PRL 94, 057008 (2005)

PHYSICAL REVIEW LETTERS

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Evidence for Zero- and -Phase Order Parameters of Superconducting Nb=Co Triand Pentalayers from the Oscillatory Behavior of the Transition Temperature
Y. Obi,1 M. Ikebe,2 and H. Fujishiro2
1

Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
2
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan(Received 14 June 2004; published 10 February 2005)

The oscillation behavior of the superconducting transition temperature Tc as a function of theferromagnetic Co layer thickness (dCo ) has been examined for Nb=Co superconductor(S)/ferromagnetic(F) trilayer series (F/S/F) and pentalayer series (F/S/F/S/F). Tc of the pentalayer series
takes a local maximum between dCo  2:0–3:2 nm, where Tc of the trilayer shows a local minimum. Thisdifference in the Tc versus dCo curves provides a clear evidence for the occurrence of the  phase in the
pentalayers, which has been theoretically predicted by Buzdin et al., Radović et al., and Tagirov.
DOI: 10.1103/PhysRevLett.94.057008

PACS numbers: 74.78.Fk, 74.20.Rp, 74.62.–c

For several decades, the superconducting properties ofsuperconductor(S)/ferromagnet(F) layered materials haveattracted much interest due to the peculiar oscillation
behavior of the superconducting transition temperature
Tc when plotted as a function of the ferromagnetic layer
thickness dF . This phenomenon has not been observed in
S/nonmagnetic material(N) layered systems, where Tc
monotonically decreases as a function of the N-layer thickness dN . In S/F layers the superconducting pair distributionfunction Fz is modulated by the exchange potential Eex
in the ferromagnetic layer. At the boundary of S and F
layers, Fz penetrating into the F layer interferes with thatreflected from the opposite boundary of the F layer. As aresult, the modulation of Fz in the F layer depends on dF ,yielding the oscillation of Tc [1]. Buzdin et al. [2] predicted
the appearance of the -phase superconducting state for
the S/F multilayer; the pairing function Fz can stably
take a phase factor difference of  between the neighboring superconducting layers (we call this state ‘‘-phasestate’’ or ‘‘-phase scheme,’’ hereafter), that is,Fz    Fzexpi;(1)z being the coordination vector directed perpendicular tothe layer plane and  dS  dF .
Several experimental works on the Tc oscillation effectof S/F layers have been reported so far. These experiments
can be classified into two categories on the basis of thenumber of the constituent S layers. One is the case in which
only a single S layer exists and the other is the case inwhich two or more S layers are included. The bilayers
or trilayers (F/S/F) such as Nb=Fe [3,4], Nb=Ni [1],Nb=Fe=Cu [5], V=Fe [6], Pb=Fe [7] correspond to theformer case, while the trilayers (S/F/S) with surface S
layers or multilayers such as Nb=Gd [8,9], Nb=CuMn[10,11], Nb=Co [12,13], V=Co [12] correspond to the lattercase. All the S/F layers listed above show the oscillation
phenomenon with at least one minimum and one peak,
0031-9007=05=94(5)=057008(4)$23.00

except for the case of V=Fe which shows the reentrant
behavior.
As for the Nb=Fe bilayer belonging to the former type,Mühge et al. attributed the origin of the Tc oscillation to
the change of the magnetic nature of the Fe sublayer correlated with the sublayer thickness. Sidorenko et al. [1]analyzed the oscillation of the Nb=Ni bilayer based on theinhomogeneous superconducting pairing like the FuldeFerrell-Larkin-Ovchenko state, where the proximity effect[14,15] was assumed to be essential for the oscillation. In
the Nb=Fe=Cu bilayer, Vélez et al. [5] reported that, bychanging dF , Fe structure in the (Fe=Cu) sublayer transforms from fcc to bcc, which results in the oscillation of Tc
due to the change in the spin flip scattering strength. For the
V=Fe bilayer [6], the reentrant behavior was also explainedby the proximity effect. For the Fe=Pb=Fe trilayer [7],Tagirov’s proximity theory was applied.
As for the latter type systems with two or more S layers,
Jiang et al. [8] analyzed their experimental curve of Nb=Gd
based on the Radović’s proximity effect theory taking
account of the -phase scheme. They did not fully agreewith the -phase scheme, emphasizing the importance ofthe magnetic fluctuations. Mercaldo et al. [10] andAttanasio et al. [11] found the Tc oscillation for Nb=
CuMn with the CuMn sublayer being in a spin glass stateand supported the -phase scheme. In our previous paper[12], we observed the double minimum in Tc versus dCo for
the Nb=Co and V=Co multilayers, where experimental
results were explained to come from the following threedifferent type origins: (i) inelastic electron scattering by
the virtual level in intermixed nonferromagnetic NbCo
alloy layers, (ii) magnetic spin flip scattering by ferromagnetic Co layers which are not in so rigidly ferromagnetic
due to the two-dimensional character of magnetization,and (iii) the strong pair breaking mechanism due to themean-field exchange potential. In the paper, we concluded
that the first minimum may originate from the mechanisms(i) and (ii), and the second minimum may be due to the057008-1

