Y. Obi et al.: Synthetic Analyses of T c and H c2 of NbTi/Nb Superlattice799

phys. stat. sol. (b) 223, 799 (2001)
Subject classification: 74.50.+r; 74.60.Ec; 74.62.–c; 74.80.Dm; S1.1

Synthetic Analyses of Tc and Hc2
of NbTi/Nb Superconductor/Superconductor SuperlatticeY. Obi1) (a), M. Ikebe (b), H. Fujishiro (b), K. Takanaka (c), and H. Fujimori (a)(a) Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku,
Sendai 980-8577, Japan(b) Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan(c) Faculty of Engineering, Tohoku University, Aramaki-aza-aoba, Aoba-ku,
Sendai 980-8579, Japan(Received March 21, 2000; in revised form August 16, 2000)
Based on an improved calculation formalism of the proximity effect, the transition temperature Tc,
the parallel (Hc2k) and perpendicular (Hc2?) critical fields of the Nb50Ti50/Nb superlattice havebeen analyzed as a function of the structural modulation wavelength ls. Tc, Hc2k and H c2? havebeen calculated using a unified set of the material parameters and allowing for the transition of
the center of the order parameter between the NbTi and Nb layers. The calculation has systematically reproduced the characteristics of Hc2k and H c2? including the two-step dimensional crossover of Hc2k(T) and the anomalous upward curvature of H c2?(T).

1. Introduction
It is now well known that in superconductor/normal metal (S/N) superlattices, quasitwo-dimensional (q-2D) superconductivity is realized and the critical field parallel tothe layer (H c2k) exhibits a dimensional crossover at TD from three-dimension (3D) for
T > T D to q-2D for T < T D [1–6]. Takahashi and Tachiki [7–9] predicted that, underappropriate conditions, superconductor/superconductor (S/S0 ) superlattices with different electron diffusion constants D exhibit a more complicated two-step dimensionalcrossover associated with a jump of the center position of the order parameter. Withdecreasing temperature T, the dimensionality of the superconductivity at first changes
0from 3D to q-2D at TD. In this q-2D state (2D0 ) the center of the order parameter issituated in the S0 layer with the larger D value. With the further decrease of T, thecenter of the order parameter shifts from the S0 layer to the S layer with the smaller D
value at T *D and another q-2D state (2D*) is realized. Experimentally, the dimensionalcrossover appears most clearly as an inflection point in the parallel critical field Hc2k
versus T curves [10–13].
In our previous papers [14, 15], we reported an anomalous depression of the superconducting transition temperature Tc of NbTi/Nb (S/S0 ) superlattices with decreasing
structural modulation wavelength ls. The depression of Tc was attributed to the Tc drop0
) with decreasing thickness and analyses were made for typical twoof the Nb layer (Tcs0drop was attributed to the decrease in the electron density
cases. In the case (i) the Tcs
1) Corresponding author; Fax: 81-22-215-2096; Tel.: 81-22-215-2097; e-mail: obi@imr.tohoku.ac.jp

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Y. Obi et al.

of state N of the Nb layer and in the case (ii) it was attributed to the decrease of the
BCS potential V of Nb. The calculations were performed based on an improved formalism of the proximity effect, which reproduced the experimental Tc versus ls dependence equally well for both cases and we could not conclude which model was more
likely to correspond to the actual case.
In the present paper, we try to analyze Tc(ls), Hc2k(T) and the perpendicular criticalfield Hc2?(T) self-consistently based on the advanced formalism of the proximity effect[16–19] using a unified set of material parameters. Hc2k(T) of the present Nb50Ti50/Nb
A shows the characteristic two-step crossover. Hc2?(T) shows
superlattice with ls ¼ 120 a clear upward curvature for large ls values, which also seems to be peculiar to S/S0
superlattices with different D values [20, 21]. It is meaningful to examine if such peculiar behavior of the upper critical fields can be reproduced together with the ls dependence of Tc on a same footing.

