Journal of Superconductivity and Novel Magnetism
https://doi.org/10.1007/s10948-019-05333-z

ORIGINAL PAPER

Highly Isotropic In-plane Upper Critical Field in the Anisotropics -Wave Superconductor 2H -NbSe2
Syuma Yasuzuka1 · Shinya Uji2 · Shiori Sugiura2,5 · Taichi Terashima2 · Yoshio Nogami3 · Koichi Ichimura4 ·Satoshi Tanda4
Received: 12 June 2019 / Accepted: 1 October 2019
© Springer Science+Business Media, LLC, part of Springer Nature 2019

Abstract
Resistivity measurements for a layered transition metal dichalcogenide 2H -NbSe2 were performed to investigate the inplane anisotropy of the upper critical field (Hc2 ) for magnetic fields up to 14.5 T at temperatures down to 2.0 K. For fields
rotated within the basal plane, the in-plane anisotropy of a characteristic field H ∗ (defined as the field strength at which theresistivity reaches zero) showed twofold symmetry at low temperatures. On the basis of this result, the in-plane anisotropyof Hc2 and the superconducting gap structure in 2H -NbSe2 are discussed.
Keywords 2H -NbSe2 · Upper critical field · Anisotropic s-wave superconductivity · Superconducting gap

1 Introduction
In quasi-two-dimensional (Q2D) transition metal dichalcogenides (TMDs), the physical properties originating fromstrong electron-phonon interactions and low dimensionality are of great interest. Among the many layered TMDs,
2H -NbSe2 (hereafter referred to as “NbSe2 ”) is one of themost intriguing systems [1–7] because a 2D charge-densitywave (CDW) phase transition occurs at TCDW = 33.5 K [8],followed by a phonon mediated superconducting (SC) transition at Tc = 7.2 K [9]. According to band-structure calculations [10], there are several FS sheets: a small Γ -centeredFermi pocket derived from the Se 4p band and larger nearly
2D FS sheets from the Nb 4d bands centered on the Γ (A)and K(H ) points. Experimental results of the de Haas−vanAlphen (dHvA) effect [10] and angle-resolved photoemission
 Syuma Yasuzuka

yasuzuka@cc.it-hiroshima.ac.jp
1

Research Center for Condensed Matter Physics, Hiroshima
Institute of Technology, Hiroshima 731-5193, Japan

2

National Institute for Materials Science, Tsukuba,Ibaraki 305-0003, Japan3

Department of Physics, Okayama University, Okayama
700-8530, Japan

4

Department of Applied Physics, Hokkaido University,
Kita 13, Nishi 8, Kita-ku Sapporo 060-8628, Japan

5

Present address: Institute for Materials Research,Tohoku University, Sendai 980-8577, Japan

spectroscopy (ARPES) [3] are consistent with these bandstructure calculations. Studies of superconductivity in
NbSe2 have been performed via various experimental methods, such as the tunneling spectroscopy [11], specific heat
[12, 13], heat transport [14], London penetration depth measurements [15], scanning tunneling microscopy (STM) [16,17], and ARPES [18, 19] to investigate deviations fromthe conventional Bardeen-Cooper-Schrieffer (BCS) behavior. A sixfold tail structure has been observed around thevortex cores in STM spectra [11], indicating the presence ofan anisotropic s-wave gap structure with sixfold symmetry,
which is consistent with ARPES measurements [18].
The dependence of the upper critical field (Hc2 )on the magnetic field orientation is known to provide
valuable information into the SC gap anisotropy. In thiscase, the observed anisotropy of Hc2 is often comparedwith that deduced from the linearized Ginzburg-Landau
equation assuming the symmetry of the gap function [20].
For a d-wave SC order parameter, Hc2 maxima appear
when the magnetic field in the real space is parallelto the antinodal directions [20–23]. Similar behavior isfound for an anisotropic s-wave superconductivity: theupper critical fields for anisotropic s-wave superconductors
LuNi2 B2 C and YNi2 B2 C exhibit clear fourfold symmetry
within the basal plane, which is reminiscent of d-wave
superconductors [24, 25]. In spite of extensive studies on
the superconductivity of NbSe2 , it remains unclear whether
or not the in-plane anisotropy of Hc2 shows the sixfold
symmetry within the basal plane. Thus, it is of great interest
to investigate the in-plane anisotropy of Hc2 for NbSe2 .

