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Effects of pressure and magnetic field onsuperconductivity in ZrTe3: local pair-induced
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New J. Phys. 19 (2017) 063004

https://doi.org/10.1088/1367-2630/aa6d48

PAPER

OPEN ACCESS

Effects of pressure and magnetic ﬁeld on superconductivity in ZrTe3:
local pair-induced superconductivity

RECEIVED

23 January 2017
REVISED

3 April 2017

S Tsuchiya1,2 , K Matsubayashi2,3, K Yamaya1, S Takayanagi1, S Tanda1,4 and Y Uwatoko2
1
2

ACCEPTED FOR PUBLICATION

3

13 April 2017

4

PUBLISHED

2 June 2017

Department of Applied Physics, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Department of Engineering Science, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan

E-mail: satoshi.tsuchiya@eng.hokudai.ac.jp
Keywords: superconductivity, magnetic ﬁeld, pressure

Original content from thiswork may be used under
the terms of the Creative
Commons Attribution 3.0
licence.

Abstract
Any further distribution of In this work, the origin of the highly anisotropic superconducting transition in ZrTe3, where thethis work must maintainresistance along the a axis, Ra, is reduced at 4 K but those along the b axis, Rb, and c ¢ axis, Rc′, areattribution to theauthor(s) and the title ofreduced at 2 K, was explored with the application of a magnetic ﬁeld and pressure by the electricalthe work, journal citation
resistance measurements. We found that the behavior of the upper critical ﬁeld and its anisotropy as
and DOI.
well as the pressure dependence determined by the Ra measurements are quite similar to those of Rb.
Moreover, the excess conductivity for Rb indicates anomalous behavior. These results support an
unconventional origin for the anisotropic transition rather than conventional superconducting
ﬂuctuation. The reduction in Ra is due to ﬁlamentary superconductivity(SC) induced by locallybound electron pairs (local pairs), which correspond to bi-polarons, and the transition of Rb
corresponds to the emergence of bulk SC originating from the Cooper pairs triggered by the transferof the local pairs.

1. Introduction
Up to now, the coexistence and competition between charge density waves (CDWs) and superconductivity (SC)have been a major research topic in the ﬁeld of condensed matter physics. Indeed, CDW transitions have been
observed in low-dimensional materials such as transition-metal di- and trichalcogenides [1–4], which havechain and layered crystal structures, as well as cuprates [5–8]. Many compounds exhibit SC behavior at low
temperatures in the CDW ordering state [9, 10]. In some cases, the CDW and SC states are suppressed orenhanced by pressure, indicating that their peculiar electronic structures due to low dimensionality can play a
crucial role with regard to CDW and SC coexistence and competition. It is of interest and value to reveal themechanism of such CDWs and SC.
Among these compounds, ZrTe3 has attracted considerable attention owing to its unique electronic
structures, which originate from the low dimensionality of the crystal structure [11, 12]. Strikingly, when thetemperature is reduced below the CDW transition temperature TCDW (=63 K), a highly anisotropic resistivetransition in the SC has been observed. The resistance Ra along the a axis starts to decrease at 4 K, whereas theresistances Rb and Rc′ along the b and c ¢ axis, respectively exhibit relatively rapid reductions at 2 K [13, 14]. Since
magnetic measurements have shown that a diamagnetic signal is not observed down to 2 K [14], the reduction of
Ra can arise from ﬁlamentary SC rather than bulk SC. This unusual behavior can be explained by two scenarios,
conventional superconducting ﬂuctuation and SC induced by locally bound electron pairs (local pairs). In theformer, the reduction in Ra is attributed to a ﬂuctuation in the bulk SC enhanced by the low dimensionality. In
the latter, the transitions of Ra and Rb, c ¢ correspond to ﬁlamentary and bulk SC, which originate from theformation of local pairs, namely bi-polarons, and Cooper pairs induced by the transfer of the local pairs,
respectively, owing to a peculiar electronic structure after the CDW transition [15]. At this moment, it remains
unclear which interpretation is acceptable. Moreover, although formation of the local pairs has been discussed as© 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