2. Theoretical Background [18]Takahashi and Tachiki [7] made an eigenfunction expansion for the de Gennes kernel
and rewrote the equations to determine Hc2 of the superconducting superlattices withspatial variations of the BCS interaction constant V, the density of states N and the
diffusion constant D. They showed that 1. for the case with the spatial variation of D
and V, one can obtain Hc2k taking into account only the lowest eigenstate and eigenfunction for the secular equation. 2. For the case with the spatial variation of N, it isnecessary to take account of all the eigenvalues and eigenfunctions. For the case 2,
Takahashi and Tachiki actually made an approximation to use only the lowest eigenstate, which is valid only for small values of Ns0 =Ns. In the present calculations of Hc2k
and Hc2?, the finite element method is applied to the equations derived on the basis of
the Takahashi-Tachiki formalism. The finite element method permits us to take accountof the higher eigenvalues and eigenfunctions besides the lowest ones. In the following,
we describe an outline of the theoreticals used in the present calculation.
The eigenfunctions Y l ðrÞ to expand the de Gennes kernel satisfy the equation [7, 22]22ieAðrÞ Y l ðrÞ ¼ El Y l ðrÞ ;ð1ÞhDðrÞ r 
hcwhere A(r) is the vector potential. The boundary conditions postulate that the quantities
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Y l ðrÞ
2ie
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
andNðrÞ DðrÞ r AðrÞ Y l ðrÞ ;
ð2Þ
hc
NðrÞ
are continuous at the interfaces of the superlattice. We assume that the BCS potential
V(r), the density of states N(r) and the diffusion constant D(r) are constant inside eachlayer.
The order parameter D(r) and the pair function F(r) are connected with Y l ðrÞ by the
relationpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PDðrÞ
¼ FðrÞ ¼ NðrÞ Cl Y l ðrÞ :VðrÞl
The pair function F(r) then satisfies the same boundary conditions as

ð3Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃNðrÞ YðrÞ [22].

Synthetic Analyses of T c and H c2 of NbTi/Nb Superconductor/Superconductor Superlattice801

The transition temperature and the upper critical field of the superconducting superlattice are obtained by solving the following determinant raised from the de Gennes kernel
for the coefficient cl [7, 17]:Phlj VN jl0 i 
 0ð4Þ
det dll  2pkB T
 ¼ 0;
h jwn j þ El 
jwn j WD 2
where hwn ¼ ð2n þ 1Þ pkB T, and WD is the Debye frequency. For the wn summation in
the above expression we take into account the effect of the Debye cutoff wn WD,
pointed out by Auvil and Ketterson [17] and approximate the sum as




P
1
2hWD þ El
1ElþIðEl Þ  2pkB T
¼ log
Y
;
4pkB Th jwn j þ El
2 4pkB T
jwn j WD 2ð5Þ
where we assume that the Debye cutoff frequency is same for both layers. By the useof Eqs. (4) and (5), Tc of superlattices in the small ls limit (the Cooper limit) is correctly reproduced. In the Werthamer [23] and the original Takahashi-Tachiki calculation[7], the small ls Cooper limit cannot be reproduced.
To have the eigenvalues and eigenfunctions of the lowest and the excite states, thefollowing functional is considered:
"
#
2




Ð
2ie
2
AðrÞ fl ðrÞ  El jfl ðrÞj :ð6Þ
dn NðrÞ hDðrÞ  r 
hc
The variation of (6) with respect to fl ðrÞ gives an equation of the same form as Eq.
(1),

2
2ie
AðrÞ fl ðrÞ ¼ El fl ðrÞ :hDðrÞ r 
ð7Þ
hc
We impose the conditions on fl ðrÞ that

2ieAðrÞ fl ðrÞ ;fl ðrÞ; and NðrÞDðrÞ r 
hc

ð8Þ

are continuous everywhere. The latter condition comes from the partial integral whichappears at the interfaces by the variation of fl ðrÞ. We relate the eigenfunction fl ðrÞ
with Y l ðrÞ by the relationpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Y l ðrÞ ¼ NðrÞ fl ðrÞ :ð9Þ
Then the finite element method can be applied to the functional (6). Numerical calculations in this study were performed on the computer program due to Takanaka.

