PHYSICAL REVIEW B 79, 014304 共2009兲

Evidence of circulating charge density wave current: Shapiro interferencein NbSe3 topological crystals
Toru Matsuura, Katsuhiko Inagaki, and Satoshi Tanda
Department of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan
共Received 6 October 2008; published 28 January 2009兲
We investigate the topological effect of charge density wave 共CDW兲. We found a “beat” Shapiro step
structure in NbSe3 topological 共ring兲 crystals. We interpreted it as an evidence of the circulating CDW current
that flows in the rings and modifies the CDW fundamental frequency, namely, narrow-band-noise frequency.
This result also implied that a precursor of Fröhlich’s CDW supercurrent emerged in the ring system.
DOI: 10.1103/PhysRevB.79.014304

PACS number共s兲: 71.45.Lr, 72.15.Nj, 73.23.Ra, 74.50.⫹r

I. INTRODUCTION

The charge density wave 共CDW兲 was originally proposed
as a model of superconductivity by Fröhlich1 in 1954. The
sliding CDW state was discovered in quasi-one-dimensional
conductors such as NbSe3 crystals in the 1970s;2 however
Fröhlich’s CDW supercurrent has yet to be observed. Recently, the Hokkaido University group discovered topologically nontrivial loops of the NbSe3 crystals, including rings,
Möbius strips, figure 8s,3,4 and Hopf links of crystals.5 The
discovery of topological crystals of the CDW materials completely changed the situation. Since one-dimensional chainsof these crystals have no edges, they allow the CDW to flow
without electrical contacts. Both edges of crystals and electrical contacts have been considered as strong pinning centers that inhibit Fröhlich’s supercurrent. Therefore, topological crystals are a candidate system for realizing Fröhlich’s
CDW supercurrent.
In this paper, we report a phenomenon that is a precursor
to Fröhlich’s CDW supercurrent. Shapiro interference measurement is a very useful approach for investigating macroscopic quantum states because a Shapiro spectroscopy provides detailed information about the CDW current.6–8 We
found the beat structure of the Shapiro step spectrum only in
the NbSe3 rings. The beat signal results from the existence ofa circulating CDW current, which modifies the fundamental
frequencies of sliding CDW 共narrow-band noise兲 in the rightand left arms. The phason damping time of the circulating
CDW must be longer than at least 10−8 s, which is threeorders longer than that of the conventional CDWs in NbSe3
whiskers.9 The elongation of the damping time suggests thatthe circulating current is the Fröhlich CDW supercurrent.
II. EXPERIMENTThe experimental setup was as follows. Ring-shaped crystals of NbSe3 were synthesized by the chemical vapor transportation method.4 The ribbonlike quasi-one-dimensional
crystals of NbSe3 form seamless loop crystals by bending
and joining naturally under controlled conditions. A ringshaped crystal with an outer diameter of 120 ␮m, an innerdiameter of 10 ␮m, and a thickness of 40 ␮m was fixed ona sapphire insulative plate by epoxy adhesive. Then two electrical contacts were attached to the surface of the ring using
the gold evaporation method and silver conductive paste.
1098-0121/2009/79共1兲/014304共5兲

Figure 1共a兲 shows an optical microscopy image of a typical
thick ring-shaped crystal of NbSe3 equipped with two electrical contacts. The sample was placed on a copper block inan evacuated chamber. We confirmed that the first CDW
transition occurred at 144 K, which corresponded to the results of previous research.10,11 For simplicity we measured
the Shapiro step spectrum at 120 K because there are two
types of CDWs below the second transition temperature
共TC2 = 59 K兲. Moreover, the Shapiro step spectrum can beobserved clearly around 120 K since the dc resistanceanomaly is at its maximum value. A feedback heater control
system kept the temperature of the sample within120⫾ 0.1 K.
The Shapiro interference is a common phenomenon observed with various condensed-matter systems including superconducting Josephson junctions,12 vortex flows,13 weaklinked superfluids,14 and CDW systems. When ac and dc
voltages are applied to CDW materials, it is well known that
harmonic and subharmonic Shapiro steps are observed in dc
current-voltage characteristics.15,16 The condition underwhich a step appears ispIdc= f ex ,
2eN q