New J. Phys. 19 (2017) 063004

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Figure 1. (a) Schematic of ZrTe3 crystal structure. The unit cell parameters are a=5.8948 Å, b=3.9264 Å and c = 10.109 Å andβ=97.32° [11]. (b) Schematic of ZrTe3 Fermi surface [12].

a possible mechanism of high-temperature SC for a long time [16–18], experimental indications are still limited.
In this sense, investigations on SC in ZrTe3 can provide important insights regarding the presence of local-pair
induced SC.
To understand SC in more detail, we discuss the inﬂuence of pressure and the magnetic ﬁeld on the SC in
ZrTe3, as determined via Ra and Rb measurements. The experimental results are shown in section 3. In
section 3.1, the pressure–temperature (P–T) phase diagram is constructed. The behaviors of the upper criticalﬁelds Hc2 are discussed using the Ginzburg–Landau (GL) theory in sections 3.2 and 3.3. We found that thetemperature and pressure dependences of Hc2 obtained by Ra measurement is similar to those obtained by Rb
measurement. Moreover, to explore the peculiar properties of the ﬁlamentary SC, the Hc2 curves and Hc2
anisotropy are assessed with several models based on the GL theory. Additionally, in the analysis of thesuperconducting ﬂuctuation with the Aslamazov and Larkin (AL) theory (section 3.4), the excess conductivity of
Rb seems to have a one-dimensional (1D) character, although the bulk SC is observed in this current
conﬁguration. As discussed in section 4, by taking into consideration previous studies, we conclude that our
results indicate unconventional origins for the anisotropic superconducting transition, which is attributed to
local-pair-induced SC rather than conventional superconducting ﬂuctuation.

2. Experimental
Single crystals of ZrTe3 were prepared using the iodine (I2) vapor transport method [15]. The crystal structure isshown in ﬁgure 1(a). Chains consisting of ZrTe3 prisms formed along the b axis. In the ﬁgure, upright andinverted chains are alternately arranged along the a axis, forming layers. The layers are stacked perpendicularly
to the ab plane (along the c ¢ axis), where the c ¢ axis is tilted slightly from the c axis because of the monoclinic
crystal structure. Band structure calculations predicted quasi-one-dimensional (Q1D) Fermi surface (FS) sheets,a three-dimensional (3D) sheet, and a van Hove singularity (vHS), which consists of the intersection of Q1D and
3D (Q1D+3D) FSs, [12] as shown in ﬁgure 1(b).
The electrical resistivity was measured using a standard four-terminal method. The typical crystal sizes usedin the experiments conducted in this study were 1×0.5×0.01 mm3. Four gold wires 20 μm in diameter wereattached to the cleaved surfaces of the samples using carbon paste. To detect the anisotropic SC transition, Ra
and Rb were measured with 0.1 mA of AC current directed along the a axis for samples 1 and 3 and along the baxis for 2 and é4, respectively. For the measurements conducted under ambient pressure, 1 and 2 were
mounted on the same sample holder and the data were recorded simultaneously. For the pressure
measurements, samples 3 and 4 and lead (Pb), which was used as the manometer [19], were set in a pistoncylinder-type pressure cell with the pressure medium (Daphne 7373). The experiments were carried out using
3
He and dilution refrigerators with an SC magnet.

2

New J. Phys. 19 (2017) 063004

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Figure 2. (a), (b) Temperature dependence of resistance for various pressures in I  a (1 and 3) and I  b (2 and 4), respectively. (c),(d) Temperature dependence of resistance in low temperature region on logarithmic scales.

