PHYSICAL REVIEW B 81, 184506 共2010兲

Topological effects of the superconducting vortex state in a TaSe3 ring crystal: Observation ofmagnetic torque oscillations
Genki Kumagai,1 Toru Matsuura,1 Koichi Ichimura,1,2 Kazuhiko Yamaya,2 Katsuhiko Inagaki,1,2 and Satoshi Tanda1,2
1Department of Applied Physics, Hokkaido University, Kita 13, Nishi 8 Kita-ku, Sapporo 060-8628, Japan
Center of Education and Research for Topological Science and Technology, Kita 13, Nishi 8 Kita-ku, Sapporo 060-8628, Japan
共Received 21 January 2010; revised manuscript received 9 April 2010; published 11 May 2010兲

2We measured the magnetic torque of the ring-shaped crystals of TaSe3 by using piezoresistive cantilevers toinvestigate the superconducting topological properties. We measured three ring samples and we observed thatthe magnetic torque of the ring crystals oscillates with increasing external magnetic field. The magnetic periodsof Sample A 共radius: 37.9 ␮m兲, Sample B 共24.5 ␮m兲, and Sample C 共15.6 ␮m兲 were 2.16⫾ 6.6⫻ 10−2 G,
4.69⫾ 1.8⫻ 10−1 G, and 5.44⫾ 3.8⫻ 10−1 G, respectively. We found that hundreds of vortices collectively
and simultaneously penetrated the ring crystal. When such a phenomenon occurs, it is natural that the vortices
in the ring crystal would be positioned along the circumference. From these results, we suggest that vortices
exist as cylinder vortices in the rings, and this phenomenon provides unambiguous experimental evidence ofthe topological effect in a superconductor.
DOI: 10.1103/PhysRevB.81.184506

PACS number共s兲: 74.25.Wx, 74.62.BfI. INTRODUCTION

the mixture was sealed in a quartz ampoule and air wasevacuated from the ampoule to the order of 10−6 Torr. Next,
the ampoule was placed in a furnace. The temperature of thestarting material at the end of the ampoule was increased tothe reaction temperature, typically 700 ° C. The mixture wasthen kept at this temperature for a few hours. By using thismethod, thousands of ring crystals were synthesized per ampoule. The b axis is parallel to the circumferential direction
and the highest conductivity is expected along this direction.
To investigate the effect of the topology we need a measurement that requires no electrode because the topology ischanged by the presence of an electrode. In this study, wemeasured the magnetic torque by using microcantilevers.12The magnetic torque ␶ = M ⫻ B is produced on superconducting samples by an applied field B and sample magnetization
M. In a homogeneous applied field, the transverse component of the sample magnetization will cause a deflection of
the cantilever ⌬z, which is proportional to the magnetic
torque ␶. The sample is mounted at the extremity of the leveras shown in Fig. 1. ⌬z is measured here as the change in thepiezoresistance ⌬R. To detect a small change in the resistance, we used a Wheatstone bridge circuit measured with alock-in amplifier. We compared the magnetic torque at temperatures below and above Tc to confirm the effect of thesuperconducting properties. The cantilever and the sample
were cooled with a 3He cryostat. All the measurements wereperformed in a magnetic shield room.

Topology and geometry can have various effects on
physical properties. For example, when a type II superconductor is in the mixed state and the system size is small, thepenetration and arrangement of vortices are influenced by thegeometrical effect of the shape of the superconductors.1 The
system topology is also expected to affect the superconducting properties. For example, in nonorientable systems suchas a Möbius strip crystal,2 the possibility of an unusual vortex state has been predicted.3 We should treat the system
topology as one of the important parameters for investigatingsuperconducting properties.
In this research, we measured the magnetic torque of thering-shaped crystals of TaSe3 to investigate the effects oftopology on superconducting properties. The ring-shaped
crystals of the quasi-one-dimensional superconductor TaSe3
共Tc = 2.0 K兲 共Ref. 4兲 are suitable for use in investigatingthese topological effects because ring crystals are expected tohave a topological effect on its chain-loop structure.5–11 In
addition, measurements performed without an electrode areappropriate for investigating these effects because thismethod conserves the system topology. Therefore, we employed magnetic torque measurements with microcantilevers
as a method without an electrode.12 We found that the magnetic torque of the ring crystals oscillates periodically below
the transition temperature. We suggest that a cylinder vortex
is formed in the rings as a result of the topological effects ofthe superconductive ring-shaped crystals.
II. EXPERIMENTAL

We measured the magnetic torque of the ring-shaped crystals of TaSe3 共Tc = 2.0 K兲 共Ref. 4兲 to investigate the effectsof topology on superconducting properties. The TaSe3 ring
crystals were synthesized by the chemical-vapor transportation method.2,4 The starting material was a mixture of Ta andSe with about 5% excess Se as a transport agent. Ta andquartz ampoules were baked at around 900 ° C. After baking,1098-0121/2010/81共18兲/184506共4兲

FIG. 1. 共Color online兲 Schematic image of cantilever and ring
crystal. Inset: scanning electron microscope image of TaSe3 ring
crystal 共Sample A兲 and cantilever.

