Physica C 317–318 Ž1999. 142–148

On anomalous temperature dependence of HC2 H in natural andartificial layer superconductors
M. Ikebe a,) , H. Fujishiro a , Y. Obi b, H. Fujimori b, S. Morohashi caDepartment of Materials Science, Faculty of Engineering, Iwate UniÕersity, 4-3-5 Ueda, Morioka 020-8551, Japanb
Institute for Materials Research, Tohoku UniÕersity, 2-1-1 Katahira, Sendai 980-8577, Japanc
Faculty of Engineering, Yamaguchi UniÕersity, 2557 Tokiwadai, Ube 755-8611, Japan

Abstract
Anomalous temperature dependence of the perpendicular critical field HC2 H ŽT . of layered transition metal dichalcogenide MX 2 with and without pyridine intercalation and that of NbrAl 2 O 3 superlattice are consistent with the theory ofMaekawa, Ebisawa and Fukuyama ŽMEF. wS. Maekawa, H. Ebisawa, H. Fukuyama, J. Phys. Soc. Jpn. 52 Ž1983. 1352x
based on the weak Anderson localization. For the pure conductor 2H-NbSe 2 , however, the anomalous HC2 H ŽT . is tooriginate from the anisotropies in the Fermi surface and energy gap through the effect of nonlocality. q 1999 Elsevier
Science B.V. All rights reserved.
Keywords: HC2 H ; Artificial layer superconductor; Natural layer superconductor

1. Introduction
Superconductors with layered structures have long
been model systems to investigate anisotropic andtwo-dimensional Ž2D. properties of superconductivity. Historically, the transition metal dichalcogenides
MX 2 ŽM s Nb, Ta: X s S, Se, etc.. w1x were firstly
and most extensively studied among the various
layer-metal systems because they are rich in physicalproperties, having various polytypes of the crystal
structure. Another attractive point of MX 2 is theirability to accept various organic molecules as anintercalant between layers. As a result of intercalation, the anisotropy is generally markedly increased,
and quasi-2D superconductivity was first establishedin this system w2–5x. The most characteristic feature)

Corresponding author. E-mail: ikebe@iwate-u.ac.jp

of quasi-2D superconductivity is the phenomenon of
the dimensional crossover, which causes a clear upturn at the crossover temperature TD in the temperature dependence of the parallel critical field Ž HC2 5 ..
With recent progress in the film deposition techniques, on the other hand, superconductivity of artificial multilayers has also been widely investigated
and the dimensional crossover phenomenon inHC2 5 ŽT . associated with quasi-2D superconductivity
has been reported for superconductorrinsulator ŽSrI.
w6,7x, superconductorrnormal–metal ŽSrN. w8,9x andsuperconductorrsuperconductor ŽSrS. w10,11x multilayers. A very useful theoretical study on the superlattice superconductor was given by Takahashi andTachiki w12x based on their original formalism. At
present, the anomalous behavior of HC2 5 ŽT . is quite
well-known and widely accepted as a fundamental
property of superconductors with a layered structure.

0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 0 5 4 - 4

M. Ikebe et al.r Physica C 317–318 (1999) 142–148

The anomalous behavior in temperature dependence, however, is not a monopoly of HC2 5 ŽT . andthe critical field perpendicular to the layer Ž HC2 H .
also shows anomalous behavior. HC2 H ŽT . of layer
superconductors, quite often, does not follow theclassical curves of Werthamer–Hefand–Hohenberg
ŽWHH. w13x in pure and dirty limits but almost
always deviates upward from the WHH curves.
In this paper, we present typical examples ofanomalous HC2 H of natural and artificial layer superconductors and discuss the origins which cause
the anomalies. We will not discuss HC2 H ŽT . of
high-Tc oxides which also have layer structures, inwhich the upper critical field is hard explicitly todefine and determine because of thermal activationof fluxoids. However, it is worthwhile to realize thatthe underlying HC2 H of high-Tc oxides can appreciably deviate from the WHH curve from the analyses of HC2 H of low-Tc layer superconductors.

