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2012 EPL 97 57011(http://iopscience.iop.org/0295-5075/97/5/57011)
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March 2012
EPL, 97 (2012) 57011
doi: 10.1209/0295-5075/97/57011

www.epljournal.org

Aharonov-Bohm eﬀect in charge-density wave loops
with inherent temporal current switching
M. Tsubota1 , K. Inagaki1,2 , T. Matsuura1 and S. Tanda1,2
1

Department of Applied Physics, Hokkaido University - Sapporo 060-8628, Japan
Center of Education and Research for Topological Science and Technology, Hokkaido University
Sapporo 060-8628, Japan2

received 28 July 2011; accepted in ﬁnal form 7 February 2012published online 5 March 201271.45.Lr – Charge-density-wave systems
73.23.-b – Electronic transport in mesoscopic systems
PACS 03.65.Vf – Phases: geometric; dynamic or topological
PACS
PACS

Abstract – We report evidence for Aharonov-Bohm (AB) oscillations with temporal current
switching at 5.1 and 79 K by using charge-density wave (CDW) loops in TaS3 ring crystals. The
periodicity of the AB oscillations is h/2e, and this is conﬁrmed by employing four samples with
diﬀerent diameters. This means that sliding CDWs carry 2e charge quanta. We observe current
switching between the two states, namely the high and low current states, as a result of theseparation of ﬁeld-dependent and time-dependent ﬂuctuations. The probability distribution of theswitching oscillates also with a period of h/2e. We discover that the observed oscillations, which
are a superposition of the current ﬂuctuations and the switching behavior, have a period h/2e.
c EPLA, 2012
Copyright Introduction. – Fifty years ago, Aharonov and Bohm
challenged our understanding of vector potential, which
had not been considered a physical entity [1,2]. Since then,the concept of the AB (Aharonov and Bohm) phase has
been accepted in many areas of modern physics [3] including condensed-matter physics [4–8], particle physics [9],non-Abelian gauge theories [10], gravitational physics [11,12] and laser dynamics [13]. To observe the AB phaseusing metallic rings, the ring must be 1 µm in diameterand measured at a low temperature (typically below 1 K)
to maintain quantum phase coherence [5].
The existence of the AB eﬀect of charge-density wave
(CDW) condensates has remained an open question.
CDWs have not been considered to exhibit the AB eﬀect
because an electron-hole pair of a CDW has neutral chargethat cannot couple to vector potential, and the CDW
phase is usually treated as a classical coordinate. However,the AB eﬀect of a CDW was ﬁrst predicted by Bogachek
et al. [14]. Their model includes a term representing
the interaction of the CDW with the vector potential
ﬁeld, and then estimates the oscillation of h/2e. Anothertheory states that quasi-particle interference is generated
by changing the CDW ground state [15–17]. Latyshev
et al. tried to observe the AB eﬀect of CDWs in NbSe3
crystals with tiny random holes pierced by heavy-ion
irradiation [18]. They revealed oscillation in the nonlinear

CDW conductivity as a function of the magnetic ﬁeld.
Visscher et al., however, insisted that this could be theAB eﬀect of a quasi-particle passing through a metallic
region, where the CDW order was destroyed near the tinyholes [19]. For this reason, a CDW loop, over which thecontinuity of the CDW order is maintained, should be
employed to settle the argument.
Topological crystals (ring and Möbius strip shaped
seamless crystals) [20,21] are the only materials providing
such a CDW loop. The chain axis of a CDW directs itself
along the circumference of the ring without losing the
CDW order, thereby forming a CDW loop. This cannot berealized simply by piercing a hole in a conventional single
crystal, because a continuous CDW chain is not formedby the hole.
We report the ﬁrst evidence for the AB interference
eﬀect with temporal current switching at a high temperature of 79 K by using CDW loops in TaS3 ring crystals
whose circumferences are as long as 85 µm. We investigated orthorhombic TaS3 , which has a Peierls transitionat 218 K, and no uncondensed electrons at low temperatures. Current switching between two states, namely highand low current states, occurs in a time-dependent ﬂuctuation. The CDW condensate maintained macroscopic phasecoherence, which extended over the ring system [22]. The
periodicity of the AB eﬀect of the CDW was half a ﬂux