3. Results
3.1. P–T phase diagram
At room temperature, the resistivities of all ZrTe3 samples were roughly estimated under 0 kbar. as 1.5, 1.0, 0.4
and 1.0×10−6 Ω m−1 for é1–é4, which are in good agreement with those of the previous report [20].
Figures 2(a) and (b) show the temperature dependence of the Ra and Rb under various pressures, respectively.
For I  a , the resistance reduced with decreasing temperature and then suddenly increased at approximately
70 K. No visible anomaly was observed for I  b . This behavior is attributed to the CDW transition. Further, as
the CDW forms along the direction between the a and c axes [21], the anomaly in the Rb is extremely small [14].
In this study, when pressure was applied, TCDW shifted to the high-temperature side.
Figures 2(c) and (d) show the Ra and Rb at various pressures in the low-temperature region, respectively,
which are plotted as a function of temperature on a log scale. For 0 GPa, the Ra drops to zero at 2.2 K, while the
Rb decreases to zero at 1.4 K. The highly anisotropic SC transitions are reﬂected by the ﬁlamentary and bulk
nature of the SC. Note that the ﬁlamentary SC occurs along the a axis and no visible anomaly in the speciﬁc heat[15]. On the other hand, the zero resistance for I  b at 1.4 K corresponds to the bulk SC transition, where a
speciﬁc heat anomaly has previously been observed [15]. For I  a , the small drops of resistance were observed
at 0.50 GPa. At 0.82 GPa, however, no reductions in resistance were observed, even at temperatures lower than
0.1 K. On the other hand, for I  b , no resistance drops were observed at pressures higher than 0.50 GPa.
A P–T phase diagram can be constructed from the above data, as shown in ﬁgure 3. Note that TCDW wasdeﬁned as the onset temperature of the resistance anomaly. For the SC transition, we deﬁned Tc (onset) andTc (R=0) as the intersection between the extrapolation from the high temperatures and the tangent line at halfof the normal state resistance, and the extrapolation of the tangent line to zero resistance, respectively, as shownin the inset of ﬁgure 4(c). As the pressure was increased, TCDW also increases, while the Tc of the ﬁlamentary SC
reduced; these results conform to behavior reported in the previous study [22]. This behavior and the resistivity
agreement at room temperature ensure measurement reproducibility of the measurements. Strikingly, the Tc of
the bulk SC also decreases in the same manner as that of the ﬁlamentary SC. This means that the origin of thebulk SC can be related to that of the ﬁlamentary SC. Moreover, as Tc seems to be negatively correlated with the
TCDW , both the SC phases can compete with the CDW phase.
3New J. Phys. 19 (2017) 063004

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Figure 3. Temperature-pressure phase diagram of ZrTe3. The triangles and squares indicate Tc and TCDW , respectively. The closed andopen symbols indicate the Tc determined by the onset and resistance drop temperatures, respectively.

Figure 4. (a), (b) Temperature dependence of resistance under various magnetic ﬁelds for I  a (é1) and I  b (é2), respectively. (c),(d) Magnetic-ﬁeld dependence of resistance under various magnetic ﬁelds for I  a (é1) and I  b (é2), respectively. The inset shows
the deﬁnition of Hc2 (onset) and Hc2 (R=0).

3.2. Upper critical ﬁelds
Figures 4 (a) and (b) show the temperature dependence of Ra and Rb, respectively, for various magnetic ﬁelds,
which were applied with respect to the ab plane (along the c ¢ axis) under ambient pressure. When the magnetic
ﬁeld increases above 0.2 T for I  a , Tc is shifted to the low temperature side and the transition became broad.
4

New J. Phys. 19 (2017) 063004

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Figure 5. (a), (b), (c) Phase diagram of ZrTe3 for the magnetic ﬁeld H // c under P=0, 0.12 and 0.37 GPa, respectively. Circles anddiamonds indicate Hc2 (R=0) and Hc2 (onset) for I  a , respectively. Triangles indicate Hc2 (R=0) and Hc2 (onset) for I  b . Closed
and open symbols mean that Hc2 is determined by use of the magnetic ﬁeld and temperature dependence measurements, respectively.
Dashed, dashed-one dotted, and dashed-two dotted lines indicate the results of ﬁtting of the GL, coupled ﬁlament SC, and isolated
ﬁlament SC model, respectively.