共1兲

where e is the electron charge, N is the number of onedimensional chains in the crystal, f ex is the frequency of theac voltage, and both p and q are integers. The left side corresponds to the fundamental frequency of the voltage oscillation, namely, narrow-band noise produced by the sliding
CDW. We measured differential resistance dVdc / dIdc as afunction of dc current 共time-averaged current兲 at a constanttemperature 共T = 120 K兲 and with an ac voltage. Figure 1共b兲shows the electrical circuit. A Keithley 6220 current source
and a Hewlett-Packard HP8712ET network analyzer 共frequency range of 300 kHz to 1.3 GHz兲 were used as dc and ac
current sources, respectively. We used the capacitive highpass filters and inductive low-pass filters to apply both dc
and ac electric fields to the sample. Coaxial cables and microstrip lines were employed to achieve impedance matchingof the ac lines. The temperature dependence of the circuit
was sufficiently small during the measurement. Differential
resistance 共dVdc / dIdc兲 was measured using a lock-in amplifier 共Stanford Research SR 830DSP兲 to observe a clear
Shapiro step spectrum. The frequency and amplitude of the014304-1©2009 The American Physical Society

PHYSICAL REVIEW B 79, 014304 共2009兲

MATSUURA, INAGAKI, AND TANDA

0.06 p/q = 1/1

Shapiro step width of main peaks

δV [mV]

0.04(b)
Iacsin 2π fext

Idc

00.06 p/q = 2/1
0.04
0.02
0
0.06 p/q = 3/1
0.04
0.02

Idc+Iacsin 2π fext

V

0.02

0

NbSe3 ring

0

20

40

60

80

Ac voltage Vac [mV]L

fex = 100 MHz

C

FIG. 1. 共Color兲 共a兲 Optical microscopy image of a typical ringshaped crystal of NbSe3 equipped with two electrical contacts attached by silver paste. 共b兲 Electrical circuit for measuring the Shapiro steps 共ac-dc interference effect兲. The thick ring at the center ofthe circuit is the ring-shaped crystal of NbSe3. Both dc and ac
currents are induced into the crystal via the two electrical contacts.
Inductive low-pass filters and capacitive high-pass filters separate
the dc and ac lines. Differential resistance is measured as a functionof dc current. The sample and filters are placed in an evacuatedchamber and cooled to 77 K with liquid nitrogen.

FIG. 3. ac voltage amplitude dependence of Shapiro step width␦V of main peaks of 共a兲 p / q = 1 / 1, 共b兲 2/1, and 共c兲 3/1. The step

width is estimated as the product of peak height and half width ofhalf maxim of each peak of Fig. 2. Solid lines are calculation resultsof amplitudes of Bessel functions 兩J1共x兲兩, 兩J2共x兲兩, and 兩J3共x兲兩,
respectively.

0.01

p/q−

p/q = 1/1

p/q+

1Ω

δV [mV]T = 120 K
fex = 100 MHz

Differential resistance dVdc / dIdc

1/3 1/2 p/q = 1/1
1/1− 1/1

12 mV

+

1/2
1/3 1/1−

1/1+
1/1

1/2

1/1−
1/1

3/2 2/1−2/1

3/2 2/1

1/1−

−

1/1
1/1
1/2 − 1/1

+ 3/2 2/1
−

1/1
1/2

0

21 mV

3/2 2/1 2/1
−

1/1
1/2

Shapiro step width of beat peaks

Vac = 0 mV

2/1
2/1+
5/2

3/1
3/1
3/1−

2/1
2/1+ 3/1−

3/1+

36 mV
4/1− 4/1

3/1
3/1+ 4/1−

5/1−

5/1

4/1
5/1
4/1+ 5/1−

4/1
3/1
5/1
4/1+ 5/1−
3/1+ 4/1−
2/1
2/1− 2/1+ 3/1−

2
dc bias current Idc [mA]

65 mV
73 mV

0
0.01

p/q = 2/1

00.01

p/q = 3/1

82 mV
0

4

0

FIG. 2. 共Color兲 dc bias current dependence of differential resistance dVdc / dIdc with applied various ac voltages of f ex = 100 MHz
at T = 120 K. The ac voltages Vac are 0, 12, 21, 36, 65, 73, and 82
mV from top to bottom. Harmonic and subharmonic Shapiro steps
are indicated by arrows and index p / q, where p and q are integers.
Moreover, there are small beat peaks 共p / q− and p / q+兲 on both sidesof the main Shapiro peaks.

20

40

60

80

Ac voltage Vac [mV]fex = 100 MHz

FIG. 4. 共Color兲 ac voltage amplitude dependence of Shapiro
step width ␦V of negative and positive beat peaks of 共a兲 p / q = 1 / 1,
共b兲 2/1, and 共c兲 3/1. The calculation curves of amplitudes of Bessel
functions 兩J1共x兲兩, 兩J2共x兲兩, and 兩J3共x兲兩 are the rescaled ones in Fig. 3.