3. Procedure of Experiments and Analyses
Nb50Ti50/Nb (S/S0 ) superlattice samples were fabricated by an rf sputtering method under symmetrical condition, i.e., ds ¼ d0s ¼ ls =2, ls being the structural modulation wavelength and ds and d0s the layer thicknesses of NbTi and Nb, respectively. The total thick-802

Y. Obi et al.
Ta b l e 1
Parameters for the present calculation of Tc and Hc2parameter

Ns0 varied

fixed

Vs0 varied

qD (K)Tcs (K)0
Tcs(K)
Vs (1023 eV cm3)
Vs0 (1023 eV cm3)
Ns (1023 states/eV cm3)
Ns0 (1023 states/eV cm3)
Ds (cm2/sec)
D0s (cm2/sec)
xsGL ð0ÞðAÞsHc2ð0Þ(kOe)

2501)
8.4
variable
0.26842)0.343)
1.05884)
variable
1.36
variable
58.9
85

2501)
8.4
8.1
0.26842)0.343)
1.05884)
0.82735)
1.36
variable
58.9
85

2501)
8.4
variable
0.26842)variable
1.05884)0.84705)
1.36
variable
58.9
85

1

) Referred to Sasaki [24], qD ¼ 250 K is assumed for Nb and NbTi alloys as an average.
) Determined from the relation Vs ¼ ½Ns ln ðTcs =1:134 qD Þ1 .
3) Referred to Gladstone et al. [25].
4) Deduced from the NbTi composition dependence of g [24, 26–29] and the density ofstate of Nb [26–28, 30].
50
) Determined from the relation Ns0 ¼ ½Vs0 ln ðTcs=1:134 qD Þ1 using Vs0 ¼ 0:34 and00¼ 8:1 for fixed case and Tcs¼ 8:8 for varied case.
Tcs
0
Non-marked: Experimental values. For Tcs
of fixed case, the value of thick film is used.
Ds is determined from the relation Ds ¼ ð4ckB =peÞ ð@Hc2? ðTÞ=@TÞ1 jT¼Tc using
ð@Hc2? ðTÞ=@TÞjT¼Tc ¼ 8060
2

ness of each superlattice was designed to be 5000 
A. Tc, Hc2k and Hc2? were determined by a four-terminal resistive method using a 10 T superconducting magnet. The
small-angle X-ray diffraction patterns were taken to confirm the consistency between
the designed ls and the actual one.
The calculations of Tc(ls), Hc2k(T) and Hc2?(T) were performed by an ACOS systemcomputer at Tohoku University. For the calculations of Tc and Hcsk, ?, we need thevalues of several parameters of each layer, such as the GL coherence length at 0 K,
0xsGL ð0Þ of Nb50Ti50, the Debye temperatures qsD and qsD , the transition temperatures Tcs
00
and Tcs , the densities of states Ns and Ns , the BCS potentials Vs and Vs0 , and the ratioof the diffusion constant DD ¼ D0s =Ds . The values of xsGL (0) and Tcs were experimen0
tally determined from the data of the sputtered Nb50Ti50 monolayer. qsD and qsD were
set to be 250 K and other values of the material parameters were taken from references. For the fitting of the experimental data, we chose the ratio of the diffusion con0
stant, DD ¼ D0s =Ds , and either Ns0 or Vs0 of the Nb layer as adjustable parameters. Tcs
s00
0 0
was determined from the relation, Tcs ¼ 1:134 qD exp (1=N V ). The material parameters used for the fitting are summarized in Table 1.