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PRL 94, 057008 (2005)

X/ξF

mechanism (iii). More recently, Bagrets et al. [16] reinterpreted Tc oscillation of the Nb=Co multilayers [12] basedon the Gor’kov equation without taking account of the phase.
Theoretically, Buzdin [2], Radović [17,18], and also
Tagirov [19] have led the Tc oscillation behavior as a function of dF , taking account of the possibility of the  phase.
The basic theoretical approach is that in S/F layers theferromagnetic spin of the F layer yields the pair breaking effect on the Cooper pairs which penetrate into the
F layer. The pair breaking parameter is introduced into
the equation, from which Tc is determined [2,17,19],lntc   1=2  Re1=2  =tc ;tanhkF dF iF kF  2=TM  tanhkF dF 

1

0

1

2

4

5

dNb=40nm
π–phase

0.8

Tc/Tc03

67

dNb=40nm
0–phase0.6
dNb=30nm
0–phase0.4
0.2

dNb=30nm π–phase
0

0

1

2

3

4

5

X (nm)(2)

where tc is the reduced superconducting transition temperature (  Tc =Tc0 ) with Tc0 being the transition temperature for the isolated S layer,  the digamma function, Remeans the real part of , and  k2S 2S =2 with kS being
the propagation momentum of the pairing wave function in
the superconducting layer, and S is the superconducting
coherence length (  hD
 S =2kB Tc0 1=2 with DS being
the electron diffusion constant of the S layer). The kS value
is determined from the matching condition for the pairing
functions between S and F layers at the interface [1,19] as 
3dS NF vF
kS dS tankS dS  lS NS vS

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PHYSICAL REVIEW LETTERS(3)

for the 0 phase, where lS is the electron mean free path in
the S layer, NF (NS ) the density of state of the conduction
electron at the Fermi level in the F (S) layer, vF (vS ) thep
Fermi velocity in the F (S) layer, kF  iF =lF  1=F is
the propagation momentum of the pairing wave function in
the F layer with lF being the electron mean free path in the
F layer, F  hv
 F =Eex the magnetic stiffness length in the
F layer, and TM is the interface transparency parameter. For
the  phase, the matching condition is also obtained byusing the relation of Eq. (1) [18,20]. For the Tc versus dF
curve, the calculation predicts the different maximum position of Tc between the 0-phase (where the neighboring S
layer’s pair functions take an identical phase) and the-phase schemes. In the -phase scheme, the Tc maximum occurs at dF =F ’ 1, while in the 0-phase scheme theTc minimum takes place at dF =F ’ 1 and the Tc maximum locates at much larger dF (dF =F ’ 3, see Fig. 1)[21]. If the Tc maximum of the -phase scheme is higher
than the Tc minimum of the 0-phase scheme, the  phase is
to be actualized. It is difficult, however, to draw theoretically Tc versus dF curve without ambiguity because of
several necessary material parameters which are not easy
to estimate experimentally.
The purpose of the present study is to find the differencein the Tc behavior between the zero and  phases. The
guideline is given in Fig. 1. This figure presents the ex-

FIG. 1. Tc oscillation in the Nb=Co multilayer [12] plotted
as Tc =Tc0 versus x, together with the several fitting curves ofTagirov’s calculation [14]. x is equal to the Co sublayer thickness dCo for the multiplayer system. The fitting is based on
the available parameters, Tc0  9:0 K, NF vF =NS vS  0:28,
S =0  0:182, TM  2:0, and lF =F  2:0 for dS  40 and S=
30 nm, where 0 is the BCS coherence length (0:18hv
kB Tc0 ). The thin full line and the thin broken line stand fordNb  40 nm with the 0 and  phase, respectively. The thick fullline and the thick broken line stand for dNb  30 nm with the 0and  phase, respectively.

