb28Ti72/Ti

Nb50Ti50/Ti

5

Nb/Ti

4

4

3

3

2

2

1

1

0

5

2

10

2

5

λ (Å)

3

10

2

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2

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3

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λ (Å)

2

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10

5

2

5

λ (Å)

3

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2

5

10

2

2

5

λ (Å)

10

3

2

8

8

Nb65Ti35/Nb

83

Fpmax (10 dyn/cm )7

7

Nb50Ti50/Nb

Nb28Ti72/Nb65Ti35

Fp⊥maxFp//max

6
5
4

0

6
5

(b)

4

3

3

2

2

1
0

1

Nb28Ti72/Nb
5

2

10

2

λ (Å)

5

10

3

2

10

2

5

λ (Å)

10

3

5

2

10

2

5

λ (Å)

3

10

5

10

2

2

5

λ (Å)

3

10

0

Fig. 3. Maximum pinning force for the transverse direction Fp⊥max (closed circles) and longitudinal direction Fpmax(open circles) as a function of λ for (a) S/N series and (b) S/S series. Lines are the guide to the eyes.

following equation of the GL free energy Fs .
∗Fs = Fn (0) + α|| + (β/2)|| + (1/2m )24|(−ih∇ − 2eA)|2 + (1/2µ0 )(rotA)2 (1)where Fn (0) is the free energy of the normal state in
the absence of magnetic field,  the order parameter, m∗ the mass of superconducting electron, µ0 thepermeability of vacuum, A the vector potential, and
α and β are the numerical coefficients. The calculation for the pinning force has been performed by
Matsushita [8] for S/N multilayer taking account of
the difference of potential energy between the normal and the superconducting layers. The elementary transverse pinning force f p⊥ by a boundary isgiven byf p⊥ ∼ 0.305πµ0 Hc2 ξS(2)where Hc is the thermodynamical critical field. In thecase of S/S multilayer with nearly the same Hc , wehave calculated the elementary pinning force by aboundary due to the difference of kinetic energy between S and S layer in a previous paper [7] and have
got the following relation,f p⊥ ∼ 0.612πµ0 Hc2 (ξS − ξS ),(3)

where ξS is ξGL of the S layer (Nb and Nb65 Ti35
in the present case). Thus in the S/N multilayers,
the transverse elementary pinning force should be
proportional to ξS , while it is to be proportional to(ξS − ξS ) in the S/S layer. In Fig. 6, we plot M(F )Analyses of the Pinning Mechanism for the Superconducting Multilayers

121

Nb/Ti
Nb65Ti35/Ti
Nb50Ti50/Ti
Nb28Ti72/Ti

(a)

3
2

8

3

FpLmax (10 dyn/cm )

4

1
0
–1

5

10–1

2

5

100

2

5

101

2

2

5

101

2

d / 2ξS
Nb65Ti35/Nb
Nb50Ti50/Nb
Nb28Ti72/Nb
Nb28Ti72/Nb65Ti35

3
2

8

3

FpLmax (10 dyn/cm )

4

(b)
1
0
–1

5

10–1

2

5

100

d / 2ξS
Fig. 4. FpL max (= Fp⊥max − Fpmax ) for (a) S/N and (b) S/S multilayers as a function of d/2ξS , where ξS is ξGL of the S layer. Lines
are the guide to the eyes.

versus ξS (Fig. 6a) for all the S/N layers and M(F )versus (ξS − ξS ) (Fig. 6b) for all the S/S’ layers studied. In these figures, M(F ) can be confirmed to beroughly proportional to ξS for the Nb/Ti and NbTi/Ti
multilayers and to (ξS − ξS ) for the NbTi/Nb andNbTi/NbTi multilayers. These results suggest that forM(F ), satisfying the ideal size matching condition between the layer thickness and the vortex core, theelementary pinning force f p⊥ is of vital importancefor the observed pinning force strength and that themain origin for the vortex pinning is actually the condensation energy for S/N and the kinetic energy for
the S/S system. In both systems, the boundary pin
plays the main role for the pinning mechanism to determine the maximum pinning force density M(F ).
Next, let us briefly argue the macroscopic pinning strength as a function of the layer thickness d.
As mentioned above, the effective transverse pinning

Fig. 5. Schematic representation of the layer structure and the vortex core size. The arrows indicate the direction of the elementarytransverse pinning force f p⊥ . The vortex energy as a function of
the vortex center position is qualitatively given by the solid lines
for S/N and by dashed lines for S/S multilayers below each figure.

force density FpL makes the maximum at d ∼ 2ξS .
Even in the region of d > 2ξS or d < 2ξS , the S-N
or S-S boundary should vitally influences on FpL(= Fp⊥ − Fp ) because FpL remains positive irrespective of the d/2ξS values. FpL is naturally expected to
be proportional to the pin density, i.e., the number
density N of the boundary. On the other hand, theelementary pinning force f p⊥ is to become not fully
effective for d < 2ξS because of the layer size of thevortex core extending over several sublayers. We introduce the efficiency factor ε(0 ≤ ε ≤ 1). The effective transverse pinning force density is written as
FpL = Nεf p⊥ .

