PHYSICAL REVIEW B 73, 165118 共2006兲

Topological effects of charge density waves in ring-shaped crystals of NbSe3
Toru Matsuura,1 Taku Tsuneta,2 Katsuhiko Inagaki,1 and Satoshi Tanda1
1Department of Applied Physics, Hokkaido University, Kita-ku, Sapporo 060-8628, Japan
2Low Temperature Laboratory, Helsinki University of Technology, Otakaari 3A, Espoo, Finland

共Received 29 November 2005; revised manuscript received 10 March 2006; published 19 April 2006兲
We studied the differences of the charge density wave 共CDW兲 dynamics in closed-ring and open-ringtopologies by measuring the frequency-dependent resistance in the radio-frequency range. We performed atwo-probe ac resistance measurement for a ring-shaped crystal of niobium triselenide 共NbSe3兲 in the300 kHz to 1.3 GHz frequency range. We compared the frequency dependence of the resistance of closed-ringand open-ring crystals and found that the pinning potential in the former is smaller than that in the latter. The
open-ring crystal was obtained by changing of the topology of the closed-ring crystal. We believe that thereduction in the pinning effect is caused by a CDW circulating current in the closed-ring topology.
DOI: 10.1103/PhysRevB.73.165118

PACS number共s兲: 71.45.Lr, 02.40.⫺kI. INTRODUCTION

have also been observed in topological crystals.15,16 They
have provided new experimental systems for investigating
the effects of topology on the CDW properties. Since theCDW phase-phase correlation length ␰CDW is comparable tothe circumference of the CDW ring, the topological crystals
are one-dimensional closed rings. The nontrivial topologiesof their forms have been expected to change the CDW properties. Some topological effects have already been observedin the electrical properties including an anomaly in the nonlinear I-V characteristic in a NbSe3 ring18 and a reduction in
the transition temperatures in rings and figure-eight loops.19
Here we report a difference between the CDW dynamics
of two topologies: namely, a closed-ring topology and anopen-ring topology. We measured the frequency-dependent
resistance of a ring crystal of NbSe3 to investigate the CDWdynamics. First, we measured a ring crystal with the closedring topology. Then, we measured a crystal with the openring topology, which we obtained by performing topologychange surgery on the ring crystal. The closed-ring topology
of a ring crystal can be surgically changed to an open-ring
topology by using a focused ion beam 共FIB兲, as shown inFig. 1.20 An open ring is topologically different from a closed
ring because it has two edges. Using this method, we canstudy two CDW’s, which are identical except for their topology, and avoid the sample dependence.21 The frequencydependent resistance measurement was performed in the300 kHz to 1.3 GHz range below the first CDW transition
temperature TC1. We found that the pinning potential of theCDW in the closed-ring topology was smaller than that in theopen-ring topology. The reduction of the pinning effect in theclosed-ring topology suggests that the circulating current
mode plays an important role. This effect was revealed by
the topology-change surgery.

Nontrivial real-space topology plays an important role inphysical systems.1,2 Topology deals with invariant properties
in continuous transformation. For example, a ring is a fundamental topological structure, because the hole in the ringcannot be eliminated by continuous transformation. The
physical systems with a ring topology should exhibit topological effects, such as the Aharonov-Bohm 共AB兲 effect,3 dueto the nontrivial real-space topology. It is possible to change
physical properties by changing the real-space topology.
A charge density wave 共CDW兲 is the best system withwhich to investigate the effects of real-space topology, because of the interaction between a CDW and a real space, inthis case a crystal, through electron-phonon coupling. The
CDW-ordered state is a macroscopic quantum state resulting
from real-space ordering.4 A CDW has rich electrical properties such as the nonlinear current-voltage 共I-V兲
characteristic,5 narrow-band noise,6 and frequencydependent resistance.7,8 The CDW was proposed as a modelof superconductivity by Fröhlich in 1954, because an incommensurate CDW can flow without dissipation in an ideal
crystal.9 In reality, a CDW supercurrent has never been observed because the crystal edges or impurities act as pinningcenters. If the edges of a CDW system could be connected, itwould become an edgeless CDW ring. The supercurrent
would then exist in the ring as a result of the change intopology. Moreover, the CDW ring has been expected to exhibit topological effects associated with Berry’s phase.10–13
The pioneering experimental study of the AB effect in theCDW system was undertaken by Latyshev et al.14 They measured a quasi-one-dimensional conductor with many columnar defects since no one could make a CDW ring. We need aCDW ring to enable us to study the topological effects inCDW’s.
The Hokkaido group succeeded in synthesizing exotic topological crystals of NbSe3, which form rings, Möbius strips
共␲ twisted rings兲, and figure-eight loops 共2␲ twisted rings兲
with diameters of 10– 200 ␮m.1,15,16 NbSe3 is a typical CDW
material that shows two metal-incommensurate CDW transitions at TC1 = 144 K and TC2 = 59 K,17 and CDW transitions
1098-0121/2006/73共16兲/165118共5兲/$23.00