2. Experimental
The alloy single crystals of the Nb1y X Ta X Se 2
and Ta 1y X Nb X S 2 were grown by the iodine vapor
transport method. The growth zone temperatures wereset between 685 and 7508C. Pyridine ŽC 5 H 5 N. intercalation was carried out for Ta 1y X Nb X S 2 by sealing
the samples and pyridine in Pyrex ampoules withexcess sulfur and reacting them at 2008C for about aweek. Superlattice samples of NbrAl 2 O 3 ŽSrI. were
fabricated by an rf sputtering method. The thicknessof each sublayer Ž d Nb and dAl 2 O 3 . was designed over˚ The multilayer structures
the range from 5 to 100 A.
were confirmed by an X-ray diffraction. The superconducting transition temperature Tc and the upper
critical field HC2 were resistively determined by afour terminal method. The magnetic field up to 100
kOe were applied by a superconducting magnet.

3. Crystal structures and physical properties oftransition metal dichalcogenides
The layered structure of transition metal dichalcogenides consists of X–M–X unit sandwich sheets.
The bonding within the sandwich is covalent andstrong, while that between them is weak van der143

Waals type. Intercalated organic molecules take thesites between the unit layer of sandwich structure.
There are two types of the sandwich layers withdifferent electronic properties, which are characterized by the octahedral or trigonal prismatic coordination of chalcogen X around the transition metal M.
The trigonal prismatic layers generally show metallic
and superconductive properties, while octahedral layers show semiconductive behavior. The 2H polytype,which consists of only trigonal prismatic layers,
show metallic conduction in both parallel and perpendicular directions. The 4Hb-polytype is of special
interest because it consists of alternative accumulations of the trigonal prismatic Žmetallic. and theoctahedral Žsemiconductive. layers. The electronic
conduction of 4Hb-TaS 2 , which is not superconductive above 1 K, was found to be metallic along thelayer and semiconductive perpendicular to be layer
probably as a result of the peculiar stacking of thetwo types layers. We found that the mixed crystal
4Hb-Ta 0.95 Nb 0.05 S 2 is superconductive with Tc s
2.92 K w14x.

4. Results and discussion
4.1. PositiÕe curÕature anomaly in 4Hb polytype andpyridine intercalated MX2 (Py)1r 2
For an example of parallel and perpendicularcritical field of MX 2 before and after the intercalation of organic molecules, Fig. 1 shows HC2 5 andHC2 H of 4Hb-Ta 0.95 Nb 0.05 S 2 and 4Hb-Ta 0.95 Nb 0.05 S 2 ŽPy.1r2 . As a result of pyridine intercalation, HC2 5 is greatly enhanced, while HC2 H is significantly reduced. The enhancement of HC2 5 accompanied with the reduction of HC2 H is a common
feature of the intercalation in all types of MX 2
compounds and, consequently, 2D character of superconductivity is enhanced. After the theory of
Klemm, Ruther and Beakley ŽKLB. w5x, the dimensionality of superconductivity is determined by aparameter g which is defined bygs

4

j H Ž 0.

p

Sr2

2

,

Ž 1.

144

M. Ikebe et al.r Physica C 317–318 (1999) 142–148

Fig. 1. HC2 5 ŽT . and HC2 H ŽT . of 4Hb-Ta 0.95 Nb 0.05 S 2 with Žv .
and without Ž`. pyridine intercalation.

where j H Ž0. is the zero temperature Ginzburg–
Landau coherence length perpendicular to the layer
and S is the distance between spacings of the M
˚ . by ŽPyridine.
layer. S is nearly doubled Ž6 ™ 12 AŽ.
intercalation and j H 0 is given byj H Ž 0 . s j 5 Ž 0 . H C 2 H rH C 2 5 TfTc ,Ž 2.