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KUMAGAI et al.

TABLE I. Characteristic values of the ring crystals.
Radius
Effective
Effective
Inner area inner area 共45°兲 Outer Inner Circumference circumference 共45°兲 Width共 ␮ m 2兲共␮m兲共␮m兲 共␮m兲共␮m兲
共␮m兲
Sample name
共 ␮ m 2兲
37.9
24.5
15.6

634
424

We measured three Samples A, B, and C. The sample
dimensions are summarized in Table I. With Sample A, theangle between the applied field and the ring crystal surface
was 90°. This sample was directly measured in liquid 3He.
With Samples B and C, the angle between the applied field
and the ring crystal surface was 45°. This sample was measured in a vacuum. No difference was found between themeasurement in 3He and that in a vacuum. This shows thatthe thermal conduction of the sample to the cold head issufficiently high. Moreover, since this is not a dynamical
method we need not worry about the damping factor. First,
we swept the field in the increasing direction from 0 G. Thenwe swept the field in the decreasing direction. The fieldsweep speeds of Samples A, B, and C were 4.4⫻ 10−2 G / s,4.3⫻ 10−2 G / s, and 1.1⫻ 10−2 G / s, respectively. In addition, we performed measurements at different rates for
Samples A and C of 4.4⫻ 10−1 G / s and 3.3⫻ 10−3 G / s, respectively. With both Samples A and C, we observed thesame results for measurements at different speeds. Therefore,
we concluded that the speed of the field sweep has no influence on the results.
III. RESULTSFigures 2共a兲 and 2共b兲 are the typical results for Samples A

05Field (Gauss)

i4TaSe3Sample B

6−20(b)0−2000Field (Gauss)

005
i

1020FIG. 2. 共Color online兲 共a兲 Magnetic torque of Sample A as afunction of the applied magnetic field. Arrows show the peak andvalley positions. The blue line represents the magnetic torque measured at T = 0.56 K. The red line represents the torque measured atT = 4.3 K. Inset: open and filled circles show the valley and peak
positions, respectively. 共b兲 Magnetic torque of Sample B. Arrows
show the peak and valley positions. The blue line represents thetorque measured at T = 0.38 K. The red line represents the torquemeasured at T = 10.9 K. Inset: open and filled circles show the valley and peak positions, respectively. The magnetic torques below
Tc = 2.0 K of both samples oscillate.

TaSe3
Sample C−1 T=0.68 K
10−200Field (Gauss)

T=0.68 K−10−100(a)TaSe3
Sample C’(cut ring)Nm)

2−4Nm)−5020and B, respectively. For each sample, we obtained the same
result for several measurements when the experimental conditions were the same. The horizontal axis shows the appliedmagnetic field and the vertical axis shows the magnetic
torques translated from measured voltages. Offset voltages,
which may be caused by residual magnetic fields, were carefully removed. Below Tc, the magnetic torques in all of thesamples oscillated as a function of the applied magnetic
field. These oscillations were not observed in the normal
state as shown by the red line in Fig. 2. This shows that theobserved oscillations originated from the superconducting
state.
To demonstrate the periodicity of the oscillations, we plot
their peak and valley positions as indicated by the arrows.
We identify each field with indices. We plot the magnetic
field of the peak and valley positions Bi as a function ofindex i. We found that Bi is linear to i 关inset of Figs. 2共a兲,2共b兲, and 3共a兲兴. The periods of Samples A, B, and C were
2.16⫾ 6.6⫻ 10−2 G, 4.69⫾ 1.8⫻ 10−1 G, and 5.44⫾ 3.8⫻ 10−1 G, respectively.
To confirm the topological effect, we performed a control
experiment on a cut ring Sample C⬘, after measuring Sample
C. Sample C⬘ was obtained by cutting Sample C with afocused ion beam.13,14 Sample cutting is an effective methodfor changing the topology. Figures 3共a兲 and 3共b兲 show thetypical results of these measurements. In Sample C, the magnetic torque oscillated in the same way as in Samples A andB. In Sample C⬘, the magnetic oscillation did not appear buta broad antisymmetric peak was observed. In Sample C⬘,
when the applied magnetic field is removed, vortices may be
trapped by a pinning effect at the edge. Therefore, the magnetic torque reverses when the applied magnetic field re-−100Bi (Gauss)5

T=0.38 K−2

1.5
7.6
1.8

91.2
73.7−10Nm)
TaSe3Sample ATorque ∆τz (10T=0.56 K−5−5(a)0−110Bi (Gauss)