57011-p1

M. Tsubota et al.

(a)Ring A

10 mm
(c)
(C)

10 mm
mm
10

10 mm
mm
10

Fig. 1: (Colour on-line) (a) Scanning electron microscope(SEM) image of a TaS3 tube-shaped crystal. (b) Contact
conﬁguration for transport measurement. Two gold electrodes(100 nm thick, 5 µm wide) were evaporated. (c) SEM imageof a TaS3 tube/ring crystal before cutting, (left) after cutting(right).

Ring A

quantum h/2e with h and e being Planck’s constant andthe elementary charge, respectively.
Experiments and results. – The samples weresynthesized by the chemical vapor transport method [14].
The materials (tantalum and sulfur) were reacted in an
evacuated quartz tube to produce TaS3 crystals (whisker,
ring and other kinds of topological crystals). We prepareda ring/tube crystal reproducibly (ﬁg. 1(a)), and then cut
one with a focused ion beam (ﬁg. 1(c)). We fabricatedgold electrodes on the sample using electron-beam lithography [23], after coating it with polymethylmethacrylate(ﬁg. 1(b)). We measured the temperature dependence ofthe resistivity to identify the ring crystal. As a result,the properties of our sample agreed with the previously reported properties of a conventional whisker oforthorhombic TaS3 [24].
We measured the dc electric current at 5.1 K while
applying a magnetic ﬁeld in the vertical direction to thering crystal, which was 27 µm in diameter and 1 × 0.1 µm2
in cross-sectional area. The magnetic ﬁeld was swept
at a sweep rate of 50 µG/s which is slow enough toprevent the generation of eddy current. Each data set
consisted of an average of 10 raw current values measured
every ﬁve seconds with a time constant of 100 ms. The
magnetoresistance of a TaS3 ring crystal is shown atvoltages of 100, 200 and 300 mV, in ﬁg. 2(a). We observed
periodic oscillations in these voltages that increased inamplitude when we increased the voltage.
We calculated the power spectra of the observed oscillations using the discrete Fourier transform (DFT) and the
maximum entropy method (MEM) for an accurate structural analysis. In particular, MEM analysis can detect thepeaks of a signal that includes noise. Figure 2(b) showsthe evolution of the power spectra using DFT (left) andMEM (right), 100, 200 and 300 mV, from bottom to top.
The oscillation period ∆B estimated from the power spectra is 40 mG for all voltages. By assuming that the origin ofFig. 2: (Colour on-line) (a) Change in current value as magnetic
ﬁeld at 5.1 K, applied voltage 100, 200 and 300 mV. (b) The
power spectra of the observed oscillation using DFT (left) andMEM (right). The red lines show the sinusoidal oscillation
estimated by IDFT and MEM. (c) Current depending on themagnetic ﬁeld and (d) the evolution of the power spectra DFT(left) and MEM (right), 25, 50 and 75 mV, from bottom to top
at 79 K.

the oscillation is the AB eﬀect, we estimated the eﬀective
charge e∗ using the following formula:
∆B =

Φ0
h
1
= ∗ · 2,
S
e πr

(1)

where h is Planck’s constant and r is the radius of thering crystal. The estimated value of an eﬀective charge e∗is 3.0 × 10−19 C, and this is approximately equal to twotimes the elementary charge (2e = 3.2 × 10−19 C).
We conﬁrmed the period h/2e of the main peak by thefollowing analyses. The position of the main peak wasdetermined with MEM, and then the amplitude and thephase of the corresponding peak were obtained from the
DFT. This was why many peaks were observed caused
by narrow-band noise or extrinsic noise using DFT, andwhy the main peak was blurred. Using MEM, we carefully
analyzed the magnetoresistance oscillation compared with
DFT for the disappearance of false peaks [25]. The redlines in ﬁg. 2(a) show the sinusoidal oscillation with theperiod estimated by MEM and the amplitude estimatedby the inverse DFT (IDFT) of the main peak. The major
contribution of the observed oscillations is well expressedby the obtained spectra. Other peaks, such as h/e or h/4e,
were not observed.