and the zero temperature parallel coherence lengthj 5 Ž0. is determined byTc Ž yd HC 2 H rdT . Tf Tc s f 0r2pj 52 Ž 0 . ,Ž 3.

where f 0 is the flux quantum. Estimated parametervalues for the present samples of MX 2 are summarized in Table 1. For g - 1.76, superconductivity
becomes quasi-2D and the dimensional crossovertakes place at TD . At TD , HC2 5 limited by the orbital
effect diverges and HC2 5 of MX 2 is determined
mainly by the Pauli-paramagnetism. In Fig. 1 wenotice that H C2 5 of 4Hb-Ta 0.95 Nb 0.05 S 2 ŽPy.1r2 behaves anomalously on temperature T, while H C2 5 of
4Hb-Ta 0.95 Nb 0.05 S 2 shows linear dependence on T.
But it must not be overlooked that H C2 H of bothintercalated and non-intercalated 4Hb-Ta 0.95 Nb 0.05 S 2shows quite anomalous dependence on T. In order to
see the anomalous behavior of HC2 H of MX 2 moreclearly, Fig. 2 presents the normalized critical fieldh C2 H s HC2 H Ž t .rŽyd HC2 H rdt . vs. the reduced
temperature t s TrTc together with examples of 2H
polytypes. In this figure, a distinct upward curvature
is noticed for h C2 H of intercalated compounds andnon-intercalated 4Hb polytypes. We denote this typeof anomaly as positive curvature anomaly ŽPCA..
The h C2 H curve of 2H-Ta 0.7 Nb 0.3 S 2 does not show
clear PCA but it shows a minor anomaly, i.e., h C2 H
ŽT . follows almost T-linear dependence down tovery low temperatures Žlinearity anomaly: LA., exceeding the WHH curves in low temperature regions.
First, we discuss the origin of PCA. Now it is
widely recognized that the Anderson localization
damages superconductivity mainly through the enhancement of the coulomb repulsion uU between
electrons and the depression of the electron densityof state N Ž0. w15x. Taking account of quantum corrections due to the Anderson localization, Maekawa,
Ebisawa and Fukuyama ŽMEF. w16x calculated HC2 H
of dirty 2D systems and predicted the occurrence of
the PCA anomaly. According to MEF, PCA is caused
mainly by the enhanced coulomb repulsion uU in the
weak localization regime, which is weakened in theperpendicular magnetic field. Fig. 3 shows the temperature dependence of the electrical resistance of
2H-Ta 0.7 Nb 0.3 S 2 ŽPy.1r2 which was reacted for two
weeks in an ampoule. The two-week reaction resulted in significant destruction of the sealed compounds and the resistance measurement was performed on a fragment of the obtained samples. In
Fig. 3, the resistance minimum is discernible atabout 15 K, which is consistent with the occurrenceof the Anderson localization. Although PCA occurred for all the Py intercalation compounds, theresistance minimum was observed only in the fullyintercalated fragments of partially destroyed compounds with small residual resistance ratio ŽRRR 2.. It seems that very slight imperfect intercalation
quite easily masked out the resistance minimum.
However, the existence of the resistance minimum
supports the Anderson localization as the origin of
PCA in HC2 H .
From the viewpoint of the Anderson localization
as the origin of PCA, the behavior of HC2 H andHC2 5 of nonintercalated 4Hb-Ta 0.95 Nb 0.05 S 2 isworthwhile to reconsider. In Fig. 1, PCA is observable only in H C2 H of 4Hb-Ta 0.95 Nb 0.05 S 2 , which is