014304-2

EVIDENCE OF CIRCULATING CHARGE DENSITY WAVE…

PHYSICAL REVIEW B 79, 014304 共2009兲

ac signal for the differential resistance measurement were set
at 33 MHz and 1 ␮A, respectively, which were sufficiently
smaller than that of the ac field for producing Shapiro steps.

The solid curves are results of numerical calculation of amplitudes of Bessel functions Jn共x兲. The experimental data ofp / q = 1 / 1, 2/1, and 3/1 follow the calculated curves of J1共x兲,
J2共x兲, and J3共x兲, respectively. These results correspond to theprevious researches.
Figures 4共a兲–4共c兲 show ac voltage dependence of the Shapiro step width of negative and positive beat peaks of p / q= 1 / 1, 2/1, and 3/1, respectively. Both beat peaks have similar magnitudes and they also follow the curves of amplitudesof Bessel functions, which are the rescaled ones in Fig. 3.
These results imply that there is a common origin of thenegative and positive beat peaks.
Figure 5 shows differential resistance dVdc / dIdc as a function of dc current without ac voltage and with an ac voltageof amplitude Vac = 65 mV and frequency f ex = 40, 80, 100,
200, and 300 MHz. The CDW current ICDW is roughly proportional to Ip = Idc − Ith, where Idc is the dc current value at
the Shapiro peak and Ith is the threshold current of each ac
voltage; hence the peak position Ip must be proportional to
the frequency. From Eq. 共1兲, we can roughly estimate theeffective number of the one-dimensional chains N from thecurrent value at the first peak 共p / q = 1兲. Then we find thatabout 2 ⫻ 107 chains contribute to the current. This value
corresponds to the order of the effective area estimated fromthe resistivity.
Figure 6 shows frequency dependence of the peak position Ip, the harmonic and subharmonic peaks, and the negative and positive beat peaks. The peak positions increase
with increasing frequency. These features correspond to
those of Shapiro steps observed on usual NbSe3 crystals.16
The beat peak positions Ip are proportional to the frequency
in the same way as the harmonic and subharmonic peaks andare independent of Vac. Although some peak heights were
small, the positions of beat peaks fall on the same lines inFig. 6.
What is the origin of the beat peaks? Since the ring
sample has two current paths, an imbalance between the currents flowing through the left and right arms would makeShapiro peaks split. However, this model can explain only
the presence of two peaks because there are only two kindsof CDW current. It is not consistent with the observation ofboth one main peak and two beat peaks.
Another idea is that the CDW current flows inhomogeneously in the samples. For example, difference in current
density, the inner and outer circumferences made a differencein current density along radial direction and derived several
Shapiro peaks. However, it is not consistent with our results
because the beat peaks were also commonly observed in allother three samples with different dimensions, although theinhomogeneity must be strongly dependent on thickness andradius. Moreover, the magnitude of the step widths of bothbeat peaks 共see Fig. 4兲 supports that the pairs of the beatpeaks originate from a common mechanism. Therefore, it isreasonable to consider that the beat peaks are caused by aninterference effect associated with the ring topology and notwith inhomogeneous current flow.
We propose a plausible model that the beat structures of
the Shapiro step spectrum result from a circulating current of
the CDW as shown in the inset of Fig. 7. When the circulating CDW current flows in the ring, the resonance conditionof the Shapiro step spectrum is modified asIII. RESULTS AND DISCUSSIONSThe Shapiro step spectrum of the CDW ring was clearlyobserved. Figure 2 shows differential resistance dVdc / dIdc asa function of dc bias current at applying various ac voltagesof 100 MHz at 120 K. In the absence of ac voltage 共Vac
= 0 mV兲, the value of the differential resistance at Vdc= 0 mV was 34.7 ⍀, which included the contact resistance,since this was a two-contact measurement. Since uncondensed electrons remain below the first transition temperature, the resistance of NbSe3 crystals is ohmic at low voltage.
The contact resistance is estimated to be about 8 ⍀ from thetemperature coefficient of resistance above the first transition
temperature 共metallic state兲. Using the sample dimensions,
we estimated that resistivity is of the order of 10−3 ⍀ m; onthe other hand, the usual resistivity of NbSe3 is of the orderof 10−6 ⍀ m at T = 120 K. Hence only 0.1% of the onedimensional chains contributes to conduction. This result
suggests that the current is concentrated only near the surfaceof the crystal.
Increasing dc bias current, a clear threshold voltage andcurrent were observed at Vth = 9.4 mV and Ith = 87 ␮A, respectively. The threshold electric field was estimated at 520
mV/cm, which is similar of order to that of usual NbSe3
crystals 共whiskers兲.17 In a previous paper, two threshold voltages were observed in a ring-shaped crystal with electricalcontacts.18 However, we observed only one threshold current. This result reflects the fact that the two electrical contacts were symmetrically attached to the ring. When the ac
voltage increases, the ohmic region decreases and theShapiro steps appear as dVdc / dIdc peaks.
Almost all the large peaks can be classified by index p / q,
where p and q are integers. When p / q is an integer, the peakcorresponds to a harmonic Shapiro step. On the other hand,when p / q is a rational number the peaks are called subharmonic steps. We also observed subharmonic peaks at p / q= 1 / 3, 1/2, 3/2, and 5/2.
Moreover, we found small beat peaks on both sides of themain Shapiro peaks with only the CDW loops. The beatpeaks are completely different from usual subharmonicpeaks because the distance between the main peak and thebeat peak is increased when the dc bias current is increased.
Therefore, the peak cannot be classified as subharmonic. Thebeat peaks at negative and positive sides of the main peaks
are indicated by p / q− and p / q+, respectively. The beat structures were reproduced in other three ring-shaped crystals
with different diameters and thicknesses.
Peak heights of both the main Shapiro peaks and the beatpeaks depend on Vac. It is well known that the Shapiro step
widths, which are voltage step width in current-voltage characteristics, follow Bessel functions of ac voltage.7 We estimated the Shapiro step width ␦V as the product of the peakheight and the half width at half maximum of each dVdc / dIdc
peak. Figures 3共a兲–3共c兲 show ac voltage dependence of the
Shapiro step width of p / q = 1 / 1, 2/1, and 3/1, respectively.