57011-p2

Aharonov-Bohm eﬀect in charge-density wave loops with inherent temporal current switching
The oscillation period with data at 79 K agreed withthat at 5.1 K (ﬁg. 2(c)). Figure 2(d) shows the evolutionof the power spectra, 25, 50 and 75 mV, from bottom totop. The estimated value of an eﬀective charge e∗ is 3.1 ×
10−19 C, which corresponds to the value at 5.1 K. The
periodic oscillation h/2e remained in magnetoresistanceat 79 K. It is noteworthy that at such a high temperatureusually thermal ﬂuctuation destroys quantum coherence.
In addition, four samples with diﬀerent diameters
showed the periodic oscillation h/2e. We measured themagnetoresistance of the ring crystals, whose diameters
were 4.9, 12 and 17 µm, respectively. The observed periods
were 1.3, 1.8 × 10−1 and 9.9 × 10−2 G, which were 1.2,
1.1 and 1.1 in h/2e units, respectively. Four samples
showed a periodicity of h/2e. The periodic oscillation
depended only on the magnetic ﬂux penetrating thering. Our previous report [26] showed the voltage-current
characteristics of the Ring-A sample. In fact, the CDW
current and the amplitude of the oscillation behaved
similarly, in particular, there was a signiﬁcant increase
in VT . All four samples behaved in the same way. The
increase in the amplitude with respect to h/2e oscillation
can be plausibly attributed the development of a sliding
CDW. These results are summarized in table 1.
To consider the observed current ﬂuctuations without the period, it is necessary to rule out the possibility of an experimental artifact. The resistance may varysigniﬁcantly with a tiny change of temperature because
the temperature dependence of the sample obeys the
Arrhenius-type activation. In our experiment, the temperature varied no more than 10−3 K, so the current ﬂuctuation was not caused by a temperature change. In addition,
the narrow-band noise (NBN) in sliding CDWs should
be considered. Our experimental circuit includes low-passﬁlters with a time constant of 10−3 s, as well as the numerical averaging of obtained signals. Hence the measurement
is insensitive to NBN signals, typically, ﬂuctuations with a10−6 s period. Thus the h/2e periodic oscillation is inherent in the sample, and should not be attributed to experimental noises or to thermal ﬂuctuations.
We note that the magnetoresistance seems to include
some unspeciﬁed components rather than the sinusoidal
oscillation corresponding to the peak in the power spectrum, as shown in ﬁg. 2(c). The conventional ﬁeld sweep
measurement may have been smeared by time-domain
ﬂuctuations with a long time constant. Hence we separated
the ﬁeld-dependent ﬂuctuations from the time-dependentﬂuctuations by using the following procedure.

Fig. 3: (Colour on-line) Measurements and analysis technique.
Time dependence of magnetic ﬁeld (a) and the electric current(b). (c) Relation between the current and the magnetic ﬁeld.
(d) Current data applied with high magnetic ﬁeld. (e) DFT
performed on the time-averaged data I(Bi ) to extract periodic
components which depend only on the magnetic ﬁeld.

3) Consequently, time-domain current data I(Bi , k) at
B2 = B1 + δB, B3 = B2 + δB, · · · , and B49 = B48 +δB were obtained (ﬁg. 3(b)).
4) The time-averaged
current value was calculatedby I(Bi ) = k I(Bi , k)/500 for each ﬁxed ﬁeld Bi(shown with red squares in ﬁg. 3(b), (c) and (d)). By
employing this averaging, time-domain ﬂuctuations
were canceled if their time constants were faster than
500 seconds. Since the averaging is performed onlyin the time domain, we can separate ﬁeld-domain
ﬂuctuations from time-domain ﬂuctuations.
5) Finally, DFT was performed on the time-averaged
data I(Bi ) (ﬁg. 3(d)) to extract periodic componentsthat depend only on the magnetic ﬁeld (ﬁg. 3(e)).