–
0
2D
–

–
0
2D
–

–
0
2D
–

145

Fig. 2. The normalized perpendicular critical field h C2 H s
HC2 H Ž t .rŽyd HC2 H rd t . t ; 0.95 vs. reduced temperature t s
T r Tc for sulfide compounds. `: 4Hb-Ta 0.95 Nb 0.05 S 2 , \: 4HbTa 0.95 Nb 0.05 S 2 Ž Py . 1 r 2 , v : 2H-Ta 0.7 Nb 0.3 S 2 , B: 2HTa 0.7 Nb 0.3 S 2 ŽPy.1r2 . The solid and dotted curves are the WHH
theoretical curves for the dirty and pure limit. The bold dash-dotted. is the calculation curves of Maekawa etline Ž
–––al. w16x for l s 0.05 and l1 s 0.02 Žsee text..

23
62
3D
4.9
16
37
3D
5.5
See text.
a

2.1
0.16
q-2D
4.1

˚.
j H Ž0. ŽAgDimension
RRR

3.4
0.42
q-2D
3.4

3.2
0.35
q-2D
1.9

13
24
3D
4.6

25
;100
3D
7.6

33
)100
3D
33

21
43
3D
6.9

–
0
2D
–

–
–
6.30
`
4.4
–
LA
5.02
`
5.6
–
PCA
3.51
`
5.3

50r30
40r40
30r20

–
PCA
3.01
`
5.0
0.2
2H
LA
5.37
3.3
8.0
0.0
2H
LA
7.29
2.4
7.5
0.3
2H
LA
4.55
7.5
5.55
0.05
2H
LA
4.02
4.0
7.30
0.05
2H
PCA
3.14
70
1.65
0.05
4Hb
PCA
3.16
135
1.35

d Nb r dAl 2 O 3X
Polytype
HC2 H anomaly a
TC ŽK.
Ž HC2 5 r HC2 H . t ; 0.9
Žyd HC2 H rdT . t ; 0.9ŽkOerK.

Table 1
Material parameters

Ta 1y X Nb X S 2 ŽPy.1r2

0.3
2H
PCA
3.08
80
1.50

0.05
4Hb
PCA
2.92
14
3.30

NbrAl 2 O 3
Nb1yX Ta X Se 2
Ta 1yX Nb X S 2

0.1
2H
LA
6.42
3.0
8.3

70r30

M. Ikebe et al.r Physica C 317–318 (1999) 142–148

a 3D superconductor Žsee Table 1.. HC2 5 ŽT . of thiscompound follows T-linear relation in the measured
temperature range and does not show any anomaly.
This fact can be qualitatively understood on the basisof the Anderson localization. Under the magnetic
field parallel to the layer, the electron orbital motionin the plane perpendicular to the field is subject tosevere restriction because of the very large electric
resistivity perpendicular to the layer plane. The delocalization effect due to the magnetic field which is
the origin of PCA is far easy to occur in the HC2 H
geometry.
As we have mentioned in Section 3, 4Hb polytypecontains similar octahedral layers as 1T-TaS 2 which
consists of only octahedral layers. 1T polytype of
TaS 2 is a typical material which shows the Anderson
localization w17x. It is not unreasonable to assume
that the octahedral layer is responsible for the Anderson localization also in 4Hb-Ta 0.95 Nb 0.05 S 2 . The experimental data of the 3D superconductor, 4Hb-

146

M. Ikebe et al.r Physica C 317–318 (1999) 142–148

Fig. 3. The resistance minimum observed in 2H-Ta 0.7 Nb 0.3 S 2 ŽPy.1r2 . The temperature dependence of the resistance is consistent with the log T behavior for the case of the 2D weakAnderson localization between 6 K and 12 K.

Fig. 4. h C2 H vs. t for NbrAl 2 O 3 multilayers. A clear positive
˚ The bold
curvature is noticed for Nb layer thickness d Nb F 40 A.
. is the calculation curve ofdash-dotted line Ž–––
Maekawa et al. w16x Ž l s 0.05, l1 s 0.02..