Moreover, for I  b , Tc decreases with increasing magnetic ﬁeld. Figures 4 (c) and (d) show themagnetoresistance results for several temperatures for I  a and I  b , respectively. For I  a , the resistancecurve exhibits a steep increase at magnetic ﬁelds 0.2 T at 0.4 K, while for I  b , this increase occurs for magnetic
ﬁelds above 0.04 T. Using the criteria shown in the inset of ﬁgure 4(c), we deﬁned the upper critical ﬁeld
Hc2(onset) and Hc2(R=0). Figure 5(a) shows the phase diagram for H  c ¢. The open and closed symbols
represents the Hc2 derived from the temperature and the magnetic-ﬁeld dependence measurements,
respectively.
The upper critical ﬁeld can be associated with two different types of suppression mechanism of SC, i.e. thePauli paramagnetic and orbital effects. In the former mechanism, the SC states are broken by Zeeman splitting
for spin singlet pairings [23]. For Bardeen–Cooper–Schrieffer (BCS) superconductors, the Pauli paramagnetic
limit can be obtained as HP = D 2 m B = 1.84Tc , where m B is the Bohr magneton and Δ=1.76kB Tc is the
BCS energy gap in the weak coupling limit. For example, by substituting Tc = 3.4 K for Hc2(onset) with I  a ,m0 HP is estimated as 6.07 T, which is signiﬁcantly larger than the Hc2(onset) value of ∼0.9 T obtained from linear
extrapolation to T=0 K. On the other hand, in the latter case, the SC is suppressed by increased kinetic energy
5New J. Phys. 19 (2017) 063004

S Tsuchiya et al

orb
Table 1. Slopes of Hc2 curve near Tc (dH/dT), Tc , and Hc2(0) for P=0, 0.12, and 0.37 GPa, respectively.

0 GPa

Tc
(K)
I // b
I // aHc2 (R=0)
Hc2 (onset)
Hc2 (R=0)
Hc2 (onset)

1.37
1.87
2.2
3.4

m0 dH
dT T = Tc−1( )0.12 GPa(T K )

orb
Hc2
(0)
(T)

HP
(T)

Tc
(K)

−0.043−0.081−0.11−0.341

0.04
0.11
0.18
0.80

2.52
3.44
4.14
6.07

0.9
1.4
1.4
2.8

m0 dH
dT T = Tc−1( )0.37 GPa(T K )

orb
Hc2
(0)
(T)

Tc
(K)

−0.021−0.074−0.064−0.241

0.02
0.096
0.10
0.55

—
0.63
0.6
1.5

m0 dH
dT T = Tc−1( )(T K )

orb
Hc2
(0)
(T)

—
−0.043−0.041−0.106

—
0.019
0.017
0.11

due to the orbital current to exclude the magnetic ﬁeld. The upper critical ﬁeld determined by the orbital effect atT=0 K is given by [24]⎛ m dHc2 ⎞orb
m 0 Hc2(0) = - 0.69Tc ⎜ 0⎟,⎝ dT ⎠T = Tc
m dH

(1)

m dH

where 0dT c2 is the initial slope at Tc . For Hc2(onset) with I  a , 0dT c2 ~ -0.341, which provides
Hc2orb (T=0)∼0.8 T. The Hc2 curve was ﬁtted using this value and the empirical expression based on the GL
theory [25], which provides the phase boundary determined by the orbital effect. The GL expression is expressedasorb
Hc2 (T ) = Hc2(0)