4. Experimental Results
Figure 1a shows Tc data of Nb50Ti50/Nb superlattices as a function of the structuralmodulation wavelength ls , together with the results of the Tc calculations. In spite ofalmost the same Tc values (&8 K) of both Nb and Nb50Ti50, Tc(ls) of the superlattice
substantially decreases with decreasing ls . If we calculate Tc ðls Þ of the Nb50Ti50/Nbsuperlattice sticking to the standard set of the material parameters in Table 1 and keep-

Synthetic Analyses of T c and H c2 of NbTi/Nb Superconductor/Superconductor Superlattice803

Fig. 1. a) Tc vs. ls of Nb50Ti50/Nb superlattices. The closed circles are experimental points. The
0
dotted line almost independent of ls represents the fitting with a constant Tcs(Tc of the Nb layer)
0value. The full line is the case (i) fitting changing the Ns value as a fitting parameter with the ratioof the electron diffusion constant DD  D0s =Ds ¼ 8. The dashed line in the case (ii) fitting changing
0
the Vs0 value as a fitting parameter with DD ¼ 9 (see text). b) ls dependence of Tcs, Ns0 (for case0(i)) and Vs0 (for case (ii)) used in the present calculation. The full line is Tcs
vs. ls for the case (i)fitting and the dashed one for the case (ii) fitting0
ing Tcs and Tcs
constant, the calculated Tc(ls) curve derivates far higher from the ob0
served one as shown by the dotted line in Fig. 1a. Then we must vary either Tcs or Tcs
as a ls dependent parameter to achieve the fitting. In a previous paper [15] we pointed
0
out that Tc of the Nb layer (Tcs
) should decrease with decreasing ls .
0
(ls Þ depression; a) Ns0 varied
Here we consider two typical cases of origin of the Tcs
case, which we denoted as case (i) and b) Vs0 varied case, which we denoted as case (ii).
The full line in Fig. 1a represents the calculated line for the case (i) and the dashed line
represents that of case (ii). In order to keep consistency with the calculations of Hc2k
and Hc2? to be given later, the ratio of the diffusion constants DD (¼ D0s =Ds ) is set to
be DD ¼ 8 for the case (i) and DD ¼ 9 for the case (ii). Both calculated curves reproA: For ls ¼ 60, and 40 A, however,duce well the experimental Tc(ls ) down to ls ¼ 86 

804

Y. Obi et al.

Fig. 2. Typical a) Hc2k and b) Hc2? data as a function of T. The origin of the ordinate is shifted foreach sample to get a clear view

Tc of NbTi/Nb again increases having a minimum around ls ’ 80 
A. As we will mention later in connection with the Hc2k fitting, the non-neglectable alloying effect and/orinterdiffusion of Nb and Ti atoms through the interface are considered to be the main0
origin of the Tc minimum. Figure 1b shows the ls dependences of Tcs
and N0s (case (i))
00and Tcs and Vs (case (ii)) necessary to achieve the Tc fitting given in Fig. 1a.
Figures 2a and 2b show the temperature dependences of Hc2k and Hc2? of six typical
NbTi/Nb samples. In both figures, the origins of the ordinates (Hc2 ¼ 0) are shifted toget clearer views. For ls ¼ 60 
A in Fig. 2a, Hc2k shows the usual 3D behavior without
A, the crossover is seen in Hc2k(T) at T ’ 5:0 K.
any anomalous behavior. For ls ¼ 86 
A the two-step crossovers of Hc2k(T) occur at T ’ 5:8 K and 3.7 K. For
For ls ¼ 120 ls ¼ 160 A, the dimensional crossover from 3D to 2D is barely noticed in the very
vicinity of Tc, followed by another clear crossover at T ’ 5:2 K. For ls ¼ 300 and500 A, Hc2k shows a 2D behavior following the relation Hc2k / ð1  tÞ1=2 , and no crossover phenomenon is observable.
In Fig. 2b, the temperature dependence of Hc2? is not so drastically different from
A. What is to be noticed in Fig. 2bthat of Hc2 of usual 3D superconductors for ls < 300 
A. The exisis that a clear positive curvature anomaly (PCA) is noticed for ls  300 
tence of PCA in S/S0 multilayer systems with different D values of the sublayers was
first predicted by Takahashi and Tachiki [7] in their pioneering paper.