(4)

For d ≥ 2ξS , ε is constant (ε = 1). For d < 2ξS , ε waspointed out to decrease with decreasing d as ε ∝(d/2ξS )2 by Kramer et al. [9]. The pin density N may

122

Obi, Ikebe, and Fujishiro

20

40

60

80

100

Nb65Ti35/Ti
Nb28Ti72/Ti

(a)

120
4
3
2

Nb/Ti
1
0

5

(b)

3

M(F)(10 dyn/cm )Nb50Ti50/Ti

M(F)(108dyn/cm3)

ξS (Å)
0

4

Nb50Ti50/Nb
Nb28Ti72/Nb65Ti35

8

3
2

Nb28Ti72/Nb

1
0

Nb65Ti35/Nb
0

10

20

30

40

50

60

70

80

ξS’–ξS (Å)
Fig. 6. M(F ) (= [FpL max ]max ) as a function of (a) ξS for S/N and as
a function of (b) ξS − ξS for S/S multilayer. Straight lines are theresult of the least square fitting.

be equal to 1/d over the entire region of d/2ξS . Therefore, the following relations are expected to hold:FpL ∝ df p⊥

for d < 2ξGL

(5a)

FpL ∝f p⊥ /dford ≥ 2ξGL(5b)

The effective transverse pinning force density
has the peak around the typical quasi-2D superconductivity region in accord with Eq. (5), where the vortex size just fits the S layer thickness (d = 2ξS ). In
1.2

FpLmax / M(F)

1
0.8
0.6
0.4
0.2
0
–0.2

0

0.2 0.4 0.6 0.8

1 2 3 4 5 6 7 8 9 10

d / 2ξS
Fig. 7. Normalized FpLmax value (= FpL max /M(F )) as a functionof d/2ξS for all the present multilayers. Each symbol has the same
meaning as that in Fig. 4. Two solid lines show FpL max /M(F ) =d/2ξS for d < 2ξS and FpL max /M(F ) = 2ξS /d for d ≥ 2ξS . The
dashed line for d < 2ξS presents FpL max /M(F ) = (d/2ξS )2 .

Fig. 7, we present the normalized maximum pinning
force FpL max /M(F ) as a function of d/2ξS for all thepresent S/N and S/S multilayers. In the figure, thesolid linear line for d ≤ 2ξS and 1/d curve for d ≥ 2ξS
correspond to Eqs. (5a) and (5b), respectively. The
experimental results for both the N/S (open symbols) and S/S (closed symbols) systems follow pretty
well the predicted 1/d line of Eq. (5b) for d ≥ 2ξS .
In the region of d < 2ξS , the experimental points for
the S/N system seem to roughly follow the d-linear
line of Eq. (5a). However, the data points for the
S/S system are more scattered and seem to follow
rather the (d/2ξS )2 dashed line. The calculation of
Kramer et al. which obtained ε = (d/2ξS )2 relation isbased on the proximity effect and assumes the S/Nlayer structure and might not be applicable to the
S/S system. Another possibility for the failure of theEq. (5a) for the S/S system might come from the estimation of the pin number density N. In the S/S multilayers for d < 2ξS , N might not be well-defined interms of the layer boundary and the relation N = 1/dmight not be valid, making FpL directly proportional
to εf p⊥ ∝ d2 f p⊥ .

4. SUMMARY
We have investigated and analyzed the mechanisms of pinning force for the superconductor/
normal-metal (S/N: S = Nb, Nb65 Ti35 , Nb50 Ti50 ,Nb28 Ti72 , N = Ti) and superconductor/superconductor (S/S : S = Nb65 Ti35 , Nb50 Ti50 , Nb28 Ti72 , S =Nb, Nb65 Ti35 ) multilayers having equal thickness dof each sublayer for various structural wavelength λ(=2d). For all the multilayer samples of S/N and S/S ,the effective transverse pinning force density FpL max
makes the peak at d ∼= 2ξS (=GL coherence lengthof the S layer) and decreases as FpL max ∝ 2ξS /d ford > 2ξS in accord with the decrease of the transverse
pin density N (=number density of the layer boundary). For d < 2ξS , FpL max of the S/N multilayers decreased roughly as FpL max ∝ d/2ξS in accord with theefficiency factor ε(∝ (d/2ξ)2 ) calculated by Kramer
et al. for the S/N multilayer system based on the proximity effect. The data points of FpL max of the S/S system are somewhat scattered but seems to decrease
rather as FpLmax ∝ (d/2ξS )2 , i.e., directly proportionalto ε, making a marked contrast to the S/N system.
M(F ), defined as the maximum value of FpL max
for each series has the linear relation with respect to ξS for the S/N and to the ξS − ξS for
the S/S multilayers. These results clearly support

Analyses of the Pinning Mechanism for the Superconducting Multilayersthat the boundary pin arising from the difference in the potential energy is of essential importance for the S/N system and the boundary
pin arising from the difference in the kinetic energy
is of essential importance for the S/S system as a pinning mechanism.

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