1 - t2,
1 + t2(2)where t is the reduced temperature T Tc . The results of the calculations are in good agreement with the
Hc2(onset) and Hc2(R=0) curves for I  a and I  b over the entire temperature range, as shown in ﬁgure 5(a).
The m0 HP and m0 Hc2orb(T=0) results for all the Hc2(onset) and Hc2(R=0) values are summarized in table 1.
These results indicate that the orbital effect is dominant for suppression of both the bulk and ﬁlamentary SC,
when a magnetic ﬁeld is applied.
To investigate the ﬁlamentary SC further, the Hc2 curves were compared with those predicted by a model
based on GL theory [26]. This theory proposes two types of ﬁlamentary SC, where the SC ﬁlaments are either
coupled or isolated. In the former case, the SC ﬁlaments interact via the Josephson coupling and the I paths canbe enclosed. As a result, Hc2 is expected to be proportional to ∣(T - Tc ) Tc∣ near Tc . On the other hand, SC
ﬁlaments are decoupled in the latter case. Thus, the screening I path is not enclosed, leading to suppression of theorbital effect. In that case, Hc2 is proportional to ∣(T - Tc ) Tc∣1 2. Therefore, the difference between these two
types of ﬁlamentary SC is most signiﬁcantly pronounced near Tc . As shown by the plot in ﬁgure 5(a), the coupledmodel ﬁts the obtained data more closely than the isolated model. Further, the deviation between the coupledmodel and our data at low temperatures is irrelevant because the models are valid near Tc only.
The H–T phase diagrams obtained under 0.12 and 0.37 GPa are shown in ﬁgures 5(b) and (c), respectively.
The Hc2 curves are quite similar to that for 0 GPa and ﬁt the GL expression (equation (2)) well. Therefore, them dH
suppression mechanism of SC for application of the magnetic ﬁeld is unchanged by pressure. All the 0dT c2 andHc2orb (0) values are also summarized in table 1.

3.3. Angular dependence of Hc2
For the ﬁlamentary SC, the angular dependence of Hc2 was measured. Figure 6(a) shows the magnetic-ﬁeld
dependence of the resistance for various angles at 0.4 K under 0 GPa. The ﬁeld angle f was deﬁned as the angle
between the b axis and magnetic ﬁeld, as shown in the inset of ﬁgure 6(b). When the magnetic ﬁeld was tilted
from the 90° direction (H  c ¢), the resistive transition was shifted to the high-ﬁeld side. Figure 6(b) shows thef dependence of Hc2(R=0) for I  a . The Hc2(R=0) values in the vicinity of 0° are approximately twotimes greater than those at high f, indicating anisotropic SC. We tentatively compared the data with two
different theoretical models, the anisotropic GL (AGL) model [27] and the two-dimensional (2D) model. For
the AGL model, the anisotropy of SC is included as that of Hc2 or the effective mass. The f dependence of Hc2
is described as⎛ H (f) sin (f) ⎞2⎛ Hc2 (f) cos (f) ⎞2⎟⎟ = 1,⎜⎟ + ⎜⎜ c2 
^
Hc2
⎝
⎠
Hc2
⎝
⎠

(3)


^
where Hc2
and Hc2
are the upper critical ﬁelds for the perpendicular and parallel ﬁeld directions relative to SC
layers, respectively. On the other hand, in the case of 2D superconductors, the layers of which are connected via
Josephson coupling, the f dependence of the critical ﬁeld is given by [28]6New J. Phys. 19 (2017) 063004

S Tsuchiya et al

Figure 6. (a) Magnetic-ﬁeld dependence of resistance for various f at 0.4 K under 0 GPa. (b) Angular dependence of Hc2 (R=0) forI  a . The inset shows the deﬁnition of f.

Hc2 (f) cos (f)^Hc2⎛ H (f) sin (f) ⎞2⎟⎟ = 1.
+ ⎜⎜ c2 
Hc2
⎝
⎠

(4)