Ta 0.95 Nb 0.05 S 2 , clarify that PCA is related to theorbital motion of electrons under magnetic fields andis not directly related to the dimensionality of superconductivity.
4.2. HC2 H anomalies in Nb r Al 2 O3 superlatticeAnomalous behavior of HC2 H is not peculiar
only to naturally occurring layer compounds MX 2 ,but artificially prepared multilayer superconductors
also show anomalous behaviors. In this subsection,
we show and discuss HC2 H anomalies of the superconductorrinsulator multilayer NbrAl 2 O 3 w7,18x.
Fig. 4 shows the renormalized perpendicular criticalfield h C2 H vs. the reduced temperature t. Withdecreasing Nb layer thickness Ž d Nb ., the anomalous
temperature dependence of HC2 H is enhanced and˚ and d Nb s 30 A˚ shows PCA
h C2 H of d Nb s 40 AŽ.
similar to MX 2 Py 1r2 . With increase of d Nb , PCA
disappears and linearity anomaly ŽLA. appears for˚ Fig. 5 shows the sheet conductivity G I
d Nb s 50 A.
per Nb sublayer as a function of d Nb . G I vs. d Nb

Fig. 5. The sheet conductance G I vs. Nb layer thickness d Nb .
˚ and G I is very
G I shows an inflection point at around 40 A˚small for d Nb F 40 A.

M. Ikebe et al.r Physica C 317–318 (1999) 142–148

˚ and thehas a inflection point at around d Nb ; 40 A
sheet resistance R I Žs 1rG I . is very large for˚ This fact suggests that PCA ofd Nb F 40 A.
˚ alsoNbrAl 2 O 3 , which occurs only for d Nb F 40 A,
originates from the Anderson localization. Here, it
must be mentioned that PCA was observed only inthe fully decoupled 2D NbrAl 2 O 3 samples with an
infinite value of Žd HC 2 5rdT .T f Tc. For example, in
˚ .rAl 2 O 3 Ž20 A˚ . superlattice with quasi-2DNbŽ40 Asuperconductivity w7x, HC2 H showed only the linearity anomaly ŽLA..
4.3. Theoretical aspects of the Anderson localizationand HC2 H
According to the theory and calculation due to
Maekawa et al. w16x, HC2 H of 2D superconductors in
the presence of the Anderson localization is determined by the following equations:
ln

T
Tc 0

sc

1

1

ž / ž
yc

2

q

eD 0 HC 2

2

2p Tc

ž

1 y llnq R HF q R V ,R HF s y

l1
2

1

= c

RV sy

l1
3

= c

2

ln
1
2

y l1 ln

Tt 0

yc

1

1

q

yc

g 1 N Ž 0.
2p´ F t 0,

Ž 8.

where N Ž0. is the electron density of state at ´ F andg 1 is the long ranged Coulomb interaction Ž g 1 ) 0..
The Tc depression in the absence of the magnetic
field is determined by the l1 value. In Figs. 2 and 4,
the result of the calculation with l s 0.05 and l1 s
0.02 is presented for comparison. The calculation
curve reproduces the observed PCA quite well. However, the validity to use the parameter values, l s
0.05 and l1 s 0.02 has not been established andmust be examined carefully. Owing to complex distribution of the electrical current, to obtain unambiguous data on transport properties is usually very
difficult in layered compounds and in superlattices.
4.4. On the origin of the linearity anomaly (LA)
Almost all nonintercalated MX 2 compounds w4xŽexcept 4Hb-Ta 0.95 Nb 0.05 S 2 with PCA. and some of
the NbrAl 2 O 3 multilayer samples w7x show linearity

y l1 ln

q

2

,

2p Tc

2

1

Tt 0

eD 0 HC 2

3

Tt 0

//

l1 s

1

ž / ž /
ž / ž/
ž / ž /
ž / ž
/
ln

Tt 0

where ´ F is the Fermi energy and

Ž 4.

2

1

1

147

1

Ž 5.

