Physica B 329–333 (2003) 1423–1424

Pauli paramagnetic effect and spin-orbit scattering time in
Nb=Al2O3 superconducting multilayers
Yoshihisa Obia,*, Manabu Ikebeb, Hiroyuki Fujishirob
aInstitute for Materials Research, Tohoku University, 2-1-1 Katahira, Sendai 980-8577, Japanb
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, JapanAbstract
The parallel critical ﬁeld Hc2 jjðTÞ of the Nb=Al2 O3 two-dimensional (2D) multilayer superconductors has been
analyzed by developing the 2D-version of the WHH theory and by utilizing the Takahashi–Tachiki formalism. A
reliable method to estimate the spin-orbit scattering time tS0 of thin ﬁlms has been provided.
r 2003 Elsevier Science B.V. All rights reserved.
Keywords: Nb=Al2 O3 multilayers; Parallel critical ﬁeld; Spin-orbit scattering time

The parallel orbital critical ﬁeld Hc2jj ðTÞ of thin ﬁlms
is inversely proportional to the thickness d and can bemade arbitrary large in principle. Then instead, the spinparamagnetic effect counteracted by spin-orbit scattering limits Hc2jj : Accordingly, a reasonable estimation ofthe spin-orbit scattering time tS0 can be made if weproperly analyze the Hc2jj ðTÞ of thin ﬁlms. In this paper,
we determine tS0 of Nb by analyzing Hc2jj of theNb=Al2 O3 multilayers.
The Nb=Al2 O3 multilayers were prepared by a dual
sputtering method [1]. The transition temperature Tc
and the critical ﬁelds Hc2jj;> were determined resistively.
The list of the multilayers analyzed in this report is given
in Table 1.
Fig. 1 shows the conductance per square, G& ; at 10 K
as a function of the Nb layer thickness dNb : G& exhibitsa linear dependence on Nb in both the large and thesmall dNb regions, but the slope of the linear dependence( A similar beharadically changes around dNb ¼ 30 A:vior was ﬁrst observed in Nb/Ge multilayers [2] and wasunderstood on the basis of a trilayer model. We assume
that thin layers of disordered and degraded highresistivity Nb are formed on both interfacial sides ofeach Nb layer (see the inset in Fig. 1). From the slopes of*Corresponding author. Tel.: +81-22-215-2097; fax: +8122-215-2096.
E-mail addresses: obi@imr.tohoku.ac.jp (Y. Obi),
ikebe@iwate-u.ac.jp (M. Ikebe).

the linear dependences of the G& vs. dNb line, theresistivities of the degraded layers (rdeg ) and the genuine
Nb layer (rNb ) are estimated to be rdeg ¼ 49:5 mOcm
and rNb ¼ 7:3 mOcm; respectively. From the turning
point of the slopes, the thickness of each degraded Nbdeg( (totally 28 A
(layer, dNbis estimated to be about 14 A
per Nb layer).
Fig. 2 presents Hc2jj ðTÞ and Hc2> ðTÞ as a function ofT.Hc2jj ðTÞ rises with an inﬁnite slope at Tc ; showing thetwo-dimensional (2D) character. The present Nb=Al2 O3
multilayers are surely in the dirty limit but not dirty
enough to satisfy the condition, lS0 ¼ 2_=3pkB Tc tS0
b1 ðlS0 B0:5; for present samples). Then the simpliﬁed
Maki’s theory [3] is not applicable and the 2D versionof the full WHH theory [4] must be invoked. The 2D
formalism of the WHH theory has been accomplishedby Ebisawa [5] and Hc2jj ðTÞ of a thin ﬁlm with thethickness d is given by the following equations:  
1
1lS0
1 h2 lS0 Gþc þ þcnðtÞ ¼ c

1
2G
t
4t 2t
222
 

lS0
1 h2 lS0 Gc þ þ;
ð1Þ
þ 1þ
2G
t
4t 2t
2
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
where G ¼ ðlS0 =2Þ2  4fh2 ; c is the di-gamma function, f ¼ a2 ðd0 =dÞ2 ; d0 ¼ p_D=2kB Tc ; a ¼ _=2mD; h2 ¼
p_d 2 H 2 =12F20 kB Tc ; t ¼ T=Tc ; F0 ¼ p_=e; D ¼electron
diffusion constant, m ¼electron mass, H ¼applied ﬁeld(other notations are of standard style). Eq. (1) allows us

0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0921-4526(02)02351-7

Y. Obi et al. / Physica B 329–333 (2003) 1423–1424

1424

Table 1
Material and superconducting parameters of Nb=Al2 O3
dNb =dAl2 O3 G&( AÞ(ðA=ðO1&Þ
10/10
40/40
50/50
60/40
70/70
80/40
100/1009:52  104
1:07  102
1:77  102
2:92  102
4:90  102
5:71  102
8:93  102Tc xðT ¼ 0KÞ dHc2> =dT D((K) ðAÞ(kG/K)
ðcm2 =sÞ
1.87 87
3.51 101
4.60 1106.18 109
6.06 107
6.55 107
6.92 107

16.2
6.55
6.25
4.50
4.90
4.11
4.27

0.60
1.50
1.47
2.26
2.00
2.38
2.31

Fig. 3. (a) Calculated effective superconducting layer thickness
ds for a unit Nb trilayer as a function of dNb : (b) Calculatedspin-orbit scattering time tS0 vs. dNb : Total scattering time t is7
shown by the dashed line ðvNbF ¼ 2:6  10 cm=sÞ:Fig. 1. G& vs. dNb : The inset schematically shows the trilayer
structure inside of Nb layers. The shaded parts indicate thedegraded Nb layer. ds is the effective superconducting layer.

Fig. 2. Examples of measured critical ﬁelds and numericalﬁttings for Hc2jj ðTÞ on the basis of Eq. (1) and ds :to calculate Hc2jj ðTÞ of thin ﬁlms in the dirty limit forarbitrary tS0 on the basis of Tc ; D; tS0 and d values.
Because of the trilayer structure of the Nb sublayer,
however, we cannot directly regard dNb as d in Eq. (1).
To estimate the effective superconducting ﬁlm thickness
ds ; we utilized the Takahashi–Tachiki formalism [6] andcalculated Hc2jj ðTÞ of the sandwiched Nb ﬁlms (withoutthe paramagnetic effect). The material parameters ofNb ðyD ¼ 277 K; N ¼ 0:9  1023 =eVcm3 ; V ¼ 0:315 1023 eVcm3 ; D ¼ 3:0 cm2 =sÞ were used for the central
intact Nb layer and those for degraded Nb layer were
modiﬁed ðyD ¼ 277 K; N ¼ 0:77  1023 =eVcm3 ; V ¼
0:252  1023 eVcm3 ; D ¼ 0:60 cm2 =sÞ to best reproduce
the Tc vs. dNb curve [1]. The obtained Hc2jj ðTÞ curvepﬃﬃﬃﬃﬃwasapproximated by the G–L result, Hc2jj ðTÞ ¼ 12F0 =2pxðTÞds ðxðTÞ ¼coherent length), to get the effective ds
which is given in Fig. 3(a). Fig. 3(b) presents our ﬁnal
results, tS0 as a function of dNb : We can conclude thatthe spin-orbit scattering accounts for 1:2–3% of the total
scattering rate in the sputtered Nb. The authors wish tothank Prof. H. Ebisawa for the derivation of Eq. (1).

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