The solid and dotted curves in ﬁgure 6(b) correspond to the calculated results of the AGL and 2D models,
respectively. At low f, the Hc2(R=0) values are close to those of the AGL model (equation (3)), but tend toagree with the 2D model (equation (4)) at high f. Remarkably, the anisotropy direction, as well as the^g = Hc2
Hc2~ 2.2 in the ﬁlamentary SC, agree reasonably well with the result of γ∼3 obtained for the bulk
SC [14], indicating similarity between the ﬁlamentary and bulk SCs.
To explore the peculiar properties of the ﬁlamentary SC, we compared the azimuthal f dependence datawith the theoretical model [26]. As the ﬁlamentary SC model is very simple [26], it can provide useful clues
concerning the behavior of the ﬁlamentary SC. In the case that SC ﬁlaments are arranged in a rectangular lattice,
Hc2 is expected to exhibit a unique f dependence of Hc2, with a minimum value for the azimuthal angle f=45°
well below Tc . This behavior corresponds to variation of Hc2 with a 90° period. On the other hand, near Tc , the fdependence is identical to that of the AGL model (equation (3)). In our results for T Tc ∼0.4 (ﬁgure 6), Hc2varied with a 180° period which is consistent with the AGL model. This can indicate that the SC ﬁlaments cannot
be arranged regularly.
3.4. Excess conductivity
To investigate the SC ﬂuctuation, we extracted the excess conductivity from the resistance data. Figures 7 (a) and(b) show the temperature dependence of the normalized excess conductivity Ds s0 =(σ– s0)/ s0, where s0 is
the conductivity under various pressures, for I  a (at 5 K) and I  b (3 K), respectively. Tc (R=0) values wereused. The Ds s0 curves were found to be qualitatively similar for pressure variations in both current
conﬁgurations. According to the AL theory [29], the excess conductivity due to SC ﬂuctuation is given by⎛ T - Tc ⎞- 2
Ds3D µ ⎜⎟ ,⎝ Tc ⎠(5)⎛ T - Tc ⎞- 2
Ds1D µ ⎜⎟ ,⎝ Tc ⎠(6)

1

3

for 3D and 1D superconductors, respectively. The dashed lines in ﬁgure 7 are calculations for the 3D and 1D
cases, which have been included to facilitate a qualitative comparison in the temperature dependence. For I  a ,the experimental results agree with the 1D AL model, which is consistent with a previous report [30]. For I  b ,the temperature dependence corresponds to the 1D rather than the 3D model, although the bulk SC is observedin this I conﬁguration.
7

New J. Phys. 19 (2017) 063004

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Figure 7. (a), (b) Temperature dependence of normalized excess conductivity for I  a and I  b , respectively. The dashed lines
indicate the results calculated for 1D and 3D cases using the AL theory.

4. Discussion
We now discuss the origins of the bulk and ﬁlamentary SCs from the perspective of the electronic structure.
Recent ARPES measurements have shown that the Q1D FSs vanish as a results of the CDW formation (seeﬁgure 1(b)) and, simultaneously, a large enhancement in the electronic density of states near the vHS can beobserved at the Fermi energy (EF ) below TCDW [31]. As has been shown in ﬁgure 3, TCDW increases when the Tc
values of both SCs decrease with increasing pressure. This indicates that the remaining Q1D+3D FSs are related
to the emergence of the SC, rather than the 3D FSs.
In the band calculations [12], the energy dispersions of the Q1D+3D FSs in the vicinity of the vHS are almost
ﬂat, corresponding to vF∼0, where vF is the Fermi velocity. It is known that a vHS at EF can have FS instabilities
that introduce a SC state [32, 33]. If electron pairs of Q1D parts of the FSs are formed, the spatial size of thevcorresponding wave packet can be estimated based on the Pippard coherence length x0 ~ k TF , where ÿ and kB
B cdenote the Planck and Boltzmann constants, respectively [25]. As vF∼0, x0 is expected to be very small, yielding
electron pairs with very short coherence length that tend to be localized. Such local pairs correspond to bipolarons consisting of electrons bound by strong electron–phonon interaction [16–18]. As they have Bosecharacters, the local pairs can condense into a SC state analogous to the superﬂuid state of 4He II [34].
In ZrTe3, the local pairs form along the a axis; this is because Q1D FS consists of electrons in the narrow band
originating from the px orbitals of the Te(2)–Te(3) chain parallel to the a axis, as has been shown in ﬁgure 1(a).
With decreasing temperature, local pairs are induced and phase coherence develops gradually, causing broad
resistive transition along the a axis. In this manner, the unique electronic structure of ZrTe3 favors SC as a resultsof the local pair formation.
Furthermore, several theories have predicted that, in the case of narrow-and wide-band hybridization, thelocal pairs in the narrow band can induce Cooper pairs spontaneously among the electrons of the wide band
[16, 17]. In ZrTe3, as the 3D FS is derived from wide-band electrons, such hybridization is realized in the vicinityof vHS [12]. If a large number of local pairs are induced as the temperature decreases, the local pairs of thenarrow band are transfered to the wide-band states of the Q1D+3D FSs, inducing Cooper pairs. This behavior
yields the bulk SC. In fact, this picture corresponds to a crossover between the ﬁlamentary and bulk SC withvarying temperature, which explains the anomalous behavior of the speciﬁc heat well, as was discussed in aprevious report [15].
The inseparable relationship between the two types of SC induced by the local and Cooper pairs is consistent
with the implications of our results. As shown in ﬁgure 5, the Hc2 curves of the ﬁlamentary SC were quite similar
to those of the bulk SC. Moreover, the ﬁlamentary SC phase was decreased in the same manner as the bulk phase,when the pressure was increased. As regards the anisotropy of Hc2, the direction and γ value of the ﬁlamentary
SC are in agreement with those of the bulk SC.
In ZrTe3, the onset temperature of Ra is much higher than that of Rb by ∼3 K. This behavior indicate thatbulk SC coherence is not easy to establish because coupling between the SC ﬁlaments is very weak because of a8