J Supercond Nov Magn

In this paper, we report in-plane anisotropy of acharacteristic field H ∗ (where the resistivity reaches zero)of NbSe2 at magnetic fields up to 14.5 T and temperatures
down to 2.0 K. For fields rotated within the basal plane,
we observed the in-plane anisotropy of H ∗ , with twofoldsymmetric periodicity at low temperatures. On the basis ofthese results, the in-plane anisotropy of Hc2 and its interplay
with the anisotropic gap structure are discussed.

2 Experimental Methods
Single crystals of NbSe2 were synthesized by the iodine
vapor transport method in a sealed quartz tube using Nb
wire with a purity of 99.9% and Se shots of 99.999%.
The single crystals were cut into a rectangular form
(0.90 × 0.25 × 0.02 mm3 ) with a razor blade. In ourexperiments, the crystal orientation was carefully verified
by X-ray diffraction. In-plane resistivity was measured bya conventional four-probe AC technique with an electric
current along the a-axis. We used a two-axis rotator in a
4 He cryostat with a 17-T SC magnet. The field direction is
specified by the polar angle θ and the azimuthal angle φ,where θ is the angle between the field and the c-axis and φ isthe inclination from the b -axis perpendicular to the a-axis
within the basal plane [inset of Fig. 2]. Two single crystals
denoted as samples A and B were examined to assess thereproducibility. The residual resistivity ratio (RRR), definedas ρ300 K /ρ8 K , was 31 for sample A, and 34 for sample B.
The sample properties are summarized in Table 1.

3 Results
Figure 1 shows the temperature dependence of the resistivity
for sample B at magnetic fields of up to 14.5 T. At μ0 H =0 T, the resistivity near Tc varies from 10 to 90% of thenormal-state value over a narrow range of the approximate± 0.1 K centered at Tc = 7.3 K. With increasing magnetic
field, the resistivity curves shift to lower temperatures.
Resistivity anomaly is found at around the CDW transition,
TCDW = 33.5 K. For simplicity, the dip position in thedρ/dT is defined as TCDW , as shown by the inset of Fig. 1.
Similar sharp SC and CDW transitions were observed forsample A (not shown here). In the magnetic fields, theresistivity is enhanced below TCDW but TCDW is independent
Table 1 Properties of two single crystals of NbSe2Sample

ρ8 K (μ cm)

RRR

Tc (K)

TCDW (K)

AB

10.6
8.6

31
34

7.3
7.3

33.5
33.5

Fig. 1 Temperature dependence of the resistivity for sample B inmagnetic fields ranging from 0 to 14.5 T. The inset shows thetemperature dependence of the dρ/dT at 0 T and 14.5 T. The dips
indicated by arrows define the CDW transition temperatures, TCDWof the magnetic fields of up to 14.5 T. All the behavior isconsistent with those reported by others [26–28]. We notethat positive magnetoresistance is evident up to relatively
high temperatures, which also suggests the high quality of
our single-crystal samples.
Figure 2a shows the resistivity of sample B as a functionof the field angle θ under the magnetic field of 10.5
T, where the field is rotated from positive to negative
angles at φ = −110◦ . We observe deep minima dueto the superconductivity for the magnetic field nearlyparallel to the basal plane (θ = ± 90◦ ), while a large
magnetoresistance is observed for fields perpendicular tothe layers (θ = 0◦ ). This behavior is consistent with theprevious report by Morris et al [29]. We do not find dip
structures and shallow minimum arising from flux pinningat around θ = 0◦ , which is another good sign of the high-quality samples [29]. These features in Fig. 2a show that
our single-crystal samples are suitable for magnetotransport
studies. At around θ = ± 90◦ , asymmetric angulardependence of the resistivity is observed. Origin of theasymmetric behavior is discussed later.
Figure 2b shows the resistivity as a function of thefield angle θ at various magnetic fields in the c-bplane (φ = 0◦ ). With increasing magnetic field, thedeep minimum due to the SC transition is gradually