5. Discussion
In order to perform systematic calculations of Tc, Hc2k and Hc2?, we must determine aself-consistent set of material parameters. For the Hc2 fitting of S/S0 multilayers, theSynthetic Analyses of T c and H c2 of NbTi/Nb Superconductor/Superconductor Superlattice805

ratio of the electron diffusion constants DD ¼ D0s =Ds is very important to decide theshape of the Hc2k(T) curves; the crossover temperature from 3D to 2D0 or from 2D0 to
2D* crucially depends on the ratio DD. Let us first discuss Hc2k(T) of Nb50Ti50/Nb withls ¼ 120 A, which shows the typical two-step crossover. The results of the calculations
using the material parameters of Table 1 and with DD as a variable parameter are
shown in Figs. 3a and b. In Fig. 3a Vs0 is kept the same as that of bulk Nb, while N0s is
0varied to give the reduced Tcs
for ls ¼ 120 
A. The calculation in Fig. 3a corresponds tothe case (i) of the Tc calculation in Fig. 1b. In Fig. 3b, on the other hand, Ns0 is kept thesame as that of bulk Nb, while Vs0 is varied to give the reduced Ts0 which corresponds tothe case (ii) in Fig. 1b. In these figures, the thick lines represent the calculated resultswith the center of the order parameters situated in the S layer (NbTi layer) and thethin lines represent those with the center of the order parameter situated in the S0 layer(Nb layer). According to the calculations for ls ¼ 120 A, the two-step crossover is expected for both case (i) and (ii) for DD & 6. In Figs. 3a and b, all thick and thin lines
are almost indistinguishable between Tc =Tcs ’ 0:78 and T=Tcs ð t 0 Þ ’ 0:7: This factAmeans that the dimensionality of the superconductivity of Nb50Ti50/Nb with ls ¼ 120 

Fig. 3. a) Case (i) fitting for Hc2k of thels ¼ 120 A sample. The Ns0 value is set
to be 0.6976  1023 states/eV cm3
based on the result of Ns0 -calculation(see Fig. 1b), while the bulk Nb value
is
used
for
the
Vs0
value
(¼ 0:34  1023 eV cm3 ). The thickand thin lines represent the calculated
results with the center of the orderparameter in the NbTi(S) and Nb(S0 )layer, respectively b) Case (ii) fittingfor Hc2k of the ls ¼ 120 A sample. The
Vs0 is set to be 0:2858  1023 eV cm3
based on the result of Vs0 -calculation(see Fig. 1b), while the bulk Nb value is used for the Ns0 value(¼ 0:8470  1023 states/eV cm3 ). The
thick and thin lines represent the calculated results with the center of theorder parameter in the NbTi and Nb
layer, respectively