New J. Phys. 19 (2017) 063004

S Tsuchiya et alvery short coherence length. This is consistent with their extremely short coherence length of less than 10 Å forthe ﬁlamentary SC, which suggested by the speciﬁc heat measurements [15].
On the other hand, a similar two-stage resistive transition of SC has been reported in the Q1D materials
family such as Tl2Mo6Se6 and Na2Mo6Se6 [35, 36]. As temperature decreases, the resistivity reduces but does not
become zero in Tl2Mo6Se6. This behavior can be explained by appearance of SC ﬂuctuation owing to a highly 1D
crystal structure. When the temperature is lowered further, such 1D SC bundles become coupled with eachother and establish 3D phase coherence, showing the zero-resistive transition due to the bulk SC. However, theonset temperature of the SC ﬂuctuation is higher than that of the bulk SC by only ∼1 K, indicating that 3D SC
correlation develops more easily than it does in ZrTe3. Therefore, the coherence length of the SC ﬂuctuation is
expected to be longer than that in ZrTe3. This ﬁnding suggests that the ﬁlamentary SC in ZrTe3 can arise fromthe local pairs as a limit for a very short coherence length rather than the ordinary Cooper pairs.
Our results regarding the excess conductivity are in agreement with the 1D AL model rather than the 3Dmodel, for both the I conﬁgurations. This agreement is reasonable for I  a , because the SC channel develops
along the a axis, indicating that the ﬂuctuation is derived from the highly 1D SC. On the other hand, for I  b ,the disagreement with the 3D model is unexpected, as the bulk SC can originate from the 3D FS. Similar
behaviors have been observed in the cuprates and suggest the existence of 1D conducting channels in thecompound [37]. However, as such a 1D conducting channel is expected along the a axis in ZrTe3, this ﬁnding
cannot explain our results. Instead, this anomalous behavior may indicate that the conventional ﬂuctuation
theory is not appropriate for our data. According to the local-pair induction scenario, the bulk SC emerges in
addition to the ﬁlamentary SC. Thus, we speculate that such this mixing of two kinds of SC, that is, theunconventional mechanism of the bulk SC increases the complexity of the ﬂuctuation phenomena. Thisbehavior can yeild unusual excess conductivity, corresponding to deviation from the 3D AL model.
Here, we summarize the above discussions and consider the origin of the ﬁlamentary SC. In our results, thepressure dependencies of Tc (ﬁgure 3) and Hc2 (ﬁgure 5), as well as the magnetic ﬁeld angular dependence of Hc2(ﬁgure 6) of the ﬁlamentary SC are similar to those of the bulk SC, indicating a connection between these types ofSC. These discussions are not sufﬁcient to conclude whether the ﬁlamentary SC can be regarded as a ﬂuctuationin the bulk SC enhanced by low dimensionality alone or local-pair-induced SC. However, the discussion of theelectronic structure based on the band structure calculation [12] and ARPES measurements [31] indicates that
the Q1D+3D FSs after the CDW transition are relevant to an SC emergence that favors local-pair formation.
The comparison with the SC transition in Tl2Mo6Se6 suggests that the ﬁlamentary SC in ZrTe3 can be attributed
to Cooper pairs characterized by extremely short coherence lengths. Moreover, the excess conductivity observedin this study (ﬁgure 7) and the anomalous speciﬁc heat [15] suggest that bulk SC ﬂuctuation does not agree withconventional ﬂuctuation theory. Therefore, from a comprehensive perspective, it is reasonable to conclude that