014304-3

Differential Resistance dVdc / dIdc [Ω]Vac = 0 mV

1/1

p/q = 1/3
1/2
1/2−

1/2+
2/3

Vac = 65 mV
fex = 300 MHz

1/1+

1/1−

Vac = 65 mV
3

1/1+
1/1−

1

1/1+

3/2

2/1−

5/2

2/1+

1/1 +
1/1 1/1 −

2/3
1/2 +
1/2 1/2 −
1/3

1

1/1
1/2

3/1+
3/1−

4/1+
4/1−

5/1−

3/1
4/1 5/1
5/2−
5/1
3/2 2/1
4/1+
3/1+
4/1−
2/1− 2/1
+ 3/1
−

5/1

100 MHz0

3/1−

80 MHz

8/1 9/1

2/1 3/1
2/1−

5/1

3/1+ 4/1
4/1−

100

200

300

FIG. 6. 共Color兲 Frequency dependence of peak position Ip = Idc− Ith of each step. The peak positions of the harmonic, subharmonic,
and beat peaks are proportional to frequency. The slopes of eachline are shown as a function of the index p / q in Fig. 7.

4/1

3/1

1/1

1/2

−

3/2

2

0
0

3/1

2/1

Frequency fex [MHz]
2/1

1/1−

2/1+

7/1
6/1

4/1+

2
dc bias current Idc [mA]

40 MHz4

FIG. 5. 共Color兲 Differential resistance dVdc / dIdc versus dc current Idc at T = 120 K without ac voltage and with ac voltage of
amplitude Vac = 65 mV and frequency f ex = 40, 80, 100, 200, and
300 MHz, respectively.

Idc ⫾ ␦I p
= f ex ,2eN
qResults of the previous researches for the dynamical property of CDWs in NbSe3 rings suggested the existence of acirculating current mode. Okajima et al.18 found several
anomalous dVdc / dIdc steps in the sliding state. In our previous paper,11 we found that the pinning frequency of a ring issmaller than that of a cut ring from the investigation of thefrequency dependence of ac conductivity of a CDW ring in
the pinning state. These results implied that the CDW dynamics in NbSe3 rings is different from conventional dynamics due to the circulating CDW current mode.
0.04

共2兲

where ⫾␦I is the additional current flowing through the rightand left arms. Since a temporal circulating current ␦I flows
through the ring, the CDW current flowing in the right arm
must decrease 共or increase兲 and that flowing in the left arm
must increase 共or decrease兲. Then the beat peaks appear on
both sides of the main peaks. Figure 7 shows the slope ofeach line in Fig. 6 as a function of index p / q. The slopesof the main and two beat peaks are proportional to p / q.
The coefficient of p / q of the main peak is 5.964⫻ 10−3 mA/ MHz, and the coefficients of the beat peaks are
5.348⫻ 10−3 and 6.256⫻ 10−3 mA/ MHz. Since the slopesof the beat peaks fall on the same lines, the circulating current ␦I is proportional to dc bias current Idc. The difference
between the coefficients of the negative beat peaks and themain peaks is greater than that between the positive beatpeaks and the main peaks. In the model, the beat peaks must
appear symmetrically. Since we observed symmetric beatpeaks with other ring samples, the asymmetry may not be an
essential property. Although this simple model does not completely explain the asymmetry of peak position of negativeand positive beat peaks, it is sufficiently consistent with experimental results of frequency dependence and ac voltage
dependence of the beat peaks.