perimental results of the Nb=Co multilayer by our previous study [12] together with Tagirov’s fitting curves [14]based on the parameter values listed in the figure caption.
It can be seen that our data for the Nb=Co multilayer with
the Nb sublayer thickness dNb  40 nm are satisfactorily
fitted on the -phase line (thin broken line) in the region ofx (dCo for the multilayer) about 1.2–3.1 nm. Accordingly, we should be able to make a clear distinction between the two schemes from the Tc versus x curves for thetrilayer (F/S/F) with one S layer and the pentalayer (F/S/F/
S/F) with two S layers. We have chosen dNb  30 nm as
the S-layer thickness, because for dNb  30 nm, a clearer
difference between the 0 and  phase is expected than fordNb  40 nm, as shown by the thick lines in the figure.
The prepared layered specimens are Cox=2 nm=Nb30 nm=Cox=2 nm trilayers and Cox=2 nm=Nb30 nm=Cox nm=Nb30 nm=Cox=2 nm
pentalayers where x was varied from about 0.8 nm to 5.2 nm.
The thickness of the Co-edge layer was taken as x=2 so as
to satisfy the boundary condition for the pair function [14].
(Note that dCo  x in the pentalayer, while dCo  x=2 in
the trilayer) The layered specimens were fabricated by
rf-sputtering onto quartz-substrate with a very smoothsurface. Target materials were Nb (purity is 99.95%) andCo (purity is 99.9%). After sputtered to the designed
sequences, the samples were coated by Nb(2 nm) in orderto protect the surface layer from oxidation. This Nb layer is
not superconductive down to 2 K. The base pressures were
under 2  107 Torr and sputtering was carried out in
20 mTorr Ar-pressure. The layer structure was confirmedby the low angle x-ray diffraction. The typical diffraction

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6

10

5

1000

105

104

800

4

10
Co(1.3)/Nb(30)/Co(1.3)

2

10

10

102

101

1

0

10

2

3

4

5

3

penta–layer
tri–layer

600
400
200

10

Co(1.3)/Nb(30)/Co(2.6)/Nb(30)/Co(1.3)
1

M (emu/cm )3

10

Intensity (counts/sec)

Intensity (counts/sec)

10

3

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PHYSICAL REVIEW LETTERSPRL 94, 057008 (2005)

0

6

7

0

1

2

8

2θ (deg)

FIG. 2. The low angle x-ray diffraction profiles for theCo1:3=Nb30=Co1:3 trilayer and Co1:3=Nb30=Co2:6=Nb30=Co1:3 pentalayer.

R/Rn

3.8

5
4

2.2

3

3.0
3.4

0.4

penta–layer
tri–layer

6

1.8

0.6

6

7

1
4.2

5

tral Co layer is subject to the mixing or alloying with theboth side Nb layers. Therefore, both tri- and pentalayer
series may have a magnetically dead layer of about 0.7 nm.
Usually thin Co layer retains the magnetic moment down
to 1  2 monolayer, as reported, for example, for the
Cu=Co multilayer [22]. The large lattice mismatch between Nbh110i and Coh0002i of about 17% and/or formingof a mixture region may result in quenching of the magnetic moment up to as large dCo  0:7 nm in the present
layer samples.
Figure 5 shows Tc as a function of x for the tri- andpentalayers. The symbols are the same as Fig. 4. The
curves of each two series of the trilayer and the pentalayer
are slightly scattered, but several characteristic points canbe seen. (i) In the trilayers, in which only the 0 phase ispossible, Tc takes a local broad minimum around x 
2:6 nm. Nearly at the same position (x  2:3 nm), thepentalayers take a local maximum, taking local minima

Co(x/2)/Nb(30)/Co(x)/Nb(30)/Co(x/2)

0.8

4FIG. 4. The magnetic moment at 10 K versus x for the trilayers
Cox=2=Nb30=Cox=2 and the pentalayers Cox=2=Nb30=Cox=Nb30=Cox=2. Each symbol represents the specimens
sputtered in the same run.