our results favor the local-pair-induced SC scenario rather than conventional SC.
The Hc2 of the ﬁlamentary SC is worth noting as it can reﬂect the characteristics of the local pair-induced SC.
Usually, for Josephson-coupled superconductors, the Hc2 values at low temperatures are large. In previous
studies, they exceeded Hp [38, 39] when a magnetic ﬁeld was applied to the Josephson-coupled SC chains orlayers, because the orbital current path was enclosed by the Josephson junctions, suppressing an increase in the
kinetic energy. In contrast to the behavior reported in those studies, although the SC ﬁlaments were coupled by
the Josephson effect in this study, the Hc2 curves indicate that the orbital pair-breaking effect played a dominantrole in the SC. The weakness of the ﬁlamentary SC under the inﬂuence of a magnetic ﬁeld may be unique to
ZrTe3. As the local pairs are conﬁned within narrow SC ﬁlaments having extremely small cross-sectional area (asa result of highly 1D electronic structure of ZrTe3), a critical value for the screening current can be obtained
more easily than for other high-dimensional superconductors. This may be the reason why the Hc2 curves of theﬁlamentary SC are dominated by the orbital pair-breaking effect.
Several theoretical models of local pair-induced SC have predicted that Hc2 is determined by the Pauliparamagnetic effect [40] and that the Hc2 curve has upward curvature and shows no saturation behavior at low
temperatures [41]. However, our results show that the Hc2 curves are in good agreement with those expectedbased on consideration of the orbital effect. This discrepancy may be due to the difference in dimensionality. In
all the theoretical models, a 2D electron system is assumed. However, in ZrTe3, FSs in which electrons have Q1D
characteristics play a crucial role. As has been discussed above, the small critical current in the ﬁlamentary SC
can make a signiﬁcant contribution to the behavior of the corresponding Hc2.
The small critical current of the ﬁlamentary SC may have an effect on the magnetic response. Previous
magnetic measurements have shown that a diamagnetic signal is not observed down to 2 K [14]. This may
indicate that SC ﬁlaments are not coupled, which seems to contradict our results. In the ﬁlamentary SC, thecritical value of the current can be very small, and the screening current will be small even if SC ﬁlaments arecoupled. This will cause difﬁculty in detecting the diamagnetic response in the ﬁlamentary SC.
9New J. Phys. 19 (2017) 063004

S Tsuchiya et al

5. Conclusion
By measuring the resistance of ZrTe3 with different I conﬁgurations under an applied pressure and magnetic
ﬁelds, we showed the ﬁlamentary and bulk SC nature, which shows resistive transition along the a axis at 4 K andalong the other axes at 2 K. The Hc2 curves, Hc2 anisotropy, and pressure evolution obtained from themeasurement for I  a were quite similar to those for I  b , indicating that the origin of the ﬁlamentary SC canbe linked to that of the bulk SC. Moreover, the bulk SC ﬂuctuation exhibited unusual behavior. The results
suggest that the emergence of ﬁlamentary and bulk SC is a consequence of local-pair formation and local-pair
induced Cooper pairs.

Acknowledgments
The authors would like to thank S Yasuzuka, K Ichimura and T Matsuura for helpful discussion.

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