II. EXPERIMENTThe ring crystals of NbSe3 are synthesized by a chemical
vapor transportation method. Nb and Se powders with a 1:3
molar ratio are enclosed in an evacuated quartz tube andsoaked at 740 ° C for a few hours using a furnace with anappropriate temperature gradient. The crystals acquire their165118-1©2006 The American Physical Society

PHYSICAL REVIEW B 73, 165118 共2006兲

MATSUURA et al.

FIG. 2. Electrical circuit for the frequency-dependent resistancemeasurement. The ring at the center of the circuit indicates a ring
crystal, which is attached to two contacts.

age is small and the frequency f is much smaller than ␻0 / 2␲,conductivity is expressed by the Drude law

␴CDW共␻兲 = ␴⬁FIG. 1. 共a兲 Scanning ion beam microscopy 共SIM兲 image of atypical ring crystal with a radius of 18.2 ␮m and a thickness of0.47 ␮m. It is a closed ring. 共b兲 SIM image of the ring crystal after
focused ion beam 共FIB兲 surgery. The FIB surgery provided twoedges in the ring crystal. The forms in 共a兲 and 共b兲 are referred to in
the text as a closed-ring topology and an open-ring topology, respectively. This is a useful method for investigating topology dependence alone, without sample dependence.

topological forms by a previously reported mechanism.1
We performed a two-probe ac resistance measurement fora ring-shaped crystal of NbSe3 in the closed-ring and openring topologies. The ac resistance measurement is a usefulway to investigate the CDW dynamics in topological crystals
because it does not require a current injection via contacts.
Current injection causes a phase slip near the contacts,22 andthis acts as a pinning center.23 The current injection may
change the topological property of the CDW ring. On theother hand, frequency-dependent resistance measurement
does not require the current injection. The CDW dynamics is
expressed by the damped oscillation.7,8 The single-coordinate
model isd␾n␲ed 2␾
+ ␻20 sin ␾ = − * E共t兲,2 +⌫dtdtm共1兲

where ␾ is the CDW phase, ⌫ is the damping constant, and␻20 is the pinning potential. The right-hand side indicates anexternal force. The CDW current is expressed as
ICDW = −e d␾.
␲ dt

共2兲

The CDW contributes electrical conduction as a displacement current in the ac electric field. When the applied volt-i␻/␻C.
1 + i␻/␻C共3兲

Here we neglect the inertial term d2␾ / dt2 and assumesin ␾ ⬇ ␾ and we use the characteristic frequency ␻C= ␻20 / ⌫. The CDW dynamical parameters can be investigatedby measuring the frequency-dependent resistance. While themeasurement voltage remains smaller than the thresholdvoltage of a CDW sliding state, we can avoid the effect ofcurrent injection. Therefore, the frequency-dependent resistance measurement is advantageous in terms of investigating
the CDW in the topological crystals.
The frequency-dependent resistance was measured as follows. A NbSe3 ring crystal, with a diameter of 40 ␮m and awidth and thickness both less than 1 ␮m, was fixed on asapphire insulator plate. We attached two contacts to thesample to enable to supply an ac voltage. The plate wasplaced on a Cu block in the vacuum chamber of a cryostat.
The crystal was cooled to 77 K by liquid nitrogen, and wemeasured the temperature dependence of the resistance whileheating the crystal from 77 K to room temperature. We usedthe conventional transmission method for the frequencydependent resistance measurement with a Hewlett-Packard
HP8712ET network analyzer in a frequency range of f
= 300 kHz to 1.3 GHz. The measurement circuit is shown inFig. 2. The voltage amplitude was kept below the thresholdvoltage of the sample. The sample impedance Z is measuredby using the following equation:
Z = Z0

1/V − 1/Vs,
1/V0 − 1/Vs

共4兲

where ␻ = 2␲ f, Z0 is a standard impedance, and V= Vb(Z共␻ , T兲) = 兩V兩ei␾ is the complex voltage at b in Fig. 2,
which is a function of the sample impedance Z共␻ , T兲
= R共␻ , T兲 + iX共␻ , T兲. Other parameters Vs = Vb共0兲 and V0= Vb共Z0兲 are used to calibrate the frequency dependence ofthe finite loss and phase delay in the cables. Since the calibration parameters depended on the temperature sweep, wecalibrated both frequency and temperature. After measuring
the frequency-dependent resistance of the closed-ring topology, we performed FIB surgery on the ring crystal to change
to be the open-ring topology. We then again measured thefrequency-dependent resistance of the pinning state and compared the results we obtained for the two topologies.