J Supercond Nov MagnFig. 2 a Resistivity of sample B
as a function of the field angle θunder the magnetic field of 10.5
T, where the field is rotated frompositive to negative angles at
φ = −110◦ . The angles θ and φ
are defined in the inset. b
Resistivity of sample B as afunction of the field angle θ inthe c-b plane (φ = 0◦ ) at
various fields and c at various
currents

suppressed. We find that the angular dependence showsslightly asymmetric around θ = 90◦ . It is known that sample
inhomogeneity (geometrical, compositional, structural, etc.)
acts as flux pinning centers [30]. Among various origins,some randomness at the sample edges form large energy
barriers against the entry and exit of magnetic flux [30]. In
the field rotation, the flux pinning at the sample edges likely
leads to asymmetric behavior.
Figure 2c shows the resistivity as a function of the field
angle θ at various currents. Below 1 mA, the resistivity
shows a quasi-sinusoidal angular dependence and the SC
state is observed only when the field is nearly parallel tothe basal plane (θ ≈ 90◦ ). As the current increases further(I ≥ 2 mA), the asymmetric behavior is enhanced and fluxflow resistance appears even at θ = 90◦ . Above 5 mA,the R(θ ) curve shows a sharp dip at θ = 90◦ . When themagnetic field of 10.5 T is applied parallel to the basal
plane, the critical current is roughly estimated to be Ic ∼2 mA (40 A/cm2 ), below which the flux-flow resistance isstrongly suppressed. For sample A, the critical current isestimated as 1 mA (33 A/cm2 ).
Figure 3a shows the in-plane resistivity of sample B as afunction of H for various φ values within the basal plane at2.8 K. The current used here is 0.2 mA which is one orderof magnitude smaller than the estimated critical current (Ic∼ 2 mA) at φ = 0◦ as shown by Fig. 2c. In Fig. 3a, wenote that the resistive transition is very sharp, whose widthamounts to only ∼ 0.4 T. Figure 3b shows the φ dependenceof the resistivity at 2.8 K derived from the field dependenceof the resistivity for a fixed value of φ. For the fieldrotated within the basal plane, a highly isotropic normalstate magnetoresistance is found at 14.5 T. As the magnetic
field strength decreases, the field orientation anisotropyof the resistivity increases, whereas the zero resistivity isobserved for all φ values at 10.6 T. The larger resistivity forI ⊥ H (φ = 0◦ ) than that for I H (φ = 90◦ ) show that
the Lorentz force plays a dominant role in the origin of thetwofold symmetry.

J Supercond Nov MagnFig. 4 Characteristic field H ∗ for both samples derived from theresistive transition vs angle φ from b -axis (see the inset of Fig. 3) at
2.0 and 2.8 K. The solid lines present fits of the date based on a sinefunction with a periodicity of 180◦

as μ0 H ∗ = 0.17 T and μ0 H ∗ = 10.8 T, respectively.
Similarly, we obtained μ0 H ∗ = 0.15 T and μ0 H ∗ = 12.6
T at T = 2.0 K.

4 Discussion

Fig. 3 a Resistivity of sample B as a function of the magnetic field H
at 2.8 K for various values of φ. The characteristic field, H ∗ , at φ = 0◦
is indicated by the arrow. b Angular dependence of the resistivity whenthe magnetic field is rotated within the most highly conducting plane,
derived from a. The solid curves provide a guide for the eye. The inset
shows the definition of magnetic field orientation relative to φ