2

Tt 0

eD 0 HC 2
2p Tc

.

Ž 6.

Here, R HF and R V are self-energy and vertex
corrections of the Cooper pair propagator. c Ž z . isthe digamma function, t 0 the electron relaxation
time, D 0 the electron diffusion constant, and TC0 is
the transition temperature in the absence of the Anderson localization. The two parameters l and l1
are given byls

1
2p´ F t 0,

Ž 7.

Fig. 6. h C2 H vs. t for selenide compounds. `: 2H-NbSe 2 , e:
2H-Nb 0.9Ta 0.1 Se 2 , ^: 2H-Nb 0.8Ta 0.2 Se 2 . The bold dash-dotted. represents the calculation of Maekawa
curve Ž–––et al. w16x for l s 0.05 and l1 s 0.01 Žsee text..

148

M. Ikebe et al.r Physica C 317–318 (1999) 142–148

anomaly. Examples of LA of selenide 2H polytypes
are shown in Fig. 6 and LA in 2H-Ta 0.7 Nb 0.3 S 2 and˚ .rAl 2 O 3 Ž30 A˚ . can be seen in Figs. 2 andNbŽ50 A4, respectively. According to the calculation of MEF,
the HC2 H ŽT . curve based on Eqs. Ž2., Ž4. – Ž6. reproduces LA if the parameter values l ; 0.05 andl1 ; 0.01 are adopted. Thus LA in alloyed MX 2
compounds and NbrAl 2 O 3 multilayers, which arepretty dirty superconductors, can be naturally understood as also an effect of the weak Anderson localization in anisotropic 3D superconductivity regime.
Then the main difference between PCA and LA
comes from the difference in l1 value, i.e., strengthof the Coulomb interaction g 1 N Ž0. in Eq. Ž8.. For
2H-Ta 0.7 Nb 0.3 S 2 , the depression of Tc caused byintercalation Ž4.55 ™ 3.08 K. is also consistent with
the increase of l1Ž0.01 ™ 0.02.. In Fig. 6, however,it must be noticed that 2H-NbSe 2 which is a pretty
pure 3D superconductor w4x also shows LA. In caseof 2H-NbSe 2 , it seems more reasonable to ascribe
LA to the anisotropies in the Fermi surface and theenergy gap as a result of the nonlocal effect w19,20x.
The origin of LA in layered superconductors is notunique and may depend on respective cases.

5. Summary
The anomalous behavior of the perpendicular critical field HC2 H of natural and artificial superconductors with layered structure was analyzed mainly fromthe viewpoint of the weak Anderson localization.
The conspicuous upward curvature ŽPCA. found in
Ta 1y X Nb X S 2 ŽPy.1r2 with quasi-2D superconductivity, 4Hb-Ta 0.95 Nb 0.05 S 2 with 3D superconductivity
˚ . with 2D
and NbrAl 2 O 3 multilayers Ž d Nb F 40 A
superconductivity is to come from the Anderson
localization and is consistent with the theory due to
Maekawa et al. w16x. The main origin of the less
spectacular linearity anomaly ŽLA. found in nonintercalation alloy compounds, 2H-Ta 1y X Nb X S 2 and2H-Nb1y X Ta X Se 2 with anisotropic 3D superconduc˚ can also
tivity and LA in NbrAl 2 O 3 for d Nb ; 50 A
be understood as a remnant effect of the Anderson
localization. But LA of 2H-NbSe 2 , which is a pureanisotropic 3D superconductor, should be understoodas to come from the anisotropies of the Fermi sur-face and the superconducting energy gap through theeffect of nonlocality. The origin of LA which is very
ubiquitous in layered superconductors is not uniqueand competition and interplay of several mechanisms
must be taken into account.

Acknowledgements
One of the present authors ŽM.I.. wishes to thankProfessor Maekawa and Professor Ebisawa for usefulconversations.

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