This procedure revealed the existence of current ﬂuctuations in both the time and ﬁeld domains (ﬁg. 3(b)).
The measurement current switched between two states,
namely a high current state and a low current state.
The switching occurred at intervals of a few seconds toseveral hundred seconds. The current of the ring had abimodal distribution. Our discovery of temporal switching1) 500 current data, I(B0 , 1), I(B0 , 2), I(B0 , 3), · · · , and phenomena is crucial in terms of verifying the observed
I(B0 , 500), were measured once per second with a oscillations in a CDW.
Since the switching interval was shorter than the averagtime constant of 200 ms in a ﬁxed magnetic ﬁeld B0 .
ing time (500 seconds), the averaged current was expected
2) Next, 500 current data, I(B1 , 1), I(B1 , 2), I(B1 , 3), to exclude time-domain ﬂuctuations, as shown in ﬁg. 3(c).
· · · , and I(B1 , 500), were obtained at B1 = B0 + δB, Figure 3(d) shows the relationship between the current
which corresponded to ∼1/10 step of a ﬂux quantum, and the high magnetic ﬁeld. This conﬁrms unambiguh/2e, as shown in ﬁg. 3(a).
ously the existence of magneto-oscillations. The DFT of
57011-p3

M. Tsubota et al.
Table 1: List of the periodic oscillation of four samples at 5.1 and 79 K. D is the diameter of the ring crystals, ∆B is the period
in the oscillation of magnetoresistance, Φ is ∆B multiplied by the area of the ring crystal, Φ0 is h/2e and ipp are the amplitudesof the periodic oscillation resulting from each voltage.

T = 79 K

D (µm)∆B (G)−2

Φ/Φ0

VT (mV)ipp [VT ] (A)ipp [3VT ] (A)

27
17
12
4.9

4.1 × 10
9.9 × 10−2
1.8 × 10−1
1.3

1.1
1.1
1.0
1.2

25
50
24
32

3.9 × 10
6.0 × 10−11
7.0 × 10−8
5.0 × 10−10

2.8 × 10
3.0 × 10−10
3.0 × 10−7
2.1 × 10−9

7.0 × 10−10
9.1 × 10−10
1.1 × 10−6–

T = 5.1 K
Ring A
Ring B
Ring C
Ring DD (µm)
27
17
12
4.9

∆B (G)
4.0 × 10−2
9.5 × 10−2
–
1.3

Φ/Φ0
1.1
1.0
–
1.2

VT (mV)
110
250
–
190

ipp [VT ] (A)
3.6 × 10−13
1.5 × 10−11–
5.1 × 10−12

ipp [3VT ] (A)
2.6 × 10−11
6.5 × 10−11–
3.0 × 10−11

ipp [5VT ] (A)
8.0 × 10−11
2.0 × 10−10–
3.9 × 10−10

Discussion. – In our results, we discovered threephenomena in the ring crystal, namely, the periodic
oscillation of the CDW current, the switching behavior,and the oscillation of the distribution of the two states ofthe electric current. The period was h/2e corresponding to
the area of the ring crystals. We discuss these phenomena,
which were not observed in the whisker crystal, below.
The periodic oscillations are unlikely to originate fromnormal carriers generated by the conversion process. Whena normal current is converted into a CDW, a CDW phaseslip is induced [27]. The CDW strain energy F ∝ (∂θ/∂x)2
near the electrode is larger than that at the center.
Hence, the energy is increased near the electrodes anda phase-slip is induced near the electrodes. Therefore,
regardless of the size, the conversion process only takes−10