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TOPOLOGICAL EFFECTS OF CHARGE DENSITY WAVES¼

FIG. 3. Temperature dependence of resistance in a closed-ring
topology and in an open-ring topology. The line thickness indicatesthe frequencies, and the dashed lines are linear fitting lines, R共T兲
= ␣T + R0, fitted to the metallic region in the 160– 200- K range. The
inset shows the temperature dependence of the normalized resistance, R̃共␻ , T兲 = 关R共␻ , T兲 − R0兴 / ␣ − T. The first CDW transition temperature TC1 is ⬃145 K in both topologies. The resistance peak
below TC1 is maximum at the same temperature.
III. RESULTS AND DISCUSSIONS

We observed a CDW transition with a resistance increaseat ⬃145 K in both topologies. Figure 3 shows the temperature dependence of the resistance R共␻ , T兲 关real part of impedance Z共␻ , T兲兴 in the closed-ring and open-ring topologies
in the 77-K–to–room-temperature range, at frequencies of300 kHz, 300 MHz, and 600 MHz. The sample is metallic
above 150 K because the resistance in each topology is proportional to temperature. Below the CDW transition temperature, the resistance increases because the electron density
decreases as a result of the metal-CDW transition. The resistance shows a peak at 120 K and decreases proportionallybelow 100 K, because uncondensed electrons remain evenbelow TC1 in NbSe3. The second transition was not observedin this measurement because TC2 was below 77 K.
In the CDW state R decreases with increases in frequency.
This behavior is explained by the CDW collective displacement around the pinning center with a finite damping timeand corresponds to the results of a previous experiment for
NbSe3 whiskers.7,8 There is almost no frequency dependence
in the low-frequency region below 1 MHz. Therefore, weassumed the temperature dependence at 300 kHz to be thesame as that at a dc voltage. The dashed lines seen in Fig. 3
are fitting lines for Rfit共T兲 = ␣T + R0 at 300 kHz, where T is
the temperature. We fit lines in the 160– 200-K range forboth sets of data. This is just above the transition temperature. The temperature coefficient ␣ is 0.0208 in the closedring topology and 0.0407 in the open-ring topology. ␣doubled as a result of the topology change. This corresponds
to one of the two current paths being eliminated by the FIBsurgery. On the other hand, the intercept R0 is 2.93 ⍀ in theFIG. 4. The frequency dependence of the normalized resistance,R̃共␻ , T兲 = 关R共␻ , T兲 − R0兴 / ␣ − T, at T = 100, 110, and 120 K below thefirst CDW transition temperature TC1 = 145 K. The black dots areexperimental data for the closed-ring topology, and the gray dots aredata for the open-ring topology. The solid and dashed lines indicatethe fitting curves given in the text for the closed-ring and open-ringtopologies, respectively. The fitting parameter ␤ is set at 0.8 in bothtopologies and for all temperatures. The characteristic frequency
␻C / 2␲ is 0.85 GHz in the closed-ring topology and 1.2 GHz in theopen-ring topology at all temperatures.

closed-ring topology and 2.98 ⍀ in the open-ring topology.
R0 does not depend on the topology change and so must
represent the contact resistances. These results are consistent
with the elimination of one path by topology-change surgery.
Therefore, we considered that the surgery was successful. We
then normalized the resistance using ␣ and R0 to compare thefrequency dependences. We disregard the temperature dependence of R0 because it is sufficiently smaller than that of thesample. Thus, we can define the normalized resistanceR̃共␻ , T兲 = 关R共␻ , T兲 − R0兴 / ␣ − T. The inset shows R̃ as a function of temperature T. The temperature dependence of R̃ at
300 kHz corresponds in both topologies. However, it does
not correspond at high frequencies.
We found that the frequency dependence of the normalized resistance R̃ is different for the two topologies. Figure 4
shows the frequency dependence of R̃ at 100, 110, and120 K. The black dots are experimental data for the closedring topology, and the gray dots are data for the open-ringtopology. Each R̃ value decreases with an increase of frequency above 100 MHz. We found that the frequency dependence was changed by changing the topology. The resistanceof the closed-ring topology starts to decrease at a lower frequency than that of the open-ring topology. We fitted theresults using the following empirical curve, which was used
for NbSe3 by Wu et al.:24165118-3

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MATSUURA et al.