To further characterize the in-plane anisotropy in theSC state, we consider the characteristic field H ∗ which isdefined as the field strength at which the resistivity reaches
zero. The in-plane anisotropy of H ∗ is plotted as a functionof φ in Fig. 4. At 2.8 K, H ∗ displays twofold symmetry witha maximum value at φ = ± 90◦ , corresponding to I H .
The same twofold anisotropy is observed at 2.0 K. From thefitting results based on a sine function, the amplitude andaverage of the H ∗ anisotropy at T = 2.8 K are estimatedLet us discuss the in-plane anisotropy of the intrinsic
Hc2 on the basis of observed H ∗ (φ) with twofold
symmetry. Because the highly isotropic normal-statemagnetoresistance is found at 14.5 T as shown in Fig. 3b,
the twofold symmetry in H ∗ can not be ascribed to the inplane field dependence of the normal-state resistance [31].
When the magnetic field is not parallel to the current in thevortex liquid regime, the vortices are driven by the Lorentz
force, which gives rise to energy dissipation. Because themagnitude of the Lorentz force changes with the angle φin our experimental setup, the angular dependence of theresistance may be affected by the angular dependence ofthe energy dissipation. It is well known that the Lorentz
force picture gives the twofold anisotropy of the resistivityas ρ(φ) ∝ sin2 (φ). This sin2 (φ) dependence can certainly
describe the ρ(φ) behavior of 90 K YBCO [32] and isexplained in terms of SC fluctuation effects in the vortex
liquid regime [33]. In this situation, it is unlikely thatthe anisotropies of both H ∗ and ρ(φ) represent that ofHc2 . After subtracting the twofold symmetry component inFig. 4, the sixfold symmetry component is less than 2%.
Thus, we conclude that the intrinsic Hc2 may be highly
isotropic within the basal plane.

J Supercond Nov Magn

The relationship between the in-plane isotropic natureof the intrinsic Hc2 and the anisotropic s-wave SC gap
structure is next discussed in the following three viewpoints.
One is the anisotropy of Hc2 in the basal plane of ahexagonal crystal [34, 35]. Burlachkov [34] theoreticallyshowed in the basal plane of a hexagonal lattice that there
is no anisotropy of Hc2 near Tc , regardless of the nature of
the pairing symmetry. Thus, the obtained isotropic nature in
Hc2 (φ) may be consistent with the prediction that the sixfold
symmetry part of Hc2 is absent in hexagonal systems.
However, we should note that the theory is not directly
applicable to NbSe2 because the SC phase coexists with theincommensurate CDW phase. It should be also noted thata clear anisotropy of Hc2 (φ) more than 30% in hexagonalmaterial Csx WO3 [36] is inconsistent with the theory.
The second viewpoint is related to the absence of nodes
in anisotropic s-wave SC gap structure. According to theARPES experiment by Kiss et al [18], superconductivity
occurs over the Nb-derived FS, with maximal gaps
along Γ − K and smaller gaps along Γ − M (i.e.,sixfold symmetry), indicating the anisotropic s-wavesuperconductivity. Although Hc2 with the sixfold symmetryis expected [20, 21] from the ARPES measurements [18],only the twofold in-plane anisotropy of H ∗ is appreciable.
The small sixfold symmetry component less than 2% inFig. 4 shows that the amplitude of the Hc2 oscillations
with sixfold symmetry is significantly reduced in NbSe2 ascompared with a d-wave SC gap with nodes [22, 23]. This isbecause the ARPES experiments [18] suggest the existenceof gap minima (i.e., no nodes) in the anisotropic s-wave SC
gap for NbSe2 .
The third viewpoint is the multiband nature in NbSe2 .
The multiband effect is neglected in the theories for Hc2
in d-wave superconductors [20, 21] although the ARPES
measurements [19] suggest an anisotropic SC gap withdifferent gap amplitudes in different FS sheets, i.e., two or
more types of anisotropic gaps in each FS sheet. Thus, thein-plane anisotropy of Hc2 in NbSe2 will be averaged out by
the different SC gaps.

5 Conclusion
The in-plane angular dependence of H ∗ showed only
twofold symmetry associated with the angular dependenceof the Lorentz force. This behavior is likely ascribed toSC fluctuation effects in the vortex liquid regime. On thebasis of the in-plane anisotropy of H ∗ , the amplitude of
the Hc2 oscillation with sixfold anisotropy, if it exists, ismuch smaller than μ0 H ∗ ∼ 0.2 T. These results suggest
a highly isotropic Hc2 within the basal plane. Contrary toexpectations, no sixfold symmetry was observed for Hc2 .
The experimental results reported herein demonstrate thatthe characteristics of the anisotropic s-wave gap in NbSe2 ,that is, the minimum gap (instead of nodes in a d-wavesuperconductor) and/or the multiband effect significantly
suppress the amplitude of Hc2 oscillations related to the SC
gap anisotropy.
Acknowledgments The authors thank T. Konoike for useful discussion and experimental supports.

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