ipp [5VT ] (A)Ring A
Ring B
Ring C
Ring Dthe oscillations is also shown in ﬁg. 3(e). The oscillation
period was h/2e, which is consistent with the results inﬁg. 2 and our previous report [26].
We investigated the switching phenomenon in detail.
Figure 4(a) shows the time dependence of the electric
current when constant voltages are applied for 3000
seconds at 79 K. As a result, we observed two values of
the current in a ring crystal. The panels in the ﬁgureshow the applied magnetic ﬁeld Φ/Φ0 = 0, −1/4, −1/2,−3/4 and −1 from top to bottom (Φ0 is h/2e). The
distribution of the two states changed according to theapplied ﬁeld. Figure 4(c) shows the peak currents of thetwo states as a function of the magnetic ﬁeld. The peakcurrent of the high current state oscillated signiﬁcantly,
whereas that of the low current state remained almost
constant. Figure 4(d) shows the product of the full widthat half-maximum (FWHM) and the peak height P , whichcorrespond to the probability of the states. The probabilityof the high current state oscillated in a similar mannerto the peak current. The oscillation period was 170 ±10 mG. Surprisingly, this period corresponds to the ﬂux
quantum h/2e. Conformity between the peak current andthe probability of the high current state mean that theperiodic oscillation shown in ﬁg. 2 was a superposition of
the current ﬂuctuations and the switching behavior.

−11

Fig. 4: (Colour on-line) (a) Time dependence of the electric
current for applying constant voltages for 3000 seconds at 79 K.
(b) Top panel of (a) zoomed in. (c) Peak currents of the twostates as a function of the magnetic ﬁeld. (d) The product of
the full width at half-maximum (FWHM) and the peak heightPmax , which correspond to the areas of the states.

place near the electrodes [28,29]. This scenario is valid
unless the spacing between the electrodes is too small,namely a few micrometers [30]. With our sample, which
was 85 µm in circumference, the region is estimated to be
several micrometers from the electrodes. In this context,
the carriers generated by the process do not exist over
the ring crystal. Next, we discuss the carrier lifetime τto determine whether the carrier travel over the ring
crystal. τ is represented by kB T = h/τ ∼ ∆E since the57011-p4

Aharonov-Bohm eﬀect in charge-density wave loops with inherent temporal current switching
quantum state of the carrier is blurred with the thermal
ﬂuctuation kB T , where kB is the Boltzmann constant, T istemperature and h is Planck’s constant, respectively. The
traveling length l of the carrier in the system is estimatedas l = vF τ , where vF is the Fermi velocity estimated as 106
m/s. l is calculated to be 10 µm at 5.1 K and about 600 nm
at 79 K. Since the circumference is 85 µm, the carrier will
not travel and its phase is unlikely to be preserved over
the whole system at 79 K. With our sample, the possibilityof the Al’tshuler-Aronov-Spivak (AAS) eﬀect [6], namely
the h/2e oscillation of a quasi-particle, is ruled out.
With a sliding CDW, AB interference occur in our
sample length because the correlation length is long. This
length was reported to be longer than 100 µm from anestimation of the size eﬀect in the sliding CDW [31,32].
Another estimation provided a value of about 1 µm byusing the X-ray scattering technique [33,34]. The reason
for the diﬀerence between these lengths remains unclear.
We consider that the phase correlation length in our
samples reaches about 100 µm from the circumference.
The NbSe3 ring crystals also have a correlation lengthof about 100 µm as determined by the Shapiro step
experiment [35]. These results are consistent with length
results reported in the sliding state [31,32].
Our study shows that the periodicity of CDW interference is half a ﬂux quantum h/2e, which is the same as for a
superconductor resulting from Cooper pairing. Bogachek
et al. [14] suggested that the charge quanta of CDWs are
multiples of the chain number. For example, the interference of a single CDW chain results in the period h/2e,while a bundle of CDWs with N chains provides an oscillation with a period of h/2N e. If a transverse correlation
between CDWs forms CDW bundles, a period of h/4e or
h/8e must be observed. In fact, the power spectra only
gives the fundamental h/2e oscillation. Thus, we proposeda model to describe the h/2e oscillation in our previous
report [26]. This model showed a CDW soliton conﬁned ina single chain encircled by a dislocation loop, carrying acharge of 2e per chain [36–38]. The existence of the dislocation loop makes the soliton move freely along the chain
accompanying the 2e charge. Then the charge neutrality ispreserved by the creation of soliton-antisoliton pairs. The
AB amplitude represents the total of these conﬁned chains
and about 8% of all CDW chains. In the context, the soliton excitation is the origin of the 2e charge in a CDW.
Now we consider the two states indicated by the switching of the electric current. Assume that a soliton pair iscreated when the CDW slides [36,39]. When a soliton pairis created, a voltage ∆V is generated between the solitons.
Thus, the soliton pair is accompanied by a voltage diﬀerent from the initial voltage due to the relaxation of CDW
strain, which may drive a circulating current in the CDW
loop [35]. Figure 5(a) and (b) are schematics showing cases
in which a soliton pair was absent and present, respectively. The ammeter measures the sum of the currents thatﬂow in the left and the right arms, namely, I = I1 + I2 ,(I1 = V0 /RR , I2 = V0 /RL ), where RR and RL are the