806

Y. Obi et al.

000
is 3D above TD=Tcs ’ 0:7 and changes to q-2D below TD. In this q-2D state below TD,
0the center of the order parameter is located in the S layer, which we denote as the 2D00
state. In Figs. 3a and b, we notice that TD
is not so much sensitive to the DD value andis nearly independent of DD for 7 DD 10: With further decrease of temperature,
the thick lines cross the thin lines at TD
and surpass the thin lines for T . TD
. Thismeans that the center of the order parameter shifts to the S layer (NbTi layer) at TD
and the superconductivity of the superlattice enters into another q-2D state, which wedenote as 2D* state. Experimentally, the transition from the 2D0 to 2D* state occurs at=Tcs ’ 0:4 for ls ¼ 120 
A and the calculation line with DD ¼ 8 of the case (i) inTD
Fig. 3a and that with DD ¼ 9 of the case (ii) in Fig. 3b are both consistent with theexperimental results. Comparing the DD ¼ 8 line in the case (i) and DD ¼ 9 line in thecase (ii), the case (i) lines give somewhat a better fitting with experimental data, especially in the lower temperature region (t 0 < 0:3).
A. TheFigure 4 shows the fitting of Hc2k(T) of NbTi/Nb for ls ¼ 160, 300 and 500 
results are presented for both case (i) and (ii) calculations. The same parameter values
A were used except for the Ns0 and Vs0 values. The Ns0 values for the case (i)as ls ¼ 120 
0and Vs values for the case (ii) were set to be consistent with Ns0 ðls ) and Vs0 ðls ) lines inFig. 1b, to reproduce the experimental Tc values for each modulation wavelength ls .
A sample, for which Hc2k shows apparently one crossover at t 0 ’ 0:65,
For the ls ¼ 160 the calculation indicates that the observed crossover actually corresponds to TD
from
0
0
0
2D to 2D*, and TD exists just below Tc (TD =Tcs ’ 0:81, Tc =Tcs ’ 0:84Þ. Though thecalculation lines somewhat derivate downward from experimental Hc2k(T) near Tc andthe calculated TD=Tc (’ 0:68Þ is slightly higher than the experimental one, both fittings
for the cases (i) and (ii) are satisfactorily equall as a whole.
The dimensionality of superconductivity is q-2D for ls ¼ 300 and
500 
A and the Hc2k calculations suggest that the superconductivity of
those samples is in the 2D* stateover the whole temperature range

Fig. 4. Hc2k fitting for ls ¼ 160, 300 and500 
A samples. The full lines represent case(i) fitting, where Ns0 is set to be
0:7140  1023
(160 
A),
0:7511  1023
(300 A) and 0:7801  1023 (500 A) states/eV
cm3 with DD ¼ 8. The dashed lines represent the case (ii) fitting, where Vs0 is set to beA), 0:3050  1023
0:2918  1023 (160 (300 A) and 0:3150  1023 eV cm3 (500 A)with DD ¼ 9. The thick and thin lines are
the calculated results with the center of theorder parameter in the NbTi and Nb layer,
respectively

Synthetic Analyses of T c and H c2 of NbTi/Nb Superconductor/Superconductor Superlattice807

Fig. 5. Hc2k fitting for the ls ¼ 86 Asample. The full and dashed lines represent case (i) (Ns0 ¼ 0:6800  1023
states/eV cm3; DD ¼ 8) and case (ii)
(Vs0 ¼ 0:2791  1023 eV cm3 ; DD ¼ 9)
fittings, respectively. The thick andthin lines represent the calculated
Hc2k(T) with the center of the orderparameter in the NbTi and Nb layer,
respectively

below Tc with the center of the order parameter in the NbTi layer. Comparing thefitting lines for the cases (i) and (ii), the line of the case (i) is more satisfactorily forls ¼ 300 A.
A. In this sample
Figure 5 shows the Hc2k fitting results for the NbTi/Nb with ls ¼ 86 
the one-step crossover occurs at around t 0 ’ 0:65 and the calculation indicates that the0crossover corresponds to TD
from 3D to 2D0. With this small layer thickness
(dNbTi ¼ dNb ¼ ls =2 ¼ 43 A), the center of the order parameter is situated in the Nb
0and does not shift to the NbTi layer down to 0 K. The present calculalayer below TD
tion assumes that xsGL ð0Þ, the GL coherence length at 0 K, is 58.9 
A. The calculation
A is not thick enough to localize the order parameter in thesuggests that dNbTi ¼ 43 
NbTi layer to take full advantage of its smaller xGL ð0Þ value. Comparing the lines forthe cases (i) and (ii) in Fig. 5, we notice that the lines for the case (i) again result in asomewhat better fitting than for the case (ii).
Figure 6 shows the result of the case (i) fitting for Hc2k(T) of NbTi/Nb withls ¼ 60 
A. As we have mentioned already in Fig. 1a, Tc for this sample deviates upwards from the extrapolation of the calculated line and we suggest an interdiffusion