Torque ∆τz (10−10Nm)

T=10.9 K

228.7
106.2
86.9

Torque ∆τz (10T=4.3 K

36.4
16.9
13.8

Bi (Gauss)

4162
897599Torque ∆τz (10Sample A
Sample B
Sample C2i

4

206(b)−20020Field(Gauss)

FIG. 3. 共Color online兲 共a兲 Magnetic torque of Sample C at T
= 0.68 K as a function of the applied magnetic field. Arrows show
the peak and valley positions. Inset: open and filled circles show thevalley and peak positions, respectively. 共b兲 Magnetic torque of thecut Sample C⬘ at T = 0.68 K.

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TOPOLOGICAL EFFECTS OF THE SUPERCONDUCTING…
4AC

2Area S (µm )10210010 −1
10x111

Bx144x434

flux quantum h2e−11/B (Gauss )100FIG. 4. 共Color online兲 The ring area as a function of 1 / B. B is
the period of the magnetic torque peaks. Open circles show the ringarea. Filled circles show the effective ring area for 45°. The solidline represents S = ␾0 / B. The broken line is a guide to the eye.

verses. This is in contrast to the case with Sample C, where
the direction of the magnetic torque remains unchanged
when the applied magnetic field is reversed. The difference
between Samples C and C⬘ suggests that vortex state ischanged by the effect of ring topology.
IV. DISCUSSION

Here we discuss the magnetic torque oscillation 共Figs. 2and 3兲. We assume that vortices penetrate at each period ofmagnetic field B. We plot the inner area S and the effective
inner area 共for 45°兲 S⬘ as a function of 1 / B in Fig. 4. The
solid line represents S = ␾0 / B, that is, the field corresponds toa flux quantum ␾0 = h / 2e per inner area. As shown in Fig. 4,we found a significant discrepancy between the solid line andthe data. We estimate the number of vortices for 90° fromN = BS / ␾0 and the number of vortices for 45° from N⬘= BS⬘ / ␾0: N = 434, 203, and 157 for Samples A, B, and C,respectively, and N⬘ = 144 and 111 for Samples B and C,respectively. N for Samples A and N⬘ for Samples B and C
suggest that hundreds of the vortices penetrated the samples
collectively and simultaneously. This is an unprecedented
phenomenon that is in contrast to the known topological effects accompanied by flux quantization.15,16
We propose a cylinder vortex model to explain the oscillation of the magnetic torque. We mean that the cylinder
vortex is a cylinder formed by connecting penetrated vortices. While the magnetic field applied to the ring crystal issmall, vortices cannot penetrate the ring crystal to realize theMeissner effect. Then, when the applied field increases as thevortices fill the inner area of the ring crystal, the vortices
penetrate the ring crystal because of the effect of the ringshaped topology. On the basis of the above discussion, sincea number of vortices penetrate the ring crystal, we expect
them to penetrate the entire ring crystal collectively and position themselves along a circumferential line rather than as

FIG. 5. 共Color online兲 Schematic model of cylinder vortex. 共a兲
Vortices positioned in line. 共b兲 Vortices extending along the circle
direction. 共c兲 Vortices connected with each other and forming acylinder vortex.

an Abrikosov lattice 关Fig. 5共a兲兴. Moreover, vortices extend
along the tangential direction owing to the strong anisotropyof the crystal structure of TaSe3.17 And, the ring crystals
have a b axis connecting structure. Therefore, extended vortices are stable in ring crystals and the supercurrents flowing
around the vortices overlap each other 关Fig. 5共b兲兴. This property probably causes the state whereby the vortices positioned in line connect with each other. Hence, it is naturalthat vortices in the ring crystal form a cylinder vortex 关Fig.
5共c兲兴. This laminar structure was anticipated by de Gennes.18
Furthermore, the length of each ring circumference divided
by each N and N⬘ is on the order of 10−1 ␮m. This length isof the same order as the coherence length ␰ of a typicalsuperconductor. The cylinder vortex model is consistent withthe fact that the magnetic torque oscillates.
V. SUMMARY

In this research, we measured magnetic torque by using
piezoelectric microcantilevers. As a result, we found that themagnetic torque oscillates as the applied magnetic field increases in ring crystals and hundreds of the vortices collectively and simultaneously penetrate the ring crystal. Theseresults indicate that vortices penetrate ring crystals as a cylinder vortex. The cylinder vortex forms as a result of theclosed ring topology of the ring crystals of TaSe3.
ACKNOWLEDGMENTS

We are grateful to N. Matsunaga, K. Nomura, E. Ohmichi,Y. Nogami, and S. Takayanagi, for experimental support anddiscussions. This work was partially supported by the 21stcentury COE program on “Topological Science and Technology” from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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