(a)

(b)

V0

V0

RR

RL

IL

IR

Path 2RR

RL

Path 3A

Path 3IL

Path 1DV

Path 2IR

A

Path 1

Fig. 5: (Colour on-line) Schematic picture of two current states.
(a) Absence and (b) presence of the voltage ∆V generated bya soliton pair.

resistances of the right and left arms, respectively. Itshould be noted that the ammeter does not measure thecirculating current through path 3. The sign of ∆V is
opposite to that of the applied voltage V0 , because thesoliton pair generation relaxes the strain energy of the
CDW. Within this closed circuit, according to the Kirchhoﬀ’s law,
I1 RR = V0 − ∆V,(2)

I2 RL = V0 ,(3)

−I1 RR + I2 RL = ∆V.

(4)

As a result, I1 and I2 are equal to (V0 − ∆V )/RR and
V0 /RL . Consequently, the total current for the latter caseis ∆V /RR smaller than that of the former case. Hence thecurrent can be either of the two values representing theswitching between the two states, namely the value showswhether or not the phase slip has occured.
Next, we consider the point distribution of two states.
The probability of high current state behaves similarly
to the current peak as a function of a magnetic ﬁeld.
When we consider whether or not a phase slip occurs,
the probability distribution of the soliton-antisoliton paircreated by the phase slip appears to oscillate depending on
the ﬁeld. The period can be realized as it ﬁts the AB phase.
The switching interval of the order of several seconds is
not explained by the NBN or other CDW phenomenain the whisker [40,41]. The enhancement of the periodic
boundary condition originating from the ring topology
may play an important role in relation to the CDW
phase [35,42]. The distribution change is obviously related
to the CDW loop, and the resolution of the dynamics
requires further experiments.
Our result implies that a persistent CDW current should
be present since the electric charge in a sliding CDW iscoupled with the vector potential. Our group has alreadysynthesized NbS3 ring crystals [43] in which a Peierls
transition occurs at room temperature. In the context, it
will be necessary to reconsider the possibility of Fröhlichtype superconductivity [44,45] and a room temperature
operating quantum interference device.

57011-p5

M. Tsubota et al.
∗∗∗
The authors are grateful to T. Toshima, T. Tsuneta,
K. Matsuda, K. Ichimura, and K. Yamaya, for experimental support and Y. Asano, M. Hayashi, N.
Hatakenaka, and P. Monceau for stimulating discussions. This research has been supported by Grant-inAid for the 21st Century COE program on “Topological
Science and Technology” and the Japan Society for the
Promotion of Science.
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