Fig. 6. Case (i) fitting for Hc2k of thels ¼ 60 A sample. Ns0 is set to be
0:7272  1023 states/eV cm3 to givethe observed Tc . The dashed lines represent the calculated results for
DD ¼ 8 and the full ones those for
DD ¼ 4. The thick and thin lines are
the results with the center of the order parameter in the NbTi and Nb
layer, respectively

808

Y. Obi et al.
Fig. 7. Hc2? fittings for the typical Nb50Ti50
superlattices. The full lines represent thecase (i) fitting (DD ¼ 8) and the dashedlines show case (ii) fitting (DD ¼ 9)through the interface as a possible origin [15]. In order to remedy the deviated Tc value, we adopted a higher
value of Ns0 (¼ 0.7272  1023 states/eV cm3) than that of Fig. 1b. If westick to DD ¼ 8, however, the calculated lines do not reproduce the experimental data as shown in the figure.
By scanning the DD values we foundthat DD ¼ 4 lines reproduce the experimental Hc2k(T). If the mixing of
NbTi and Nb components through theinterface becomes important, the diffusion constant D0s of the Nb layer may
become smaller due to the impurity
scattering by immigrant Ti atoms.
Thus, the smaller effective value of
DD ¼ D0s =Ds may be consistent with the interdiffusion model for this small ls superlattice.
Finally, let us discuss the ls dependence of the perpendicular critical field Hc2?(T) of
NbTi/Nb superlattices. Figure 7 shows the fitting lines of Hc2?(T) for ls ¼ 86, 120, 160,
300 and 500 A which were calculated using the same set of parameter values as for Tc
and Hc2k(T) fittings. The overall agreement between the theoretical lines and the experimental data is satisfactory for the case (i) calculations. The case (ii) calculation lines
with varying Vs0 deviate from the experimental data for ls ¼ 300 and 500 A.
The Hc2? versus t 0 lines in Fig. 7 show a clear positive curvature anomaly (PCA) for
A and 500 
A. As Takahashi and Tachiki [7] suggested for S/S0 superlattices
ls ¼ 300 with different D values of sublayers, the PCA anomaly occurs because the pair function
for the order parameter, which are widely spread over several layers near Tc, become
gradually localized in the S layers with smaller D as temperatures decrease. Our present Hc2?(T) calculations well reproduce the upward curvature of Hc2? for ls ¼ 500 A,
A does not fully account for the observed very
though the calculation line for ls ¼ 300 clear PCA anomaly.

6. Summary
The superconductor/superconductor (S/S0 ) superlattices, Nb50Ti50/Nb multilayers, were
fabricated by a dual sputtering method over a wide range of structural modulation
wavelenghths ls. The transition temperature Tc, the parallel critical field Hc2k(T) andthe perpendicular critical field Hc2?(T) were resistively measured. The experimental

Synthetic Analyses of T c and H c2 of NbTi/Nb Superconductor/Superconductor Superlattice809

results were synthetically analyzed based on an improved formalism of the proximity
effect. From the anomalous depression of Tc of the superlattice, it was concluded that
the Tc depression of the Nb sublayer with decreasing ls is responsible for the Tc depression of Nb50Ti50/Nb. The analyses of Hc2k(T) and Hc2?(T) indicated that the degradation of Tc of the Nb sublayer should be mainly caused by the reduction in the elctron
A, thedensity of state N rather than by reduction of the BCS potential V. For ls  86 
calculated lines using a unified set of material parameter values minutely reproduced
the behavior of Hc2k(T) and Hc2?(T), including the two step crossover in Hc2k(T) charA, the Hc2k analyses provide an evidence ofacteristic of S/S0 superlattices. For ls ¼ 60 
the interdiffusion effect as a reduction of the diffusivity ratio D0s =Ds of the NbTi andNb sublayers. The calculation also reproduced the anomalous upward curvature in
Hc2?(T) semi-quantitatively.
Acknowledgement The authors are grateful to the Center for Low Temperature
Science of Tohoku University for the use of the 10 T superconducting magnet.

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