␴CDW共␻兲 = ␴⬁

共i␻/␻C兲␤.
1 + 共i␻/␻C兲␤

共5兲This expression is Eq. 共2兲 modified by the empirical parameter ␤, which represents the random distribution of weakpinning centers by the impurities.24 We assumed that the total
conduction is the sum of that of the CDW and quasiparticles.
Hence,

␴total共␻兲 = ␴metal + ␴CDW共␻兲

共6兲

−1共␻兲兴.
R共␻兲 ⬀ Re关␴total

共7兲

andThe solid and dashed curves in Fig. 4 indicate the fittedcurves for the closed-ring and open-ring topologies, respectively. We set R̃共␻ = 0兲 = 62, 64, and 60 at T = 100, 110, and120 K, respectively. When the system follows the singlecoordinate model, ␤ = 1. In reality, the random distribution of
weak pinning centers modulates the CDW phase locally. In
our sample, ␤ = 0.8 in both topologies and at 100, 110, and120 K. The distributions of the pinning centers in both paths
are the same. These parameters have the same values in bothtopologies, except for the characteristic frequency ␻C / 2␲.
We found that the characteristic frequency was changed bythe surgery. ␻C / 2␲ = 0.85 GHz in the closed-ring topology
and 1.2 GHz in the open-ring topology at all three temperatures. ␻C became about 1.4 times larger as a result of thetopology change. ␻C = ␻20 / ⌫, where ⌫ is the damping constant of the CDW, and ␻20 is the pinning potential. Since ⌫ isproportional to ␴⬁−1, which was unchanged by the surgery,this result suggests that the pinning potential in the closedring topology is greatly suppressed.
Why does the topology of the crystal affect the pinning?
The result cannot be explained by the simplest parallel circuit model. When we assume that a closed-ring topology
corresponds to a parallel circuit of two CDW materials, thepinning potential is unchanged by the surgery, because thenormalized resistance does not depend on the number ofpaths. It is possible that the strain energy of the crystal waschanged by the change of topology.25 However, the strainenergy of the remaining path was not changed by the surgery
because the sample was fixed at two points on a sapphire
plate by silver paste. We therefore considered the effect ofthe edges or the interaction between the two current paths.

1 S. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya, and N.

Next, we discuss the elastic energy of the CDW phase,
which must differ depending on the topology. In the openring topology, the edges act as strong pinning centers. In anelectric field, the CDW is compressed near one edge andexpanded near the other edge. The elastic energy increases
the pinning potential energy. On the other hand, there are twoCDW current paths in the closed-ring topology. Therefore,
the CDW is compressed and expanded near the contacts under the ac electric field in an ac closed-ring topology. Theincreases in the pinning potential in the closed-ring topology
must be larger than those for the open-ring topology becauseof the existence of the two current paths. This is inconsistent
with the experimental result and suggests that we need another explanation.
A possible explanation is that a circulating current in thering affects the pinning. The Fröhlich supercurrent can flowin an impurity-free ring crystal without dissipation becauseof the absence of edges.9 We believe that the CDW circulating current mode is excited by the ac electric field in theclosed-ring topology. The phase-phase correlation length isgreater than the average impurity length. Impurities act as
weak pinning centers that modulate phase locally. Therefore,
it is reasonable to consider that the CDW correlation length
exceeds the circumference of the ring crystal. Nevertheless,the circulating current mode certainly does not exist in theopen-ring topology. We believe that the circulating currentmode is the key to the decrease in the pinning potential in theclosed-ring topology.
IV. CONCLUSION

We found that the pinning potential of a CDW in a closedring topology was smaller than that in an open-ring topology
by measuring ac resistance of a NbSe3 ring crystal. The difference is caused by the circulating current mode in theclosed-ring topology.
ACKNOWLEDGMENTS

The authors are grateful to K. Yamaya, T. Toshima, and K.
Shimatake for stimulating discussions. This research wassupported by the 21 COE program on Topological Scienceand Technology from the Ministry of Education, Culture,Sport, Science and Technology of Japan and the Japan Society for the Promotion of Science.

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