Tc (K)

patterns are shown in Fig. 2 for Co1:3 nm=Nb30 nm=Co1:3 nm trilayer and Co1:3 nm=Nb30 nm=Co2:6 nm=Nb30 nm=Co1:3 nm pentalayer. The satellite peaks characteristic of the layer structure can be seen.
Analyses of the x-ray diffraction confirmed that deviationof the layer thickness from the designed one is less than
5%. Tc was decided as a middle point between 10% and
90% of the resistive transition measured by the four terminal method.
Figure 3 presents the typical results of the reduced
electrical resistance R=Rn versus temperature T curve forone of the present pentalayer samples Cox=2=Nb30=Cox=Nb30=Cox=2, where Rn is resistance in the normal state just above the transition temperature. The superconducting transitions take place within the width of 0.2 K
between 10% and 90% change of R=Rn for most of thesamples, suggesting the good sample quality. The data
exhibit a clear nonmonotonic dependence of Tc on x.
Figure 4 shows the magnetic moment at 10 K as afunction of x for the present tri- and pentalayers. In the
figure, each symbol represents the layer samples sputteredin the same run. The magnetization for both series becomeszero around x  1:4 nm. In the pentalayer series, the cen1.2

3

X (nm)

X=1.4

2.6

2

0.2

1

1.5

2

2.5

3

3.5

4

X (nm)

0
4

4.4

4.8

5.2

5.6

6

6.4

T (K)

FIG. 3. The reduced resistance R=Rn versus T curve for one ofthe pentalayer series Cox=2=Nb30=Cox=Nb30=Cox=2.

FIG. 5. Tc as a function of x for the trilayers Cox=2=Nb30=Cox=2 and the pentalayers Cox=2=Nb30=Cox=Nb30=Cox=2. The respective symbols stand for the data points of thesamples sputtered in the same run as in Fig. 4. Lines are guides tothe eyes.

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PHYSICAL REVIEW LETTERS

at around x  2:0 nm and 3.2 nm. (ii) In the region between the two minimum points, the -phase superconductor is to be realized for the pentalayers, as predicted by
Radović et al. (see Fig. 3 of Ref. [18]) and Tagirov (seeFig. 1). The different behavior between the tri- and pentalayer in the region between x  2:0 and 3.2 nm provides aclear evidence for the occurrence of the  phase in the
pentalayer. (iii) As mentioned in Fig. 4, the magnetic moment disappears nearly at about x  1:4 nm for both triand pentalayers. This means that the Tc anomaly due to thespin flip scattering (which was attributed to the origin of
the Tc oscillation of Nb=Fe case [3]) may be excluded forthe origin of the present different oscillation behavior
between the tri- and pentalayers.
According to Tagirov’s calculation for our multiplayer
data on the basis of parameters listed in the caption ofFig. 1, Tc of the 0 phase should degrade below 0 K for thetrilayer for dNb  30 nm, and the reentrant behavior of Tc
should be realized. The observed Tc depression for thepresent trilayers is not so drastic. This may come fromthe several factors which restrain the Tc depression.
Garifullin et al. [6] suggested that interface roughness
introduces random phase shifts for the interfering pairwave function, smearing out the interference pattern andthe oscillation amplitude. On the basis of the Tagirov’s
formalism [19], the Tc oscillation amplitude is also controlled by the interface transparency parameter TM in
Eq. (3). TM prescribes the mixing of the Cooper pairs
with the ferromagnetic electron system and the smaller
TM value results in the smaller Tc oscillation amplitude.
According to Garifullin et al., the interface roughness
should reduce the mixing of the Cooper pairs with theferromagnetic electron system. Tagirov introduced the numerical value of TM  2 for the fitting of the multiplayer
data in Fig. 1. The present Nb=Co trilayer may actually
have a TM value smaller than 2 because of the enhanced
interface roughness compared to the multiplayer system.
In conclusion, we have studied in detail the Tc oscillation behavior for Co=Nb=Co trilayers and Co=Nb=Co=
Nb=Co pentalayers with the superconducting Nb layer
thickness dNb  30 nm. The behavior of the Tc versus x
curve is different between the tri- and pentalayer constructions; at around x  2:6 nm the trilayer has a local minimum in Tc versus x, while the pentalayer has a local
maximum around x  2:3 nm. This difference shouldcome from the different superconducting phase factor of
the pair function Fz. Fz of the trilayer consists of asingle phase over the entire Co layer thickness dCo , whilein the pentalayer Fz of neighboring superconducting
layers take a -phase shift between x  2:0 and 3.2 nm.
From the appearance of the different Tc behavior between
the trilayer and pentalayer Nb=Co systems, a clear evidence of the  phase of superconducting phase of superconductor/ferromagnet layered systems has been provided
experimentally.

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We would like to greatly thank Professor L. R. Tagirov
for valuable discussions and the fitting of our multilayer’s
data based on his theory.

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