Idc + Iac sin 2π fext

Ip / fex [mA/MHz]

2/3

2/1−

1/1

1/2

200 MHz

2/1

1/2

2/

2/

1/1

1/2−
1/3

1+

5/
2

Peak Position Ip = Idc − Ith [mA]1Ω

4/1 4/1+
4/1−
3/1 3/ 3/1
1 +−PHYSICAL REVIEW B 79, 014304 共2009兲

MATSUURA, INAGAKI, AND TANDA

δI

0.02
Coefficient of p/q−36.256x10−35.904x10−35.348x100024Shapiro Index p/q
FIG. 7. 共Color兲 Slopes of each peak position versus frequency
Ip / f ex as a function of index p / q. Circles, squares, and triangles
indicate the main peaks 共harmonic and subharmonic peaks兲, negative beat, and positive beat peaks, respectively. Each Ip / f ex is proportional to p / q. The coefficients of p / q for each line are shown inthe box. They were calculated by the least-squares method. The
inset shows a model of the circulating current to explain the beatstructure of the Shapiro step spectrum. If the circulating current is
temporarily excited by an external electric field, it modifies thecurrents through the right and left arms as I ⫾ ␦I. As a result, fundamental frequency of each arm is modified and two beat peaks areproduced around the main peaks.

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EVIDENCE OF CIRCULATING CHARGE DENSITY WAVE…

PHYSICAL REVIEW B 79, 014304 共2009兲

The circulating current is possibly Fröhlich’s supercurrent. The experimental results show that the circulating current is proportional to the external dc current because thepeak positions of the negative and positive beat peaks areproportional to the index p / q. A possible mechanism for exciting the circulating current is as follows. When dc voltage
is applied, a CDW phase slip occurs at the electrical contacts.
Since the electrical contacts were actually not completely
symmetrically attached, the electric-field imbalance induces
a circulating current of CDW. The current must flow for atleast 10−8 s, which is the ratio of the circumference of thering-shaped crystal and the CDW phase velocity. This time isthree orders longer than the phason damping time of NbSe3
whiskers 共␶ ⬃ 10−11 s兲.9 Experimental10,19 and theoretical20,21
researches of the CDW transition temperatures in topologicalcrystals concluded that phase fluctuation is enhanced by theclosed-loop topology. The enhanced fluctuation allows a
CDW in each one-dimensional chain to move independently.
Hence, the damping time becomes longer than that observedwith conventional measurements, and the CDW reverts tothe type that Fröhlich1 considered.
Shapiro step spectrum measurement allows us to detect
the circulating current. A conventional dc current measurement cannot detect the circulating current because, althougha superconducting current circulates in the rings, the total
time-averaged current does not change. Moreover, since a
CDW is a kind of electron crystal, the electrical contacts
essentially change the dynamic property of the CDW. In
principle, a CDW system has the inertial and elastic factors

and quantum tunneling property.22 Those factors play important roles in the dynamics of a CDW supercurrent system.
Investigations of the CDW dynamics with the topologicalcrystals will reveal the hidden intrinsic properties of the
CDWs.

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IV. CONCLUSION

In conclusion, we observed clear harmonic and subharmonic Shapiro step spectra and a beat spectrum around themain Shapiro peaks in differential resistance-dc current characteristics. The beat spectrum suggests that a circulating
CDW current flows in the ring. This is a sign of Fröhlich’s
CDW supercurrent. The realization of Fröhlich’s CDW supercurrent will open a different field of condensed-matterphysics that may be associated with the supersolid state of4
He 共Ref. 23兲 and have applications to macroscopic quantum
coherent devices working at high temperatures.
ACKNOWLEDGMENTS

We thank K. Yamaya, K. Ichimura, M. Oda, H. Orihara,
O. Matsuda, V. Osipov, D. Churochkin, M. Tsubota, J. Hara,
T. Tsuneta, T. Toshima, and K. Shimatake for fruitful discussions, helpful suggestions, and technical support. This research was supported by the Japan Society for the Promotionof Science and the 21 COE program on “Topological Scienceand Technology” from the Ministry of Education, Culture,
Sport, Science and Technology of Japan.

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